Datasets:

haritzpuerto

/

the_pile_00_arxiv

like 0

--- abstract: 'We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton-Jacobi equations on a general metric space. As a first consequence, we show in full generality that the log-Sobolev inequality is equivalent to an hypercontractivity property of the Hamilton-Jacobi semi-group. As a second consequence, we prove that Talagrand’s transport-entropy inequalities in metric space are characterized in terms of log-Sobolev inequalities restricted to the class of $c$-convex functions.' address: - 'Université Paris Est Marne la Vallée - Laboratoire d’Analyse et de Mathématiques Appliquées (UMR CNRS 8050), 5 bd Descartes, 77454 Marne la Vallée Cedex 2, France' - 'Université Paris Ouest Nanterre la Défense, MODAL’X, EA 3454, 200 avenue de la République 92000 Nanterre, France' author: - 'N. Gozlan, C. Roberto, P-M. Samson' title: Hamilton Jacobi equations on metric spaces and transport entropy inequalities --- [^1] Introduction ============ Let $L:{\mathbb{R}}^m\to {\mathbb{R}}$ be a convex function with super linear growth, in the sense that $L(h)/\|h\|\to \infty$, when $\|h\|\to \infty$, where $\|\cdot\|$ is any norm on ${\mathbb{R}}^m$. It is well known that if $f$ is some Lipschitz function on ${\mathbb{R}}^m$, the function $Q_tf$ defined by $$\label{HLO} Q_tf(x)=\inf_{y\in {\mathbb{R}}^m}\left\{f(y)+tL((x-y)/t)\right\},\quad t\geq 0, x\in {\mathbb{R}}^m,$$ is a solution, in different weak senses, of the following Hamilton-Jacobi equation $$\label{HJ classical} \partial_t u(t,x)=-L^*(\partial_{x}u(t,x))$$ with initial condition $u(0,x)=f(x)$, where $L^*(v)=\sup_{u\in {\mathbb{R}}^m}\{u\cdot v -L(u)\}$ is the Fenchel-Legendre transform of $L$ (see for instance [@Evans-book]). It can be shown, for example, that the function $(t,x) \mapsto Q_tf(x)$ is almost everywhere differentiable in $(0,\infty)\times {\mathbb{R}}^m$ and that is verified at every such point of differentiability (see e.g [@Evans-book Chapter 3]). Formula is usually referred to as the Hopf-Lax-Oleinik formula for Hamilton-Jacobi equations. The objective of this paper is twofold: - generalize the Hopf-Lax-Oleinik (HLO) formula to a class of Hamilton-Jacobi equations in a metric space framework; - use this aforementioned HLO formula to establish different connections between logarithmic Sobolev type inequalities and transport-entropy inequalities. General framework. ------------------ In this section we give the general setting of this article. ### Assumptions on the space In all the paper, $(X,d)$ will be a complete and separable metric space in which closed balls are compact. This latter assumption could be removed at the expense of additional standard technicalities. We will sometimes assume that $(X,d)$ is a *geodesic space*, meaning that for every two points $x,y \in X$ there is at least one curve $(\gamma_t)_{t\in [0,1]}$ with $\gamma_0=x$, $\gamma_1=y$ and such that $d(\gamma_s,\gamma_t)=|t-s|d(x,y)$ for all $s,t \in [0,1].$ Such a curve is called a geodesic between $x$ and $y$. ### The sup and inf convolution “semigroups". In all the paper, $\alpha:{\mathbb{R}}^+\to{\mathbb{R}}^+$ will be an increasing convex function of class $\mathcal{C}^1$ such that $\alpha(0)=0.$ If $f:X\to{\mathbb{R}}$ is a bounded function, we define for all $t>0$ the functions $P_{t}f$ and $Q_{t}f$ as follows: $$\label{Pt} P_{t}f(x)=\sup_{y\in X}\left\{f(y)-t\alpha\left(\frac{d(x,y)}{t}\right)\right\},\qquad \forall x\in X,$$ and $$\label{Qt} Q_{t}f(x)=\inf_{y\in X}\left\{f(y)+t\alpha\left(\frac{d(x,y)}{t}\right)\right\},\qquad \forall x\in X.$$ The operators $P_{t}$ and $Q_{t}$ are connected by the following simple relation $$Q_{t}f= -P_{t}(-f).$$ When the space $(X,d)$ is geodesic, the families of operators $\{Q_t\}_{t>0}$ and $\{P_t\}_{t>0}$ form non-linear semigroups acting on bounded functions: $$Q_{t+s}f=Q_t\left(Q_sf\right)\qquad \text{and}\qquad P_{t+s}f=P_t\left(P_s f\right),\qquad \forall t,s>0,$$ for all bounded function $f:X\to{\mathbb{R}}.$ When $(X,d)$ is not geodesic, only half of this property is preserved: $$Q_{t+s}f\leq Q_t\left(Q_sf\right)\qquad \text{and}\qquad P_{t+s}f\geq P_t\left(P_s f\right),\qquad \forall t,s>0.$$ Now we present our main results. An Hopf-Lax-Oleinik formula on a metric space. ---------------------------------------------- Our objective is to show that the Hamilton-Jacobi equation is still verified by $Q_tf$ in the metric space framework introduced above. To that purpose we first need to give a meaning to the state space partial derivative $\partial_x$ in this context. We will adopt the following classical measurements $|\nabla^+ f|(x)$ and $|\nabla ^- f|(x)$ of the local slope of a function $f:X\to{\mathbb{R}}$ around $x\in X$ defined by $$\label{nabla pm} |\nabla^+ f|(x)=\limsup_{y\to x} \frac{[f(y)-f(x)]_+}{d(x,y)}, \quad |\nabla^- f|(x)=\limsup_{y\to x} \frac{[f(y)-f(x)]_-}{d(x,y)},$$ (by convention, we set $|\nabla^\pm f|(x)=0$, if $x$ is an isolated point in $X$). If $f$ is locally Lipschitz, then $|\nabla^{\pm} f|(x)$ are finite for every $x\in X$. Moreover, if $f$ is Lipschitz continuous with Lipschitz constant denoted by $\mathrm{Lip}(f)$, then $|\nabla^{\pm}f|(x)\leq \mathrm{Lip}(f)$ for all $x\in X.$ Finally, when $X$ is a Riemannian manifold and $f$ is differentiable at $x$, it is not difficult to check that $|\nabla^\pm f|(x)$ is equal to the norm of the vector $\nabla f(x) \in T_x X$ (the tangent space at $x$). One of our main result is the following theorem. \[HJ\] If $f:X\to {\mathbb{R}}$ is an upper semicontinuous function bounded from above, then the following Hamilton-Jacobi differential inequalities hold $$\label{eq HJ} \frac{d}{dt_+} P_tf(x) \geq \alpha^*\left( |\nabla ^+ P_tf|(x)\right)\qquad \forall t>0,\quad \forall x\in X,$$ and $$\frac{d}{dt_-} P_tf(x) \geq \alpha^*\left( |\nabla ^- P_tf|(x)\right)\qquad \forall t>0,\quad \forall x\in X,$$ where $\alpha^*(u)=\sup_{h\geq 0}\left\{ hu-\alpha(h)\right\}$, $u\geq 0$, and where $d/dt_+$ and $d/dt_-$ denote respectively the right and left time derivatives.\ Moreover, when the space $(X,d)$ is geodesic, it holds $$\label{eq HJ bis} \frac{d}{dt_+} P_tf(x) = \alpha^*\left( |\nabla ^+ P_tf|(x)\right)\qquad \forall t>0,\quad \forall x\in X.$$ The interesting feature of Theorem \[HJ\] is that there is no measure theory in its formulation: the conclusion holds for *all* $t>0$ and *all* $x\in X$. Theorem \[HJ\] extends previous results by Lott and Villani [@LV07; @Villani-book], where was obtained on compact *measured* geodesic spaces $(X,d,\mu)$ provided the measure $\mu$ verifies some additional assumptions. More precisely, it is proved in [@LV07] that if $\mu$ verifies a doubling condition together with a local Poincaré inequality, then holds true, for all $t$ and for all $x$ outside a set $N_t$ of $\mu$ measure $0$. Under the geometric assumption that $(X,d)$ is finite dimensional with Aleksandrov curvature bounded below, Lott and Villani obtained the validity of for all $t$ and $x$. In [@Villani-book Theorem 22.46], Villani proves for all $t$ and $x$ on a Riemannian manifold. We indicate that, during the preparation of this work, we learned that Theorem \[HJ\] has also been obtained by Ambrosio, Gigli and Savaré in their recent paper [@AGS12] (see also [@AGS12bis]), with a very similar proof. Let us underline that the inequality $$\label{AGS} \frac{d}{dt_{+}} Q_{t}f(x) \leq -\alpha^*\left(|\nabla^- Q_t f|(x)\right),$$ which is equivalent to , is an important ingredient in their study of gradient flows of entropic functionals over general metric spaces. The main source of inspiration of the present paper is the seminal work by Bobkov, Gentil and Ledoux [@BGL01] establishing the equivalence between the logarithmic Sobolev inequality and hypercontractivity properties of Hamilton-Jacobi solutions. The main tool in the proof of Theorem \[HJ\] is the following result of independent interest. \[HJ 1\] Let $f:X\to {\mathbb{R}}$ be an upper semicontinuous function bounded from above. For all $t>0$ and $x\in X$, denote by $m(t,x)$ the set of points where the supremum defining $P_tf(x)$ is reached: $$m(t,x)=\left\{\bar{y}\in X : P_tf(x)=f(\bar{y})-t\alpha\left(\frac{d(x,\bar{y})}{t}\right)\right\}.$$ These sets are always non empty and compact and it holds $$\frac{d}{dt_{+}} P_{t}f(x)=\beta\left(\frac{1}{t}\max_{\bar{y}\in m(t,x)} d(x,\bar{y})\right),\qquad \forall t>0,\quad \forall x\in X$$ and $$\frac{d}{dt_{-}} P_{t}f(x)=\beta\left(\frac{1}{t}\min_{\bar{y}\in m(t,x)} d(x,\bar{y})\right),\qquad \forall t>0,\quad \forall x\in X,$$ where $\beta(h)=h\alpha'(h)-\alpha(h)$, $h\geq 0$. Hypercontractivity of $Q_t$ and the log-Sobolev inequality {#sec:hyp} ---------------------------------------------------------- Let $\mu$ be a Borel probability measure on $X$. Recall that the entropy functional ${\operatorname{Ent}}_\mu(\,\cdot\,)$ is defined by $${\operatorname{Ent}}_\mu(g) = \int g \log\left(\frac{g}{\int g\,d\mu}\right)\,d\mu,\qquad \forall g>0.$$ In order to introduce the log-Sobolev inequality, and for technical reasons, define, for $r>0$, $$\mathrm{Lip}(f,r) = \sup_{\genfrac{}{}{0pt}{}{x,y:}{d(x,y)\leq r}} \frac{|f(y)-d(x)|}{d(x,y)}$$ and observe that the usual Lipschitz constant is $\mathrm{Lip}(f)=\sup_r \mathrm{Lip}(f,r)$. Then, we denote by $\mathcal{F}_\alpha$ the set of bounded functions $f \colon X \to \mathbb{R}$ such that $\mathrm{Lip}(f,r)<\infty$ for some $r>0$ and $$\mathrm{Lip}(f) \leq \lim_{h\to \infty}\frac{\alpha(h)}{h}$$ (observe that if $\alpha(h)/h \to \infty$ when $h\to\infty$, this last condition is empty). The probability measure $\mu$ is said to satisfy the *modified log-Sobolev inequality minus* ${\bf LSI}_\alpha^-(C)$ for some $C>0$ if $$\tag{${\bf LSI}_\alpha^-(C)$} {\operatorname{Ent}}_\mu(e^f)\leq C\int \alpha^*(|\nabla^-f|) e^f\,d\mu \qquad \forall f \in \mathcal{F}_\alpha .$$ In particular, when $\alpha(h)= h^p/p$, $h\geq 0$, with $p>1$, it holds $\alpha^*(h)=h^q/q$, $h\geq 0$ with $1/p+1/q=1$. In this case, we write ${\bf LSI}^-_{q}$ for ${\bf LSI}_\alpha^-$. If $X$ is a Riemannian manifold and $\mu$ is absolutely continuous with respect to the volume element, the inequality ${\bf LSI}^-_{2}$ is the usual logarithmic Sobolev inequality introduced by Gross [@Gr75]. Following Bobkov, Gentil and Ledoux [@BGL01] we relate ${\bf LSI}_\alpha^-(C)$ to hypercontractivity properties of the family of operators $\{Q_t\}_{t>0}$. To perform the proof, we need to make some restrictions on the function $\alpha.$ We will say that $\alpha$ verifies the *$\Delta_2$-condition* [@RR] if there is some positive constant $K$ such that $$\alpha(2x)\leq K\alpha(x),\qquad \forall x\geq0.$$ \[thm-hyper\] Suppose that $\alpha$ verifies the $\Delta_2$-condition. Then the exponents $r_\alpha\leq p_\alpha$ defined by $$r_\alpha = \inf_{x>0} \frac{x\alpha'(x)}{\alpha(x)} \geq 1\qquad\text{and}\qquad 1<p_\alpha=\sup_{x>0} \frac{x\alpha'(x)}{\alpha(x)}$$ are both finite. Moreover, the measure $\mu$ satisfies ${\bf LSI}_\alpha^-(C)$ if and only if for all $t>0$, for all $t_o\leq C(p_\alpha -1)$ and for all bounded continuous function $f:X \to \mathbb{R}$, $$\begin{aligned} \label{hypercont} \left\|e^{Q_tf}\right\|_{k(t)} \leq \left\| e^f\right\|_{k(0)},\end{aligned}$$ with $$k(t)=\left\{\begin{array}{ll} \left(1+\frac{C^{-1}(t-t_o)}{p_\alpha-1}\right)^{p_\alpha-1}\mathbf{1}_{t\leq t_o} + \left(1+\frac{C^{-1}(t-t_o)}{r_\alpha-1}\right)^{r_\alpha-1}\mathbf{1}_{t> t_o}& \text{ if } r_{\alpha}>1 \\ \min\left(1; \left(1+\frac{C^{-1}(t-t_o)}{p_\alpha-1}\right)^{p_\alpha-1}\right) & \text{ if } r_{\alpha}=1\end{array}\right.,$$ where $\|g\|_k= \left(\int |g|^k d\mu\right)^{1/k}$ for $k\neq0$ and $\|g\|_{0}= \exp\left( \int \log g \,d\mu\right)$. Our proof follows the line of [@BGL01]. Let us explain in few words how to derive from ${\bf LSI}^-_\alpha$. Since $Q_tf\to f$ when $t\to 0$, it is enough to show that $H :t\mapsto \log\left\|e^{Q_tf}\right\|_{k(t)}$ is non-increasing. The left derivative of $H$ has an expression involving ${\operatorname{Ent}}_\mu (e^{k(t)Q_t f})$ and $\int \frac{d}{dt_+}Q_tf e^{k(t)Q_t f}\,d\mu$ (see Proposition \[deriv\]). To bound the first term from above, we apply the inequality ${\bf LSI}^-_\alpha$. To bound the second term, we use the inequality which is precisely in the right direction to prove that the left derivative of $H$ is negative. From log-Sobolev to transport-entropy inequalities -------------------------------------------------- Following [@BGL01; @LV07], a byproduct of the above hypercontractivity result is a metric space extension of Otto-Villani’s theorem [@OV00] that indicates that log-Sobolev inequalities imply transport-entropy inequalities. Let $c:X\times X \to {\mathbb{R}}$ be a continuous function; recall that the optimal transport cost $\mathcal{T}_{c}(\nu_{1},\nu_{2})$ between two Borel probability measures $\nu_{1},\nu_{2} \in \mathcal{P}(X)$ (the set of all Borel probability measures on $X$) is defined by $$\mathcal{T}_{c} (\nu_{1},\nu_{2}) =\inf_{\pi \in P(\nu_{1},\nu_{2})} \iint c(x,y)\,\pi(dxdy),$$ where $P(\nu_{1},\nu_{2})$ is the set of all probability measures $\pi$ on $X\times X$ such that $\pi(dx\times X)=\nu_{1}(dx)$ and $\pi(X\times dy)=\nu_{2}(dy).$ The probability measure $\mu$ is said to satisfy the *transport-entropy inequality ${\bf T}_c(C)$*, for some $C>0$ if $$\tag{${\bf T}_c(C)$} \mathcal{T}_c(\mu,\nu)\leq C H(\nu| \mu), \qquad\forall \nu \in \mathcal{P}(X),$$ where $$H(\nu|\mu)= \left\{ \begin{array}{ll} \int \log \frac{d\nu}{d\mu} \,d\nu & \mbox{if } \nu \ll \mu \\ +\infty & \mbox{otherwise } \end{array} \right.$$ is the relative entropy of $\nu$ with respect to $\mu$. This class of inequalities was introduced by Marton and Talagrand [@Ma86; @Ta96]. When $c(x,y)=\alpha(d(x,y))$ we denote the optimal transport cost by $\mathcal{T}_\alpha(\,\cdot\,,\,\cdot\,)$ and the corresponding transport inequality by ${\bf T}_\alpha.$ In the particular case, when $\alpha(x)=x^p/p,$ $p\geq2$ we use the notation $\mathcal{T}_p$ and ${\bf T}_p$. The first point of the next theorem will appear to be an easy consequence of Theorem \[thm-hyper\] and of Bobkov and Götze dual formulation of the inequality ${\bf T}_\alpha$ (which roughly speaking corresponds to the hypercontractivity with $t_o=C(p_\alpha-1)$ or equivalently $k(0)=0$). \[Otto-Villani\] Suppose that $\alpha$ verifies the $\Delta_2$-condition. If $\mu$ verifies ${\bf LSI}_{\alpha}^-(C)$, then it verifies ${\bf T}_{\alpha}(A)$, with $$A=\max\left( ((p_\alpha-1)C)^{r_\alpha-1};((p_\alpha-1)C)^{p_\alpha-1}\right),$$ where the numbers $r_\alpha,p_\alpha$ are defined in Theorem \[hypercont\]. In a Riemannian framework and for the quadratic function $\alpha(t)=t^2/2$, Theorem \[Otto-Villani\] was first obtained by Otto and Villani in [@OV00], closely followed by Bobkov, Gentil and Ledoux [@BGL01]. Extensions to other functions $\alpha$ were provided in [@BGL01; @GGM05]. The path space case was treated by Wang in [@W04]. In [@LV07], Lott and Villani extended to certain geodesic measured spaces $(X,d,\mu)$ the Hamilton-Jacobi approach of [@BGL01] in the quadratic case. They proved Theorem \[Otto-Villani\] under additional assumptions on $\mu$ (doubling property and local Poincaré). Under the same assumptions Balogh, Engoulatov, Hunziker and Maasalo [@BEHM09] treated the case of ${\bf LSI}_q^-$ for all $q\leq2$. The first proofs of Otto-Villani theorem valid on any complete separable metric space appeared in [@Go09] and [@GRS12]. Their common feature is the use of the stability of the log-Sobolev inequality under tensor products of the reference probability measure. In a recent paper [@GL12], Gigli and Ledoux give another quick proof of Otto-Villani theorem on metric spaces. It is based on calculations along gradient flows in the Wasserstein space. Using some rough properties of the operators $Q_t$, we also provide a metric space generalization of another result by Otto and Villani [@OV00] relating transport-entropy inequalities to Poincaré inequality. \[TransversPoincare\] Let $\theta:{\mathbb{R}}^+\to{\mathbb{R}}^+$ be any function such that $\theta(x)\geq \min(x^2,a^2)$ for some $a>0$. If $\mu$ verifies ${\bf T}_\theta(C)$ for some $C>0$, then it verifies the following Poincaré inequality: $$\mathrm{Var}_\mu (f) \leq \frac{C}{2} \int |\nabla^- f|^2\,d\mu,$$ for all bounded function $f$ such that $\mathrm{Lip}(f,r)<\infty$, for some $r>0$. Transport-entropy inequalities as restricted log-Sobolev inequalities --------------------------------------------------------------------- A second consequence of the Hamilton-Jacobi approach on metric spaces is a characterization of transport-entropy inequalities in terms of log-Sobolev inequalities restricted to a certain class of functions depending on the cost function $\alpha$. To be more precise, let us say that a function $f$ is *$c$-convex* with respect to a cost function $(x,y)\mapsto c(x,y)$ defined on $X\times X$ if there is a function $g:X\to{\mathbb{R}}\cup\{\pm\infty\}$ such that $$f(x)=P_c g(x) = \sup_{y\in X} \{g(y)-c(x,y)\} \in {\mathbb{R}}\cup\{\pm\infty\},\qquad \forall x\in X.$$ The class of $c$-convex functions is intimately related to optimal-transport, via for instance the Kantorovich duality theorem (see e.g [@Villani-book]). An important case is when $c(x,y)=\frac{1}{2}\|x-y\|_2^2$ on ${\mathbb{R}}^m$ (see Proposition \[examples\] below). In this case, a function $f:{\mathbb{R}}^m \to{\mathbb{R}}$ is $c$-convex if and only if the function $x\mapsto f(x)+\|x\|_{2}^2/2$ is convex on ${\mathbb{R}}^m$. If $f$ is of class $\mathcal{C}^2$, this amounts to say that $\mathrm{Hess}\, f \geq -\mathrm{Id}$.\ In what follows, we consider the cost $c_p(x,y)=d^p(x,y)/p,$ $p\geq 2$. The second main result of this paper is the following \[main result\] Let $\mu$ be a probability measure on a geodesic space $(X,d)$ and $p\geq 2$. The following properties are equivalent: 1. There is some $C>0$ such that $\mu$ verifies ${\bf T}_p(C)$.\ 2. There is some $D>0$ such that $\mu$ verifies the following $(\tau)$-log-Sobolev inequality: for all bounded continuous $f$ and all $0<\lambda<1/D$, it holds $${\operatorname{Ent}}_\mu(e^f)\leq \frac{1}{1-\lambda D} \int (f-Q^\lambda f)e^f\,d\mu,$$ where for all $\lambda>0$, $Q^\lambda f(x)=\inf_{y\in X}\left\{f(y) +\lambda c_{p}(x,y)\right\}.$\ 3. There is some $E>0$ such that $\mu$ verifies the following restricted log-Sobolev inequality: for all $Kc_p$-convex function $f$, with $0<K<1/E$ it holds $${\operatorname{Ent}}_{\mu}(e^f)\leq \frac{\beta_p(u)-1}{pK^{q-1} (1-KEu)}\int |\nabla^+ f|^{q}e^f\,d\mu,\qquad \forall u\in(1,1/(KE)),$$ where $q=p/(p-1)$ and $\beta_p(u)= \frac u{[u^{1/(p-1)}-1]^{p-1}}$ for all $u>1.$ The optimal constants $C_\mathrm{opt},D_\mathrm{opt},E_\mathrm{opt}$ are related as follows $$E_\mathrm{opt} \leq D_\mathrm{opt}\leq C_\mathrm{opt}\leq \kappa_p E_\mathrm{opt},$$ where $\kappa_p$ is some universal constant depending only on $p.$ For $p=2$, one can take $\kappa_2 = e^2.$ Let us make some comments on Theorem \[main result\]. - The implication $(1)\Rightarrow (2)$ is true for any cost function $c$. It was first proved in [@GRS11]. - In [@GRS12], we proved that (1) is equivalent to (2) for cost functions $c(x,y)=\alpha(d(x,y))$ as soon as $\alpha$ verifies the $\Delta_2$-condition. Our proof (in [@GRS12]) makes use of a tensorization technique and is thus rather different from the one presented here. - In [@GRS11], we proved that (1) is equivalent to (3) in a framework essentially Euclidean: $X={\mathbb{R}}^m$ and $c(x,y)=\frac{1}{2}\|x-y\|_2^2$. Theorem \[main result\] thus provides a wide extension of the results in [@GRS11] and unifies nicely the results of [@GRS11] and [@GRS12]. Let us mention that Theorem \[main result\] as stated above is not as general as possible. Indeed, we will see in Section 5 that this equivalence is still true when the space is not geodesic (Theorem \[main result improved\]). In this more general framework (3) has to be replaced by a slightly weaker version of the restricted log-Sobolev inequality. The main tool to prove this extension is Theorem \[HJ 1\]. It would also be possible to consider more general costs of the form $c(x,y)=\alpha(d(x,y))$ with $\alpha$ satisfying the $\Delta_2$-condition but, to avoid some lengthy developpements, this will not be treated here. We end this introduction with a short roadmap of the paper. Section 2 is devoted to $c$-convex functions. In particular, we will recall and prove some well known facts about the subdifferential $\partial_cf(x)$ of a $c$-convex function. In Proposition \[Gradients comparisons\], we will relate their gradients $|\nabla^\pm f|(x)$ to the minimal or maximal distance between $x$ and the subdifferential $\partial_cf(x).$ Section 3 contains the proof of the HLO formula. In Section 4, we prove the hypercontractivity property of Theorem \[hypercont\], and deduce as a corollary the Otto-Villani Theorem \[Otto-Villani\]. Section 5 contains the proof of an improved version of our main result Theorem \[main result\]. Finally, the appendix gathers some technical results. About $c$-convex functions ========================== In this section we introduce the somehow classical notions of $c$-convex (and $c-$concave) functions and of $c$-subdifferential. We will also give several useful facts about these notions. The interested reader may find more results and comments, and some bibliographic notes, in [@Villani-book Chapter 5]. Definition of $c$-convex functions and first results ---------------------------------------------------- Let $X,Y$ be two polish spaces and $c:X\times Y\to{\mathbb{R}}$ be a general cost function and set $\overline{{\mathbb{R}}}={\mathbb{R}}\cup\{\pm \infty\}.$ For any function $f:X\to \overline{{\mathbb{R}}}$, we define $Q_cf:Y\to \overline{{\mathbb{R}}}$ by $$Q_c f(y):= \inf_{x\in X}\{f(x)+c(x,y)\}.$$ For any function $g:Y\to \overline{{\mathbb{R}}}$, we define $P_cg:X\to \overline{{\mathbb{R}}}$, by $$P_c g(x):= \sup_{y\in Y}\{g(y)-c(x,y)\}.$$ A function $f:X\to\overline{{\mathbb{R}}}$ is said to be *$c$-convex* if there is some function $g:Y\to \overline{{\mathbb{R}}}$ such that $f=P_c g.$ A function $g:Y\to \overline{{\mathbb{R}}}$ is said to be *$c$-concave* if there is some function $f:X \to \overline{{\mathbb{R}}}$ such that $g = Q_c f.$ In the definition above, we follow the convention of Villani’s book for $c$-convex functions [@Villani-book]. Other authors as Rachev and Rüschendorf [@RR-book] define $c$-convex functions as those functions $f:X\to\overline{{\mathbb{R}}}$ such that there is some function $g:Y \to \overline{{\mathbb{R}}}$ such that $f(x)=\sup_{y\in Y}\{g(y) + c(x,y)\}.$ For any function $f:X\to\overline{{\mathbb{R}}}$, the inequality $P_cQ_cf\leq f$ holds. Moreover, $f:X \to \overline{{\mathbb{R}}}$ is $c$-convex if and only if $P_c Q_cf =f.$ For the first point observe that; for $z=x$, $$P_c Q_c f(x)=\sup_{y\in Y} \inf_{z\in X} \{f(z)+c(z,y)-c(x,y)\}\leq f(x) .$$ Let us prove the second point. Trivially, a function $f$ such that $f=P_cQ_cf$ is $c$-convex. Conversely, if $f:X\to \overline{{\mathbb{R}}}$ is $c$-convex, then there is some function $g$ on $Y$ such that $f(x)=\sup_{y\in Y}\{g(y)-c(x,y)\}=Q_c g(y)$. Hence $g$ verifies $g(y)\leq \inf_{x\in X}\{f(x) + c(x,y)\}.$ Plugging this inequality into $f=P_{c}g$ gives $f\leq P_cQ_cf$. Since the other direction always holds, the proof is complete. Recall that a function $f:{\mathbb{R}}^m \to \overline{{\mathbb{R}}}$ is said to be *closed* (see [@Rock-book]) if either $f=-\infty$ everywhere or $f$ takes its values in ${\mathbb{R}}\cup\{+\infty\}$ and is lower semicontinuous. It is said to be *convex* if its epigraph $\{(x,\alpha)\in {\mathbb{R}}^m\times {\mathbb{R}}: \alpha \geq f(x)\}$ is a convex subset of ${\mathbb{R}}^m \times {\mathbb{R}}.$ Let us denote by $\Gamma ({\mathbb{R}}^m)$ the set of all closed and convex functions on ${\mathbb{R}}^m.$ \[examples\] Assume that $X=Y={\mathbb{R}}^m,$ $m\in{\mathbb{N}}^*$, equipped with its standard Euclidean structure and let $f:{\mathbb{R}}^m\to \overline{{\mathbb{R}}}$. Then, 1. If $c(x,y)=x\cdot y,$ $f$ is $c$-convex if and only if $f\in \Gamma({\mathbb{R}}^m)$. 2. If $c(x,y)=\frac{1}{2}\|x-y\|_2^2$, $f$ is $c$-convex if and only if $f+\|\cdot\|_2^2/2 \in \Gamma({\mathbb{R}}^m)$. In particular, if $f:{\mathbb{R}}^m \to {\mathbb{R}}$ is of class $\mathcal{C}^2$ then it is $c$-convex if and only if $\mathrm{Hess}\, f (x) \geq -\mathrm{Id},$ for all $x\in {\mathbb{R}}^m$.  \ (1) By definition, a function $f$ is $c$-convex for $c(x,y)=x\cdot y$ if and only if $f=h^*$ for some function $h:{\mathbb{R}}^m \to \overline{{\mathbb{R}}}$. It is well known (and easy to check) that $h^* \in \Gamma({\mathbb{R}}^m)$ for all $h$. Conversely, if $f\in \Gamma({\mathbb{R}}^m)$ then $f=f^{**}$ (see e.g [@Rock-book]) and so $f$ is $c$-convex.\ (2) The function $f$ is a $c$-convex function for $c(x,y)=\|x-y\|_2^2/2$ if and only if $f=P_c g$, for some $g:{\mathbb{R}}^m\to\overline{{\mathbb{R}}}.$ Since $$f(x) + \frac{\|x\|_{2}^2}{2}=\sup_{y\in {\mathbb{R}}^m}\left\{x\cdot y - \left(\frac{\|y\|_{2}^2}{2}-g(y)\right)\right\},$$ the conclusion follows from the first point. The $c$-subdifferential of a $c$-convex function ------------------------------------------------ In this section we define the notion of $c$-subdifferential of a $c$-convex function and derive some facts that will appear to be useful later. \[subdiff\] Let $f:X\to\overline{{\mathbb{R}}}$ be a $c$-convex function and $x\in X$; the $c$-subdifferential of $f$ at point $x$ is the set, denoted by $\partial_{c}f(x)\subset Y$, of the points $\bar{y} \in Y$ such that $$f(z)\geq f(x)+c(x,\bar{y}) - c(z,\bar{y}),\qquad \forall z\in X.$$ The next lemma gives a characterisation of the $c$-subdifferential. \[attainment\] For all $x\in X$, $\partial_cf(x)$ is the set of points $y\in Y$ achieving the supremum in $f(x)=P_cQ_c f(x)$. More precisely, $$\partial_c f(x) = \{ y\in Y : f(x)= Q_cf(y) - c(x,y)\}.$$ More generally, if $f=P_c g$, for some function $g:Y\to \overline{{\mathbb{R}}}$, then $$\{y\in Y : f(x)=g(y) - c(x,y)\} \subset \partial_c f(x).$$ The first part of the lemma is simple and left to the reader. Let us prove the second part. Since $f(x)=\sup_{y\in Y}\{g(y) - c(x,y)\}$, $x\in X$, we have $g\leq Q_c f$. So if, $f(x)=g(\bar{y})-c(x,\bar{y})$ then $f(x)\leq Q_cf(\bar{y})-c(x,\bar{y}) \leq f(z) +c(z,\bar{y}) -c(x,\bar{y})$, for all $z\in X$ which proves that $\bar{y}\in \partial_cf(x).$ \[not empty\] Suppose that the function $c:X\times Y\to {\mathbb{R}}$ is continuous and bounded from below and that, for all $x\in X$, the level sets $\{y\in Y; c(x,y) \leq r\}$, $r\in {\mathbb{R}}$, are compact. If $f:X\to {\mathbb{R}}\cup\{-\infty\}$ is a $c$-convex function bounded from above, then $\partial_cf(x)\neq \emptyset$ for all $x\in X.$ The function $Q_cf$ is an infimum of continuous functions on $Y$, so it is upper semicontinuous on $Y$. For all $x\in X$, the function $\varphi_x: y\mapsto Q_cf(y)-c(x,y)$ is thus upper semicontinuous on $Y$. Since $f$ is bounded from above and $c$ from below, the function $\varphi_x$ is bounded from above. Finally if $y\in \{\varphi_x\geq r\}$ then $c(x,y)\leq \sup f + \inf_{z} c(x,z) - r$. Hence $\{\varphi_x\geq r\}$ is compact. From this follows that $\varphi_x$ achieves its supremum at some point $\bar{y}$ which, according to Lemma \[attainment\], necessarily belongs to $\partial_cf(x)$. For a better understanding of the notion, in the next lemma we express the $c$-subdifferential of a $c$-convex function $f$ in term of its gradient in some simple cases. \[bysarko1\] Suppose that $X=Y={\mathbb{R}}^m$ and that $c(x,y)=L(x-y)$ where $L:{\mathbb{R}}^m\to{\mathbb{R}}^+$ is a differentiable convex function with superlinear growth, i.e $L(x)/\|x\|\to +\infty$ when $x\to \infty$, where $\|\cdot\|$ denotes any norm on ${\mathbb{R}}^m$. Let $f$ be a $c$-convex function bounded from above differentiable at some point $x$. Then $$\partial_c f(x)= \{x-\nabla (L^*)(-\nabla f(x))\},$$ where $L^*(y)=\sup_{x\in {\mathbb{R}}^m}\{x\cdot y -L(y)\}$ is the Fenchel-Legendre transform of $L$. We recall that if $L$ is strictly convex and has a superlinear growth, then its Fenchel-Legendre transform is differentiable everywhere [@Rock-book]. Lemma \[bysarko1\] is well known. However, for the sake of completeness, we will recall its proof in the appendix. Comparisons of gradients ------------------------ In this last section, as in the rest of the paper, we will assume that $(X,d)$ is a complete separable metric space in which closed balls are compact. We take $Y=X$ and we consider a cost function $c$ on $X \times X$ of the form $$c(x,y)=\alpha (d(x,y)),$$ where $\alpha:{\mathbb{R}}^+\to{\mathbb{R}}^+$ is an increasing convex function of class $\mathcal{C}^1$ such that $\alpha(0)=0$. If $f:X\to{\mathbb{R}}$ is $c$-convex for the cost $c(x,y)=\alpha(d(x,y))$, we introduce the following quantities $$|\nabla_{c}^-f|(x)=\alpha'\left( \inf_{\bar{y}\in \partial_{c}f(x)} d(x,\bar{y})\right)\qquad \text{and}\qquad|\nabla_{c}^+f|(x)=\alpha'\left( \sup_{\bar{y}\in \partial_{c}f(x)} d(x,\bar{y})\right).$$ The following proposition compares $|\nabla^\pm_{c}f|$ to $|\nabla^\pm f|$ defined in . \[Gradients comparisons\] Let $f:X\to{\mathbb{R}}$ be a $c$-convex function for the cost $c(x,y)=\alpha(d(x,y))$. Suppose that $f=P_{c}g$ for some upper semicontinuous function $g:X\to {\mathbb{R}}$ bounded from above and consider for all $x\in X$ the set $m(x)$ defined by $m(x)=\{ y \in X : f(x)=g(y)-\alpha(d(x,y))\}.$ 1. The following inequalities hold $$|\nabla^+ f|(x)\leq \alpha'(\max_{\bar{y} \in m(x)} d(x,\bar{y}))\leq |\nabla_{c}^+f|(x).$$ 2. If $(X,d)$ is a geodesic space, then $$|\nabla^+ f|(x)= \alpha'(\max_{\bar{y} \in m(x)} d(x,\bar{y}))= |\nabla_{c}^+f|(x).$$ 3. The following inequalities hold $$|\nabla^-f|(x)\leq |\nabla_{c}^-f|(x)\leq \alpha' (\min_{\bar{y}\in m(x)} d(x,\bar{y})).$$ We do not know if there is equality in (3) when the space is geodesic.   (1) First observe that, since $f=P_c g$ with $g$ bounded above, $f$ is locally Lipschitz (see [@GRS12 Lemma 3.8]), so that $|\nabla^+f|$ is finite everywhere. The second inequality is an immediate consequence of the definition of $|\nabla_{c}^+f|(x)$ and the fact that, according to Lemma \[attainment\], $m(x)\subset \partial_{c}f(x)$. Let us prove the first inequality. Let $(x_n)_{n\in {\mathbb{N}}}$ be a sequence of points converging to $x$, with $x_n\neq x$ for all $n$. For all $n$, fix $y_n\in m(x_n)$. It holds $$\begin{aligned} f(x_n)-f(x)&\leq g(y_n)-\alpha(d(x_n,y_n))-\left(g(y_n)-\alpha(d(x,y_n))\right)\\ &\leq d(x,x_n)\alpha'\left(\max(d(x_n,y_n);d(x,y_n))\right),\end{aligned}$$ where the last inequality follows from the mean value theorem, the triangle inequality, the non-negativity and the monotonicity of $\alpha'$. Since the function $t\mapsto [t]_+$ is non-decreasing, we get $$\frac{[f(x_n)-f(x)]_+}{d(x_n,x)}\leq \alpha'(\max(d(x_n,y_n);d(x,y_n))).$$ So letting $n\to\infty$, $$\begin{aligned} \limsup_{n\to\infty}\frac{[f(x_n)-f(x)]_+}{d(x_{n},x)}&\leq\alpha'\left(\limsup_{n\to \infty}d(x,y_n)\right)\\&= \alpha'\left(\max\{d(x,\bar{y}) : \bar{y} \text{ limit point of } (y_n)_{n\in {\mathbb{N}}}\}\right).\\ & \leq \alpha'\left( \max_{\bar{y} \in m(x)} d(x,\bar{y})\right),\end{aligned}$$ where the last inequality comes from Lemma \[Kuratowski\] bellow. \(2) To prove the second point it is enough to show that $|\nabla_{c} ^+f|(x) \leq |\nabla ^+ f|(x)$ for all $x\in X$. Let $\bar{y}\in \partial_cf(x)$. According to the definition of the $c$-subdifferential, $$f(z)-f(x)\geq \alpha(d(x,\bar{y}))-\alpha(d(z,\bar{y})),\qquad \forall z\in X.$$ From the definition of $|\nabla^+ f|(x)$, it follows that $$|\nabla^+ f|(x)\geq \limsup_{z \to x} \frac{ \alpha(d(x,\bar{y}))-\alpha(d(z,\bar{y}))}{d(x,z)}.$$ Let $(z_t)_{t\in [0,1]}$ be a geodesic connecting $x$ to $\bar{y}$, it holds $d(x,z_t)=td(x,\bar y)$, $d(z_t,\bar y)=(1-t)d(x,\bar y)$ and therefore $$|\nabla^+ f|(x)\geq \limsup_{t \to 0} \frac{ \alpha(d(x,\bar{y}))-\alpha((1-t)d(x,\bar y))}{td(x,\bar y)}=\alpha'(d(x,\bar y)).$$ Optimizing over all $\bar y \in \partial_{c}f(x)$ completes the proof. \(3) Let $(x_n)_{n\in {\mathbb{N}}}$ be a sequence of points converging to $x$, with $x_n\neq x$ for all $n$. If $\bar{y}\in \partial_cf(x)$, then it holds $$\begin{aligned} f(x_{n})-f(x)&\geq \alpha\left(d(x,\bar{y})\right)-\alpha\left(d(x_n,\bar{y})\right)\\ &\geq -{d(x,x_{n})}\alpha'\left(\max(d(x_{n},\bar{y});d(x,\bar{y}))\right),\end{aligned}$$ where the second inequality follows from the mean value theorem and the triangle inequality. Since the function $t\mapsto[t]_{-}$ is non-increasing, it holds $$\limsup_{n\to+\infty} \frac{[f(x_{n})-f(x)]_{-}}{d(x,z_{n})}\leq \alpha'\left(d(x,\bar{y})\right).$$ Optimizing over all $\bar{y}\in \partial_cf(x)$ leads to the first bound in (3). As above, the second inequality in $(3)$ is an immediate consequence of the definition of $|\nabla^-_c f|(x)$ together with the fact that, according to Lemma \[attainment\], $m(x)\subset \partial_{c}f(x)$. This achieves the proof. During the proof we have used the following simple lemma whose proof can be found in the appendix. \[Kuratowski\] Let $X$ be a complete separable metric space with compact balls and $g:X\to {\mathbb{R}}$ be an upper semicontinuous function bounded from above. Define, for all $x \in X$, $P_{t}g(x)=\sup_{y\in X}\left\{ g(y) -t\alpha\left(\frac{d(x,y)}{t}\right) \right\}$ and $m(t,x)$ as the set of points $y \in X$ where this supremum is reached. Then, 1. The set $m(t,x)$ is a non empty compact set of $X$. 2. Let $x_{n}\to x\in X$ and $t_{n}\to t>0$ be two converging sequences and consider a sequence $(y_{n})_{n\in {\mathbb{N}}}$ such that $y_{n}\in m(t_{n},x_{n})$ for all $n$. Then $(y_{n})_{n\in {\mathbb{N}}}$ is bounded and all its limit points belong to $m(t,x).$ Proof of the Hamilton-Jacobi equations ====================================== This part is devoted to the proof of Theorem \[HJ\] and \[HJ 1\]. According to Lemma \[Kuratowski\], $m(t,x)$ is a non empty compact set of $X$. We treat the case of the right derivative; the other case is completely analogous. Let $t>0$, $x\in X$ and $(h_{n})_{{n\in {\mathbb{N}}}}$ a sequence of positive numbers converging to $0$. For all $n\in {\mathbb{N}}$, we consider $z_{n}\in m(t+h_{n},x).$ Then, $$\begin{aligned} \frac{1}{h_{n}}\left(P_{t+h_{n}}f(x)-P_{t}f(x)\right) &\leq \frac{1}{h_{n}}\left[ f(z_{n})-(t+h_n)\alpha\left(\frac{d(x,z_n)}{t+h_n}\right)-\left(f(z_{n})-t\alpha\left(\frac{d(x,z_n)}t\right)\right)\right]\\ &= \frac{1}{h_{n}}\left[t\alpha\left(\frac{d(x,z_n)}t\right) -(t+h_n)\alpha\left(\frac{d(x,z_n)}{t+h_n}\right)\right] .\end{aligned}$$ Define $D= \limsup_{k\rightarrow \infty} d(x,z_k)$ and take $\varepsilon>0$. For all $n$ large enough, $$d(x,z_n)\leq D+\varepsilon .$$ For all $h\geq 0$, all $t >0$, by the convexity assumption on $\alpha$, the map $$d\mapsto t\alpha\left( \frac d t \right)-(t+h)\alpha\left( \frac d {t +h}\right)$$ is non-decreasing. Hence $$\begin{aligned} \limsup_{n\to\infty} \frac{1}{h_{n}} & \left[ t\alpha \left( \frac{d(x,z_n)}t\right) -(t+h_n)\alpha\left(\frac{d(x,z_n)}{t+h_n}\right) \right] \\ & \quad \leq \lim_{n\to\infty} \frac{1}{h_{n}} \left[ t\alpha\left(\frac{D+\varepsilon}t\right) -(t+h_n)\alpha\left(\frac{D+\varepsilon }{t+h_n}\right)\right] = \beta \left(\frac{D+\varepsilon}t \right)\end{aligned}$$ where we recall that $\beta(h)=h\alpha'(h)-\alpha(h)$, $h \geq 0$. Since $\alpha$ is of class ${\mathcal C}^1$, as $\varepsilon$ goes to $0$ we get $$\limsup_{n\to+\infty}\frac{1}{h_{n}}\left(P_{t+h_{n}}f(x)-P_{t}f(x)\right)\leq \beta \left(\frac{D}t \right).$$ Applying Lemma \[Kuratowski\], it is not difficult to check that $$D=\limsup_{n\to\infty} d(x,z_n)=\max\{d(x,\bar{z}) : \bar{z}\text{ limit point of } (z_n)_{n\in {\mathbb{N}}}\}\leq \max_{\bar{y}\in m(t,x)} d(x,\bar{y}).$$ The conditions on $\alpha$ ensure that $\beta $ is non-decreasing and therefore $$\label{liminf} \limsup_{n\to+\infty}\frac{1}{h_{n}}\left(P_{t+h_{n}}f(x)-P_{t}f(x)\right)\leq \beta \left(\frac{\max_{\bar{y}\in m(t,x)} d(x,\bar{y})}t \right).$$ Analogously, if $\bar{y} \in m(t,x)$ then $$\begin{aligned} \frac{1}{h_{n}}\left(P_{t+h_{n}}f(x)-P_{t}f(x)\right)&\geq \frac{1}{h_{n}} \left(t\alpha\left(\frac{d(x,\bar{y})}t\right) -(t+h_n)\alpha\left(\frac{d(x,\bar{y})}{t+h_n}\right)\right)\end{aligned}$$ So, letting $n$ go to $\infty$, and optimizing over $\bar{y}$ yields $$\label{limsup} \liminf_{n\to \infty} \frac{1}{h_n}\left(P_{t+h_{n}}f(x)-P_{t}f(x)\right)\geq\beta \left(\frac{\max_{\bar{y}\in m(t,x)} d(x,\bar{y})}t \right).$$ We conclude from and that $$\lim_{n\to\infty} \frac 1{h_n}\left(P_{t+h_{n}}f(x)-P_{t}f(x)\right) = \beta \left(\frac{\max_{\bar{y}\in m(t,x)} d(x,\bar{y})}t \right).$$ This completes the proof of proposition \[HJ 1\]. According to Theorem \[HJ 1\], $$\frac{d}{dt_+} P_t f(x) = \beta\left(\frac{\max_{\bar{y} \in m(t,x)}d(x,\bar{y})}{t}\right),$$ with $\beta(u)=u\alpha'(u)-\alpha(u)$, for all $u\geq0.$ By definition of the $c$-convexity, the function $x\mapsto P_tf(x)$ is $c$-convex for the cost $c(x,y)=t\alpha\left(\frac{d(x,y)}{t}\right)$. Applying the point (1) of Proposition \[Gradients comparisons\], it holds $$|\nabla^+ P_tf|(x)\leq \alpha'\left(\frac{\max_{\bar{y} \in m(t,x)}d(x,\bar{y})}{t}\right).$$ Observing that $\beta(u)=\alpha^*(\alpha'(u))$ gives the result. According to point (3) of Proposition \[Gradients comparisons\], equality holds in the geodesic case. The proof of the inequality involving the left derivative of $P_tf$ is similar. log-Sobolev inequality and hypercontractivity on a metric space {#sectionlogsob} =============================================================== In this section, following [@BGL01], we show that log-Sobolev inequalities on metric spaces are equivalent to some hypercontractivity property of the “semigroup" $Q_t$. The proof of Theorem \[thm-hyper\] relies on the differentiation of the left hand side of . To that purpose, we use the next technical proposition whose proof is postponed to the appendix. \[deriv\] Let $f$ be a bounded and continuous function on $X$ and $k:(a,b)\to (0,+\infty)$ be a function of class $\mathcal{C}^1$ defined on an open interval $(a,b)\subset (0,\infty)$ and such that $k'(t)\neq 0$ for all $t$. Define $$H(t)=\frac{1}{k(t)}\log\left(\int e^{k(t)Q_{t}f}\,d\mu \right) \quad \mbox{and} \quad K(t)=\frac{1}{k(t)}\log\left(\int e^{k(t)P_{t}f}\,d\mu \right),\qquad t\in (a,b) .$$ The functions $H$ and $K$ are continuous and differentiable on the right and on the left on $(a,b)$. Moreover, for all $t\in (a,b)$, it holds $$\frac{dH}{dt_+ } (t)=\frac{k'(t)}{k(t)^2} \frac{1}{\int e^{k(t)Q_{t}f}\,d\mu}\left[{\operatorname{Ent}}_{\mu}\left( e^{k(t)Q_{t}f}\right)+ \frac{k(t)^2}{k'(t)} \int \left(\frac{d}{dt_+ } Q_{t}f\right) e^{k(t)Q_{t}f}\,d\mu\right].$$ The same formula holds for $dH/dt_-$, $dK/dt_+$ and $dK/dt_-$ (replacing $Q_t$ by $P_t$). Let us first show that the log-Sobolev inequality implies the hypercontractivity property: $$\label{hypercont-proof} \left\|e^{Q_tf}\right\|_{k(t)}\leq \left\|e^f\right\|_{k(0)},$$ for all bounded continuous function $f:X\to {\mathbb{R}},$ with $$\label{k(t)} k(t)=\left(1+\frac{C^{-1}(t-t_o)}{p_\alpha-1}\right)^{p_\alpha-1}\mathbf{1}_{t\leq t_o} + \left(1+\frac{C^{-1}(t-t_o)}{r_\alpha-1}\right)^{r_\alpha-1}\mathbf{1}_{t> t_o},$$ with the convention that $k(t)=\min\left(1;\left(1+\frac{C^{-1}(t-t_o)}{p_\alpha-1}\right)^{p_\alpha-1} \right),$ if $r_\alpha=1.$ The exponents $r_\alpha$ and $p_\alpha$ have the following property (see [@GRS12 proof of Lemma A.3]): $$\begin{aligned} \alpha^*(sx)&\leq s^{\frac{p_\alpha}{p_\alpha-1}}\alpha^*(x),\qquad \forall x\geq0,\forall s\in[0,1]\\ \alpha^*(sx)&\leq s^{\frac{r_\alpha}{r_\alpha-1}}\alpha^*(x),\qquad \forall x\geq0,\forall s>1.\end{aligned}$$ Let $H(t)=\log \left\|e^{Q_tf}\right\|_{k(t)}$, with $f:X\to {\mathbb{R}}$ bounded and continuous. According to Proposition \[deriv\], we have for all $t>0$ $$\frac{dH}{dt_+ } (t)\leq \frac{k'(t)}{k^2(t)} \frac{1}{\int e^{k(t)Q_{t}f}\,d\mu}\left[{\operatorname{Ent}}_{\mu}\left( e^{k(t)Q_{t}f}\right)+\frac{k^2(t)}{k'(t)} \int \frac{d}{dt_{+}}Q_{t}f e^{k(t)Q_{t}f}\,d\mu\right].$$ Applying ${\bf LSI}_\alpha^-(C)$ to the function $k(t)Q_{t}f$ (which belongs to $\mathcal{F}_\alpha$ thanks to Lemma \[Falpha\] below), it follows that for all $t>0$ (or all $0<t\leq t_o$ if $r_\alpha=1$), $$\begin{aligned} {\operatorname{Ent}}_{\mu}\left(e^{k(t)Q_{t}f}\right)&\leq C\int \alpha^*\left(k(t)|\nabla^- Q_{t}f|\right)e^{k(t)Q_{t}f}\,d\mu\\ &\leq C\left(k(t)^{\frac{p_\alpha}{p_\alpha-1}}\mathbf{1}_{t\leq t_o} + k(t)^{\frac{r_\alpha}{r_\alpha-1}}\mathbf{1}_{t> t_o}\right)\int \alpha^*\left(|\nabla^- Q_{t}f|\right)e^{k(t)Q_{t}f}\,d\mu\\ & \leq -C\left(k(t)^{\frac{p_\alpha}{p_\alpha-1}}\mathbf{1}_{t\leq t_o} + k(t)^{\frac{r_\alpha}{r_\alpha-1}}\mathbf{1}_{t> t_o}\right) \int \frac{d}{dt_{+}} Q_{t}fe^{k(t)Q_{t}f}\,d\mu,\end{aligned}$$ where the last inequality follows from the Hamilton-Jacobi differential inequality . Therefore, $$\frac{dH}{dt_+ } (t)\leq \frac{1-Ck'(t)\left(k(t)^{\frac{2-p_\alpha}{p_\alpha-1}}\mathbf{1}_{t\leq t_o} + k(t)^{\frac{2-r_\alpha}{r_\alpha-1}}\mathbf{1}_{t> t_o}\right)}{\int e^{k(t)Q_{t}f}\,d\mu} \int \frac{d}{dt_{+}} Q_{t}fe^{k(t)Q_{t}f}\,d\mu =0$$ where the last equality is a consequence of the very definition of $k$. Hence $H$ is non-increasing on $(0,+\infty)$ (or on $(0,t_o]$ if $r_\alpha=1$). When $\alpha(h)/h \to\infty$, when $h\to\infty,$ then according to point (3) of Proposition \[prop Q\_[t]{}\] and the dominated convergence theorem, it holds $$\log \left\|e^{Q_{t}f}\right\|_{k(t)} = H(t)\leq \lim_{s \to 0^+} H(s) = \log \left\|e^f\right\|_{k(0)}.$$ If $\alpha(h)/h \to \ell \in {\mathbb{R}}^+$, when $h\to\infty$, then according to point (3) of Proposition \[prop Q\_[t]{}\], the same conclusion holds if $\mathrm{Lip}(f)<\ell.$ Consider now a bounded continuous function $f : X \to \mathbb{R}$ and fix $\varepsilon \in (0,1)$. Thanks to Lemma \[Falpha\] below, $\mathrm{Lip}((1-\varepsilon)Q_s f)\leq (1-\varepsilon)\ell$ for all $s>0$. Since $Q_sf\leq f$, we can conclude that $$\left\|e^{Q_t ((1-\varepsilon) Q_s f)}\right\|_{k(t)}\leq \left\|e^{(1-\varepsilon)Q_sf}\right\|_{k(0)}\leq \left\|e^{(1-\varepsilon)f}\right\|_{k(0)}.$$ Using Lebesgue’s Theorem and Lemma \[Falpha\], as $\varepsilon \to 0$, we get $$\left\|e^{Q_t ( Q_s f)}\right\|_{k(t)} \leq \left\| e^{f}\right\|_{k(0)}.$$ Since $Q_{t+s}f\leq Q_t(Q_sf)$ and thanks to point (2) of Proposition \[prop Q\_[t]{}\], we have $\lim_{s \to 0} Q_{t+s} f = Q_t f$ so that (using Lebesgue’s theorem) the hypercontractivity property still holds when $f$ is bounded and continuous, as expected. Now we prove that if holds for all bounded continuous $f$ and all $t>0$ with $k$ defined by , then $\mu$ verifies ${\bf LSI}_\alpha^-(C)$. Observe that in the case $\alpha(h)/h\to \ell\in {\mathbb{R}}^+$, it is enough to show that ${\bf LSI}_\alpha^-$ holds for functions with $\mathrm{Lip}(f)<\ell.$ Let $H(t)=\log \left\|e^{Q_tf}\right\|_{k(t)}$, for all $t>0$, with $f\in \mathcal{F}_\alpha$ and $\mathrm{Lip} (f)< \ell$ when $\alpha(h)/h \to\ell \in {\mathbb{R}}^+$ as $h\to\infty.$ By assumption, it holds $$\limsup_{t\to 0^+} \frac{H(t)-H(0^+)}t\leq 0.$$ Let us choose $t_o<C(p_\alpha-1)$ in the definition of $k(t)$ so that $k(0)$ and $k'(0)>0$. It is not difficult to check that $$\limsup_{t\to 0^+} \frac{H(t)-H(0^+)} t = \frac{k'(0)}{k(0)^2} \frac{{\operatorname{Ent}}_{\mu}\left( e^{k(0)f}\right)}{\int e^{k(0)f}\,d\mu} - \frac{1}{k(0)\int e^{k(0)f}\,d\mu} \liminf_{t\to 0^+}\int \frac{e^{k(t)f}-e^{k(t)Q_tf}}t \,d\mu.$$ According to the mean value theorem, there exists a function $\varphi:(0,\infty)\times X \to {\mathbb{R}}$ taking values in the interval $[k(t)e^{k(t)Q_{t}f(x)} ; k(t)e^{k(t)f(x)}]$ such that $$\frac{e^{k(t)f}-e^{k(t)Q_tf}}t = \frac{f-Q_tf}t \varphi(t,x),\qquad \forall t>0,x\in X.$$ Applying point (4) of Proposition \[prop Q\_[t]{}\], we get $$\begin{aligned} \liminf_{t\to 0^+} \int \frac{e^{k(t)f}-e^{k(t)Q_tf}}t \,d\mu &\leq \limsup_{t\to 0^+} \int \frac{e^{k(t)f}-e^{k(t)Q_tf}}t \,d\mu\\ &\leq k(0 ) \int \alpha^*(|\nabla^- f|) \,e^{k(0)f}\,d\mu .\end{aligned}$$ So $${\operatorname{Ent}}_{\mu}\left( e^{k(0)f}\right) \leq \frac{k(0)^2} {k'(0)}\int \alpha^*\left(|\nabla^-f|\right) e^{k(0)f}\,d\mu.$$ Since $k(0)=\left(1-\frac{C^{-1}t_o}{p_\alpha-1}\right)^{p_\alpha-1}\to 1$ and $k(0)^2/k'(0) = C \left(1-\frac{C^{-1}t_o}{p_\alpha-1}\right)^{p_\alpha-2}\to C$, when $t_o\to0^+,$ we conclude that ${\bf LSI}_\alpha^-(C)$ holds. This completes the proof. During the proof above, we used the following technical lemma whose proof is postponed to the appendix for the clarity of the exposition. \[Falpha\] Set $\ell = \lim_{h \to \infty}\frac{\alpha(h)}{h} \in \mathbb{R}\cup\{+\infty\}$. Let $f:X\to {\mathbb{R}}$ be a bounded and continuous function. Then, 1. For all $t >0$, $Q_t f \in \mathcal{F}_\alpha$ and $\mathrm{Lip}(Q_t f) \leq \ell$. 2. For all $t>0$ and all $x \in X$, $\lim_{\varepsilon \to 0} Q_t ((1-\varepsilon) f)(x) = Q_t f(x)$. We are now in position to derive the Otto-Villani Theorem from Theorem \[thm-hyper\]. Recall that, according to Bobkov and Götze characterization [@BG99], $\mu$ verifies the transport-entropy inequality ${\bf T}_\alpha(C)$ if and only if $$\label{BG} \int e^{C^{-1}Q_1f}\,d\mu \leq \exp\left(C^{-1}\int f\,d\mu\right),$$ for all bounded continuous function $f:X\to {\mathbb{R}}.$ Since $\mu$ verifies ${\bf LSI}_\alpha^{-}(C)$, it verifies the hypercontractivity property of Theorem \[thm-hyper\]. Take $t_o=C(p_\alpha-1)$ in the definition of $k(t)$, the hypercontractivity inequality yields for all bounded continuous function $f$, $$\int e^{k(t)Q_{t}f} d\mu\leq e^{k(t)\int f d\mu},\qquad \forall t>0.$$ According to , this means that $\mu$ verifies the following family of transport-entropy inequalities $$\mathcal{T}_{\alpha(\,\cdot\, /t)}(\mu,\nu)\leq \frac{1}{tk(t)} H(\nu|\mu),\qquad \forall \nu \in \mathcal{P}(X),$$ where $\alpha(\,\cdot\, /t)$ denotes the function $x\mapsto \alpha(x/t)$. According to [@GRS12 Proof of Lemma A.3], $$\alpha(x) \leq \max(t^{r_\alpha} ; t^{p_\alpha}) \alpha(x/t),\qquad \forall t>0.$$ Therefore, $\mu$ verifies ${\bf T}_\alpha(A)$, with the constant $$A=\inf_{t>0} \frac{\max(t^{r_\alpha-1} ; t^{p_\alpha-1})}{k(t)}.$$ Taking $t=C(p_\alpha-1)$ for which $k(t)=1$, we see that $$A\leq \max\left( ((p_\alpha-1)C)^{r_\alpha-1};((p_\alpha-1)C)^{p_\alpha-1}\right),$$ which ends the proof. Define for all $t>0$ the operators $$R_t f(x)=\inf_{y\in X}\left\{f(y) + \frac{1}{t} \theta(d(x,y))\right\}\quad\text{and}\quad Q_t f(x)=\inf_{y\in X}\left\{f(y) + \frac{1}{t} d^2(x,y)\right\}$$ According to Bobkov and Götze dual formula and by homogeneity, it holds for all $t>0$ $$\int e^{C^{-1}tR_t f}\, d\mu \leq e^{C^{-1}t\int f\,d\mu},$$ for all bounded continuous function $f.$ Take a function $f$ such that $|f|\leq M$ and $\mathrm{Lip}(f,r)<\infty$ for some $r>0$. If $d(x,y)\geq a$, and $t\leq a^2/(2M)$, then it holds $$f(y)+\frac{1}{t}\theta(d(x,y)) \geq -M + \frac{(2M)}{a^2} a^2=M \geq f(x) \geq R_tf(x).$$ It follows that if $t\leq a^2/2M$, then $$R_t f(x)\geq \inf_{ y : d(x,y)\leq a}\left\{f(y) + \frac{1}{t}d^2(x,y))\right\} \geq Q_tf(x).$$ So the following inequality holds $$\int e^{C^{-1}tQ_tf}\,d\mu \leq e^{C^{-1}t\int f\,d\mu},\qquad \forall t\leq a^2/(2M).$$ Applying Taylor formula, we see that $$e^{C^{-1}tQ_t f(x)} = 1 + C^{-1}tQ_tf(x) + \frac{C^{-2}(tQ_tf)^2(x)}{2}e^{\varphi(t,x)},$$ where $|\varphi(t,x)| \leq tM$, for all $t,x$. So, for all $t\leq a^2/(2M)$, $$C^{-1}\int \frac{Q_tf-f}{t}\,d\mu + \frac{C^{-2}}{2}\int (Q_tf)^2(x)e^{\varphi(t,x)}\,\mu(dx)\leq \frac{e^{C^{-1}t\int f\,d\mu}-1 -tC^{-1}\int f\,d\mu}{t^2}.$$ Letting $t$ go to $0$ and using points (3) and (4) of Proposition \[prop Q\_[t]{}\] together with the dominated convergence theorem yields to $$-\frac{C^{-1}}{4}\int |\nabla^-f|^2\,d\mu + \frac{C^{-2}}{2} \int f^2\,d\mu \leq \frac{C^{-2}}{2} \left(\int f\,d\mu\right)^2,$$ which is the announced Poincaré inequality. Transport-entropy inequalities as restricted log-Sobolev inequalities {#transpotlogsob} ===================================================================== In this section, we show that a transport-entropy inequality can be characterized as a modified log-Sobolev inequality restricted to a class of $c$-convex functions. Actually we will prove the following improved version of Theorem \[main result\] which holds even if the space is not geodesic. \[main result improved\] Let $\mu$ be a probability measure on $(X,d)$ and $p\geq 2$. Define the function $\beta_p$ as follows: $$\label{beta_p} \beta_p(u)= \frac u{[u^{1/(p-1)}-1]^{p-1}},\qquad \forall u>1.$$ The following properties are equivalent: 1. There is some $C>0$ such that $\mu$ verifies ${\bf T}_p(C)$.\ 2. There is some $D>0$ such that $\mu$ verifies the following $(\tau)$-log-Sobolev inequality: for all bounded continuous $f$ and all $0<\lambda<1/D$, it holds $${\operatorname{Ent}}_\mu(e^f)\leq \frac{1}{1-\lambda D} \int (f-Q^\lambda f)e^f\,d\mu,$$ where for all $\lambda>0$, $Q^\lambda f(x)=\inf_{y\in X}\left\{f(y) +\lambda c_{p}(x,y)\right\}.$\ 3. There is some $E>0$ such that $\mu$ verifies the following restricted log-Sobolev inequality: for all $Kc_p$-convex function $f$, with $0<K<1/E$ it holds $${\operatorname{Ent}}_{\mu}(e^f)\leq \frac{\beta_p(u)-1}{(1-KEu)pK^{q-1}}\int |\nabla_{Kc_p}^- f|^{q}e^f\, d\mu,\qquad \forall u\in(1,1/(KE))$$ where $q=p/(p-1)$ and $|\nabla_{Kc_p}^-f|(x)= K\left(\inf_{\bar{y}\in \partial_{Kc_p}f(x)} d(x,\bar{y})\right)^{p-1}$ (see Proposition \[Gradients comparisons\]). Moreover, when the space $(X,d)$ is geodesic these properties are equivalent to the following - There is some $F>0$ such that $\mu$ verifies the following restricted log-Sobolev inequality: for all $Kc_p$-convex function $f$, with $0<K<1/F$ it holds $${\operatorname{Ent}}_{\mu}(e^f)\leq \frac{\beta_p(u)-1}{(1-KFu)pK^{q-1}}\int |\nabla^+ f|^{q}e^f\, d\mu,\qquad \forall u\in(1,1/(KF))$$ The optimal constants $C_\mathrm{opt},D_\mathrm{opt},E_\mathrm{opt},F_\mathrm{opt}$ are related as follows $$F_\mathrm{opt} \leq E_\mathrm{opt} \leq D_\mathrm{opt}\leq C_\mathrm{opt}\leq \kappa_p F_\mathrm{opt},$$ where $\kappa_p$ is some universal constant depending only on $p.$ For $p=2$, one can take $\kappa_2 = e^2.$ From transport-entropy inequalities to $(\tau)$-log-Sobolev inequalities ------------------------------------------------------------------------ Let us recall the following proposition from [@GRS11] whose proof relies on a simple Jensen argument. If $\mu$ verifies the transport-entropy property $\mathbf{T}_{c}(C)$, for some continuous cost function $c$ on $X^2$, then the following ($\tau$)-log-Sobolev property holds: for all function $f$, for all $0<\lambda<1/C$, $$\begin{aligned} \label{taulogsob} {\operatorname{Ent}}_{\mu}(e^f)\leq \frac{1}{1-\lambda C} \int (f-Q^\lambda f) e^f\,d\mu,\end{aligned}$$ where for all $x\in X$, $Q^\lambda f(x)=\inf\{f(y)+\lambda c(x,y)\}.$ This proves the step $(1) \Rightarrow (2)$ in Theorem \[main result improved\]. From transport entropy inequalities to log-Sobolev inequalities for $c_{p}$-convex functions -------------------------------------------------------------------------------------------- The general link between the ($\tau$)-log-Sobolev property and the restricted log-Sobolev inequality is the following: if the function $f$ is $c$-convex then the quantity $f-Q^\lambda f$ in the right-hand side of $\eqref{taulogsob}$ can be bounded by a function of $|\nabla_c^- f|$ (see Lemma \[adieupec\] below). From now on, let us assume that $c=c_p$ is the cost function defined by: for all $x,y$ in $X$, $c_p(x,y)=d^p(x,y)/p,$ for some $p>1$. \[adieupec\] Let $\lambda>0$. If $f$ is a $Kc_p$-convex function bounded from above, and if $0<K<\lambda$, then for all $x\in X$ and all $\bar y$ in the $Kc_p$-subdifferential of $f$ at point $x$, $\partial_{Kc_p} f(x)$, $$f(x)-Q^\lambda f(x)\leq K \left(\beta_p\left( \lambda /K\right) -1\right)c_p(x,\bar y),$$ where $Q^\lambda f(x)=\inf_{y\in X}\{f(y)+\lambda c_{p}(x,y)\}$ and for all $u>1$, $\beta_p(u)= \frac u{[u^{1/(p-1)}-1]^{p-1}}.$\ Equivalently, with the notation of Proposition \[Gradients comparisons\], $$f(x)-Q^\lambda f(x) \leq (\beta_{p}(\lambda/K)-1) \frac{1}{pK^{q-1}} |\nabla^-_{Kc_{p}}f|^q(x),$$ where $q=\frac{p}{p-1}.$ According to Definition \[subdiff\] of $\partial_{Kc_p} f(x)$ and using the triangular inequality we get, for all $\bar y\in \partial_{Kc_p} f(x)$ $$\begin{aligned} f(x)-Q^\lambda f(x)&=\sup_{z\in X} \{f(x)-f(z)-\lambda c_p(z,x)\}\\ &\leq\sup_{z\in X} \{ Kc_p(z,\bar y)- Kc_p(x,\bar y) -\lambda c_p(z,x)\}\\ &\leq \sup_{z\in X} \{ Kc_p(z,\bar y) -\lambda c_p(z,x)\} - Kc_p(x,\bar y)\\ &\leq \frac 1p\sup_{z\in X}\{ K(d(z,x)+ d(x,\bar y))^p -\lambda d^p(z,x)\} - Kc_p(x,\bar y)\\ &\leq \frac 1p\sup_{r\geq 0 }\{ K(r+ d(x,\bar y))^p -\lambda r^p\} - Kc_p(x,\bar y)\\ &= Kc_p(x,\bar y) \left(\beta_p\left( \lambda /K\right) -1\right). \end{aligned}$$ Thus optimizing over all possible $\bar{y} \in \partial_{Kc_{p}}f(x)$ yields to the expected result $$\begin{aligned} f(x)-Q^\lambda f(x)&\leq \left(\beta_p\left( \lambda /K\right) -1\right) \inf_{\bar{y} \in \partial_{Kc_{p}}f(x)} Kc_p(x,\bar y) =(\beta_{p}(\lambda/K)-1) \frac{1}{pK^{q-1}} |\nabla^-_{Kc_{p}}f|^q(x) .\end{aligned}$$ From this lemma the ($\tau$)-log-Sobolev property provides immediately the first part of the following statement by setting $u=\lambda/C$. \[transverslog\] If $\mu$ verifies the $(\tau)$-log-Sobolev with the cost $c=c_{p}$, $p>1$, then for all $K\in(0,1/C)$ and all function $f$ bounded from above and $Kc_{p}$-convex, it holds $${\operatorname{Ent}}_{\mu}(e^f)\leq \frac{\beta_p(u)-1}{(1-KCu)pK^{q-1}} \int |\nabla^-_{Kc_{p}}f|^q(x)\,e^{f(x)}\,\mu(dx),\quad \forall u\in(1,1/(KC)).$$ Moreover, when $(X,d)$ is geodesic, the same inequality holds with $|\nabla^+ f|$ instead of $|\nabla_{K_{c_{p}}}^-f|$ in the right-hand side. This proves the steps $(2)\Rightarrow (3)$ and $(2)\Rightarrow (3')$ (in the geodesic case) in Theorem \[main result improved\]. Let us justify the statement in the geodesic case. According to Proposition \[Gradients comparisons\] (applied with the function $\theta(x)=Kx^p/p$), it holds $|\nabla_{Kc_{p}}^- f|\leq |\nabla_{Kc_{p}}^+ f|$ and when the space is geodesic, $|\nabla_{Kc_{p}}^+f |=|\nabla^+f|$, which completes the proof. From log-Sobolev inequalities for $c_{p}$-convex functions to transport-entropy inequalities -------------------------------------------------------------------------------------------- In this part we prove that a modified log-Sobolev inequality restricted to the class of $Kc_{p}$-convex functions also implies a transport entropy-inequality. One of the main ingredient of the proof is Theorem \[HJ 1\]. \[logverstrans\] Let $p\geq 2$. Suppose that for all $K\in( 0,1/C)$ and all $Kc_p$-convex function $f:X\to {\mathbb{R}}$ bounded from above, it holds $$\label{eq logverstrans} {\operatorname{Ent}}_{\mu}(e^f)\leq \frac{\beta_p(u)-1}{(1-KCu)pK^{q-1}} \int |\nabla^-_{Kc_{p}}f|^q(x)\,e^{f(x)}\,\mu(dx),\quad \forall u\in(1,1/(KC)).$$ then $\mu$ verifies the inequality $\mathbf{T}_{p}(\kappa_{p}C)$, where $\kappa_{p}$ is some numerical constant depending only on $p.$ For $p=2$, $\kappa_{2}=e^2.$ Moreover, if the space is geodesic, the same conclusion holds if $|\nabla^-_{Kc_{p}}f|$ is replaced by $|\nabla^+f|$ in the right hand side of . This proves the steps $(3)\Rightarrow (1)$ and $(3')\Rightarrow (1)$ (in the geodesic case) and completes the proof of Theorem \[main result improved\]. For any bounded continuous function $g$, we define the function $P_{t}g$ as follows $$P_{t}g(x)=\sup_{y\in X}\left\{ g(y) - \frac{1}{t^{p-1}} c_{p}(x,y)\right\}.$$ Let $\ell : [a,1]\to (0,+\infty)$ be a decreasing function of class $\mathcal{C}^1$ defined on some interval $[a,1]$ with $a>0$ and such that $\ell (1)=0.$ For all bounded continuous $g$ define $H_{g}(t)=\frac{C}{\ell(t)} \log\left(\int e^{C^{-1}\ell(t)P_{t}g}\,d\mu\right)$, $t\in [a,1).$ If all the $H_{g}$’s were non-decreasing, then it would hold that $H_{g}(a)\leq \lim_{t\to 1^-} H_{g}(t)=\int P_{1}f\,d\mu.$ Since $g\leq P_{a}g$, we would get $$\int e^{C^{-1}\ell(a) g}\,d\mu \leq e^{C^{-1}\ell(a) \int P_{1} g\,d\mu}$$ which in turn, according to Bobkov and Götze characterization Theorem, would prove that $\mu$ verifies $\mathbf{T}_{p}(C/\ell(a)).$ Hence, our aim is to construct a function $\ell$ such that all the $H_{g}$’s are non-decreasing. Set $f_{t}=C^{-1}\ell(t)P_{t}g$. According to Proposition \[deriv\], $H_{g}$ is continuous and differentiable on the right and $$\frac{d}{dt_{+}} H_{g}(t)=\frac{C\ell'(t)}{\ell^2(t) \int e^{f_{t}}\,d\mu} \left[{\operatorname{Ent}}_{\mu}\left(e^{f_{t}}\right)+\frac{\ell(t)^2}{C\ell'(t)} \int \frac{dP_{t}g }{dt_{+}} e^{f_{t}}\,d\mu\right].$$ Since $\ell'<0$, all we have to show is that the term into brackets is non-positive. For all $t>0$, the function $f_{t}$ is $K(t)c_{p}$-convex, with $K(t)=\frac{\ell(t)}{Ct^{p-1}}$. Hence, for all $t$ such that $\ell(t)< t^{p-1}$ and all $u\in(1,1/(CK(t))$, $${\operatorname{Ent}}_{\mu}(e^{f_{t}})\leq \frac{\beta_p(u)-1}{(1-K(t)Cu)pK(t)^{q-1}}\int |\nabla_{K(t)c_{p}}^-(f_{t})|^q(x)e^{f_{t}(x)}\,\mu(dx) .$$ Since $f_{t}$ is $K(t)c_{p}$-convex, it follows from Proposition \[Gradients comparisons\] (applied with $\alpha(h)=K(t)h^p/p$) that $$|\nabla^-_{K(t)c_{p}} f_{t}|(x) = K(t) \left(\min_{\bar{y} \in \partial_{K(t)c_{p}}f_{t}(x)} d(x,\bar{y})\right)^{p-1} \leq K(t) \left(\max_{\bar{y} \in m(t,x)} d(x,\bar{y})\right)^{p-1},$$ denoting by $m(t,x)$ the set of points $\bar{y}$ where the supremum defining $P_{t}g$ is reached. As a result, it holds $$\frac{1}{p K(t)^{q-1}} |\nabla^-_{K(t)c_{p}} f_{t}|^q(x) \leq K(t) \max_{\bar{y}\in m(t,x)} c_{p}(x,\bar{y}).$$ On the other hand, according to Proposition \[HJ 1\], $$\frac{dP_{t}g}{dt_{+}}(x)=\frac{p-1}{t^p}\max_{\bar{y}\in m(t,x)} c_{p}(x,\bar{y}).$$ Therefore $$\label{a changer} \frac{1}{p K(t)^{q-1}} |\nabla^-_{K(t)c_{p}} f_{t}|^q(x) \leq \frac{K(t)t^p}{(p-1)} \frac{dP_{t}g}{dt_{+}}(x)=\frac{t\ell(t)}{(p-1)C}\frac{dP_{t}g}{dt_{+}}(x).$$ So, for all $t>0$ with $\ell(t)<t^{p-1}$ it holds $$\begin{gathered} \left[{\operatorname{Ent}}_{\mu}\left(e^{f_{t}}\right)+\frac{\ell(t)^2}{C\ell'(t)} \int \frac{dP_{t}g }{dt_{+}} e^{f_{t}}\,d\mu\right] \leq \frac{\ell(t)}{C} \left[\theta_{p}\left(\frac{\ell(t)}{t^{p-1}}\right) \frac{t}{p-1}+\frac{\ell(t)}{\ell'(t)}\right]\int\frac{dP_{t}g }{dt_{+}} e^{f_{t}}\,d\mu,\end{gathered}$$ where the function $\theta_{p}$ is defined by $\theta_{p}(x)= \inf_{1<u<1/x}\left\{\frac{\beta_p(u)-1}{1- x u}\right\}$, for $x<1$. Observe that $\theta_{p}$ is finite on $[0,1[.$ Consider the function $$\Psi_{p}(r)=\frac{1}{p-1}\int_{0}^{r} \frac {\theta_p(s)}{s(\theta_p(s)+1)}\,ds,\qquad \forall r\in [0,1].$$ According to Lemma \[legueanvichy\] below, since $p\geq 2$, the function $\Psi_{p}$ is well defined, increasing and of class $\mathcal{C}^1$ on $(0,1).$ Define $v(t)=\Psi_{p}^{-1} (-\ln(t)),$ for all $t\in[ a_{p}, 1]$, with $a_{p}= \exp\left(-\Psi_{p}(1)\right)$. The function $v$ is increasing and $v(t)\in[0,1]$ for all $t\in[ a_{p}, 1]$. Finally, define $\ell_{p}(t) = t^{p-1}v(t)$, for all $t\in [a_{p},1].$ A simple calculation shows that $$\theta_{p}\left(\frac{\ell_{p}(t)}{t^{p-1}}\right) \frac{t}{p-1}+\frac{\ell_{p}(t)}{\ell_{p}'(t)} = 0,\qquad \forall t\in (a_{p},1).$$ We conclude that $\mu$ verifies the inequality $\mathbf{T}_{p}$ with the constant $$\frac{C}{\ell_{p}(a_{p})}=C\exp\left(\int_{0}^1\frac {\theta_p(s)}{s(\theta_p(s)+1)}\,ds\right)=C\kappa_p.$$ In the particular case $p=2$, one has $\theta_{2}(x)=\frac{4x}{\left(1-x\right)^2}$, and it is easy to check that $\kappa_{2}=e^{2}$. It remains to consider the geodesic case. In this case, the inequality is replaced by the equality $$\frac{1}{p K(t)^{q-1}} |\nabla^+ f_{t}|^q(x) = \frac{K(t)t^p}{(p-1)} \frac{dP_{t}g}{dt_{+}}(x),$$ and the rest of the proof remains unchanged. \[legueanvichy\] The function $s \mapsto \phi(s)=\frac {\theta_p(s)}{s(\theta_p(s)+1)}$ is continuous on $(0,1)$. Moreover, $\phi(s)$ goes to $1$ as $s$ goes to 1 and $$\phi(s)=\frac {p^{p/(p-1)}}{ s^{(p-2)/(p-1)}}(1+\varepsilon(s)),$$ with $\varepsilon(s)\to 0$ as $s\to 0$. After some computations, it is easily to check that for $s\in(0,1)$, the infimum $\theta_p(s)$ is reached at some unique point $u=u(s)\in (1,1/s)$ such that $$\beta'_p(u)(1-su)+s(\beta_p(u)-1)=0,$$ or equivalently $$u(s)^{p/(p-1)}-\left( u(s)^{1/(p-1)}-1\right)^p=1/s.$$ It follows from this equality that $u(s)$ is continuous on (0,1), $u(s)\to 1$ as $s\to 1$ and $u(s)\to +\infty $ as $s\to 0$. As a first consequence, $\phi$ is continuous on $(0,1)$. By a Taylor expansion at point 0, one has $$\frac{1}{su(s)^{p/(p-1)}}= 1-\left(1-\frac{1}{u(s)^{1/(p-1)}}\right)^p=\frac{p}{u(s)^{1/(p-1)}}(1+\varepsilon(s)),$$ with $\varepsilon(s)\to 0$ as $s\to 0$. It follows that $su(s)\to 1/p$ as $s\to 0$. From all this observations, we get $$\phi(s)=\frac{1-\left(1-u(s)^{-1/(p-1)}\right)^{p-1}}{s\left(1-su(s)\left(1-u(s)^{-1/(p-1)}\right)^{p-1}\right)}=\frac {p^{p/(p-1)}}{ s^{(p-2)/(p-1)}}(1+\varepsilon(s)),$$ with $\varepsilon(s)\to 0$ as $s\to 0$. Since $u(s)\to 1$ as $s\to 1$ we easily get that $\phi(s)\to 1$ as $s\to 1$. Proof of Lemma \[bysarko1\], Lemma \[Kuratowski\], Proposition \[deriv\] and Lemma \[Falpha\] ============================================================================================= In this appendix we collect all the technical proofs of Lemmas \[bysarko1\], \[Kuratowski\] and \[Falpha\] and of Proposition \[deriv\]. Let $\bar{y}\in \partial_cf(x)$. According to the definition of the $c$-subdifferential, $$f(z)-f(x)\geq L(x-\bar{y})-L(z-\bar{y}),\qquad \forall z\in {\mathbb{R}}^m.$$ Let $z= x+\varepsilon u$ with $\varepsilon >0$ and $u\in {\mathbb{R}}^m$. Since $L$ and $f$ are smooth functions at point $x$, we get as $\varepsilon$ tends to $0$, for all $u\in {\mathbb{R}}^m$, $$u\cdot\nabla f(x) \geq -u\cdot \nabla L(x-\bar{y}),$$ and therefore $\nabla f(x)=- \nabla L(x-\bar{y}).$ Let $v_o=x-\bar{y}$ and $u_o=\nabla L(v_0)$, by the convexity property of $L$, $$\label{eq:subgradient} L(v)\geq L(v_o) +u_o\cdot(v-v_o),\qquad \forall v\in {\mathbb{R}}^m,$$ or equivalently $L(v_o)\leq u_o\cdot v_o-L^*(u_o)$. Since $L(v_o)=\sup_{u\in {\mathbb{R}}^m}\{u\cdot v_o-L^*(u)\}$, it follows that the derivative of $u\mapsto u\cdot v_o - L^*(u)$ vanishes at $u_o$, and so $v_o=\nabla L^*(u_o)$. Finally, $x-\bar{y}=\nabla L^*(u_o)=\nabla L^*(-\nabla f(x))$, which completes the proof. \(1) The function $h:y\mapsto g(y)-t\alpha\left(d(x,y)/t\right)$ is upper semicontinuous, bounded from above and its level sets $\{h\geq r\}$ $r\in {\mathbb{R}}$ are compact. It follows that $h$ reaches its supremum and so $m(t,x)=\{h\geq \sup h\}$ is not empty and compact.\ (2) Let $h_n(y)=g(y)-t_n\alpha\left(\frac{d(x_n,y)}{t_n}\right),$ $y\in X.$ The sequence of functions $h_n$ converges pointwise to the function $h$, and the convergence is uniform on each bounded set. Since $g$ is bounded from above by some constant $r\in {\mathbb{R}}$, it holds $$\label{eq Kuratowski} r-t_n\alpha\left(\frac{d(x_n,y_n)}{t_n}\right)\geq g(y_n)-t_n\alpha\left(\frac{d(x_n,y_n)}{t_n}\right)\geq g(y)-t_n\alpha\left(\frac{d(x_n,y)}{t_n}\right),\qquad \forall y\in X.$$ Since $(x_n)_{n\in {\mathbb{N}}}$ is bounded and $\lim_{n\rightarrow \infty}t_n=t>0$, we conclude that $(y_n)_{n\in {\mathbb{N}}}$ is a bounded sequence. As balls are supposed to be compact, $(y_n)_{n\in {\mathbb{N}}}$ has converging subsequences. Passing to the limit into the inequality along a converging subsequence of $(y_n)_{n\in {\mathbb{N}}}$, yields to the conclusion that any limit point $\bar{y}$ of $(y_n)_{n\in {\mathbb{N}}}$ belongs to $m(t,x).$ Let us turn to the proof of Proposition \[deriv\]. The proof requires some regularity properties of $Q_tf$ in the $t$ variable that are gathered in the following proposition. \[prop Q\_[t]{}\] Let $f$ be a bounded lower semicontinuous function on $X$; define for all $t>0$ and $x\in X$ $Q_tf (x) = \inf\left\{f(y)+t\alpha\left(\frac{d(x,y)}{t}\right)\right\}$ and let $m(t,x)$ denote the set of points where this infimum is attained. The following properties hold 1. For all $x\in X$, $$m(t,x)\subset B\left(x, t\alpha^{-1}\left({\mathrm{Osc}(f)}/t\right)\right).$$ 2. For all $t,h>0$, $$\frac{1}{h}\sup_{x\in X} |Q_{t+h}f(x)-Q_{t}f(x)| \leq \beta\left(\alpha^{-1}\left({\mathrm{Osc}(f)}/t\right)\right).$$ 3. If $\alpha(h)/h\to\infty$, when $h\to \infty$, then for all bounded continuous function $f$ and for all $x\in X$, $$\lim_{t\to 0^+} Q_{t}f(x)= f(x).$$ and $$\liminf_{t\rightarrow 0^+} \frac{Q_tf(x)-f(x)}t \geq -\alpha^*(|\nabla^- f|(x)).$$ If $\alpha(h)/h\to\ell \in {\mathbb{R}}^+$, when $h\to \infty$, the same conclusions hold for all function $f$ with $\mathrm{Lip}(f)<\ell.$ 4. Let $\mu$ be a probability measure and $\varphi:(0,+\infty)\times X \to {\mathbb{R}}$ be such that $|\varphi|\leq M$ for some $M>0$ and $\lim_{t\to0+}\varphi(t,x)=\psi(x)$ for all $x\in X$. If $\alpha(h)/h \to\infty$ when $h\to\infty$ and if $f$ is such that $\mathrm{Lip}(f,r)<+\infty$ for some $r>0$, then $$\limsup_{t \to 0} \int \frac{f-Q_t f}{t}\varphi(t,x)\,d\mu \leq \int \alpha^*(|\nabla^- f|(x))\psi(x)\,d\mu.$$ The same conclusion holds if $\alpha(h)/h \to \ell \in {\mathbb{R}}^+,$ when $h\to\infty,$ and $\mathrm{Lip}(f)<\ell.$ \(1) Let $M=\sup (f)$ and $m=\inf(f)$. If $\bar{y}\in m(t,x)$, it holds $$m+t\alpha\left(\frac{d(x,\bar{y})}t\right) \leq f(\bar{y})+t\alpha\left(\frac{d(x,\bar{y})}t\right)=Q_{t}f(x)\leq M,$$ which proves the first claim. \(2) Since $t\mapsto Q_{t}f(x)$ is non-increasing, $|Q_{t+h}f(x)-Q_{t}f(x)|=Q_{t}f(x)-Q_{t+h}f(x).$ If $\bar{y}\in m(t+h,x)$, then $$\begin{aligned} \frac 1h \left(Q_{t}f(x)-Q_{t+h}f(x)\right) & \leq \frac 1h \left(t\alpha\left(\frac{d(x,\bar{y})}t\right)-(t+h)\alpha\left(\frac{d(x,\bar{y})}{t+h}\right)\right) \leq \beta\left(\alpha^{-1}\left({\mathrm{Osc}(f)}/t\right)\right),\end{aligned}$$ where the last inequality comes from the mean value theorem, the monotonicity of the function $\beta$ and point (1). \(3) Let us first assume that $\lim_{h\rightarrow \infty}\alpha(h)/h=+\infty$. In this case, $\lim_{t\rightarrow 0}t\alpha^{-1}\left(\frac{\mathrm{Osc}(f)}t\right)=0$ and so, according to the first point, $$\inf_{y\in B\left(x, t\alpha^{-1}\left({\mathrm{Osc}(f)}/t\right)\right)} \{f(y)\}\leq Q_{t}f(x)\leq f(x).$$ Since $f$ is lower semicontinuous, the limit when $t$ goes to $0$ of the left hand side is greater than or equal to $f(x)$. This guarantees that $\lim_{t\to 0^+} Q_{t}f(x)= f(x)$. Moreover, for all $\bar y_t\in m(t,x)$, $f(\bar y_t)\leq f(x)$ and therefore $$\begin{aligned} \label{mamamia} \frac {f(x)-Q_{t}f(x)}t &= \frac{f(x)-f(\bar y_t)}{t} -\alpha\left(\frac{d(x,\bar y_t)}t\right) = \frac{[f(\bar y_t)-f(x)]_-}{d(x,\bar y_t)}\,\frac{d(x,\bar y_t)}t -\alpha\left(\frac{d(x,\bar y_t)}t\right) \nonumber \\ &\leq \alpha^*\left(\frac{[f(\bar y_t)-f(x)]_-}{d(x,\bar y_t)}\right).\end{aligned}$$ Arguing as before, we see that $\bar y_t\to x$ as $t\to 0$ so that $$\limsup_{t\to 0^+} \frac {f(x)-Q_{t}f(x)}t\leq \alpha^*\left(|\nabla^-f|(x)\right).$$ Now let us assume that $\alpha(h)/h\to\ell \in {\mathbb{R}}^+$ when $h\to\infty$. According to what precedes, it is enough to show that there is a constant $r>0$ such that $$m(t,x)\subset B(x; rt),\qquad \forall t>0,x\in X.$$ Let $\bar{y} \in m(t,x)$. Then it holds $f(\bar{y})-f(x)+t\alpha\left(d(x,\bar{y})/t\right)\leq 0.$ Since $f$ is assumed to be Lipschitz, we conclude that $\mathrm{Lip}(f) d(x,\bar{y})/t\geq \alpha\left( d(x,\bar{y})/t\right).$ Since $\mathrm{Lip} (f)<\ell = \lim_{h\to+\infty} \alpha(h)/h$, this implies that $d(x,\bar{y})\leq rt$ where $r=\sup\{h : \alpha(h)/h \leq \mathrm{Lip}(f)\}<+\infty$, which proves the claim. \(4) We already know, by point (3), that $\limsup_{t\to 0^+} \frac {f(x)-Q_{t}f(x)}t\leq \alpha^*\left(|\nabla^-f|(x)\right)$. Hence the result of point (4) will follow from Fatou’s Lemma (in its limsup version) as soon as for some $t_0>0$, it holds $\sup_{x} \sup_{t \in (0,t_o)}\frac {f(x)-Q_{t}f(x)}t < \infty$. Assume first that $\lim_{h \to \infty} \alpha(h)/h=\infty$ and let $r>0$ be such that $\mathrm{Lip}(f,r)< \infty$. Observe that $\lim_{t\rightarrow 0}t\alpha^{-1}\left(\frac{\mathrm{Osc}(f)}t\right)=0$ so that, by point (1), there exists $t_o>0$ such that, for all $t \in (0,t_o)$, all $x\in X$ and all $\bar y_t\in m(t,x)$, $d(x,\bar y_t) \leq r$. Using , we conclude that $\sup_{x} \sup_{t \in (0,t_o)}\frac {f(x)-Q_{t}f(x)}t \leq \alpha^*\left(\mathrm{Lip}(f,r)\right) < \infty$. Assume now that $\alpha(h)/h\to\ell\in {\mathbb{R}}^+$, when $h\to\infty.$ Then, since $\mathrm{Lip}(f) < \ell$, implies that $\sup_{x,t}\frac{f(x)-Q_{t}f(x)}t \leq \alpha^*\left(\mathrm{Lip}(f)\right) < \infty$. This ends the proof of point (4) and of the proposition. We will prove that $H$ is right differentiable, the proof of the left-differentiability being similar. By formally differentiating under the sign integral yields for all $t>0$, $$\begin{aligned} \label{dif int} \frac{dH}{dt_+ }(t) & = -\frac{k'(t)}{k(t)^2}\log\left(\int e^{k(t)Q_{t}f}\,d\mu \right) \nonumber\\ & \quad + \frac{1}{k(t)\int e^{k(t)Q_{t}f}\,d\mu} \left[\int k'(t)Q_{t}fe^{k(t)Q_{t}f}\,d\mu + \int k(t)\frac{d}{dt_+ }Q_{t}fe^{k(t)Q_{t}f}\,d\mu\right],\end{aligned}$$ which easily gives the desired identity. Hence, it remains to justify the above calculation. Define $F(t)=\int e^{k(t) Q_{t}f }\,d\mu.$ To obtain , it is enough to show that $F$ is right differentiable and that $$\frac{dF}{dt_+ } (t)=\int k'(t)Q_{t}f e^{k(t)Q_{t}f}\,d\mu + \int k(t) \frac{d}{dt_+ }Q_{t}f e^{k(t) Q_{t}f}\,d\mu.$$ For all $s>0$, $\frac{1}{s}\left(F(t+s)-F(t)\right)= \int G_{s} \,d\mu,$ with $G_{s}=\frac{1}{s}\left(e^{k(t+s)Q_{t+s}f} -e^{k(t)Q_{t}f}\right).$ Since $t\mapsto Q_{t}f(x)$ is right differentiable for $t>0$, $$G_{s}(x) \underset{s\to 0}{\longrightarrow} k'(t)Q_{t}f(x) e^{k(t)Q_{t}f(x)} + k(t) \frac{d}{dt_+ }Q_{t}f(x) e^{k(t)Q_{t}f(x)} .$$ For a given $t\in (a,b)$, let $\eta_{t}>0$ be any number such that $t+\eta_{t}<b$. Then, using the mean value Theorem together with point (2) of Proposition \[prop Q\_[t]{}\], it is not difficult to prove that $\sup_{x\in X}\sup_{s\leq \eta_{t}} |G_{s}| (x)<+\infty.$ Applying the dominated convergence theorem completes the proof. Let $f:X\to {\mathbb{R}}$ be a bounded and continuous function. Fix $t>0$. \(1) First, following [@GRS12 Lemma 3.8], we will prove that there exists $r>0$ such that $\mathrm{Lip}(Q_t f,r) < \infty.$ Set $r=t \alpha^{-1}(\mathrm{Osc}(f)/t)$. From point (1) of Proposition \[prop Q\_[t]{}\], it holds $$Q_tf(u)=\inf_{d(y,u)\leq r} \left\{f(y)+t\alpha(d(u,y)/t)\right\}, \qquad \forall u \in X.$$ Fix $u,v \in X$ with $d(u,v) \leq r$. Then, given $y_o \in X$ such that $d(v,y_o) \leq r$, it follows from the mean value theorem that $$\begin{aligned} |t\alpha(d(u,y_o)/t)-t\alpha(d(v,y_o)/t)| & \leq |d(v,y_o)-d(u,y_o)|\max_{s\in [0,1]} \alpha'([sd(u,y_o)+(1-s)d(v,y_o)]/t)\\ &\leq \alpha'(2r/t)d(u,v) .\end{aligned}$$ Now, let $y_o$ be such that $Q_tf(v) = f(y_o)+t\alpha(d(v,y_o)/t)$ and observe that, thanks to the previous observation, $d(v,y_0) \leq r$. It follows that (choosing $y=y_o$), $$\begin{aligned} Q_t f(u) - Q_t f(v) & = \inf_{y} \left\{f(y)+t\alpha(d(u,y)/t)\right\} - f(y_o) -t \alpha(d(v,y_o)/t) \\ & \leq t \alpha(d(u,y_o)/t) - t \alpha(d(v,y_o)/t) \\ & \leq \alpha'(2r/t)d(u,v) ,\end{aligned}$$ which proves that $\mathrm{Lip}(Q_t f, r)< \infty$. Now assume that $\alpha(h)/h\to\ell \in {\mathbb{R}}^+$, when $h\to \infty$ and let us prove that $Q_{t}f$ is $\ell$-Lipschitz. The convexity of $\alpha$ implies that $$\frac{\alpha(h)}{h}\leq \alpha'(h)\leq \frac{\alpha(2h)-\alpha(h)}{h},\qquad \forall h>0.$$ So $\sup_h \alpha'(h) = \lim_{h\to \infty} \alpha'(h)=\ell$ and it follows that $Q_{t}f$ is $\ell$-Lipschitz as an infimum of $\ell$-Lipschitz functions. \(2) Let $(\lambda_n)_{n \geq 0}$ be a sequence of real numbers converging to $1$. For any $x \in X$, let $m(t,x)$ be the set of points $y \in X$ such that $Q_tf(x)=\inf_{z \in X} \{f(z)+t\alpha(d(x,z)/t)\}=f(y)+t\alpha(d(x,y)/t)$. For any $n$, let $y_n$ be such that $Q_t (\lambda_n f)(x)= \lambda_n f(y_n) + t\alpha(d(x,y_n)/t)$. We have, for all $z \in X$, $$\lambda_n \inf f + t\alpha(d(x,y_n)/t) \leq \lambda_n f(y_n) + t\alpha(d(x,y_n)/t) \leq \lambda_n f(z) + t\alpha(d(x,z)/t) .$$ Since $(\lambda_n)_n$ converges, we deduce that the sequence $(y_n)_n$ is bounded. Let $y$ be a limit point of a converging subsequence of $(y_n)_n$. Passing to the limit in the latter leads to $$f(y) + t\alpha(d(x,y)/t) \leq f(z) + t\alpha(d(x,z)/t) \qquad \forall z \in X .$$ Hence, $y \in m(t,x)$. In turn, after easy considerations left to the reader, $Q_t (\lambda_n f)(x)\to Q_t f(x)$, when $n\to\infty$ as expected. The conclusion of point (2) follows and the proof is complete. [10]{} L. [Ambrosio]{}, N. [Gigli]{}, and G. [Savar[é]{}]{}. . , June 2011. L. [Ambrosio]{}, N. [Gigli]{}, and G. [Savar[é]{}]{}. Density of lipschitz functions and equivalence of weak gradients in metric measure spaces. Preprint., 2012. Z. Balogh, A. Engoulatov, L. Hunziker, and O. E. Maasalo. Functional inequalities and [H]{}amilton-[J]{}acobi equations in geodesic spaces. Preprint. Available on the ArXiv http://arxiv.org/abs/0906.0476, 2009. S. G. Bobkov, I. Gentil, and M. Ledoux. Hypercontractivity of [H]{}amilton-[J]{}acobi equations. , 80(7):669–696, 2001. S. G. Bobkov and F. G[ö]{}tze. Exponential integrability and transportation cost related to logarithmic [S]{}obolev inequalities. , 163(1):1–28, 1999. L. C. Evans. , volume 19 of [*Graduate Studies in Mathematics*]{}. American Mathematical Society, Providence, RI, 1998. I. Gentil, A. Guillin, and L. Miclo. Modified logarithmic [S]{}obolev inequalities and transportation inequalities. , 133(3):409–436, 2005. N. Gigli and M. Ledoux. From log sobolev to talagrand: a quick proof. Submitted, 2011. N. Gozlan. A characterization of dimension free concentration in terms of transport inequalities. , 37(6):2480–2498, 2009. N. Gozlan, C. Roberto, and P.M. Samson. A new characterization of [T]{}alagrand’s transport-entropy inequalities and applications. , 39(3):857–880, 2011. N. Gozlan, C. Roberto, and P.M. Samson. Characterization of [T]{}alagrand’s transport-entropy inequalities in metric spaces. To appear in Annals of Probability, 2012. L. Gross. Logarithmic [S]{}obolev inequalities. , 97(4):1061–1083, 1975. J. Lott and C. Villani. Hamilton-[J]{}acobi semigroup on length spaces and applications. , 88(3):219–229, 2007. K. Marton. A simple proof of the blowing-up lemma. , 32(3):445–446, 1986. F. Otto and C. Villani. Generalization of an inequality by [T]{}alagrand and links with the logarithmic [S]{}obolev inequality. , 173(2):361–400, 2000. S. T. Rachev and L. R[ü]{}schendorf. . Probability and its Applications (New York). Springer-Verlag, New York, 1998. Theory. M. M. Rao and Z. D. Ren. . Marcel [D]{}ekker [I]{}nc., 1991. R. T. Rockafellar. . Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original, Princeton Paperbacks. M. Talagrand. Transportation cost for [G]{}aussian and other product measures. , 6(3):587–600, 1996. C. Villani. , volume 338 of [*Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]*]{}. Springer-Verlag, Berlin, 2009. Old and new. F. Y. Wang. Probability distance inequalities on [R]{}iemannian manifolds and path spaces. , 206(1):167–190, 2004. [^1]: The authors were partially supported by the “Agence Nationale de la Recherche” through the grants ANR 2011 BS01 007 01 and ANR 10 LABX-58; the second author was partially supported by the European Research Council through the “Advanced Grant” PTRELSS 228032.

--- abstract: 'We present an integrated source of counterpropagating entangled states based on a coupled resonator optical waveguide that is pumped by a classical pulsed source incident from above the waveguide. We investigate theoretically the generation and propagation of continuous variable entangled states in this coupled-cavity system in the presence of intrinsic loss. Using a tight-binding approximation, we derive analytic time-dependent expressions for the number of photons in each cavity, as well as for the correlation variance between the photons in different pairs of cavities, to evaluate the degree of quantum entanglement. We also derive simple approximate expressions for these quantities that can be used to guide the design of such systems, and discuss how pumping configurations and physical properties of the system affect the photon statistics and the degree of quantum correlation.' author: - Hossein Seifoory - 'L. G. Helt' - 'J. E. Sipe' - 'Marc M. Dignam' bibliography: - 'mybib.bib' title: 'Counterpropagating continuous variable entangled states in lossy coupled-cavity optical waveguides ' --- \[sec:level0\]Introduction ========================== Entangled quantum states have potential applications in quantum teleportation [@1997/12/11/online; @PhysRevLett.88.017903], quantum computation, and quantum information [@cerf2007quantum; @bouwmeester2013physics]. They can either involve discrete variables (DVs), such as the polarization of a photon, or continuous variables (CVs), such as the quadratures of a beam of light. Although DV systems provide high-fidelity operations, photonic-based DV entanglement is currently limited by the difficulties of single-photon generation and detection, and by high sensitivity to optical losses. In contrast, CV entanglement is more robust to loss, and can be more efficiently created and used for the implementation of quantum protocols [@Huang2016; @PhysRevA.83.042312; @PhysRevX.5.041010; @PhysRevLett.93.250503; @PhysRevA.80.050303]. Spontaneous parametric down conversion (SPDC), a second order nonlinear process in which a pump photon is converted into a signal and an idler photon, is one of the processes that can be used to generate quantum correlated states [@PhysRevLett.97.223602; @PhysRevA.74.013815; @Yang:07]. It has been implemented in both bulk media and integrated photonic structures. However, as the size and complexity of quantum information processing systems increase, the limitations in achieving stability, precision, and small physical size with bulk optical systems become significant. Systems for on-chip SPDC, which are integrable with other photonic elements and could be used to generate CV entangled states involving two spatially separated sites, are therefore very promising [@PhysRevA.83.062310; @PhysRevA.82.012105; @PhysRevA.83.032102]. One such platform involves the use of waveguides made of materials with a large second order nonlinear optical response, such as AlGaAs, to generate counterpropagating, quantum correlated photons [@doi:10.1080/09500340802192431; @Orieux:11]. The particular system we consider here is the coupled-resonator optical waveguide (CROW), in which the waveguide consists of optical cavities weakly coupled in one dimension. By adjusting the nature of the cavities and the coupling between them, the dispersive properties of the propagating modes can be controlled [@Yariv:99]. Loss, which can destroy the nonclassical properties of light [@PhysRevLett.55.2409; @Jasperse:11; @Seifoory:17; @PhysRevA.97.023840; @PhysRevA.85.052330], can also be controlled to some extent, allowing at least a partial optimization for particular applications. CROW structures have been shown to have potential in generating CV entangled states between two side cavities coupled to the CROW, and as well between spatially separated sites [@PhysRevA.83.062310]. It is the latter application we study here. Our integrated source of entangled states is schematically shown in Fig. \[fig:schematic\]. A pump pulse is incident on a set of central cavities from above. Consequently, in order for the phase matching condition to be fulfilled, the generated signal and idler modes propagate in opposite directions in the CROW structure. An important advantage of such a configuration is the absence of the pump mode in the guided direction. Moreover, it has also been shown that the properties of the counterpropagating guided signal and idler modes can be tuned using the spectral and spatial properties of the pump [@Orieux:11; @PhysRevA.67.053810; @PhysRevA.70.052317]. ![Schematic picture of the particular CROW structure with period $D$ formed from defects in a slab photonic crystal with a square lattice of period $d$ and height $h$. The blue region shows the region covered by the pump. The origin of the coordinate system is at the center of the slab; i.e., the center of the central cavity.[]{data-label="fig:schematic"}](1c){width="\linewidth"} The tight-binding (TB) method [@ashcroft2011solid; @doi:10.1063/1.2737430], which uses localized single-cavity modes as a basis, can be applied to model the evolution of light in such a coupled structure. Assuming that all the cavities are identical and support the same mode with complex frequency $\tilde{\omega}_F$, it has been shown [@complex] that in the nearest-neighbor tight-binding (NNTB) approximation the dispersion relation can be written as $$\begin{aligned} \tilde{\omega}_{Fk}&\approx\tilde{\omega}_F[1-\tilde{\beta}_1 \cos(kD)]\nonumber\\ &\equiv\omega_{Fk}-i\gamma_{Fk}, \label{eq:dispersion_NNTB}\end{aligned}$$ where $ \tilde{\beta}_1 $, $D$, and $k$ are respectively the complex coupling parameter, the periodicity of the CROW, and the Bloch vector component. The imaginary part of the complex frequency is associated with the loss of the Bloch modes in the CROW. It is clear from Eq. (\[eq:dispersion\_NNTB\]) that these modes experience different loss rates; it has been shown that the rates can differ by an order of magnitude or more [@MohsenThesis; @Fussell:07; @PhysRevA.97.023840]. In our previous work, we focused on the time evolution of a state generated in a coupled-cavity system, and studied the evolution and propagation of squeezing and entanglement [@PhysRevA.97.023840]. We presented analytic expressions for a general initial state, but only presented detailed results for an initial state that was a squeezed vacuum state in one of the cavities. In this work, we investigate both the generation and propagation of entangled states in coupled-cavity systems. In addition, we engineer the pump parameters to produce counterpropagating pulses of the generated signal and idler modes, which are entangled but are not individually squeezed. Including the effects of intrinsic propagation loss, we calculate the number of photons in each cavity and the CV correlation variance of photons in different cavities. Previous approaches for generating counterpropagating entangled states have focused on photon pairs and been based either on ridge waveguides with vertical pumping [@doi:10.1080/09500340802192431; @Orieux:11], or on periodic waveguides with horizontal pumping [@PhysRevLett.118.183603]. The new approach of using a CROW has a number of advantages. First, the CROW allows us to control the group velocity and the frequency at which there is zero group velocity dispersion. Second, because a CROW can be modelled using a TB method, we are able to specify and model the effects of intrinsic scattering loss on the generated CV entanglement as a function of propagation distance, which is important for any application. Finally, using vertical pumping leads to counterpropagating entangled states, with no co-propagating pump at the outputs. This paper is organized as follows. In Sec. \[sec:level1\] we present the general theory of the generation and the evolution of the generalized two mode squeezed state in lossy CROWs via SPDC. In Sec. \[sec:result\_for\_gaussian\], we consider the special case of a pump that is Gaussian in time and space, and derive analytic expressions for the time dependence of the number of photons and the CV correlations. In Sec. \[sec:level4\], we present our results for a particular CROW in a slab photonic crystal, and discuss how they might be affected by the pumping configuration and the physical properties of the structure. Finally, in Sec. \[sec:conclusion\], we present our conclusions. \[sec:level1\]General Theory ============================ In order to determine both the generation and evolution of the entangled squeezed states in the system, we divide the analysis into two separate tasks. First, we study the creation of the entangled photons via SPDC using the backward Heisenberg method [@PhysRevA.77.033808], which is intrinsically a lossless approach. Having determined the initial entangled states created by the pump, we then include the loss to see how the generated state evolves in time and how loss affects it. Note that this two-step approach is valid because, for the parameters considered in this paper, the pump pulse is short enough that the signal loss is negligible over its duration. Generation ---------- There are two mode types that are relevant here, the fundamental modes and the pump modes; we indicate them by $ F $ and $ S $ respectively. In the SPDC process, two photons are generated in $ F $ modes from one pump photon in mode $ S $. Expanding the full displacement field $ \bm{D}(\bm{r}) $ in terms of the modes of interest, we have $$\begin{aligned} \mathbf{D}\left(\mathbf{r}\right)&=\left(\int\text{d}k\,\sqrt{\frac{\hbar\omega_{Fk}}{2}}\mathbf{D}_{Fk}\left(\mathbf{r}\right)\hat{a}_{Fk}\right.\nonumber\\ &\quad\left.+\sum_{m}\int\text{d}\mathbf{q}\,\sqrt{\frac{\hbar\omega_{Sm\mathbf{q}}}{2}}\mathbf{M}_{Sm\mathbf{q}}\left(\mathbf{r}\right)\hat{a}_{Sm\mathbf{q}}\right)+\text{H.c.}, \label{eq:D_field_tot} \end{aligned}$$ where $\mathbf{M}_{Sm\mathbf{q}}$, $\omega_{Sm\mathbf{q}}$, and $\hat{a}_{Sm\mathbf{q}}$ are the modes, eigenfrequencies and annihilation operators of the pump field, respectively, and $\mathbf{D}_{Fk}$, $\omega_{Fk}$, and $\hat{a}_{Fk}$ are the corresponding quantities for the generated signal and idler fields. Note that the integral over $k$ in Eq. (\[eq:D\_field\_tot\]) and in the rest of the paper (except where explicitly noted) only ranges from $-\pi/D$ to $\pi/D$. The continuous index, $\mathbf{q}$, is to identify the different pump modes in 3D while $m$ identifies the polarization state. The normalization conditions for the modes are presented in Appendix \[appendix\_normalization\]. Here $$\begin{aligned} \left[\hat{a}_{Fk},\hat{a}^\dagger_{Sm\mathbf{q}}\right]&=\left[\hat{a}_{Fk},\hat{a}_{Sm\mathbf{q}}\right]=0,\nonumber\\ \left[\hat{a}_{Sm\mathbf{q}},\hat{a}^\dagger_{Sm\mathbf{q^\prime}}\right]&=\delta_{m\,m^\prime}\delta(\mathbf{q}-\mathbf{q^\prime}),\nonumber\\ \left[\hat{a}_{Fk},\hat{a}^\dagger_{Fk^\prime}\right]&=\delta(k-k^\prime).\end{aligned}$$ For convenience we put $$\hat{a}_{Fk} \rightarrow \hat{b}_{k},\quad \hat{a}_{Sm\mathbf{q}} \rightarrow \hat{c}_{m\mathbf{q}},$$ and the linear Hamiltonian is then given by $$H_{L}=\int\text{d}k\,\hbar\omega_{Fk}\hat{b}_{k}^{\dagger}\hat{b}_{k}+\sum_{m}\int\text{d}\mathbf{q}\,\hbar\omega_{Sm\mathbf{q}}\hat{c}_{m\mathbf{q}}^{\dagger}\hat{c}_{m\mathbf{q}},$$ where we neglect the zero point energy and use only the real part of $\tilde{\omega}_{Fk}$ for the mode frequency. The nonlinear Hamiltonian that should be added to $H_{L}$ to construct the full Hamiltonian is [@PhysRevA.77.033808] $$H_{NL}=-\sum_{m}\int\text{d}k_{1}\text{d}k_{2}\text{d}\mathbf{q}\,S\left(k_{1},k_{2},m,\mathbf{q}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}\hat{c}_{m\mathbf{q}}+\text{H.c.}, \label{H_NL}$$ where $ S\left(k_{1},k_{2},m, \mathbf{q}\right) $ is the coupling coefficient, which is given by $$\begin{aligned} \label{eq:S_function01} S\left(k_{1},k_{2},m, \mathbf{q}\right) &=\frac{1}{\varepsilon_{0}}\sqrt{\frac{\hbar\omega_{Fk_1}\hbar\omega_{Fk_2}\hbar\omega_{Sm\mathbf{q}}}{8}}\int\text{d}\mathbf{r}\,\chi_{2}^{ijk}\left(\mathbf{r}\right)\nonumber\\ &\quad\times\frac{\left[D_{Fk_{1}}^{i}\left(\mathbf{r}\right)D_{Fk_{2}}^{j}\left(\mathbf{r}\right)\right]^{*}M^k_{Sm\mathbf{q}}\left(\mathbf{r}\right)}{\varepsilon_{0}n^{2}\left(\mathbf{r};\omega_{Fk_1}\right)n^{2}\left(\mathbf{r};\omega_{Fk_2}\right)n^{2}\left(\mathbf{r};\omega_{Sm\mathbf{q}}\right)},\end{aligned}$$ where $n\left(\mathbf{r};\omega\right)$ is a real, position and frequency dependent refractive index and $\chi_{2}^{ijk}\left(\mathbf{r}\right)$ is the position-dependent second-order nonlinear susceptibility. We take the pump to be a classical pulse incident on the slab, which we can expand as a superposition of the pump modes $\bm{M}_{Sm\mathbf{q}}\left(\bm{r}\right)$. We borrow a strategy from Yang *et al*. [@PhysRevA.77.033808] and define asymptotic-in and -out states to be respectively the input and output states of the nonlinear region at $t=0$, taking $t=0$ to be the time when the pump is centred on the slab. For the asymptotic-in state $\left|\psi_{\text{in}}\right\rangle$ that describes the classical pump pulse as a coherent state, we have $$\left|\psi_{\text{in}}\right\rangle =e^{\alpha\sum_{m}\int\text{d}\mathbf{q}\,\phi_{P}\left(m,\mathbf{q}\right)\hat{c}^\dag_{m\mathbf{q}}-\text{H.c.}}\left|\text{vac}\right\rangle , \label{eq:psi_in}$$ where $\alpha$ is a complex number, and we normalize the complex function $\phi_{P}\left(m,\mathbf{q}\right)$ according to $$\label{eq:normalization_condition} \sum_{m}\int \text{d}\mathbf{q}\,|\phi_P\left(m,\mathbf{q}\right)|^{2}=1.$$ The expectation value of the displacement field of the pump pulse is then $$\begin{aligned} \langle \psi_{\text{in}}|\mathbf{D}\left( \mathbf{r}\right)|\psi_{\text{in}}\rangle=&\alpha\sum_{m}\int\text{d}\mathbf{q}\,\sqrt{\frac{\hbar\omega_{Sm\mathbf{q}}}{2}}\phi_{P}\left(m,\mathbf{q}\right)\mathbf{M}_{Sm\mathbf{q}}\left(\mathbf{r}\right)\nonumber\\ &+\text{c.c.}\end{aligned}$$ and since $$\langle \psi_{\text{in}}|\hat{c}^\dag_{m\mathbf{q}}\hat{c}_{m\mathbf{q}}|\psi_{\text{in}}\rangle=|\alpha|^2|\phi_P(m,\mathbf{q})|^2$$ we can identify $|\alpha|^2$ as the expectation value of the number of photons in the pump pulse. Following the backward Heisenberg picture approach [@PhysRevA.77.033808], the asymptotic-out state for the *generated* photons in the first approximation is then $$\label{assym_out} \left|\psi_{\text{out}}^{F}\right\rangle =e^{\frac{\beta}{\sqrt{2}}\int\text{d}k_{1}\text{d}k_{2}\,\phi\left(k_{1},k_{2}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}-\text{H.c.}}\left|\text{vac}\right\rangle ,$$ where $\phi\left(k_{1},k_{2}\right)$ is the biphoton wave function, which from Yang *et al*. [@PhysRevA.77.033808] is given by $$\begin{aligned} \label{eq:biphoton_sipe} \phi\left(k_{1},k_{2}\right)&=\frac{2i\sqrt{2}\pi\alpha}{\beta\hbar}\sum_{m}\int\text{d}\mathbf{q}\,\phi_{P}\left(m,\mathbf{q}\right)\nonumber\\ &\quad\times S\left(k_{1},k_{2},m, \mathbf{q}\right) \delta (\omega_{Sm\mathbf{q}}-\omega_{F_{k_{1}}}-\omega_{F_{k_{2}}}),\end{aligned}$$ where $\beta$ is a real and positive normalization constant chosen to ensure that $\int \text{d}k_1\,\text{d}k_2\,|\phi\left(k_{1},k_{2}\right)|^{2}=1$. While the biphoton wave function in general obeys the symmetry $\phi\left(k_{1},k_{2}\right)=\phi\left(k_{2},k_{1}\right)$, it can sometimes be useful to work with a function that breaks this symmetry to focus attention on a particular quadrant of $\left(k_{1},k_{2}\right)$ space. In this work, we will always choose the pump parameters such that to a very good approximation, $\phi(k_1,k_2)$ is nonzero only when $k_1$ and $k_2$ have opposite signs, as we detail in Section \[sec:result\_for\_gaussian\] below. Then, we can write $$\begin{aligned} &\int_{-\pi/D}^{\pi/D}\text{d}k_{1}\int_{-\pi/D}^{\pi/D}\text{d}k_{2}\,\phi\left(k_{1},k_{2}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}\nonumber\\ &\approx\int_{0}^{\pi/D}\text{d}k_{1}\int_{-\pi/D}^{0}\text{d}k_{2}\,\phi\left(k_{1},k_{2}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}\nonumber\\ &\quad+\int_{-\pi/D}^{0}\text{d}k_{1}\int_{0}^{\pi/D}\text{d}k_{2}\,\phi\left(k_{1},k_{2}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}\nonumber\\ &=2\int_{0}^{\pi/D}\text{d}k_{1}\int_{-\pi/D}^{0}\text{d}k_{2}\,\phi\left(k_{1},k_{2}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}.\end{aligned}$$ We then define $$\label{eq:Luke_Phi} \Phi\left(k_{1},k_{2}\right)\equiv\sqrt{2}\phi\left(k_{1},k_{2}\right)\Theta\left(k_{1}\right)\Theta\left(-k_{2}\right),$$ where $ \Theta\left(k\right) $ is the Heaviside function, such that we may rewrite Eq.  as $$\label{eq:assym_out_2} \left|\psi_{\text{out}}^{F}\right\rangle =e^{\beta\int\text{d}k_{1}\text{d}k_{2}\,\Phi\left(k_{1},k_{2}\right)\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}-\text{H.c.}}\left|\text{vac}\right\rangle.$$ Employing a Schmidt decomposition [@peres2006quantum; @doi:10.1119/1.17904], we have $$\label{eq:phi_inermsof_mu_nu} \Phi\left(k_{1},k_{2}\right)=\sum_{\lambda}\sqrt{p_{\lambda}}\mu_{\lambda}\left(k_{1}\right)\nu_{\lambda}\left(k_{2}\right),$$ for $p_{\lambda} > 0$ with $\sum_{\lambda}p_{\lambda}=1$, where the Schmidt functions are orthonormal, $$\int\text{d}k\mu_{\lambda}(k)\mu^*_{\lambda^\prime}(k)=\int\text{d}k\nu_{\lambda}(k)\nu^*_{\lambda^\prime}(k)=\delta_{\lambda,\lambda^{\prime}}.$$ We extend the sets of ${\mu_\lambda(k)}$ and ${\nu_\lambda(k)}$ associated with $p_\lambda>0$ to form complete sets with $$\label{eq:orthonormal} \sum_\lambda \mu_\lambda(k)\mu^*_\lambda(k^\prime)=\sum_\lambda \nu_\lambda(k)\nu^*_\lambda(k^\prime)=\delta(k-k^\prime),$$ and with some of the $p_\lambda$ appearing in Eq. (\[eq:phi\_inermsof\_mu\_nu\]) then equal to zero. Using Eqs.  and , the generated squeezed state can be written as $$\left|\psi_{\text{out}}^{F}\right\rangle =\hat{S}\left|\text{vac}\right\rangle ,$$ where the squeezing operator, $ \hat{S} $, can be written as $$\begin{aligned} \hat{S}&=\exp\left(\beta\int\text{d}k_{1}\text{d}k_{2}\,\sum_{\lambda}\sqrt{p_{\lambda}}\mu_{\lambda}\left(k_{1}\right)\nu_{\lambda}\left(k_{2}\right)\right.\nonumber\\ &\quad\times\left.\hat{b}_{k_{1}}^{\dagger}\hat{b}_{k_{2}}^{\dagger}-\text{H.c.}\vphantom{\beta\int\text{d}k_{1}\text{d}k_{2}\,\sum_{\lambda}}\right)\nonumber\\ &=\exp\left(\sum_{\lambda}r_{\lambda}\hat{B}_{\lambda}^{\dagger}\hat{C}_{\lambda}^{\dagger}-\sum_{\lambda}r_{\lambda}^{*}\hat{B}_{\lambda}\hat{C}_{\lambda}\right),\end{aligned}$$ where $r_{\lambda}=\beta\sqrt{p_{\lambda}}$ is the squeezing parameter, $$\begin{aligned} \label{eq:B_operator} \hat{B}_{\lambda}&\equiv\int\mu_{\lambda}^{*}\left(k\right)\hat{b}_{k}\text{d}k,\end{aligned}$$ and $$\begin{aligned} \label{eq:C_operator} \hat{C}_{\lambda}&\equiv\int\nu_{\lambda}^{*}\left(k\right)\hat{b}_{k}\text{d}k.\end{aligned}$$ Using Eq. (\[eq:orthonormal\]), it can be shown that $ [\hat{B}_{\lambda},\hat{B}_{\lambda^{\prime}}^{\dagger}]=[\hat{C}_{\lambda},\hat{C}_{\lambda^{\prime}}^{\dagger}]=\delta_{\lambda,\lambda^{\prime}} $ and $ [\hat{B}_{\lambda},\hat{C}_{\lambda^{\prime}}^{\dagger}]=[\hat{B}_{\lambda},\hat{B}_{\lambda^{\prime}}]=[\hat{C}_{\lambda},\hat{C}_{\lambda^{\prime}}]=[\hat{B}_{\lambda},\hat{C}_{\lambda^{\prime}}]=0$. The importance of the Schmidt decomposition and the operator transformation is that it enables us to express the generated state as a generalized two-mode squeezed state, where the modes are no longer the Bloch modes. As we shall see in the next section, this will enable us to easily determine the evolution of the state in the presence of loss. Evolution --------- As mentioned earlier, we have assumed that the loss during the generation process is negligible. However, the effect of loss cannot be ignored when calculating the evolution of the generated pulses down the CROW. Following the formalism presented in our previous work [@PhysRevA.97.023840] on lossy coupled-cavity systems, the individual single-mode cavity annihilation operator for the $ p^{th} $ cavity, $ \hat{a}_p $, can be written in terms of the $ k^{th} $ mode annihilation operator of the coupled-cavity-system, $ \hat{b}_k $, as $$\label{a_to_b} \hat{a}_{p}(t)=\sqrt{\frac{D}{2\pi}}\int\hat{b}_{k}(t)e^{ikpD}\text{d}k.$$ The time evolution of the full coupled-cavity annihilation operator can also be found by solving the adjoint master equation for this open, lossy system [@open_quantum_system]. We have previously shown that the time dependence of the individual annihilation operators is given by $$\label{b_time} \hat{b}_k(t)=\hat{b}_ke^{-i\tilde{\omega}_{Fk}t},$$ where $ \hat{b}_k=\hat{b}_k(0) $ is the corresponding operator in the Schrödinger representation [@PhysRevA.85.013809]. Using Eqs. (\[a\_to\_b\]), (\[b\_time\]), and their complex conjugates, the time dependent average photon number in the $ p^{th} $ cavity can be written as $$\begin{aligned} \label{eq:apdap} \left\langle \hat{a}_{p}^{\dagger}\left(t\right)\hat{a}_{p}\left(t\right)\right\rangle &=\frac{D}{2\pi}\int\int \text{d}k\text{d}k^{\prime}\left\langle \hat{b}_{k}^{\dagger}\hat{b}_{k^{\prime}}\right\rangle e^{-i\left(k-k^{\prime}\right)pD}\nonumber\\ &\times\left(e^{i\tilde{\omega}_{F}^{*}\left(1-\tilde{\beta}_{1}^{*}\cos(kD)\right)t}e^{-i\tilde{\omega}_{F}\left(1-\tilde{\beta}_{1}\cos(k^{\prime}D)\right)t}\right),\end{aligned}$$ where we have used the lossy dispersion relation of the CROW structure \[Eq. \]. To facilitate the evaluation of $\left\langle \hat{b}_{k}^{\dagger}\hat{b}_{k^{\prime}}\right\rangle$, we introduce the restricted operators, $$\begin{aligned} \hat{b}_{k,+}&\equiv\Theta\left(k\right)\hat{b}_{k}\nonumber\\ \hat{b}_{k,-}&\equiv\Theta\left(-k\right)\hat{b}_{k}.\end{aligned}$$ Using these operators, we can write $$\label{eq:4_terms} \left\langle \hat{b}_{k}^{\dagger}\hat{b}_{k^{\prime}}\right\rangle= \left\langle \hat{b}_{k,+}^{\dagger}\hat{b}_{k^{\prime},-}^{}+ \hat{b}_{k,-}^{\dagger}\hat{b}_{k^{\prime},-}^{}+ \hat{b}_{k,+}^{\dagger}\hat{b}_{k^{\prime},+}^{}+ \hat{b}_{k,-}^{\dagger}\hat{b}_{k^{\prime},+}^{}\right\rangle.$$ To evaluate each of these terms, we use the following Bogoliubov transformations $$\begin{multlined} \label{bogol1} \hat{S}^{\dagger}\hat{b}_{k,+}\hat{S}=\hat{S}^{\dagger}\sum_{\lambda}\mu_{\lambda}\left(k\right)\hat{B}_{\lambda}\hat{S}\\=\sum_{\lambda}\mu_{\lambda}\left(k\right)\left[\hat{B}_{\lambda}\cosh\left(r_\lambda\right)-\hat{C}_{\lambda}^{\dagger}\sinh\left(r_\lambda\right)\right], \end{multlined}$$ $$\begin{multlined} \label{bogol2} \hat{S}^{\dagger}\hat{b}_{k,-}\hat{S}=\hat{S}^{\dagger}\sum_{\lambda}\nu_{\lambda}\left(k\right)\hat{C}_{\lambda}\hat{S}\\=\sum_{\lambda}\nu_{\lambda}\left(k\right)\left[\hat{C}_{\lambda}\cosh\left(r_\lambda\right)-\hat{B}_{\lambda}^{\dagger}\sinh\left(r_\lambda\right)\right]. \end{multlined}$$ Using these in Eq. , we obtain $$\begin{aligned} \label{eq:bdkbk} \left\langle \hat{b}_{k}^{\dagger}\hat{b}_{k^{\prime}}\right\rangle &=\sum_{\lambda}\bigg(\mu_{\lambda}^{*}\left(k\right)\mu_{\lambda}\left(k^{\prime}\right)\nonumber+\nu_{\lambda}^{*}\left(k\right)\nu_{\lambda}\left(k^{\prime}\right)\bigg)\\ &\times\sinh^{2}\left(r_\lambda\right).\end{aligned}$$ To study the degree of entanglement between the photons in cavities $ p $ and $ p^\prime $ in a CROW, we use the correlation variance, which is defined as $$\Delta_{p,p^\prime}^2=\left\langle [\Delta(\hat{X}_p-\hat{X}_{p^\prime})]^2\right\rangle+ \left\langle [\Delta(\hat{Y}_p+\hat{Y}_{p^\prime})]^2\right\rangle, \label{eq:correlation_nm_def}$$ where $$\label{X_and_Y} \begin{aligned} \hat{X}_p&\equiv\hat{a}_p+\hat{a}_p^{\dagger},\\ \hat{Y}_p&\equiv-i(\hat{a}_p-\hat{a}_p^{\dagger}). \end{aligned}$$ It has been shown that $ \Delta_{p,p^\prime}^2 <4$ can be considered as the inseparability criterion for entanglement [@PhysRevLett.84.2722; @PhysRevLett.84.2726; @Masada2015; @Zhang2015]. Using Eq. (\[X\_and\_Y\]) in Eq. (\[eq:correlation\_nm\_def\]), the time-dependent correlation variance can be written as $$\label{eq:entanglement_01} \Delta_{pp^\prime}^{2}=4+4\left(\langle \hat{a}_{p}^{\dagger}\hat{a}_{p}\rangle +\langle \hat{a}_{p^{\prime}}^{\dagger}\hat{a}_{p^{\prime}}\rangle -\langle \hat{a}_{p}\hat{a}_{p^{\prime}}\rangle-\langle \hat{a}_{p}^{\dagger}\hat{a}_{p^{\prime}}^{\dagger}\rangle \right).$$ Following a procedure similar to that used to arrive at Eqs. (\[eq:apdap\]) and (\[eq:bdkbk\]), one can derive the other expectation values that are needed to evaluate the variances of the quadrature operators and the correlation variance in the CROW structure. Results for a Gaussian pump pulse {#sec:result_for_gaussian} ================================= The results of the previous sections are general and independent of the temporal and spatial form of the pump pulse, as long as $\phi(k_1,k_2)$ is nonzero only when $k_1$ and $k_2$ have opposite signs. However, in this section we consider the special case of a Gaussian pump pulse incident on the slab, and brought to a Gaussian focus there. We assume that the slab does not have a significant effect on the pump pulse, and so take the pump modes to be plane waves in free space and set $n(\mathbf{r},\omega_{Sm\mathbf{q}})=1$ in Eq. (\[eq:S\_function01\]). Thus, we have $$\mathbf{M}_{Sm\mathbf{q}}\left(\mathbf{r}\right)=\frac{\sqrt{\varepsilon_{0}}\mathbf{e}_{m,\mathbf{q}}}{\left(2\pi\right)^{3/2}}e^{i\mathbf{q}\cdot\mathbf{r}}, \label{M}$$ where $\mathbf{e}_{m,\mathbf{q}}$ is the polarization unit vector. In what follows, we assume that in the vicinity of the CROW, the pump is polarized in the $y$-direction. Thus, we obtain $$\phi_{P}\left(m,\mathbf{q}\right)=\delta_{my}\varphi\left(q_{x}\right)F\left(q_{y},q_{z}\right),$$ where $$F\left(q_{y},q_{z}\right)=\frac{1}{2\pi}\int\text{d}y\text{d}z\,f\left(y,z\right)e^{-i\left(q_{y}y+q_{z}z\right)},$$ is the Fourier transform of the transverse profile $$\label{eq:f_yz_text} f\left(y,z\right)=\frac{\sqrt{2}}{\sqrt{\pi}W_{S}}e^{-\frac{y^{2}+z^{2}}{W_{S}^{2}}}e^{iq_{P}\frac{y^{2}+z^{2}}{2R_{P}}},$$ and $q_P$ is the value of $q_x$ at which $\phi(q_x)$ peaks. Here $R_{P}$ and $W_{S}$ are the radius of curvature and spot size, respectively, evaluated at $x=0$ and $q_{x}=q_{P}$. We have assumed the Rayleigh range to be much larger than the slab thickness, which justifies the neglect of the Gouy phase. The prefactors have been chosen so that the normalization condition Eq. (\[eq:normalization\_condition\]) becomes $$\int\text{d}q_{x}\left|\varphi\left(q_{x}\right)\right|^{2}=1.$$ Neglecting the dependence of the indices of refraction and the frequencies under the square root on $k_{1}$, $k_{2}$, and $\mathbf{q}$, based on the small frequency range of the input pump pulse and the limited range of the signal and idler photons in the CROW, we can rewrite the biphoton wave function of Eq.  as $$\begin{split} \label{eq:bithoton_approx} \Phi &\left(k_{1},k_{2}\right)=\frac{i\alpha\sqrt{\pi}}{\beta\hbar\sqrt{\varepsilon_0}}\sqrt{\left(\hbar\omega_{F}\right)^{2}\hbar\omega_{Sy}}\int\text{d}q_{x}\varphi(q_{x})\\ &\times\int\text{d}\mathbf{r}\,\chi_{2}^{ijy}\left(\mathbf{r}\right)\frac{\left[D_{Fk_{1}}^{i}\left(\mathbf{r}\right)D_{Fk_{2}}^{j}\left(\mathbf{r}\right)\right]^{*}f(y,z)e^{iq_{x}x}}{\varepsilon_{0}n^{4}\left(\mathbf{r};\omega_{F}\right)}\\ &\times \delta (cq_{x}-\omega_{F_{k_{1}}}-\omega_{F_{k_{2}}})\Theta\left(k_{1}\right)\Theta\left(-k_{2}\right). \end{split}$$ Because we have made the approximation that the transverse profile of the pump does not depend on frequency (for the frequencies of interest) in Eq.  we set $\omega_{Sy\mathbf{q}}=cq_{x}$ in the Dirac delta function. We now employ the nearest-neighbour tight-binding approximation [@doi:10.1063/1.2737430] and expand the $D_{Fk}^{i}\left(\mathbf{r}\right)$ modes in terms of the single-cavity quasimodes, $N_{Fp}^{i}\left(\mathbf{r}\right)$, as $$D_{Fk}^{i}\left(\mathbf{r}\right)=\sqrt{\frac{D}{2\pi}}\sum_{p}N_{Fp}^{i}\left(\mathbf{r}\right)e^{ikpD}, \label{eq:D_in terms of N}$$ which leads to the lossy frequency dispersion given in Eq. (\[eq:dispersion\_NNTB\]). The single-cavity quasimodes and frequencies are calculated in the standard way using finite difference time domain calculations [@doi:10.1063/1.2737430]. Assuming that the cavity modes are well-localized [@well_localized], we obtain $$\label{eq:biphoton_sigma_and_integral} \begin{multlined} \Phi\left(k_{1},k_{2}\right)=\frac{i \alpha D}{2\beta\hbar\sqrt{ \varepsilon_0\pi}}\sqrt{\left(\hbar\omega_{F}\right)^{2}\hbar\omega_{S}}\int\text{d}q_{x}\varphi(q_{x})\\ \times \sum_p \int\text{d}\mathbf{r}\,\chi_{2}^{ijy}\left(\mathbf{r}\right)\frac{\left[N_{Fp}^{i}\left(\mathbf{r}\right)N_{Fp}^{j}\left(\mathbf{r}\right)\right]^{*}f(y,z)e^{iq_{x}x}}{\varepsilon_{0}n^{4}\left(\mathbf{r};\omega_{F}\right)}\\ \times e^{-i(k_1 + k_2 )pD} \delta (cq_{x}-\omega_{F_{k_{1}}}-\omega_{F_{k_{2}}})\Theta\left(k_{1}\right)\Theta\left(-k_{2}\right). \end{multlined}$$ We define $$\label{PHI_qx_text} \varphi(q)=\sqrt{W_T/\sqrt{2\pi}}\exp{\left[-\left(\frac{\left(q-q_{P}\right)W_T}{2}\right)^{2}\right]},$$ and, because and spatial extent of the single-cavity quasimodes is small relative to that of the pump field, in the integral in Eq.  we replace $f\left(y,z\right)$ by $$f\left(y=0,z=pD\right)\approx\frac{\sqrt{2}}{\sqrt{\pi}W_{S}}e^{-\frac{p^{2}D^{2}}{W_{S}^{2}}},$$ to obtain the approximate expression $$\label{biphoton_01} \begin{multlined} \Phi(k_{1},k_{2})=\frac{i\alpha\bar{\chi}_{2}}{\beta}\sqrt{\frac{\hbar\omega_{F}^{2}\omega_{S}W_T}{\varepsilon_{0}\left(2\pi\right)^{3/2}}}e^{\frac{-(k_{1}+k_{2})^{2}W_S^{2}}{4}}\\ \times\int\text{d}q_{x}\,e^{-\frac{\left(q_{x}-q_{P}\right)^{2}W_T^{2}}{4}}\delta(\omega_{Sq_{x}}-\omega_{Fk_{1}}-\omega_{Fk_{2}})\Theta\left(k_{1}\right)\Theta\left(-k_{2}\right), \end{multlined}$$ where $$\label{eq:Xi2bar} \bar {\chi}_{2}\equiv \int\text{d}\mathbf{r}\,\chi_{2}^{ijy}\left(\mathbf{r}\right)\frac{\left[N_{F0}^{i}\left(\mathbf{r}\right)N_{F0}^{j}\left(\mathbf{r}\right)\right]^{*}}{\varepsilon_{0}n^{4}\left(\mathbf{r};\omega_{F}\right)}e^{iq_{P}x},$$ is the effective second order susceptibility for the system [@expiq_x] (See Appendix \[appendix\_B\] for more details). For $\omega_{Fk}\equiv\omega_{F}\left[1-\beta_{1}\cos\left(kD\right)\right]$, which we consider to be a real quantity at this point, Eq. (\[biphoton\_01\]) can be rewritten as $$\label{biphoton_02} \begin{multlined} {\Phi}(k_{1},k_{2})=Q_0\int\text{d}q_{x}\,e^{\frac{-(k_{1}+k_{2})^{2}W_S^{2}}{4}}e^{-\frac{\left(q_{x}-q_{P}\right)^{2}W_T^{2}}{4}}\delta\left\{ q_{x}-\frac{1}{c}\left[\omega_{Fk_1}+\omega_{Fk_2}\right]\right\}\Theta\left(k_{1}\right)\Theta\left(-k_{2}\right)\\ =Q_{0}\exp\left(\frac{-(k_{1}+k_{2})^{2}W_S^{2}}{4}\right)\exp\left(-\left(\frac{2\omega_{F}-\beta_{1}\omega_{F}\left[\cos\left(k_{1}D\right)+\cos\left(k_{2}D\right)\right]-\omega_{P}}{2c}\right)^{2}W_T^{2}\right)\Theta\left(k_{1}\right)\Theta\left(-k_{2}\right), \end{multlined}$$ where $$\label{eq:Q_0} Q_{0}\equiv\frac{i\alpha\bar{\chi}_{2}}{\beta c}\sqrt{\frac{\hbar\omega_{F}^{2}\omega_{S}W_T}{\varepsilon_{0}\left(2\pi\right)^{3/2}}}.$$  In order to derive analytic expressions for the photon number and correlation variance as a function of cavity index, $p$, we need to place further restrictions on the pump pulse. From the first exponential in Eq. (\[biphoton\_02\]), we see that the biphoton wave function will only be non-negligible if $k_2$ is approximately equal to $-k_1$. Thus we set $$\begin{aligned} k_{1} &\rightarrow k_{0}+\delta_{1},\\ k_{2} &\rightarrow-k_{0}+\delta_{2}, \end{aligned}$$ where the $\delta_{i}$ are small relative to $W_S$, where $k_0$ is determined by the central frequency of the pump, through the equation $$\omega_{P}=2\omega_{Fk_{0}}=2\omega_{F}-2\beta_{1}\omega_{F}\cos\left(k_{0}D\right).$$ In order to obtain a biphoton wave function for which there is an analytic Schmidt decomposition, we take $k_0=\pi/(2D)$ and choose the frequency width parameter, $W_T$, of the pulse to satisfy $c/W_T\ll\Delta$, where $\Delta=2\omega_{F}\beta_{1}$. Now, expanding the cosines in Eq. (\[biphoton\_02\]) to first order in $\delta_1$ and $\delta_2$, the biphoton wave function can be rewritten as $$\begin{aligned} \label{biphoton_form} %\begin{split} &{\Phi(k_0+\delta_1,-k_0+\delta_2)}=\nonumber\\ &\sqrt{\frac{2}{\pi\sigma_{+}\sigma_{-}}} \exp\left(-\frac{(\delta_{1}+\delta_{2})^{2}}{2\sigma_{+}^{2}}\right)\exp\left(-\frac{\left(\delta_{1}-\delta_{2}\right)^{2}}{2\sigma_{-}^{2}}\right), %\end{split}\end{aligned}$$ where $$\sigma_{+}\equiv\frac{\sqrt{2}}{W_S} \label{eq:sigma_plus}$$ and $$\label{eq:sigma_minus} \sigma_{-}\equiv\frac{\sqrt{2}c}{W_T\beta_{1}\omega_{F}D\sin\left(|k_{0}|D\right)}.$$ Strictly speaking, the biphoton wave function of Eq.  does not satisfy the restriction that it is zero unless $k_1>0$ and $k_2<0$. However, as long as $\sigma_+$ and $\sigma_-$ are chosen to be small enough, then these conditions are satisfied to a very high degree. In what follows, we shall only consider situations where this is the case. Using the normalization condition, $\int \text{d}k_{1}\,\text{d}k_{2}\left|{\Phi}\left(k_{1},k_{2}\right)\right|^{2}=1$, it can be shown that $$\label{eq:Q_0_second} Q_0=\sqrt{\frac{2}{\pi\sigma_{+}\sigma_{-}}}.$$ In Fig. \[fig:biphoton\], we plot three sample biphoton wave functions of the form given in Eq.  for different $\sigma_+$ and $\sigma_-$ for $k_0=\pi/2D$. To graphically illustrate the validity of the assumptions made to obtain Eq. (\[biphoton\_form\]), in Fig. \[fig:dispersion\] we plot the dispersion of our CROW. The physical parameters of the CROW are from Ref. [@doi:10.1063/1.2737430]. It consists of a dielectric slab of refractive index $n=3.4$ having a square array of cylindrical air voids of radius $a=0.4d$, height $h=0.8d$, and lattice vectors $\textbf{a}_1=d\hat{\textbf{x}}$ and $\textbf{a}_2=d\hat{\textbf{y}}$, where $d$ is the period. The cavities are point defects formed by periodically removing air voids in a line with $D=2d$ (see Fig. \[fig:schematic\]). The complex frequency, $\tilde{\omega}_F$, and the complex coupling parameter, $\tilde{\beta}_1$, of the structure are $(0.305-i7.71\times10^{-6})4\pi c/D$, and $9.87\times10^{-3}-i1.97\times10^{-5}$, respectively. To visualize the biphoton wave function superimposed on the CROW dispersion, we plot ${\Phi}(k,-k)$ for $W_S=3D$ and $\sigma_+=\sigma_-$ in Fig. \[fig:dispersion\] as well. As can be seen, the first order expansion of $\cos(k_1D)$ and $\cos(k_2D)$ about $k_0$ and $-k_0$ is accurate as the dispersion within this range is very close to linear. In order for our Schmidt decomposition to be valid for this structure, $\sigma_+D$ and $\sigma_-D$ cannot be increased significantly beyond the chosen value of $0.47$ otherwise the biphoton wave function will not be confined to the quadrant where $k_1>0$ and $k_2<0$. Note that increasing the pump width to higher values, $W_S>3D$, increases the accuracy of our approximation, as is evident from Figs. \[fig:biphoton\](b) and \[fig:biphoton\](c) where $W_S$ is $5.05D$ and $10.10D$, respectively. ![The CROW dispersion relation and biphoton wave function. The solid black line shows the frequency as a function of the Bloch vector for the CROW structure. The dashed horizontal line gives the pump frequency divided by two. The dashed green line gives the function $\Phi(k,-k)$ for $k_0=\pi/(2D)$. The two vertical solid blue lines indicate the FWHM in $k$, while the shaded blue region indicates the FWHM in frequency, both of which can be found from Eq.  for $\sigma_+D=\sigma_-D=0.47$.[]{data-label="fig:dispersion"}](dispersion) Before employing a Schmidt decomposition, we present the relation between $\sigma_\pm$ and the temporal and spatial full width at half maximum (FWHM) of the pump pulse. Using Eqs. (\[eq:f\_yz\_text\]) and (\[eq:sigma\_minus\]), the *temporal* FWHM of the pump can be written as $$\label{eq:delta_t} \Delta t_{FWHM}=\frac{2\sqrt{\ln2}\tau}{\sigma_{-}D},$$ where $\tau\equiv1/Re(\tilde{\omega}_F\tilde{\beta}_1)$ is the time for a pulse with Bloch vector $k=k_0=\pi/(2D)$ to travel one period. Similarly, using Eqs. (\[PHI\_qx\_text\]) and (\[eq:sigma\_plus\]), the *spatial* FWHM of the pump is found to be $$\label{eq:deltar} \Delta r_{FWHM}=\frac{2\sqrt{\ln2}}{\sigma_{+}}.$$ Using these two equations, one can obtain a clear understanding of the necessary pumping conditions. For instance, the quantities considered in Fig. \[fig:biphoton\](a) correspond to a pump with $3.54\tau$ and $3.54D$ as the temporal and spatial FWHM, respectively. As we show later, for our CROW, the propagation loss in the system is very small while the squeezed light is being generated, which validates the neglect of loss during the generation process. The special form of the biphoton wave function, Eq. (\[biphoton\_form\]), allows us to perform a Schmidt decomposition analytically [@U'Ren:2003:PEQ:2011564.2011567; @doi:10.1080/09500340600777805; @PhysRevA.96.053842] for $\sigma_{-}\geq\sigma_{+}$ as $$\label{Schmidt} \begin{multlined} \sqrt{\frac{2}{\pi\sigma_{+}\sigma_{-}}}\exp\left(-\frac{(\delta_{1}+\delta_{2})^{2}}{2\sigma_{+}^{2}}\right)\exp\left(-\frac{\left(\delta_{1}-\delta_{2}\right)^{2}}{2\sigma_{-}^{2}}\right)\\ =\sum_{\lambda}\sqrt{p_{\lambda}}\psi_{\lambda}\left(\delta_{1}\right)\psi_{\lambda}\left(\delta_{2}\right), \end{multlined}$$ where $$\label{p_lambda} p_{\lambda}=4\sigma_{+}\sigma_{-}\frac{\left(\sigma_{+}-\sigma_{-}\right)^{2\lambda}}{\left(\sigma_{+}+\sigma_{-}\right)^{2\left(\lambda+1\right)}},$$ $$\label{eq:psi_lambda} \begin{multlined} \psi_{\lambda}\left(\delta\right)=\left(-i\right)^{\lambda}\sqrt{\frac{\sqrt{2}}{2^{\lambda}\lambda!\sqrt{\pi\sigma_{+}\sigma_{-}}}}\\ \times\exp\left(-\frac{\delta^{2}}{\sigma_{+}\sigma_{-}}\right)H_{\lambda}\left(\frac{\sqrt{2}\delta}{\sqrt{\sigma_{+}\sigma_{-}}}\right), \end{multlined}$$ and the $H_{\lambda}\left(x\right)$ are Hermite polynomials of order $\lambda$. Note that the Schmidt number is given by [@Zhukovsky:12] $$K=\frac{1}{\sum_{\lambda}p_{\lambda}^{2}}=\frac{\sigma_{+}^{2}+\sigma_{-}^{2}}{2\sigma_{+}\sigma_{-}}.$$ Using Eq. (\[p\_lambda\]) for the special case where $ \sigma_-=\sigma_+ =\sigma$, it can be shown that the only nonzero term in Eq. (\[ph\_number\_01\]) is for $ \lambda=0 $. However, in general, one needs to include several of the Schmidt modes (up to $\lambda=\lambda_{max}$) to accurately represent the biphoton wave function. To quantify the accuracy of the Schmidt decomposition used in our calculations, we define an error function as $$Err=\sqrt{\frac{\int\text{d}k_{1}\,\text{d}k_{2}\,|{\Phi}\left(k_1,k_2\right)-{\Phi_{App}}\left(k_1,k_2\right)|^2}{\int\text{d}k_{1}\,\text{d}k_{2}\,|{\Phi}(k_1,k_2)|^2}},$$ where [$\Phi$ and $\Phi_{App}$]{} are the exact and approximate expressions, respectively, given by Eq. (\[biphoton\_form\]) and $${\Phi_{App}(k_0+\delta_1,-k_0+\delta_2)}=\sum_{\lambda=0}^{\lambda_{max}}\sqrt{p_{\lambda}}\psi_{\lambda}\left(\delta_{1}\right)\psi_{\lambda}\left(\delta_{2}\right).$$ In Fig. \[fig:max\_lambda\] we plot the index of the maximum Schmidt mode needed to be included in ${\Phi_{App}}$, in order to ensure $Err < 0.1\%$. For example, as can be seen, when $\sigma_{-}=2\sigma_{+}$, one needs to include $6$ terms ($\lambda_{max}=6$) to achieve the desired accuracy. ![Maximum number of Schmidt modes required to be considered in $\Phi$ as a function of $\sigma_{+}/\sigma_{-}$ to ensure $Err < 0.1\%$. []{data-label="fig:max_lambda"}](lambda_max) Using these results in Eqs. (\[eq:apdap\]) and (\[eq:bdkbk\]), the time-dependent average photon number in the $ p^{th} $ cavity is found to be $$\begin{multlined} \label{ph_number_01} \left\langle \hat{a}_{p}^{\dagger}\left(t\right)\hat{a}_{p}\left(t\right)\right\rangle = \frac{D}{2\pi}e^{-2Re\left(i\tilde{\omega}_{F}\left(1-\tilde{\beta}_{1}\cos(|k_{0}|D)\right)t\right)}\\\times\sum_{\lambda}\sinh^{2}\left(r_\lambda\right)\frac{\sqrt{2\sigma_{+}\sigma_{-}\pi}}{2^{\lambda}\lambda!}\times \left(\left|H_{\lambda}\left(\tilde{S}_{p-}\right)\right|^{2}e^{-\left(\frac{\left(\tilde{S}_{p-}^{*2}+\tilde{S}_{p-}^{2}\right)}{2}\right)} +\left|H_{\lambda}\left(\tilde{S}_{p+}\right)\right|^{2}e^{-\left(\frac{\left(\tilde{S}_{p+}^{*2}+\tilde{S}_{p+}^{2}\right)}{2}\right)}\right), \end{multlined}$$ where $$\begin{aligned} \tilde{S}_{p\pm}=\left(\tilde{\omega}_{F}\tilde{\beta}_{1}D\sin\left(|k_{0}|D\right)t\pm pD\right)\sqrt{\frac{\sigma_{+}\sigma_{-}}{2}}. \end{aligned}$$ The time-dependent average photon number in the $ p^{th} $ cavity when $\sigma_{+}=\sigma_{-}$ can be simplified to $$\begin{split} \label{ph_number_02} \left\langle \hat{a}_{p}^{\dagger}\left(t\right)\hat{a}_{p}\left(t\right)\right\rangle &= \frac{\sigma D}{\sqrt{2\pi}}e^{-2Re\left(i\tilde{\omega}_{F}\left(1-\tilde{\beta}_{1}\cos(|k_{0}|D)\right)t\right)}\sinh^{2}\left(r_{0}\right)\\ &\times\left(e^{-\left(\frac{\left(\tilde{S}_{p-}^{*2}+\tilde{S}_{p-}^{2}\right)}{2}\right)} +e^{-\left(\frac{\left(\tilde{S}_{p+}^{*2}+\tilde{S}_{p+}^{2}\right)}{2}\right)}\right). \end{split}$$ We note that using Eq. (\[ph\_number\_02\]) for a lossless system, one finds that the total number of photons in the CROW is $2\sinh^2\left(r_{0}\right)$, independent of $\sigma$, which agrees with the total number of generated photons in any two-mode squeezed state. Following a procedure similar to that used to arrive at Eq. (\[ph\_number\_01\]), one can derive the following expectation value for $\left\langle \hat{a}_{p}(t)\hat{a}_{p^{\prime}}(t)\right\rangle$, which is needed in Eq. (\[eq:entanglement\_01\]) to evaluate the variances of the quadrature operators and the correlation variance in the CROW structure: $$\begin{multlined} \left\langle \hat{a}_{p}(t)\hat{a}_{p^{\prime}}(t)\right\rangle =\frac{D}{2\pi}\sum_{\lambda}(-1)^{\lambda}\left(\cosh(r_\lambda)\sinh\left(r_\lambda\right)\right)e^{-i2\tilde{\omega}_{F}\left(1-\tilde{\beta}_{1}\cos(|k_{0}|D)\right)t}\frac{\sqrt{2\sigma_{+}\sigma_{-}\pi}}{2^{\lambda}\lambda!}\\ \times\left[e^{-i|k_{0}|\left(p-p^{\prime}\right)D}H_{\lambda}\left(\tilde{S}_{p+}\right)H_{\lambda}\left(\tilde{S}_{p^{\prime}-}\right)e^{-\left(\frac{\left(\tilde{S}_{p+}^{2}+\tilde{S}_{p^{\prime}-}^{2}\right)}{2}\right)}\right.+\left.e^{i|k_{0}|\left(p-p^{\prime}\right)D}H_{\lambda}\left(\tilde{S}_{p-}\right)H_{\lambda}\left(\tilde{S}_{p^{\prime}+}\right)e^{-\left(\frac{\left(\tilde{S}_{p-}^{2}+\tilde{S}_{p^{\prime}+}^{2}\right)}{2}\right)}\right]. \end{multlined}\label{Eq:apapp}$$ Note that $\left\langle \hat{a}_{p}^{\dagger}(t)\hat{a}_{p^{\prime}}^{\dagger}(t)\right\rangle$ is simply the complex conjugate of Eq. (\[Eq:apapp\]). In the next section, we will use these equations to determine the photon number and correlation variance under a variety of different pump conditions. \[sec:level4\]Results ===================== Using the expectation values derived in Sec. \[sec:result\_for\_gaussian\], we can now study the photon evolution and inseparability criteria for the generalized two-mode squeezed light inside the CROW structure. In Fig. \[fig:ph\_number\], we plot the average number of photons in the $p$th cavity for $p=0,40$ as a function of time for both a lossy (solid green line) and lossless (dashed grey line) system. The propagation of light between the coupled cavities and effect of loss on the number of photons in each cavity is evident in Figs. \[fig:ph\_number\](a) and \[fig:ph\_number\](b), where $\sigma_+\,D=\sigma_-\,D=0.47$. Because the system and pump are spatially symmetric, the results are identical for $p\rightarrow-p$. In Fig. \[fig:ph\_number\](c), we plot the time-dependent average photon number in the $40$th cavity with $\beta=2.2$ still, but with the $\sigma_+\,D=0.14$ and $\sigma_-\,D=0.28$. As can be seen, the pulse width is wider for this smaller $\sigma_+$, as expected. Since our input pump state is a coherent state, the number of pump photons is $N_P=|\alpha|^2$. We now examine what the pump parameters will be for a specific case of interest. We again consider the case where $\sigma_{+}D=\sigma_{-}D=0.47$; using Eqs. (\[eq:delta\_t\]) and (\[eq:deltar\]) this gives temporal and spatial FWHM for the pump of $295\,\text{fs}$ and $3.3\,\mu \text{m}$, respectively. We choose the CROW material to be $\text{Al}_{0.35}\text{Ga}_{0.65}\text{As}$ due to its high nonlinearity and relatively large bandgap. In addition, we choose the pump wavelength to be $\lambda_S=775\,\text{nm}$, which not only results in generating counterpropagating signal and idler photons at the telecommunication wavelength, $\lambda_F=1550\,\text{nm}$, but also ensures operation below the band gap of $\text{Al}_{0.35}\text{Ga}_{0.65}\text{As}$. Choosing the periodicity of the CROW structure to yield a signal central wavelength of $1550\,\text{nm}$, gives $D\approx0.9\,\mu \text{m}$. Using Eq. (\[eq:Xi2bar\]) and the normalization condition given in Appendix \[appendix\_normalization\], $\bar{\chi}_2$ for our structure is approximately given by $\bar{\chi}_2\approx\chi_2/n^2(\omega_F)$, where $\chi_2\approx 100~\text{pm/V}$, appropriate for AlGaAs alloys [@Yang:07; @Gili:16; @Carletti:15], and $n\approx3.4$ at $\omega_{F}$. We now seek to determine the approximate number of pump photons under the above conditions that will give a squeezing parameter of $2.2$. Employing Eqs. (\[eq:Q\_0\]) and (\[eq:Q\_0\_second\]), the average number of photons in the pump is found to be $7.4\times10^{10}$, which gives a total pump pulse energy of approximately $19~\text{nJ}$. We note that all of the above pump characteristics are easily achievable from a Ti:Sapphire laser. ![Average photon number in the (a) central and (b) fortieth cavities of the CROW as a function of time for $\sigma_+\,D=\sigma_-\,D=0.47$ and $\beta=2.2$. (c) The time dependent average photon number for $\sigma_+\,D=0.14$, $\sigma_-\,D=0.28$, and $\beta=2.2$. The dashed grey lines show the case in which the effect of loss is ignored. []{data-label="fig:ph_number"}](Ph_number) In Fig. \[fig:Entanglement\] we plot the time-dependent correlation variances for different sets of lossy and lossless cavities in blue and grey, respectively. Note that there are fast oscillations that are not observable on this time scale. The dashed lines in the insets show the inseparability criteria below which the light is considered to be entangled. Here we only focus on cases where the two cavities considered are located the same distance from the central cavity, as this will yield the maximum entanglement; however, using Eq. (\[eq:entanglement\_01\]) one can explore the entanglement between any two cavities of the CROW. As can be seen in Fig. \[fig:Entanglement\](a) and (b), due to the loss in the system, the degree of entanglement decreases as the system evolves in time, whereas for a lossless system (the grey color) the degree of entanglement does not change as the light propagates. ![(a-b) Correlation variance between different pairs of cavities in the CROW as a function of time for $\sigma_+\,D=\sigma_-\,D=0.47$ and $\beta=2.2$. (c) The time dependent correlation variance for $\sigma_+\,D=0.14$, $\sigma_-\,D=0.28$, and $\beta=2.2$. The results for a lossless system are shown in gray. []{data-label="fig:Entanglement"}](entanglement) ![Maximum number of photons (left axis) as a function of the cavity index and minimum correlation variances (right axis) between different symmetrically displaced pairs of cavities for $\sigma_+\,D=\sigma_-\,D=0.47$ and $\beta=2.2$. The solid black and dashed grey lines represent the results from Eqs. (\[eq:ph\_num\_max\_p\]) and (\[eq:ent\_max\_p\]) with and without including $\exp[{\left(V_{I}t_{max}\right)^{2}\frac{\sigma^{2}}{2}}]$, respectively. []{data-label="fig:maximums"}](maximums) In Fig. \[fig:maximums\] we plot the maximum number of photons for a lossless and lossy CROW as a function of the cavity index, $p$. As expected, when loss is included, the number of photons decreases as we move away from the central cavity. In Fig. \[fig:maximums\] we also plot the minimum correlation variance for the lossless and lossy CROW as a function of cavity index, $p$. As can be seen, due to the reduction in the number of photons in the lossy case, there is a decrease in the degree of entanglement as a function of $p$. For instance, the minimum correlation variance at the tenth cavity is $0.9$ times the corresponding value at the one hundredth cavity. For a general pump, the evolution equations for the photon number and correlation variance are quite complicated and it is difficult to discern the general behaviour or the effects of loss from the full equations. However, for the special case where $\sigma_+=\sigma_{-}=\sigma$ and $k_0=\pi/2D$, approximate analytic expressions can be obtained. We begin by defining the complex quantity, $\tilde{V}=V_{R}+iV_{I}=\tilde{\omega}_{F}\tilde{\beta}_{1}D$, which enables us to rewrite $\tilde{S}_{p\pm}^{2}$ as $$\begin{multlined} \tilde{S}^2_{p\pm}=\left(\left(V_{R}+iV_{I}\right)t\pm pD\right)^{2}\frac{\sigma^{2}}{2}\\=\left(\left(V_{R}t\pm pD\right)^{2}+2i\left(V_{R}t\pm pD\right)V_{I}t-\left(V_{I}t\right)^{2}\right)\frac{\sigma^{2}}{2}. \end{multlined}$$ Considering only the dominant terms in Eq. (\[ph\_number\_02\]), to a very good approximation one can show that the time at which the photon number in the $p^{th}$ cavity peaks in a lossless system is $t_{max}=p\tau\approx p/\omega_{F}\beta_{1}$. As can be seen in Fig. \[fig:ph\_number\], the photon number peaks at essentially the same time in both lossy and lossless system. Using $t_{max}$, we are able to derive the following approximate expression for the maximum photon number in the $p^{th}$ cavity (for $p>0$) in a lossy system: $$\begin{aligned} \left\langle a_{p}^{\dagger}a_{p}\right\rangle_{max} \approx \frac{\sigma D}{\sqrt{2\pi}}\sinh^{2}\left(r_{0}\right)e^{-2\gamma_{F}p\tau}e^{\left(V_{I}p\tau\right)^{2}\frac{\sigma^{2}}{2}}. \label{eq:ph_num_max_p}\end{aligned}$$ Following the same procedure and using Eqs. (\[eq:entanglement\_01\]) and (\[Eq:apapp\]), one obtains $$\begin{aligned} \left(\Delta_{p,-p}^{2}\right)_{min}\approx & 4+4\frac{\sigma D}{\sqrt{2\pi}}\left(2\sinh^{2}\left(r_{0}\right)-\sinh\left(2r_{0}\right)\right)\nonumber\\ &\quad\times e^{-2\gamma_{F}p\tau}e^{\left(V_{I}p\tau\right)^{2}\frac{\sigma^{2}}{2}}. \label{eq:ent_max_p}\end{aligned}$$ We note first that both the photon number and the deviation of the correlation variance from $4$ depend linearly on $\sigma$. Thus, as expected, the separability is largest when the pump is short in time and narrow in space (for a fixed squeezing parameter, $r_{0}$). Now, according to Eqs. (\[eq:ph\_num\_max\_p\]) and (\[eq:ent\_max\_p\]), under the above-mentioned pumping conditions the effect of loss on the maximum number of photons and on the entanglement as a function of $p$ is given by two exponential factors. The first factor accounts for the intrinsic loss in an individual cavity (which is also the intrinsic loss of the Bloch mode with $k=\pi / 2D$). The second factor accounts for the loss dispersion in the CROW, and results in a reduction in the loss. Of course, these analytic results are only valid when the effect of the dispersion of the loss is small. We now consider how well these approximate expressions reproduce the exact results. In Fig. \[fig:maximums\], we plot the results of Eqs. (\[eq:ph\_num\_max\_p\]) and (\[eq:ent\_max\_p\]) with (black solid line) and without (red solid line) including the factor, $\exp[{\left(V_{I}p\tau\right)^{2}\frac{\sigma^{2}}{2}}]$. As can be seen, for this CROW the full approximate analytic expressions very accurately reproduces the exact results. Moreover, to a very good approximation, one can evaluate Eqs. (\[eq:ph\_num\_max\_p\]) and (\[eq:ent\_max\_p\]) neglecting the loss-dispersion factor, $\exp[{\left(V_{I}p\tau\right)^{2}\frac{\sigma^{2}}{2}}]$. For example, for $\sigma_-D$=$\sigma_+D=0.47$ and $p=350$, the first and the second exponential factors in Eqs. (\[eq:ph\_num\_max\_p\]) and (\[eq:ent\_max\_p\]) are $0.18$ and $1.10$, respectively, showing that loss dispersion only changes the results by $10\%$. In general, it can be shown that in order to have less than $10\%$ error in evaluating the photon number and the difference of the correlation variance from $4$, the range of $p$ must be limited to $p\le\sqrt{2}/(10V_I\tau\sigma)$. Finally, we now consider the more general cases in which $\sigma_{-}$ is not necessarily equal to $\sigma_{+}$. Under these conditions, we cannot derive simple expressions for the maximum photon number and entanglement as a function of $p$. We present the results of the full calculations for a *lossless* system in Table \[table1\] for a number of different pump durations and spatial widths; all other parameters are the same as in the previous plots. As can be seen, the maximum entanglement is obtained when the pump duration is as short as possible and the pump width is as narrow as possible (i.e. $\sigma_+D=\sigma_-D=0.47$ for our system). We have also performed full calculations for the evolution in the presence of loss for different pump configurations. In Fig. \[fig:maximums\_not\_equal\_sigma\_appendix\], we compare the results of the full calculations with the results when the loss is incorporated approximately using only the exponential factor $\exp{(-2\gamma_{F}p\tau)}$. We plot the maximum number of photons and the deviation of the minimum of the correlation variance from the inseparability threshold of 4 for a lossy system as a function of $p$ for different pumping configurations. As can be seen, for short distances from the central cavity (small $p$), the effect of loss on the result can, to a very good approximation, still be explained by only including the exponential factor $\exp{(-2\gamma_{F}p\tau)}$. However, for the cavities far from the central cavity (large $p$), although the general trends can still be predicted by such an approximation, the difference between the exact and approximation results becomes pronounced and the validity of this approximation becomes questionable. In order to show the difference between exact and approximate results for large $p$, in Fig. \[fig:Frac\_diff\_350\] we plot the relative difference between the results from the full calculations and those from approximating the loss in the system by only considering the exponential factor $\exp{(-2\gamma_{F}p\tau)}$ as a function of $\sigma_+/\sigma_-$, for $p=350$ and $\sigma_-D=0.47$. As expected, when $\sigma_+=\sigma_-$, we obtain a relative error of approximately $5.4\%$, which is simply due to the dispersion factor, $\exp[{\left(V_{I}p\tau\right)^{2}\frac{\sigma^{2}}{2}}]$. However, in general, the error depends on $\sigma_+/\sigma_-$, and is different for the photon number and correlation variance, due to the different way in which loss dispersion affects these two quantities. As can be seen in this specific example, evaluating $4-\left(\Delta_{p,-p}^{2}\right)_{min}$ using the approximation method results in a $9 \%$ deviation from the results of the full calculations when $\sigma_{+}=0.5\sigma_{-}$, while, the relative difference for the maximum number of photons is always less than $5.4\%$. It is thus evident that a simple exponential factor will capture the general effect of loss on the correlation, but it may not be very accurate depending on the structure and the pump conditions. -------------------------------------------------------- ------ ------ ------ ------ ------ \[1ex\] $\left\langle a_{p}^{\dagger}a_{p}\right\rangle_{max}$ 1.23 2.24 0.77 2.51 3.74 \[1ex\] $4-\left(\Delta_{p,-p}^{2}\right)_{min}$ 0.38 0.44 0.58 0.65 0.74 \[1ex\] -------------------------------------------------------- ------ ------ ------ ------ ------ : The maximum number of photons and the deviation of the minimum of the correlation variance from the inseparability threshold of $4$ in a lossless system for $\beta=2.2$ and different pumping configurations[]{data-label="table1"} ![(a) Maximum number of photons and (b) the deviation of the minimum of the correlation variance from the inseparability threshold of $4$ for a lossy system and selected range of cavities for $\beta=2.2$ and different pumping configurations. The solid lines represent the results from data given in Table. \[table1\] and the exponential factor $\exp{(-2\gamma_{F}p\tau)}$, and the markers show the results from the full calculations. []{data-label="fig:maximums_not_equal_sigma_appendix"}](maximums_ph_different_sigma "fig:")![(a) Maximum number of photons and (b) the deviation of the minimum of the correlation variance from the inseparability threshold of $4$ for a lossy system and selected range of cavities for $\beta=2.2$ and different pumping configurations. The solid lines represent the results from data given in Table. \[table1\] and the exponential factor $\exp{(-2\gamma_{F}p\tau)}$, and the markers show the results from the full calculations. []{data-label="fig:maximums_not_equal_sigma_appendix"}](maximums_ent_different_sigma "fig:") ![Relative deviation of the results by the exponential factor $\exp{(-2\gamma_{F}p\tau)}$ from the results of the full calculation as a function of $\sigma_+/\sigma_-$ for $p=350$ and $\sigma_-D=0.47$.[]{data-label="fig:Frac_diff_350"}](Frac_diff_350) Conclusion {#sec:conclusion} ========== In this work we studied the generation and propagation of entangled states in lossy coupled-cavity systems. We applied the tight-binding method to evaluate the fields and complex frequencies for the leaky modes of lossy coupled-cavity system, and presented analytic time-dependent expressions for the photon number and correlation variance in a lossy CROW structure. We showed how properties such as the average number of photons in each cavity and the correlations between cavities are affected by loss. For the CROW structure considered in this work, we found that as the light gets far from the central cavity, the effects of loss become more significant and cannot be ignored. Moreover, we obtained simple, approximate analytic expressions for the effects of loss on the propagation of the generated light in the CROW, and have shown that they can be used to predict general trends. However, depending on the details of the pumping conditions and the CROW structure itself, the accuracy of this approximation varies, and to get an accurate result, specifically for cavities far from the central cavity, the effect of loss cannot be well-described using a simple exponential factor that is given by the loss in an individual cavity. Using full numerical results is suggested for optimization. Yet these analytic results allow researchers to easily explore the spectral and spatial pumping configurations needed to generate the counterpropagating entangled states in a CROW. {#appendix_normalization} In this appendix we present the normalization condition for the modes. Considering the refractive index of the material to be nondispersive within the range of frequency considered here, following Yang *et al*. [@PhysRevA.77.033808], the normalization conditions can be written as $$\int\text{d}\mathbf{r}\frac{\mathbf{D}^*_{Fk}(\mathbf{r})\cdot\mathbf{D}_{Fk^\prime}(\mathbf{r})}{\varepsilon_0 n^2(\mathbf{r};\omega_{F})}=\delta(k-k^\prime) \label{eq:D_norm}$$ and $$\int\text{d}\mathbf{r}\frac{\mathbf{M}^*_{Sm\mathbf{q}}(\mathbf{r})\cdot\mathbf{M}_{Sm^\prime\mathbf{q^\prime}}(\mathbf{r})}{\varepsilon_0n^{2}\left(\mathbf{r};\omega_{Sm}\right)}=\delta_{mm^\prime}\delta(\mathbf{q}-\mathbf{q^\prime}).$$ Using Eq. (\[eq:D\_in terms of N\]) in Eq. (\[eq:D\_norm\]) we obtain the normalization condition for the single-cavity modes as $$\int\text{d}\mathbf{r}\frac{N^*_{Fp}(\mathbf{r})\cdot N_{Fp^\prime}(\mathbf{r})}{\varepsilon_0 n^2(\mathbf{r};\omega_F)}=\delta_{pp^\prime}.$$ {#sec:commutation_appendix} In this appendix, we evaluate the commutation relation between the operators $\hat{B}_\lambda$ and $\hat{B}^\dagger_\lambda$. Using Eq. (\[eq:B\_operator\]) and its conjugate we have $$\begin{multlined} \Big[B_{\lambda},B_{\lambda^{\prime}}^{\dagger}\Big]=\Big[\int\mu_{\lambda}^{*}\left(k\right)\Theta\left(k\right)b_{k}\text{d}k,\\ \int\mu_{\lambda^{\prime}}\left(k^{\prime}\right)\Theta\left(k^{\prime}\right)b_{k^{\prime}}^{\dagger}\text{d}k^{\prime}\Big]=\int\int\left[b_{k},b_{k^{\prime}}^{\dagger}\right]\\\times\mu_{\lambda}^{*}\left(k\right)\mu_{\lambda^{\prime}}\left(k^{\prime}\right)\Theta\left(k\right)\Theta\left(k^{\prime}\right)\text{d}k\text{d}k^{\prime}\\ =\int\int\mu_{\lambda}^{*}\left(k\right)\mu_{\lambda^{\prime}}\left(k^{\prime}\right)\Theta\left(k\right)\delta\left(k-k^{\prime}\right)\text{d}k\text{d}k^{\prime}\\ =\int\mu_{\lambda}^{*}\left(k\right)\mu_{\lambda^{\prime}}\left(k\right)\Theta\left(k\right)\text{d}k \approx\delta_{\lambda,\lambda^{\prime}}, \label{eq:q:comuutation_B} \end{multlined}$$ where the last line is valid as long as one the two Schmidt functions in the integral is very small when $k <0$. As we show in section \[sec:result\_for\_gaussian\], this can be accomplished through careful choice of the pump parameters. {#appendix_bplus_bminus} In this appendix we present $\hat{b}_{k,+}$ and $\hat{b}_{k,-}$ operators in terms of $\hat{B}_{\lambda}$ and $\hat{C}_{\lambda}$, respectively. Multiplying Eq. (\[eq:B\_operator\]) by $\mu_{\lambda}\left(k\right)$ and summing over $\lambda$ we have $$\begin{multlined} \sum_{\lambda}\mu_{\lambda}\left(k\right)\hat{B}_{\lambda}\\=\sum_{\lambda}\mu_{\lambda}\left(k\right)\int\mu_{\lambda}^{*}\left(k^{\prime}\right)\hat{b}_{k^{\prime},+}\text{d}k^{\prime}\\=\sum_{\lambda}\mu_{\lambda}\left(k\right)\int\mu_{\lambda}^{*}\left(k^{\prime}\right)\Theta\left(k^{\prime}\right)\hat{b}_{k^{\prime}}\text{d}k^{\prime}\\=\Theta\left(k\right)\hat{b}_{k} =\hat{b}_{k,+}. \end{multlined}$$ Following the same procedure we have $$\hat{b}_{k,-}=\sum_{\lambda}\nu_{\lambda}\left(k\right)\hat{C}_{\lambda}.$$ {#appendix_B} In this appendix we present an accurate approximate analytic result for the summation in Eq. (\[eq:biphoton\_sigma\_and\_integral\]). Starting from Eq. (\[eq:biphoton\_sigma\_and\_integral\]) we have $$S=\sum_{p}e^{-\frac{p^{2}D^{2}}{W_S^{2}}}e^{-ip(k_{1}+k_{2})D}.$$ Approximating this sum as an integral, we obtain $$S\approx\int dp\,e^{-\frac{p^{2}D^{2}}{W_S^{2}}}e^{-ip(k_{1}+k_{2})D} = \frac{1}{D}\int dp^{\prime}\,e^{-\frac{p^{\prime2}}{W_S^{2}}}e^{-i2\pi p^{\prime}k^{\prime}},$$ where $p^{\prime}=pD$ and $k^{\prime}=\frac{k_{1}+k_{2}}{2\pi}$. Now using $$\int_{-\infty}^{\infty}e^{-ax^{2}}e^{-2\pi ikx}dx=\sqrt{\frac{\pi}{a}}e^{-\frac{\pi^{2}k^{2}}{a}},$$ we can write $$S=\frac{W_S\sqrt{\pi}}{D}\,\exp\left(-\left(\frac{\left(k_{1}+k_2\right)W_S}{2}\right)^{2}\right).$$ One can show numerically that for $W_S\geq2D$, the approximation made here is accurate to within $0.01 \%.$

--- abstract: 'For the microwave equivalent of “light shining through the wall” (LSW) experiments, a sensitive microwave detector and very high electromagnetic shielding is required. The screening attenuation between the axion generating cavity and the nearby detection cavity should be greater than 300 dB, in order to improve over presently existing exclusion limits. To achieve these goals in practice, a “box in a box" concept was utilized for shielding the detection cavity, while a vector signal analyzer was used as a microwave receiver with a very narrow resolution bandwidth in the order of a few micro-Hz. This contribution will present the experimental layout and the results to date.' author: - 'M. Betz, F. Caspers, CERN, Geneva, Switzerland' title: 'A microwave paraphoton and axion detection experiment with 300 dB electromagnetic shielding at 3 GHz[^1]' --- Motivation {#lbl:intro} ========== The axion is a hypothetical elementary particle, which emerged originally from a proposal by Peccei and Quinn, intended to solve the strong CP problem [@src:WISPy] in theoretical physics. The axion is neutral, only interacts weakly with matter, has a low mass ($\approx 10^{-4} eV/c^2$), spin zero, and a natural decay constant (to 2 photons) in the order of $10^{17}$ years. The axion belongs to the family of Weakly Interacting Sub-eV Particles (WISP). Another WISP, closely related to the axion is the paraphoton or hidden photon. The existence of these WISPs could not be confirmed yet and all experimental efforts to date have so far produced only exclusion results. Nevertheless there is strong motivation to advance the experimental “low energy frontier” as the axion is the most popular solution for the strong CP-problem. Many WISPs are also excellent candidates for dark matter and explain numerous astrophysical phenomena. Experimental setup ================== WISPs can be probed in the laboratory by “Light Shining through the Wall” (LSW) experiments. They exploit the very weak coupling to photons, allowing an indirect proof of the otherwise hidden particles without relying on any cosmological assumptions. Previous LSW experiments have been carried out with optical laser light at DESY (ALPS), CERN (OSQAR) and Fermilab (GammeV). The concept of an optical LSW experiment can be adapted to microwaves [@src:hoogUW; @src:moiMykonos]. A block diagram of the setup is shown in Fig. \[fig:ovrBlock\], it consists of two identical low loss microwave cavities with a diameter of 140 mm, a height of 120 mm and a spacing between them of 150 mm. One serves as WISP emitter and is excited by an external microwave source. It develops a strong electromagnetic (EM) field, which corresponds to a large amount of microwave photons $\gamma$. Theory predicts that some of these photons convert to paraphotons $\gamma'$ by kinetic mixing (similar to neutrino oscillations) or – if the cavities are placed in a strong static magnetic field – to axion-like particles by the Primakoff effect [@src:WISPy]. Both particles only interact very weakly with matter (similar to neutrinos in this respect) and thereby, in contrast to the photons, can traverse the cavity walls. Some WISPs propagate towards the detection cavity, which is connected to a very sensitive microwave receiver. The reciprocal conversion process transforms WISPs to microwave photons, which can be observed as an excitation of the seemingly empty and well shielded detection cavity. ![Block diagram of the experiment[]{data-label="fig:ovrBlock"}](ovr2.pdf){width="49.00000%"} Since there is no energy loss associated with the WISP conversion process, the regenerated photons in the detecting cavity have exactly the same energy as the photons in the emitting cavity. Thus, the signal which is coupled out from the detection cavity has the same frequency as the one which is generated on the emitting side, making a narrowband receiving concept feasible. This paper will focus on the latest exclusion results for **paraphotons** from the microwave WISP search at CERN. In a future upgrade, an additional magnet will allow the search for axions. Considering current exclusion limits, it takes $> 10^{24}$ photons on the emitting side to generate one photon on the detection side, making this the most challenging aspect of an LSW experiment. The expected output power (or photon flux) from the detecting cavity towards the microwave receiver due to paraphotons is given by Eq. \[equ:power\], $$\begin{aligned} \label{equ:power} P_{\mathrm{det}} &= \chi^4 \left(\frac{m_{\gamma'} c^2}{f_{\mathrm{sys}} h}\right)^8 |G|^2 Q_{\mathrm{em}} Q_{\mathrm{det}} P_{\mathrm{em}}\end{aligned}$$ where $Q_{\mathrm{em}}$ and $Q_{\mathrm{det}}$ are the loaded Q factors of emitting and detection cavity, $f_{\mathrm{sys}}$ is the frequency where the experiment is carried out (and to which the cavities are tuned), $h$ is Planck’s constant and $G$ is a dimensionless geometric form factor in the order of 1, describing the position, shape and resonating mode of the cavities [@src:JaCaRi]. The rest mass of hidden photons is a priori unknown and given by $m_{\gamma'}$. The kinetic mixing parameter $\chi$ describes the likeliness of paraphoton - photon oscillations. A previous examination of Coloumb’s law indicates that $\chi < 3 \cdot 10^{-8}$ in this energy range. If there is no significant signal detected, an exclusion result can be produced by determining $\chi$ from the other known values. This provides a convenient way to compare the achieved sensitivity to other experiments. The parameters of the paraphoton experiment as it has been set up and carried out at CERN in March 2012, are summarized in Table \[tbl:param\]. As no paraphotons were observed, the corresponding exclusion limit in comparison to other experiments is shown in Fig. \[fig:exclPlot\]. ---------------------------------------------------------------------------------------------- $f_{\mathrm{sys}} = 2.9565$ GHz $Q_{\mathrm{det}} = 23620$ $Q_{\mathrm{em}} = 23416$ \[0.1 cm\] $P_{\mathrm{det}} = 8.51 \cdot 10^{-25}$ W $P_{\mathrm{em}} = 37$ W $|G| = 0.222$ ---------------------------------------------------------------------------------------------- : parameters of the paraphoton run in March 2012[]{data-label="tbl:param"} ![Exclusion limit for paraphotons as a result of the measurement-run at CERN in March 2012, compared to other experiments (details in [@src:JaCaRi])[]{data-label="fig:exclPlot"}](exclusion.pdf){width="49.00000%"} Engineering aspects =================== On the left side of Fig. \[fig:ovrBlock\], a commercial microwave source is shown, which generates a signal on $f_{\mathrm{sys}}$ (see Table \[tbl:param\]) which is amplified up to 50 W and drives the emitting cavity on its resonant frequency. Power is coupled in and out of each cavity with a small inductive coupling loop, adjusted for critical coupling to the TE$_{011}$ mode. This mode has been chosen for its high Q-factor and reasonable coupling to paraphotons compared to other modes. The loaded Q-factor of the silver coated brass cavities has been determined by a network analyzer, their 3 dB bandwidth is $BW_{3 dB} \approx 126$ kHz. A tuning stub perturbing the H-field allows to compensate manufacturing tolerances within a bandwidth of $\approx 10$ MHz. Shielding is required around the detecting cavity and the microwave receiver to eliminate ambient electromagnetic interference (EMI) and to mitigate coupling to the emitting cavity by simple EM leakage. This would generate false positive results as a signal originating from leakage can not be distinguished from a signal propagating by WISP conversion. Within 15 cm, the field strength must be reduced by at least a factor of $7.7 \cdot 10^{12} = 258 \mathrm{~dB}$ to get meaningful results. The shielding box has been built from a straight piece of WR-2300 waveguide with the inside dimensions 58x29x100 cm. Feeding all RF signals over optical fibres by analog transceivers prevents the propagation of EMI trough ordinary transmission lines. An optical ethernet link is used to remote control the signal analyser. Shielding effectiveness has been measured by comparing the electric field strength in- and outside the wall with a calibrated electric field probe and a spectrum analyzer. The shielding enclosure provides $\approx 90$ dB and each of the cavities provides an additional $\approx$ 110 dB of shielding. The combined EM attenuation is $\approx 310$ dB, making thermal noise the limiting factor for the minimum detectable signal. For an exclusion result it is necessary to prove the detector is working and actually able to pick up any WISP related signals. Detection sensitivity will be limited if the resonant frequency of one or both cavities does not equal the system frequency $f_{\mathrm{sys}}$. This is especially delicate as the cavities are sensitive to temperature variations; their resonant frequency is inversely proportional to the thermal expansion coefficient of their wall material. For brass, a temperature change of $\Delta T = +2$ K leads to a detuning of $\Delta f_{\mathrm{res}} = -112$ kHz and a reflection of around half the input power back towards the amplifier. Thermal drift is more critical for the emitting cavity as it has to dissipate up to 50 W of heat by forced air cooling without any external temperature stabilization. Before data taking, the cavity was operated at full power and its resonant frequency was kept constant manually. After around 1 h, the cavity reached thermal equilibrium and no further tuning was necessary. The reflected power $P_{\mathrm{refl}}$ indicates how far the emitting cavity is off tune, and is minimized for $f_{\mathrm{res}} = f_{\mathrm{sys}}$. The relation is given in Eq. \[equ:cavReflFreq\], $$\begin{aligned} \label{equ:cavReflFreq} \frac{P_{\mathrm{refl}}}{P_{\mathrm{inc}}} = \left|\Gamma\right|^2 = \frac{f_{\mathrm{n}}^2}{4 + f_{\mathrm{n}}^2} \quad f_{\mathrm{n}} = Q_{\mathrm{em}} \left(\frac{f_{\mathrm{sys}}}{f_{\mathrm{res}}} - \frac{f_{\mathrm{res}}}{f_{\mathrm{sys}}}\right)\end{aligned}$$ where $P_{\mathrm{inc}}$ is the constant ($\pm 1 \%$) and known incident RF power at the frequency $f_{\mathrm{sys}}$. Critical coupling has been assumed, with coupling losses and impedance mismatch not considered. $P_{\mathrm{refl}}$ is measured on a directional coupler, placed between power amplifier and cavity. Tuning the detecting cavity once before the measurement run is sufficient, as it does not dissipate any power and changes in the laboratory’s ambient temperature are small enough. Its resonant frequency was estimated by evaluating the spectral noise power density $n_{\mathrm{o}}$ at the output of the low noise amplifier (LNA), given by Eq. \[eq:thNoise\] $$\begin{aligned} \label{eq:thNoise} n_{\mathrm{o}} = G k_B \left[ T_{\mathrm{cav}} \left(1 - \left|\Gamma\right|^2\right) + T_{\mathrm{LNA}} + T_{\mathrm{LNA}}' \left|\Gamma\right|^2 \right]\end{aligned}$$ where $G = 44.7$ dB is the LNA’s gain and $k_B$ is the boltzmann constant. Three noise temperature terms have to be considered, $T_{\mathrm{cav}} \left(1 - \left|\Gamma\right|^2\right)$ describes the noise from the detecting cavity itself. Its noise temperature is frequency dependent by the reflection coefficient $\Gamma$. The maximum occurs at $f_{\mathrm{res}}$ and is equal to ambient temperature. $T_{\mathrm{LNA}} = 32.4$ K describes the intrinsic noise temperature of the amplifier. $T_{\mathrm{LNA}}' \left|\Gamma\right|^2$ describes the noise temperature of the amplifier input, which emits a noise wave towards the cavity where it is reflected back. As the transmission line between amplifier and cavity is short ($\approx$ 1 cm) and as the noise temperature of the cavity walls is significantly higher than the noise wave transmitted from the amplifier’s input, a good estimate of the resonant frequency can be determined from the maximum of the noise power spectrum. This has been done before and after the actual experimental run, the results are shown in Fig. \[fig:diagTune\] and indicate no significant (bigger than $BW_{3 \mathrm{dB}}$) drift of the emitting cavity’s resonant frequency. ![Measured spectral noise power density before and after the 11.5 h measurement run, indicating $f_{\mathrm{res}}$[]{data-label="fig:diagTune"}](tune.png){width="50.00000%"} The output signal of the detecting cavity is coupled out, amplified by the LNA ($G = 44.7$ dB) and then further processed by an Agilent EXA N9010A signal analyzer. The center frequency was set to $f_{\mathrm{sys}} + 4~\mathrm{Hz}$ to avoid internal spurious signals appearing at interesting parts of the spectrum. The center frequency is shifted to baseband and the complex IQ signal is digitized with 20 Hz bandwidth. The 11.5 h long time record is stored for further processing. As the capture memory of the analyzer limits the maximum number of continuously acquired samples to $\approx 10^6$, a trade-off between recorded bandwidth and recording length had to be found. For this reason the signal analyzer will be replaced by a dedicated processing chain without this limitation for the future. For offline data processing the spectral power of the recorded noise like signals is estimated by Welch’s method implemented in a python script. The time record is read and divided into segments overlapping by $\approx 90 \%$. Each segment is multiplied by a Hann window and the complex spectra are calculated by a fast Fourier transform. Averaging these spectra trades resolution bandwidth for less variance in the noise floor. For a 11.5 h long time trace, resolution bandwidths as narrow as $BW_{\mathrm{res}} = 24~\mu$Hz can be achieved. The average noise floor is determined by $Pn = BW_{\mathrm{res}} \cdot n_o$. The frequency error of RF-source and signal analyzer needs to be within $BW_{\mathrm{res}}$ during the experimental run, otherwise the signal power will spread out over several bins in the spectrum, degrading the signal to noise ratio. While absolute frequency drifts are unavoidable, phase-locking RF-source and signal analyzer to a common 10 MHz frequency reference allows to achieve a good relative frequency stability. This has been explained and successfully demonstrated down to $BW_{\mathrm{res}} = 10~\mu$Hz in [@src:narrowband]. Immediately before the actual measurement, a test run was conducted where the shielding box was left open to provoke EM leakage between the cavities. The resulting power spectrum contained a single peak, clearly above the noise floor, spanning only one single bin. Its absolute position on the frequency axis was offset by $\approx 3$ mHz due to the finite resolution of the RF source. As no parameters were changed after the test run and only the shielding box was closed, the WISP related signal is expected to appear at the same position in the spectrum. ![Spectral noise power from the detecting cavity. The left diagram shows the min. and max. peaks of the recorded span, while the right diagram is zoomed[]{data-label="fig:spectra"}](spectra.png){width="48.00000%"} The peaks within a window of $\pm 1.5$ mHz around the expected signal do not exceed the peaks in other parts of the spectrum, as shown in Fig. \[fig:spectra\]. Therefore an exclusion result is produced by setting the minimum detectable power ($P_{\mathrm{det}} = -211$ dBm) to the maximum peak within the frequency window. Conclusion ========== No paraphotons were observed in the first measurement-run of the microwave WISP search at CERN, improving the existing exclusion limits. Several technical challenges, like $> 300$ dB EM shielding between the cavities, keeping them on tune during the 11.5 h measurement run and filtering the signal with a bandwidth of $BW_{\mathrm{res}} = 24~\mu$Hz, had to be overcome.\ [We are grateful for the practical hints and assistance from M. Gasior and M. Thumm. Thanks to R. Jones, E. Jensen and the BE department management for support. ]{} [9]{} P. Arias et al., *“Illuminating WISPs with photons”*, proc. PHOTON2011, [arXiv:1110.2126v1](http://lanl.arxiv.org/abs/1110.2126v1) F. Hoogeveen, *“Terrestrial axion production and detection using RF cavities”*, [Physics B288 (1992) 195-200](http://www.sciencedirect.com/science/article/pii/037026939291977H) M. Betz et al., *“Status report of the CERN light shining through the wall experiment with microwave axions and related aspects”*, to appear in proc. 7th Patras axion workshop, Mykonos 2011, [http://arxiv.org/abs/arXiv:1204.4370](arXiv:1204.4370) F. Caspers et al., *Feasibility, engineering aspects and physics reach of microwave cavity experiments searching for hidden photons and axions*, JINST 4 P11013 (Nov. 2009) F. Caspers et al., *Demonstration of $10^{-22}$ W Signal Detection Methods in the Microwave Range at Ambient Temperature*, CERN-BE-Note-2009-026 [^1]: Work supported by the Wolfgang-Gentner-Programme of the Bundesministerium für Bildung und Forschung (BMBF).

Introduction ============ The interplay between disorder and electron-electron interactions has been a subject of intense activity in the last few years. Recent experiments have shown the existence of an apparent metal-insulator transition (MIT) in two-dimensional (2D) semiconductor devices [@2DMIT]. This observation came as a surprise to the community, since the scaling theory of localization, developed for disordered, non-interacting systems predicts insulating states even for infinitesimal disorder strengths [@lee]. Therefore, these experiments have promoted intense theoretical activity. There is at present no consensus and considerable controversy surrounds this problem [@2DMIT]. A novel feature of the experimental high mobility samples investigated is their exceptionally low electronic density $n_s$. This leads to an unusually high value of the dimensionless parameter $r_s \propto n_s^{-1/2}$ of up to 80. The parameter $r_s$ sets the scale of electron-electron interaction $E_{e-e}$ as compared to the Fermi energy $E_F$ through $r_s \approx E_{e-e}/E_F$. Thus the question of electron-electron interaction effects becomes important in these disordered systems. The analytical treatment of the problem of disordered, interacting electrons is possible only in some limiting cases. The effects of weak interactions in disordered systems have been studied in great detail in the metallic (delocalized) regime [@altshuler]. The other extreme, corresponding to the strongly localized system can be treated by mean-field methods [@efros]. Although these analytical approaches have provided many useful physical results, the general treatment of disordered quantum many-body systems remains an unsolved problem. Indeed, one of the most powerful methods developed for the analytical treatment of disordered systems, namely, the supersymmetry approach [@supersym], cannot handle the effects of electron-electron interactions. Given this context, numerical approaches play a crucial role in the treatment of disordered, interacting systems. Several numerical approaches have been applied to the study of 2D disordered, strongly correlated systems [@schreiber; @benenti0; @benenti; @song; @dassarma; @pichard; @benenti2; @berkovits]. Among the approximate approaches, Hartree-Fock calculations with residual interactions similar to the configuration interaction approaches of quantum chemistry have been applied to systems with spinless fermions and electrons with spin [@schreiber; @benenti]. Exact diagonalization approaches have been useful but suffer from severe limitations in the accessible system size and number of particles [@benenti0; @dassarma; @pichard; @berkovits]. From these studies, it has been observed that repulsive electron-electron interactions can have a delocalizing effect in small systems. In addition, it has been shown experimentally [@2DMIT] that the spin degrees of freedom play a crucial role in the physics of these strongly interacting systems. The inclusion of the spin degrees of freedom renders the numerical calculations even more difficult and strongly reduces the number of fermions accessible (see e.g. [@schreiber; @dassarma; @pichard; @benenti2; @berkovits]). Quantum Monte Carlo (QMC) approaches provide a powerful alternative to the treatment of quantum many-body systems. These methods are in principle exact, apart from statistical errors and allow the treatment of much larger system sizes with many particles. The finite-temperature determinantal QMC approach has been applied to the two-dimensional disordered Hubbard model and signatures of a metal-insulator transition were obtained [@trivedi]. However, these calculations were carried out at finite temperature and could not access the ground state of the system. In previous work, we studied the ground state properties of the disordered 2D Hubbard model by the projector quantum Monte Carlo (PQMC) method [@caldara]. We studied the properties of the Green’s function, charge density and inverse participation ratio against model parameters and system size. While we observed some local charge reorganization, we could not detect any significant delocalizing influence of the Hubbard repulsion $U$ on the many-body ground state. However, it should be noted that all the physical quantities studied in Ref. [@caldara] were obtained by integrating over all particles and effectively, the entire energy spectrum. Here we introduce a new approach which allows us to study the properties of a single added particle and therefore emphasizes the physical effects of interactions in the proximity of the Fermi edge. This approach uses the inverse participation ratio (IPR) extracted from the charge density differences of two many-body ground states, as described below. The IPR, $\xi$, for a normalized single-particle wavefunction $\psi(i)$ is given by $\xi = [\sum_i |\psi(i)|^4]^{-1}$, where $i$ is the site index of the system. Clearly, $|\psi(i)|^2$ can be identified as the one-particle charge density at the site $i$. This definition is usually carried over to many-body systems by renormalizing the total charge density at site $i$, $\rho(i,N_p)$, which is obtained in the standard way from the ground state many-body wavefunction for $N_p$ particles. The IPR $\xi$ is then calculated with the effective charge density $\rho(i,N_p)/N_p$. We found that the IPR obtained from such a definition showed slight variations with model parameters and practically no variation with system size [@caldara]. We believe that the physical reason for this weak variance is the fact that this procedure effectively integrates over all energies and therefore the dominant contribution comes from the states deeply below the Fermi energy, which remain strongly localized even in the presence of interactions. Therefore, it is necessary to find a method which is more sensitive to the contribution of states in the vicinity of the Fermi energy. Recently, we studied the localization properties of the disordered, [*attractive*]{} Hubbard model in two- and three-dimensions [@attractive]. In Ref. [@attractive], we showed that an IPR calculated for an added pair of particles is a very sensitive and relevant quantity to study the localization properties of the wavefunction. Based on our results for the attractive Hubbard model, we have introduced a related quantity in the repulsive case, the IPR for a single added particle, which turns out to be much more sensitive to the effects of interaction. This quantity has certain similarities to the single-particle tunneling amplitude. The latter has recently been studied by exact diagonalization methods for small clusters with spin-1/2 fermions [@berkovits]. This work provided indications for a delocalizing effect induced by repulsive interactions in disordered systems. However, the number of particles studied was rather restricted due to the limitations of the exact diagonalization methods. With our method and extensive, highly accurate PQMC simulations, we study considerably larger numbers of particles in presence of strong electron-electron interactions. In this study we report a significant delocalizing influence of the Hubbard repulsion on 2D disordered electronic systems. This paper is organized as follows : after this Introduction, we describe the method used and the tests performed in the next section. Our results and discussion are presented in detail in the third section. Model and method ================ The Hamiltonian studied in this paper is the disordered, repulsive Hubbard model given by: $$\label{hamiltonian} \begin{array}{l} H = H_A + H_I \\= \Bigl( -t\sum\limits_{\langle ij \rangle, \sigma} c^{\dagger}_{i,\sigma} c_{j,\sigma} + \sum\limits_{i,\sigma} \epsilon_i c^{\dagger}_{i,\sigma} c_{i,\sigma} \Bigr) + U \sum\limits_{i} n_{i\uparrow}n_{i\downarrow} \end{array}$$ where $c_{i\sigma}^\dagger$ ($c_{i\sigma}$) creates (destroys) an electron at site $i$ with spin $\sigma$ and $n_{ i\sigma}=c_{i\sigma}^{\dagger} c_{ i\sigma}$ is the corresponding occupation number operator. The hopping term $t$ between nearest neighbor lattice sites characterizes the kinetic energy and the random site energies $\epsilon_{i}$ are taken from a box distribution over $[-W/2,W/2]$. The parameter $U$ measures the strength of the screened, repulsive Hubbard interaction ($U>0$). We have considered both the one- and two-dimensional cases, with periodic boundary conditions in all directions. In the 2D case, the sites $i$ lie on a rectangular lattice of linear dimension $N_x,N_y$. The system size $N = N_x \times N_y$ then follows accordingly in one- and two-dimensions. In the limit $U$ = 0, this Hamiltonian reduces to the Anderson model (given by $H_A$) which is a standard model for the study of disordered systems [@kramer]. In the absence of disorder, $W$ = 0, this Hamiltonian reduces to the usual Hubbard model, which is one of the best-studied models for correlated electronic systems [@dagotto]. We have studied this Hamiltonian by the projector quantum Monte Carlo (PQMC) method [@imada] and exact diagonalization calculations. The PQMC method was initially developed to study the ground state of the Hubbard model (clean limit of Eqn. (1)). The method can be generalized, in principle quite simply, to incorporate disorder via the random site energies. However, the actual implementation and convergence of the algorithm [@caldara] is highly non-trivial compared to the pure case, as will be discussed in detail below. The PQMC method consists in filtering out the true ground state $|\psi_{0} \rangle $ of the many-body system from an appropriately chosen trial function $|\phi\rangle$: $$|\psi_{0} \rangle = \lim\limits_{\Theta \rightarrow \infty} { {e^{- \Theta H} |\phi\rangle} \over {\sqrt{\langle \phi| e^{- 2 \Theta H} |\phi \rangle}}}. \label{projec}$$ This method is exact in principle, apart from statistical errors and the sign problem which appears for fermions at $U > 0$. The Hamiltonian plays the role of the projection operator through the term $e^{- \Theta H}$, where $\Theta$ plays the role of the projection parameter. The trial wave-function is usually formed as a product of up and down spin states from the eigenstates of the non-interacting Hamiltonian. In our case, we chose the Fermi sea of $H_A$ as the trial wave-function. In the PQMC procedure, the projection operator $\exp(-\Theta H)$ is first Trotter decomposed as $\Bigl( \exp(- \Delta \tau H_{A}) \exp(- \Delta \tau H_{I}) {\Bigr)}^{L}$ with $\Theta = \Delta\tau \times L$. This introduces a systematic error of order $(\Delta \tau)^2$ due to non-commutation of $H_A$ and $H_I$. We have used the symmetric Trotter decomposition, which introduces a systematic error of $(\Delta \tau)^3$. The interaction is then decoupled by a discrete Hubbard-Stratonovich transformation, by the introduction of $N \times L$ Ising-like fields. This Ising model with complicated effective interactions is then treated by a Monte Carlo (MC) procedure to obtain the ground state properties of the system. The quantity calculated during the simulation is the zero-temperature, equal-time Green’s function, $G_{ij} =\sum_{\sigma} \langle \psi_0 | c^{\dagger}_{i,\sigma} c_{j,\sigma}| \psi_0 \rangle$. During the simulation this Green’s function can be used to obtain all the other static correlation functions. In the algorithm used $O(N^2)$ operations are required to update the Green’s function after a MC step. This procedure introduces cumulative errors and therefore the Green’s function has to be recalculated from scratch regularly during the simulation (every $L_C$ steps), which requires $O(N^3)$ operations. When the projection parameter $\Theta$ becomes large, which is necessary for good convergence, the different components of the wavefunction tend to become parallel during the projection process. Therefore, it is necessary to reorthogonalise the components of the wavefunction regularly, every $L_R$ time steps. In addition, it is well known that quantum simulations of fermionic systems suffer from the sign problem except in some special cases. However, it has been observed that disorder in fact diminishes the magnitude of the sign problem. In our simulations, the sign problem is well under control, with the number of negative signs being less than $1\%$ of the total number of steps considered. We have studied systems with particle number ($N_p$) up to 25 fermions on lattice sizes of up to $6\times8$ sites, with particle density always around quarter-filling. The simulations were carried out in the $S_z = 0$ sector for even numbers of particles and the $S_z = 1/2$ sector for odd number of particles. The specific quantity studied in this paper is the difference of charge densities $\delta\rho(i)$ between systems with $N_p$ and $N_p+1$ particles (see the next section for more details). Here, we discuss the convergence of the PQMC calculations. The charge densities involved in the calculation $\delta\rho(i)$ are obtained from two independent simulations of the same disorder realization with different particle numbers, $N_p$ and $N_p +1$. Therefore, it becomes necessary to measure very accurately the distribution of this added particle over the lattice. Clearly, it is harder to measure $\delta\rho(i)$ accurately than measuring the total charge densities $\rho(i,N_p)$ in the ground state with $N_p$ particles. We carried out extensive tests to verify the quality of our $\delta\rho(i)$ data, by varying the PQMC parameters until convergence was obtained. We have tested for convergence by varying the following PQMC parameters: the projection parameter $\Theta$, the Trotter time-step $\Delta\tau$, the Monte Carlo parameters, the reorthogonalization interval $L_R$ and the interval to recalculate the Green’s function $L_C$. We have gone up to $\Theta = 15$ and $\Delta\tau = 0.05$. For the physical parameters used in this paper, we find that $\Theta = 10$ with $\Delta \tau = 0.1$ are sufficient. This corresponds to a systematic error of $10^{-3}$. After varying the parameters $L_R$ and $L_C$ we decided to recalculate the Green’s function with reorthogonalization of the components, after every 5 time-steps ( $L_R = L_C = 5$). We have also checked the MC parameters and find that 3000 sweeps are adequate for convergence, with 1000 sweeps for equilibration. The disorder average was carried out over $N_R$ different disorder realizations with $N_R = 16$ in the PQMC simulations and $N_R = 100$ for most exact calculations. The site-energies are randomly chosen for $N_R$ disorder realizations at $W/t = 1$ and then scaled proportionally to $W/t$ for stronger values of disorder. While our main motivation is the study of the 2D case, there are many useful reasons to study 1D systems. The 1D case can be studied conveniently by exact diagonalization methods, while changing the system size. Thus, we can study the variation of various properties such as the inverse participation ratio (IPR) $\xi$, with system size, exactly. These calculations provide a strong independent check on the PQMC data at small sizes. Further, the localization effect is much stronger in 1D and $\delta\rho$ is usually strongly peaked in 1D as compared to 2D. Therefore, if the algorithm is capable of reproducing a localized peak, this provides a strong check on the method, at the given range of physical parameters (examples provided in the next section). Results and Discussion ====================== In order to study the localization properties of the system, we use the charge density distribution for an added particle at the Fermi level, given by: $\delta\rho(i)=\rho(i,N_p+1)-\rho(i,N_p)$, where $\rho(i)$ is the ground state charge density at site $i$. The values of $\rho(i,N_p)$, $\rho(i,N_p+1)$ are obtained from two independent PQMC simulations for the same disorder realization. At $U$ = 0, this $\delta\rho(i)$ is identically equal to the one-particle probability distribution (added particle) at the Fermi edge. In the interacting case, this charge density distribution ($\delta\rho(i)$) is not generally equal to a probability distribution. For example, it is not necessarily positive definite. However, in all the cases studied we find that this quantity is always positive. Therefore, $\delta\rho(i)$ can be considered to be similar to a one-particle probability distribution. Thus, we can associate an inverse participation ratio (IPR) for an added particle, for a given disorder realization as: $\xi=(\sum_{i}\delta\rho(i)^2)^{-1}$. The disorder averaged IPR is given by $ \langle \xi \rangle$. At $U=0$, the IPR is a standard tool used to obtain the number of sites over which the particle is localized [@supersym]. We have successfully used this approach in previous work on the disordered, attractive Hubbard model for a quantitative description of the localization properties of the ground state [@attractive]. =8.cm =8.cm In Figs. 1-2, we study the behavior of the IPR as a function of $U/t$ and $W/t$ to try and establish the interesting range of physical parameters in this system. In Fig. 1, we compare the IPR $\langle\xi\rangle$ for a 1D ring of 12 sites and a $4\times3$ lattice in 2D as a function of interaction strength $U/t$. From the figure we see that increasing $U/t$ from 0 tends to increase the IPR up to intermediate strengths of the interaction. Thus, the optimal value appears to be around $U/t=4$ in both 1D and 2D. Indeed, for $U/t > 4$, the value of IPR starts to decrease. In Fig. 2, we see the variation of $\langle\xi\rangle$ for small system sizes for $5 \le W/t \le 12$. Since the ratio $\langle\xi(U/t = 4)\rangle/ \langle\xi(U/t = 0)\rangle$ is practically constant, this justifies our choice of parameter range for $W/t$. Disorder strength $W/t < 5$ would lead to states with localization length larger than the accessible system size even at $U/t =0$. Convergence of the PQMC calculations becomes progressively more difficult for larger disorder strengths. Therefore, we studied the variation of the IPR for $W/t = 5, 7$ in the 2D case. =8.cm Thus, it would seem ideal to study the IPR of 1D and 2D systems for $5 \le W/t \le 7$ and $U/t \approx 4$. In 1D we could access these parameter values conveniently for the system sizes studied by exact calculations. However, this becomes much more difficult in 2D. This is because we study a small difference of large total charge densities in the PQMC simulations. We found that the accuracy of our method for $\delta\rho(i)$ is good for for values of the interaction $U/t \leq 2$. While we are not exactly at the optimal value of $U/t$, we are nevertheless in a region with sufficiently strong interactions. Indeed, it can be seen from Fig. 1 that $U/t = 2$ already has a substantial delocalizing influence on the system. In Fig. 3, we present the PQMC calculation of $\delta\rho(i)$ in 1D at $U/t = 2$ as compared to exact calculations. We note that the $\delta\rho(U=0)$ is strongly peaked. Increasing $U/t$ to 2 radically changes the picture and shifts the peak completely. The PQMC curve shown reproduces the quantitative picture of charge density difference. It should be noted that the calculation begins with a trial wavefunction corresponding to the $U/t =0$ data and changes completely to give the correct physical picture. Quantitatively, the usual errors seen at these values of the physical parameters were around $3 - 5 \%$. The 2D case is more delocalized compared to 1D and we have observed that convergence is better in 2D. For example, we have $\langle \xi \rangle$ = 7.19 (exact) and 7.43 (QMC) for a $4\times3$ lattice at $U/t = 2$ and $W/t = 7$, averaged over 16 disorder realizations. This corresponds to an error of about $3\%$ for the most extreme parameter values studied. However, for $U/t = 4$, the error increases to $8 - 10\%$ and therefore we restrict our studies to $U/t \leq 2$ in 2D. =8.cm Since we study a model of electrons with spin, it is interesting to consider even-odd effects in the particle number $N_p$ and system size $N$. In Fig. 4, we consider the effect of progressively adding one and two particles to a 12 site ring with 6 electrons. From Fig. 3 and Fig. 4, we see that the first added electron at $U/t$ = 2 and 4, has an entirely different peak as compared to the non-interacting case. In Fig. 4, we see that the second added electron also occupies a different region, as seen from the position of the peak. This leads to the following physical image for the 1D case : each added electron finds its own location in space with optimal energy and it is well localized by external disorder and random distribution of charges of other electrons. This is in contrast with the non-interacting case, where the non-interacting orbital remains the same for odd and next even electron. The situation is qualitatively different in 2D. Indeed, in Fig. 5, we show the charge density difference for one and two added particles on a $6 \times 6$ lattice with 18 electrons. This quantity is obtained from PQMC simulations of the system for a particular disorder realization, taken at $W/t$ = 7. The data show that the initial $U/t = 0$ configuration is very clearly localized for the given disorder realization and lattice size. It is seen from the figure that the introduction of a repulsive Hubbard interaction ($U/t = 2$) leads to a substantial delocalization of the added particle. This is also borne out quantitatively, since the IPR increases practically by a factor of 3, as compared to $U/t = 0$. In the non-interacting case, both added particles occupy the same orbital. Therefore, the peak for the second added particle, is trivially identical to the first. It is remarkable that in the interacting case, the peak is transformed to a much more extended distribution over the lattice. Furthermore, the second added particle has practically the same distribution $\delta\rho(i)$ as the first. Thus, there appears to be almost zero effective repulsion between the two added particles, despite the interaction strength $U/t = 2$. This is completely in contrast to the scenario in 1D with interactions, where we observe significant repulsion between the two added particles. We note that this is the case for every disorder realization studied. =4.2cm =4.2cm =4.2cm =4.2cm We now turn to a more quantitative picture of the one and two dimensional systems. For this, it is necessary to average the data over different disorder realizations and to vary the system size. We start the analysis from the data for the 1D case. In Fig. 6, we present the IPR for interacting particles on 1D rings of 4–12 sites at constant density of particles (quarter filling) and compare against the variation of the IPR at $U/t = 0$. We find a delocalization effect even for 1D rings. We note that the curve for $U/t = 0$ is well saturated in 1D even at 8–12 sites. The $U/t = 2$ curve shows signs of saturation. The delocalization effect is most visible for $U/t = 4$, as expected from Fig. 1, where unfortunately, we do not have access to PQMC data for comparisons. The ratio $\langle\xi(U/t)\rangle/\langle\xi(U/t =0)\rangle$ goes to 1.64 for $U/t = 2$ and 1.94 for $U/t = 4$. =8.cm In 2D previous exact diagonalization studies have gone up to 6 electrons on a $4\times4$ lattice, or 4 electrons on a $6\times6$ lattice [@dassarma; @pichard; @berkovits]. Therefore, it is of great importance to access larger system sizes with more particles. In Fig. 7, we present the IPR for different lattice sizes in 2D, obtained from PQMC simulations. We have gone up to 48 sites ($6 \times 8$ lattice) and 24-25 particles, which goes beyond any existing study of the ground state in the literature. We observe a delocalizing effect of the Hubbard repulsion manifested by an increase of the IPR with system size and interaction strength. There is a remarkable difference between the present result and the IPR for the full many-body ground state wavefunction obtained in Ref. [@caldara]. In the previous work, we could notice no size effects over a large range of $W$ and lattice sizes. In fact, the curves for different lattice sizes against $U/t$ at a given value of $W/t$ tended to collapse as seen in Figs. 8 and 9 of Ref. [@caldara]. On the contrary, in the present work, we see from Fig. 7 that effects of lattice size on the IPR for an added particle are considerable. Furthermore, we see that the Hubbard repulsion has a strong delocalizing influence, compared to the data for the non-interacting case. As explained in the previous sections, we attribute the difference between the present results and those presented in Ref. [@caldara] to the fact that the IPR from $\delta\rho$ measures directly the response in the vicinity of the Fermi level. =8.cm The ratio $\langle\xi(U/t)\rangle/\langle\xi(U/t =0)\rangle$ can be considered as a quantitative measure of delocalization effect induced by repulsive interaction. From the inset of Fig. 7, it can be seen that this ratio rises with system size for a given density and we do not observe saturation at the system sizes studied. Our data for this ratio for small clusters in 2D are comparable to the enhancement ratio obtained in Ref. [@berkovits] though the quantity studied in Ref. [@berkovits] was slightly different i.e. the one-particle tunneling amplitude. From Figs. 6 and 7 we can compare the values of the ratio $\langle\xi(U/t)\rangle/\langle\xi(U/t =0)\rangle$ in 1D and 2D. At the system sizes studied these values are comparable for $U/t = 2$. However, it would necessary to analyze the behavior of this ratio for larger system sizes in order to get a clear answer to whether the localization properties are indeed different in 1D and 2D. At the same time, we note that there seem to be qualitative differences based on even-odd effects for added particles in 1D and 2D (see Fig. 4 and Fig. 5 and discussion there). This difference favors the picture of stronger delocalization in 2D as compared to 1D. Even if our data clearly show a repulsion induced delocalization effect, they do not permit us to draw a definite answer about the existence of a metal-insulator transition for this system in the thermodynamic limit. Indeed, even though we have a significant number of fermions, we cannot go to larger system sizes because of the accuracy constraint of calculating charge differences. It should also be pointed out that our calculations done at $U/t \leq 2$ are below the optimal value of the interaction strength. We expect that the delocalizing effect is even stronger at $U/t=4$. In conclusion, we have studied the ground state properties of the repulsive Hubbard model with disorder through the powerful PQMC method. Highly accurate simulations permit us to obtain the difference of charge density between two ground states with $N_p$ and $N_p+1$ fermions respectively. The analysis of this characteristic clearly shows that the Hubbard repulsion has a delocalizing effect in the system. We have observed some qualitative differences between 1D and 2D for this characteristic. However, the restrictions on system size and interaction strength do not permit us to draw a definite conclusion about the existence of a MIT in the thermodynamic limit in 2D. We thank the IDRIS at Orsay for access to their supercomputers. [99]{} E. Abrahams, S.V. Kravchenko, and M.P. Sarachik, Rev. Mod. Phys. [**73**]{}, 251 (2001). P.A. Lee, and T.V. Ramakrishnan, Rev. Mod. Phys. [**57**]{}, 287 (1985). B.L. Altshuler and A.G. Aronov, in [*Electron-Electron Interactions in Disordered Systems*]{}, edited by A.L. Efros and M. Pollak (North Holland,Amsterdam, 1985). A.L. Efros and B.I. Shklovskii, J. Phys. C [**8**]{}, L49 (1975); M. Pollak, Phil. Mag. B [**65**]{}, 657 (1992); see also [*Electron-Electron Interactions in Disordered Systems*]{}, edited by A.L. Efros and M. Pollak, (North-Holland, Amsterdam, 1985). K.B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997); A.D. Mirlin, Phys. Rep. [**326**]{}, 259 (2000). T. Vojta, F. Epperlein, and M. Schreiber, Phys. Rev. Lett. [**81**]{}, 4212 (1998); T. Vojta, F. Epperlein, S. Kilina, and M. Schreiber, Phys. Status Solidi B [**218**]{}, 31 (2000). X. Waintal, G. Benenti, and J.-L. Pichard, Phys. Rev. Lett. [**83**]{}, 1826 (1999). G. Benenti, X. Waintal, J.-L. Pichard, and D.L. Shepelyansky, Eur. Phys. J B [**17**]{}, 515 (2000). P.H. Song and D.L. Shepelyansky, Phys. Rev. [**61**]{}, 15546 (2000). R. Kotlyar and S. Das Sarma, Phys. Rev. Lett. [**86**]{}, 2388 (2001). F. Selva and J.L. Pichard, Europhys. Lett. [**55**]{}, 518 (2001). G.Benenti, G.Caldara and D.L.Shepelyansky, Phys. Rev. Lett. [**86**]{}, 5333 (2001). R. Berkovits and J.W. Kantelhardt, Phys Rev. B [**65**]{}, 125308 (2002). P.J.H. Denteneer, R.T. Scalettar, and N. Trivedi, Phys. Rev. Lett. [**83**]{}, 4610 (1999). G. Caldara, B. Srinivasan, and D.L. Shepelyansky, Phys. Rev. B [**62**]{}, 10680 (2000). B. Srinivasan and D.L. Shepelyansky, Eur. Phys. J. B [**24**]{}, 469 (2001); B. Srinivasan, G. Benenti, and D.L. Shepelyansky, Phys. Rev. B [**66**]{}, 172506 (2002). A. MacKinnon and B. Kramer, Rep. Prog. Phys. [**56**]{}, 1469 (1993). E. Dagotto, Rev. Mod. Phys. [**66**]{}, 763 (1994). M. Imada and Y. Hatsugai, J. Phys. Soc. Jpn. [**58**]{}, 3752 (1989).

--- abstract: 'Let ${{\mathbb G}}$ be a split reductive group over a finite extension $L$ of ${{\mathbb Q}}_p$ and let $G = {{\mathbb G}}(L)$. In this paper we prove that formal models ${{\mathfrak X}}$ of the flag variety of ${{\mathbb G}}$ are ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-affine for certain sheaves of arithmetic differential operators ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$. Given a $G$-equivariant system ${{\mathcal F}}$ of formal models ${{\mathfrak X}}$, we deduce that the category of admissible locally analytic $G$-representations with trivial central character is naturally equivalent to a full subcategory of the category of $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}_{\infty,{{\mathbb Q}}}^\dagger$-modules on the projective limit ${{\mathfrak X}}_\infty = {{\mathfrak X}}_\infty({{\mathcal F}})$ of this system of formal models. When ${{\mathcal F}}$ is cofinal in the system of all formal models, the space ${{\mathfrak X}}_\infty({{\mathcal F}})$ can be identified with the adic flag variety of ${{\mathbb G}}$.' address: - 'IRMA, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg cedex, France' - 'Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, U.S.A.' - 'Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany' - 'Indiana University, Department of Mathematics, Rawles Hall, Bloomington, IN 47405, U.S.A.' author: - Christine Huyghe - Deepam Patel - Tobias Schmidt - Matthias Strauch bibliography: - 'mybib.bib' title: '${{\mathscr D}}^\dagger$-affinity of formal models of flag varieties' --- [^1] Introduction ============ Let $L/{{\mathbb Q_p}}$ be a finite extension with ring of integers ${{\mathfrak o}}= {{\mathfrak o}}_L$. In [@PSS4] we have introduced certain sheaves of differential operators ${{\widetilde{{{\mathscr D}}}}}^\dagger_{n,k,{{\mathbb Q}}}$ on a family of semistable models ${{\mathfrak X}}_n$ of the projective line and have shown that ${{\mathfrak X}}_n$ is ${{\widetilde{{{\mathscr D}}}}}^\dagger_{n,k,{{\mathbb Q}}}$-affine. In this paper we generalize this approach to (not necessarily semistable) formal models of general flag varieties of split reductive groups. So let ${{\mathbb G}}_0$ be a reductive group scheme over ${{\mathfrak o}}$ with generic fibre ${{\mathbb G}}$. Denote by $X^{\rm rig}$ the rigid analytic space attached to the flag variety of ${{\mathbb G}}$. We consider a formal model ${{\mathfrak X}}$ of $X^{\rm rig}$ which we assume to be an admissible blow-up of a smooth formal model ${{\mathfrak X}}_0$. On such a model ${{\mathfrak X}}$ we introduce certain sheaves of differential operators ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$, for $k \ge k_{{\mathfrak X}}$, where $k_{{\mathfrak X}}$ is a non-negative integer depending on ${{\mathfrak X}}$. Our first main result is then [**Theorem 1.**]{} [*For all $k \ge k_{{\mathfrak X}}$ the formal scheme ${{\mathfrak X}}$ is ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-affine.*]{} This means that the global sections functor furnishes an equivalence of categories between coherent modules over ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ and finitely presented modules over the ring $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$ (cf. [@BB81],[@BK80], [@BK81] for the classical setting). It is shown that $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$ can be identified with a central reduction of Emerton’s analytic distribution algebra ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)$ of the wide open rigid analytic $k^{\rm th}$ congruence subgroup of ${{\mathbb G}}_0$, cf. [@EmertonA]. As in [@PSS4] our main motivation for this result concerns locally analytic representations. The category of admissible locally analytic representations of the $p$-adic group $G:={{\mathbb G}}(L)$[^2] with trivial infinitesimal character $\theta_0$ is anti-equivalent to the category of coadmissible modules over $D(G)_{\theta_0}$, the central reduction of the locally $L$-analytic distribution algebra of $G$ at $\theta_0$. On the geometric side, we consider a projective system ${{\mathcal F}}$ of formal models ${{\mathfrak X}}$. We assume that each model in this system is an admissible blow-up of some smooth formal model of $X^{\rm rig}$, and that the system ${{\mathcal F}}$ is equipped with an action of $G$. Then we can form the limit of the sheaves ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ and obtain a sheaf of (infinite order) differential operators ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ on ${{\mathfrak X}}_\infty = {{\mathfrak X}}_\infty({{\mathcal F}}) = \varprojlim_{{\mathcal F}}{{\mathfrak X}}$. In this situation, the localization functors for the various ${{\mathfrak X}}$ assemble to a functor ${{\mathscr Loc}}^\dagger_\infty$ from the coadmissible modules over $D(G)_{\theta_0}$ into the $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules on ${{\mathfrak X}}_\infty$. As in [@PSS4] we tentatively call its essential image the coadmissible (equivariant) ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules. We then obtain [**Theorem 2.**]{} [*The functors ${{\mathscr Loc}}^\dagger_\infty$ and $H^0({{\mathfrak X}}_\infty,\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(G)_{\theta_0}$-modules and coadmissible equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules.*]{} When the projective system ${{\mathcal F}}$ is cofinal in the system of all formal models of $X^{\rm rig}$, then ${{\mathfrak X}}_\infty = {{\mathfrak X}}_\infty({{\mathcal F}})$ is the Zariski-Riemann space attached to $X^{\rm rig}$. The latter space is in turn isomorphic (as a ringed space, after inverting $p$ on the structure sheaf) to the adic space attached to $X^{\rm rig}$, cf. [@SchnVPut]. With Theorem 1 as key ingredient, the proof of Theorem 2 is a straightforward generalization of the ${\rm GL}_2$-case treated in [@PSS4], but we give all the details. In this paper we only treat the case of the central character $\theta_0$, but there is an extension of this theorem available for characters more general than $\theta_0$ by using twisted versions of the sheaves ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$. Moreover, the construction of the sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ carries over to general smooth rigid (or adic) spaces over $L$. We will discuss these addenda in a sequel paper. We would also like to mention that K. Ardakov and S. Wadsley are developing a theory of ${{\mathcal D}}$-modules on general rigid spaces, cf. [@ArdakovICM], [@AW_Dcap_I]. In their work they consider deformations of the sheaves of crystalline differential differential operators (as in [@AW]), whereas we take as a starting point deformations of Berthelot’s rings of arithmetic differential operators. Insight from prior studies of logarithmic arithmetic differential operators (cf. [@PSS2], [@PSS3]) suggested to consider these latter deformations, because their global sections turned out to be (central reductions of) the analytic distribution algebras mentioned above. [*Notation.*]{} $L$ denotes a finite extension of ${{\mathbb Q_p}}$, with ring of integers ${{\mathfrak o}}$ and uniformizer ${{\varpi}}$. We put ${{\mathfrak S}}= {{\rm Spf}}({{\mathfrak o}})$ and $S = {{\rm Spec}}({{\mathfrak o}})$. Let $q$ denote the cardinality of the residue field ${{\mathfrak o}}/({{\varpi}})$ which we also denote by ${{\mathbb F_q}}$. ${{\mathbb G}}_0$ denotes a split reductive group scheme over ${{\mathfrak o}}$ and ${{\mathbb B}}_0 {\subset}{{\mathbb G}}_0$ a Borel subgroup scheme. The Lie algebra of ${{\mathbb G}}_0$ is denoted by ${{\mathfrak g}}_{{{\mathfrak o}}}$. For any scheme $X$ over ${{\mathfrak o}}$, we denote by ${{\mathcal T}}_{X}$ its relative tangent sheaf. A coherent sheaf of ideals ${{\mathcal I}}\subset {{\mathcal O}}_{X}$ is called [*open*]{} if it contains a power $\varpi^N{{\mathcal O}}_{X}$. A scheme which arises from blowing up an open ideal sheaf on $X$ will be called [*an admissible blow-up*]{} of $X$. We always denote by ${{\mathfrak X}}$ the completion of $X$ along its special fiber. The sheaves ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ and ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$ {#new_sheaves} ======================================================================================================================= Definitions ----------- In this section, $X_0$ denotes a smooth $S$-scheme, and ${{\mathfrak X}}_0$ its formal completion. Let ${\rm pr}:X\rightarrow X_0$ be an admissible blow-up of the scheme $X_0$, defined by a sheaf of ideals ${{\mathcal I}}$, containing ${{\varpi}}^N$ and ${{\mathfrak X}}$ be the formal completion of $X$. For any integer $k\geq N$, and $m$ a fixed positive integer, we will define a $p$-adically complete sheaf of arithmetic differential operators ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$ over the non-smooth formal scheme ${{\mathfrak X}}$. To achieve this construction, we will first define a sheaf of differential operators ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ on $X$ for every $k\geq N$. The scheme $X_{0}$ being a smooth $S$-scheme, one can use the sheaves of arithmetic differential operators of Berthelot defined in [@BerthelotDI]. In particular, for a fixed $m\in{{\mathbb N}}$, ${{\mathcal D}}^{(m)}_{X_{0}}$ will denote the sheaf of differential operators over $X_{0}$ of level $m$. The usual sheaf of differential operators (EGA IV) over $X_{0}$ will be denoted by ${{\mathcal D}}_{X_{0}}$. Let $U$ be a smooth affine open scheme of $X_{0}$ endowed with coordinates $x_1,\ldots,x_N$ and ${{\mathfrak U}}$ its formal completion. One denotes ${\partial}_l$ the derivation relative to $x_l$, ${\partial}_l^{[\nu_l]}\in {{\mathcal D}}_{U}$ defined by $\nu_i!{\partial}_l^{[\nu_l]}={\partial}_l^{\nu_l}$, and finally $${\partial}_{l}^{{\langle}\nu_l {\rangle}}=q_{\nu_l}!{\partial}_{l}^{[\nu_l]}.$$ Here, $q_{\nu_l}$ denotes the quotient of the euclidean division of $\nu_l$ by $p^m$. For $(\nu_1,\ldots,\nu_N)\in{{\mathbb N}}^N$, let us define ${\underline{\partial}}^{{\langle}{\underline{\nu}}{\rangle}}=\prod_{l=1}^N {\partial}_{l}^{{\langle}\nu_l {\rangle}}$, and ${\underline{\partial}}^{[{\underline{\nu}}]}=\prod_{l=1}^N {\partial}_{l}^{[\nu_l]}$. Restricted to $U$, the sheaf ${{\mathcal D}}^{(m)}_{X_{0},k}$ is a sheaf of free ${{\mathcal O}}_U$-algebras, with basis the elements ${\underline{\partial}}^{{\langle}{\underline{\nu}}{\rangle}}$. Following  [@PSS4], given two positive integers, $k,m$, one introduces the subsheaves of algebras ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}}$ (resp. ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$) of differential operators of congruence level $k$, which are locally free ${{\mathcal O}}_{U_i}$-algebras (resp. ${{\mathcal O}}_{{{\mathcal U}}}$-algebras), with basis the elements ${{\varpi}}^{k|{\underline{\nu}}|}{\underline{\partial}}^{{\langle}{\underline{\nu}}{\rangle}}$, meaning that $${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}}(U)=\left\{\sum_{{\underline{\nu}}}{{\varpi}}^{k|{\underline{\nu}}|}a_{{\underline{\nu}}}{\underline{\partial}}^{{\langle}{\underline{\nu}}{\rangle}}\,|\, a_{{\underline{\nu}}}\in {{\mathcal O}}_{X_0}(U)\right\}.$$ Let $m$ be a fixed non negative integer, if $\nu$ is a non negative integer, one denotes by $q$ the quotient of the euclidean division of $\nu$ by $p^m$. Let $\nu\geq \nu'$ be two nonnegative integers and $\nu'':=\nu-\nu'$, then we introduce for the corresponding numbers $q,q'$ and $q''$ the following notation $${\genfrac{\{}{\}}{0pt}{}{\nu}{\nu'}}=\frac{q!}{q'!q''!}.$$ We also define $${{\widetilde{{{\mathscr D}}}}}_{{{\mathfrak X}}_0,k}^{(m)}=\varprojlim_i {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_{0},k}/p^i \hskip5pt \textrm{ and }\hskip5pt {{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}}_0,k,{{\mathbb Q}}}=\varinjlim_m {{\widetilde{{{\mathscr D}}}}}_{{{\mathfrak X}}_0,k}^{(m)}{\otimes}{{\mathbb Q}}.$$ As explained in  [@PSS4], these sheaves of rings have the same finiteness properties as the usual sheaves of Berthelot (corresponding to $k=0$). In particular, over an affine smooth scheme $U_{0}$ there is an equivalence of categories between the coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{U_{0},k}$-modules and the finite type modules over the algebra ${\Gamma}(U_{0},{{\widetilde{{{\mathcal D}}}}}^{(m)}_{U_{0},k})$, for every $k\geq 0$. Let us now explain how to construct the desired sheaves over $X$. Imitating 2.3.5 of  [@BerthelotDI], if we can prove that ${{\mathcal O}}_{X}$ can be endowed with a structure of ${\rm pr}^{-1}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_{0},k}$-module, then the sheaf of ${{\mathcal O}}_{X}$-modules, ${\rm pr}^{*}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_{0},k}$, will be a sheaf of rings over ${{\mathcal O}}_{X}$. In order to prove this, we need the following proposition. Let $k\geq N$. Then the sheaf of rings ${{\mathcal O}}_{X}$ can be endowed with a structure of ${\rm pr}^{-1}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,k}$-module, such that the map ${\rm pr}^{-1}{{\mathcal O}}_{X_0}{\rightarrow}{{\mathcal O}}_{X}$ is ${\rm pr}^{-1}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,k}$-linear. It is enough to prove the statement in the case where $X_0$ is affine, say $X_0={{\rm Spec}}A$. Denote then $I={\Gamma}(X_0,{{\mathcal I}})$, which contains ${{\varpi}}^N$, and $B$ the $A$-graded algebra $B=Sym_{A}\;I$, $B=\bigoplus_n I_n$ (meaning that $I_n$ equals as an abelian group $I^n$), so that $X=Proj\; B$. Let $t\in I_d$ an homogeneous element of degree $d>0$ of $B$, and $B[1/t]_0$ the algebra of degree $0$-elements in the localization $B[1/t]$. It is enough to prove that $B[1/t]_0$ can be endowed with a structure of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}}(X_0)$-module. We can even assume that $X_0$ is affine endowed with local coordinates $x_1,\ldots,x_N$. In this case, we let $D^{(m)}_{k}={\Gamma}(X_0,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}})$ which can be described in the following way $$D^{(m)}_{k}= \left\{\sum_{{\underline{\nu}}}{{\varpi}}^{k|{\underline{\nu}}|}a_{{\underline{\nu}}}{\underline{\partial}}^{{\langle}{\underline{\nu}}{\rangle}}\,|\, a_{{\underline{\nu}}}\in A\right\}.$$ Let us observe that the algebra $B$ is a subgraded algebra of $A[T]$. Indeed, there is a graded ring morphism $$\xymatrix{B \ar@{->}[r]^(.4){\varphi}& A[T]\\ x_n \in I_n \ar@{->}[r]& x_n T^n. }$$ We can identify ${{\rm Spec}}\; A[T]$ with $Y=\mathbb{A}^1_{X_0}$ and consider ${\rm pr}_2$: $Y {\rightarrow}X_0$. By copying the classical proof, one can check that ${\rm pr}_2^{-1}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}}$ is a subsheaf of the sheaf ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{Y,k}}$, as sheaf of rings. Let $U_t$ be the affine open set of $Y$ where $\varphi(t)$ is nonzero. and $C_t={\Gamma}(U_t, {{\mathcal O}}_Y)$. By flatness of localization, there is an injective map $B_0[1/t]\subset C_t$. Moreover, $C_t$ and $A[T]$ are ${\Gamma}(Y,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{Y,k}})$-modules and thus $D^{(m)}_{k}$-modules. Take $P\in D^{(m)}_{k}$ and $a\in A$, then $P(aT^n)=P(a)T^n$. This shows that the action of $D^{(m)}_{k}$ over $A[T]$ is graded. To prove that $B_0[1/t]$ is a $D^{(m)}_{k}$-module, we will first prove that $B$ is a $D^{(m)}_{k}$-submodule of $A[T]$ and then that $B_0[1/t]$ is a $D^{(m)}_{k}$-submodule of $C_t$. The fact that the defined actions will be compatible with the action of $D^{(m)}_{k}$ over $A$, will come from the fact that this is true for the action of $D^{(m)}_{k}$ over $A[T]$. Let us prove now that $B$ is a $D^{(m)}_{k}$-submodule of $A[T]$. To avoid too heavy notations, we identify $B$ (resp. $B_t$) with their image into $A[T]$ (resp. $C_t$) via $\varphi$. The ideal $I$ is trivially stable by the action of $A$ by left multiplication, and $A=I_0$ is certainly a $D^{(m)}_{k}$-module. Recall that the algebra $D^{(m)}_{k}$ is generated by the operators ${{\varpi}}^{k\nu_i} {\partial}_{i}^{{\langle}\nu_i {\rangle}}$ for all $\nu_i \leq p^m$. Since $k\geq N$, it is clear that if $\nu_i\geq 1$, ${{\varpi}}^{k\nu_i} {\partial}_{i}^{{\langle}\nu_i {\rangle}}(IT)\subset {{\varpi}}^N AT \subset IT $, since the action of $D^{(m)}_{k}$ over $A[T]$ is graded, so that we can define an action of $D^{(m)}_{k}$ : $I_1 {\rightarrow}I_1$. We next prove by induction on $d$ that $I_d$ is stable by the action of $D^{(m)}_{k}$. Let $$x= {{\varpi}}^{N(d-b)}y_1\ldots y_b\,\in I_d.$$ Let $\nu_i\geq 1$ and $\nu_i\leq p^m$, then one computes $$\begin{aligned} {{\varpi}}^{N\nu_i}{\partial}_{i}^{{\langle}\nu_i {\rangle}}(y_1\ldots y_b) &= \sum_{\mu_i\leq \nu_i} \left({{\varpi}}^{N\mu_i}{\genfrac{\{}{\}}{0pt}{}{\mu_i}{\nu_i}} {\partial}_{i}^{{\langle}\mu_i {\rangle}}(y_1)\right){{\varpi}}^{N(\nu_i-\mu_i)}{\partial}_{i}^{{\langle}\nu_i-\mu_i {\rangle}}(y_2\cdots y_b)\end{aligned}$$ by [@BerthelotDI 2.2.4(iv)]. By induction, we know that ${{\varpi}}^{N(\nu_i-\mu_i)}{\partial}_{i}^{{\langle}\nu_i-\mu_i {\rangle}}(y_2\cdots y_b)\in I_{b-1}$, that implies that $${{\varpi}}^{N\nu_i}{\partial}_{i}^{{\langle}\nu_i {\rangle}}(x)\in I_d.$$ It remains now to prove that $B[1/t]_0$ is a $D^{(m)}_{k}$-submodule of $C_t$. We have to check the following statement $$\begin{gathered} \label{stat-inter} \forall \nu_i\leq p^m, \forall c\in {{\mathbb N}},\forall g\in I_{dc},\, {{\varpi}}^{N\nu_i}{\partial}_{i}^{{\langle}\nu_i {\rangle}}\left(\frac{g}{t^c}\right)\in B_0[1/t].\end{gathered}$$ Let us first prove by induction on $\nu_i$ that $$\begin{gathered} {{\varpi}}^{N\nu_i}{\partial}_{i}^{{\langle}\nu_i {\rangle}} \left(t^{-1}\right)\in \frac{I_{\nu_id}}{t^{\nu_i+1}}.\end{gathered}$$ This is true for $\nu_i=0$. Consider then the formula $${{\varpi}}^{N(\nu_i+1)}{\partial}_{i}^{{\langle}\nu_i +1{\rangle}}(t^{-1})=-\sum_{\mu =0}^{\nu_i}{\genfrac{\{}{\}}{0pt}{}{\nu_i+1}{\mu }}t^{-1} {{\varpi}}^{N(\nu_i+1-\mu )}{\partial}_{i}^{{\langle}\nu_i+1 -\mu {\rangle}}(t) {{\varpi}}^{N\mu }{\partial}_{i}^{{\langle}\mu {\rangle}}(t^{-1}).$$ By induction on $\nu_i$, one knows that for any integer $\mu\leq \nu_{i}$, $${{\varpi}}^{N(\nu_i+1)}t^{-1}{\partial}_{i}^{{\langle}\nu_i+1 -\mu {\rangle}}(t){\partial}_{i}^{{\langle}\mu {\rangle}}(t^{-1}) \in \frac{1}{t^{\mu +2}}I_{d}I_{\mu d}\, \subset \frac{I_{(\nu_i+1) d}}{t^{\nu_i+2}},$$ which proves our claim. Applying this claim to $t^c$ gives for $\mu\leq p^m$ $${{\varpi}}^{N\mu }{\partial}_{i}^{{\langle}\mu {\rangle}}(t^{-c})\in \frac{I_{\mu dc}}{t^{c(\mu +1)}}.$$ Then, we have the following formula $${{\varpi}}^{N\nu_i}{\partial}_{i}^{{\langle}\nu_i {\rangle}}\left(\frac{g}{t^c}\right)=\sum_{\mu =0}^{\nu_i} {\genfrac{\{}{\}}{0pt}{}{\nu_i}{\mu }}{{\varpi}}^{N(\nu_i- \mu)}{\partial}_{i}^{{\langle}\nu_i- \mu {\rangle}}(g){{\varpi}}^{N \mu}{\partial}_{i}^{{\langle}\mu {\rangle}}(t^{-c}),$$ whose right-hand terms are contained in $$I_{dc}\frac{I_{ \mu dc}}{t^{c(\mu +1)}}\subset B_0[1/t],$$ and this completes the proof of \[stat-inter\] and the proposition. From this, we deduce as in 2.3.5 of [@BerthelotDI] the following. Under the hypothesis of the section, for $k\geq N$, the sheaf ${\rm pr}^*{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}}$ is a sheaf of rings over $X$. This allows us to introduce the following sheaves of rings over the admissible blow-up ${{\mathfrak X}}$ of ${{\mathfrak X}}_0$ $${{\widetilde{{{\mathscr D}}}}}_{{{\mathfrak X}},k}^{(m)}=\varprojlim_i {\rm pr}^*{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_{0},k}/p^i \hskip5pt \textrm{ and }\hskip5pt {{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}},k,{{\mathbb Q}}}=\varinjlim_m {{\widetilde{{{\mathscr D}}}}}_{{{\mathfrak X}},k}^{(m)}{\otimes}{{\mathbb Q}}.$$ We abbreviate ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}:={\rm pr}^*{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_{0},k}}$ in the following. Finiteness properties of the sheaves ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X,k}}$ ----------------------------------------------------------------------------------- We keep here the hypothesis of the beginning of the section. For a given natural number $k\geq 0$ we let $${{\widetilde{{{\mathcal T}}}}}_{X,k} := {{\varpi}}^k ({\rm pr})^*({{\mathcal T}}_{X_0}).$$ \[tcT\_lemma1\] (i) ${{\widetilde{{{\mathcal T}}}}}_{X,k}$ is a locally free ${{\mathcal O}}_{X}$-module of rank equal to the relative dimension of $X_0$ over $S$, \(ii) Suppose $\pi: X'\rightarrow X$ is a morphism of admissible blow-ups of $X_0$. Let $k',k\geq 0 $. One has $${{\widetilde{{{\mathcal T}}}}}_{X',k'} = {{\varpi}}^{k'-k} {\rm \pi}^*({{\widetilde{{{\mathcal T}}}}}_{X,k})$$ as subsheaves of ${{\mathcal T}}_{X'}\otimes L$. [[*Proof.* ]{}]{}The assertion (i) follows directly from the definition. The assertion (ii) follows since $\pi$ is a morphism over $X_0$. We also have \[finite\_tDm\] 1. The sheaves ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X,k}}$ are filtered by the order of differential operators and there is a canonical isomorphism of graded sheaves of algebras $${\rm gr}\left({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}\right) \simeq {{\rm Sym}}({{\widetilde{{{\mathcal T}}}}}_{X,k})^{(m)} = \bigoplus_{d \ge 0} {{\rm Sym}}^d({{\widetilde{{{\mathcal T}}}}}_{X,k})^{(m)} \;.$$ 2. There is a basis of the topology ${{\mathscr B}}$ of $X$, consisting of open affine subsets, such that for any $U \in {{\mathscr B}}$, the ring ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(U)$ is noetherian. In particular, the sheaf of rings ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ is coherent. 3. The sheaf ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{{\mathfrak X}},k}}$ is coherent. [[*Proof.* ]{}]{}Denote by ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,k;d}$ the sheaf of differential operators of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X_0,k}}$ of order less than $d$. It is straightforward that we have an exact sequence of ${{\mathcal O}}_{X_0}$-modules on $X_0$ \[fil\_gr\_ex\_seq\_n0\] 0 \^[(m)]{}\_[X\_0,k;d-1]{} \^[(m)]{}\_[X\_0,k;d]{} [[Sym]{}]{}\^d(\_[X\_0,k]{})\^[(m)]{} 0 . Now we apply ${\rm pr}^*$ and get an exact sequence since ${{\rm Sym}}^d({{\widetilde{{{\mathcal T}}}}}_{X_0,k})^{(m)}$ is a free ${{\mathcal O}}_{X_0}$-module of finite rank. This gives (i). As we work with quasi-coherent sheaves of ${{\mathcal O}}_X$-modules, it is enough to show that ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X,k}}(U)$ is noetherian in the case where $U={\rm pr}^{-1}(U_0)$, and $U_0\subset X_0$ has some coordinates $x_1,\ldots,x_N$. In this case one has the following description $${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X},k}(U)=\left\{\sum_{{\underline{\nu}}}{{\varpi}}^{k|{\underline{\nu}}|}a_{{\underline{\nu}}}{\underline{\partial}}^{{\langle}{\underline{\nu}}{\rangle}}\,|\, a_{{\underline{\nu}}}\in {{\mathcal O}}_{X}(U)\right\}.$$ By (i), the graded algebra ${\rm gr}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X},k}(U)$ is isomorphic to ${{\rm Sym}}_{{{\mathcal O}}(U)}({{\widetilde{{{\mathcal T}}}}}_{X,k}(U))^{(m)}$, which is known to be noetherian [@Huyghe97 Prop. 1.3.6]. This gives the noetherianity of the algebras ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X},k}(U)$. As ${{\mathscr B}}$ we may take the set of open subsets of $X$ that are contained in some ${\rm pr}^{-1}(U_0)$, for some open $U_0\subset X_0$ endowed with global coordinates. We finally note that, since the sheaf ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X,k}}$ is ${{\mathcal O}}_X$-quasicoherent and has noetherian sections of over open affines, it is actually a sheaf of coherent rings [@BerthelotDI 3.1.3(i)]. The last assertion is a direct consequence of (ii), as in [@BerthelotDI]. We do not redo the proof here. From these considerations, we give theorems A and B for affine schemes $X$, whose proofs follows readily by the proofs given by Berthelot in [@BerthelotDI]. Let $U\subset X$ be an affine subscheme of $X$, ${{\mathfrak U}}$ its formal completion. 1. The algebra ${\Gamma}(U,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{U,k}})$ is noetherian and the functor ${\Gamma}(U,.)$ establishes an equivalence of categories between coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{U,k}}$-modules and finite ${\Gamma}(U,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{X,k}})$-modules. 2. The algebra ${\Gamma}({{\mathfrak U}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{{\mathfrak U}},k}})$ is noetherian and the functor ${\Gamma}({{\mathfrak U}},.)$ establishes an equivalence of categories between coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{{\mathfrak U}},k}}$-modules and finite ${\Gamma}({{\mathfrak U}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{{\mathfrak U}},k}})$-modules. Formal models of flag varieties {#models} =============================== From now on $X_0$ denotes the smooth flag scheme ${{\mathbb G}}_0/{{\mathbb B}}_0$ of ${{\mathbb G}}_0$ and ${{\mathfrak X}}_0$ denotes its $p$-adic completion. Congruence group schemes {#groups} ------------------------ We let ${{\mathbb G}}(k)$ denote the $k$-th scheme-theoretic congruence subgroup of the group scheme ${{\mathbb G}}_0$ [@WW80 1.], [@YuSmoothModels 2.8]. So ${{\mathbb G}}(0)={{\mathbb G}}_0$ and ${{\mathbb G}}(k+1)$ equals the dilatation (in the sense of [@BLR 3.2]) of the trivial subgroup of ${{\mathbb G}}(k)\otimes_{{{\mathfrak o}}}{{\mathbb F}}_q$ on ${{\mathbb G}}(k).$ In particular, if ${{\mathbb G}}(k) = {{\rm Spec}}\; {{\mathfrak o}}[t_1,...,t_N]$ with a set of parameters $t_i$ for the unit section of ${{\mathbb G}}(k)$, then ${{\mathbb G}}(k+1)={{\rm Spec}}\; {{\mathfrak o}}[\frac{t_1}{\varpi},...,\frac{t_N}{\varpi}]$. The ${{\mathfrak o}}$-group scheme ${{\mathbb G}}(k)$ is again smooth, has Lie algebra equal to $\varpi^k{{\mathfrak g}}_{{{\mathfrak o}}}$ and its generic fibre coincides with the generic fibre of ${{\mathbb G}}_0$. A very ample line bundle on $X$ ------------------------------- Let ${\rm pr}: X {\rightarrow}X_0$ be an admissible blow-up, and let ${{\mathcal I}}{\subset}X_0$ be the ideal sheaf that is blown up. Since ${{\mathcal I}}$ is open, there is $N \in {{\mathbb Z}}_{>0}$ such that \[invert\_p\_1\] p\^N [[O]{}]{}\_[X\_0]{} [[I]{}]{}[[O]{}]{}\_[X\_0]{} . Put ${{\mathcal S}}= \bigoplus_{s \ge 0} {{\mathcal I}}^s$, then $X$ is glued together from schemes ${\bf Proj}({{\mathcal S}}(U))$ for affine open subsets $U {\subset}X$. On each ${\bf Proj}({{\mathcal S}}(U))$ there is an invertible sheaf ${{\mathcal O}}(1)$, and these glue together to give an invertible sheaf ${{\mathcal O}}(1)$ on $X$ which we will denote ${{\mathcal O}}_{X/X_0}(1)$ (cf. the discussion in [@HartshorneA ch. II,§7].) This invertible sheaf is in fact the inverse image ideal sheaf ${\rm pr}^{-1}({{\mathcal I}}) \cdot {{\mathcal O}}_{X}$, cf. [@HartshorneA ch. II, 7.13]. From \[invert\_p\_1\] conclude that \[invert\_p\_2\] p\^N [[O]{}]{}\_[X]{} [[O]{}]{}\_[X/X\_0]{}(1) [[O]{}]{}\_[X]{} [[O]{}]{}\_[X]{} [[O]{}]{}\_[X/X\_0]{}(-1) p\^[-N]{}[[O]{}]{}\_[X]{}. And for any $r \ge 0$ we get \[invert\_p\_3\] p\^[rN]{} [[O]{}]{}\_[X]{} [[O]{}]{}\_[X/X\_0]{}(1)\^[r]{} [[O]{}]{}\_[X]{} [[O]{}]{}\_[X]{} [[O]{}]{}\_[X/X\_0]{}(-1)\^[r]{} p\^[-rN]{}[[O]{}]{}\_[X]{}. \[direct\_im\_str\_sh\] Suppose $X$ is normal. Let $X'$ be the blow-up of a coherent open ideal on $X$ and let ${\rm pr}: X' \rightarrow X$ be the blow-up morphism. Then $({\rm pr})_*({{\mathcal O}}_{X'}) = {{\mathcal O}}_{X}$. [[*Proof.* ]{}]{}The morphism ${\rm pr}: X' {\rightarrow}X$ is a birational projective morphism of noetherian integral schemes. The assertion then follows exactly as in the proof of Zariski’s Main Theorem as given in [@HartshorneA ch. III, Cor. 11.4]. \[v\_ample\_sh\_lemma\] There is $a_0\in {{\mathbb Z}}_{>0}$ such that the line bundle \[v\_ample\_sh\_disp\] [[L]{}]{}\_[X]{} = [[O]{}]{}\_[X/X\_0]{}(1) \^\*([[O]{}]{}\_[X\_0]{}(a\_0)) on $X$ is very ample over ${{\rm Spec}}({{\mathfrak o}})$, and it is very ample over $X_0$. [[*Proof.* ]{}]{}By [@HartshorneA ch. II, ex. 7.14 (b)], the sheaf \[v\_ample\_sh\] [[L]{}]{}= [[O]{}]{}\_[X/X\_0]{}(1) \^\*([[O]{}]{}\_[X\_0]{}(a\_0)) is very ample on $X$ over ${{\rm Spec}}({{\mathfrak o}})$ for suitable $a_0 > 0$. We fix such an $a_0$. By [@EGA_II 4.4.10 (v)] it is then also very ample over $X_0$. \[tcT\_lemma\] Suppose $\pi: X'\rightarrow X$ is a morphism of admissible blow-ups of $X_0$. \(i) In the case $\pi_*{{\mathcal O}}_{X'}={{\mathcal O}}_X$, one has $${{\widetilde{{{\mathcal T}}}}}_{X,k} = {{\varpi}}^{k-k'} {\rm \pi}_*({{\widetilde{{{\mathcal T}}}}}_{X',k'})$$ as subsheaves of ${{\mathcal T}}_{X}\otimes L$. \(ii) There is a ${{\mathcal O}}_{X}$-linear surjection ${{\mathcal O}}_{X}\otimes_{{{\mathfrak o}}} \varpi^k{{\mathfrak g}}_{{{\mathfrak o}}}\twoheadrightarrow{{\widetilde{{{\mathcal T}}}}}_{X,k}$. [[*Proof.* ]{}]{} The assertion (i) follows from (ii) of the previous lemma \[tcT\_lemma1\] together with the projection formula. Finally, (ii) follows from [@NootHuyghe09 1.6.1] by applying $\pi^*$. . \[graded\_tcD\] In the case $\pi_*{{\mathcal O}}_{X'}={{\mathcal O}}_X$, one has $ \pi_*\left({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X',k}\right) = {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$. [[*Proof.* ]{}]{}The sheaves ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,k;d}$ are locally free of finite rank, and so are the sheaves ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k;d}$, by construction. We can thus apply the projection formula and get $$\pi_*\Big({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X',k;d}\Big) = {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k;d} \;.$$ The claim follows because the direct image commutes with inductive limits on a noetherian space. [*Twisting by ${{\mathcal L}}_X$.*]{} Recall the very ample line bundle ${{\mathcal L}}_X$ from \[v\_ample\_sh\_lemma\]. In the following we will always use this line bundle to ’twist’ ${{\mathcal O}}_{X}$-modules. If ${{\mathcal F}}$ is a ${{\mathcal O}}_{X}$-module and $r \in {{\mathbb Z}}$ we thus put $${{\mathcal F}}(r) = {{\mathcal F}}\otimes_{{{\mathcal O}}_{X}} {{\mathcal L}}_X^{\otimes r} \;.$$ Some caveat is in order when we deal with sheaves which are equipped with both a left and a right ${{\mathcal O}}_{X}$-module structure (which may not coincide). For instance, if ${{\mathcal F}}_d = {{\widetilde{{{\mathcal D}}}}}_{X,k;d}^{(m)}$ then we let $${{\mathcal F}}_d(r) = {{\widetilde{{{\mathcal D}}}}}_{X,k;d}^{(m)}(r) = {{\widetilde{{{\mathcal D}}}}}_{X,k;d}^{(m)} \otimes_{{{\mathcal O}}_{X}} {{\mathcal L}}_X^{\otimes r} \;,$$ where we consider ${{\mathcal F}}_d = {{\widetilde{{{\mathcal D}}}}}_{X,k;d}^{(m)}$ as a [*right*]{} ${{\mathcal O}}_{X}$-module. Similarly we put $${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)} \otimes_{{{\mathcal O}}_{X}} {{\mathcal L}}_X^{\otimes r} \;,$$ where we consider ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}$ as a [*right*]{} ${{\mathcal O}}_{X}$-module. Then we have ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r) = \varinjlim_d {{\mathcal F}}_d(r)$. When we consider the associated graded sheaf of ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r)$, it is with respect to the filtration by the ${{\mathcal F}}_d(r)$. The sheaf ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r)$ is a coherent left ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}$-module according to part (iii) of \[graded\_tcD\]. Global sections of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$, ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$, and ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ {#global_sec} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [*Divided power enveloping algebras.*]{} We denote by $D^{(m)}({{\mathbb G}}(k))$ the distribution algebra of the smooth ${{\mathfrak o}}$-group scheme ${{\mathbb G}}(k)$ of level $m$ [@HS13]. It is noetherian and admits the following explicit description. Let ${{\mathfrak g}}_{{\mathfrak o}}={{\mathfrak n}}^{-}_{{\mathfrak o}}\oplus{{\mathfrak t}}_{{\mathfrak o}}\oplus{{\mathfrak n}}_{{\mathfrak o}}$ be a triangular decomposition of ${{\mathfrak g}}_{{\mathfrak o}}$. We fix basis elements $(f_i),(h_j)$ and $(e_i)$ of the ${{\mathfrak o}}$-modules ${{\mathfrak n}}^{-}_{{\mathfrak o}}, {{\mathfrak t}}_{{\mathfrak o}}$ and ${{\mathfrak n}}_{{\mathfrak o}}$ respectively. Then $D^{(m)}({{\mathbb G}}(k))$ equals the ${{\mathfrak o}}$-subalgebra of $U({{\mathfrak g}}) = U({{\mathfrak g}}_{{\mathfrak o}}) \otimes_{{\mathfrak o}}L$ generated by the elements \[generators\_n\] q\^[(m)]{}\_! q\^[(m)]{}\_[’]{}! \^[k |’|]{}[ ’]{} q\^[(m)]{}\_[”]{}! . In the case of the group ${\rm GL}_2$ we considered the same algebra in [@PSS4]. We denote by $\widehat{D}^{(m)}({{\mathbb G}}(k))$ the $p$-adic completion of $D^{(m)}({{\mathbb G}}(k))$. It is noetherian. \[global\_sections\_tcD\] There is a canonical homomorphism of ${{\mathfrak o}}$-algebras \[xi\_uncompl\] Q\^[(m)]{}\_[X,k]{}: D\^[(m)]{}([[G]{}]{}(k))H\^0(X, \^[(m)]{}\_[X,k]{}) , [[*Proof.* ]{}]{}Recall that a filtered ${{\mathfrak o}}$-algebra (or sheaf of such algebras) $A$ with positive filtration $F_iA$ and ${{\mathfrak o}}\subseteq F_0A$ yields the graded subring $R(A):=\oplus_{i\geq 0}F_iAt^{i}\subseteq A[t]$ of the polynomial ring over $A$, its associated Rees ring. Specialising $R(A)$ in an element $\lambda\in{{\mathfrak o}}$ yields a filtered subring $A_{\lambda}$ of $A$. For fixed $\lambda$, the formation of $A_{\lambda}$ is functorial in $A$. We apply this remark to the filtered homomorphism $$D^{(m)}({{\mathbb G}}(0)) {\rightarrow}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,0}$$ appearing in [@HS13 4.4.4] (and denoted by $Q_m$ in loc.cit.) which comes by functoriality from the ${{\mathbb G}}(0)$-action on $X_0$. Passing to Rees rings and specialising the parameter in $\varpi^k$ yields a filtered homomorphism $$D^{(m)}({{\mathbb G}}(k)) {\rightarrow}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,k}.$$ Taking global sections and using $H^0(X, {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}) = H^0(X_0, {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_0,k})$ by \[graded\_tcD\] yields a homomorphism $$Q_{X,k}^{(m)}: D^{(m)}({{\mathbb G}}(k)) {\rightarrow}H^{0}(X,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k})$$ as claimed. Let ${{\mathcal A}}^{(m)}_{X,k}={{\mathcal O}}_{X}\otimes_{{{\mathfrak o}}}D^{(m)}({{\mathbb G}}(k))$ with the twisted ring multiplication coming from the action of $D^{(m)}({{\mathbb G}}(k))$ on ${{\mathcal O}}_{X}$ via $Q^{(m)}_{X,k}$. It has a natural filtration whose associated graded equals the ${{\mathcal O}}_{X}$-algebra ${{\mathcal O}}_{X}\otimes_{{{\mathfrak o}}}{{\rm Sym}}^{(m)}({{\rm{Lie}}}({{\mathbb G}}(k)))$ [@HS13 Cor. 4.4.6]. In particular, ${{\mathcal A}}^{(m)}_{X,k}$ has noetherian sections over open affines. \[prop-auxiliaryI\] The homomorphism ${{\mathcal A}}^{(m)}_{X,k}\rightarrow {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ induced by $\xi^{(m)}_{X,k}$ is surjective. In particular, ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{{{\mathfrak X}},k}$ has noetherian sections over open affines. [[*Proof.* ]{}]{}We adapt the argument of [@HS13 4.4.7.2(ii)]: The homomorphism is filtered. Applying ${{\rm Sym}}^{(m)}$ to the surjection in (iv) of \[tcT\_lemma\] we obtain a surjection ${{\mathcal O}}_{X}\otimes_{{{\mathfrak o}}}{{\rm Sym}}^{(m)}({{\rm{Lie}}}({{\mathbb G}}(k))\rightarrow{{\rm Sym}}^{(m)}({{\widetilde{{{\mathcal T}}}}}_{X,k})$ which equals the associated graded homomorphism by \[graded\_tcD\]. Hence the homomorphism is surjective as claimed. \[prop-auxiliaryII\] Let ${{\mathcal M}}$ be a coherent ${{\mathcal A}}^{(m)}_{X,k}$-module. \(i) $H^0(X,{{\mathcal A}}^{(m)}_{X,k})=D^{(m)}({{\mathbb G}}(k))$. \(ii) There is a surjection ${{\mathcal A}}_{X,k}^{(m)}(-r)^{\oplus s}\rightarrow{{\mathcal M}}$ of ${{\mathcal A}}_{X,k}^{(m)}$-modules for suitable $r,s\geq 0$. \(iii) For any $i\geq 0$ the group $H^{i}(X,{{\mathcal M}})$ is a finitely generated $D^{(m)}({{\mathbb G}}(k))$-module. \(iv) The ring $H^{0}(X,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k})$ is a finitely generated $D^{(m)}({{\mathbb G}}(k))$-module and hence noetherian. [[*Proof.* ]{}]{}The points (i)-(iii) follow exactly as in [@HS13 A.2.6.1]. Statement (iv) is a special case of (iii) by \[prop-auxiliaryI\]. Now let ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$ be the $p$-adic completion of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$, which we will always consider as a sheaf on the formal scheme ${{\mathfrak X}}$. We also put ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}} = {{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k} \otimes_{{\mathbb Z}}{{\mathbb Q}}$, and \[tsD\_dagger\] \^\_[[[X]{}]{},k]{} = \_m \^[(m)]{}\_[[[X]{}]{},k]{} , and \[tsD\_dagger\_Q\] \^\_[[[X]{}]{},k,[[Q]{}]{}]{} = \^\_[[[X]{}]{},k]{} \_[[Z]{}]{}[[Q]{}]{}= \_m \^[(m)]{}\_[[[X]{}]{},k,[[Q]{}]{}]{} . We denote by $\widehat{D}^{(m)}({{\mathbb G}}(k))_{{{\mathbb Q}},\theta_0}$ the quotient of $\widehat{D}^{(m)}({{\mathbb G}}(k))\otimes {{\mathbb Q}}$ modulo the ideal generated by the center of the ring $U({{\mathfrak g}})$ [@HS13 A.2.1]. This is the same central reduction considered in [@PSS4] for the group ${\rm GL}_2$. In the proposition below, and in the remainder of this paper, certain rigid analytic ‘wide open’ groups ${{\mathbb G}}(k)^\circ$ will be used repeatedly. To define them, consider first the formal completion ${{\mathfrak G}}(k)$ be of the group scheme ${{\mathbb G}}(k)$ along its special fiber, which is a formal group scheme (of topologically finite type) over ${{\mathfrak S}}= {{\rm Spf}}({{\mathfrak o}})$. Then let $\widehat{{{\mathfrak G}}}(k)^\circ$ be the completion of ${{{\mathfrak G}}}(k)$ along its unit section ${{\mathfrak S}}{\rightarrow}{{{\mathfrak G}}}(k)$, and denote by ${{\mathbb G}}(k)^\circ$ its associated rigid space. \[global\_sec\_tsD\] (i) There is a basis of the topology ${{\mathfrak U}}$ of $X$, consisting of open affine subsets, such that for any $U \in {{\mathfrak U}}$, the ring $H^0(U,{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})$ is noetherian. \(ii) The transition map ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}} {\rightarrow}{{\widetilde{{{\mathscr D}}}}}^{(m+1)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$ is flat. \(iii) The sheaf of rings ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ is coherent. \(iv) The homomorphism $Q^{(m)}_{X,k}$ induces an algebra isomorphism $$\widehat{D}^{(m)}({{\mathbb G}}(k))_{{{\mathbb Q}},\theta_0}{\stackrel{\simeq}{\longrightarrow}}H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}).$$ \(v) The ring $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$ is canonically isomorphic to the coherent $L$-algebra ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}$, where ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)$ is the analytic distribution algebra in the sense of Emerton, cf. [@EmertonA ch. 5]. [[*Proof.* ]{}]{}(i) This follows from [@BerthelotDI 3.3.4(i)] together with \[prop-auxiliaryI\]. \(ii) This can be proved as in [@BerthelotDI sec. 3.5]. \(iii) Follows from (i) and (ii). \(iv) This follows from statement (iv) in \[prop-auxiliaryII\] together with [@HS13 Lem. A.3]. \(v) Follows from (iv) and the description of the analytic distribution algebra as given in [@EmertonA ch. 5], compare also [@HS13 Prop. 5.2.1]. Localization on ${{\mathfrak X}}$ via ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ {#loc_n} ================================================================================================================ The general line of arguments follows fairly closely [@NootHuyghe09]. As usual, ${\rm pr}: X {\rightarrow}X_0$ denotes an admissible blow-up of $X_0$. The number $k\geq 0$ is fixed throughout this section and chosen large enough so that the sheaf of coherent rings ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ is defined for all $m$. Cohomology of coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-modules --------------------------------------------------------------------------- \[van\_coh\] Let ${{\mathcal E}}$ be an abelian sheaf on $X$. For all $i >\dim X$ one has $H^i(X,{{\mathcal E}}) =0$. [[*Proof.* ]{}]{}Since the space $X$ is noetherian the result follows from Grothendieck’s vanishing theorem [@HartshorneA Thm. 2.7]. \[vanishing\_coh\_gr\_Dnk\] There is a natural number $r_0$ such that for all $r \ge r_0$ and all $i \ge 1$ one has \[vanishing\_coh\_gr\_Dnk\_disp\] H\^i(X,[gr]{}(\_[X,k]{}\^[(m)]{})(r)) = 0 . [[*Proof.* ]{}]{}Since ${{\mathcal L}}_X$ is very ample over ${{\mathfrak o}}$ by \[v\_ample\_sh\_lemma\], the Serre theorems [@HartshorneA II.5.17/III.5.2] imply that there is a number $u_0$ such that for all $u\geq u_0$ the module ${{\mathcal O}}_X(u)$ is generated by global sections and has no higher cohomology. After this remark we prove the proposition along the lines of [@NootHuyghe09 Prop. 2.2.1]. There is an ${{\mathcal O}}_{X_0}$-linear surjection $({{\mathcal O}}_{X_0})^{\oplus a}\rightarrow{{\mathcal T}}_{X_0}$ for a suitable natural number $a$. Applying $({\rm pr})^*$ and multiplication by $\varpi^k$ gives an ${{\mathcal O}}_X$-linear surjection $({{\mathcal O}}_X)^{\oplus a}\simeq \varpi^k ({{\mathcal O}}_X)^{\oplus a} \rightarrow {{\widetilde{{{\mathcal T}}}}}_{X,k}$. By functoriality we get a surjective morphism of algebras $${{\mathcal C}}:={{\rm Sym}}(({{\mathcal O}}_{X})^{\oplus a})^{(m)}\longrightarrow {{\rm Sym}}({{\widetilde{{{\mathcal T}}}}}_{X,k})^{(m)}.$$ The target of this map equals ${\rm gr}\left({{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}\right)$ according to \[graded\_tcD\]. It therefore suffices to prove the following: given a coherent ${{\mathcal C}}$-module ${{\mathcal E}}$, there is a number $r_0$ such that for all $r\geq r_0$ and $i\geq 1$, one has $H^{i}(X,{{\mathcal E}}(r))=0$. To start with, $X$ is noetherian, hence ${{\mathcal E}}$ is quasi-coherent and so equals the union over its ${{\mathcal O}}_X$-coherent submodules ${{\mathcal E}}_i$. Since ${{\mathcal E}}$ is ${{\mathcal C}}$-coherent and ${{\mathcal C}}$ has noetherian sections over open affines [@Huyghe97 1.3.6], there is a ${{\mathcal C}}$-linear surjection ${{\mathcal C}}\otimes_{{{\mathcal O}}_{X}} {{\mathcal E}}_i\rightarrow {{\mathcal E}}$. Choose a number $s_0$ such that ${{\mathcal E}}_i(-s_0)$ is generated by global sections. We obtain a ${{\mathcal O}}_{X}$-linear surjection ${{\mathcal O}}_{X}(s_0)^{\oplus a_0}\rightarrow {{\mathcal E}}_i$ for a number $a_0$. This yields a ${{\mathcal C}}$-linear surjection $${{\mathcal C}}_0:={{\mathcal C}}(s_0)^{\oplus a_0}\longrightarrow {{\mathcal E}}.$$ The ${{\mathcal O}}_X$-module ${{\mathcal C}}_0$ is graded and each homogeneous component equals a sum of copies of ${{\mathcal O}}_X(s_0)$. It follows that $H^{i}(X,{{\mathcal C}}_0(r)=0$ for all $r\geq u_0-s_0$ and all $i\geq 1$. The rest of the argument proceeds now as in [@NootHuyghe09 2.2.1]. \[vanishing\_coh\_Dnk\] Let $r_0$ be the number occuring in the preceding proposition. For all $r \ge r_0$ and all $i \ge 1$ one has \[vanishing\_coh\_Dnk\_disp\] H\^i(X,\_[X,k]{}\^[(m)]{}(r)) = 0 . [[*Proof.* ]{}]{}For $d\geq 0$ we let ${{\mathcal F}}_d = {{\widetilde{{{\mathcal D}}}}}_{X,k;d}^{(m)}$. We consider the exact sequence \[ex\_fil\_seq\_1\] 0 [[F]{}]{}\_[d-1]{} [[F]{}]{}\_d [gr]{}\_d(\_[X,k]{}\^[(m)]{}) 0 (where ${{\mathcal F}}_{-1}:=0$) from which we deduce the exact sequence \[ex\_fil\_seq\_2\] 0 [[F]{}]{}\_[d-1]{}(r) [[F]{}]{}\_d(r) [gr]{}\_d(\_[X,k]{}\^[(m)]{})(r) 0 because tensoring with a line bundle is an exact functor. Since cohomology commutes with direct sums, we have for all $r\geq r_0$ and $i\geq 1$ that $$H^{i}(X,{\rm gr}_d\left({{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}\right)(r))=0$$ according to the preceding proposition. Using the sequence \[ex\_fil\_seq\_2\] we can then deduce by induction on $d$ that for all $r\geq r_0$ and $i\geq 1$ $$H^i(X, {{\mathcal F}}_d(r)) = 0 \;.$$ Because cohomology commutes with inductive limits on a noetherian scheme we obtain the asserted vanishing result. \[surjection\] Let ${{\mathcal E}}$ be a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module. \(i) There is a number $r=r({{\mathcal E}}) \in {{\mathbb Z}}$ and $s \in {{\mathbb Z}}_{\ge 0}$ and an epimorphism of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-modules $$\Big({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r)\Big)^{\oplus s} \twoheadrightarrow {{\mathcal E}}\;.$$ \(ii) There is $r_1({{\mathcal E}}) \in {{\mathbb Z}}$ such that for all $r \ge r_1({{\mathcal E}})$ and all $i >0$ $$H^i\Big(X, {{\mathcal E}}(r)\Big) = 0 \;.$$ [[*Proof.* ]{}]{}(i) As $X$ is a noetherian scheme, ${{\mathcal E}}$ is the inductive limit of its coherent subsheaves. There is thus a coherent ${{\mathcal O}}_{X}$-submodule ${{\mathcal F}}{\subset}{{\mathcal E}}$ which generates ${{\mathcal E}}$ as a ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module, i.e., an epimorphism of sheaves $${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k} \otimes_{{{\mathcal O}}_{X}} {{\mathcal F}}\stackrel{\alpha}{{\longrightarrow}} {{\mathcal E}}\;,$$ where ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ is considered with its right ${{\mathcal O}}_{X}$-module structure. Next, there is $r>0$ such that the sheaf $${{\mathcal F}}(r) = {{\mathcal F}}\otimes_{{{\mathcal O}}_{X}} {{\mathcal L}}_X^{\otimes r}$$ is generated by its global sections. Hence there is $s > 0$ and an epimorphism ${{\mathcal O}}_{X}^{\oplus s} \twoheadrightarrow {{\mathcal F}}(r)$, and thus an epimorphism of ${{\mathcal O}}_{X}$-modules $$\left({{\mathcal O}}_{X}(-r)\right)^{\oplus s} \twoheadrightarrow {{\mathcal F}}\;.$$ From this morphism we get an epimorphism of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-modules $$\left({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r)\right)^{\oplus s} = {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k} \otimes_{{{\mathcal O}}_{X_n}} \left({{\mathcal O}}_{X}(-r)\right)^{\oplus s} \twoheadrightarrow {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k} \otimes_{{{\mathcal O}}_{X}} {{\mathcal F}}\stackrel{\alpha}{{\longrightarrow}} {{\mathcal E}}\;.$$ \(ii) Consider for $i\geq 1$ the following assertion $(a_i)$: for any coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module ${{\mathcal E}}$, there is a number $r_i({{\mathcal E}})$ such that for all $r\geq r_i({{\mathcal E}})$ and all $i\leq j$ one has $H^{j}(X,{{\mathcal E}}(r))=0$. For $i>\dim X$ the assertion holds, cf. \[van\_coh\]. Suppose the statement $(a_{i+1})$ holds. Using (i) we find an epimorphism of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-modules $$\beta: {{\mathcal C}}_0:=\Big({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(s_0)\Big)^{\oplus s} \twoheadrightarrow {{\mathcal E}}$$ for numbers $s_0\in{{\mathbb Z}}$ and $s\geq 0$. By \[graded\_tcD\], the kernel ${{\mathcal R}}= \ker(\beta)$ is a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module. Recall the number $r_0$ of the preceding corollary. For any $r\geq \max (r_0-s_0, r_{i+1}({{\mathcal R}}))$ we have the exact sequence $$0=H^{i}(X,{{\mathcal C}}_0(r))\longrightarrow H^{i}(X,{{\mathcal E}}(r))\longrightarrow H^{i+1}(X, {{\mathcal R}}(r))=0$$ which shows $H^{i}(X,{{\mathcal E}}(r))=0$ for these $r$. So we may take as $r_i({{\mathcal E}})$ any of these $r$ which is larger than $r_{i+1}({{\mathcal E}})$ and obtain the statement $(a_i)$. In particular, $(a_1)$ holds which proves (ii). \[finite\_power\] \(i) Fix $r \in {{\mathbb Z}}$. There is $c_2 = c_2(r) \in {{\mathbb Z}}_{\ge 0}$ such that for all $i>0$ the cohomology group $H^i(X,{{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r))$ is annihilated by $p^{c_2}$. \(ii) Let ${{\mathcal E}}$ be a coherent ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}$-module. There is $c_3 = c_3({{\mathcal E}}) \in {{\mathbb Z}}_{\ge 0}$ such that for all $i>0$ the cohomology group $H^i(X,{{\mathcal E}})$ is annihilated by $p^{c_3}$. [[*Proof.* ]{}]{}(i) Since the blow-up morphism ${\rm pr}: X \rightarrow X_0$ becomes an isomorphism over $X_0\times_{{{\mathfrak o}}} L$ any coherent module over ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}\otimes {{\mathbb Q}}$ induces a coherent module over the sheaf of usual differential operators on $X_0\times_{{\mathfrak o}}L$. By [@BB81] we conclude that the global section functor on $X$ is exact for coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}\otimes{{\mathbb Q}}$-modules. In particular, the cohomology group $H^i(X,{{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r))$ is $p$-torsion. To see that the torsion is bounded, we deduce from \[prop-auxiliaryI\] that ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r)$ is a coherent module over ${{\mathcal A}}^{(m)}_{X,k}$. According to \[prop-auxiliaryII\], $H^i(X,{{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r))$ is therefore finitely generated over $D^{(m)}({{\mathbb G}}(k))$. This implies the claim. \(ii) We consider for any $i\geq 1$ the following assertion $(a_i)$: for any coherent ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}$-module ${{\mathcal E}}$, there is a number $r_i({{\mathcal E}})$ such that the groups $H^{j}(X,{{\mathcal E}}), i\leq j$ are all annihilated by $p^{r_i({{\mathcal E}})}$. For $i>\dim X$ the assertion holds, cf. \[van\_coh\]. Let us assume $(a_{i+1})$ holds and consider an arbitrary coherent ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}$-module ${{\mathcal E}}$. Acccording to \[surjection\] we have a ${{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}$-linear surjection $${{\mathcal E}}_0:={{\widetilde{{{\mathcal D}}}}}_{X,k}^{(m)}(r)^{\oplus s}\longrightarrow {{\mathcal E}}$$ for numbers $r\in{{\mathbb Z}}$ and $s\geq 0$. Let ${{\mathcal E}}'$ be the kernel. We have an exact sequence $$H^{i}(X,{{\mathcal E}}_0)\stackrel{\iota}{\rightarrow}H^{i}(X,{{\mathcal E}})\stackrel{\delta}{\rightarrow}H^{i+1}(X,{{\mathcal E}}').$$ Then $p^{c_2(r)}$ annihilates the image of $\iota$ according to (i) and $p^{r_{i+1}({{\mathcal E}}')}$ annihilates the image of $\delta$ according to $(a_{i+1})$. So we may take as $r_i({{\mathcal E}})$ any number greater than the maximum of $r_{i+1}({{\mathcal E}})$ and $c_2(r)+r_{i+1}({{\mathcal E}}')$ and obtain the statement $(a_i)$. In particuar, $(a_1)$ holds which proves (ii). Cohomology of coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}}, k,{{\mathbb Q}}}$-modules {#coh_coh_tsD_mod} -------------------------------------------------------------------------------------------------------- We denote by $X_{j}$ the reduction of $X$ modulo $p^{j+1}$. \[completion\] Let ${{\mathcal E}}$ be a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module on $X$ and $\widehat{{{\mathcal E}}} = \varprojlim_j {{\mathcal E}}/p^{j+1}{{\mathcal E}}$ its $p$-adic completion, which we consider as a sheaf on ${{\mathfrak X}}$. \(i) For all $i \ge 0$ one has $H^i({{\mathfrak X}},\widehat{{{\mathcal E}}}) = \varprojlim_j H^i\left(X_{j},{{\mathcal E}}/p^{j+1}{{\mathcal E}}\right)$. \(ii) For all $i>0$ one has $H^i({{\mathfrak X}},\widehat{{{\mathcal E}}}) = H^i(X,{{\mathcal E}})$. \(iii) $H^0({{\mathfrak X}},\widehat{{{\mathcal E}}}) = \varprojlim_j H^0(X,{{\mathcal E}})/p^{j+1}H^0(X,{{\mathcal E}})$. [[*Proof.* ]{}]{}Put ${{\mathcal E}}_j = {{\mathcal E}}/p^{j+1}{{\mathcal E}}$. Let ${{\mathcal E}}_t$ be the subsheaf defined by $${{\mathcal E}}_t(U) = {{\mathcal E}}(U)_{\rm tor} \;,$$ where the right hand side denotes the group of torsion elements in ${{\mathcal E}}(U)$. This is indeed a sheaf (and not only a presheaf) because $X$ is a noetherian space. Furthermore, ${{\mathcal E}}_t$ is a ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-submodule of ${{\mathcal E}}$. Because the sheaf ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$ has noetherian rings of sections over open affine subsets of $X$, cf. \[graded\_tcD\], the submodule ${{\mathcal E}}_t$ is a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module. ${{\mathcal E}}_t$ is thus generated by a coherent ${{\mathcal O}}_{X}$-submodule ${{\mathcal F}}$ of ${{\mathcal E}}_t$. The submodule ${{\mathcal F}}$ is annihilated by a fixed power $p^c$ of $p$, and so is ${{\mathcal E}}_t$. Put ${{\mathcal G}}= {{\mathcal E}}/{{\mathcal E}}_t$, which is again a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module. Using \[finite\_power\], we can then assume, after possibly replacing $c$ by a larger number, that $$\begin{array}{cl} (a) & p^c{{\mathcal E}}_t = 0 \;,\\ (b) & \mbox{for all } i>0: p^cH^i(X,{{\mathcal E}}) = 0 \;,\\ (c) & \mbox{for all } i>0: p^cH^i(X,{{\mathcal G}}) = 0 \;.\\ \end{array}$$ From here on the proof of the proposition is exactly as in [@PSS4 4.2.1]. \[surj\_compl\] Let ${{\mathscr E}}$ be a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-module. \(i) There is $r_1({{\mathscr E}}) \in {{\mathbb Z}}$ such that for all $r \ge r_1({{\mathscr E}})$ there is $s \in {{\mathbb Z}}_{\ge 0}$ and an epimorphism of ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-modules $$\Big({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}(-r)\Big)^{\oplus s} \twoheadrightarrow {{\mathscr E}}\;.$$ \(ii) There is $r_2({{\mathscr E}}) \in {{\mathbb Z}}$ such that for all $r \ge r_2({{\mathscr E}})$ and all $i >0$ $$H^i\Big({{\mathfrak X}}, {{\mathscr E}}(r)\Big) = 0 \;.$$ [[*Proof.* ]{}]{}(i) Because ${{\mathscr E}}$ is a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-module, and because $H^0(U,{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})$ is a noetherian ring for all open affine subsets $U {\subset}{{\mathfrak X}}$, cf. \[global\_sec\_tsD\], the torsion submodule ${{\mathscr E}}_t {\subset}{{\mathscr E}}$ is again a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-module. As ${{\mathfrak X}}$ is quasi-compact, there is $c \in {{\mathbb Z}}_{\ge 0}$ such that $p^c {{\mathscr E}}_t = 0$. Put ${{\mathscr G}}= {{\mathscr E}}/{{\mathscr E}}_t$ and ${{\mathscr G}}_0 = {{\mathscr G}}/p{{\mathscr G}}$. For $j \ge c$ one has an exact sequence $$0 {\rightarrow}{{\mathscr G}}_0 \stackrel{p^{j+1}}{{\longrightarrow}} {{\mathscr E}}_{j+1} {\rightarrow}{{\mathscr E}}_j {\rightarrow}0 \;.$$ We note that the sheaf ${{\mathscr G}}_0$ is a coherent module over ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}/p{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$. We view ${{\mathfrak X}}$ as a closed subset of $X$ and denote the closed embedding temporarily by $i$. Because the canonical map of sheaves of rings \[iso\_uncompl\_compl\] \^[(m)]{}\_[X,k]{}/p\^[(m)]{}\_[X,k]{} i\_\*(\^[(m)]{}\_[[[X]{}]{},k]{}/p\^[(m)]{}\_[[[X]{}]{},k]{}) is an isomorphism, $i_*{{\mathscr G}}_0$ can be considered a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module via this isomorphism. Hence we can apply \[surjection\] to $i_*{{\mathscr G}}_0$ and deduce that there is $r_2({{\mathscr G}}_0)$ such that for all $r \ge r_2({{\mathscr G}}_0)$ one has $$H^1({{\mathfrak X}},{{\mathscr G}}_0(r))= H^1(X,i_*{{\mathscr G}}_0(r))=0.$$ The canonical maps \[surj\_H0\] H\^0([[X]{}]{},[[E]{}]{}\_[j+1]{}(r)) H\^0([[X]{}]{},[[E]{}]{}\_j(r)) are thus surjective for $r \ge r_2({{\mathscr G}}_0)$ and $j \ge c$. Similarly, ${{\mathscr E}}_c$ is a coherent module over ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}/p^c{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module, in particular a coherent ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-module. By \[surjection\] there is $r_1({{\mathscr E}}_c)$ such that for every $r \ge r_1({{\mathscr E}}_c)$ there is $s \in {{\mathbb Z}}_{\ge 0}$ and a surjection $$\lambda: \Big({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}/p^c{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}\Big)^{\oplus s} \twoheadrightarrow {{\mathscr E}}_c(r) \;.$$ Let $r_1({{\mathscr E}}) = \max\{r_2({{\mathscr G}}_0),r_1({{\mathscr E}}_c)\}$, and assume from now on that $r \ge r_1({{\mathscr E}})$. Let $e_1, \ldots, e_s$ be the standard basis of the domain of $\lambda$, and use \[surj\_H0\] to lift each $\lambda(e_t)$, $1 \le t \le s$, to an element of $$\varprojlim_j H^0({{\mathfrak X}},{{\mathscr E}}_j(r)) \simeq H^0({{\mathfrak X}},\widehat{{{\mathscr E}}(r)}) \;,$$ by \[completion\] (i). But $\widehat{{{\mathscr E}}(r)} = \widehat{{{\mathscr E}}}(r)$, and $\widehat{{{\mathscr E}}} = {{\mathscr E}}$, as follows from [@BerthelotDI 3.2.3 (v)]. This defines a morphism $$\Big({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}\Big)^{\oplus s} {\longrightarrow}{{\mathscr E}}(r)$$ which is surjective because, modulo $p^c$, it is a surjective morphism of sheaves coming from coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{X,k}$-modules by reduction modulo $p^c$, cf. [@BerthelotDI 3.2.2 (ii)]. \(ii) We deduce from \[vanishing\_coh\_Dnk\] and \[completion\] that for all $i>0$ $$H^i\Big({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}(r)\Big) = 0 \;,$$ whenever $r \ge r_0$, where $r_0$ is as in \[v\_ample\_sh\_lemma\]. Since the sheaf ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$ is coherent, cf. \[global\_sec\_tsD\], and ${{\mathfrak X}}$ is a noetherian space of finite dimension, the statement in (ii) can now be deduced by descending induction on $i$ exactly as in the proof of part (ii) of \[surjection\]. \[compl\_finite\_power\_torsion\] Let ${{\mathscr E}}$ be a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-module. \(i) There is $c=c({{\mathscr E}}) \in {{\mathbb Z}}_{\ge 0}$ such that for all $i>0$ the cohomology group $H^i({{\mathfrak X}},{{\mathscr E}})$ is annihilated by $p^c$. \(ii) $H^0({{\mathfrak X}},{{\mathscr E}}) = \varprojlim_j H^0({{\mathfrak X}},{{\mathscr E}})/p^jH^0({{\mathfrak X}},{{\mathscr E}})$. [[*Proof.* ]{}]{}(i) Let $r\in{{\mathbb Z}}$. By \[completion\] we have for $i>0$ that $$H^{i}({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}(-r)) = H^{i}(X,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r)) \;,$$ and this is annihilated by a finite power of $p$, by \[finite\_power\]. The proof now proceeds by descending induction exactly as in the proof of part (ii) of \[finite\_power\]. \(ii) Let ${{\mathscr E}}_t {\subset}{{\mathscr E}}$ be the subsheaf of torsion elements and ${{\mathscr G}}= {{\mathscr E}}/{{\mathscr E}}_t$. Then the discussion in the beginning of the proof of \[completion\] shows that there is $c \in {{\mathbb Z}}_{\ge 0}$ such that $p^c{{\mathscr E}}_t = 0$. Part (i) gives that $p^cH^1({{\mathfrak X}},{{\mathscr E}}) = p^cH^1({{\mathfrak X}},{{\mathscr G}}) = 0$, after possibly increasing $c$. Now we can apply the same reasoning as in the proof of \[completion\] (iii) to conclude that assertion (ii) is true. Let ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})$ (resp. ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})$) be the category of coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-modules (resp. ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules). Let ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})_{{\mathbb Q}}$ be the category of coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-modules up to isogeny. We recall that this means that ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})_{{\mathbb Q}}$ has the same class of objects as ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})$, and for any two objects ${{\mathcal M}}$ and ${{\mathcal N}}$ one has $$Hom_{{{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})_{{\mathbb Q}}}({{\mathcal M}},{{\mathcal N}}) = Hom_{{{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})}({{\mathcal M}},{{\mathcal N}}) \otimes_{{\mathbb Z}}{{\mathbb Q}}\;.$$ \[integral\_models\] (i) The functor ${{\mathcal M}}\mapsto {{\mathcal M}}_{{\mathbb Q}}= {{\mathcal M}}\otimes_{{\mathbb Z}}{{\mathbb Q}}$ induces an equivalence between ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k})_{{\mathbb Q}}$ and ${{\rm Coh}}({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})$. \(ii) For every coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module ${{\mathscr M}}$ there is $m \ge 0$ and a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module ${{\mathscr M}}_m$ and an isomorphism of ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules $${{\varepsilon}}: {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr M}}_m \stackrel{\simeq}{{\longrightarrow}} {{\mathscr M}}\;.$$ If $(m', {{\mathscr M}}_{m'},{{\varepsilon}}')$ is another such triple, then there is $\ell \ge \max\{m,m'\}$ and an isomorphism of ${{\widetilde{{{\mathscr D}}}}}^{(\ell)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules $${{\varepsilon}}_\ell: {{\widetilde{{{\mathscr D}}}}}^{(\ell)}_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathcal D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr M}}_m \stackrel{\simeq}{{\longrightarrow}} {{\widetilde{{{\mathscr D}}}}}^{(\ell)}_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathcal D}}}}}^{(m')}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr M}}_{m'}$$ such that ${{\varepsilon}}' \circ \Big({\rm id}_{{{\widetilde{{{\mathcal D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}} \otimes {{\varepsilon}}_\ell\Big) = {{\varepsilon}}$. [[*Proof.* ]{}]{}(i) This is [@BerthelotDI 3.4.5]. Note that the sheaf ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$ satisfies the conditions in [@BerthelotDI 3.4.1], by \[global\_sec\_tsD\]. We point out that the formal scheme ${{\mathcal X}}$ in [@BerthelotDI sec. 3.4] is not supposed to be smooth over a discrete valuation ring, but only locally noetherian, cf. [@BerthelotDI sec. 3.3]. \(ii) This is [@BerthelotDI 3.6.2]. In this reference the formal scheme is supposed to be noetherian and quasi-separated, but not necessarily smooth over a discrete valuation ring. \[acycl\_tsD\] Let ${{\mathscr E}}$ be a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module (resp. ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module). \(i) There is $r({{\mathscr E}}) \in {{\mathbb Z}}$ such that for all $r \ge r({{\mathscr E}})$ there is $s \in {{\mathbb Z}}_{\ge 0}$ and an epimorphism of ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules (resp. ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules) $$\Big({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}(-r)\Big)^{\oplus s} \twoheadrightarrow {{\mathscr E}}\; \hskip16pt (\; \mbox{resp.} \; \Big({{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}(-r)\Big)^{\oplus s} \twoheadrightarrow {{\mathscr E}}\;) \;.$$ \(ii) For all $i >0$ one has $H^i({{\mathfrak X}}, {{\mathscr E}}) = 0$. [[*Proof.* ]{}]{}(a) We first show both assertions (i) and (ii) for a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module ${{\mathscr E}}$. By \[integral\_models\] (i) there is a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}$-module ${{\mathscr F}}$ such that ${{\mathscr F}}\otimes_{{\mathbb Z}}{{\mathbb Q}}= {{\mathscr E}}$. We use \[surj\_compl\] to find for every $r \ge r_1({{\mathscr F}})$ a surjection $$\Big({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k}(-r)\Big)^{\oplus s} \twoheadrightarrow {{\mathscr F}}\;,$$ for some $s$ (depending on $r$). Tensoring with ${{\mathbb Q}}$ gives then the desired surjection onto ${{\mathscr E}}$. Hence assertion (i). Furthermore, for $i>0$ $$H^i({{\mathfrak X}},{{\mathscr E}}) = H^i({{\mathfrak X}},{{\mathscr F}}) \otimes_{{\mathbb Z}}{{\mathbb Q}}= 0 \;,$$ by \[compl\_finite\_power\_torsion\], and this proves (ii). \(b) Now suppose ${{\mathscr E}}$ is a coherent ${{\widetilde{{{\mathcal D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module. By \[integral\_models\] (ii) there is $m \ge 0$ and a coherent module ${{\mathscr E}}_m$ over ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$ and an isomorphism of ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules $${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr E}}_m \stackrel{\simeq}{{\longrightarrow}} {{\mathscr E}}\;.$$ Now use what we have just shown for ${{\mathscr E}}_m$ in (a) and get the sought for surjection after tensoring with ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$. This proves the first assertion. We have $${{\mathscr E}}= {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr E}}_m = \varinjlim_{\ell \ge m} {{\widetilde{{{\mathscr D}}}}}^{(\ell)}_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr E}}_m = \varinjlim_{\ell \ge m} {{\mathscr E}}_\ell$$ where ${{\mathscr E}}_\ell = {{\widetilde{{{\mathscr D}}}}}^{(\ell)}_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}} {{\mathscr E}}_m$ is a coherent ${{\widetilde{{{\mathscr D}}}}}^{(\ell)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module. Then we have for $i>0$ $$H^i({{\mathfrak X}},{{\mathscr E}}) = \varinjlim_{\ell \ge m} H^i({{\mathfrak X}},{{\mathscr E}}_\ell) = 0 \;,$$ by part (a). And this proves assertion (ii). ${{\mathfrak X}}$ is ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-affine and ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-affine ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ \[prop-genglobal\] (i) Let ${{\mathscr E}}$ be a coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module. Then ${{\mathscr E}}$ is generated by its global sections as ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module. Furthermore, ${{\mathscr E}}$ has a resolution by finite free ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules. \(ii) Let ${{\mathscr E}}$ be a coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module. Then ${{\mathscr E}}$ is generated by its global sections as ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module. $H^0({{\mathfrak X}},{{\mathscr E}})$ is a $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$-module of finite presentation. Furthermore, ${{\mathscr E}}$ has a resolution by finite free ${{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules. [[*Proof.* ]{}]{}(i) Using \[acycl\_tsD\] it remains to see that any ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module of type ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}(-r)$ admits a linear surjection $({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})^{\oplus s}\rightarrow{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}(-r)$ for suitable $s\geq 0$. We argue as in [@Huyghe97 5.1]. Let $M:=H^0(X,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r))$, a finitely generated $D^{(m)}({{\mathbb G}}(k))$-module by \[prop-auxiliaryII\]. Consider the linear map of ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}$-modules equal to the composite $${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}\otimes_{D^{(m)}({{\mathbb G}}(k))} M \rightarrow {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}\otimes_{H^0(X,{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k})} M\rightarrow {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r)$$ where the first map is the surjection induced by the map $Q^{(m)}_{X,k}$ appearing in \[global\_sections\_tcD\]. Let ${{\mathcal E}}$ be the cokernel of the composite map. Since $D^{(m)}({{\mathbb G}}(k))$ is noetherian, the source of the map is coherent and hence ${{\mathcal E}}$ is coherent. Moreover, ${{\mathcal E}}\otimes{{\mathbb Q}}=0$ since ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r)\otimes{{\mathbb Q}}$ is generated by global sections [@BB81]. All in all, there is $i$ with $p^{i}{{\mathcal E}}=0$. Now choose a linear surjection $(D^{(m)}({{\mathbb G}}(k)))^{\oplus s}\rightarrow M$. We obtain the exact sequence of coherent modules $$({{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k})^{\oplus s}\rightarrow {{\widetilde{{{\mathcal D}}}}}^{(m)}_{X,k}(-r)\rightarrow {{\mathcal E}}\rightarrow 0.$$ Passing to $p$-adic completions (which is exact in our situation [@BerthelotDI 3.2]) and inverting $p$ yields the linear surjection $$({{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})^{\oplus s}\rightarrow {{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}(-r).$$ This shows (i). \(ii) This follows from (i) exactly as in [@Huyghe97]. [*The functors ${{\mathscr Loc}}^{(m)}_{{{\mathfrak X}},k}$ and ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}$.*]{} Let $E$ be a finitely generated $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})$-module (resp. a finitely presented $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$-module). Then we let ${{\mathscr Loc}}^{(m)}_{{{\mathfrak X}},k}(E)$ (resp. ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}(E)$) be the sheaf on ${{\mathfrak X}}$ associated to the presheaf $$U \mapsto {{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}(U) \otimes_{H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{n,k,{{\mathbb Q}}})} E \hskip16pt ({\rm resp.} \;\; U \mapsto {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}(U) \otimes_{H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})} E \;) \;.$$ It is obvious that ${{\mathscr Loc}}^{(m)}_{{{\mathfrak X}},k}$ (resp. ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}$) is a functor from the category of finitely generated $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})$-modules (resp. finitely presented $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$-modules) to the category of sheaves of modules over ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$ (resp. ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$). \[thm-equivalence\] (i) The functors ${{\mathscr Loc}}^{(m)}_{{{\mathfrak X}},k}$ and $H^0$ (resp. ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}$ and $H^0$) are quasi-inverse equivalences between the categories of finitely generated $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}})$-modules and coherent ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules (resp. finitely presented $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$-modules and coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules). \(ii) The functor ${{\mathscr Loc}}^{(m)}_{{{\mathfrak X}},k}$ (resp. ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}$) is an exact functor. [[*Proof.* ]{}]{}The proofs of [@Huyghe97 5.2.1, 5.2.3] for the first and the second assertion, respectively, carry over word for word. Localization of representations of ${{\mathbb G}}(L)$ {#loc} ===================================================== Modules over ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}$ ------------------------------------------------------------------------- Let as before ${{\mathfrak X}}$ be the $p$-adic completion of an admissible blow up $X$ of $X_0$. We recall that the algebra $H^0({{\mathfrak X}},{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}})$ is canonically isomorphic to the coherent $L$-algebra ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}$, cf. \[global\_sec\_tsD\]. According to [@PSS4 Lem. 5.2.1] the algebras ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)$ and ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}$ are compact type algebras with noetherian defining Banach algebras in the sense of loc.cit. In the following we extend some notions appearing in [@PSS4 5.2] to the more general situation considered here. We consider the locally $L$-analytic compact group ${G_0}={\rm {{\mathbb G}}}_0({{\mathfrak o}})$ with its series of congruence subgroups $G_{k+1}={{\mathbb G}}(k)^\circ(L)$. The group ${G_0}$ acts by translations on the space $C^{\rm cts}({G_0},K)$ of continuous $K$-valued functions. Following [@EmertonA (5.3)] let ${D({{\mathbb G}}(k)^\circ,{G_0})}$ be the strong dual of the space of ${{\mathbb G}}(n)^\circ$-analytic vectors $${D({{\mathbb G}}(k)^\circ,{G_0})}:= (C^{\rm cts}({G_0},K)_{{{\mathbb G}}(k)^\circ-\rm an})'_b \;.$$ It is a locally convex topological $L$-algebra naturally isomorphic to the crossed product of the ring ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)$ with the finite group ${G_0}/G_{k+1}$. In particular, \[equ-finitefree\][D([[G]{}]{}(k)\^,[G\_0]{})]{}=\_[g[G\_0]{}/G\_[k+1]{}]{} [[D]{}]{}\^[an]{}([[G]{}]{}(k)\^)\*\_g is a finitely generated free topological module over ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)$. Denoting by $C^{\rm la}({G_0},K)$ the space of $K$-valued locally analytic functions and dualizing the isomorphism $$\varinjlim_k C^{\rm cts}({G_0},K)_{{{\mathbb G}}(k)^\circ-\rm an}{\stackrel{\simeq}{\longrightarrow}}C^{\rm la}({G_0},K)$$ yields an isomorphism of topological algebras $$D({G_0}){\stackrel{\simeq}{\longrightarrow}}\varprojlim_k {D({{\mathbb G}}(k)^\circ,{G_0})}\;.$$ This is the weak Fréchet-Stein structure on the locally analytic distribution algebra $D({G_0})$ as introduced by Emerton in [@EmertonA Prop. 5.3.1]. In an obviously similar manner, we may construct the ring ${D({{\mathbb G}}(k)^\circ,{G_0})_{\theta_0}}$ and obtain an isomorphism ${D({G_0})_{\theta_0}}{\stackrel{\simeq}{\longrightarrow}}\varprojlim_k {D({{\mathbb G}}(k)^\circ,{G_0})_{\theta_0}}.$ We consider an admissible locally analytic ${G_0}$-representation $V$, its coadmissible module $M:=V'_b$ and its subspace of ${{{\mathbb G}}(k)^\circ}$-analytic vectors $V_{{{{\mathbb G}}(k)^\circ}-\rm an}\subseteq V$. The latter subspace is naturally a nuclear Fréchet space [@EmertonA Lem. 6.1.6] and we let $(V_{{{{\mathbb G}}(k)^\circ}-\rm an})'_b$ be its strong dual. It is a space of compact type and a topological ${D({{\mathbb G}}(k)^\circ,{G_0})}$-module which is finitely generated [@EmertonA Lem. 6.1.13]. According to [@EmertonA Thm. 6.1.20] the modules $M_k:= (V_{{{{\mathbb G}}(k)^\circ}-\rm an})'$ form a $({D({{\mathbb G}}(k)^\circ,{G_0})})_{k\in{{\mathbb N}}}$-sequence, in the sense of [@EmertonA Def. 1.3.8], for the coadmissible module $M$ relative to the weak Fréchet-Stein structure on $D({G_0}).$ This implies that one has \[equ-weakfamily\] M\_k=[D([[G]{}]{}(k)\^,[G\_0]{})]{}\_[D([G\_0]{})]{} M as ${D({{\mathbb G}}(k)^\circ,{G_0})}$-modules for any $k$. Here, the completed tensor product is understood in the sense of [@EmertonA Lem. 1.2.3], as in [@PSS4]. \[lem-refine\] (i) The ${D({{\mathbb G}}(k)^\circ,{G_0})}$-module $M_k$ is finitely presented. \(ii) There are natural isomorphisms $$D({{\mathbb G}}(k-1)^\circ,{G_0})\otimes_{{D({{\mathbb G}}(k)^\circ,{G_0})}} M_k{\stackrel{\simeq}{\longrightarrow}}M_{k-1}.$$ [[*Proof.* ]{}]{}This can be proved exactly as [@PSS4 Lem. 5.2.4]. Remark: These results have obvious analogues when the character $\theta_0$ is involved. $G_0$-equivariance and the functor ${{\mathscr Loc}}_{\infty}^\dagger$ {#subsec_G0} ---------------------------------------------------------------------- A $p$-adic completion ${{\mathfrak X}}$ of an admissible blow-up $X$ of $X_0$ will be called an [*admissible formal blow-up*]{} of ${{\mathfrak X}}_0$. We note here that any formal scheme ${{\mathfrak X}}$ which is obtained from ${{\mathfrak X}}_0$ by blowing-up a coherent open ideal on ${{\mathfrak X}}_0$ is an admissible blow-up in this sense. Indeed, if ${{\mathcal I}}\subset{{\mathfrak X}}_0$ is the ideal which is blown-up, then ${{\mathcal I}}\cap{{\mathcal O}}_{X_0}$ is a coherent open ideal on $X_0$. Blowing-up this ideal on $X_0$ and completing $p$-adically gives back ${{\mathfrak X}}$. If the coherent open ideal ${{\mathcal I}}\subset X_0$ that is blown-up is $G_0$-stable, then there is an induced $G_0$-action on $X$ and ${{\mathfrak X}}$. In this case, we will say that $X$ and ${{\mathfrak X}}$ are $G_0$-stable. In the following we suppose that $k$ is large enough for ${{\mathfrak X}}$, so that the sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ is defined on ${{\mathfrak X}}$. \[prop-exactdirectimage\] Let $\pi:{{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ be a morphism between admissible formal blow-ups of ${{\mathfrak X}}_0$ which is an isomorphism on corresponding rigid analytic spaces. If ${{\mathscr M}}'$ is a coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}$-module, then $R^j\pi_*{{\mathscr M}}'=0$ for $j>0$. Moreover, $\pi_*{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}={{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}$, so that $\pi_*$ induces an exact functor between coherent modules over ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}$ and ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k',{{\mathbb Q}}}$ respectively. [[*Proof.* ]{}]{}We denote the associated rigid analytic space of ${{\mathfrak X}}$ by ${{\mathfrak X}}_{{\mathbb Q}}$. Coherent modules over ${{\mathfrak X}}_{{\mathbb Q}}$ are equivalent to coherent modules over ${{\mathcal O}}_{{{\mathfrak X}},{{\mathbb Q}}}$ [@BerthelotDI 4.1.3] and similarly for ${{\mathfrak X}}'$. This implies $R^j\pi_*{{\mathcal O}}_{{{\mathfrak X}}',{{\mathbb Q}}}=0$ for $j>0$ and $\pi_*{{\mathcal O}}_{{{\mathfrak X}}',{{\mathbb Q}}}={{\mathcal O}}_{{{\mathfrak X}},{{\mathbb Q}}}$. In particular, there is $N\geq 0$ such that $p^NR^j\pi_*{{\mathcal O}}_{{{\mathfrak X}}'}=0$ for $j>0$ and such that the kernel and cokernel of the natural map ${{\mathcal O}}_{{{\mathfrak X}}}\rightarrow\pi_*{{\mathcal O}}_{{{\mathfrak X}}'}$ are killed by $p^N$. For any $i\geq 0$, let $X_i$ be the reduction of ${{\mathfrak X}}$ mod $p^{i+1}$ and similarly for ${{\mathfrak X}}'$ and denote by $\pi_i: X'_i\rightarrow X_i$ the morphism induced by $\pi$. For any $i$ we have then $p^NR^j\pi_{i*}{{\mathcal O}}_{X_i}=0$ for $j>0$ and such that the kernel and cokernel of the natural map ${{\mathcal O}}_{X_i}\rightarrow\pi_{i*}{{\mathcal O}}_{X_i'}$ are killed by $p^N$. We now prove the assertions. According to \[prop-genglobal\] the module ${{\mathscr M}}'$ is generated by global sections. Induction on $j$ reduces us therefore to the case ${{\mathscr M}}={{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}$. Since $R^j\pi_*$ commutes with inductive limits, it suffices to prove the claim for ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}}',k',{{\mathbb Q}}}$. Abbreviate ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_i,k'}$ for ${{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k'}/p^{i+1}{{\widetilde{{{\mathscr D}}}}}^{(m)}_{{{\mathfrak X}},k'}$ and similarly for ${{\mathfrak X}}'$. Then ${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X'_i,k'}=\pi_i^*{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_i,k'}$ by \[graded\_tcD\] and the projection formula implies $$R^j\pi_{i*}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X'_i,k'}={{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_i,k'}\otimes R^j\pi_{i*}{{\mathcal O}}_{X'_i}.$$ We see for any $i\geq0$ that $$p^N R^j\pi_{i*}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X'_i,k'}=0$$ for all $j>0$ and that the natural map $${{\widetilde{{{\mathcal D}}}}}^{(m)}_{X_i,k'}\rightarrow \pi_{i*}{{\widetilde{{{\mathcal D}}}}}^{(m)}_{X'_i,k'}$$ has kernel and cokernel killed by $p^N$. Taking inverse limits over $i$, arguing as in \[completion\] and finally inverting $p$ yields the first claim and $\pi_*{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}={{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k',{{\mathbb Q}}}$. If ${{\mathfrak X}}$ is $G_0$-stable, then there is an induced (left) action of ${G_0}$ on the sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$. Given $g \in {G_0}$ and a local section $s$ of ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$, there is thus a local section $g.s$ of ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$. We can then consider the abelian category ${\rm Coh}({{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}},{G_0})$ of (left) ${G_0}$-equivariant coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules. Furthermore, the group $G_{k+1}$ is contained in ${{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)$ as a set of delta distributions, and for $h \in G_{k+1}$ we write $\delta_h$ for its image in $H^0({{\mathfrak X}}, {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}) = {{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}$. For $g \in {G_0}$, $h \in G_{k+1}$, we have $g.\delta_h = \delta_{ghg^{-1}}$, and for a local section $s$ of ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ we have then the relation \[non\_commuting\] g.(s\_h) = (g.s)(g.\_h) = (g.s) \_[ghg\^[-1]{}]{} . Suppose now that $\pi:{{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ is a $G_0$-equivariant morphism between admissible formal blow-ups of ${{\mathfrak X}}_0$ which is an isomorphism on corresponding rigid analytic spaces and that $k'\geq k$ are sufficiently large. According to \[prop-exactdirectimage\] there is then a natural morphism of sheaves of rings \[equ-transit\_sheaf\] \_\*\^\_[[[X]{}]{}’,k’,[[Q]{}]{}]{}=\^\_[[[X]{}]{},k’,[[Q]{}]{}]{} \^\_[[[X]{}]{},k,[[Q]{}]{}]{} which is ${G_0}$-equivariant. Given a coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}$-module ${{\mathscr M}}_{{{\mathfrak X}}'}$, the ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module $${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{\pi_* {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}} \pi_*{{\mathscr M}}_{{{\mathfrak X}}'}$$ is ${G_0}$-equivariant via $g.(s\otimes m)=(g.s) \otimes (g.m)$ for local sections $s,m$ and $g \in {G_0}$. Consider its submodule ${{\mathcal R}}_{{{\mathfrak X}}}$ locally generated by all elements $s \delta_h \otimes m - s\otimes (h.m)$ for $h \in G_{k'}$. Because of \[non\_commuting\] the submodule ${{\mathcal R}}_{{{\mathfrak X}}}$ is ${G_0}$-stable. We put $${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}},G_{k'}} {{\mathscr M}}_{{{\mathfrak X}}'}:= \left({{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{\pi_* {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}} \pi_*{{\mathscr M}}_{{{\mathfrak X}}'}\right)/\;{{\mathcal R}}_{{{\mathfrak X}}} \;.$$ We let $X={{\mathbb G}}/{{\mathbb B}}$ be the flag variety of ${{\mathbb G}}$ and denote by $${{\mathcal X}}:=X^{\rm ad}$$ the associated adic space. For simplicity, an admissible formal blow-up ${{\mathfrak X}}$ of ${{\mathfrak X}}_0$ will be called a [*formal model for ${{\mathcal X}}$ over ${{\mathfrak X}}_0$*]{}. This set of formal models is a projective system if it is indexed by the directed family of coherent open ideals on ${{\mathfrak X}}_0$, cf. [@BoschLectures 9.3]. Any morphism in the projective system is an isomorphism on corresponding rigid analytic spaces. Given a subsystem ${{\mathcal F}}$ of this projective system, we will denote the corresponding projective limit by ${{\mathfrak X}}_\infty({{\mathcal F}})=\varprojlim_{{\mathcal F}}{{\mathfrak X}}$ or simply ${{\mathfrak X}}_\infty$. In the following we will work relative to such a fixed system ${{\mathcal F}}$. A [*family of congruence levels for ${{\mathcal F}}$*]{} is an increasing family ${\underline{k}}$ of natural numbers $k=k_{{\mathfrak X}}$ for each ${{\mathfrak X}}\in{{\mathcal F}}$ (increasing means that $k'\geq k $ whenever there is a morphism ${{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ in ${{\mathcal F}}$). In the following, we fix such a family ${\underline{k}}$ with the additional property that the sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k_{{\mathfrak X}},{{\mathbb Q}}}$ is defined for each ${{\mathfrak X}}\in {{\mathcal F}}$. We finally assume that all models in ${{\mathcal F}}$ are $G_0$-stable and that all morphisms in ${{\mathcal F}}$ are $G_0$-equivariant. \[prop-cofinal\] If ${{\mathcal F}}$ equals the set of [*all*]{} $G_0$-equivariant formal models of ${{\mathcal X}}$ over ${{\mathfrak X}}_0$, then ${{\mathfrak X}}_\infty={{\mathcal X}}$. [[*Proof.* ]{}]{}According to [@SchnVPut] it suffices to see that any admissible formal blow-up ${{\mathfrak X}}$ of ${{\mathfrak X}}_0$ is dominated by one which is $G_0$-stable. If ${{\mathcal I}}$ is the ideal which is blown-up and if $p^k{{\mathcal O}}_{{{\mathfrak X}}_0}\subset{{\mathcal I}}$ for some $k$, then $G_k:={{\mathbb G}}(k)({{\mathfrak o}})$ stabilizes ${{\mathcal I}}$ and ${{\mathcal I}}':={{\mathcal I}}\cdot {{\mathcal O}}_{{{\mathfrak X}}}$. Let $g_1,...,g_N$ be a system of representatives for $G_0/G_k$ and let ${{\mathcal J}}$ be the product of the finitely many ideals $g_i{{\mathcal I}}'$. Then ${{\mathcal J}}$ is $G_0$-stable and blowing-up ${{\mathcal J}}$ on ${{\mathfrak X}}$ yields a $G_0$-stable model over ${{\mathfrak X}}$. \[dfn-coadmod\] A [*${G_0}$-equivariant coadmissible module*]{} on ${{\mathfrak X}}_\infty$ consists of a family ${{\mathscr M}}:=({{\mathscr M}}_{{\mathfrak X}})_{{{\mathfrak X}}\in{{\mathcal F}}}$ of objects ${{\mathscr M}}_{{\mathfrak X}}\in {\rm Coh}({{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}},{G_0})$ together with isomorphisms \[equ-transit\_mod\] \^\_[[[X]{}]{},k,[[Q]{}]{}]{} \_[\^\_[[[X]{}]{}’,k’,[[Q]{}]{}]{},G\_[k’]{}]{} [[M]{}]{}\_[[[X]{}]{}’]{}[[M]{}]{}\_[[[X]{}]{}]{} of ${G_0}$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-modules whenever there is a morphism ${{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ in ${{\mathcal F}}$. The isomorphisms are required to satisfy the obvious transitivity condition whenever there are morphisms ${{\mathfrak X}}''\rightarrow{{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ in ${{\mathcal F}}$. A morphism ${{\mathscr M}}\rightarrow{{\mathscr N}}$ between two such modules consists of morphisms ${{\mathscr M}}_{{\mathfrak X}}\rightarrow{{\mathscr N}}_{{\mathfrak X}}$ in ${\rm Coh}({{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}},{G_0})$ compatible with the isomorphisms above. Let ${{\mathscr M}}$ be a ${G_0}$-equivariant coadmissible module on ${{\mathfrak X}}_\infty$. The isomorphisms \[equ-transit\_mod\] induce morphisms $\pi_*{{\mathscr M}}_{{{\mathfrak X}}'}\rightarrow {{\mathscr M}}_{{{\mathfrak X}}}$ having global sections $H^0({{\mathfrak X}}',{{\mathscr M}}_{{{\mathfrak X}}'})\rightarrow H^0({{\mathfrak X}},{{\mathscr M}}_{{{\mathfrak X}}})$. We let $$H^0({{\mathfrak X}}_\infty,{{\mathscr M}}):=\varprojlim_{{\mathfrak X}}H^0({{\mathfrak X}},{{\mathscr M}}_{{{\mathfrak X}}}) \;.$$ On the other hand, we consider the category of coadmissible ${D({G_0})_{\theta_0}}$-modules. Given such a module $M$ we have its associated admissible locally analytic ${G_0}$-representation $V=M'_b$ together with its subspace of ${{{\mathbb G}}(k)^\circ}$-analytic vectors $V_{{{{\mathbb G}}(k)^\circ}-\rm an}$. The latter is stable under the ${G_0}$-action and its dual $M_k:= (V_{{{{\mathbb G}}(k)^\circ}-\rm an})'$ is a finitely presented ${D({{\mathbb G}}(k)^\circ,{G_0})_{\theta_0}}$-module, cf. \[lem-refine\]. Now consider a model ${{\mathfrak X}}$ in ${{\mathcal F}}$ and let $k=k_{{\mathfrak X}}$. According to Thm. \[thm-equivalence\] we have the coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$-module $${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}(M_k)= {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}\otimes_{{{{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}}} M_k$$ on ${{\mathfrak X}}$. Using the contragredient ${G_0}$-action on the dual space $M_k$, we put $$g.(s \otimes m) := (g.s) \otimes (g.m)$$ for $g\in G_0, m\in M_k$ and a local section $s$. In this way, ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}(M_k)$ becomes an object of ${\rm Coh}({{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}},{G_0})$. \[prop-equivalenceII\] (i) The family ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k_{{\mathfrak X}}} (M_{k_{{{\mathfrak X}}}})$ forms a ${G_0}$-equivariant coadmissible module on ${{\mathfrak X}}_\infty$. Call it ${{\mathscr Loc}}^\dagger(M)$. The formation of ${{\mathscr Loc}}^\dagger(M)$ is functorial in $M$. \(ii) The functors ${{\mathscr Loc}}^\dagger$ and $H^0({{\mathfrak X}}_\infty,\cdot)$ are quasi-inverse equivalences between the categories of coadmissible ${D({G_0})_{\theta_0}}$-modules and ${G_0}$-equivariant coadmissible modules on ${{\mathfrak X}}_\infty$. [[*Proof.* ]{}]{}Assume $k'\geq k$. We let $H:=G_{k+1}/G_{k'+1}$ and we denote a system of representatives in $G_{k+1}$ for the cosets in $H$ by the same symbol. For simplicity, we abbreviate in this proof $$D(k):={{{\mathcal D}}^{\rm an}({{\mathbb G}}(k)^\circ)_{\theta_0}}\hskip10pt {\rm and}\hskip10pt D(k,G_0):={D({{\mathbb G}}(k)^\circ,{G_0})_{\theta_0}}$$ and similarly for $k'$. We have the natural inclusion $D(k)\hookrightarrow D(k,G_0)$ from \[equ-finitefree\] which is compatible with variation in $k$. Now suppose $M$ is a $D(k',G_0)$-module. We then have the free $D(k)$-module $D(k)^{\oplus M\times H}$ on a basis $e_{m,h}$ indexed by the elements $(m,h)$ of the set $M\times H$. Its formation is functorial in $M$: if $M'$ is another module and $f: M\rightarrow M'$ a linear map, then $e_{m,h}\rightarrow e_{f(m),h}$ induces a linear map between the corresponding free modules. The free module comes with a linear map $$f_M: D(k)^{\oplus M\times H}\rightarrow D(k)\otimes_{D(k')} M$$ given by $$\oplus_{(m,h)}\lambda_{m,h}e_{m,h}\mapsto (\lambda_{m,h} \delta_h) \otimes m - \lambda_{m,h} \otimes (h \cdot m)$$ for $\lambda_{m,h}\in D(k)$ where we consider $M$ a $D(k')$-module via the inclusion $D(k')\hookrightarrow D(k',G_0)$. The map is visibly functorial in $M$ and gives rise to the sequence of linear maps $$D(k)^{\oplus M\times H}\stackrel{f_M}{\longrightarrow} D(k)\otimes_{D(k')} M \stackrel{can_M}{\longrightarrow} D(k,G_0)\otimes_{D(k',G_0)} M\longrightarrow 0$$ where the second map is induced from the inclusion $D(k')\hookrightarrow D(k',G_0)$. The sequence is functorial in $M$, since so are both occuring maps. [*Claim 1: If $M$ is a finitely presented $D(k',G_0)$-module, then the above sequence is exact.*]{} [[*Proof.* ]{}]{}This can be proved as in the proof of [@PSS4 Prop. 5.3.5]. Let $\pi:{{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ be a morphism in ${{\mathcal F}}$ and let $k=k_{{{\mathfrak X}}}, k'=k_{{{\mathfrak X}}'}$. *Claim 2: Suppose $M$ is a finitely presented $D(k')$-module and let ${{\mathscr M}}:= {{\mathscr Loc}}^\dagger_{{{\mathfrak X}}',k'}(M)$. The natural morphism* $${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}(D(k)\otimes_{D(k')} M){\stackrel{\simeq}{\longrightarrow}}{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{\pi_*{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}} \pi_*{{\mathscr M}}$$ is bijective. [[*Proof.* ]{}]{}The functor $\pi_*$ is exact on coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}$-modules according to \[prop-exactdirectimage\]. Choosing a finite presentation of $M$ reduces to the case $M=D(k')$ which is obvious. Now let $M$ be a finitely presented $D(k',G_0)$-module. Let $m_1, \ldots ,m_r$ be generators for $M$ as a $D(k')$-module. We have a sequence of $D(k)$-modules $$\bigoplus_{i,h} D(k)e_{m_i,h}\stackrel{f'_M}{\longrightarrow} D(k)\otimes_{D(k')} M \stackrel{can_M}{\longrightarrow} D(k,G_0)\otimes_{D(k',G_0)} M\longrightarrow 0$$ where $f'_M$ denotes the restriction of the map $f_M$ to the free submodule of $D(k)^{\oplus M\times H}$ generated by the finitely many vectors $e_{m_i,h}, i=1,\ldots,r$, $h \in H_n$. Since ${\rm im}(f'_M)={\rm im}(f_M)$ the sequence is exact by the first claim. Since it consists of finitely presented $D(k)$-modules, we may apply the exact functor ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}}$ to it. By the second claim, we get an exact sequence $$({{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}},k,{{\mathbb Q}}})^{\oplus r|H|} {\rightarrow}{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{\pi_*{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}}} \pi_*{{\mathscr M}}{\rightarrow}{{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}(D(k,G_0)\otimes_{D(k',G_0)} M) {\rightarrow}0$$ where ${{\mathscr M}}= {{\mathscr Loc}}^\dagger_{{{\mathfrak X}}',k'}(M)$. The cokernel of the first map in this sequence equals by definition $${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{\pi_* {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}},G_{k'}} \pi_*{{\mathscr M}}\;,$$ whence an isomorphism $${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}},k,{{\mathbb Q}}} \otimes_{\pi_* {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}',k',{{\mathbb Q}}},G_k'} \pi_*{{\mathscr M}}{\stackrel{\simeq}{\longrightarrow}}{{\mathscr Loc}}^\dagger_{{{\mathfrak X}},k}(D(k,G_0)\otimes_{D(k',G_0)} M) \;.$$ This implies both parts of the proposition. \[para-sheaf\] Denote the canonical projection map ${{\mathfrak X}}_\infty\rightarrow{{\mathfrak X}}$ by ${\rm sp}_{{{\mathfrak X}}}$ for each ${{\mathfrak X}}$. We define the following sheaf of rings on ${{\mathfrak X}}_\infty$. Assume $V\subseteq{{\mathfrak X}}_\infty$ is an open subset of the form ${\rm sp}_{{{\mathfrak X}}}^{-1}(U)$ with an open subset $U\subseteq{{\mathfrak X}}$ for a model ${{\mathfrak X}}$. We have that $${\rm sp}_{{{\mathfrak X}}'}(V)=\pi^{-1}(U)$$ for any blow-up morphism $\pi:{{\mathfrak X}}'\rightarrow{{\mathfrak X}}$ in ${{\mathcal F}}$ and so, in particular, ${\rm sp}_{{{\mathfrak X}}'}(V)\subseteq{{\mathfrak X}}'$ is an open subset for such ${{\mathfrak X}}'$. Moreover, $$\pi^{-1}({\rm sp}_{{{\mathfrak X}}'}(V))={\rm sp}_{{{\mathfrak X}}''}(V)$$ whenever $\pi:{{\mathfrak X}}''\rightarrow{{\mathfrak X}}'$ is a blow-up morphism over ${{\mathfrak X}}$ in ${{\mathcal F}}$. In this situation, the morphism \[equ-transit\_sheaf\] induces the ring homomorphism \[equ-transit\_hom\] \^\_[[[X]{}]{}”,k”,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}”]{}(V))=\_\*\^\_[[[X]{}]{}”,k”,[[Q]{}]{}]{}(([sp]{}\_[[[X]{}]{}’]{}(V))\^\_[[[X]{}]{}’,k’,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}’]{}(V)) and we form the projective limit $${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}(V):=\varprojlim_{{{\mathfrak X}}'\rightarrow {{\mathfrak X}}} {{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}}',k',{{\mathbb Q}}}({\rm sp}_{{{\mathfrak X}}'}(V))$$ over all these maps. The open subsets of the form $V$ form a basis for the topology on ${{\mathfrak X}}_\infty$ and ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ is a presheaf on this basis. We denote the associated sheaf on ${{\mathfrak X}}_\infty$ by the symbol ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ as well. It is a $G_0$-equivariant sheaf of rings on ${{\mathfrak X}}_\infty$. Remark: In the case where ${{\mathcal F}}$ consists of [*all*]{} $G_0$-stable formal models of ${{\mathcal X}}$ over ${{\mathfrak X}}_0$, we have ${{\mathfrak X}}_\infty={{\mathcal X}}$ by \[prop-cofinal\] and we denote the sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ by ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathcal X}},{{\mathbb Q}}}$ (or simply ${{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathcal X}}}$). Suppose ${{\mathscr M}}:=({{\mathscr M}}_{{\mathfrak X}})_{{{\mathfrak X}}}$ is a $G_0$-equivariant coadmissible module on ${{\mathfrak X}}_\infty$ as defined in \[dfn-coadmod\]. The isomorphisms \[equ-transit\_mod\] induce $G_0$-equivariant maps $\pi_*{{\mathscr M}}_{{{\mathfrak X}}'}\rightarrow{{\mathscr M}}_{{{\mathfrak X}}}$ which are linear relative to the morphism \[equ-transit\_sheaf\]. In a completely analogous manner as above, we obtain a sheaf ${{\mathscr M}}_\infty$ on ${{\mathfrak X}}_\infty$. It is a $G_0$-equivariant (left) ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-module on ${{\mathfrak X}}_\infty$ whose formation is functorial in ${{\mathscr M}}$. The functor ${{\mathscr M}}\rightarrow{{\mathscr M}}_\infty$ from $G_0$-equivariant coadmissible modules on ${{\mathfrak X}}_\infty$ to $G_0$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules is a fully faithful embedding. [[*Proof.* ]{}]{}We have ${\rm sp}_{{{\mathfrak X}}}({{\mathfrak X}}_\infty)={{\mathfrak X}}$ for all ${{\mathfrak X}}$. The global sections of $M_\infty$ are therefore equal to $$\Gamma({{\mathfrak X}}_\infty,{{\mathscr M}}_\infty)=\varprojlim_{{{\mathfrak X}}}\Gamma({{\mathfrak X}},{{\mathscr M}}_{{{\mathfrak X}}})=H^0({{\mathfrak X}}_\infty,{{\mathscr M}})$$ in the notation of the previous section. Thus, the functor ${{\mathscr Loc}}^\dagger \circ \Gamma({{\mathfrak X}}_\infty,-)$ is a left quasi-inverse according to Prop. \[prop-equivalenceII\]. We denote by ${{\mathscr Loc}}^\dagger_\infty$ the composite of the functor ${{\mathscr Loc}}^\dagger$ with $(\cdot)_\infty$, i.e. $$\{ ~coadmissible~ D(G_0)_{\theta_0}-modules~\}\stackrel{{{\mathscr Loc}}^\dagger_\infty}{\longrightarrow} \{ ~G_0-equivariant~ {{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}-modules~ \}.$$ It is fully faithful. We tentatively call its essential image the [*coadmissible*]{} $G_0$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules. It is an abelian category. In the case where ${{\mathfrak X}}_\infty$ equals the whole adic flag variety ${{\mathcal X}}$ we write ${{\mathscr Loc}}^\dagger_{{\mathcal X}}$ for the functor ${{\mathscr Loc}}^\dagger_\infty$. $G$-equivariance and the main theorem ------------------------------------- Let $G:={{\mathbb G}}(L)$. Denote by ${{\mathcal B}}$ the (semi-simple) Bruhat-Tits building of the $p$-adic group $G$ together with its natural $G$-action. To each special vertex $v\in{{\mathcal B}}$ we have the associated smooth affine Bruhat-Tits group scheme ${{\mathbb G}}_v$ over ${{\mathfrak o}}$ and a smooth model $X_0(v)$ of the flag variety of ${{\mathbb G}}$. All constructions in sections \[models\]-\[loc\_n\] are associated with the group scheme ${{\mathbb G}}_0$ with vertex, say $v_0$, but can be done canonically for any other of the reductive group schemes ${{\mathbb G}}_v$. We distinguish the various constructions from each other by adding the corresponding vertex $v$ to them, i.e. we write $X(v)$ for an admissible blow-up of the smooth model $X_0(v)$ and so on. We then choose a subset ${{\mathcal F}}(v)$ of models over ${{\mathfrak X}}_0(v)$ for ${{\mathcal X}}$ as in the preceding subsection, but relative to the vertex $v$. We assume that one of these subsets, say the one belonging to $v_0$, is stable under admissible blowing-up. According to [@BoschLectures], any ${{\mathfrak X}}(v)\in{{\mathcal F}}(v), v\in{{\mathcal B}}$ is then dominated by an element from ${{\mathcal F}}(v_0)$. As before, we denote the projective limit over ${{\mathcal F}}(v_0)$ by ${{\mathfrak X}}_\infty$. According to the previous subsection, we have the $G_0$-equivariant sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ on ${{\mathfrak X}}_\infty$. An element $g\in G$ induces a morphism $X_0(v)\stackrel{g.}{\longrightarrow}X_0(gv)$ which satisfies $(gh).=(g.)\circ(h.)$ and $1.={\rm id}$ for $g,h\in G$. If $X(v)\rightarrow X_0(v)$ is an admissible blowing up of an ideal ${{\mathcal I}}\subset X_0(v)$, then the universal property of blowing-up induces an isomorphism $X(v)\stackrel{g.}{\longrightarrow}X(gv)$ onto the blowing-up of $X_0(gv)$ at the ideal $g.{{\mathcal I}}$. We make the assumption that our union of models is [*$G$-stable*]{} in the sense that $${{\mathfrak X}}(v) \in {{\mathcal F}}(v) \Longrightarrow {{\mathfrak X}}(gv)\in{{\mathcal F}}(gv)$$ for any ${{\mathfrak X}}(v)\in{{\mathcal F}}(v)$ and any $g\in G$. We also assume that $k_{{{\mathfrak X}}(v)}=k_{{{\mathfrak X}}(gv)}$ in this situation. We obtain thus a $G$-action on ${{\mathfrak X}}_\infty$. By definition of this action, there is an equality \[equivariant\][sp]{}\_[[[X]{}]{}(gv)]{}(g.V)=g.[sp]{}\_[[[X]{}]{}(v)]{}(V) in ${{\mathfrak X}}(gv)$ for $g\in G$ and $V\subseteq{{\mathfrak X}}_\infty$. The $G_0$-equivariant structure on the sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$ extends to a $G$-equivariant structure. [[*Proof.* ]{}]{}Let $g\in G$. The isomorphism ${{\mathfrak X}}(v)\stackrel{g.}{\longrightarrow}{{\mathfrak X}}(gv)$ induces a ring isomorphism \[equ-ringiso\]\^\_[[[X]{}]{}(v),k,[[Q]{}]{}]{}(U)\^\_[[[X]{}]{}(gv),k,[[Q]{}]{}]{}(g.U) for any open subset $U\subseteq{{\mathfrak X}}(v)$ where $k=k_{{{\mathfrak X}}(v)}$. In particular, for an open subset $V\subseteq{{\mathfrak X}}_\infty$ of the form $V={\rm sp}_{{{\mathfrak X}}(v)}^{-1}(U)$ with $U\subseteq{{\mathfrak X}}(v)$ open and a blow-up morphism ${{\mathfrak X}}'(v)\rightarrow{{\mathfrak X}}(v)$, this gives a ring homomorphism \[equ-action\] \^\_[[[X]{}]{}’(v),k’,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}’(v)]{}(V))\^\_[[[X]{}]{}’(gv),k’,[[Q]{}]{}]{}(g.[sp]{}\_[[[X]{}]{}’(v)]{}(V)) =\^\_[[[X]{}]{}’(gv),k’,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}’(gv)]{}(gV)) where we have used \[equivariant\] and where $k'=k_{{{\mathfrak X}}'(v)}$. A given morphism $\pi: {{\mathfrak X}}(v')\rightarrow{{\mathfrak X}}(v)$ with ${{\mathfrak X}}(v')\in{{\mathcal F}}(v')$ and ${{\mathfrak X}}(v)\in{{\mathcal F}}(v)$ which is an isomorphism on corresponding rigid analytic spaces induces a morphism of sheaves of rings \[equ-transit\_sheafII\] \_\* \^\_[[[X]{}]{}(v’),k’,[[Q]{}]{}]{}=\^\_[[[X]{}]{}(v),k’,[[Q]{}]{}]{} \^\_[[[X]{}]{}(v),k,[[Q]{}]{}]{} by \[prop-exactdirectimage\]. Here, $k=k_{{{\mathfrak X}}(v)}$ and $k'=k_{{{\mathfrak X}}(v')}$. Given $V {\subset}{{\mathfrak X}}_\infty$ of the form $V={\rm sp}_{{{\mathfrak X}}(\tilde{v})}^{-1}(U)$ with an open set $U\subseteq{{\mathfrak X}}(\tilde{v})$, the morphism \[equ-transit\_sheafII\] induces a ring homomorphism \[equ-transit\_homII\] \^\_[[[X]{}]{}(v’),k’,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}(v’)]{}(V)) =\_\*\^\_[[[X]{}]{}(v’),k’,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}(v)]{}(V))\^\_[[[X]{}]{}(v),k,[[Q]{}]{}]{}([sp]{}\_[[[X]{}]{}(v)]{}(V)) whenever the morphism $\pi:{{\mathfrak X}}(v')\rightarrow{{\mathfrak X}}(v)$ lies over ${{\mathfrak X}}(\tilde{v}).$ If we write ${{\mathfrak X}}(v')\geq {{\mathfrak X}}(v)$ in this situation, then the family of all models ${{\mathfrak X}}(v)$ over ${{\mathfrak X}}(\tilde{v})$ becomes directed, the ${{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}({\rm sp}_{{{\mathfrak X}}(v)}(V))$ become a projective system and we may form the projective limit $$\varprojlim_{{{\mathfrak X}}(v)\rightarrow{{\mathfrak X}}(\tilde{v})}{{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}({\rm sp}_{{{\mathfrak X}}(v)}(V)) \;.$$ By cofinality, this projective limit equals ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}(V)$. Since the homomorphism \[equ-action\] is compatible with varying ${{\mathfrak X}}'(v')$ in the directed family, we deduce for a given $g\in G$ a ring homomorphism $${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}(V)=\varprojlim_{{{\mathfrak X}}(v)}{{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}({\rm sp}_{{{\mathfrak X}}(v)}(V))\stackrel{g.}{\rightarrow} \varprojlim_{{{\mathfrak X}}(gv)}{{\widetilde{{{\mathscr D}}}}}^{\dagger}_{{{\mathfrak X}}(gv),k,{{\mathbb Q}}}({\rm sp}_{{{\mathfrak X}}(gv)}(gV))={{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}(gV) \;.$$ It implies that the sheaf ${{\widetilde{{{\mathscr D}}}}}^{\dagger}_{\infty,{{\mathbb Q}}}$ is $G$-equivariant. It is clear from the construction that the $G$-equivariant structure extends the $G_0$-structure. A coadmissible $G_0$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-module whose equivariant structure extends to the full group $G$, will simply be called a coadmissible $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-module. The functors ${{\mathscr Loc}}^\dagger_\infty$ and $\Gamma({{\mathfrak X}}_\infty,\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(G_0)_{\theta_0}$-modules and coadmissible $G_0$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules. The subcategories of coadmissible $D(G)_{\theta_0}$-modules and coadmissible $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-modules correspond to each other. [[*Proof.* ]{}]{}We only need to show the second statement. It is clear that a coadmissible $D(G_0)_{\theta_0}$-module which comes from a coadmissible $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-module is a $D(G)_{\theta_0}$-module. For the converse, we consider a special vertex $v\in{{\mathcal B}}$ and a model ${{\mathfrak X}}(v)$ and the corresponding localisation functor ${{\mathscr Loc}}^\dagger_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}$ (where $k=k_{{{\mathfrak X}}(v)}$) which is an equivalence between finitely presented $D^{an}({{\mathbb G}}_v(k)^\circ)_{\theta_0}$-modules and coherent ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}$-modules on ${{\mathfrak X}}(v)$. Here, ${{\mathbb G}}_v$ denotes as before the reductive Bruhat-Tits group scheme over ${{\mathfrak o}}$ associated with the special vertex $v$. The adjoint action of $G$ on its Lie algebra induces a ring isomorphism \[equ-ringisoII\] D\^[an]{}([[G]{}]{}\_[v]{}(k)\^)D\^[an]{}([[G]{}]{}\_[gv]{}(k)\^) for any $g\in G$. Now consider a coadmissible $D(G)_{\theta_0}$-module $M$ with dual space $V=M'$. We have the family $({{\mathscr M}}_{{{\mathfrak X}}(v)})_{{{\mathfrak X}}(v)}$ where $${{\mathscr M}}_{{{\mathfrak X}}(v)} = {{\mathscr Loc}}^\dagger_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}(M_{k,v}) = {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}\otimes_{D^{an}({{\mathbb G}}_v(k)^\circ)_{\theta_0}} M_{k,v}$$ and $M_{k,v}=(V_{{{\mathbb G}}_v(k)^\circ-\rm an})'$ with $k=k_{{{\mathfrak X}}(v)}$. Let $g\in G$. The map $m \mapsto gm$ on $M$ induces a map $M_{k,v}\rightarrow M_{k,gv}$ which is linear relative to \[equ-ringisoII\]. We therefore have for any open subset $U\subseteq{{\mathfrak X}}(v)$ a homomorphism $${{\mathscr M}}_{{{\mathfrak X}}(v)}(U)\stackrel{g.}{\longrightarrow}{{\mathscr M}}_{{{\mathfrak X}}(gv)}(g.U)$$ which is induced by the map $$s \otimes m \mapsto (g.s) \otimes gm \;.$$ for $s\in {{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}(v),k,{{\mathbb Q}}}(U), m\in M_{k,v}$ and where $g.$ is the ring isomorphism \[equ-ringiso\]. In particular, for an open subset $V\subseteq{{\mathfrak X}}_\infty$ of the form $V={\rm sp}_{{{\mathfrak X}}(v)}^{-1}(U)$ with $U\subseteq{{\mathfrak X}}(v)$ open, this gives a homomorphism for \[equ-actionII\] [[M]{}]{}\_[[[X]{}]{}(v)]{}([sp]{}\_[[[X]{}]{}(v)]{}(V)) [[M]{}]{}\_[[[X]{}]{}(gv)]{}(g.[sp]{}\_[[[X]{}]{}(v)]{}(V)) = [[M]{}]{}\_[[[X]{}]{}(gv)]{}([sp]{}\_[[[X]{}]{}(gv)]{}(gV)) which is linear relative to the ring homomorphism \[equ-action\]. A given morphism $\pi: {{\mathfrak X}}(v')\rightarrow{{\mathfrak X}}(v)$ with ${{\mathfrak X}}(v')\in{{\mathcal F}}(v')$ and ${{\mathfrak X}}(v)\in{{\mathcal F}}(v)$ which is an isomorphism on corresponding rigid analytic spaces induces a morphism \[equ-mor\]\_\*[[M]{}]{}\_[[[X]{}]{}(v’)]{}[[M]{}]{}\_[[[X]{}]{}(v)]{} compatible with the morphism of rings \[equ-transit\_sheafII\] as follows. First of all, one has an isomorphism $$\pi_*\left({{\mathscr Loc}}^\dagger_{{{\mathfrak X}}(v'),k',{{\mathbb Q}}}(M_{k',v'})\right){\stackrel{\simeq}{\longrightarrow}}\left(\pi_*{{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathfrak X}}(v'),k',{{\mathbb Q}}}\right)\otimes_{D^{an}({{\mathbb G}}_{v'}(k')^\circ)_{\theta_0}} M_{k',v'}.$$ Indeed, $\pi_*$ is exact by \[prop-exactdirectimage\] and we may argue with finite presentations as usually. Moreover, we have inclusions ${{\mathbb G}}_{v'}(k')\subseteq {{\mathbb G}}_{v}(k)$ and thus $$V_{{{\mathbb G}}_{v}(k)^\circ-\rm an}\subseteq V_{{{\mathbb G}}_{v'}(k')^\circ-\rm an} \;.$$ The dual map $M_{k',v'}\rightarrow M_{k,v}$ is linear relative to the natural inclusion $$D^{an}({{\mathbb G}}_{v'}(k')^\circ)\rightarrow D^{an}({{\mathbb G}}_{v}(k)^\circ) \;.$$ The latter inclusion is compatible with the morphism of rings \[equ-transit\_sheafII\] via taking global sections. Hence, we have a morphism \[equ-mor\] as claimed. We now have everything at hand to follow the arguments in the proof of the preceding proposition word for word and to conclude that the projective limit ${{\mathscr M}}_\infty$ has a $G$-action which extends its $G_0$-action and which makes it a $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{\infty,{{\mathbb Q}}}$-module. This completes the proof of the theorem. We finally look at the special case of the whole adic flag variety ${{\mathcal X}}$ with its sheaf of infinite order differential operators ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathcal X}},{{\mathbb Q}}}$, cf. \[para-sheaf\]. Recall the functor ${{\mathscr Loc}}^\dagger_{{\mathcal X}}$ from the end of \[subsec\_G0\]. \[thm-main\] Let ${{\mathcal X}}$ be the adic analytic flag variety of ${{\mathbb G}}$ with its sheaf ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathcal X}},{{\mathbb Q}}}$ of infinite order differential operators. The functors ${{\mathscr Loc}}^\dagger_{{\mathcal X}}$ and $\Gamma({{\mathcal X}},\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(G_0)_{\theta_0}$-modules and coadmissible $G_0$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathcal X}},{{\mathbb Q}}}$-modules. The subcategories of coadmissible $D(G)_{\theta_0}$-modules and coadmissible $G$-equivariant ${{\widetilde{{{\mathscr D}}}}}^\dagger_{{{\mathcal X}},{{\mathbb Q}}}$-modules correspond to each other. [[*Proof.* ]{}]{}Taking each ${{\mathcal F}}(v)$ to be the set of [*all*]{} equivariant formal models, we obtain ${{\mathfrak X}}_\infty={{\mathcal X}}$ according to \[prop-cofinal\] and the result follows from the preceding theorem. [^1]: C.H. and M.S. benefited from an invitation to MSRI during the Fall 2014 and thank this institution for excellent working conditions. D.P. would like to acknowledge support from IHÉS and the ANR program $p$-adic Hodge Theory and beyond (ThéHopaD) ANR-11-BS01-005. T.S. would like to acknowledge support of the Heisenberg programme of Deutsche Forschungsgemeinschaft (SCHM 3062/1-1). M.S. would like to acknowledge the support of the National Science Foundation (award DMS-1202303). [^2]: considered as a locally $L$-analytic group

--- abstract: 'We apply the framework of imperfect empirical coordination to a two-node setup where the action $X$ of the first node is not observed directly but via $L$ agents who observe independently impaired measurements $\hat X$ of the action. These $L$ agents, using a rate-limited communication that is available to all of them, help the second node to generate the action $Y$ in order to establish the desired coordinated behaviour. When $L<\infty$, we prove that it suffices $R_i\geq I\left(\hat X;\hat{Y}\right)$ for at least one agent whereas for $L\longrightarrow\infty$, we show that it suffices $R_i\geq I\left(\hat X;\hat Y|X\right)$ for all agents where $\hat Y$ is a random variable such that $X-\hat X-\hat Y$ and $\|p_{X,\hat Y}\left(x,y\right)-p_{X,Y}\left(x,y\right)\|_{TV}\leq \Delta$ ( $\Delta$ is the pre-specified fidelity).' author: - - - bibliography: - 'string.bib' - 'references.bib' title: | Remote Empirical Coordination\ [^^]{} --- Introduction ============ The development of machine to machine communication and the Internet of Things has enabled a renewed interest in further investigating heterogeneous network topologies where various objects are allowed to be interconnected. Such objects may be for instance computers with different operating systems and protocols, embedded sensors, medical devices, smart meters, and autonomous vehicles. A key factor to elucidate further insights of such network topologies is to study the cooperation and coordination of the different devices in the network on the level of information theory. In many practical scenarios, there is no direct access to the source data of some phenomenon due to possible technical limitations. In this case, multiple agents can be deployed to collect noisy measurements of the source. Examples include the [*remote source coding problem*]{} introduced in [@dobrushin:1962] (see also [@berger:1971; @witsenhausen:1980]) and the [*CEO problem*]{} introduced in [@berger:1996]. Here, we adopt the concept of the “remote source” to the framework of [*“imperfect” empirical coordination*]{} [@mylonakis:2019] using also ideas from the framework of [*“perfect” empirical coordination*]{} [@cuff:2010]. The notion of empirical coordination in information theory was formalized in [@cuff:2010]. According to [@cuff:2010], when we are given the actions of some nodes by nature, empirical coordination is achieved if the joint type, measured by total variation distance, of the actions of all nodes in a network is close to the desired distribution, in probability. The literature on empirical coordination is vast. For instance, the authors in [@bereyhi:2013; @letreust:2015a] studied empirical coordination for various network topologies, whereas in [@chou:2018] empirical coordination was established using polar coding and distributed approximation. This type of coordination is also used with ideas from other fields, such as game theory [@letreust:2016], optimal control [@letreust:2018] and networked control systems [@sharieepoorfard:2018]. The framework of empirical coordination of [@cuff:2010] was recently extended to the more general framework of imperfect empirical coordination in [@mylonakis:2019] who was inspired by [@kramer:2007]. According to [@mylonakis:2019], imperfect empirical coordination is established if the total variation between the joint type of the actions in a network comes close, on average, to a desired distribution within distance pre-specified by a threshold $\Delta$. The choice of $\Delta$ regulates the coordination rates between the agents and therefore the system’s designer can choose to coordination in a range of rates depending on the available rate budget. Clearly, if we choose $\Delta=0$, then, we obtain as a special case the perfect empirical coordination of [@cuff:2010]. The result in [@mylonakis:2019] was applied to a multiple description problem with two channels in [@mylonakis:2019b]. ![System model.[]{data-label="fig1"}](noisy2.png){width="5cm" height="4cm"} In this work, we consider the setup illustrated in Fig. \[fig1\]. In this setup, the action of the first node, which is distributed according to $p_{0}$, is partially observed via multiple agents who then communicate via multiple rate-limited links to the second node. In particular, the $L$ agents collect independently noisy versions of the action, distributed according to $p_{\hat X}$, and, by applying the coordination code, communicate to the second node. Based on the messages that it receives, the second node produces the action $Y$. Through our framework, we claim that imperfect empirical coordination is an appropriate approach to study coordination of nodes which do not directly communicate. This is because, by definition, the metric to achieve perfect empirical coordination can only be satisfied if the desired distributions satisfy the Markov chain $X-\hat{X}-Y$ which is not a necessary requirement in imperfect empirical coordination due to the flexibility of our achievability performance criterion. Our achievability results rely on [@mylonakis:2019 Theorem 1] and we break our derivations in two parts. First, in sections III and IV, we give a lower bound of the coordination capacity region for the problem of perfect empirical coordination [@cuff:2010]. Second, in section V, we apply [@mylonakis:2019 Theorem 1] to get a lower bound of the rate-distortion-coordination region. It is noteworthy to point out that our results are obtained for $L<\infty$ and when $L\longrightarrow\infty$. General Definitions =================== We begin with some basic mathematical concepts and the definition of the coordination code i.e., the protocol which is used to coordinate the nodes of the network. We denote as $\mathbb X$ the (common) alphabet of random varibales $X$ and $\hat X$ and as $\mathbb Y$ the (common) alphabet of $Y$ and $\hat Y$. The joint type $P_{x^n,y^n}$ of a tuple of sequences $\left(x^n,y^n\right)$ is the empirical probability mass function, given by $$P_{x^n,y^n}\left(x,y\right)\triangleq \frac{1}{n}\sum_{i=1}^n{\mathbf 1\big(\left(x_i,y_i\right)=\left(x,y\right)\big)},$$ for all $\left(x,y\right)\in \mathbb{X}\times \mathbb{Y}$, where $\mathbf 1$ is the indicator function. The total variation between two probability mass functions (PMF) is given by $$\|p\left(x,y\right)-q\left(x,y\right)\|_{TV}\triangleq\frac{1}{2}\sum_{x,y}{|p\left(x,y\right)-q\left(x,y\right)|}.$$ The $\Delta$-neighborhood of a PMF $p\left(x,y\right)$ is defined as [rCl]{} N\_(p(x,y))}{q(x,y):p(x,y)-q(x,y)\_[TV]{}}. The $\left(2^{nR_1},2^{nR_2},\cdots,2^{nR_L},n\right)$ coordination code for our set-up consists of $L+1$ functions-L encoding functions $$i_l:\mathbb X^n \rightarrow\left\{1,\dots,2^{nR_l}\right\}, l=1,\dots,L,$$ and a decoding function $$y^n:\left\{1,\dots,2^{nR_1}\right\}\times \dots \times \left\{1,\dots,2^{nR_L}\right\} \rightarrow \mathbb Y^n.$$ In our set-up, the actions $X^n$ and ${\hat{X}_l}^n$ for $l=1,\dots,L$ are chosen by nature to be i.i.d according to $p_{X,\hat X_1,\dots,\hat X_L}\left(x,\hat x_1,\dots,\hat x_L\right)=p_0\left(x\right)\prod_{l=1}^{L}{p_{\hat X|X}\left(\hat x_l|x\right)}$. Thus, $X^n$ and ${\hat{X}_l}^n$ for $l=1,\dots,L$ are distributed according to a product distribution $\left(X^n,\hat X_1^n,\dots,\hat X_L^n\right)\sim \prod_{i=1}^{n}{p_0\left(x_i\right)\prod_{l=1}^{L}{p_{\hat X|X}\left(\hat x_{li}|x_i\right)}}$. The action $Y^n$ is function of $\hat X_1^n,\dots,\hat X_L^n $ given by $Y^n=y^n\bigg(i_1\left(\hat X_1^n\right),\dots,i_L\left(\hat X_L^n\right)\bigg)$. Finite number of agents ======================= In this section, we give and discuss an inner bound of the coordination capacity region for the case of finite $L$. For a proof, see Appendix A. We begin with the required definitions. A desired PMF $p_{X,Y}\left(x,y\right)\triangleq p_{0}\left(x\right)p_{Y|X}\left(y|x\right)$ is achievable for empirical coordination with the rates $\left(R_1,\dots,R_L\right)$ if there exists a sequence of $\Big(2^{nR_1},\cdots,2^{nR_L},n\Big)$ coordination codes such that as $n\to \infty$ $$\|P_{x^n,y^n}\left(x,y,z\right)-p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\|_{TV}\to 0, \label{eq:empcord}$$ in probability. \[def:perfect\] The coordination capacity region $C_{p_{X,\hat X_1,\dots,\hat X_L}}^P$ for the source-agent joint PMF $p_{X,\hat X_1,\dots,\hat X_L}\left(x,\hat x_1,\dots,\hat x_L\right)=p_0\left(x\right)\prod_{l=1}^{L}{p_{\hat X|X}\left(\hat x_l|x\right)}$ is the closure of the set of rate-coordination tuples $\big(R_1,R_2,\dots,R_L,p_{Y|X}\left(y|x\right)\big)$ that are achievable: $$C_{p_{X,\hat X_1,\dots,\hat X_L}}^P\triangleq \mathbf {Cl}\left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \big(R_1,\dots,R_L,p_{Y|X}\left(y|x\right)\big):\\ p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\\\text{is achievable at rates $\left(R_1,\dots,R_L\right)$} \end{IEEEeqnarraybox}\right\}.$$ $$C_{p_{X,\hat X_1,\dots,\hat X_L}}^P\supseteq \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \big(R_1,\dots,R_L,p_{Y|X}\left(y|x\right)\big): \quad X-\hat X-Y,\\ \exists l \quad \text{such that} \quad R_l\geq I\left(\hat X;Y\right) \end{IEEEeqnarraybox}\right\}.$$ \[th:fiag\] See Appendix A. According to Theorem \[th:fiag\], in the case of $L< \infty$, the PMFs $p_{X,Y}\left(x,y\right)\triangleq p_{0}\left(x\right)p_{Y|X}\left(y|x\right)$ which form a markov chain $\quad X-\hat X-Y$, are achievable if the rate of at least one agent exceeds the mutual information between $\hat X$ and $\hat Y$. Although we do not prove an outer bound, it seems to us that, if the number of agents is finite and the rate of all of them is under the thresold of $I\left(\hat X;Y\right)$, the establishment of perfect empirical coordination is impossible i.e., that is optimal to deactivate all but one agent with rate at least equal to $I\left(\hat X;Y\right)$. On the other hand, as we will see in the next section, if the number of agents allowed to become arbitrarily large, then, the joint decoding is becoming gainful and we can distribute the rate among the different agents in order to satisy the coordination criterion. Infinite number of agents ========================= In this section, we give and discuss an inner bound of the coordination capacity region for the case of $L\longrightarrow\infty$. For a proof, see Appendix A. We begin with the required definitions. A desired PMF $p_{X,Y}\left(x,y\right)\triangleq p_{0}\left(x\right)p_{Y|X}\left(y|x\right)$ is achievable for empirical coordination with the rate per agent $R_{\text{ag}}$ if there exists a sequence of $\Big(\underbrace{2^{nR_{\text{ag}}},\dots,2^{nR_{\text{ag}}}}_{L \quad \text{times}},n\Big)$ coordination codes such that as $L\to \infty$ and $n\to\infty$ $$\|P_{x^n,y^n}\left(x,y,z\right)-p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\|_{TV}\to 0, \label{eq:empcord1}$$ in probability. \[def:perfect\] The double convergence in Definition \[def:perfect\] should be interpreted as $L\to \infty$ first, followed by $n\to \infty$. See proof of Theorem \[th:infiag\] (in Appendix A). The coordination capacity region $C_{p_{X,\hat X}}^P$ for the source-agent PMF $p_{X,\hat X}\left(x,\hat x\right)=p_0\left(x\right){p_{\hat X|X}\left(\hat x|x\right)}$ is the closure of the set of rate-coordination tuples $\big(R_{\text{ag}},p_{Y|X}\left(y|x\right)\big)$ that are achievable: $$C_{p_{X,\hat X}}^P\triangleq \mathbf {Cl}\left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \big(R_{\text{ag}},p_{Y|X}\left(y|x\right)\big): p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\\\text{is achievable at rate per agent $R_{\text{ag}}$} \end{IEEEeqnarraybox}\right\}.$$ $$C_{p_{X,\hat X}}^P\supseteq \mathbf {Cl}\left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \big(R_{\text{ag}},p_{Y|X}\left(y|x\right)\big): X-\hat X-Y,\\\quad \quad \quad \quad \quad \quad \quad \quad R_{\text{ag}}\geq I\left(\hat X;Y|X\right) \end{IEEEeqnarraybox}\right\}.$$ \[th:infiag\] See Appendix A. According to Theorem \[th:infiag\], in the case of $L\to \infty$, the PMFs $p_{X,Y}\left(x,y\right)\triangleq p_{0}\left(x\right)p_{Y|X}\left(y|x\right)$ which form a markov chain $\quad X-\hat X-Y$, are achievable if every agent has rate at least equal to $ I\left(\hat X;Y|X\right)$, which of course is smaller or equal to $I\left(\hat X; Y\right)$ due to the markovian property. In other words, the arbitrarily large number of agents allows us to get rid of the constraint $R_i\geq I\left(\hat X;Y\right)$ for at least one agent. Imperfect empirical coordination ================================ In this section, we combine the inner bounds from the previous two sections with [@mylonakis:2019 Theorem 1] in order to get inner bounds for the rate-distortion-coordination region, both in the cases of $L$ finite and $L\longrightarrow\infty$. Finite number of agents ----------------------- A desired PMF $p_{X,Y}\left(x,y\right)\triangleq p_{0}\left(x\right)p_{Y|X}\left(y|x\right)$ is achievable for $\Delta$-empirical coordination with the rate-pair $\left(R_1,\dots,R_L\right)$ if there is an $N$ such that for all $n>N$, there exists a coordination code $\Big(2^{nR_1},\dots,2^{nR_L},n\Big)$ such that $$ \mathbb{E}\big\{\|P_{x^n,y^n}\left(x,y\right)-p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\|_{TV}\big\}\leq \Delta.$$\[def:imperfect\] The rate-distortion-coordination region $R_{p_{X,\hat X_1,\dots,\hat X_L}}^I$ for the source-agent PMF $p_{X,\hat X_1,\dots,\hat X_L}\left(x,\hat x_1,\dots,\hat x_L\right)=p_0\left(x\right)\prod_{l=1}^{L}{p_{\hat X|X}\left(\hat x_l|x\right)}$ and for a fixed conditional distribution $p_{Y|X}\left(y|x\right)$ is defined as: $$\begin{gathered} R_{p_{X,\hat X_1,\dots,\hat X_L}}^I\big(p_{Y|X}\left(y|x\right)\big)\\\quad \triangleq \mathbf{Cl}\left. \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \left(R_1,\dots,R_L,\Delta\right): p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\\ \text{is achievable for $\Delta$-empirical coordination}\\\text{ at rates} \left(R_1,\dots,R_L\right) \end{IEEEeqnarraybox}\right\}. \right. \end{gathered}$$ For every source-agent PMF $p_{X,\hat X_1,\dots,\hat X_L}\left(x,\hat x_1,\dots,\hat x_L\right)=p_0\left(x\right)\prod_{l=1}^{L}{p_{\hat X|X}\left(\hat x_l|x\right)}$ and for every fixed conditional PMF $p_{Y|X}\left(y|x\right)$: $$\begin{gathered} R_{p_{X,\hat X_1,\dots,\hat X_L}}^I\big(p_{Y|X}\left(y|x\right)\big)\\\quad= \left. \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{2pt} {2pt}][c]{l} \left(R_1,\dots, R_L,\Delta\right):\\ \left(R_1,\dots, R_L,q_{\hat{Y}|X}\right)\in C_{p_{X,\hat{X}_1,\dots,\hat{X}_L}}^P\quad\\ \text{for some $\hat Y$ which satisfy}\\ p_0\left(x\right)q_{\hat{Y}|X}\left(y|x\right)\in N_{\Delta}\big(p_0\left(x\right)p_{Y|X}\left(y|x\right)\big) \end{IEEEeqnarraybox}\right\}. \right. \end{gathered}$$ \[th:maintheorem\] This Lemma is a direct consequence of a more general result which is explained and proved in [@mylonakis:2019]. See also, Fig. \[fig2\] and Fig. \[fig3\]. For every source-agent PMF $p_{X,\hat X_1,\dots,\hat X_L}\left(x,\hat x_1,\dots,\hat x_L\right)=p_0\left(x\right)\prod_{l=1}^{L}{p_{\hat X|X}\left(\hat x_l|x\right)}$ and for every fixed conditional PMF $p_{Y|X}\left(y|x\right)$: $$\begin{gathered} R_{p_{X,\hat X_1,\dots,\hat X_L}}^I\big(p_{Y|X}\left(y|x\right)\big)\supseteq\\ \bigcup_{\substack{q_{\hat{Y}|X}: \quad X-\hat{X}-\hat{Y},\\ p_0\left(x\right)q_{\hat{Y}|X}\left(y|x\right)\in N_{\Delta}\left(p_{X,Y}\left(x,y\right)\right)\\ }} R\left(p_0q_{\hat{Y}|X}\right),\end{gathered}$$ where $$\begin{gathered} R\left(p_0 q_{\hat{Y}|X}\right) \triangleq \left. \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \left(R_1,\dots,R_L,\Delta\right):\\ \exists i \quad \text{such that} \quad R_i\geq I\left(\hat X;\hat Y\right) \end{IEEEeqnarraybox}\right\}. \right.\end{gathered}$$ From Theorem \[th:fiag\] and Lemma \[th:maintheorem\], we obtain the characterization of the theorem. For $L=1$, the previous theorem together with a simple converse give [@kramer:2007 Theorem 1]. Infinite number of agents ------------------------- A desired PMF $p_{X,Y}\left(x,y\right)\triangleq p_{0}\left(x\right)p_{Y|X}\left(y|x\right)$ is achievable for $\Delta$-empirical coordination with the rate per agent $R_{\text{ag}}$ if there is an $\bar{L}$ and an $N$ such that for all $L>\bar{L}$ and $n>N$, there exists a coordination code $\Big(\underbrace{2^{nR_{\text{ag}}},\dots,2^{nR_{\text{ag}}}}_{L \quad \text{times}},n\Big)$ such that $$ \mathbb{E}\big\{\|P_{x^n,y^n}\left(x,y\right)-p_{0}\left(x\right)p_{Y|X}\left(y|x\right)\|_{TV}\big\}\leq \Delta.$$\[def:imperfect\] ![Interpretation of Lemmas \[th:maintheorem\] and \[th:maintheorem1\] - Achievability part: Every good coordination code designed for achieving perfect empirical coordination according to some distribution $p_0\left(x\right)q_{\hat{Y}|X}\left(y|x\right)\in N_{\Delta}\big(p_0\left(x\right)p_{Y|X}\left(y|x\right)\big)\cap C^P$ (blue) achieves $\Delta$-empirical coordination according to $p_0\left(x\right)p_{Y|X}\left(y|x\right)$.[]{data-label="fig2"}](noisy3.png){width="7cm" height="6cm"} ![Interpretation of Lemmas \[th:maintheorem\] and \[th:maintheorem1\] - Converse part: For every coordination code that achieves $\Delta$-empirical coordination according to $p_0\left(x\right)p_{Y|X}\left(y|x\right)$, there is a coordination code with the same rates which achieves perfect empirical coordination according to some distribution in $N_{\Delta}\big(p_0\left(x\right)p_{Y|X}\left(y|x\right)\big)\cap C^P$ (purple).[]{data-label="fig3"}](noisy4.png){width="6cm" height="6cm"} The rate-distortion-coordination region $R_{p_{X,\hat X}}^I$ for the source-agent PMF $p_{X,\hat X}\left(x,\hat x\right)=p_0\left(x\right){p_{\hat X|X}\left(\hat x|x\right)}$ and for a fixed conditional PMF $p_{Y|X}\left(y|x\right)$ is defined as: $$\begin{gathered} R_{p_{X,\hat X}}^I\big(p_{Y|X}\left(y|x\right)\big) \triangleq\\ \mathbf{Cl}\left. \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt} {1pt}][c]{l} \left(R_{\text{ag}},\Delta\right): p_{0}\left(x\right)p_{Y|X}\left(y|x\right) \text{is achievable}\\\text{for $\Delta$-empirical coordination}\text{ at rate per agent $R_{\text{ag}}$} \end{IEEEeqnarraybox}\right\}. \right. \end{gathered}$$ For every source-agent PMF $p_{X,\hat X}\left(x,\hat x\right)=p_0\left(x\right){p_{\hat X|X}\left(\hat x|x\right)}$ and for every fixed conditional PMF $p_{Y|X}\left(y|x\right)$, $$\begin{gathered} R_{p_{X,\hat X}}^I\big(p_{Y|X}\left(y|x\right)\big)\\\quad= \left. \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{2pt} {2pt}][c]{l} \left(R_{\text{ag}},\Delta\right):\\ \left(R_{\text{ag}},q_{\hat{Y}|X}\right)\in C_{p_{X,\hat{X}}}^P\quad\\ \text{for some $\hat Y$ which satisfy}\\ p_0\left(x\right)q_{\hat{Y}|X}\left(y|x\right)\in N_{\Delta}\big(p_0\left(x\right)p_{Y|X}\left(y|x\right)\big) \end{IEEEeqnarraybox}\right\}. \right. \end{gathered}$$ \[th:maintheorem1\] This Lemma is a direct consequence of a more general result which is explained and proved in [@mylonakis:2019]. See also, Fig. \[fig2\] and Fig. \[fig3\]. For every source-agent PMF $p_{X,\hat X}\left(x,\hat x\right)=p_0\left(x\right){p_{\hat X|X}\left(\hat x|x\right)}$ and for every fixed conditional PMF $p_{Y|X}\left(y|x\right)$: $$\begin{gathered} R_{p_{X,\hat X}}^I\big(p_{Y|X}\left(y|x\right)\big)\supseteq \min_{\substack{q_{\hat{Y}|X}:X-\hat{X}-\hat{Y},\\ p_0\left(x\right)q_{\hat{Y}|X}\left(y|x\right)\in N_{\Delta}\left(p_{X,Y}\left(x,y\right)\right) }} I\left(\hat X;\hat Y|X\right). \end{gathered}$$ From Theorem \[th:infiag\] and Lemma \[th:maintheorem1\], we obtain the characterization of the theorem. Appendix A Proofs of Theorem 1 and Theorem 2 {#appendix-a-proofs-of-theorem-1-and-theorem-2 .unnumbered} ============================================ - [*Setup*:]{} We assume that $\epsilon_l>0$ are given for every $l$. We fix some rates $R_l$, some blocklength $n$, and some $\epsilon>0$ and for every PMF $p_{X,\hat X,Y}=p_Xp_{\hat X|X}p_{Y|\hat X}$ compute the marginal $p_{Y}$. - [*Codebook design*:]{} Generate $\lfloor e^{n\left(R_l+\epsilon_l\right)} \rfloor$ length-$n$ codewords $\mathbf Y^{\left(l\right)}\left(w^{\left(l\right)}\right), w^{\left(l\right)}=1,\dots, \lfloor e^{n\left(R_l+\epsilon_l\right)} \rfloor$, by choosing each of the $n\lfloor e^{n\left(R_l+\epsilon_l\right)} \rfloor$ symbols $Y^{\left(l\right)}_k\left(w^{\left(l\right)}\right), \quad k=1,\dots,n$ independently at random according to $p_{Y}$ for $l=1,\dots,L$. - [*Encoder Design*:]{} For given sequences $\mathbf{x},\mathbf{\hat x_1},\dots,\mathbf{\hat x_L}$, the $l$-th encoder tries to find a $w^{\left(l\right)}$ such that $$\begin{gathered} \bigg( \hat {\mathbf x}_l,\mathbf Y^{\left(l\right)}\left(w^{\left(l\right)}\right)\bigg) \in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{\hat{X},Y}\big).\label{eq:jtypicality} \end{gathered}$$ If it finds several possible choices, they pick the first. If it finds none, it declares an error. The $l$-th encoder puts out $w^{\left(l\right)}$. Name $\mathbb L$ the set of indices $l$ for which the $l$-th encoder does not declare an error. - [*Decoder Design*:]{} The decoder $y^n$ puts out $\mathbf Y^{\left(l\right)}\left(w^{\left(l\right)}\right)$ for some $l\in \mathbb L$. - [*Performance Analysis*:]{} We define $\epsilon^\prime=\frac{\epsilon}{2|\mathbb X|}$ and partition the error space into three disjoint cases: (a) $\left(\mathbf x, \hat {\mathbf x}_l\right) \notin \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for some $l$ (b) $\left(\mathbf x, \hat {\mathbf x}_l\right) \in \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for every $l$ but $\mathbb L$ is empty (c) $\left(\mathbf x, \hat {\mathbf x}_l\right) \in \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for every $l$, $\mathbb L$ is not empty but is not satisfied. By the Union Bound and Lemma \[lem:TA\] (in Appendix B), we can bound the probability of Case (a) as $\Pr\left(\text{Case a}\right)\leq \sum_{l=1}^{L}\delta_{t}\left(n,\epsilon^\prime,\mathbb{X\times X}\right)$. In case (b), we get [rCl]{}&& ()\_[l=1]{}\^[L]{} ({(x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{})}\ &&{w\^[(l)]{}: (\_l,Y\^[(l)]{}(w\^[(l)]{}))\_[\^]{}\^[(n)]{}(p\_[X,Y]{}) })\ &&= \_[l=1]{}\^[L]{}\_[1]{}\ &&( w\^[(l)]{}: (\_l,Y\^[(l)]{}(w\^[(l)]{}))\_[\^]{}\^[(n)]{}(p\_[X,Y]{})\ && |(x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{}))\ && \_[l=1]{}\^[L]{}\_[w\^[(l)]{}=1]{}\^[e\^[n(R\_l+\_l)]{} ]{} ( (,Y\^[(l)]{}(w\^[(l)]{}))\_[\^]{}\^[(n)]{}(p\_[X,Y]{})\ &&|(x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{}))\ && = \_[l=1]{}\^[L]{}\_[w\^[(l)]{}=1]{}\^[e\^[n(R\_l+\_l)]{} ]{} ( (\_l,Y\^[(l)]{}(w\^[(l)]{}))\_[\^]{}\^[(n)]{}(p\_[X,Y]{})\ && |{ \_l \_[\^]{}\^[(n)]{}(p\_[X]{})}{x\_\^[(n)]{}(p\_[X,X]{}|\_l)})\[caseb1\]\ && = \_[l=1]{}\^[L]{}\_[w\^[(l)]{}=1]{}\^[e\^[n(R\_l+\_l)]{} ]{} ( Y\^[(l)]{}(w\^[(l)]{})\_[\^]{}\^[(n)]{}(p\_[X,Y]{}|x\_l)\ &&| \_l \_[\^]{}\^[(n)]{}(p\_[X]{}))\[caseb2\]\ &&=\_[l=1]{}\^[L]{}\_[w\^[(l)]{}=1]{}\^[e\^[n(R\_l+\_l)]{} ]{}( 1-\ &&(1-\_t(n,\^/2,XY)) e\^[-n(I(X;Y)+2\_m)]{})\[caseb3\]\ &&=\_[l=1]{}\^[L]{}(1-\ &&((1-\_t(n,\^/2,XY)) e\^[-n(I(X;Y)+2\_m)]{}))\^[e\^[n(R\_l+\_l)]{}]{}\[caseb4\] [rCl]{}&&=\_[l=1]{}\^[L]{}(-e\^[n(R\_l+\_l)]{}\ && (1-\_t(n,\^/2,XY)) e\^[-n(I(X;Y)+2\_m)]{})\[caseb5\]\ &&=\_[l=1]{}\^[L]{}(-e\^[n(R\_l-I(X;Y)+\_l-\_l)]{}), where $\delta_l$ accounts for the rounding mistake and includes the $2\epsilon_m$-term and the $\left(1-\delta_t\right)$-factor. So, we see that as long as $n$ is large enough, $R_l\geq I\left(\hat X;Y\right)$ for at least one $l$ and $\epsilon$ small enough such that $\delta_l<\epsilon_l$ for this $l$, the probability $\Pr\left(\text {Case b}\right)$ tends to zero double-exponentially fast in $n$. Here, results from Lemma \[lem:chtypsets\] (in Appendix B), follows again from Lemma \[lem:chtypsets\] (in Appendix B) and because we discard irrelevant information, follows from Lemma \[lem:TC\] (in Appendix B), holds because the factor in the product does not depend on $w$ anymore and follows from Lemma \[lem:expineq\] (in Appendix B). In case (c), [rCl]{} &&()\ &&({(x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{})l}{L }\ &&{lL:([x]{},Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[ X,Y]{})} )\ &&(l:{(x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{})}\ && {( \_[l]{},Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[X,Y]{})}\ &&{([x]{},Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[ X,Y]{})} )\ &&(l: {(x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{})}\ && {([x]{},\_l,Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[ X,,Y]{})} )\[caseb6\]\ &&\_[l=1]{}\^[L]{}\_[1]{}\ && ( ([x]{},\_l,Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[ X,,Y]{})\ &&| (x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{}) )\ &&\_[l=1]{}\^[L]{} ( ([x]{},\_l,Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[ X,,Y]{})\ &&| (x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{}) )\ &&= \_[l=1]{}\^[L]{} 1-( ([x]{},\_l,Y\^[(l)]{}(w\^[(l)]{})) \_\^[(n)]{}(p\_[ X,,Y]{}) [rCl]{}&&| (x, \_l) \_[\^]{}\^[(n)]{}(p\_[X,X]{}) )\ &&\_[l=1]{}\^[L]{}\_[t]{}(n,/2,XY) , where follows because, due to the fact that (strong) joint typicality implies pairwise typicality, we enlarge the set and the last step follows from Lemma \[lem:markov\] (in Appendix B). <!-- --> - [*Setup*:]{} We assume that $\epsilon_{\text{ag}}>0$ is given. We fix some rates per agent $R_{\text{ag}}$ and $R^{\prime}_{\text{ag}}$, some blocklength $n$, some $\epsilon>0$, $\epsilon_0>0$ and for every PMF $p_{X,\hat X,Y}=p_Xp_{\hat X|X}p_{Y|\hat X}$ compute the marginal $p_{Y}$. - [*Codebook design*:]{} Generate $\lfloor e^{n\left(R_{\text{ag}}+\epsilon_{\text{ag}}\right)} \rfloor \lceil e^{n\left(R^{\prime}_{\text{ag}}-\epsilon_0\right)} \rceil$ length-$n$ codewords $\mathbf Y^{\left(l\right)}\left(w^{\left(l\right)}, v^{\left(l\right)}\right), w^{\left(l\right)}=1,\dots, \lfloor e^{n\left(R_{\text{ag}}+\epsilon_{\text{ag}}\right)} \rfloor,v^{\left(l\right)}=1,\dots, \lceil e^{n\left(R^{\prime}_{\text{ag}}-\epsilon_0\right)} \rceil$, by choosing each of the $n\lfloor e^{n\left(R_{\text{ag}}+\epsilon_{\text{ag}}\right)} \rfloor \lceil e^{n\left(R^{\prime}_{\text{ag}}-\epsilon_0\right)} \rceil $ symbols $Y^{\left(l\right)}_k\left(w^{\left(l\right)},v^{\left(l\right)}\right)$ independently at random according to $p_{Y}$ for $l=1,\dots,L$. - [*Encoder Design*:]{} For given sequences $\mathbf{x},\hat{\mathbf{x}}_1,\dots,\hat{\mathbf{x}}_L$, the $l$-th encoder tries to find a pair $\left(w^{\left(l\right)},v^{\left(l\right)}\right)$ such that $$\begin{gathered} \bigg( \hat {\mathbf x}_l,\mathbf Y^{\left(l\right)}\left(w^{\left(l\right)},v^{\left(l\right)}\right)\bigg) \in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{\hat{X},Y}\big).\label{eq:jtypicality2} \end{gathered}$$ If it finds several possible choices, they pick the first. If it finds none, it declares an error. The $l$-th encoder puts out $w^{\left(l\right)}$. - [*Decoder Design*:]{} The decoder $y^n$ based on the bin numbers $\Big(w^{\left(1\right)},\dots,w^{\left(L\right)}\Big)$ that receives, it tries to find a tuple $\Big(v^{\left(1\right)},\dots,v^{\left(L\right)}\Big)$ and an $\mathbf x$ such that $\mathbf Y^{nL}= \bigg(\mathbf Y^{\left(1\right)}\left(w^{\left(1\right)},v^{\left(1\right)}\right),\dots,\mathbf Y^{\left(L\right)}\left(w^{\left(L\right)},v^{\left(L\right)}\right)\bigg)$ and $\mathbf x^{nL}=\left(\underbrace{\mathbf x,\dots,\mathbf x}_{L \quad \text{times}}\right)$ to be jointly typical i.e., [r]{}(x\^[nL]{},Y\^[nL]{})\_\^[(nL)]{}(p\_[X,Y]{}).\[eq:jtypicality3\] If it finds more than one $\big(v^{\left(1\right)},\dots,v^{\left(L\right)}\big)$ or none, it declares an error. Otherwise, it chooses some $j$ and puts out $\mathbf Y^{\left(j\right)}\left(w^{\left(j\right)},v^{\left(j\right)}\right)$. - [*Performance Analysis*:]{} We define $\epsilon^\prime=\frac{\epsilon}{2|\mathbb X|}$ and partition the error space into four disjoint cases: (a) $\left(\mathbf x, \hat {\mathbf x}_l\right) \notin \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for some $l$ (b) $\left(\mathbf x, \hat {\mathbf x}_l\right) \in \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for every $l$ but at least one encoder declares an error (c) $\left(\mathbf x, \hat {\mathbf x}_l\right) \in \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for every $l$, all encoders do not declare an error but the decoder finds none $\big(v^{\left(1\right)},\dots,v^{\left(L\right)}\big)$ (event $C_{\text{a}}$) or more than one (event $C_{\text{b}}$) (d) $\left(\mathbf x, \hat {\mathbf x}_l\right) \in \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for every $l$, all encoders do not declare an error and the decoder finds exactly one $\big(v^{\left(1\right)},\dots,v^{\left(L\right)}\big)$ but is not satisfied. By the Union Bound and Lemma 5 (in Appendix 5), we get $\Pr\left(\text{Case a}\right)\leq \sum_{l=1}^{L}\delta_{t}\left(n,\epsilon^\prime,\mathbb{X\times X}\right)$. Easily, it follows that $\Pr\left(\text{Case b}\right)\leq \sum_{l=1}^{L}\exp\Big(-e^{n\left(R_{\text{ag}}+R^\prime_{\text{ag}}-I\left(\hat X;Y\right)+\epsilon_{\text{ag}}-\delta\right)}\Big)$, where $\delta$ accounts for the rounding mistake and includes the $2\epsilon_m$-term and the $\left(1-\delta_t\right)$-factor. Hence, we see that as long as $n$ is large enough, $R_{\text{ag}}+R^\prime_{\text{ag}}\geq I\left(\hat X;Y\right)$ and $\epsilon$ small enough such that $\delta<\epsilon_{\text{ag}}$, the probability $\Pr\left(\text {Case b}\right)$ tends to zero double-exponentially fast in $n$. In case (c), we have $\Pr\left(\text{Case c}\right) =\Pr\left(C_{\text{a}}\cup C_{\text{b}}\right) =\Pr\left( C_{\text{a}}\right)+ \Pr\left( C_{\text{b}}-C_{\text{a}} \right)$. By , the fact that $\left(\mathbf x, \hat {\mathbf x_l}\right) \in \mathbb{A}_{\epsilon^\prime}^{\ast\left(n\right)}\big(p_{X,\hat X}\big)$ for every $l$ and the simple properties $P_{\hat{x}^{nL},Y^{\left(nL\right)}}=\frac{1}{L}\sum_{l=1}^{L}{P_{\hat{x}_{l},Y^{\left(l\right)}}}$, $P_{x^{nL},\hat{x}^{nL}}=\frac{1}{L}\sum_{l=1}^{L}{P_{x,\hat{x}_{l}}}$, it follows that $\left(\hat{\mathbf{x}}^{nL},\mathbf Y^{nL}\right)\in \mathbb{A}_{\epsilon}^{\ast\left(nL\right)}\big(p_{\hat{X},Y}\big)$ and $\left(\mathbf{x}^{nL},\hat{\mathbf{x}}^{nL}\right)\in \mathbb{A}_{\epsilon}^{\ast\left(nL\right)}\big(p_{X,\hat{X}}\big)$ where $ \hat{\mathbf{x}}^{nL}=\left(\hat {\mathbf{ x}}_1,\dots,\hat {\mathbf{ x}}_L\right)$. Lemma \[lem:markov\] (in Appendix B) gives us that $\Pr\left(C_{\text{a}}\right)\leq\delta_t\left(nL,\epsilon/2,\mathbb X \times \mathbb { X}\times \mathbb Y\right)$. We proceed with the event $C_{\text{b}}-C_{\text{a}}$. The cardinality of the set $\tilde{\mathbb Y}^{nL}\subseteq\mathbb Y^{nL}$ of the codewords which satisfy is bounded as $|\tilde{\mathbb Y}^{nL}|\leq|\mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{X}\big)|\max_{\substack{\mathbf x^{nL}\in \mathbb{A}_{\epsilon}^{\ast\left(nL\right)}\big(p_{X}\big)}}\big|\mathbb{A}_{\epsilon}^{\ast\left(nL\right)}\big(p_{X,Y}|\mathbf x^{nL}\big)\big|\leq e^{n\big(H\left(X\right)+\epsilon_m\big)}e^{nL\big(H\left(Y|X\right)+\epsilon_m\big)}\leq e^{nL\big(H\left(Y|X\right)+4\epsilon_m\big)}$, where the second inequality follows from Lemma \[lem:TA\] and Lemma \[lem:TB\] (in Appendix B) and the last inequality is true for $L\geq H\left(X\right)/{\epsilon_m}$. The probability for each element of this set to be chosen to a specific bin-tuple $\Big(w^{\left(1\right)},\dots,w^{\left(L\right)}\Big)$ is due to Lemma \[lem:TA\] (in Appendix B) at most $e^{-nL\big(H\left(Y\right)-\epsilon_m\big)}e^{nL\left(R^{\prime}_{\text{ag}}-\epsilon_0\right)}=e^{-nL\big(H\left(Y\right)-R^{\prime}_{\text{ag}}-\epsilon_m+\epsilon_0\big)}$. Hence, combining these two give us that $\Pr\left( C_{\text{b}}-C_{\text{a}}\right)\leq e^{-nL\big(I(X;Y)-R^{\prime}_{\text{ag}}-5\epsilon_m+\epsilon_0\big)}$. Therefore, we see that as long as $n$ is large enough, $L\geq H\left(X\right)/{\epsilon_m}$, $R^{\prime}_{\text{ag}}\leq I\left( X;Y\right)$ and $\epsilon$ small enough such that $\epsilon_m<\epsilon_0/5$, the probability $\Pr\left(\text {Case c}\right)$ tends to zero exponentially. Lemma \[lem:markov\] (in Appendix B) guarantees again that $\Pr\left(\text {Case d}\right)$ decays. So, collecting all the cases together gives us the desired result. Appendix B Typical Sets {#appendix-b-typical-sets .unnumbered} ======================= $\forall \theta>0, \quad \forall \xi\leq 1: \left(1-\xi\right)^\theta\leq e^{\theta\xi}$. \[lem:expineq\] $$\begin{gathered} \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\left(p_{X,Y}\right)\triangleq\\ \left. \left\{ \, \begin{IEEEeqnarraybox}[ \IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{2pt} {2pt}][c]{l} \left(\mathbf x,\mathbf y\right)\in \mathbb X^n\times \mathbb Y^n:\\ |P_{\mathbf x,\mathbf y}\left(a,b\right)-p_{X,Y}\left(a,b\right)|<\frac{\epsilon}{|\mathbb X||\mathbb Y|},\forall \left(a,b\right)\in \mathbb X\times \mathbb Y, \end{IEEEeqnarraybox}\right\}, \right. \label{def:typsets} \end{gathered}$$ [rCl]{}&&\_\^[(n)]{}(p\_[X,Y]{}|x){yY\^n:(x,y)\_\^[(n)]{}(p\_[X,Y]{}) }. \[def:ctypsets\] $\Big\{\left(\mathbf X,\mathbf Y\right)\in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\left(p_{X,Y}\right) \Big\} \iff \left\{\mathbf X\in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\left(p_{X}\right) \right\}\cap \left\{\mathbf Y\in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{X,Y}|\mathbf X\big)\right\}$. \[lem:chtypsets\] $\epsilon_m\big(p_{X,Y}\left(x,y\right)\big)\triangleq-\epsilon\log\left(p_{X,Y}^{\min}\right)$, $\delta_{t}\left(n,\epsilon,\mathbb{X}\times \mathbb{Y} \right)\triangleq \left(n+1\right)^{|\mathbb X||\mathbb Y|}e^{-n\frac{\epsilon^2}{2|\mathbb X|^2|\mathbb Y|^2}\log e}$, where $p_{X,Y}^{\min}$ is the smallest value of $p_{X,Y}\left(x,y\right)$. \[def:epsilondelta\] Let $\left(\mathbf x,\mathbf y\right) \in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\left(p_{X,Y}\right)$. Then, $e^{-n\Big(H\left(X,Y\right)+\epsilon_m\big(p_{X,Y}\left(x,y\right)\big)\Big)}< p_{X,Y}^n\left(\mathbf x,\mathbf y\right)< e^{-n\Big(H\left(X,Y\right)-\epsilon_m\big(p_{X,Y}\left(x,y\right)\big)\Big)}$. Moreover, $1-\delta_{t}\left(n,\epsilon,\mathbb{X}\times \mathbb{Y}\right)\leq\Pr\left[\left(\mathbf x, \mathbf y\right) \in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\left(p_{X,Y}\right)\right]\leq 1$ and $|\mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{X,Y}\big)|< e^{n\Big(H\left(X,Y\right)+\epsilon_m\big(p_{X,Y}\left(x,y\right)\big)\Big)}$. \[lem:TA\] $|\mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{X,Y}|\mathbf x\big)|< e^{n\Big(H\left(Y|X\right)+\epsilon_m\big(p_{X,Y}\left(x,y\right)\big)\Big)}$. \[lem:TB\] Let a PMF $p_{U,V,W}\left(u,v,w\right)$ with a Markov stracture $U-V-W$ and let $\left(\mathbf u,\mathbf v\right)\in \mathbb{A}_{\epsilon ^\prime}^{\ast\left(n\right)}\big(p_{U,V}\big)$ with $\epsilon ^ \prime \triangleq \frac{\epsilon}{2|\mathbb W|}$. Then, $p_{W|V}^n\left(\mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{U,V,W}|\mathbf u,\mathbf v\big)|\mathbf v\right)\geq 1-\delta_t\left(n,\epsilon/2,\mathbb U \times \mathbb V \times \mathbb W \right)$. \[lem:markov\] Let $p_{X,Y}\left(x,y\right)$ be a joint PMF with marginals $p_X\left(x\right), p_Y\left(y\right)$. Let $\left(\mathbf x,\mathbf y\right)$ be generated:$\left\{\left(x_k,y_k\right)\right\}_{k=1}^n \text{IID} \sim p_X\left(x\right)p_Y\left(y\right).$ If $\mathbf x \in \mathbb{A}_{\frac{\epsilon}{2|\mathbb Y|}}^{\ast\left(n\right)}\left(p_{X}\right)$, then, we obtain $\Pr\left[\mathbf Y\in \mathbb{A}_{\epsilon}^{\ast\left(n\right)}\big(p_{X,Y}|\mathbf x\big)\right]>\big(1-\delta_t\left(n,\frac{\epsilon}{2},\mathbb X\times \mathbb Y\right)\big) e^{-n\big(I\left(X;Y\right)+\epsilon_3\big)}$ where $\epsilon_3\triangleq \epsilon_m\big(p_{X,Y}\left(x,y\right)\big)+\epsilon_m\big(p_Y\left(y\right)\big)\leq 2\epsilon_m\big(p_{X,Y}\left(x,y\right)\big)$. \[lem:TC\]

--- abstract: 'We investigated the dissipative dynamics of quantum discord for correlated qubits under Markovian environments. The basic idea in the present scheme is that quantum discord is more general, and possibly more robust and fundamental, than entanglement. We provide three initially correlated qubits in pure Greenberger-Horne-Zeilinger (GHZ) or W state and analyse the time evolution of the quantum discord under various dissipative channels such as: Pauli channels $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$, as well as depolarising channels. Surprisingly, we find that under the action of Pauli channel $\sigma_{x}$, the quantum discord of GHZ state is not affected by decoherence. For the remaining dissipative channels, the W state is more robust than the GHZ state against decoherence. Moreover, we compare the dynamics of entanglement with that of the quantum discord under the conditions in which disentanglement occurs and show that quantum discord is more robust than entanglement except for phase flip coupling of the three qubits system to the environment.' author: - | M. Mahdian$^a$[^1], R. Yousefjani$^b$[^2] and S. Salimi$^b$[^3]\ [$^a$*Department of Theoretical Physics and Astrophysics, University of Tabriz,*]{}\ [*P.O.Box 51664 , Tabriz , Iran*.]{}\ [$^b$*Department of Physics, University of Kurdistan,*]{}\ [*P.O.Box 66177-15175 , Sanandaj, Iran*.]{} bibliography: - 'plain.bib' title: '**Quantum discord evolution of three-qubit states under noisy channels** ' --- -0.75in \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Remark]{} Introduction ============ One of the most remarkable properties of quantum mechanics is represented by the quantum correlation. Entanglement, which is a prominent feature of quantum correlation plays an important role in quantum computing and informational processing [@52; @53; @54; @55]. Recently, it has been perceived that entanglement is not the only kind of quantum correlation. In this context, others suitable measures of quantum correlations such as quantum discord [@20], quantum deficit [@60; @61; @62], quantumness of correlations [@63] and quantum dissonance [@64] have been proposed. Among them, quantum discord as a measure that is based on the difference between two quantum extensions of classically equivalent concepts has received considerable attention. This measure which quantifies the all nonclassical correlations between parts of a quantum system, actually supplements the measure of entanglement. For pure entangled states quantum discord coincides with the entropy of entanglement. It can also be nonzero for some mixed separable state. It is worth mentioning that quantum discord is related to other concepts such as Maxwell’s demons [@31; @32], quantum phase transitions [@33; @34], completely positive maps [@35], and relative entropy [@36]. Moreover, the characteristics of quantum discord have been studied in some physical models and information processing. It was shown that quantum discord can be considered as a more universal quantum resource than quantum entanglement in some sense. It offers new prospects for quantum information processing [@22; @24; @25; @26; @27]. Studying of the quantum discord evolution exposed to noisy environments has led to the surprising result that it may obviously differ from entanglement evolution. In fact, the unavoidable interaction of the systems with their environment, based on completely positive quantum dynamical semi groups, can be modelled by means of the noisy quantum channels. Quantum channels are completely positive and trace preserving maps between spaces of operators. The study of those can be broadly divided into Markovian and non-Markovian channels depend on the interaction of the system and environment. Markovian channels describes memoryless environments. The prototype of it is given by a quantum dynamical semi group, that is by solving a master equation for the reduced density matrix with Lindblad structure [@15; @37; @38]. For non-Markovian channels, environmental memory plays an important role, so the master equations describing their dynamics are often complicated integro-differential equations which are rarely exactly solvable [@37; @38].\ \ When a system of qubits with quantum correlation is exposed to noisy channels disentanglement can occur suddenly, but the quantum discord mostly decays in the asymptotic time [@57; @58; @59; @65; @66]. This points to a controversial fact that quantum discord may be more robust against decoherence than entanglement. Hence quantum algorithms that are based only on the quantum discord correlations can be more robust than those based on entanglement. The aim of this paper is to illustrate the mentioned subject for a system of three-qubit which is initially prepared in pure state by Greenberger-Horne-Zeilinger (GHZ) [@44] or W [@45] state as $$\begin{aligned} \label{1-1} |\psi ^{GHZ}\rangle&=&\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)\cr |\psi ^{W}\rangle&=&\frac{1}{2}(|100\rangle+|010\rangle+\sqrt{2}|001\rangle).\end{aligned}$$ Jung et al. [@1] showed that these pure states will be mixed due to transmission through some of the common channels for qubits. Moreover, it was shown that under these noisy channels the sudden death of entanglement of three qubits occurs in a finite time [@51]. The question is, what happens to quantum discord under the same conditions in which disentanglement can occur?\ In reply to this question we lead to remarkable results. The decoherence induced by the Pauli channel $\sigma_{x}$, can not affect the quantum discord of GHZ state contrary to the W state. We also show that the W state lose less quantum discord than the GHZ state due to transmission through the Pauli channels $\sigma_{y}$, $\sigma_{z}$ and the depolarising channel. Comparison of entanglement and quantum discord demonstrates that for the Pauli channels $\sigma_{x}$ and $\sigma_{y}$ and the depolarising channel, quantum discord is more robust against decoherence than entanglement.\ \ The organisation of this paper is as follows: Section $2$ is devoted to the necessary theoretical background to describe the time evolution of the system and introduces the global quantum discord. Evolution of quantum discord in transmission through the Pauli and the depolarising channels for the GHZ and W states is calculated in section $3$ and $4$, respectively. Finally, we summarise our results. Method ====== Time evolution of states under Markovian channels ------------------------------------------------- In an open quantum system the prototype of a Markov process is given by a quantum dynamical semigroup of a completely positive and trace preserving map $\Phi(t)=\exp[\pounds t]$, $t\geq 0$. In this case, quantum evolution of the system is given by the solution of a Markovian master equation with Lindblad structure for the reduced density matrix [@15], $$\begin{aligned} \label{1-2} \frac{d}{dt}\rho(t)&=&\pounds \rho(t)\cr &=& -i[H_{s},\rho(t)]+\sum_{i}L_{i}\rho(t)L_{i}^{\dag}-\frac{1}{2}\{L_{i}^{\dag}L_{i},\rho(t)\}.\end{aligned}$$ For any Pauli channel $\sigma_{\alpha}$ ($\alpha=x,y,z$), the decoherence dynamic is described by Lindblad operators $L_{A_{j},\alpha}\equiv\sqrt{\kappa_{A_{j},\alpha}}\,\sigma_{\alpha}^{A_{j}}$, $j=1,2,3$, which act independently upon the $j$-th qubit. In these operators, $\sigma_{\alpha}^{A_{j}}$ denote the Pauli matrices of the $j$-th qubit and the constants $\kappa_{A_{j},\alpha}$ are relaxation rates. For depolarising channel nine of these operators are needed. Here we assume that the Hamiltonian of the system is zero $H_{s}=0$, and the strength of the coupling between each of the qubits and channels is equal. Recently, Jung et al. [@1] analysed time evolution of three qubit GHZ and W states in the presence of noisy channels. Here we use their results. Quantum discord --------------- For a bipartite system AB quantum discord is given by [@20] $$\begin{aligned} \label{2-2} \textit{D}(\rho^{AB})=\textit{I}(\rho^{AB})- \textit{C}(\rho^{AB}),\end{aligned}$$ where $\textit{I}(\rho^{AB})=S(\rho^{A})+S(\rho^{B})-S(\rho^{AB})$, is quantum mutual information which includes the total correlation between A and B. The last section on the right represents classical correlation $\textit{C}(\rho^{AB})=max_{\{\Pi_{k}\}}[S(\rho^{A})-S(\rho^{AB}|\{\Pi_{k}\})]$ with $S(\rho)=-Tr[\rho \log_{2}\rho]$ as the von-Neumann entropy. Notice that the maximum is taken over the set of projective measurements $\{\Pi_{k}\}$ [@4].\ By definition the conditional density operator $\rho^{AB}_{k}=\frac{1}{p_{k}}\{(I^{A}\otimes \Pi_{k}^{B})\rho^{AB}(I^{A}\otimes \Pi_{k}^{B})\}$ with $p_{k}=Tr[(I^{A}\otimes \Pi_{k}^{B})\rho^{AB}]$ as the probability of obtaining the outcome $k$. We can define the conditional entropy of $A$ as $S(\rho^{AB}|\{\Pi_{k}\})=\sum_{k}p_{k}S(\rho_{k}^{A})$. This entropy includes the knowledge of subsystem $B$, with $\rho_{k}^{A}=Tr_{B}[\rho_{k}^{AB}]$ and $S(\rho_{k}^{A})=S(\rho_{k}^{AB})$. It has been shown that $\textit{D}(\rho^{AB})\geq 0$ with the equal sign, only for classical correlation [@5].\ Very recently, Rulli et al. [@3] have proposed a global measure of quantum discord based on a systematic extension of the bipartite quantum discord. Global quantum discord (GQD) which satisfy the basic requirements of a correlation function, for an arbitrary multipartite state $\rho^{A_{1}...A_{N}}$ under a set of local measurement $\{\Pi_{j}^{A_{1}}\otimes ... \otimes \Pi_{j}^{A_{N}}\}$ is defined as $$\begin{aligned} \label{2-3} \textit{D}(\rho^{A_{1}...A_{N}})=\min_{\{\Pi_{k}\}}\,[S(\rho^{A_{1}...A_{N}}\|\Phi(\rho^{A_{1}...A_{N}}))-\sum_{j=1}^{N}S(\rho^{A_{j}}\|\Phi_{j}(\rho^{A_{j}}))].\end{aligned}$$ Where $\Phi_{j}(\rho^{A_{j}})=\sum_{i}\Pi_{i}^{A_{j}}\rho^{A_{j}}\Pi_{i}^{A_{j}}$ and $\Phi(\rho^{A_{1}...A_{N}})=\sum_{k}\Pi_{k}\rho^{A_{1}...A_{N}}\Pi_{k}$ with $\Pi_{k}=\Pi_{j_{1}}^{A_{1}}\otimes ... \otimes\Pi_{j_{N}}^{A_{N}}$ and $k$ denoting the index string ($j_{1}...j_{N}$). We could eliminate dependence on measurement by minimising the set of projectors $\{\Pi_{j_{1}}^{A_{1}}, ... ,\Pi_{j_{N}}^{A_{N}}\}$.\ With these remarks about the global quantum discord (4), one can describe the time evolution of the quantum discord for three-qubit GHZ and W states when they are passed through a noisy channel. By selecting a set of von-Neumann measurements as $$\begin{aligned} \Pi_{1}^{A_{j}}=\left(\matrix{\cos^{2}(\frac{\theta_{j}}{2})& \,\,\,\,\,\,\,\,\,e^{i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2})\cr e^{-i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2})& \,\,\,\,\,\,\,\,\,\sin^{2}(\frac{\theta_{j}}{2})}\right), \cr \cr \cr \Pi_{2}^{A_{j}}=\left(\matrix{\sin^{2}(\frac{\theta_{j}}{2})& -e^{-i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2})\cr -e^{i\varphi_{j}}\cos(\frac{\theta_{j}}{2})\sin(\frac{\theta_{j}}{2})&\cos^{2}(\frac{\theta_{j}}{2})}\right),\end{aligned}$$ the quantum discord for $\rho=\rho^{GHZ}(t)$ or $\rho^{W}(t)$ can be obtained from $$\begin{aligned} \label{2-3} \textit{D}(\rho)=\min_{\{\theta_{j},\varphi_{j}\}}\,[S(\rho\|\Phi(\rho))-\sum_{j=1}^{3}S(\rho^{A_{j}}\|\Phi_{j}(\rho^{A_{j}}))].\end{aligned}$$ Where $\theta_{j}\in[0,\pi)$ and $\varphi_{j}\in[0,2\pi)$, for $j=1,2,3$, are azimuthal and polar angles, respectively. Quantum discord of three-qubit system with initial GHZ state under Markovian noise channel ========================================================================================== Pauli channel $\sigma_{x}$ -------------------------- When a three-qubit system with initial GHZ state is coupled to a shift-flip noise channel, in which each qubit is coupled to its own channel, the time evolution is obtained by the solution of the master equation (2) with Lindbald operators $L_{j,x}\equiv\sqrt{\kappa_{j,x}}\,\sigma_{x}^{A_{j}}, (j=1,2,3$). For this coupling, the master equation reduces to 8 diagonal and 28 off-diagonal coupled linear differential equations. So after transmission of the GHZ through the Pauli channel $\sigma_{x}$ the density matrix is given by [@1] $$\begin{aligned} \label{8-3} \rho_{x}^{GHZ}(t)=\frac{1}{8}\left(\matrix{\alpha_{+}&0&0&0&0&0&0&\alpha_{+}\cr 0&\alpha_{-}&0&0&0&0&\alpha_{-}&0 \cr 0&0&\alpha_{-}&0&0&\alpha_{-}&0&0 \cr 0&0&0&\alpha_{-}&\alpha_{-}&0&0&0 \cr 0&0&0&\alpha_{-}&\alpha_{-}&0&0&0 \cr 0&0&\alpha_{-}&0&0&\alpha_{-}&0&0 \cr 0&\alpha_{-}&0&0&0&0&\alpha_{-}&0 \cr \alpha_{+}&0&0&0&0&0&0&\alpha_{+}}\right),\end{aligned}$$ with $$\begin{aligned} \label{9-3} \alpha_{+}&\equiv & 1+3e^{-4\kappa t},\cr \alpha_{-}&\equiv & 1-e^{-4\kappa t}.\end{aligned}$$ For this case, the lower bound to the concurrence is [@51] $$\begin{aligned} \label{10-3} \tau_{3}(\rho_{x}^{GHZ}(t))=e^{-4\kappa t}.\end{aligned}$$ In order to find the analytical expression for the quantum discord with $\rho_{x}^{GHZ}(t)$ we must consider equation (6). By tracing out two qubits, the one qubit density matrices representing the individual subsystems are proportional to the identity operator $$\begin{aligned} \label{12-3} \rho_{x}^{(A_{1})}(t)=\rho_{x}^{(A_{2})}(t)=\rho_{x}^{(A_{3})}(t)=\frac{\alpha_{+}+3\alpha_{-}}{8}I.\end{aligned}$$ Therefore, we have $S(\rho_{x}^{(j)}(t)\| \Phi_{j}(\rho_{x}^{(j)}(t)))=0$ $(j=1,2,3)$ and the equation(6) reduces to $$\begin{aligned} \label{13-3} \textit{D}(\rho_{x}^{GHZ}(t))&=&\min_{\{\theta_{j},\varphi_{j}\}}\{ S(\rho_{x}^{GHZ}(t)\Vert \Phi(\rho_{x}^{GHZ}(t)))\}\cr &=&\min_{\{\theta_{j},\varphi_{j}\}}\{ S(\Phi(\rho_{x}^{GHZ}(t)))-S(\rho_{x}^{GHZ}(t))\}.\end{aligned}$$ The entropy $S(\rho_{x}^{GHZ}(t))$ can be obtained as $$\begin{aligned} \label{13-3} S(\rho_{x}^{GHZ}(t))=2-\frac{3\alpha_{-}}{4}\log_{2}\alpha_{-}-\frac{\alpha_{+}}{4}\log_{2}\alpha_{+}.\end{aligned}$$ In order to obtain the maximum classical correlation among the parts of $\rho_{x}^{GHZ}(t)$, by varying the angles $\theta_{j}$ and $\varphi_{j}$, we must find the measurement bases that minimise $\textit{D}(\rho_{x}^{GHZ}(t))$. After some calculation we have perceived that, by adopting local measurements in the $\sigma_{z}$ eigenbases for each particle, the value of quantum discord will be minimised. So the von-Neumann entropy, after completing such measurements, can be written as $$\begin{aligned} \label{13-3} S(\Phi(\rho_{x}^{GHZ}(t)))=3-\frac{3\alpha_{-}}{4}\log_{2}\alpha_{-}-\frac{\alpha_{+}}{4}\log_{2}\alpha_{+}.\end{aligned}$$ By substituting equations (12) and (13) into equation (11) quantum discord is readily found to be $$\begin{aligned} \label{14-3} \emph{D}(\rho_{x}^{GHZ}(t))=1.\end{aligned}$$ As can be seen, the Pauli channel $\sigma_{x}$ can not change the quantum discord of the three qubit systems with the initial GHZ state. We have plotted the dynamic evolution of the quantum discord for density matrix (7) versus the dimensionless scaled time $\kappa t$ in Fig. 1 (dashed violet ). Pauli channel $\sigma_{y}$ -------------------------- If the three-qubit GHZ state are transmitted through the Pauli channel $\sigma_{y}$, its density matrix takes the following form [@1] $$\begin{aligned} \label{16-3} \hspace{-7mm} \rho_{y}^{GHZ}(t)=\frac{1}{8}\left(\matrix{\alpha_{+}&0&0&0&0&0&0&\beta_{1}\cr 0&\alpha_{-}&0&0&0&0&-\beta_{2}&0 \cr 0&0&\alpha_{-}&0&0&-\beta_{2}&0&0 \cr 0&0&0&\alpha_{-}&-\beta_{2}&0&0&0 \cr 0&0&0&-\beta_{2}&\alpha_{-}&0&0&0 \cr 0&0&-\beta_{2}&0&0&\alpha_{-}&0&0 \cr 0&-\beta_{2}&0&0&0&0&\alpha_{-}&0 \cr \beta_{1}&0&0&0&0&0&0&\alpha_{+}}\right),\end{aligned}$$ where $\alpha_{\pm}$ are given in equation (8) and, $\beta_{1}$ and $\beta_{2}$ are defined as $$\begin{aligned} \label{17-3} \beta_{1}&\equiv&3e^{-2\kappa t}+e^{-6\kappa t},\cr \beta_{2}&\equiv&e^{-2\kappa t}-e^{-6\kappa t}.\end{aligned}$$ For this matrix, the entanglement vanishes after some finite time due to the condition [@51] $$\begin{aligned} \label{18-3} \tau_{3}(\rho_{y}^{GHZ}(t))=max\{0,\frac{1}{4}(3e^{-2\kappa t}+e^{-4\kappa t}+e^{-6\kappa t}-1)\}.\end{aligned}$$ The reduced density matrices of subsystem of (15), is the same as equation (10). Hence, the quantum discord becomes $$\begin{aligned} \label{20-3} \textit{D}(\rho_{y}^{GHZ}(t))=\min_{\{\theta_{j},\varphi_{j}\}}\{ S(\Phi(\rho_{y}^{GHZ}(t)))-S(\rho_{y}^{GHZ}(t))\}.\end{aligned}$$ When the projective measurement in the eigenprojectors of $\sigma_{z}$ is performed on any one of the remaining qubits of $\rho_{y}^{GHZ}(t)$, the minimum quantum discord is obtained as $$\begin{aligned} \label{21-3} \textit{D}(\rho_{y}^{GHZ}(t))&=&\frac{(\alpha_{+}-\beta_{1})}{8}\log_{2}(\alpha_{+}-\beta_{1})+\frac{(\alpha_{+}+\beta_{1})}{8}\log_{2}(\alpha_{+}+\beta_{1})\cr &+&\frac{3(\alpha_{-}-\beta_{2})}{8}\log_{2}(\alpha_{-}-\beta_{2})+\frac{3(\alpha_{-}+\beta_{2})}{8}\log_{2}(\alpha_{-}+\beta_{2})\cr &-&\frac{\alpha_{+}}{4}\log_{2}(\alpha_{+})-\frac{3\alpha_{-}}{4}\log_{2}(\alpha_{-}).\end{aligned}$$ Quantum discord is reduced due to the Pauli noisy channel $\sigma_{y}$ and disappears with low speed, as shown in Fig. 1 (solid blue ). Pauli channel $\sigma_{z}$ -------------------------- Transmission of the GHZ state through the Pauli channel $\sigma_{z}$ result in the master equation (2) reduces to 8 diagonal and 28 off-diagonal first order differential equations with a simply trivial solution. So $\rho^{GHZ}(0)=|GHZ \rangle\langle GHZ|$ is evolved into [@1] $$\begin{aligned} \label{1-3} \hspace{-9mm}\rho_{z}^{GHZ}(t)=\frac{1}{2}(\vert 000 \rangle\langle 000 \vert + \vert 111 \rangle\langle 111 \vert ) + \frac{1}{2}e^{-6\kappa t}(\vert 000 \rangle\langle 111 \vert + \vert 000 \rangle\langle 111 \vert ). \nonumber \\\end{aligned}$$ For this mixed state, the lower bound to concurrence is a mono-exponential function of time $$\begin{aligned} \label{1-3} \tau_{3}(\rho_{z}^{GHZ}(t))=e^{-6\kappa t}.\end{aligned}$$ Let us now focus on the quantum discord for $\rho_{z}^{GHZ}(t)$ as defined by (6). After tracing out two qubits, the three reduce density matrices are equal, given by $\rho_{z}^{(A_{1})}(t)=\rho_{z}^{(A_{2})}(t)=\rho_{z}^{(A_{3})}(t)=\frac{I}{2}$. Hence, we have $S(\rho_{z}^{(A_{j})}(t)\Vert \Phi_{j}(\rho_{z}^{(A_{j})}(t)))=0$ ($j=1,2,3$) and the equation (6) reduces to $$\begin{aligned} \label{5-3} \textit{D}(\rho_{z}^{GHZ}(t))=\min_{\{\theta_{j},\varphi_{j}\}}\{ S(\Phi(\rho_{z}^{GHZ}(t)))-S(\rho_{z}^{GHZ}(t))\}.\end{aligned}$$ The von-Neumann entropy of $\rho_{z}^{GHZ}(t)$ is $$\begin{aligned} \label{5-3} \hspace{-10mm}S(\rho_{z}^{GHZ}(t))=\{1-\frac{1+e^{-6\kappa t}}{2}\log_{2}(1+e^{-6\kappa t})-\frac{1-e^{-6\kappa t}}{2}\log_{2}(1-e^{-6\kappa t})\}.\nonumber \\\end{aligned}$$ We have found that for the density matrix (20), measurements in the eigenprojectors of $\sigma_{z}^{j}$, that is boundary values $\varphi_{j}=0$ and $\theta_{j}=0$, minimise $\Phi(\rho_{z}^{GHZ}(t))$ as $$\begin{aligned} \label{5-3} \Phi(\rho_{z}^{GHZ}(t))=\frac{1}{2}\{|+++\rangle\langle +++|+ |---\rangle\langle ---|\},\end{aligned}$$ which cause $S(\Phi(\rho_{z}^{GHZ}(t)))=1$. In these circumstances, the quantum discord is expressed as $$\begin{aligned} \label{6-3} \textit{D}(\rho_{z}^{GHZ}(t))=\frac{1+e^{-6\kappa t}}{2}\log_{2}(1+e^{-6\kappa t})+\frac{1-e^{-6\kappa t}}{2}\log_{2}(1-e^{-6\kappa t}).\end{aligned}$$ In Fig. 1, we have plotted the dynamic evolution of the quantum discord for the density matrix (20) as a function of $\kappa t$ (solid red). It can be seen that the quantum discord decreases from its maximal value $\textit{D}(\rho_{z}^{GHZ}(t))=1$ and vanishes after some finite time. Depolarising channel --------------------- For depolarising noise which is described by nine Lindblad operators, $L_{j,z}$, $L_{j,x}$ and $L_{j,y}$ ($j=1,2,3$) the state of three-qubit system that were initially described by the GHZ state replaces $$\begin{aligned} \label{23-3} \rho_{d}^{GHZ}(t)=\frac{1}{8}\left(\matrix{\tilde{\alpha}_{+}&0&0&0&0&0&0&\gamma\cr 0&\tilde{\alpha}_{-}&0&0&0&0&0&0 \cr 0&0&\tilde{\alpha}_{-}&0&0&0&0&0 \cr 0&0&0&\tilde{\alpha}_{-}&0&0&0&0 \cr 0&0&0&0&\tilde{\alpha}_{-}&0&0&0 \cr 0&0&0&0&0&\tilde{\alpha}_{-}&0&0 \cr 0&0&0&0&0&0&\tilde{\alpha}_{-}&0 \cr \gamma&0&0&0&0&0&0&\tilde{\alpha}_{+}}\right),\end{aligned}$$ where $$\begin{aligned} \label{24-3} \tilde{\alpha}_{+}&\equiv & 1+3e^{-8\kappa t}\cr \tilde{\alpha}_{-}&\equiv & 1-e^{-8\kappa t}\cr \gamma&\equiv &4e^{-12\kappa t}.\end{aligned}$$ Since $\rho_{d}^{(A_{j})}(t)=\frac{\tilde{\alpha}_{+}+3\tilde{\alpha}_{-}}{8}I$, then $S(\rho_{d}^{(A_{j})}(t)\Vert \Phi_{j}(\rho_{d}^{(A_{j})}(t)))=0$. The quantum discord, due to the condition of depolarizing channel may be written as $$\begin{aligned} \label{27-3} \textit{D}(\rho_{d}^{GHZ}(t))=\min_{\theta_{j},\varphi_{j}}\{ S(\Phi(\rho_{d}^{GHZ}(t)))-S(\rho_{d}^{GHZ}(t))\}.\end{aligned}$$ We find that measurements in the eigenprojectors of $\sigma_{z}^{j}$ minimise the quantum discord. After some algebraic calculation it is found that $$\begin{aligned} \label{28-3} \textit{D}(\rho_{d}^{GHZ}(t))&=&\frac{(\alpha_{+}+\gamma)}{8}\log_{2}(\alpha_{+}+\gamma)\cr &+&\frac{(\alpha_{+}-\gamma)}{8}\log_{2}(\alpha_{+}-\gamma)-\frac{\alpha_{+}}{4} \log_{2}(\alpha_{+}).\end{aligned}$$ The time evolution of quantum discord (6) as a function of $\kappa t$ in the case of the depolarising channel is plotted in Fig. 1 (solid black). Due to the noisy channel, quantum discord decays from its maximum value $\textit{D}(\rho_{z}^{GHZ}(0))=1$ and disappears after some finite time.\ \ As we have pointed out previously, transmission of the GHZ state through the Pauli channel $\sigma_{x}$ contrary to other channels, could not disturb the quantum discord. We also observed that the quantum discord of the initial state due to transmission of the GHZ state through the Pauli channel $\sigma_{y}$ decayed less quickly than $\sigma_{z}$ and the depolarising channel, as shown in Fig. 1. In order to comparse quantum discord and entanglement we have used the results of reference [@51]. In Fig. 2, the time dependent entanglement of the GHZ state under the noisy channels is shown. In all cases, one can see decrease of the entanglement due to the noise of the channels. Clearly, under the dissipative dynamics considered here, except for phase flip, coupling of the three qubits system to the environment, quantum discord is more robust than entanglement.\ In the next section we will discuss the effects of noisy channels when we prepare three qubits with initial W state. Quantum discord of three-qubit systems with initial W state under Markovian noise channel ========================================================================================= Pauli channels $\sigma_{x}$ and $\sigma_{y}$ -------------------------------------------- If the three-qubit system is initially prepared in the W state, a similar analysis as in the previous section to compute the quantum discord can be made. Under the bit flip or bit-phase flip coupling to environment the density matrix of three-qubit W state at time t has the following analytical expression [@1] $$\begin{aligned} \label{8-3} \hspace{-25mm} \rho_{\pm}^{W}(t)=\frac{1}{16}\left(\matrix{ 2\alpha_{2}&0&0&\pm\sqrt{2}\alpha_{2}&0&\pm\sqrt{2}\alpha_{2}&\pm\alpha_{2}&0 \cr 0&2\alpha_{1}&\sqrt{2}\alpha_{1}&0&\sqrt{2}\alpha_{1}&0&0&\pm\alpha_{3} \cr 0&\sqrt{2}\alpha_{1}&2\beta_{+}&0&\alpha_{1}&0&0&\pm\sqrt{2}\alpha_{3} \cr \pm\sqrt{2}\alpha_{2}&0&0&2\beta_{-}&0&\alpha_{4}&\sqrt{2}\alpha_{4} &0 \cr 0&\sqrt{2}\alpha_{1}&\alpha_{1}&0&2\beta_{+}&0&0&\pm\sqrt{2}\alpha_{3} \cr \pm\sqrt{2}\alpha_{2}&0&0&\alpha_{4}&0&2\beta_{-}&\sqrt{2}\alpha_{4}&0 \cr \pm\alpha_{2}&0&0&\sqrt{2}\alpha_{4}&0&\sqrt{2}\alpha_{4}&2\alpha_{4}&0 \cr 0&\pm\alpha_{3}&\pm\sqrt{2}\alpha_{3}&0&\pm\sqrt{2}\alpha_{3}&0&0&2\alpha_{3}} \right),\cr\end{aligned}$$ where the $+$ sign refers to the $\sigma_{x}$ and $-$ to the $\sigma_{y}$ channel, respectively. The density matrix elements are given by $$\begin{aligned} \label{8-3} \alpha_{1}&=&1+e^{-2\kappa t}+ e^{-4\kappa t}+e^{-6\kappa t}\cr \alpha_{2}&=&1+e^{-2\kappa t}- e^{-4\kappa t}-e^{-6\kappa t}\cr \alpha_{3}&=&1-e^{-2\kappa t}- e^{-4\kappa t}+e^{-6\kappa t}\cr \alpha_{4}&=&1-e^{-2\kappa t}+ e^{-4\kappa t}-e^{-6\kappa t}\cr \beta_{\pm}&=&1\pm e^{-6\kappa t}.\end{aligned}$$ It worth mentioning that since two density matrices $\rho_{\pm}^{W}(t)$ have the same structure of matrix elements, we expect that the quantum discord dynamic of these density matrices coincide. After tracing out any two qubits, the reduced density matrices are given by $$\begin{aligned} \label{12-3} \rho_{\pm}^{(A_{j})}(t)=\frac{1}{4}\left(\matrix{2+e^{-2\kappa t}&0\cr 0&2-e^{-2\kappa t}}\right) \,\,\,\,\, j=1,2,3.\end{aligned}$$ To find the measurement bases that minimise quantum discord, we used the same procedure as in the previous section. We perceived that the maximum classical correlation for any reduced density matrix (32) was obtained by completing the projective measurements in the eigenprojectors of $\sigma_{z}$. It leads to $S(\rho_{\pm}^{(A_{j})}(t)\Vert \Phi_{j}(\rho_{\pm}^{(A_{j})}(t)))=0$. The minimum of $\textit{D}(\rho_{x}^{W}(t))$ and $\textit{D}(\rho_{y}^{W}(t))$ is obtained after the two project measurements in the eigneprojectors of $\sigma_{x}$ and $\sigma_{z}$ on $\rho_{x}^{W}(t)$ and $\rho_{y}^{W}(t)$, respectively. We do not show the achieved analytic expressions for the quantum discord here because its form are not compact. We have plotted the quantum discord for density matrix (30) as a function of the dimensionless scaled time $\kappa t$ at Fig. 3 (dashed violet). Here, we can see that quantum discord due to the transmission through $\sigma_{x}$ and $\sigma_{y}$ channels, as expected, coincide and decrease at a low rate of speed from its initial value $\textit{D}(\rho_{\pm}^{W}(0))=1.5$ and the asymptotic limit select the exact value $0.813$. Pauli channel $\sigma_{z}$ -------------------------- After transmission through the Pauli channel $\sigma_{z}$, the time evolution of the W state is described by [@1] $$\begin{aligned} \label{8-3} \rho_{z}^{W}(t)=\frac{1}{4}\left(\matrix{0&0&0&0&0&0&0&0\cr 0&2&\sqrt{2}e^{-4\kappa t}&0&\sqrt{2}e^{-4\kappa t}&0&0&0 \cr 0&\sqrt{2}e^{-4\kappa t}&1&0&e^{-4\kappa t}&0&0&0 \cr 0&0&0&0&0&0&0&0 \cr 0&\sqrt{2}e^{-4\kappa t}&e^{-4\kappa t}&0&1&0&0&0 \cr 0&0&0&0&0&0&0&0 \cr 0&0&0&0&0&0&0&0 \cr 0&0&0&0&0&0&0&0}\right).\end{aligned}$$ The density matrices representing each of the one qubit subsystem of (33) are given by $\rho_{z}^{(A_{j})}(t)=\frac{1}{4}\{3|0\rangle\langle0|+|1\rangle\langle1|\}$ ($j=1,2,3$). By using the same procedure as above to find the measurement bases that minimise $\textit{D}(\rho_{z}^{W}(t))$, we find that by performing the projective measurement in the eigenprojectors of $\sigma_{z}^{j}$ quantum discord reaches its lowest point. Thus one can verify that $S(\rho_{z}^{(A_{j})}(t)\Vert \Phi_{j}(\rho_{z}^{(A_{j})}(t)))=0$ and $S(\Phi(\rho_{z}^{W}(t)))=\frac{3}{2}$. The von-Neumann entropy of $\rho_{z}^{W}(t)$ density matrix is found as $$\begin{aligned} \label{27-3} S(\rho_{z}^{W}(t))&=&\frac{1}{4}(11+e^{-4\kappa t})-\frac{1}{4}(1-e^{-4\kappa t})\log_{2}(1-e^{-4\kappa t})\cr &-&\frac{1}{8}\{(3+e^{-4\kappa t}-\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\cr &&\log_{2}(3+e^{-4\kappa t}-\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\cr &+&(3+e^{-4\kappa t}+\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\cr && \log_{2}(3+e^{-4\kappa t}+\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\}.\end{aligned}$$ Therefore, the expression for quantum discord $\textit{D}(\rho_{z}^{W}(t))$ is given by $$\begin{aligned} \label{27-3} \textit{D}(\rho_{z}^{W}(t))&=&\frac{-1}{4}(5+e^{-4\kappa t})+\frac{1}{4}(1-e^{-4\kappa t})\log_{2}(1-e^{-4\kappa t})\cr &+&\frac{1}{8}\{(3+e^{-4\kappa t}-\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\cr &&\log_{2}(3+e^{-4\kappa t}-\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\cr &+&(3+e^{-4\kappa t}+\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\cr &&\log_{2}(3+e^{-4\kappa t}+\sqrt{1-2e^{-4\kappa t}+17e^{-8\kappa t}})\}.\end{aligned}$$ With the above channels, quantum discord of the selected system with the initial W state disappear under Pauli channel $\sigma_{z}$ after a finite time. This is displayed in Fig. 3 (solid red). Depolarising channel -------------------- When three-qubite W state is transmitted through the depolarising channel, its time evolution is described by [@1] $$\begin{aligned} \label{8-3} \hspace{-15mm} \rho_{d}^{W}(t)=\frac{1}{8}\left(\matrix{ \tilde{\alpha}_{2}&0&0&0&0&0&0&0 \cr 0&\tilde{\alpha}_{1}&\sqrt{2}\tilde{\gamma}_{+}&0&\sqrt{2}\tilde{\gamma}_{+}&0&0&0 \cr 0&\sqrt{2}\tilde{\gamma}_{+}&\tilde{\beta}_{+}&0&\tilde{\gamma}_{+}&0&0&0 \cr 0&0&0&\tilde{\beta}_{-}&0&\tilde{\gamma}_{-}&\sqrt{2}\tilde{\gamma}_{-} &0 \cr 0&\sqrt{2}\tilde{\gamma}_{+}&\tilde{\gamma}_{+}&0&\tilde{\beta}_{+}&0&0&0 \cr 0&0&0&\tilde{\gamma}_{-}&0&\tilde{\beta}_{-}&\sqrt{2}\tilde{\gamma}_{-}&0 \cr 0&0&0&\sqrt{2}\tilde{\gamma}_{-}&0&\sqrt{2}\tilde{\gamma}_{-}&\tilde{\alpha}_{4}&0 \cr 0&0&0&0&0&0&0&\tilde{\alpha}_{3}} \right),\cr\end{aligned}$$ with $$\begin{aligned} \label{8-3} \tilde{\alpha}_{1}&=&1+e^{-4\kappa t}+ e^{-8\kappa t}+e^{-12\kappa t}\cr \tilde{\alpha}_{2}&=&1+e^{-4\kappa t}- e^{-8\kappa t}-e^{-12\kappa t}\cr \tilde{\alpha}_{3}&=&1-e^{-4\kappa t}- e^{-8\kappa t}+e^{-12\kappa t}\cr \tilde{\alpha}_{4}&=&1-e^{-4\kappa t}+ e^{-8\kappa t}-e^{-12\kappa t}\cr \tilde{\beta}_{\pm}&=&1\pm e^{-12\kappa t}\cr \tilde{\gamma}_{\pm}&=&e^{-8\kappa t}\pm e^{-12\kappa t}.\end{aligned}$$ For density matrix (36), the single qubit density matrix is given by $$\begin{aligned} \label{12-3} \rho_{d}^{(A_{j})}(t)=\frac{1}{4}\left(\matrix{2+e^{-4\kappa t}&0\cr 0&2-e^{-4\kappa t}}\right) \,\,\, j=1,2,3.\end{aligned}$$ Our results show that the minimum quantum discord for the depolarising coupling of the three-qubit W state is attained at boundary values $\varphi_{j}=0$ and $\theta_{j}=0$, that is for a local measurement along the eigenstates of Pauli matrix $\sigma_{z}$. The state of the single qubit is not induced by such measurement, therefore $S(\rho_{d}^{(A_{j})}(t)\Vert \Phi_{j}(\rho_{d}^{(A_{j})}(t)))=0$. The von-Neumann entropy of $\rho_{d}^{W}(t)$ after measurement is given explicitly by $$\begin{aligned} \label{12-3} S(\Phi(\rho_{d}^{W}(t)))&=&\frac{1}{4}\{18-\tilde{\alpha}_{1}\log_{2}\tilde{\alpha}_{1}-\tilde{\alpha}_{2}\log_{2}\tilde{\alpha}_{2}\cr &-&\tilde{\alpha}_{3}\log_{2}\tilde{\alpha}_{3}-\tilde{\alpha}_{4}\log_{2}\tilde{\alpha}_{4}-\tilde{\beta}_{+}\log_{2}\tilde{\beta}_{+} -\tilde{\beta}_{-}\log_{2}\tilde{\beta}_{-}\}.\end{aligned}$$ In this case, the time evolution of quantum discord is obtained as $$\begin{aligned} \label{12-3} \hspace{-15mm} \textit{D}(\rho_{d}^{W}(t))&=&\frac{1}{2}(3-e^{-8\kappa t})-\frac{1}{4}\{\tilde{\beta}_{+}\log_{2}\tilde{\beta}_{+} +\tilde{\beta}_{-}\log_{2}\tilde{\beta}_{-}\cr&+&\tilde{\alpha}_{1}\log_{2}\tilde{\alpha}_{1}+2\tilde{\alpha}_{2}\log_{2}\tilde{\alpha}_{2} +2\tilde{\alpha}_{3}\log_{2}\tilde{\alpha}_{3}+\tilde{\alpha}_{4}\log_{2}\tilde{\alpha}_{4}\}\cr&+&\frac{1}{8}\{(\tilde{\beta}_{-}-\tilde{\gamma}_{-})\log_{2}(\tilde{\beta}_{-}-\tilde{\gamma}_{-})+ (\tilde{\beta}_{+}-\tilde{\gamma}_{+})\log_{2}(\tilde{\beta}_{+}-\tilde{\gamma}_{+})\}\cr&+&\frac{1}{16}\{(\tilde{\beta}_{+}+\tilde{\gamma}_{+}+\tilde{\alpha}_{1} -\sqrt{\tilde{\beta}_{+}^{2}+\tilde{\gamma}_{+}(2\tilde{\beta}_{+}+17\tilde{\gamma}_{+})-2\tilde{\alpha}_{1}(\tilde{\beta}_{+}+\tilde{\gamma}_{+}) +\tilde{\alpha}_{1}^{2}})\cr&&\log_{2}(\tilde{\beta}_{+}+\tilde{\gamma}_{+}+\tilde{\alpha}_{1} -\sqrt{\tilde{\beta}_{+}^{2}+\tilde{\gamma}_{+}(2\tilde{\beta}_{+}+17\tilde{\gamma}_{+})-2\tilde{\alpha}_{1}(\tilde{\beta}_{+}+\tilde{\gamma}_{+}) +\tilde{\alpha}_{1}^{2}})\cr&+& (\tilde{\beta}_{+}+\tilde{\gamma}_{+}+\tilde{\alpha}_{1} +\sqrt{\tilde{\beta}_{+}^{2}+\tilde{\gamma}_{+}(2\tilde{\beta}_{+}+17\tilde{\gamma}_{+})-2\tilde{\alpha}_{1}(\tilde{\beta}_{+}+\tilde{\gamma}_{+}) +\tilde{\alpha}_{1}^{2}})\cr&&\log_{2}(\tilde{\beta}_{+}+\tilde{\gamma}_{+}+\tilde{\alpha}_{1} +\sqrt{\tilde{\beta}_{+}^{2}+\tilde{\gamma}_{+}(2\tilde{\beta}_{+}+17\tilde{\gamma}_{+})-2\tilde{\alpha}_{1}(\tilde{\beta}_{+}+\tilde{\gamma}_{+}) +\tilde{\alpha}_{1}^{2}})\cr&+& (\tilde{\beta}_{-}+\tilde{\gamma}_{-}+\tilde{\alpha}_{4} -\sqrt{\tilde{\beta}_{-}^{2}+\tilde{\gamma}_{-}(2\tilde{\beta}_{-}+17\tilde{\gamma}_{-})-2\tilde{\alpha}_{4}(\tilde{\beta}_{-}+\tilde{\gamma}_{-}) +\tilde{\alpha}_{4}^{2}})\cr&&\log_{2}(\tilde{\beta}_{-}+\tilde{\gamma}_{-}+\tilde{\alpha}_{4} -\sqrt{\tilde{\beta}_{-}^{2}+\tilde{\gamma}_{-}(2\tilde{\beta}_{-}+17\tilde{\gamma}_{-})-2\tilde{\alpha}_{4}(\tilde{\beta}_{-}+\tilde{\gamma}_{-}) +\tilde{\alpha}_{4}^{2}})\cr&+& (\tilde{\beta}_{-}+\tilde{\gamma}_{-}+\tilde{\alpha}_{4} +\sqrt{\tilde{\beta}_{-}^{2}+\tilde{\gamma}_{-}(2\tilde{\beta}_{-}+17\tilde{\gamma}_{-})-2\tilde{\alpha}_{4}(\tilde{\beta}_{-}+\tilde{\gamma}_{-}) +\tilde{\alpha}_{4}^{2}})\cr&&\log_{2}(\tilde{\beta}_{-}+\tilde{\gamma}_{-}+\tilde{\alpha}_{4} +\sqrt{\tilde{\beta}_{-}^{2}+\tilde{\gamma}_{-}(2\tilde{\beta}_{-}+17\tilde{\gamma}_{-})-2\tilde{\alpha}_{4}(\tilde{\beta}_{-}+\tilde{\gamma}_{-}) +\tilde{\alpha}_{4}^{2}})\}. \nonumber \\\end{aligned}$$ The responses of quantum discord with the initial W state to the different noises as functions of $\kappa t$ are listed at Fig. 3. As in the case of the GHZ state, the quantum discord for the W state decays due to the noisy channels. When a three-qubit system with initial W or GHZ state is coupled to a depolarising noise channel decay of the quantum discord occurs so fast. This means that the depolarising channel has the most destructive influence on the quantum discord. However, the quantum discord decreases due to the transmission through the Pauli channels $\sigma_{x}$ or $\sigma_{y}$, but it catches the exact value $0.813$ in the long term as shown in Fig. 3. This figure also exhibits the vanishing of the quantum discord for the initial W state under the phase flip channel. These results show that GHZ-type quantum discord is more fragile under certain types of environment coupling which can be modelled by means of the Pauli channels $\sigma_{y}$ and $\sigma_{z }$ as well as the depolarising channel as compared to the quantum discord of the W state. However, in the Pauli channel $\sigma_{x}$ the situation becomes completely reversed.\ In order to show the difference between the entanglement and quantum discord under the various dissipative channels, by using the results of [@51] we have plotted the evolution of entanglement of the W state in Fig. 4. As in the case of the GHZ state, the entanglement for the W state decays exponentially due to the noisy channels. Observe that under the dissipative dynamics considered, except for the phase flip coupling of the three qubits system to the environment, discord is more robust than entanglement. SUMMARY AND CONCLUSION ====================== To sum up, we prepared the three-qubit system with the initial state formed by GHZ and W states. We then studied the dynamics of quantum discord under interaction with independent Markovian environments which can be modelled by means of the various noisy channels, namely, the Pauli channels $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ as well as the depolarising channel. It was clarified that coupling with the bit flip channel could not disorder the quantum discord of three-qubit GHZ state unlike in the W state. In other words, for this noisy channel, GHZ-type quantum discord is always more robust than quantum discord of the W state. In the Pauli channels $\sigma_{y}$, $\sigma_{z}$ and depolarising channel, the W state preserved more quantum discord than the GHZ state in the long term.\ Also, we observed that under the dissipative Markovian dynamics considered, except for the Pauli channel $\sigma_{z}$ quantum discord is more robust than entanglement. This points to the fact that quantum discord is another kind of quantum correlation different from entanglement and the absence of entanglement does not necessarily indicate the absence of quantum correlations. References {#references .unnumbered} ========== [99]{} J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. **73**, 565 (2001). Ting Yu and J. H. Eberly, Phys. Rev. Lett. **93**, 140404 (2004). B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. A **77**, 032342 (2008). A. Jamróz, J. Phys. A: Math. Gen. **39** 7727 (2006). H. Ollivier and W. H. Zurek, Phys. Rev. Lett. **88**, 017901 (2001). A. K. Rajagopal and R. W. Rendell, Phys. Rev. A **66**, 022104 (2002). M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De), U. Sen, and B. Synak-Radtke, Phys. Rev. A **71**, 062307 (2005). I. Devetak, Phys. Rev. A **71**, 062303 (2005). A. R. Usha Devi and A. K. Rajagopal, Phys. Rev. Lett. **100**, 140502 (2008). K. Modi, T. Paterek, W. Son, V. Vedral, and M. Williamson, Phys. Rev. Lett. **104**, 080501 (2010). A. Brodutch and D. R. Terno, Phys. Rev. A **81**, 062103 (2010). W. H. Zurek, Phys. Rev. A **67**, 012320 (2003). J. Maziero, H. C. Guzman, L. C. Céleri, M. S. Sarandy, and R. M. Serra, Phys. Rev. A **82**, 012106 (2010). R. Dillenschneider, Phys. Rev. B **78**, 224413 (2008). A. Brodutch and D. R. Terno, Phys. Rev. A **81**, 062103 (2010). A. Shabani and D. A. Lidar, Phys. Rev. Lett. **102**, 100402 (2009). A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett. **100**, 050502 (2008). M. S. Sarandy, Phys. Rev. A **80**, 022108 (2009). J. Cui and H. Fan, J. Phys. A: Math. Theor. **43** 045305 (2010). A. Datta and G. Vidal, Phys. Rev. A **75**, 042310 (2007). B. P. Lanyon, M. Barbieri, M. P. Almeida, and A. G. White, Phys. Rev. Lett. **101**, 200501 (2008). G. Lindblad, Commun. Math. Phys. **48**, 119 (1976). Heinz-Peter Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002). Heinz-Peter Breuer, Elsi-Mari Laine, and J. Piilo, Phys. Rev. Lett. **103**, 210401 (2009). F. F. Fanchini, T. Werlang, C. A. Brasil, L. G. E. Arruda, and A. O. Caldeira, Phys. Rev. A. **81**, 052107 (2010). B. Wang, Zhen-Yu Xu, Ze-Qian Chen, and M. Feng, Phys. Rev. A **81**, 014101 (2010). Zhao-Yu Sun, L. Li, Kai-Lun Yao, Gui-Huan Du, Ji-Wei Liu, B. Luo, N. Li, and Hai-Na Li, Phys. Rev. A **82**, 032310 (2010). T. Werlang, S. Souza, F. F. Fanchini, and C. J. Villas Boas, Phys. Rev. A **80**, 024103 (2009). A. Auyuanet and L. Davidovich, Phys. Rev. A **82**, 032112 (2010). D. M. Greenberger, M. A. Horne, A. Zeilinger, Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989) p 69. W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A **62**, 062314 (2000). E. Jung, Mi-Ra Hwang, Y. Hwan Ju, Min-Soo Kim, Sahng-Kyoon Yoo, H. Kim, D. Park, Jin-Woo Son, S. Tamaryan, and Seong-Keuck Cha, Phys. Rev. A **78**, 012312 (2008). M. Siomau and S. Fritzsche, Eur. Phys. J. D **60**, 397 (2010). C. C. Rulli and M. S. Sarandy, Phys. Rev. A **84**, 042109 (2011). L. Henderson and V. Vedral, J. Phys. A: Math. Gen. **34** 6899 (2001). A. Datta, arXiv:1003.5256. **Figure Caption** FIG. 1. (Colour online) The quantum discord (6) for the three-qubit system with the initial GHZ state as a function of $\kappa t$, if transmitted through Markovian channels: Pauli channel $\sigma_{x}$ (dashed violet), $\sigma_{y}$ (solid blue), $\sigma_{z}$ (solid red) and depolarising channel (solid black). FIG. 2. (Colour online) The entanglement for the three-qubit system with the initial GHZ state as a function of $\kappa t$, if transmitted through Markovian channels: Pauli channel $\sigma_{x}$ (dashed violet), $\sigma_{y}$ (solid blue), $\sigma_{z}$ (solid red) and depolarising channel (solid black). FIG. 3. (Colour online) The quantum discord (6) for the three-qubit system with the initial W state as a function of $\kappa t$, if transmitted through Markovian channels: Pauli channel $\sigma_{x}$ and $\sigma_{y}$ (dashed violet), $\sigma_{z}$ (solid red) and depolarising channel (solid black). FIG. 4. (Colour online) The entanglement for the three-qubit system with the initial W state as a function of $\kappa t$, if transmitted through Markovian channels: Pauli channel $\sigma_{x}$ (dashed violet), $\sigma_{y}$ (solid blue), $\sigma_{z}$ (solid red) and depolarising channel (solid black). [^1]: Mahdian@tabrizu.ac.ir [^2]: R.yousefjany@uok.ac.ir [^3]: shsalimi@uok.ac.ir

--- abstract: | We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for the two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet. Keywords: rarefaction wave; transonic shock; Riemann problem; multidimensional conservation laws; nonlinear wave system. AMS: [Primary: 76L05, 35L65; Secondary: 65M06, 35M33.]{} address: - 'Department of Mathematics and Statistics, California State University, Long Beach, CA 90840, USA. Email: EunHeui.Kim@csulb.edu ' - 'Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA. Email: tsikkou@math.wvu.edu' author: - Eun Heui Kim - Charis Tsikkou title: 'Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions' --- [^1] [^2] Introduction {#intro} ============ It is well known that the wave patterns created by initial value problems for the multidimensional compressible Euler system are complicated and challenging to study. As such, numerous works are focused on special initial data that allow us to reduce the space dimensions. In particular two-dimensional Riemann problems can be reduced to self-similar problems, which are interesting to study on their own. For instance, the flow in self-similar coordinates changes its type; it is supersonic (hyperbolic) in the far-field and becomes mixed near the origin. A seminal work by Zhang and Zheng [@ZZ] illustrated various complicated wave patterns on four sectorial Riemann data for two-dimensional self-similar Euler systems. Their conjectures are later validated numerically, see for example [@KT; @LaxLiu; @SR]. In particular Lax and Liu [@LaxLiu] attested the complicated wave patterns including Mach reflections, rolling up and instability of slip lines, and much more. They also noted that a general theory of Glimm type for one-dimensional problems would not be likely established for the multidimensional problems. While establishing a comprehensive analysis to understand the complicated wave patterns in two dimensional flows with Riemann data still has a long way to go, there is recent progress describing the solution structures of transonic regular shock reflection problems for various systems, [@CKK; @CKK:WRR; @CKKmixed; @CKK:nlwe; @Chen; @ChenFang; @EL; @JKC; @KimJDE10; @KimCPDE; @KimLeeNA13; @Morawetz; @serrefrei; @sever; @Zheng1]. This is an incomplete list of related work specifically on transonic shocks by Riemann problems or by a wedge, and interested readers can refer to the references therein. On the other hand, relatively little is known about transonic rarefaction waves. The computational result by Glimm [*et al*]{} [@Glimm] showed the shock formations from the rarefaction wave interactions. This paper addresses the understanding of rarefaction wave interactions and their shock formations in a specific self-similar problem, the nonlinear wave system in two space dimensions. More precisely, the simple wave created by the planar rarefaction wave becomes a different wave with two families of non-trivial characteristics. Both families of characteristics merge into becoming sonic near the origin and the type of the system then changes to subsonic, that is the characteristics no longer transport the data any further. We call this wave a [*transient wave*]{}. On the other hand, there exists a family emanating from this transient wave region, becoming compressive and forming a shock downstream. Thus the type changes and forms a sonic boundary in some part and a transonic shock in the other. We note that a similar configuration was studied by [@SongZheng] for the pressure gradient system. They [@SongZheng; @Zhengbook] called the region – where a family of characteristics starts on sonic curves and ends on transonic shock curves – the “semi-hyperbolic” region. [@SongZheng] established the existence of a local solution in a given semi-hyperbolic region, provided a smooth convex boundary and small Riemann data, and [@WangZheng] established the local regularity result for this solution. Our work is motivated by the work of [@Bang; @DaiZhang; @SongZheng; @WangZheng]. The nonlinear wave system, which can be considered as wave motions of shallow water and multidimensional $p$-systems, is a reduced system of the compressible Euler system for isentropic, irrotational flow in two space dimensions [@CKKmixed; @CKK:nlwe]. The nonlinear wave system can be also considered as a part of an operator splitting scheme in numerics, where the compressible Euler system can be split into the nonlinear wave system (the pressure system) and the pressure-less system (the gradient flow). In fact [@Zhengbook] noted that the Euler system can be split into the pressure-gradient system and the pressure-less system, see [@Song; @Zheng1] and the references therein. The pressure-gradient system is a special case of the nonlinear wave system. The pressure-less system is well understood by [@SZ]. Hence if one understands the solution structure of the nonlinear wave system, then one can construct the solution of the Euler system successively by using the splitting method. Furthermore, there are many similarities on the structures of both the nonlinear wave system and the Euler system, see [@CKKmixed; @serre]. As such, it is crucial to understand the nonlinear wave system in order to study the Euler system. We focus on the wave patterns created by planar rarefaction waves. For the configuration, we impose four sectorial Riemann data $U_i$, $i=1,2,3,4$ for each $ith$ quadrant, see the left figure in Figure \[fig\_Rdata\]. [0.3]{} \[\]\[\]\[0.7\]\[0\][$R_{14}$]{} \[\]\[\]\[0.7\]\[0\][$R_{34}$]{} \[\]\[\]\[0.7\]\[0\][$J_{12}$]{} \[\]\[\]\[0.7\]\[0\][$J_{23}$]{} \[\]\[\]\[0.7\]\[0\][$U_1$]{} \[\]\[\]\[0.7\]\[0\][$U_4$]{} \[\]\[\]\[0.7\]\[0\][$U_3$]{} \[\]\[\]\[0.7\]\[0\][$U_2$]{} [0.3]{} \[\]\[\]\[0.7\]\[0\][$C_1$]{} \[\]\[\]\[0.7\]\[0\][$C_4$]{} \[\]\[\]\[0.7\]\[0\][$R_{14}$]{} \[\]\[\]\[0.7\]\[0\][$R_{34}$]{} We consider planar waves with a horizontal rarefaction wave $R_{14}$ created by the constant data $U_1$ and $U_4$, and another vertical rarefaction wave $R_{34}$ created by $U_3$ and $U_4$. The contact discontinuities reside along the positive $y-$axis created by $U_1$ and $U_2$, and along the negative $x-$axis by $U_2$ and $U_3$, which have no effect on the system (the reduction of the system makes the contact discontinuities trivial), see the right figure in Figure \[fig\_Rdata\]. The data are symmetric with respect to $x=-y$ and thus it suffices to focus only on $R_{14}$. Near the locus of the sonic circle (it is the origin for the nonlinear wave system) the type of the flow changes and becomes subsonic, creating a transonic shock downstream. We formulate the boundary value problem and establish the supersonic solution for the entire hyperbolic region of this configuration. Since the change of the type is not known a priori, the problem has two different types of free boundaries: the sonic boundary and the transonic shock boundary. We show that the sonic boundary in this configuration inherits at least two types of degeneracies. The first obvious one is that the wave across the sonic boundary becomes degenerate meaning it is neither hyperbolic nor elliptic. The degeneracy of this type is well known as a Tricomi type problem, where the characteristics enter the sonic boundary perpendicularly. The Tricomi type degeneracy appears, for example, in the airflow over a wing where the steady subsonic flow creates a supersonic region over the convex surface of the wing creating a shock, see Courant and Friedrichs [@CF]. This is a long standing open problem. Numerical studies [@Tesdall1; @Tesdall2] suggest that similar wave patterns are observed for the Mach reflection problem near the Mach stem. We establish the existence of the supersonic solution, which becomes sonic, and that the solution and the sonic boundary are $C^1$. The second one is when a family of characteristics creates a compression wave downstream, the sonic boundary and the transonic shock merge into a point, and at that point, the angular derivative of the solution also disappears. To our knowledge, this is a new type of free boundary problem. We note that the Tricomi type degeneracy, since the characteristics enter the sonic boundary perpendicularly, implies that the sonic boundary is never the characteristics. Thus we utilize both the directional derivatives along the sonic boundary and the data transported along the characteristics to have the solution and the sonic boundary to be $C^1$. The solution may not be $C^1$ at the point where the sonic and shock boundaries meet, and at that point, the sonic boundary no longer has a Tricomi type degeneracy but the angular derivative of the solution becomes zero. Our results provide an insight to understand how the compressive wave is created by the expansion wave. For multidimensional conservation laws, the entropy conditions are insufficient to answer whether our solution is the physically relevant one. However, our solution captures the numerics, see Section \[numerics\]. This paper provides a framework to establish existence of the supersonic solution suggested by the numerics, and an analysis to understand how the type of the rarefaction wave changes. The complete analysis to construct the transonic wave in the entire region including the subsonic region will be discussed in our forthcoming paper. The main contributions of this paper are the following. We first formulate the boundary value problem for the self-similar nonlinear wave system. We next discuss the wave patterns, monotonicity properties, regularity and existence results. The solution will be constructed locally and then assembling the pieces together along the characteristics and the sonic boundary. We show that the sonic boundary created by the transient waves is $C^1$ and is strictly increasing radially. This sonic boundary is terminated and radially tangential when it merges to the transonic shock downstream. Numerical results by using CLAWPACK for certain pressures are presented as well. We believe our results will serve as a vehicle for understanding transonic flows in particular the long standing open problem of the flow over the convex wing, and lead to further developments of systematic theories for multi-dimensional conservation laws. Interested readers can refer to the survey paper [@SChen10] for the comprehensive references and recent progress in transonic problems. Description of the problem ========================== Nonlinear wave system: Configuration ------------------------------------ From the compressible Euler system for isentropic flow in two space dimensions, ignoring the nonlinear velocity terms (assuming low velocities) and assuming irrotational flow, we can deduce a simpler system, the nonlinear wave system [@CKKmixed]; $$\label{nlwesys} \begin{array}{rcl} \rho_t + (\rho u)_x + (\rho v)_y &=&0\\ (\rho u)_t+ p_x &=&0\\ (\rho v)_t +p_y &=&0. \end{array}$$ Here $\rho(t, x,y)$ is the density, $ u(t,x,y)$ and $v(t,x,y)$ are the $x$ and $y$ components of velocity, respectively, and $p(\rho)$ is the pressure satisfying a polytropic gas law $$\frac{dp}{d\rho} = c^2(\rho) =k \gamma \rho^{\gamma-1},$$ with constants $k$ (we let $k=1$ for simplicity), $1<\gamma<\infty$, (typically $1<\gamma<2$, $\gamma=5/3$ is air), and a local sound speed $c^2(\rho)$. This system can be considered as wave motions of shallow water and multidimensional $p$-systems. We let the momentum $(\rho u, \rho v) =(m,n)$ and use $U$ to denote $(\rho, m,n)$: $$U=(\rho, m,n)=(\rho, \rho u, \rho v).$$ Specifically we have the following Riemann data $U_i=(\rho_i, m_i, n_i)$ where $i=1,2,3,4$ for each quadrant satisfying; $$\begin{aligned} R_{14} :& \rho_1> \rho_4, \ m_1=m_4, \ n_1- n_4= \Phi_{14}\\ J_{12} : &\rho_1=\rho_2, \ m_1=m_2, \ n_1 > n_2\\ J_{23}: &\rho_2=\rho_3, \ m_2 > m_3, \ n_2=n_3\\ R_{34}: &\rho_3 > \rho_4, \ m_3 -m_4 = -\Phi_{34}, \ n_3=n_4,\end{aligned}$$ where $$\begin{aligned} \Phi_{ij} &=& \int^{\rho_i}_{\rho_j} c(s) ds = \frac{2}{\gamma+1} \left( \rho_i^{\frac{\gamma+1}{2}} - \rho_j^{\frac{\gamma+1}{2}} \right).\end{aligned}$$ That is, $$\begin{aligned} \rho_1=\rho_2=\rho_3>\rho_4, \ m_1=m_2=m_4 > m_3, \ n_2=n_3=n_4 < n_1.\end{aligned}$$ Hence we impose four sectorial data; $$\begin{aligned} U_1&=& (\rho_1, 0, \Phi_{14})^t\\ U_2 &=& (\rho_1, 0,0)^t\\ U_3&=& (\rho_1, -\Phi_{14}, 0)^t\\ U_4&=& (\rho_4,0,0)^t.\end{aligned}$$ We let $c_1=c(\rho_1)$ and $c_4=c(\rho_4)$ and denote two sonic circles $C_1=\{(c_1,\theta), 0\le \theta\le 2\pi\}$ and $C_4=\{ (c_4,\theta),0\le \theta\le 2\pi\}$ corresponding to the Riemann data $\rho_1>\rho_4$. We replace $c^2(\rho)= \gamma p^\kappa$, $\kappa = (\gamma -1)/\gamma$, and write the system in self-similar coordinates $\xi=x/t, \eta=y/t$; $$\label{selfsys} \begin{array}{rcl} -\frac{1}{\gamma} p^{-\kappa} (\xi p_\xi + \eta p _\eta) + m_\xi + n_\eta &=& 0,\\ -\xi m_\xi -\eta m_\eta+ p_\xi &=& 0,\\ -\xi n_\xi -\eta n_\eta + p_\eta &=& 0. \end{array}$$ The system can be written in a second order equation (by applying $\partial_\xi$ to the second equation and $\partial_\eta$ to the third equation in and replacing the momentum terms with their derivatives from the first equation in ) $$\label{Pself} p_{\xi\xi} + p_{\eta\eta}=\frac{1}{\gamma}p^{-\kappa}(\xi p_{\xi}+\eta p_{\eta})+\xi \bigg(\frac{1}{\gamma} p^{-\kappa} (\xi p_\xi + \eta p _\eta)\bigg)_\xi + \eta \bigg(\frac{1}{\gamma} p^{-\kappa} (\xi p_\xi + \eta p _\eta)\bigg)_\eta,$$ which can be written in polar coordinates $r^2=\xi^2 +\eta^2,\theta = \tan \eta/\xi$: $$\begin{aligned} \label{Pnd} r^2 \bigg(1 -\frac{r^2}{c^2} \bigg) p_{rr} + p_{\theta\theta} + r(1-2\frac{r^2}{c^2}) p_r + \kappa \frac{r^2}{c^2} \frac{r^2}{p} p^2_{r} =0.\end{aligned}$$ The system is hyperbolic (supersonic) when $c^2 <r^2$, sonic when $c^2 =r^2$, and elliptic (subsonic) when $c^2 >r^2$. In the following section we discuss the characteristic equations for the system in the supersonic region. Characteristic equations in the supersonic region ------------------------------------------------- When the state is hyperbolic, that is $c^2 <r^2$, we let $$\begin{aligned} \lambda_\pm &=& \pm r\sqrt{\frac{r^2 -c^2}{c^2}}, \quad {\rm or \ simply} \quad \lambda = r\sqrt{\frac{r^2 -c^2}{c^2}},\end{aligned}$$ and the positive and negative characteristics derived by integrating $\frac{dr}{d\theta} =\lambda_{\pm}=\pm\lambda,$ respectively, so that reads $$\begin{aligned} \label{Pchar} p_{\theta\theta} - \lambda^{2} p_{rr} &=& -\frac{r}{c^2}(c^2 -2r^2)p_r -\frac{\kappa r^4}{c^2 p} p_r^2.\end{aligned}$$ In addition, by letting $\partial_{\pm} =\partial_\theta \pm \lambda \partial_r$, we have, as in [@SongZheng], $$\begin{aligned} \partial_-\partial_+ p &=& \frak{h}(\partial_- p -\partial_+ p)\partial_+ p\\ \partial_+\partial_- p &=& \frak{h}(\partial_+ p -\partial_- p)\partial_- p,\end{aligned}$$ where $$\begin{aligned} \frak{h}&=&\frac{r^2 {(c^2)'}}{4c^2(r^2 -c^2)} =\frac{r^4{(c^2)'}}{4c^4\lambda^2}, \quad {(c^2)'}=\frac{dc^2}{dp} =\gamma \kappa p^{\kappa -1} = (\gamma-1)p^{\kappa -1}.\end{aligned}$$ We denote $$\begin{aligned} R=\partial_+ p, && S=\partial_-p\end{aligned}$$ so that $$\begin{aligned} p_\theta = \frac{1}{2}(R+S), && p_r =\frac{1}{2\lambda } (R-S),\end{aligned}$$ and $$\begin{aligned} {\partial_-}R &=& \frak{h}(S-R)R\\ {\partial_+}S &=& \frak{h}(R-S)S,\end{aligned}$$ which can be written as $$\begin{aligned} \label{pRS} \left( \begin{array}{c} p \\ R \\ S \end{array} \right)_\theta + \left( \begin{array}{ccc} 0 & 0& 0\\ 0 & -\lambda &0 \\ 0&0&\lambda \end{array} \right) \left( \begin{array}{c} p \\ R \\ S \end{array} \right)_r &=& \left( \begin{array}{c} \frac{1}{2}(R+S) \\ \frak{h}(S-R)R \\ \frak{h}(R-S)S \end{array} \right).\end{aligned}$$ In the following section, we discuss and formulate the boundary value problem for the system. Derivations of the boundary value problem and the main result ============================================================= \[\]\[\]\[0.7\]\[0\][$\Xi_1$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_2$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_3$]{} \[\]\[\]\[0.7\]\[0\][$C_1$]{} \[\]\[\]\[0.7\]\[0\][$C_4$]{} \[\]\[\]\[0.7\]\[0\][$\xi$]{} \[\]\[\]\[0.7\]\[0\][$\eta$]{} \[\]\[\]\[0.7\]\[0\][$\Sigma$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma$-]{} \[l\]\[\]\[0.7\]\[0\][$\Gamma_{12}$]{} \[\]\[\]\[0.7\]\[0\][$\sigma$]{} \[\]\[\]\[0.7\]\[0\][$\Sigma_{34}$]{} \[\]\[\]\[0.7\]\[0\][$R_{14}$]{} \[\]\[\]\[0.7\]\[0\][$U_1$]{} \[\]\[\]\[0.7\]\[0\][$U_4$]{} \[\]\[\]\[0.7\]\[0\][$\sigma_1$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{23}$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{24}$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_4$]{} \[\]\[\]\[0.7\]\[0\][$\eta=-\xi$]{} \[\]\[\]\[0.7\]\[0\][$\Sigma_0$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_0$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_1$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_2$]{} When the rarefaction wave $R_{14}: c^2(p)=\eta^2$, where $p_4\le \eta \le p_1$, enters the sonic circle $C_1$, a new wave pattern is created. Specifically, the wave pattern changes from the simple wave to two wave families of non-constant solutions, and these two families of characteristics merge and become sonic near the positive $\eta$-axis. We denote the sonic boundary created by these two families of characteristics by $\sigma.$ In addition, there exists a wave family connecting the sonic boundary and the transonic shock boundary. The transient wave region (consists of two wave families of non-constant solutions) is enclosed by two limiting characteristics and the sonic boundary. More precisely, let $\Gamma_{12}$ be the positive characteristic emanating from $\Xi_1=(r_1,\pi/2)$ to $\Xi_2=(r_2, \theta_2)$, which separates the simple wave $\mathcal{R}_0=\{c^2(p)=\eta^2\}$ and the transient wave. $\Gamma_{12}$ is completely determined by the simple wave $\mathcal{R}_0$. $\Xi_2=(r_2, \theta_2)$ is the point at which this characteristic crosses $\eta=c_4$, where $r_2=c_4/\sin\theta_2$ and $\theta_2= \sin^{-1}\sqrt{c_4/c_1}$. Let $\Gamma_{23}$ be the negative characteristic emanating from $\Xi_2$ to $\Xi_3$ where $\Xi_3=(r_3,\theta_3)$ is the point at which the characteristic speed becomes zero, that is, a sonic point. The transient wave region is the hyperbolic region enclosed by characteristics $\Gamma_{12}$ and $\Gamma_{23}$, and the sonic boundary $\sigma$. On the other hand, there exists yet another new simple wave created downstream, which is adjacent to the constant state $p=p_4$. More precisely, when the governing hyperbolic system is reducible to a first order homogeneous system, the new state adjacent to the constant state forms a simple wave, see Courant and Friedrichs [@CF]. We refer to Dafermos [@Dafermos] for details of the general framework of the simple wave. This simple wave becomes compressive and creates a transonic shock downstream. In summary, the transonic boundary is created by the transient wave (the sonic boundary) in part, and the simple wave (the shock boundary) in the other. Hence it is crucial to identify the corresponding characteristics first to separate these different wave regions, in order to formulate the correct boundary problems. Below we list the notation used for the characteristics and the corner points that we just discussed. Let $$\begin{aligned} \Gamma_{12} &=& \{ (r,\theta): r=c_1\sin\theta, \ \theta_2\le \theta \le \pi/2\}\\ \Gamma_{23} &=& \{(r,\theta): r=r_{23}(\theta), \ \theta_2\le \theta \le \theta_3\},\end{aligned}$$ and $$\begin{aligned} \Xi_1&=&\left(c_1, \frac{\pi}{2} \right)\\ \Xi_2 &=& \left(\frac{c_1}{\sin\theta_2}, \sin^{-1} \sqrt{\frac{c_4}{c_1}} \right) = \left(\sqrt{c_1c_4}, \sin^{-1} \sqrt{\frac{c_4}{c_1}} \right) \\ \Xi_3 &=& (r_3,\theta_3).\end{aligned}$$ The supersonic region is divided into three regions based on the characteristics where $R=\partial_+p$ and $S=\partial_-p$ are either strictly positive or zero, and denoted by; - Rarefaction wave region ${{\mathcal{R}_0}}$: $R>0$ and $S=0$, see Section \[sectionR\_0\]. - Transient wave region $\mathcal{R}_1$: Both families $R$ and $S$ are non-trivial, enclosed by the positive characteristic $\Gamma_{12}$ emanating from $\Xi_1$ to $\Xi_2$; the negative characteristic $\Gamma_{23}$ from $\Xi_2$ to $\Xi_3$; and the sonic boundary $\sigma$ on which $R=S$. See Sections  \[sectionR\_1\] and \[sectionR\_1existence\]. - Simple wave region $\mathcal{R}_2$: $R=0$ and $S>0,$ see Section \[sectionR\_2\]. The boundary of this region consists of the negative characteristic $\Gamma_{23}$ emanating from $\Xi_2$ to $\Xi_3$; the positive characteristic, denoted by $\Gamma_{24}=\{ (r_{24}(\theta),\theta)\}$, emanating from $\Xi_2$ to the shock $\Sigma$ at the point $\Xi_4$, where $r_{24}(\theta)=c(p_4)\sec\bigg(\theta+\operatorname{arcsec}\sqrt{\frac{c(p_1)}{c(p_4)}}-\arcsin\sqrt{\frac{c(p_4)}{c(p_1)}}\bigg)$; and a part of the shock $\Sigma'\subset \Sigma$ from $\Xi_3$ to $\Xi_4$. In region $\mathcal{R}_1$, we have the following Goursat boundary value problems: $$\begin{aligned} \label{eqp} p_\theta &=& \frac{1}{2}(R+S), \quad p \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} \eta^{2/\kappa}, \quad p\mid_{\Gamma_{23}}= z, \\ \label{eqR} {\partial_-}R &=& \frak{h}(S-R)R, \quad R \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} {\partial_+}\eta^{2/\kappa}, \quad R\mid_{\Gamma_{23}}= {\partial_+}z,\\ \label{eqS} {\partial_+}S &=& \frak{h}(R-S)S, \quad S \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} {\partial_-}\eta^{2/\kappa}, \quad S\mid_{\Gamma_{23}}= {\partial_-}z,\end{aligned}$$ where $$\begin{aligned} R \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} {\partial_+}\eta^{2/\kappa} = \frac{4{\lambda_+}p}{\kappa r}, \quad S \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} {\partial_-}\eta^{2/\kappa} = 0\\ S \mid_{\Gamma_{23}}= {\partial_-}z = z_\theta - z_r r\sqrt{\frac{r^2- c^2}{c^2}} \ge 0, \quad R \mid_{\Gamma_{23}}={\partial_+}z = z_\theta+ z_r r\sqrt{\frac{r^2- c^2}{c^2}}.\end{aligned}$$ We discuss the compatibility conditions in Section \[sectionR\_1\]. In $\mathcal{R}_2$, the positive characteristics, emanating from $\Gamma_{23}$, form an envelope, where the positive characteristics satisfy $$\begin{aligned} \label{pchar} \frac{dr}{d\theta} = \lambda = r\sqrt{\frac{r^2 -c^2(p)}{c^2(p)}} ,\quad r(\theta_i)=r_{23}(\theta_i), \end{aligned}$$ for each $( r_{23}(\theta_i),\theta_i)\in \Gamma_{23}, \ \theta_2\le \theta_i \le\theta_3, $ and $r=r_{23}(\theta)$ satisfies $$\label{nch} \frac{dr}{d\theta} = - \lambda = -r\sqrt{\frac{r^2 -c^2(p)}{c^2(p)}} , \quad r(\theta_2)=c_4/\sin\theta_2, \quad \theta_2<\theta<\theta_3.$$ We note that the simple wave does not create a sonic curve. More precisely if it is sonic somewhere, that is $\lambda=0$, in the simple wave region. Then $S=\partial_-p = p_\theta -\lambda p_r$ and $R=\partial_+p =p_\theta +\lambda p_r$ imply $p_\theta =S=R$, which yields a contradiction: simple wave. The family $S>0$ remains to be non-trivial and this is the one that carries the data from $\sigma$ to $\Gamma_{23}$ between the two regions $\mathcal{R}_1$ and $\mathcal{R}_2$. The other family $R$ becomes zero, and therefore carries no information in this simple wave region $\mathcal{R}_2$. Thus we have $$\label{Simple} \partial_+ S = -\frak{h} S^2, \quad S\mid_{\Gamma_{23}} = \partial_- z.$$ We can write the solution to this Riccati type equation in the following form: Let $(\hat\theta,\hat r)$ a point on the negative characteristic $\Gamma_{23}$ and integrate along the positive characteristic lines in region $\mathcal{R}_2$, then $$\label{Ssimple} S=\partial_{-}p(\theta,r)=\frac{\partial_{-}p(\hat \theta,\hat r)}{\partial_{-}p(\hat \theta,\hat r)\displaystyle\int_{\hat \theta}^\theta \frak{h}\ d_{+}\theta+1}.$$ We finally state the main theorem. \[mainth\] Let $2c(p_4)>c(p_1)$. For a convex $\Gamma_{23}\in C^2$ and the data $(p, R, S) \in C^2(\Gamma_{23})$ satisfying the compatibility conditions at $\Xi_2$ and $\Xi_3$ and on $\Gamma_{23}$, there exists a supersonic solution $(p,R, S) \in C^2({{\mathcal{R}_1}}\setminus\Gamma_{23}\cup\sigma) \cap C^1({{\mathcal{R}_1}}\cup\sigma) \cap C^{0,1}(\overline {{\mathcal{R}_1}}\setminus\{\Xi_3\})$ satisfying the Goursat boundary problems -. The solution $(p,R,S)$ creates the sonic boundary $\sigma=\{(\tau(\theta),\theta), \theta_3 <\theta<\pi/2\} \in C^1$ such that $R=S=p_\theta >0$ on $\sigma$ and $\tau'(\theta)>0$ on $\sigma$. $\tau'(\theta_3)=0$ and $\tau'(\pi/2)=0$. The sonic boundary $\sigma$ and the transonic shock $\Sigma'$ merge into a point $\Xi_3$ at which the solution $(p,R,S)$ holds $$R(\Xi)=S(\Xi) \rightarrow 0, \quad as \quad \Xi \in \sigma \rightarrow \Xi_3.$$ Furthermore there exists a simple wave creating a transonic shock $\Sigma'$ in region ${{\mathcal{R}_2}}$. Our existence result of the supersonic flow is established with the given convex negative characteristics $\Gamma_{23}$ and the data on $\Gamma_{23}$ holding the compatibility conditions. In our forthcoming paper, we establish the global transonic solution and provide the scheme to select the correct data and $\Gamma_{23}$. In what follows, we use the condition $2c(p_4)>c(p_1)$ and discuss the existence results for each region. Rarefaction wave Region $\mathcal{R}_0$ {#sectionR_0} ======================================= With $c^2(p)=\gamma p^\kappa = \eta^2$, where $p_4\le p\le p_1$, it is easy to see that $$p=\frac{1}{\gamma^{1/\kappa}}r^{2/\kappa}(\sin \theta)^{2/\kappa}$$ and $$\begin{aligned} \lambda = r\sqrt{\frac{r^2-c^2}{c^2}}=r\sqrt{\frac{\xi^2+\eta^2-c^2}{c^2}}=\frac{r\xi}{\eta}=\frac{r\cos \theta}{\sin \theta}.\end{aligned}$$ By integrating along the positive characteristic $\Gamma_{12}$ emanating from $\Xi_1,$ $(r_{1},\theta_{1})=(c(p_1),\pi/2)$, $$\begin{aligned} \frac{dr}{d\theta}=\lambda =\frac{r\cos \theta}{\sin \theta},\end{aligned}$$ we find that $$\label{G12} \Gamma_{12}: \ \ r=c(p_1)\sin \theta, \quad \frac{dr}{d\theta} = c(p_1) \cos\theta =c(p_1)\sqrt{1-\frac{r^2}{c^2(p_1)}}.$$ Note that $\Gamma_{12}$ terminates at $\eta=r\sin\theta=c(p_4)$, and thus $$(r_{2},\theta_{2})= \bigg(\sqrt{c(p_1)c(p_4)}, \arcsin\sqrt{\frac{c(p_4)}{c(p_1)}}\bigg).$$ Hence $\Gamma_{12}$ is completely determined by the rarefaction wave $R_{14}: c^2(p)=\eta^2$, where $p_4\le p\le p_1$. In addition, in region $\mathcal{R}_0$ we have $$\begin{aligned} R&= \partial_+p= \partial_\theta p +\lambda\partial_r p=\frac{4}{\kappa\gamma^{1/\kappa}}\cos\theta(\sin \theta)^{2/\kappa-1}r^{2/\kappa}>0, \quad \theta\in(\theta_{2}, \pi/2)\label{Rrare}\\ S&=\partial_-p=\partial_\theta p -\lambda\partial_r p =\frac{2}{\kappa}\frac{r^{2/\kappa}}{\gamma^{1/\kappa}}(\sin \theta)^{2/\kappa}\left[\frac{\cos \theta}{\sin \theta}-\sqrt{\frac{r^2\cos^2 \theta}{r^2 \sin^2 \theta}}\right]=0, \quad \theta\in [\theta_{2}, \pi/2].\end{aligned}$$ Note that $R=0$ when $\theta=\pi/2$. Transient wave Region $\mathcal{R}_1$ {#sectionR_1} ===================================== We first discuss many useful properties of the characteristics in the transient wave region ${{\mathcal{R}_1}}$, in the same spirit as in [@Bang]. More precisely we discuss the monotonicity and convexity properties of the characteristics, and the monotonicity of $p$ along the characteristics in polar coordinates and cartesian coordinates (different coordinates provide different aspects of the characteristics) in Lemmas \[Lemma1\_increasing\] –\[intersectionpoint\]. We also state a priori bounds at the end of the section, in Lemmas \[apriori\], \[apest\]. \[Lemma1\_increasing\] The hyperbolic solution $p \in C^1$ to the Goursat problem satisfies $$S=\partial_{-}p>0, \ \ R=\partial_{+}p>0 \ \ \text{in the interior of region} \ \ \mathcal{R}_1.$$ Hence $p_\theta = (R+S)/2>0$ in $\mathcal{R}_1$. Let $(\tilde\theta,\tilde r)$ be a point on the positive characteristic $\Gamma_{12}$, and $(\hat\theta,\hat r)$ a point on the negative characteristic $\Gamma_{23}.$ Integrate and along the negative and positive characteristic lines, respectively, to obtain $$\begin{aligned} R&=\partial_{+}p(\theta,r)=\partial_{+}p(\tilde \theta,\tilde r)\exp\bigg({\int_{\tilde \theta}^\theta \frak{h}(S-R) \ d_{-}\theta}\bigg)>0,\label{R_positive}\\ S&=\partial_{-}p(\theta,r)=\partial_{-}p(\hat \theta,\hat r)\exp\bigg({\int_{\hat \theta}^\theta \frak{h}(R-S) \ d_{+}\theta}\bigg). \label{S_positive}\end{aligned}$$ The data ensures $R>0$ only. Thus we check the positiveness of $S$ to complete the proof. From , if $S=0$ at $(\hat \theta,\hat r)$ then $S=0$ along the positive characteristic passing through $(\hat \theta,\hat r)$ in region $\mathcal{R}_1$. Thus $R=S=0$ at a point different from $\Xi_1$ on $\sigma$ which is a contradiction. Therefore $S\neq 0$ along $\Gamma_{23},$ possibly excluding the endpoints. If $S<0$ at $(\hat \theta,\hat r)$ then $R=S<0$ somewhere on the sonic curve which is again a contradiction to $R>0$. So we conclude that $S>0$ along $\Gamma_{23}$ for $\theta\neq \theta_{2}, \theta_{3}.$ Thus $R, S>0$ and consequently $p_{\theta}>0$ in the interior of region $\mathcal{R}_1.$ By Lemma \[Lemma1\_increasing\], we deduce $R=0$ in the simple wave region $\mathcal{R}_2$. We next discuss monotonicity properties of the characteristics. To ease the analysis, we write the characteristics in self-similar coordinates. From $\dfrac{dr}{d\theta}=\pm\lambda$, the characteristics $\eta=\eta(\xi)$ in the $(\xi,\eta)$-plane read $$\begin{aligned} \label{cartesiancharacteristics} \frac{d\eta}{d\xi}=\Lambda_{\pm} =\frac{\xi \eta\pm \sqrt{c^2(\xi^2+\eta^2-c^2)}}{\xi^2-c^2}.\end{aligned}$$ Let the corresponding directional derivatives in the self-similar coordinates be; $$\frac{dp}{d_{\pm}\xi}=\frac{\partial p}{\partial \xi}+\Lambda_{\pm}\frac{\partial p}{\partial\eta}, \ \ \frac{dp}{d_{\pm}\eta}=\frac{\partial p}{\partial \eta}+\Lambda_{\pm}^{-1}\frac{\partial p}{\partial\xi}.$$ We observe that in region $\mathcal{R}_1\cap \{(\xi,\eta), \xi>0, \eta>0\},$ $$\begin{aligned} \xi \Lambda_{-}-\eta&<0,\ \ \ \eta\Lambda_{-}+\xi>0,\label{fornegative}\\ \eta\Lambda_{+}^{-1}-\xi&<0, \ \ \ \xi\Lambda_{+}^{-1}+\eta>0, \label{forpositive}\end{aligned}$$ and $$(c^2-\xi^2)\Lambda_{\pm}^2+2\xi\eta \Lambda_{\pm}+c^2-\eta^2=0,$$ which can be solved for $c^2$ to get $$\begin{aligned} c^2=\frac{(\xi \Lambda_{-}-\eta)^2}{\Lambda_{-}^2+1}, \label{cfornegative} \end{aligned}$$ or $$\begin{aligned} c^2=\frac{(\xi-\eta\Lambda_{+}^{-1})^2}{\Lambda_{+}^{-2}+1} \label{cforpositive}.\end{aligned}$$ We next discuss properties of the characteristics. \[Lemma\_decreasing/increasing\] The hyperbolic solution $p\in C^2({{\mathcal{R}_1}})$ to the Goursat problem has the following properties: 1. Along the negative characteristics $\dfrac{d\eta}{d\xi} = \Lambda_-$ starting from any point on $\Gamma_{12}\setminus \Xi_1$: \(i) $\displaystyle{\frac{dp}{d_{-}\xi}<0}$, (ii) $\dfrac{d\Lambda_{-}}{d_{-}\xi}>0$, (iii) $\Lambda_-<0$, and (iv) ${\dfrac{dp}{d_{-}\eta}>0}$; 2. Along the positive characteristics $\dfrac{d\eta}{d\xi} =\Lambda_+$ starting from any point on $\Gamma_{23}$: \(v) $\dfrac{dp}{d_{+}\xi}<0$, (vi) $\dfrac{d\Lambda^{-1}_{+}}{d_{+}\eta}<0$, (vii) $\Lambda_+<0,$ and (viii) $\dfrac{dp}{d_{+}\eta}>0,$ \(i) From $$\begin{aligned}\label{S} 0< S=\partial_{-}p&=\bigg(r\cos\theta-r\sin\theta\sqrt{\frac{r^2-c^2}{c^2}}\bigg)\frac{dp}{d_{-}\eta}\\ &=-\bigg(r\sin\theta+r\cos\theta \sqrt{\frac{r^2-c^2}{c^2}}\bigg)\frac{dp}{d_{-}\xi}, \end{aligned}$$ and by Lemma \[Lemma1\_increasing\], we obtain the strict inequality $$\frac{dp}{d_{-}\xi}<0,$$ everywhere in the region $\mathcal{R}_1\cap \{(\xi,\eta), \xi\geqslant0, \eta>0\}.$ \(ii) Differentiating (\[cfornegative\]) along $\dfrac{d}{d_{-}\xi}$ gives $$\begin{aligned} \label{convexity} \gamma\kappa p^{\kappa-1}\frac{dp}{d_{-}\xi}=\frac{2(\xi\Lambda_{-}-\eta)(\xi+\eta \Lambda_{-})}{(\Lambda_{-}^2+1)^2}\frac{d\Lambda_{-}}{d_{-}\xi},\end{aligned}$$ thus using (\[fornegative\]), we conclude that $$\frac{d^2\eta}{d\xi^2}=\frac{d\Lambda_{-}}{d_{-}\xi}>0$$ in $\mathcal{R}_1\cap\{(\xi, \eta), \xi>0, \eta>0\}$, which means that the negative characteristics are convex. (iii)-(iv) Evaluate the first equation of (\[S\]) on $\sigma'=\sigma\setminus\{\Xi_1,\Xi_3\}$ where $c^2=r^2$ to get $$\begin{aligned} \label{Ssigma} 0< S=r\cos\theta \frac{dp}{d_{-}\eta}.\end{aligned}$$ This immediately implies $\dfrac{dp}{d_{-}\eta} >0$ on $\sigma'\cap \{(\xi,\eta), \xi> 0, \eta>0\}$. If $\mathcal{R}_1\cap \{(0,\eta), \eta<c(p_1)\}\neq\emptyset$ or $\sigma\cap\{(0,\eta), \eta<c(p_1)\}\neq\emptyset$ then $$\begin{aligned} \label{negativealongsigma2} \frac{dp}{d_{-}\eta}<0\end{aligned}$$ and $\Lambda_{-}=-\dfrac{\xi}{\eta}=0,$ respectively, at the points on the $\eta$-axis. On the other hand, $$\begin{aligned} 0> \dfrac{dp}{d_-\xi} = p_\xi + \Lambda_- p_\eta = \Lambda_-( p_\eta + \Lambda_-^{-1} p_\xi) = \Lambda_- \dfrac{dp}{d_-\eta}\end{aligned}$$ in region $\mathcal{R}_1\cap \{(\xi,\eta), \xi\geqslant0, \eta>0\}.$ We show that $\Lambda_-\neq 0$. If not then we have unbounded $\dfrac{dp}{d_-\eta}$ when $\Lambda_- = 0$ at some point $(r^*,\theta^*)$ on the negative characteristics. However $c^2=(\eta^*)^2$ when $\Lambda_-=0$, and at the same time $\dfrac{dp}{d_-\xi} = p_\xi<0$. At the points on $\Gamma_{12}$ we know that $c^2=\eta^2$ so by (\[cartesiancharacteristics\]) the negative characteristics satisfy $\Lambda_{-}=0.$ Hence a convex characteristic has at least two points with $\Lambda_{-}=0$ which leads to a contradiction. Thus, if $\mathcal{R}_1\cap \{(0,\eta), \eta<c(p_1)\}\neq\emptyset$ then $\Lambda_{-}>0,$ $\dfrac{dp}{d_{-}\eta} <0$ and $\dfrac{dp}{d_{-}\xi}<0$ in the region $\mathcal{R}_1\cap \{(\xi,\eta), \xi> 0, \eta>0\}$ which is again a contradiction to the convexity and the behavior of the negative characteristics along $\Gamma_{12}.$ We therefore conclude that the change of type occurs in the first quadrant, $\mathcal{R}_1$ is located in the first quadrant and $\Lambda_{-}< 0$, $\dfrac{dp}{d_{-}\eta} >0$ in the interior of the entire region $\mathcal{R}_1$. In addition, note that $\sigma\setminus\Xi_1\subseteq \{(\xi,\eta), \xi> 0, \eta>0\}.$ \(v) By Lemma \[Lemma1\_increasing\], and $$\begin{aligned} 0< R=\partial_{+}p&=\bigg(-r\sin\theta+r\cos\theta \sqrt{\frac{r^2-c^2}{c^2}}\bigg)\frac{dp}{d_{+}\xi}\\ &=\bigg(r\cos\theta+r\sin \theta\sqrt{\frac{r^2-c^2}{c^2}}\bigg)\frac{dp}{d_{+}\eta},\end{aligned}$$ we conclude that $$\frac{dp}{d_{+}\eta}>0$$ everywhere in the interior of $\mathcal{R}_1$. \(vi) In addition differentiating (\[cforpositive\]) gives $$\begin{aligned} \gamma \kappa p^{\kappa-1}\frac{dp}{d_{+}\eta}=\frac{2(\xi\Lambda_{+}^{-1}+\eta)(\eta\Lambda_{+}^{-1}-\xi)}{(\Lambda_{+}^{-2}+1)^2}\frac{d\Lambda_{+}^{-1}}{d_{+}\eta}.\end{aligned}$$ By (\[forpositive\]), we conclude that $$\frac{d^2\xi}{d\eta^2}=\frac{d\Lambda_{+}^{-1}}{d_{+}\eta}<0$$ which means that the positive characteristics are concave. (vii)-(viii) On $\sigma$ (including $\Xi_1$) we have $$\begin{aligned} \label{positivealongsigma} \frac{dp}{d_{+}\xi}<0 \ \, \ \ \text{and} \ \ \Lambda_{+}=-\frac{\xi}{\eta}\le0.\end{aligned}$$ Thus by a similar argument as before we conclude the claim. More precisely, from $$\begin{aligned} 0<\dfrac{dp}{d_{+}\eta} = p_\eta + \Lambda^{-1}_+ p_\xi = \Lambda^{-1}_+ (\Lambda_+ p_\eta + p_\xi) = \Lambda^{-1}_+ \dfrac{dp}{d_+\xi},\end{aligned}$$ we show that $\Lambda^{-1}_+\neq 0$ to obtain $\Lambda^{-1}_+<0$ and $\dfrac{dp}{d_+\xi}<0$ in the interior of ${{\mathcal{R}_1}}$ by a contradiction argument. Suppose not. Then $c^2=\xi^2$ and $\dfrac{dp}{d_{+}\eta} = p_\eta>0$ at the contradiction point $(r^*, \theta^*)$ on the positive characteristics. On the other hand for the negative characteristics passing from $(r^*, \theta^*)$ the following hold: $$\begin{aligned} \Lambda_- =-\frac{1}{\tan 2\theta^*} , \quad 0< \partial_-p = \frac{r^*\cos2\theta^*}{\cos\theta^*} \frac{dp}{d_-\eta}.\end{aligned}$$ However we have shown that $$\dfrac{dp}{d_-\eta} >0.$$ The contradiction is immediate when $\pi/4 \le \theta^* \le \pi/2$. In the following two lemmas, we discuss the properties of the sonic boundary $\sigma$; in particular the monotonicity and the corner point $\Xi_3$ with the level curve where $\{p=p(\Xi_3)\}$. \[pointxi3\] The level curve $\{p=p(\Xi_3)\}$ of the solution $p\in C^1({{\mathcal{R}_1}})$ and the sonic boundary $\sigma$ meet tangentially at $\Xi_3$ at which $$\dfrac{d\eta}{d\xi}=-\dfrac{\xi}{{\eta}}.$$ In addition $R=S$ on the sonic boundary. In particular $R=S>0$ on $\sigma\setminus\{\Xi_1,\Xi_3\}$. In region $\mathcal{R}_1\setminus\{\Xi_1,\Xi_3\},$ by Lemma \[Lemma1\_increasing\], $$\begin{aligned} R&=\partial_{+}p=p_{\xi}\bigg(-\eta +\xi\sqrt{\frac{\eta^2+\xi^2-c^2}{c^2}}\bigg)+p_{\eta}\bigg(\xi+\eta\sqrt{\frac{\eta^2+\xi^2-c^2}{c^2}}\bigg)> 0,\\ S&=\partial_{-}p=p_{\xi}\bigg(-\eta -\xi\sqrt{\frac{\eta^2+\xi^2-c^2}{c^2}}\bigg)+p_{\eta}\bigg(\xi-\eta\sqrt{\frac{\eta^2+\xi^2-c^2}{c^2}}\bigg)> 0,\end{aligned}$$ thus $$\begin{aligned} \label{conditiononsonic} R+S=2(\xi p_{\eta}-\eta p_{\xi})> 0.\end{aligned}$$ Since on the sonic boundary $c^2=r^2$ it is immediate to have $R=S=p_{\theta}=-\eta p_{\xi}+\xi p_{\eta}>0$ on $\sigma\setminus\{\Xi_1,\Xi_3\}$. On the boundary $\Gamma_{23}$ (see Section \[sectionR\_2\] for details) we have $$\begin{aligned} S=\partial_{-}p&=\frac{2r^3_-(-r'_-)p^{1/\gamma}}{\gamma \kappa}\left[\frac{r_-r''_--r^2_--2r'^2_-}{(r'^2_-+r^2_-)^2}\right]. \label{S1}\end{aligned}$$ If $r_-''$ is bounded (otherwise $S$ might maintain a positive lower bound in the neighborhood of $\Xi_3$ and by (\[Ssimple\]) that would mean that there is a singularity at $\Xi_3$) then $S\rightarrow 0$ as $\theta\rightarrow \theta_3$ on $\Gamma_{23}$ (note that $\Gamma_{23}$ is prescribed to be convex with the smooth data). Thus $\Gamma_{23}$ and the level curve $\{p=p(\Xi_3)\}$ meet tangentially at $\Xi_3$ and $\xi p_{\eta}=\eta p_{\xi}.$ The tangential derivative of $c^2=\eta^2+\xi^2$ along $\sigma =\{(\xi,\eta_\sigma(\xi) )\}$ reads $$\begin{aligned} \label{sonicboundary} c_p^2 \ p_{\xi}+c_p^2 \ p_{\eta} \frac{d\eta_\sigma}{d\xi}=2\eta \frac{d\eta_\sigma}{d\xi}+2\xi.\end{aligned}$$ On a level curve $L=\{X=(\xi,\eta_L(\xi))\}$, we have $$\begin{aligned} \label{levelcurves} \frac{dp}{d\xi} (\xi,\eta_L(\xi)) =p_{\xi}+\frac{d\eta_L}{d\xi}p_{\eta}=0.\end{aligned}$$ If $p_{\eta}=0$ at $\Xi_3$ then $p_{\xi}=0$ and thus by (\[cartesiancharacteristics\]) we have $$\Lambda_{-}=\frac{d\eta_{\sigma}}{d\xi}=-\frac{\xi}{\eta}.$$ If $p_{\eta}\neq 0$ at $\Xi_3$ then $$\Lambda_{-}=\frac{d\eta_{L}}{d\xi}=\frac{d\eta_{\sigma}}{d\xi}=-\frac{\xi}{\eta}.$$ We thus conclude that a level curve meets tangentially the sonic boundary at $\Xi_3.$ \[intersectionpoint\] The sonic boundary $\sigma=\{(\xi,\eta(\xi))\}$ of the solution $p\in C^2({{\mathcal{R}_1}})$ is strictly decreasing in the $\xi$ direction. That is $\dfrac{d\eta}{d\xi}<0$ everywhere along $\sigma,$ except at $\Xi_1.$ By Lemma \[Lemma1\_increasing\] we first note that in region $\mathcal{R}_2$ we have $R=0$ and $S>0$ along the positive characteristics and therefore the sonic boundary along which $R=S$ cannot extend below $\Xi_3.$ Let $(\xi^{\ast},\eta^{\ast})$ be a point on $\sigma =\{ (\xi(\eta),\eta)\} $ such that $c^2=(c^{\ast})^2$ and $\dfrac{d\xi(\eta^\ast)}{d\eta} =0$. Then, in the neighborhood of this point, $\dfrac{d\eta}{d\xi}$ is unbounded. Specifically, $\dfrac{d^2\eta}{d\xi^2}>0$ when $\eta<\eta{\ast}$; and $\dfrac{d^2\eta}{d\xi^2}<0$ when $\eta>\eta^{\ast}.$ The tangential derivative to $c^2=\xi^2+\eta^2$ along $\sigma=\{ (\xi, \eta(\xi) )\}$ now reads $\dfrac{d(c^2)}{d\xi}=2\xi+2\eta \dfrac{d\eta}{d\xi}.$ We deduce that $\dfrac{d(c^2)}{d\xi}$ is unbounded when $c^2=(c^{\ast})^2,$ $\dfrac{d^2(c^2)}{d\xi^2}>0$ when $c^2<(c^{\ast})^2$ and $$\frac{d^2(c^2)}{d\xi^2}=2+2\bigg(\frac{d\eta}{d\xi}\bigg)^2+2\eta\frac{d^2\eta}{d\xi^2}<0$$ when $c^2>(c^{\ast})^2.$ In the latter case, let $\dfrac{d\eta}{d\xi}=g(\eta)$ then $$\frac{d^2(c^2)}{d\xi^2}=2+2(g(\eta))^2+2\eta \frac{dg(\eta)}{d\eta}g(\eta)<0.$$ A routine integration yields $(g(\eta))^2<\frac{\text{constant}}{\eta^2}-1,$ which leads to a contradiction because close to $\eta=\eta^{\ast}$ the slope is unbounded. Since the sonic boundary lies in the region $\{(\xi,\eta), \xi\geqslant 0,\ \eta>0\}$ then $\dfrac{d\eta}{d\xi}\leqslant 0.$ Note that it would be possible to have a point on $\sigma$ such that $\dfrac{d\xi}{d\eta}=0$ if $\eta^{\ast}=0.$ However $\sigma$ should be located only where $0< c(p_4) \le \eta \le c(p_1)$. Assume now that there is a point $(\xi_{\ast},\eta_{\ast})$ on $\sigma$ such that $c^2=(c_{\ast})^2$ and $\dfrac{d\eta}{d \xi}=0$ at this contradiction point on the sonic boundary. Then from the unboundedness of $\dfrac{d\xi}{d\eta}$, we have $\dfrac{d^2\xi}{d\eta^2}<0$ when $\xi>\xi_{\ast}$ and $\dfrac{d^2\xi}{d\eta^2}>0$ when $\xi<\xi_{\ast}.$ Again the tangential derivative of $c^2=\xi^2+\eta^2$ on $\sigma=\{ (\xi(\eta),\eta)\}$ becomes $\dfrac{d(c^2)}{d\eta}=2\eta+2\xi \dfrac{d\xi}{d\eta}.$ We deduce that $\dfrac{d(c^2)}{d\eta}$ is unbounded when $c^2=(c_{\ast})^2,$ $\dfrac{d^2(c^2)}{d\eta^2}>0$ when $c^2<(c_{\ast})^2$ and $$\frac{d^2(c^2)}{d\eta^2}=2+2\bigg(\frac{d\xi}{d\eta}\bigg)^2+2\xi\frac{d^2\xi}{d\eta^2}<0$$ when $c^2>(c_{\ast})^2.$ In the latter case, let $\dfrac{d\xi}{d\eta}=f(\xi)$ then $$\frac{d^2(c^2)}{d\eta^2}=2+2(f(\xi))^2+2\xi \frac{df(\xi)}{d\xi}f(\xi)<0.$$ Closely related to the above, in spirit as well as in technique, we end up with a contradiction. Therefore $\dfrac{d\eta}{d\xi}< 0$ along $\sigma,$ except at $\Xi_1.$ In (\[sonicboundary\]), notice that if $(c^2)' \ p_{\eta}=2\eta$ then $(c^2)' \ p_{\xi}=2\xi$ and thus $R+S=0$ which does not hold for points along the sonic boundary different from $\Xi_1$ and $\Xi_3.$ We therefore conclude that $(c^2)' \ p_\eta\neq 2\eta$ and does not change sign along $\sigma$ excluding the endpoints, thus $$\begin{aligned} \label{slopeofsonic} \frac{d\eta}{d\xi}=\frac{2\xi-(c^2)' \ p_{\xi}}{(c^2)' \ p_{\eta}-2\eta}.\end{aligned}$$ Let us assume that $(c^2)' \ p_{\eta}>2 \eta$ everywhere along $\sigma,$ then $p_{\eta}$ is positive, and we also know from Lemma \[pointxi3\] that the sonic boundary satisfies $\dfrac{d\xi}{d\eta}<0,$ in the neighborhood of $\Xi_3.$ On the other hand, in the neighborhood of $\Xi_1,$ since $c^2=\eta^2$ in region $\mathcal{R}_0$ and $c^2=r^2$ on $\sigma,$ we expect that the level curves have positive slopes and thus $p_{\xi}<0$ by (\[levelcurves\]). This combined with (\[slopeofsonic\]) gives $\frac{d\xi}{d\eta}>0$ for the sonic boundary, in the neighborhood of $\Xi_1$. This implies that there must be a point along $\sigma,$ different from the endpoints, such that $\frac{d\xi}{d\eta}=0.$ which is a contradiction. We therefore conclude by (\[conditiononsonic\]) that $(c^2)' \ p_{\eta}<2\eta$ and $(c^2)' \ p_{\xi}<2\xi$ along $\sigma.$ Figure \[characteristics\] depicts the configuration of the characteristic curves in the supersonic region. \[\]\[\]\[0.7\]\[0\][$\Gamma_{12}$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_1$]{} \[\]\[\]\[0.7\]\[0\][$\eta=c(p_1)$]{} \[\]\[\]\[0.7\]\[0\][$\eta=c(p_4)$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_2$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_3$]{} \[\]\[\]\[0.7\]\[0\][$\sigma$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{+}$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{-}$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_0$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_2$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_1$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_4$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{24}$]{} \[\]\[\]\[0.7\]\[0\][$\Sigma'$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{23}$]{} We next state a priori bounds. \[apriori\] The solution $p\in C^1(\overline {\mathcal{R}_1})$ satisfies $$\label{pbd} p_4= p(\Xi_2)\le p\le p_1= p(\Xi_1), \quad in \quad \overline{\mathcal{R}_1}.$$ By Lemma \[Lemma1\_increasing\], we have $$p_\theta = (R+S)/2>0.$$ Thus $p$ is strictly increasing in the $\theta$ direction, and $p> p(\Xi_2)=p_4$ in $\mathcal{R}_1$. Note that the change of type occurs in the first quadrant as the characteristics enter the sonic circle, where $c^2(p)=r^2$, in the first quadrant, and becomes subsonic holding $c^2(p) >r^2$. Thus when $\theta\ge \pi/2$ we are now in the subsonic region. Therefore $p< p(\Xi_1)=p_1$ in $\mathcal{R}_1$. Similarly, these bounds hold in the simple wave region $\mathcal{R}_2$. While it is not straightforward to see whether these bounds remain valid in the subsonic region, we note that this lemma holds in the entire domain and refer to the forthcoming paper on the transonic problem. We next cite estimates from [@SongZheng]. \[apest\] The maximum values of $\partial_{\pm} p$ of the solution $p\in C^2$ are attained on the characteristic boundaries $\Gamma_{12}\cup\Gamma_{23}.$ Furthermore the solutions $(p, R, S)\in C^1$ satisfy $$|t^2 \partial_\pm R|, |t^2 \partial_\pm S| \le C,$$ where $t= \sqrt{ r^2 -c^2(p)}$. The existence result in the transient wave region ${{\mathcal{R}_1}}$ {#sectionR_1existence} ===================================================================== We formulate the Goursat boundary problem to construct the transient wave in $\mathcal{R}_1$. We first discuss the boundary data on the negative characteristic $\Gamma_{23}$. Let $f\in C^2(\theta_2,\pi/2)$ be convex, and $g\in C^2(\Gamma_f)$ where $\Gamma_f =\{(f(\theta),\theta): \theta_2\le \theta \le \pi/2 \}$ satisfying $$\begin{aligned} \label{G23} \frac{d f}{d\theta} &=& -\lambda(f,g) = - f \sqrt{\frac{f^2- c^2(g(f(\theta),\theta))}{c^2(g(f(\theta),\theta))}} \quad {\rm on }\ \Gamma_f,\\ f^2(\theta) &\ge& c^2(g(f(\theta),\theta)) , \label{fg}\\ f(\theta_2)& =& r_2 = \sqrt{c_1c_4}, \label{r2} \end{aligned}$$ and $$\begin{aligned} \label{pXi2} g(\Xi_2)&=&p(\Xi_2),\\ {\partial_-}g(f(\theta),\theta) &=& g_\theta(f(\theta),\theta) - g_r(f(\theta),\theta) \lambda(f(\theta), g(f(\theta),\theta)) > 0, \quad \theta\in (\theta_2,\pi/2),\label{z23}\\ {\partial_-}g(\Xi_2)&=& 0. \label{SXi2}\end{aligned}$$ The data $R$ on $\Gamma_f$, denoted by $R[f,g]$, is evaluated from $\partial_- R = \frak{h}(f,g) (S-R)R$ along $\Gamma_{f}$ with the initial value $R(\Xi_2)$ stated below, and ${\partial_-}g \mid_{\Gamma_{f}}$ from . That is $$\begin{aligned} \label{R23} R[f,g](\theta) &=& R(\Xi_2) \exp\int^\theta_{\theta_2} \frak{h}(f(z),g(f(z),z)) ({\partial_-}g(f(z),z) - R(f(z),z)) d z,\\ R(\Xi_2)&=&\frac{4}{\kappa\gamma^{1/\kappa}}\cos\theta_{2}(\sin \theta_{2})^{2/\kappa-1}r_{2}^{2/\kappa}. \label{RXi2} \end{aligned}$$ Additionally, we require that $f$ and $g$ satisfy the following compatibility condition: - There exists $\theta_2 <\theta_3<\pi/2$ such that $$\begin{aligned} \label{Xi3} \lim _{\theta\rightarrow \theta_3} R[f,g](\theta) =0,\quad \lim_{\theta\rightarrow \theta_3} {\partial_-}g(f(\theta),\theta) =0. \end{aligned}$$ The compatibility condition \[G1.\] implies $$\begin{aligned} \label{Xi3p} \lim_{\Xi \in \Gamma_f \rightarrow \Xi_3} g_\theta(\Xi) =0,\\ \lim_{\Xi \in \Gamma_f \rightarrow \Xi_3} (f^2- c^2(g))(\Xi) =0. \end{aligned}$$ We note that due to the conditions that $df/d\theta<0$, $c^2(g) \le f^2$ while $g_\theta>0$ (since $\partial_- g>0$ and $R[f,g]> 0$), we have immediate bounds for $f$ and $g$ $$\begin{aligned} \label{fgbounds} c_4^2 < c^2(g) \le f^2 \le c_1 c_4. \end{aligned}$$ We construct a solution that creates a sonic boundary where $R=S$ becomes zero as the solution approaches $\Xi_3$ on the sonic boundary. Local existence results ----------------------- We first discuss local existence results. Let $W=(p, R, S)$, and write the system to $$\begin{aligned} \label{GoursatBP} W_\theta + A W_r &=& B,\end{aligned}$$ where $A=diag (0, -\lambda, \lambda)$ and $B=(\frac{1}{2}(R+S), \frak{h} (S-R)R,\frak{h}(R-S)S)$. The eigenvalues are $\lambda_0=0$, $\lambda_{-} = -\lambda$, and $\lambda_+=\lambda$, and the corresponding eigenvectors are $l_0=(1,0,0)$, $l_-=(0,1,0)$ and $l_+=(0,0,1)$. Let $f,g$ satisfy – and \[G1.\] for some $\theta_3\in (\theta_2, \pi/2)$. Thus we have the Goursat boundary value problem with the following boundary data $$\begin{aligned} p^+=p \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} \eta^{2/\kappa}, &\quad & p^- =p\mid_{\Gamma_{23}} =g, \\ R^+ = R \mid_{\Gamma_{12}}= \gamma^{-1/\kappa} {\partial_+}\eta^{2/\kappa}=\frac{4{\lambda_+}p}{\kappa r}, &\quad &R^-= R\mid_{\Gamma_{23}} = R(\Xi_2) \exp\int^\theta_{\theta_2} \frak{h} ({\partial_-}g -R^-) d_-\theta ,\\ S^+ = S\mid_{\Gamma_{12}} =0, &\quad& S^-=S\mid_{\Gamma_{23}}= g_\theta - g_r \lambda(f, g),\end{aligned}$$ which we write $W^+ =(p^+, R^+, S^+) = W\mid_{\Gamma_{12}}$ and $W^-=(p^-,R^-, S^-)=W\mid_{\Gamma_{23}}$. Clearly $W^+(\Xi_2)= W^-(\Xi_2)$. Furthermore, we have $$\begin{aligned} \label{sonicdata} p_\theta&=& R=S, \quad {\rm on } \quad \sigma.\end{aligned}$$ By checking the compatibility conditions of the Goursat boundary value problems [@Bang; @DaiZhang; @SongZheng], we establish the local existence result. \[localexistence\] Let the Riemann data satisfy $2c(p_4)> c(p_1)$. For a given $\Gamma_{23}\in C^2$ convex, and $W=(p, R, S)\in C^1$ on $\Gamma_{23}$ satisfying – , and the compatibility condition \[G1.\], there exists a solution $W \in C^1 ({\mathcal{R}_1}(t_0))$ to the system , where ${\mathcal{R}_1}(t_0) = \{ \sqrt{r^2 -c^2(p)}\ge t_0\}$, for $t_0>0$. Let $W^{+}=W\mid_{\Gamma_{12}}$ and $W^-=W\mid_{\Gamma_{23}}$. Then the compatibility conditions at $\Xi_2$ are the following. $$\begin{aligned} \label{comp1} W^+\mid_{\Xi_2} &=& W^-\mid_{\Xi_2}\\ \frac{1}{\lambda_--\lambda_0} \left( l_0 \frac{dW^- }{d\theta} -\frac{1}{2}(R+S) \right)\mid_{\Xi_2} &=&\frac{1}{\lambda_+-\lambda_0} \left( l_0 \frac{dW^+ }{d\theta} -\frac{1}{2}(R+S) \right)\mid_{\Xi_2}.\end{aligned}$$ The second condition may be rewritten in the form $$\frac{dp^+}{d\theta}+\frac{dp^-}{d\theta} = \partial_+ p + \partial _-p.$$ Hence the Goursat problem has a local solution near $\Xi_2$. Next we establish a local solution to an initial boundary value problem. Let the initial position be $I=\{(\frak{r}(\theta),\theta)\}$ which is to be determined, and $W_I=W(\frak{r}(\theta),\theta)$. Let $X= I \cap\Gamma_{12}$ and $Y= I \cap\Gamma_{23}$ be the points where the initial and boundary positions meet. Now the compatibility conditions are as follows: for $i=0,-$, we have $$\begin{aligned} \frac{1}{\lambda_+ -\lambda_i} \left( l_i \frac{dW^+}{d\theta} - l_i B \right)\mid_X &=&\frac{1}{\frak{r}' -\lambda_i} \left( l_i \frac{dW_I}{d\theta} - l_i B \right) \mid_X;\end{aligned}$$ and for $i=0,+$, we have $$\begin{aligned} \frac{1}{\lambda_- -\lambda_i} \left(l_i \frac{dW^-}{d\theta} - l_i B \right)\mid_Y &=&\frac{1}{\frak{r}' -\lambda_i} \left( l_i \frac{dW_I}{d\theta} - l_i B \right) \mid_Y.\end{aligned}$$ The initial position $I\in C^1$ is chosen to be the constant level set of $r^2 -c^2(p)$ so that $\frak{r}'\neq \lambda_{\pm}$, which allows us to match the compatibility conditions. Thus we have the local existence result. In order to establish the global solution to the entire region $\mathcal{R}_1$, which is enclosed by $\Gamma_{12}$, $\Gamma_{23}$ and $\sigma$, we first discuss the regularity results near the sonic boundary due to [@WangZheng]. We note however that the regularity result is limited to strictly positive $R$ and $S$. Hence the estimates depend on $\delta_0$ where $R,S\ge \delta_0>0$. Equipped with these regularity results near the sonic boundary, noting that $p_\theta>0$, and the characteristics entering the sonic boundary in the radial direction and never parallel to the tangential direction of the sonic boundary, the sonic boundary $\sigma$ is estimated by the gradient of the pressure which then leads to the existence of a $C^1$ solution and $\sigma \in C^1$. Regularity results ------------------ We discuss the Hölder gradient estimates near the sonic boundary. The estimates are established by [@WangZheng] for the pressure gradient system provided that $\partial_\pm p$ are strictly positive, and under certain smoothness of the solution and the sonic boundary. While our result relies on [@WangZheng], we provide insights which signify the estimates and their consequences. We first change the coordinate system to flatten the sonic boundary as it was done in [@WangZheng]. This new coordinate system brings a couple of technical advantages. The first obvious one is that it simplifies the geometry of the sonic boundary. The next one is not immediate: the new coordinates enable us to derive the corresponding system that provides estimates on $t^\beta | R_r|, t^\beta |S_r|$, where $t=\sqrt{r^2 -c^2(p)}$, for sufficiently small $0\le t\le t_0$ and uniformly bounded $R, S$ such that $R, S\ge \delta >0$, where $1<\beta=\beta(t_0, \delta)<2$. This is essential to establish the sonic boundary to be in $C^1$. Since the estimates depend on the strict positiveness of $R, S$, we consider the region ${{\mathcal{R}_1}}$ excluding small neighborhoods of $\Xi_1$ and $\Xi_3$, where $\delta$ is the distance from these corner points, see Figure \[corners2\]. Let ${{\mathcal{R}_1}}[\delta]$ be neighborhoods of $\Xi_1$ and $\Xi_3$ with $\delta>0$ distance away from these points where $\Gamma_-^\delta$ is the negative characteristic with $\delta =dist (\Xi_1, \Gamma_-^\delta)$, and $\Gamma_+^\delta$ is the positive characteristic with $\delta =dist (\Xi_3, \Gamma_+^\delta)$. \[\]\[\]\[0.7\]\[0\][$\Gamma_{12}$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_1$]{} \[\]\[\]\[0.7\]\[0\][$\eta=c(p_1)$]{} \[\]\[\]\[0.7\]\[0\][$\eta=c(p_4)$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_2$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_3$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{+}^{\delta}$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{-}^{\delta}$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{23}$]{} The Hölder gradient estimates are established in the region ${{\mathcal{R}_1}}\setminus {{\mathcal{R}_1}}[\delta]$, in particular near the sonic boundary. Let $p$ be smooth enough to derive the characteristic equations for the derivatives (for simplicity we may assume $p \in C^3(\mathcal{R}_1)$ for now, which can be weaken). Let $t=\sqrt{r^2 -c^2}$, and consider the new coordinate system $(r,t)$. From simple calculations, $$\partial_\theta = \frac{-(c^2)' p_\theta}{2t} \partial_t, \ \ \partial_r = \partial_r + \frac{2r -(c^2)' p_r}{2t} \partial_t,$$ the system in $(r,\theta)$ coordinates becomes $$\begin{aligned} p_t &=& -\frac{2t}{(c^2)'}\\ R_t +\frac{2t \lambda}{(c^2)'S +2r \lambda} R_r &=& \frac{2t}{(c^2)' S + 2r \lambda} \frak{h}(R-S)R,\\ S_t -\frac{2t \lambda }{(c^2)' R - 2r \lambda} S_r &=& \frac{2t}{(c^2)' R - 2r \lambda} \frak{h}(S-R)S,\end{aligned}$$ where $$\lambda(t,r)=\frac{tr}{\sqrt{r^2 -t^2}}, \quad \frak{h}(t,r)= \frac{r^2 (c^2)'}{4t^2(r^2 -t^2)}.$$ The corresponding characteristics read $$\begin{aligned} \frac{dr_-}{dt} =\frac{2t \lambda}{(c^2)'S + 2r\lambda}, && \frac{dr_+}{dt} =\frac{- 2t \lambda}{(c^2)' R - 2r\lambda}.\end{aligned}$$ and the last two equations can be rewritten as $$\begin{aligned} \label{Rchar} \frac{dR}{d_{-}t}&=R_t+\dfrac{dr_{-}}{dt}R_r =& \frac{2t}{(c^2)' S + 2r \lambda}\frak{h}(R-S)R, \\ \frac{dS}{d_{+}t}&=S_t+\dfrac{dr_{+}}{dt}S_r = & \frac{2t }{(c^2)' R - 2r \lambda} \frak{h}(S-R)S.\label{Schar}\end{aligned}$$ As discussed in [@WangZheng], the following equations will be useful to establish the estimates. Let $$\begin{aligned} \label{V} V=\frac{1}{S}-\frac{1}{R}, \end{aligned}$$ and obtain $$\begin{aligned} \label{Vteq} V_t &=& \mu_0 \frac{V}{t} +\mu_1(r,t) R_r t^2 +\mu_2(r,t) S_r t^2\end{aligned}$$ where $$\begin{aligned} \mu_0(r,t)&=& \frac{ 4t^2 \frak{h} ((c^2)' + r\lambda V)}{((c^2)' + 2r\lambda S^{-1})((c^2)' - 2r\lambda R^{-1})}\\ \mu_1(r,s)&=& -\frac{2r}{R^2\sqrt{r^2 -t^2}} \frac{1}{(c^2)'S + 2r\lambda} \\ \mu_2(r,t)&=& -\frac{2r}{S^2\sqrt{r^2 -t^2}} \frac{1}{(c^2)'R - 2r\lambda}.\end{aligned}$$ Let $$\begin{aligned} \label{G} G&=&\frac{dR}{d_{+}t} - \frac{dR}{d_{-}t} =-2 t^2 e(r,t) R_r \\ H&=&\frac{dS}{d_{+}t} - \frac{dS}{d_{-}t} =-2t^2 e(r,t) S_r, \label{H}\end{aligned}$$ and $$\begin{aligned} e(r,t)&=& \frac{r}{\sqrt{r^2 -t^2}}\left( \frac{1}{(c^2)' R - 2r \lambda} + \frac{1}{(c^2)' S + 2r \lambda} \right).\end{aligned}$$ Evaluate and write $$\begin{aligned} \frac{dG}{d_{-}t} &=& \left(\frac{2}{t} +l(r,t)\right) G + \frac{\tilde f_1(r,t)}{2t} G -\frac{\tilde f_2(r,t)}{2t} H + f_3(r,t) t+ f_4(r,t)t^2,\label{Geq}\end{aligned}$$ where $$\begin{aligned} l(t) &=& \frac{t}{r^2 -t^2} + \frac{1}{((c^2)' R-2r\lambda)((c^2)' S + 2r\lambda)} \\ && \times \left( \frac{ (c^2)''}{(c^2)' (R+S)}\frac{2t}{(c^2)'} \left[ \left[ R ((c^2)' S + 2r\lambda)^2 + S((c^2)' R-2r\lambda)^2 \right] - \frac{(c^2)' r^2(R-S)^2}{\sqrt{r^2 -t^2}} \right] \right. \\ && \left. - \frac{4t^2 \frak{h} r^2 (R-S)}{\sqrt{r^2 -t^2}} -\frac{2r^4( (c^2)'(R-S) - 4r\lambda) }{(r^2 -t^2)^{3/2} } - 4t \left(\frac{r^2 t^2}{r^2 -t^2} -\frac{r^2t^4}{(r^2 -t^2)^{2}} \right) \right), \end{aligned}$$ and $$\begin{aligned} \tilde f_1(r,t)&=&\frac{4t^2\frak{h}R}{(c^2)' S + 2r \lambda} + \frac{ 4t^2 \frak{h} (R-S)}{ (c^2)' S + 2r \lambda}\\ \tilde f_2(r,t)&=& \frac{4t^2\frak{h}R}{(c^2)' S + 2r \lambda},\\ f_3(r,t) &=& -e(r,t) \frac{R(R-S)}{(c^2)' S + 2r \lambda} \left( \frac{4t^2 \frak{h} (c^2)'' S(R-S) }{2\lambda((c^2)' S + 2r \lambda)} +\frac{(c^2)'' r^2 (R-S)}{2\lambda(r^2 -t^2)} + \frac{2r (c^2)'}{ r^2 -t^2} -\frac{2r^3 (c^2)'}{(r^2 -t^2)^2}\right)\\ f_4(r,t)&=& - e(r,t) \frac{4t^2 \frak{h} R(R-S)}{((c^2)' S + 2r\lambda)^2} \left( \frac{2r}{\sqrt{r^2 -t^2}} - \frac{2rt^2}{(r^2 -t^2)^{3/2}} \right).\end{aligned}$$ Similarly we also have $$\begin{aligned} \frac{dH}{d_{+}t} &=& \left(\frac{2}{t} +l(r,t)\right) H -\frac{\tilde g_1(r,t)}{2t} G + \frac{\tilde g_2(r,t)}{2t} H + g_3(r,t) t+ g_4(r,t)t^2,\label{Heq}\end{aligned}$$ where $$\begin{aligned} \tilde g_1(r,t)&=& \frac{4t^2\frak{h}S}{(c^2)' R- 2r \lambda} \\ \tilde g_2(r,t)&=&\frac{4t^2\frak{h}S}{(c^2)' R- 2r \lambda} +\frac{4t^2\frak{h}(S-R)}{(c^2)' R- 2r \lambda},\\ g_3(r,t) &=& - e(r,t) \frac{S(S-R)}{(c^2)' R- 2r \lambda} \left( \frac{ 4t^2 \frak{h} (c^2)'' R(R-S)}{2\lambda ((c^2)' R - 2r \lambda)} +\frac{2r (c^2)'}{r^2-t^2} + \frac{(c^2)'' r^2 (R-S)}{2\lambda(r^2-t^2)} - \frac{ 2r^3(c^2)'}{ (r^2 -t^2)^2} \right) \\ g_4(r,t)&=& -e(r,t) \frac{ 4t^2 \frak{h} S(S-R)}{((c^2)' R - 2r\lambda)^2} \left(-\frac{2r}{\sqrt{r^2 -t^2}} + \frac{2r t^2}{(r^2 -t^2)^{3/2} } \right).\end{aligned}$$ We now establish, in the same spirit as in [@WangZheng], the following regularity result. \[regularity\] For given Riemann data $2c(p_4)> c(p_1)$, $t_0>0$ sufficiently small, and $\delta>0$, there exist a positive constant $C$ and $\beta\in(1,2)$ depending only on $t_0, \delta$, the Riemann data and $\max_{\Gamma_{12}\cup\Gamma_{23}} (R, S)$, such that the solutions $R, S\in C^1(\mathcal{R}_1\setminus{{\mathcal{R}_1}}[\delta])$ satisfy $$\begin{aligned} \label{Holderestimates} |R_t|, |S_t|, t^\beta |R_r|, t^\beta |S_r| \le C \delta^{-1}, \quad \forall t\le t_0.\end{aligned}$$ Moreover, $R, S$ and $t^{-1}(R-S)$ are uniformly continuous in $\mathcal{R}_1\setminus {{\mathcal{R}_1}}[\delta]$, that is, $R-S= O(\sqrt{r^2 -c^2(p)})$, and consequently $\sigma \cap \overline {\mathcal{R}_1\setminus {{\mathcal{R}_1}}[\delta]} \in C^1$. Recall that $$\begin{aligned} p_\theta = \frac{R+S}{2},&&p_r =\frac{R-S}{2\lambda^{-1}} =\frac{R-S}{t} \frac{\sqrt{r^2 -t^2}}{2r}.\end{aligned}$$ Hence in order to have the sonic line to be in $C^1$, we show that $(R-S)/t$ is uniformly bounded, and $R, S$ and $(R-S)/t$ are uniformly continuous in $\mathcal{R}_1\setminus {{\mathcal{R}_1}}[\delta]$, For each fixed $t=t_b>0$, we consider the level curve $\{ r^2 -c^2(p(r,\theta)) =t^2_b \}$ where $\theta=\theta(r)$. We then have $$\theta'(r)=\frac{2r- (c^2)' p_r}{ (c^2)' p_\theta}.$$ Since $R$ and $S$ are positive and bounded in region $\mathcal{R}_1$, we have $p_\theta =(R+S)/2$ positive and bounded. Thus $\theta'(r)$ is well-defined on each level curve. Recall that $V=1/S -1/R$ and $V$ satisfies . We have $m_i$, $i=0,1$ positive constants such that $|(\mu_0-1)/t|\le m_0 \delta^{-1} $, and $|\mu_j|\le m_1 \delta^{-3}$ where $j=1,2$, in region $\mathcal{R}_1\setminus{{\mathcal{R}_1}}[\delta]$. Hence we can write equation in the form $$\begin{aligned} &&\partial_t \left( \frac{V(r,t)}{t} \exp \left(\int^{t_b}_t \frac{\mu_0-1}{\tau} d\tau \right)\right) = \left[ \mu_1(r,t) R_r t +\mu_2(r,t) S_r t \right]\exp \left(\int^{t_b}_t \frac{\mu_0-1}{\tau} d\tau\right).\end{aligned}$$ Integrating the last equation from $t$ to $t_b$, we have $$\begin{aligned} \frac{V}{t}\mid_{t=t_b} &- \frac{V(r,t)}{t} \exp \left(\int^{t_b}_t \frac{\mu_0-1}{\tau} d\tau \right) \\ &= \int^{t_b}_t \left[ \mu_1(r,\tau ) R_r \tau +\mu_2(r,\tau) S_r \tau \right]\exp \left(\int^{t_b}_\tau \frac{\mu_0-1}{\sigma} d\sigma\right)d\tau,\end{aligned}$$ which implies $$\begin{aligned} \nonumber \exp(-m_0 \delta^{-1} & t_b) \left|\frac{V(r,t)}{t}\right| \\ &< \exp \left(-\int^{t_b}_t \frac{\mu_0-1}{\tau} d\tau \right) \left|\frac{V(r,t)}{t} \right|\\ \nonumber &\le \left|\frac{V}{t} \mid_{t=t_b} \right| + \int^{t_b}_t | \mu_1(r,\tau ) R_r \tau +\mu_2(r,\tau) S_r \tau| \exp \left(\int^{t_b}_\tau \left| \frac{\mu_0-1}{\sigma} \right| d\sigma\right)d\tau\\ &\le \left|\frac{V}{t} \mid_{t=t_b} \right| + \exp(m_0 \delta^{-1} t_b) m_1 \delta^{-3} \int^{t_b}_t \tau |R_r| + \tau | S_r| d\tau. \label{Vt}\end{aligned}$$ Thus we first establish the estimates on $ tR_r $ and $tS_r $ to obtain the bound of $V/t$. Recall $G=-2t^2 e(r,t) R_r$ and $H=-2t^2 e(r,t) S_r$, satisfying and respectively. It is important to observe that the singular terms are with $G$ and $H$ of order $O(t^{-1})$ while the remaining terms are of order $O(t)$. We first establish the estimates of $G$ and $H$ from and . We define $$M=\max\{ t^{-\beta} |G|, t^{-\beta} |H|\},$$ where $\beta>1$ and $M$ are positive constants to be determined. From we write $$\begin{aligned} \frac{d}{d_-t} \bigg( G(r,t) &\exp\int^t_{t_b} -\left(\frac{2}{\tau} + l(r_-(\tau),\tau) \right) d\tau \bigg)\\ =& \left( \frac{\tilde f_1(r_-(t),t )}{2t } G -\frac{\tilde f_2(r_-(t),t)}{2t} H + f_3(r_-(t),t) t+ f_4(r_-(t),t)t^2\right) \\ & \times\exp\int^t_{t_b} -\left(\frac{2}{\tau} + l(r_-(\tau),\tau) \right) d\tau.\end{aligned}$$ Integrate the last equation along the minus characteristic curve passing through $(r_0, {\varepsilon}_0)$ to get $$\begin{aligned} &&G(r,t) \left(\frac{t_b}{t}\right)^{2} \exp\int^{t_b}_t l(r_-(\tau),\tau) d\tau \\ &=& G(r, t_b) \\ &&+ \int^{t_b}_t \left(-\frac{\tilde f_1(r_-(\tau),\tau )}{2\tau } G + \frac{\tilde f_2(r_-(t),\tau )}{2\tau} H - f_3(r_-(t),\tau) \tau - f_4(r_-(t),\tau) \tau^2\right) \\ && \times \left(\frac{t_b}{\tau}\right)^{2} \exp\int^{t_b}_\tau l(r_-(\sigma),\sigma) d\sigma d\tau.\end{aligned}$$ We then deduce $$\begin{aligned} &&G(r,t) \left(\frac{t_b}{t}\right)^{2} \exp\int^{t_b}_t l(r_-(\tau),\tau) d\tau\\ &\le & G(r, t_b) \\ &&+ \int^{t_b}_t \left[ \left(\frac{|\tilde f_1(r_-(\tau),\tau )| }{2} + \frac{| \tilde f_2(r_-(\tau),\tau )|}{2 } \right) M \tau^{\beta-1} \right.\\ &&\left. + |f_3(r_-(\tau),\tau)| \tau+ |f_4(r_-(\tau),\tau)|\tau^2 \right] \left(\frac{t_b}{\tau}\right)^{2} \exp\int^{t_b}_\tau l(r_-(\sigma),\sigma) d\sigma d\tau.\end{aligned}$$ Observe that for $t_0>0$, there exist positive constants $L_0$ and $F_0$ such that $|l(r,t)|\le L_0 \delta^{-1} $, $\tilde f_1, \tilde f_2\le F_0$ and $ |f_3|, |f_4| \le F_1 \delta^{-1} $ for $t \le t_0$. Hence for $t\le t_b \le t_0$, we get from the last inequality $$\begin{aligned} &&G(r,t) \left(\frac{t_b}{t}\right)^{2} \exp (L_0 \delta^{-1} (t- t_b)) \\ &\le & G(r, t_b) \\ &&+ t_b^2 \int^{t_b}_t \left( F_0 M \tau^{\beta -3} + F_1 \delta^{-1} (\tau^{-1} +1) \right) \exp(L_0 \delta^{-1} (t_b-\tau)) d\tau\\ &\le & G(r, t_b) \\ &&+ t_b^{2} \exp(L_0\delta^{-1} t_b) \frac{1}{2-\beta} F_0M \left( \frac{1 }{t^{2-\beta}} -\frac{1}{t_b^{2-\beta}} \right) + t_b^{2} \exp(L_0\delta^{-1} t_b) F_1\delta^{-1} ( \ln(t_b) -\ln t +t_b-t).\end{aligned}$$ We then deduce $$\begin{aligned} G(r,t) t^{-\beta} &\le & G(r,t_b) t_b^{-2} t^{2-\beta} \exp (L_0 \delta^{-1} t_b) + \exp(2 L_0 \delta^{-1} t_b) \frac{1}{2-\beta} F_0 M \left( 1 -\frac{t^{2-\beta}}{t_b^{2-\beta}} \right) \\ &&+ \exp( 2 L_0\delta^{-1} t_b) F_1\delta^{-1} ( \ln(t_b) -\ln t +t_b-t)t^{2-\beta}.\end{aligned}$$ Thus for $1<\beta<2$, by choosing $t_b\le t_0$ sufficiently small if necessary, we then have $$\begin{aligned} \frac{ \exp(2 L_0 \delta^{-1} t_b) F_0}{2-\beta} \left( 1 -\frac{t^{2-\beta}}{t_b^{2-\beta}} \right) &\le & C_0 <1\end{aligned}$$ for all $t \le t_b$. Hence we now have established the bound of $M$; $$\begin{aligned} M &\le & \frac{C_1}{1-C_0},\end{aligned}$$ where $$C_1 =\max_{\{ t\le t_b\} }[ G(r,t_b) t_b^{-2} + F_1 ( \ln(t_b) -\ln t +t_b-t) \exp (L_0\delta^{-1} t_b)] \exp (L_0 \delta^{-1} t_b) t^{2-\beta}.$$ This uniform bound (similarly to $H$) holds for any $1<\beta<2$, and immediately gives $$t^\beta |R_r|, t^\beta|S_r| \le M.$$ Now with this uniform bound, from inequality for $V/t$, noting $|e|\le E_0 \delta^{-1}$, we have $$\begin{aligned} \left|\frac{V(r,t)}{t}\right| &\le & \exp(m_0 \delta^{-1} t_b) \left|\frac{V}{t} \mid_{t=t_b} \right| + \exp(2 m_0 t_b) m_1 \delta^{-3} \int^{t_b}_t \tau| R_r| + \tau| S_r| d\tau\\ &\le & \exp(m_0 \delta^{-1} t_b) \left|\frac{V}{t} \mid_{t=t_b} \right| + \exp(2 m_0 t_b) m_1\delta^{-3} \int^{t_b}_t \tau^{\beta -1} M \frac{1}{e(r,\tau)} d\tau\\ &\le& \exp(m_0 \delta^{-1} t_b) \left|\frac{V}{t} \mid_{t=t_b} \right|+ \exp(2 m_0 \delta^{-1}t_b) \frac{m_1 \delta^{-2} M}{\beta E_0} (t_b^{\beta} -t^{\beta})\\ &\le & M_1 \delta^{-2}.\end{aligned}$$ Hence we get $$\left|\frac{R-S}{t}\right| \le M_1 RS \delta^{-2} \le CM_1.$$ We next show that $R$, $S$ and $V/t$ are uniformly continuous. We now have $$\begin{aligned} \left|\frac{dR}{d_-t}\right| &=& \left| \frac{2t^2 \frak{h} R}{ (c^2)' S + 2r\lambda^{-1}} \frac{R-S}{t} \right| \le CM_1\\ \left|\frac{dS}{d_+t}\right| &=& \left| \frac{2t^2 \frak{h} S}{ (c^2)' R- 2r\lambda^{-1}} \frac{S-R}{t} \right| \le CM_1,\end{aligned}$$ for some constant $C>0$ uniformly in $t$. Hence integrate the last inequalities along the negative and positive characteristics respectively to obtain $$\begin{aligned} |R(r_1, 0) -R(r_0, t_0)| &\le & CM_1 t_0\\ |S(r_2, 0) -S(r_0, t_0)| &\le & CM_1 t_0,\end{aligned}$$ Since $R=S$ along the sonic line and both $R$ and $S$ are continuous inside region $\mathcal{R}_1$, we have $$\begin{aligned} |R(r_1, 0)- R(r_2, 0)|, |S(r_1, 0)- S(r_2, 0)|&\le& 2CM_1 t_0 + |R(r_0, t_0) - S(r_0, t_0)|\rightarrow 0\end{aligned}$$ as $|r_1-r_2|\rightarrow 0$. Thus $R$ and $S$ are continuous on the sonic line and uniformly continuous in $\mathcal{R}_1\setminus {{\mathcal{R}_1}}[\delta]$. Next, integrating from $0$ to $t_b$ where $t_b$ is arbitrary chosen, we have $$\begin{aligned} \frac{V}{t}\mid_{t=t_b} &- \frac{V}{t}\mid_{t=0} \exp \left(\int^{t_b}_0 \frac{\mu_0-1}{\tau} d\tau \right)\\ &= \int^{t_b}_0 \left[ \mu_1(r,\tau ) R_r \tau +\mu_2(r,\tau) S_r \tau \right]\exp \left(\int^{t_b}_\tau \frac{\mu_0-1}{\sigma} d\sigma\right)d\tau,\end{aligned}$$ which then becomes $$\begin{aligned} && \left| \frac{V}{t}\mid_{t=t_b} - \frac{V}{t}\mid_{t=0} \right|\\ &\le & \left| \frac{V}{t}\mid_{t=0} \right| \left( \exp \int^{t_b}_0 \left|\frac{\mu_0 \delta^{-1} -1}{\tau}\right| d\tau -1 \right) + m_1 \delta^{-3} M \int^{t_b}_0 \tau^{\beta-1} \exp \left(\int^{t_b}_\tau \frac{\mu_0 \delta^{-1} -1}{\sigma} d\sigma\right)d\tau\\ &\le & 2M_1 e^{m_0 \delta^{-1} t_b} t_b + \frac{m_1\delta^{-3} M e^{m_0 \delta^{-1} t_b} t_b^{\beta}}{\beta} \le M_2 t_b.\end{aligned}$$ Hence with $M_2 t_b \le {\varepsilon}/4$ (take $\eta>0$ such that if $|r_1-r_2|\le \eta$ then $\left| \frac{V}{t}(r_1, t_b) - \frac{V}{t}(r_2,t_b) \right|\le {\varepsilon}/4$) we have $$\begin{aligned} \left| \frac{V}{t}(r_1, 0) - \frac{V}{t}(r_2,0) \right| &\le& \left| \frac{V}{t}(r_1, 0) - \frac{V}{t}(r_1, t_b) \right| + \left| \frac{V}{t}(r_1, t_b) - \frac{V}{t}(r_2,t_b) \right|\\ && +\left| \frac{V}{t}(r_2, b) - \frac{V}{t}(r_2,0) \right| \le {\varepsilon}\end{aligned}$$ for $|r_1-r_2|\le \eta$. Thus $V/t$ is also uniformly continuous in $\mathcal{R}_1\setminus {{\mathcal{R}_1}}[\delta]$. Therefore we have established the claim. The supersonic solution in region ${{\mathcal{R}_1}}$ ----------------------------------------------------- The existence result in the entire region ${{\mathcal{R}_1}}$ is established in two steps. We first show Lemma \[exlemma\] to establish the sonic boundary $\sigma$ where $R=S$. We next show that $\overline\sigma\cap \Gamma_{12}=\Xi_1$ and $\overline\sigma\cap \Gamma_{23}=\Xi_3$. The proof of lemma \[exlemma\] is inspired by the work in [@Bang; @DaiZhang]. [@DaiZhang] and later [@Bang], established the global existence result for the degenerate hyperbolic system of the pressure gradient equation, where the pressure becomes zero at the origin which makes the system degenerate only at the origin and hyperbolic elsewhere. Furthermore since the pressure gradient system is quasilinear, the type of the system (whether supersonic or subsonic) must be also identified. While our system may appear to have similar technical difficulties as in [@Bang; @DaiZhang], we note that it is not straightforward to apply their result to our case. We also note that the sonic boundary will be determined by the choice of the data on $\Gamma_{23}$ and thus the compatibility conditions at $\Xi_1$ and $\Xi_3$ will play crucial roles in selecting the correct data on $\Gamma_{23}$ to find the supersonic solution in the entire region ${{\mathcal{R}_1}}$. Let $u=r^2 -c^2(p)$ so that $u=0$ on the sonic boundary $\sigma$. The characteristic equations in $u$ become $$\begin{aligned} \label{up} \partial_+ u = u_\theta + \lambda_+ u_r &=& 2\frac{r^2}{c(p)} \sqrt{u} -(c^2)' R, \\ \partial_- u = u_\theta +\lambda_- u_r &=& -2\frac{r^2}{c(p)} \sqrt{u} -(c^2)' S. \label{um}\end{aligned}$$ Thus we have $$\begin{aligned} \label{utheta} u_\theta &=& -\frac{(c^2)'}{2} (R+S),\\ u_r &=& 2r - \frac{(c^2)'c}{r} \left(\frac{R-S}{\sqrt{u}}\right).\label{ur}\end{aligned}$$ \[exlemma\] For each $d$, where $0<d \le d_m < c^2(p_1)$, there exists a level curve $l_\tau=\{(\tau(\theta), \theta): u(\tau(\theta),\theta)=d\} \subset {\mathcal{R}_1}$ and $ \tau=\tau(\theta)\in C^1$ such that the following hold: 1. there exists $U= (u, R, S) \in C^1(D_\tau)$ satisfying the Goursat boundary problem ,,, 2. Only the positive family of characteristics intersects with $\Gamma_{23}$. Only the negative family of characteristics intersect with $\Gamma_{12}$, where $D_\tau \subset {\mathcal{R}_1}$ is the closed domain enclosed by $\Gamma_{12}$, $\Gamma_{23}$ and $l_\tau$. Let $L$ be the set where $d\in L$ satisfying the assertions (1)-(2) in this lemma. Since the system for $W=(p, R, S)$ is equivalent to the new system for $U=(u,R, S)$, the local existence result from the system of $W=(p, R,S)$ ensures the existence of $d_0\in L$ (that is $L\neq \emptyset$) and $[d_0, d_m]\in L$ (since $u_\theta<0$). Hence we only need to show that $\inf L=0$ to establish the claim. We show the claim by contradiction. Suppose that $\inf L =d^*>0$. The proof consists of two main steps. By extracting a limit to first show that $d^*\in L$, and then show that this $d^*$ violates the infimum assertion. Since $d^*$ is assumed to be the infimum of $L$, there exists a monotone decreasing sequence $\{d_n\}\subset L$ satisfying $\lim_{n \rightarrow \infty} d_n = d^*$. Then for each $d_n$, there exists $\tau_n=\tau_n(\theta)$ which satisfies the assertions (1)-(2) where $l_n$ is enclosed in the domain bounded by $\Gamma_{12}$ and $\Gamma_{23}$, that is $l_n=\{(\tau_n(\theta), \theta), \theta^n_{23}\le \theta\le \theta^n_{12} \}$ where $l_n\cap \Gamma_{12}=(r^{n}_{12},\theta^n_{12})$ and $l_n\cap \Gamma_{23}=(r^n_{23}, \theta^n_{23})$. We let $D_n$ be the domain enclosed by $l_n$, $\Gamma_{12}$ and $\Gamma_{23}$. The uniqueness of the local existence results ensure the monotonicity of $D_n$ such that $D_n\subset D_{n+1}$. Thus we let $l^*$ be the graph of $\tau^*(\theta)$ where $\lim_{n \rightarrow \infty} \tau_n(\theta)= \tau^*(\theta)$, for all $\theta^*_{23}\le \theta\le \theta^*_{12}$. Let $D^*$ be the closed region enclosed by $l^*$, $\Gamma_{12}$ and $\Gamma_{23}$. Hence in $D^{*}\setminus l^*$, there exists the solution $U=(u,R,S)$ satisfying (due to Lemmas 5.1-5.6 and Theorems 6.1, 6.2) $$\begin{aligned} d^* \le u\le c^2(p_1)-c^2(p_4),\\ |u_\theta + \lambda_{\pm} u_r| \ge c_0>0,\\ u_\theta \le -c_0\\ \|U\|_{C^{1,\beta}(D^*\setminus l^*)}\le c_1,\end{aligned}$$ where $c_0, c_1>0$ depends only on $p_1,p_4$ and $d^*$. Since the tangential derivative of $u$ along $l_\tau$ is $$\begin{aligned} u_\theta + \tau' u_r=0.\end{aligned}$$ we now obtain $\|\tau\|_{C^{1,\beta}}\le C.$ By using the standard compactness argument we have $\|\tau\|_{C^{1,\alpha}}\le C_1,$ for any $0<\alpha<\beta$. Repeating a similar compactness argument we have $U=(u,R,S)\in C^{1,\alpha}$ with the uniform bound established from the regularity results. Thus we extend $U$ in $D^*$ to satisfy the governing system. Now observe that $l^*$ is the constant level set of $u$, and thus $$\label{lV} u_\theta + (\tau^*)'(\theta) u_r=0, \quad {\rm on } \quad l^*.$$ On the other hand $$u_\theta + \lambda_{\pm} u_r \neq 0\quad {\rm on } \quad l^*.$$ Thus $$(\tau^*)'(\theta)\neq \lambda_{\pm},\quad {\rm on } \quad l^*.$$ Therefore $d^*\in L$. Next we show that there exists ${\varepsilon}>0$ small such that $d^*-{\varepsilon}\in L$ to contradict $\inf L = d^*$. Since we have shown that $d^* \in L$, by the local existence result, we extend the solution in $C^1$ by solving the initial boundary value problems where the initial value on $l^*$ satisfies the compatibility conditions on $\Gamma_{23}\cap l^*$ and $\Gamma_{12}\cap l^*$. Furthermore the solution is unique (since the data is prescribed uniquely) and $u_\theta<0$, the corresponding solution determines the level curve where $u=d^*-{\varepsilon}>0$ for some small ${\varepsilon}>0$ satisfying the assertions (1)-(2) in this lemma. Hence repeating the same argument as to $d^*$ we can show $d^*-{\varepsilon}\in L$, which is a contradiction. Therefore $\inf L =0$ and this completes the proof. Lemma \[exlemma\] implies the existence of the sonic boundary $\sigma$ where $R=S$. Due to the monotonicity properties of the characteristics and the solution, the sonic boundary $\sigma$ is connected, bounded and will not form a closed loop. We now check whether this sonic boundary satisfies the compatibility conditions at $\Xi_1$ and $\Xi_3$. We first consider $\Xi_3.$ If $\sigma$ meets at $X_0\in\Gamma_{23}$ where $X_0\neq \Xi_3$. This then violates the data at $X_0$ since $S\neq R$ on $\Gamma_{23}$. So if $\sigma$ meets somewhere on $\Gamma_{23}$ it must be only at $\Xi_3$. Now if $\sigma$ does not intersect with $\Gamma_{23}$, that is the sonic boundary is terminated before it reaches to $\Gamma_{23}$ and $dist(\overline\sigma,\Gamma_{23})=\delta>0$. In that case, since $R$ and $S$ must be positive in the region ${{\mathcal{R}_1}}$, we have $R=S>0$ on $\overline\sigma$. In particular we can find the positive characteristic $\Gamma_+^\delta$ connecting the boundary point, denoted by $X_1$, of $\overline\sigma$, and $\Gamma_{23}$. Hence we can treat this as a new boundary value problem where the boundaries consist of $\Gamma_+^{\delta}$ and a segment, denoted by $\Gamma_{23}^\delta$, of $\Gamma_{23}$ from $\Xi_3$ to $Y_1=\Gamma_+^\delta \cap \Gamma_{23}$. We note that the data on $\Gamma_+^\delta$ is now prescribed with the given data on $\Gamma_{23}\setminus \Gamma_{23}^\delta$. Note that $R=S>0$ on $\overline\sigma$. Thus we may write $R(X_1) = S(X_1)=q_0>0$ and $\tau'\ge {\varepsilon}_0>0$ for some constants $q_0, {\varepsilon}_0$ (however note that these constants $q_0, {\varepsilon}_0$ depend on the choice of the data $g$ on $\Gamma_{23}$). Hence we find $0<\delta_1< \delta_0$ and ${\varepsilon}_1$ such that we can extend $\tau'(\theta) \ge {\varepsilon}_1>0$, where ${\varepsilon}_1<{\varepsilon}_0$, in a small neighborhood of $X_1$, denote $B_{\delta_1}(X_1)$ to be the circle centered at $X_1$ with radius $\delta_1$. We now find the positive characteristics $\Gamma_{+}^{\delta_2}$ connecting $Y_2\in\Gamma_{23}$ and $\partial B_{\delta_1/2}(X_1)$, see Figure \[construction\]. If there exists $\delta_1$ such that $B_{\delta_1}(X_1)$ covers the entire neighborhood of $\Xi_3$ then our choice of $g$ on the data to prescribe $\Gamma_{23}$ needs to be adjusted (scale down the strength on $S$). We next construct the solution by solving the initial and boundary value problems in the regions and enclosed by $\Gamma_+^{\delta_1}$, $\Gamma_{-}^{\delta_1}$, $\Gamma_+^{\delta_2}$, and $\Gamma_{23}$, respectively, so that we find the sonic boundary $\sigma[\delta_1]$, where $ R=S\ge q_1>0$ on $\sigma[\delta_1]$ for some constant $q_1\le q_0$. Repeat this process by finding ${\varepsilon}_n\rightarrow 0$ to construct $\sigma[\delta_n]$ for each ${\varepsilon}_n>0$ and find sequences $\{X_n\}$, $\{Y_n\}$, and $q_n$, where $R=S\ge q_n$ on $\sigma[\delta_n]$. By the construction (these sequences are monotone and bounded) we know that there exist limits for these sequences. Let $X_*, Y_*, q_*$ be the limits of these sequences respectively. Note that the tangential derivative on $r^2 -c^2(p)$ along $\sigma$ is $$\begin{aligned} (c^2)'(p_\theta +\tau' p_r) = 2r \tau'\end{aligned}$$ which implies $R=S=p_\theta$ on $\sigma$ and $$\begin{aligned} p_\theta =q_n \rightarrow q_*=0,\end{aligned}$$ as ${\varepsilon}_n \rightarrow 0$. Hence we are left to check whether $X_*=Y_*$. Suppose $ X_*\neq Y_*$. We have two possibilities either (1) $X_*$ surpasses below $\Gamma_{23}$, or (2) the sequence $\{X_n\}$ is terminated before it reaches to $\Gamma_{23}$. (We may treat this as a shooting method, that is, case (1) is overshoot, and case (2) is undershoot). We note however by the construction (we have adjusted the data $g$ so that $\delta_i$ can be chosen appropriately), $X_*$ should not be located below $\Gamma_{23}$. Hence the sonic boundary constructed by this sequence is then terminated before it reaches $\Gamma_{23}$. In other words we need to readjust the choice of the data $g$. Therefore there exists data $g$ such that the limits of $X_n$ and $Y_n$ match, and denote the limit by $\Xi_3$. For the case $\Xi_1$. As before, since $R>S=0$ on $\Gamma_{12}$, if $\sigma$ meets somewhere on $\Gamma_{12}$, it must be at $\Xi_1$. Now if $\sigma$ does not intersect with $\Gamma_{12}$, as we did for $\Xi_3$, we formulate a boundary value problem where the boundaries are now with the corresponding characteristics $\Gamma_-^\delta$ and $\Gamma_{12}^\delta$. Repeating a similar argument as we did for $\Xi_3$ and noting that the data on $\Gamma_{12}$ is given (independent of the data on $\Gamma_{23}$) while the data on $\Gamma_-^\delta$ depends on the data on $\Gamma_{23}$ we find the correct data so that it matches the compatibility condition at $\Xi_1$. Note that at $\Xi_3$ both datum on $\Gamma_+^\delta$ and $\Gamma_{23}^\delta$ depend on the choice of $g$, while at $\Xi_1$ it is only on $\Gamma_-^\delta$ that depends on $g$. Therefore we establish the following lemma. \[excorners\] There exist $f\in C^2(\theta_2,\theta_3)$ and $g\in C^2(\Gamma_{23})$ where $\Gamma_{23} =\{(f(\theta),\theta): \theta_2\le \theta \le \theta_3\}$, satisfying – and \[G1.\] with $\theta_2 <\theta_3<\pi/2$, such that the Goursat boundary value problem has the solution $(p, R, S)\in C^1({{\mathcal{R}_1}})\cap C^0({{\mathcal{R}_1}}\cup\{\Xi_1, \Xi_3\})$ that satisfies the following: 1. there exists the sonic boundary $\sigma =\{(\tau(\theta),\theta), \tau'>0 \}$ where $R=S>0$ on $\sigma$. 2. $\sigma$ is terminated at $\Xi_3=(f(\theta_3),\theta_3)$ with $\lim_{\theta \rightarrow \theta_3} \tau'(\theta)=0$. That is $\Xi_3=\overline \sigma\cap \overline \Gamma_{23}$. 3. $\sigma\in C^1$ 4. $R(\Xi)=S(\Xi) \rightarrow 0$ as $\Xi \in \sigma \rightarrow \Xi_k$, $k=1,3$. \[\]\[\]\[0.7\]\[0\][$X_1$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma^{\delta_1}_{+}$]{} \[\]\[\]\[0.7\]\[0\][$Y_1$]{} \[\]\[\]\[0.7\]\[0\][$Y_3$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_3$]{} \[\]\[\]\[0.7\]\[0\][$X_3$]{} \[\]\[\]\[0.7\]\[0\][${\tikz[baseline=(char.base)]{ \node[shape=circle,draw,inner sep=1.5pt] (char) {2};}}$]{} \[\]\[\]\[0.7\]\[0\][${\tikz[baseline=(char.base)]{ \node[shape=circle,draw,inner sep=1.5pt] (char) {1};}}$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma^{\delta_3}_{+}$]{} \[\]\[\]\[0.7\]\[0\][$X_2$]{} \[\]\[\]\[0.7\]\[0\][$B_{\delta_1}(X_1)$]{} \[\]\[\]\[0.7\]\[0\][$\delta_1$]{} \[\]\[\]\[0.7\]\[0\][$Y_2$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{23}$]{} \[\]\[\]\[0.7\]\[0\][$\delta_1/2$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma^{\delta_2}_{+}$]{} \[\]\[\]\[0.7\]\[0\][$\delta_2$]{} \[\]\[\]\[0.7\]\[0\][$B_{\delta_2}(X_2)$]{} Therefore we finally establish the existence result in the entire transient wave region ${{\mathcal{R}_1}}$. Lemma \[excorners\] provides the schematics, as illustrated in Figure \[construction\], that is, how to construct the solution near $\Xi_1$ and $\Xi_3$ by selecting the correct data on $\Gamma_{23}$ to hold the compatibility conditions at $\Xi_1$ and $\Xi_3$. Furthermore Lemmas \[exlemma\], \[excorners\] utilize the properties of the tricomi type degeneracy, that is, the sonic boundary cannot be a characteristic curve, and as a consequence $R=S=p_\theta$ remain strictly positive. Hence the natural compatibility condition where the different types of boundaries meet is $R=S=0$ which gives rise to an additional degeneracy. Simple wave in region $\mathcal{R}_2$ {#sectionR_2} ===================================== We are now left with the simple wave region ${{\mathcal{R}_2}}$. We proceed to derive certain identities that may prove useful. Note that from $$\frac{dr_+}{d\theta}=\lambda=r_+\sqrt{\frac{r^2_+ -c^2}{c^2}},$$ we can write $$\begin{aligned} c^2(p)=\gamma p^{\kappa}= \frac{r^4_+}{r '^2_++r^2_+},\end{aligned}$$ which implies $$\begin{aligned} r^2_+-\gamma p^{\kappa}&=\frac{(r'_+r_+)^2}{r '^2_++r^2_+}.\end{aligned}$$ Differentiating along the positive characteristics $r=r_+$ we have $$\begin{aligned} \partial_{+}(\gamma p^{\kappa})&=\frac{2r^3_+r'_+[r^2_++2r'^2_+-r_+r''_+]}{(r'^2_++r^2_+)^2}, \end{aligned}$$ and consequently $$\begin{aligned} R=\partial_{+}p&=\frac{2r^3_+r'_+p^{1/\gamma}}{\gamma \kappa}\left[\frac{r^2_++2r'^2_+-r_+r''_+}{(r'^2_++r^2_+)^2}\right]. \label{R1}\end{aligned}$$ Similarly from $$\frac{dr_-}{d\theta}=- \lambda=-r_-\sqrt{\frac{r^2_- -c^2}{c^2}},$$ we have $$\begin{aligned} S=\partial_{-}p&=\frac{2r^3_-(-r'_-)p^{1/\gamma}}{\gamma \kappa}\left[\frac{r_-r''_--r^2_--2r'^2_-}{(r'^2_-+r^2_-)^2}\right]. \label{S1}\end{aligned}$$ In the simple wave region $\mathcal{R}_2$, by following Lemma \[Lemma1\_increasing\], only $S=\partial_-p>0$ transfers the data across $\Gamma_{23}$, while the other family $R=\partial_+p$ becomes zero. Hence by using $r''_+=r_++\dfrac{2r'^2_+}{r_+}$ from with initial values $r_+(\theta_0)=r_0, r'_+(\theta_0)=r_0\sqrt{\frac{r_0^2-c^2(p_0)}{c^2(p_0)}},$ prescribed on $\Gamma_{23}$, we can find the equation of the positive characteristic, $\Gamma_+=\{(r_+(\theta),\theta)\}$, $$\begin{aligned} r_+(\theta)=\frac{r_0}{\left(\sin \theta_0-\cos \theta_0\sqrt{\frac{r_0^2-c^2(p_0)}{c^2(p_0)}}\right)\sin \theta+\left(\sin \theta_0\sqrt{\frac{r_0^2-c^2(p_0)}{c^2(p_0)}}+\cos\theta_0\right)\cos \theta}. \label{3}\end{aligned}$$ Solving for $c^2(p_0)$ gives $$\begin{aligned} \nonumber c^2(p_0)&=&\frac{r_+^2r_0^2[\sin^2\theta+\sin^2 \theta_0-2\sin^2\theta_0\sin^2\theta-2\sin\theta\cos\theta\sin\theta_0\cos\theta_0]}{r^2+r_0^2-2rr_0\sin\theta\sin\theta_0-2rr_0\cos\theta\cos\theta_0}\\ & = &\frac{(\eta\xi_0-\xi \eta_0)^2}{(\eta-\eta_0)^2+(\xi-\xi_0)^2}, \label{4}\end{aligned}$$ where $\xi_0=r_0\cos \theta_0, \ \ \eta_0=r_0 \sin \theta_0.$ We further find the positive characteristic emanating from $\Xi_2$, denoted by $\Gamma_{24}$, by integrating $$\begin{aligned} \frac{dr}{d\theta}=r\sqrt{\frac{r^2-c^2(p_4)}{c^2(p_4)}},\end{aligned}$$ and thus $$\Gamma_{24}: \ \ r=c(p_4)\sec\bigg(\theta+\operatorname{arcsec}\sqrt{\frac{c(p_1)}{c(p_4)}}-\arcsin\sqrt{\frac{c(p_4)}{c(p_1)}}\bigg).$$ Figure \[shock\] depicts the envelop formation by the simple wave . \[\]\[\]\[0.7\]\[0\][$\Gamma_{12}$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_1$]{} \[\]\[\]\[0.7\]\[0\][$\eta=c(p_1)$]{} \[\]\[\]\[0.7\]\[0\][$\eta=c(p_4)$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_2$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_3$]{} \[\]\[\]\[0.7\]\[0\][$\sigma$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_0$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_2$]{} \[\]\[\]\[0.7\]\[0\][$\mathcal{R}_1$]{} \[\]\[\]\[0.7\]\[0\][$\Xi_4$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{24}$]{} \[\]\[\]\[0.7\]\[0\][$\Sigma'$]{} \[\]\[\]\[0.7\]\[0\][$\Gamma_{23}$]{} Numerical Results {#numerics} ================= We conclude our paper by presenting the numerical results for the configuration. The results are produced by using the Riemann data $\rho_1=0.5$, $\rho_2=0.25$, and $\gamma =3$. The computational domain that we have implemented is $10^{-2} \le r\le 1$ and $0\le \theta \le 3\pi/2$ where $(r,\theta)$ are polar coordinates, with mesh sizes $dr = 1/2400 \approx 4.1667\times10^{-4}$ and $d\theta = 2\pi/3600 \approx 1.7\times10^{-3}$, with the final time $T=1$. These results are produced by using CLAWPACK [@Clawpack]. We implement Roe average methods [@Roe] and finite volume methods on quadrilateral grids [@Clawpack]. More precisely, we implement Roe average methods in a uniform grid in polar coordinates as our computational domain, together with a coordinate mapping and appropriate scaling of the flux differences. The scaling is done by using the area ratio “capacity” of the computational cell which is determined by the size of the corresponding physical cell [@LeVeque1]. In Figure \[fig\_cross\]: the right figure is the enlargement from the left figure near which the shock appears. In the right figure, the density flattens out near the shock, while there exists a compression (a dip in the cross section) which merges to the shock. The numeric suggests that the angle of the location of $\Xi_3$ where the sonic boundary and the shock boundary meet is between $50$ and $60$ degrees in this configuration. Acknowledgments {#acknowledgments .unnumbered} =============== The first author is grateful to Yuxi Zheng and Kyungwoo Song for helpful discussions on the Pressure gradient system. The first author is also thankful to John Hunter for a discussion on Tricomi problems. [00]{} S. Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, [*J. Differential Equations*]{} **246** (2009), no. 2, 453–481. S. [Čanić]{}, B. L. Keyfitz and E. H. Kim, Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection, **7** (2000), no. 2, 313–335. , A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks, **55** (2002), no. 1, 71–92. S. [Čanić]{}, B. L. Keyfitz, and E. H. Kim, Mixed hyperbolic-elliptic systems in self-similar flows, [*Boletim da Sociedade Brasileira de Matemática*]{} **32** (2001), no. 3, 377–399. , Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks, **37** (2006), no. 6, 1947–1977. G.-Q. Chen, X. Deng and W. Xiang, Shock diffraction by convex cornered wedges for the nonlinear wave system, [*Arch. Ration. Mech. Anal.*]{} **211** (2014), no. 1, 61–112. , Stability of transonic shocks in supersonic flow past a wedge, **233** (2007), no. 1, 105–135. S. Chen, Mixed type equations in gas dynamics, **68** (2010), no. 3, 487– 511. , [*Supersonic flow and shock waves.*]{} Springer Verlag, New York, 1948. C. M. Dafermos, [*Hyperbolic conservation laws in continuum physics.*]{} Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\], 325. Springer-Verlag, Berlin, 2000. Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics, [*Arch. Ration. Mech. Anal.*]{} **155** (2000), no. 4, 277–298. V. Elling and T.-P. Liu, Supersonic flow onto a solid wedge, [*Comm. Pure Appl. Math.*]{} **61** (2008), no. 10, 1347–1448. Transonic shock formation in a rarefaction Riemann problem for the 2D compressible Euler equations, [*SIAM J. Appl. Math.*]{} **69** (2008), no. 3, 720–742. K. Jegdic, B. L. Keyfitz, and S. [Čanić]{}, Transonic regular reflection for the nonlinear wave system, **3** (2006), no. 3, 443–474. , A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation, [*J. Differential Equations*]{} **248** (2010), 2906–2930. E. H. Kim, **37** (2012), no. 4, 610–646. E. H. Kim and C.-M. Lee, Transonic shock reflection problems for the self-similar two-dimensional nonlinear wave system, **79** (2013), 85–102. Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, [*Numer. Methods Partial Differential Equations*]{} **18** (2002), no. 5, 584–608. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, [*SIAM J. Sci. Comput.*]{} **19** (1998), no. 2, 319–340. , , 2002. CLAWPACK 4.3. . C. S. Morawetz, Potential theory for regular and Mach reflection of a shock at a wedge, **47** (1994), no. 5, 593–624. Approximate Riemann solvers, parameter vectors, and difference schemes, **43** (1981), no. 2, 357–372. Numerical solution of the Riemann problem for two-dimensional gas dynamics, [*SIAM J. Sci. Comput.*]{} **14** (1993), no. 6, 1394–1414. D. Serre, . Handbook of Mathematical Fluid Dynamics, vol. 4 . Eds: S. Friedlander, D. Serre. Elsevier, North-Holland (2007), 39–122. D. Serre and H. Freistühler, The hyperbolic/elliptic transition in the multi-dimensional Riemann problem, **62** (2013), no. 2, 465–485. M. Sever, Admissibility of self-similar weak solutions of systems of conservation laws in two space variables and time, **6** (2009), no. 3, 433–481. W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, [*Mem. Amer. Math. Soc.*]{} **137** (1999), no. 654. K. Song, The pressure-gradient system on non-smooth domains, **28** (2003), no. 1-2, 199–221. Semi-hyperbolic patches of solutions of the pressure gradient system, [*Discrete Contin. Dyn. Syst.*]{} **24** (2009), no. 4, 1365–1380. The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics, [*Indiana Univ. Math. J.*]{} **63** (2014), no. 2, 385–402. The triple point paradox for the nonlinear wave system, **67** (2006/07), no. 2, 321–336. Self-similar solutions for the triple point paradox in gasdynamics. **68** (2008), no. 5, 1360–1377. Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, [*SIAM J. Math. Anal.*]{} **21** (1990), no. 3, 593–630. Two-dimensional regular shock reflection for the pressure gradient system of conservation laws, [*Acta Math. Appl. Sin. Engl. Ser.*]{} **22** (2006), no. 2, 177–210. , Progress in Nonlinear Differential Equations and their Applications, 38. BirkhŠuser Boston, Inc., Boston, MA, 2001. [^1]: The work of Kim was supported by the National Science Foundation under the Grants DMS-1109202, 1615266 [^2]: The work of Tsikkou was supported by the National Science Foundation under the Grant DMS-1400168

--- abstract: 'Recently BERT has been adopted in state-of-the-art text summarization models for document encoding. However, such BERT-based extractive models use the sentence as the minimal selection unit, which often results in redundant or uninformative phrases in the generated summaries. As BERT is pre-trained on sentence pairs, not documents, the long-range dependencies between sentences are not well captured. To address these issues, we present a graph-based discourse-aware neural summarization model - <span style="font-variant:small-caps;">DiscoBert</span>. By utilizing discourse segmentation to extract discourse units (instead of sentences) as candidates, <span style="font-variant:small-caps;">DiscoBert</span> provides a fine-grained granularity for extractive selection, which helps reduce redundancy in extracted summaries. Based on this, two discourse graphs are further proposed: ($i$) RST Graph based on RST discourse trees; and ($ii$) Coreference Graph based on coreference mentions in the document. <span style="font-variant:small-caps;">DiscoBert</span> first encodes the extracted discourse units with BERT, and then uses a graph convolutional network to capture the long-range dependencies among discourse units through the constructed graphs. Experimental results on two popular summarization datasets demonstrate that <span style="font-variant:small-caps;">DiscoBert</span> outperforms state-of-the-art methods by a significant margin.' author: - | Jiacheng Xu^1^[^1], Zhe Gan^2^, Yu Cheng^2^, Jingjing Liu^2^\ ^1^University of Texas at Austin ^2^Microsoft Dynamics 365 AI Research\ [jcxu@cs.utexas.edu](jcxu@cs.utexas.edu); {zhe.gan, yu.cheng, jingjl}@microsoft.com bibliography: - 'jcxu.bib' title: 'Discourse-Aware Neural Extractive Model for Text Summarization' --- Conclusions =========== In this paper, we present <span style="font-variant:small-caps;">DiscoBert</span> for text summarization. <span style="font-variant:small-caps;">DiscoBert</span> uses discourse unit as the minimal selection basis to reduce summarization redundancy, and leverages two constructed discourse graphs as inductive bias to capture long-range dependencies among discourse units for better summarization. We validate our proposed approach on two popular datasets, and observe consistent improvement over baseline methods. For future work, we will explore better graph encoding methods, and apply discourse graphs to other tasks that require long document encoding. [^1]: Most of this work was done when the first author was an intern at Microsoft Dynamics 365 AI Research.

--- abstract: 'In clinical conversational applications, extracted entities tend to capture the main subject of a patient’s complaint, namely symptoms or diseases. However, they mostly fail to recognize the characterizations of a complaint such as the time, the onset, and the severity. For example, if the input is “I have a headache and it is extreme”, state-of-the-art models only recognize the main symptom entity - *headache*, but ignore the severity factor of *extreme*, that characterises *headache*. In this paper, we design a two-stage approach to detect the characterizations of entities like symptoms presented by general users in contexts where they would describe their symptoms to a clinician. We use Word2Vec and BERT to encode clinical text given by the patients. We transform the output and re-frame the task as a multi-label classification problem. Finally, we combine the processed encodings with the Linear Discriminant Analysis (LDA) algorithm to classify the characterizations of the main entity. Experimental results demonstrate that our method achieves 40-50% improvement in the accuracy over the state-of-the-art models.' author: - 'Budhaditya Saha, Sanal Lisboa, Shameek Ghosh' bibliography: - 'example.bib' title: Understanding patient complaint characteristics using contextual clinical BERT embeddings --- Introduction\[sec:intro\] ========================= Clinical Dialogue Dataset ========================== Proposed Approach \[sec:framework\] =================================== Experiments \[sec:Experiments\] =============================== Conclusion\[sec:conclusion\] ============================

--- abstract: | The characteristics of the starburst galaxies from the Pico dos Dias survey (PDS) are compared with those of the nearby UV-bright Markarian starburst galaxies, having the same limit in redshift ($v_h < 7500$ km s$^{-1}$) and absolute $B$ magnitude ($M_B < -18$). An important difference is found: the Markarian galaxies are generally undetected at 12$\mu$m and 25$\mu$m in *IRAS*. This is consistent with the UV excess shown by these galaxies and suggests that the youngest star forming regions dominating these galaxies are relatively free of dust. The FIR selection criteria for the PDS is shown to introduce a strong bias towards massive (luminous) and large size late-type spiral galaxies. This is contrary to the Markarian galaxies, which are found to be remarkably rich in smaller size early-type galaxies. These results suggest that only late-type spirals with a large and massive disk are strong emitter at 12$\mu$m and 25$\mu$m in *IRAS* in the nearby universe. The Markarian and PDS starburst galaxies are shown to share the same environment. This rules out an explanation of the differences observed in terms of external parameters. These differences may be explained by assuming two different levels of evolution, the Markarian being less evolved than the PDS galaxies. This interpretation is fully consistent with the disk formation hypothesis proposed in Coziol et al. (2000) to explain the special properties of the Markarian SBNG. date: Released 2002 Xxxxx XX title: 'The PDS vs. Markarian starburst galaxies: comparing strong and weak *IRAS* emitter at 12$\mu$m and 25$\mu$m in the nearby universe.' --- \[firstpage\] galaxies: starburst – galaxies: evolution – galaxies: formation Introduction ============ What is the origin of the nuclear star burst observed in many nearby massive galaxies? Theoretically, interaction of galaxies was shown to be one of the most efficient ways by which to fuel gas in the center of a galaxy and start a burst of star formation (Barns & Hernquist 1996; Mihos & Hernquist 1996). From the point of view of observation, however, such scenario seems to fit only a fraction of the starburst galaxies in the nearby universe. In the Markarian sample, for example, Keel & Van Soest (1992) have found that only 35% of the galaxies could be interacting with a nearby companion. A comparable fraction of non-isolated galaxies (24%) was found in the MBG survey (Coziol et al. 1997). The rapid and short duration burst predicted by many interaction models seems also in contradiction with the relatively long period of star formation observed in most starburst galaxies (Coziol 1996; Goldader et al. 1997; Coziol, Doyon & Demers 2001). These observations suggest that other mechanisms, internal to the galaxies, must play a role in triggering or regulating star formation in these systems. The effect of a bar structure was frequently proposed as a possible burst mechanism. Theoretically, the bar was shown to be capable of funnelling gas in the center of a galaxy and start a nuclear burst. This hypothesis was recently tested by Considère et al. (2000) on a sample of 16 strongly barred Markarian starburst nucleus galaxies (SBNG). The study of the oxygen and nitrogen gradients in these galaxies proved to be incompatible with the expected effect of a bar. Instead, evidence was found that the bars appeared only recently as compared to the possible ages of the bursts. In an accompanying paper, Coziol et al. (2000), it was shown that this phenomenon is typical of the whole Markarian SBNG sample. Taken at face value, these observations suggest that a large fraction of the Markarian galaxies may now be forming a disk. It would be highly significant to know if the characteristics of the Markarian SBNG, as reported in Coziol et al (2000), are typical of all massive starburst galaxies in the nearby universe. Having previously defined a new sample of such objects, the Pico Dos Dias (PDS) starburst galaxies (Coziol et al 1998a), a comparison of their characteristics with those of the Markarian SBNG seemed, therefore, like the next logical step. The plan of the article is the following. Section 2 explains the selection criteria used to define the two samples. The different particularities in the Far-Infrared (FIR) and the physical characteristics that are compared are also presented and discussed in this section. The comparison of the two samples, supported by a statistical analysis, takes place in Section 3. This is followed, in Section 4, by a discussion on the possible causes of the differences observed. The consequences of these differences for the nature of the starburst phenomenon in nearby massive galaxies and for galaxies at higher redshift are also discussed in this section. A brief conclusion is presented in section 5, followed by an appendix explaining in detail the results of the statistical tests used for this analysis. Description of the two samples ============================== Selection criteria and source of the data ----------------------------------------- The main characteristics of the PDS starburst galaxies were already presented and discussed in Coziol et al. (1998a). This sample consists of 200 galaxies, selected from the *IRAS* Point Source Catalog using the following FIR criteria: 1) all the galaxies have high or intermediate quality flux density at 12$\mu$m, 25$\mu$m, 60$\mu$m and 100$\mu$m; 2) they have an infrared spectral index $\alpha(25, 12)$ in the range $-3.00 \le \alpha(25, 12) \le +0.35$ and an infrared spectral index $\alpha(60, 25)$ in the range $-2.50 \le \alpha(60, 25) \le -1.9$ (where $\alpha(\lambda1, \lambda2)= log(S_{\lambda1}/S_{\lambda2})/log(\lambda2/\lambda1)$, and $S_{\lambda}$ is the flux in Janskys at wavelength $\lambda$). In Coziol et al. (1998a) it was shown that these criteria allow distinguishing starburst from AGN galaxies with a confidence level approaching 99%. The PDS starburst galaxies were found to be relatively luminous (M$_B < -18$). The fact that the FIR criteria introduce a strong bias against dwarf starburst galaxies (H[**II**]{} galaxies) is not surprising, considering the low metallicity and dust content of these galaxies (Salzer, MacAlpine & Gordon 1988). The physical parameters that are considered in this article were all taken from the same source: the Lyon Meudon Extragalactic Database (*LEDA*). In *LEDA*, raw data, having a different origin, are homogenized, using specific rules, to form a uniform catalogue(*LEDA* is the basis of the *RC3* catalogue of galaxies). A complete description of the normalization process used in *LEDA* can be found in de Vaucouleurs et al. (1991) and Paturel et al. (1997). Keeping only galaxies for which complete information (except for kinematics) is available in *LEDA* reduces the number of PDS starburst galaxies to 168. The comparison sample of Markarian SBNG is composed of all the galaxies with M$_B < -18$, having, like for the PDS, information in *LEDA*. This yields a sample of 505 galaxies. Comparing the redshift of these galaxies with those of the PDS in Figure 1, it is obvious that the Markarian are located at higher redshift. To eliminate the effects of Malmquist bias, a limit on the distance must be applied. For comparison, therefore, only galaxies which are nearer than 7500 km s$^{-1}$ are kept in the two samples (this corresponds to 100 Mpc, adopting H$_0=75$ km s$^{-1}$ Mpc$^{-1}$). This limit was chosen in order to exclude the smallest number of PDS galaxies, while keeping the largest number of Markarian galaxies. This reduce the final PDS and Markarian samples to 154 galaxies and 325 galaxies respectively. ![a) Distribution in redshift of the 505 barred and unbarred Markarian SBNG, with complete information in *LEDA*. b) Distribution in redshift of the 168 PDS starburst galaxies.[]{data-label="fig1"}](fig1){width="200pt"} ![a) Detected flux at 12$\mu$m as a function of the redshift for the Markarian SBNG and PDS starburst galaxies; b) Detected flux at 25$\mu$m as a function of the redshift for the Markarian SBNG and PDS starburst galaxies. The continuous lines are linear regressions on the Markarian flux values. The detected fluxes for the Markarian SBNG are always lower than those for the PDS. Note that the slope of the regression goes contrary to what is expected for an observational bias.[]{data-label="fig2"}](fig2){width="200pt"} *IRAS* M$_B$ Morph. T v$_{h}$ log(D$_{25}$) $\mu_B$ log(V$_{max}$) isolated ------------ ------- -------- ------ --------------- --------------- -------------------- ---------------- ---------- (km s$^{-1}$) (0.1 arcmin) (mag arsec$^{-2}$) (km s$^{-1}$) 00013+2028 -20.8 Sbc 4.1 2308.03 1.54 23.37 2.23 yes 00022-6220 -20.2 SBbc 3.9 4539.00 1.05 22.82 yes 00073+2538 -21.5 SBa 1.2 4564.64 1.29 22.93 2.45 no 00345-2945 -20.1 S0/a 0.3 3564.14 1.19 23.09 2.02 yes 01053-1746 -20.8 Irr 10.0 6077.20 0.84 21.84 2.32 yes The complete table is available in the electronic issue of the journal. A digital version is also available on demand to the author. *Mrk \#* M$_B$ Morph. T v$_{h}$ log(D$_{25}$) $\mu_B$ log(V$_{max}$) isolated ---------- ------- -------- ------ --------------- --------------- -------------------- ---------------- ---------- (km s$^{-1}$) (0.1 arcmin) (mag arsec$^{-2}$) (km s$^{-1}$) Mrk0002 -20.6 SBa 0.6 5543.45 0.85 22.06 2.25 no Mrk0004 -20.6 SBc 6.0 5258.60 1.28 24.14 2.11 yes Mrk0007 -19.9 Sd 7.8 3063.23 0.94 22.01 2.01 yes Mrk0008 -20.1 Sbc 4.1 3602.21 0.97 22.30 2.07 no Mrk0011 -19.8 E/S0 -3.0 3909.15 0.96 22.69 no The complete table is available in the electronic issue of the journal. A digital version is also available on demand to the author. Overlap of the samples: FIR particularity of the Markarian SBNG --------------------------------------------------------------- Because 72% of the Markarian galaxies were also detected in IRAS, a large overlap between the two samples was therefore expected. However, after comparing the galaxy names in the two samples, only 17 Markarian galaxies (5%) were found to be common to the two samples (this represents only 11% of the PDS galaxies). The reason for such low overlap is easy to understand. It is because most of the Markarian galaxies are undetected at 12$\mu$m and/or 25$\mu$m in the *IRAS* Point Source Catalogue, which is one of the condition to be part of the PDS sample. Since this is an important difference between the two samples, and an unrecognized FIR characteristics for the UV-bright Markarian galaxies, a more thorough investigation seems in order. The reason why the Markarian SBNG are not detected at 12$\mu$m and 25$\mu$m in the Point Source Catalogue is not obvious. In part, this is due to a lack of sensitivity. Using the *IRAS* Faint Source Catalogue instead of the Point Source Catalogue, for instance, increases to 29% the fraction of Markarian SBNG detected in the four bands. However, it still leaves most of the galaxies (71%) as either undetected in one (33%) or all (38%) of the four *IRAS* bands (see Table 5 in Bicay et al. 1995). Comparing the 12$\mu$m and 25$\mu$m fluxes of the galaxies detected in the Faint Source Catalogue in figure 2, it can be seen that the level of flux at these two wavebands is significantly lower in the Markarian galaxies. The continuous lines are linear regressions on the Markarian flux values. The slope of these regressions go contrary to what is expected for an observational bias, which suggests that we are still far from the observational limits. The low detection at 12$\mu$m and 25$\mu$m looks, therefore, as a physical trait of these galaxies: the Markarian SBNG are generally weak FIR emitter at 12$\mu$m and 25$\mu$m. In a sense, the above particularity may be the only one consistent with the UV-brightness of these galaxies. The presence of a UV excess implies that a high number of young star forming regions present in these galaxies are relatively free of dust. If these young and dusty free star forming regions dominate over heavily dusty ones, then the Markarian are not expected to be particularly bright at 12$\mu$m and 25$\mu$m, since only these young star forming regions can heat dust sufficiently to be visible at these two wavelengths. The lower detection rate of the Markarian SBNG at 12$\mu$m and 25$\mu$m in *IRAS* marks, therefore, an intrinsic difference between the two samples: galaxies in the Markarian sample are either less dusty or less homogeneously covered in dust than galaxies in the PDS sample. This new characteristic of the Markarian SBNG is quite important. It suggests that UV-bright and FIR bright selected samples may differ also in other important manners. This is what this analysis will now determine. For comparison sake, the 17 galaxies common to the two samples will be erased from the Markarian sample and considered as part of the PDS sample only. Table 1 lists the characteristics of the 154 PDS starburst galaxies as found in *LEDA*: column 1 gives the *IRAS* name, followed by the absolute magnitude in B, column 2, the morphology and morphological index, $T$, column 3 and 4, the mean heliocentric redshift, column 5, the logarithm of the apparent corrected diameter, $D_{25}$, column 6, the surface brightness, column 7, and the logarithm of the maximum rotation velocity, column 8. The last column indicates if the galaxy is considered to be isolated (see section 3.2). Similar characteristics for the Markarian SBNG are listed in Table 2. Comparison of the samples ========================= Differences in mean values -------------------------- ------------------------------------------------ ---------- -------- ---------- -------- PDS PDS Mrk Mrk unbarred barred unbarred barred N 83 71 247 61 $\langle$v$_{h}\rangle\ (\frac{km}{s})$ 3618 2834 4824 4129 std Dev. 1746 1406 1522 1659 $\langle$M$_B\rangle\ $ -20.27 -20.54 -19.65 -20.15 std Dev. 1.04 0.80 0.80 0.82 $\langle\mu_B\rangle\ (\frac{mag}{arsec^{2}})$ 23.01 23.11 22.99 22.94 std Dev. 0.59 0.47 0.69 0.67 $\langle$T$\rangle\ $ 2.7 3.4 1.4 3.7 std Dev. 2.9 1.5 3.7 2.1 $\langle$log D$_{25}\rangle\ (kpc)$ 22.47 25.33 17.18 20.64 std Dev. 10.38 11.49 7.74 7.53 $f(V_{max})$ 74% 87% 35% 74% $\langle$log V$_{max}\rangle\ (\frac{km}{s})$ 2.23 2.21 2.16 2.14 std Dev. 0.20 0.15 0.87 0.18 ------------------------------------------------ ---------- -------- ---------- -------- : Mean values and dispersion for the characteristics compared[]{data-label="table3"} $f(V_{max})$is the fraction of galaxies with kinematics information in *LEDA*. The mean values for the physical parameters compared are presented in Table 3. Since the presence or absence of a bar was shown to be of importance in Coziol et al. (2000), the mean values were estimated by making this distinction in each sample. In order to evaluate if the mean values are significantly different statistically, one-way ANOVA tests are run on all the samples. The kind of test applied (parametric or non-parametric) is determined by verifying if the samples have normal distribution. Once a significant difference is observed, the usual post-tests are performed in order to determine the origin and nature of this difference. Complete explanations for the tests, their interpretations and the results are presented in the appendix. Except for the surface brightness, $\mu_B$, all the other parameters were found to be significantly different, at a confidence level of 99%. The following is a summary of what Table 3 and the statistical post-tests (Tables A2) reveal. For the blue absolute magnitude, M$_B$, the unbarred Markarian SBNG are found to be less luminous than the other galaxies. The barred Markarian SBNG are also found to be marginally less luminous than the PDS barred starburst galaxies. For the morphology index, T, the unbarred Markarian SBNG are found to be more numerous in early-type galaxies than all the other galaxies, the difference being more pronounced compared to the barred galaxies (Markarian and PDS). From table 3, one can note also a large fraction of unbarred galaxies in the Markarian sample (80% compared to 46% in the PDS sample). For the velocity recession, v$_h$, it is found that the Markarian SBNG, barred and unbarred, are located at higher redshift than the PDS starburst galaxies. In both samples the unbarred galaxies seem also to be marginally farther away than the unbarred ones. For the dimension of the galaxies, D$_{25}$, the unbarred Markarian SBNG are found to be smaller in size. A marginal difference between the Markarian barred and PDS barred is also observed, the former being slightly smaller than the latter. The statistical results found for the size of the galaxies are similar to the one observed for the luminosity. Although the information about rotational velocity is still incomplete (see $f($V$_{max}$) in Table 3), the statistical tests suggest that the unbarred Markarian SBNG have smaller maximum velocity rotation than the other galaxies. The results are consistent with the size differences observed between the samples. Barred Unbarred Total ---------------- -------- ---------- ------- PDS starburst 34% 34% 34% Markarian SBNG 39% 34% 35% : Fraction of galaxies in pair or group[]{data-label="table4"} ------------------------------------------------ ---------- ---------- ---------- ---------- PDS PDS Mrk Mrk isolated pair/gr. isolated pair/gr. N 102 52 201 107 $\langle$v$_{h}\rangle\ (\frac{km}{s})$ 3256 3258 4728 4606 std Dev. 1598 1737 1566 1589 $\langle$M$_B\rangle\ $ -20.38 -20.42 -19.71 -19.82 std Dev. 1.00 0.90 0.81 0.85 $\langle\mu_B\rangle\ (\frac{mag}{arsec^{2}})$ 23.03 23.09 23.04 22.88 std Dev. 0.54 0.52 0.66 0.72 $\langle$T$\rangle\ $ 2.9 3.4 1.9 1.8 std Dev. 2.6 1.9 3.6 3.6 $\langle$log D$_{25}\rangle\ (kpc)$ 23.08 25.18 18.02 17.57 std Dev. 9.90 12.79 7.88 7.69 $f(V_{max})$ 76% 86% 43% 42% $\langle$log V$_{max}\rangle\ (\frac{km}{s})$ 2.22 2.22 2.07 2.31 std Dev. 0.17 0.18 0.20 1.19 ------------------------------------------------ ---------- ---------- ---------- ---------- : Mean values and dispersions for the samples separated by the isolation criterion[]{data-label="table5"} $f(V_{max})$ is the fraction of galaxies with kinematics information in the sample. Difference of environment ------------------------- Because the environment of a galaxy may have some influence on its evolution, it seems important to check if the two samples show any difference on this matter. One way to do this is to compare the fraction of galaxies in pair or group in the two samples. Since *LEDA* does not give information on the environment of galaxies, *NED* (the NASA/IPAC Extragalactic Database) was used to search for the presence of companion galaxies in the close environment of all the galaxies in the samples. The method followed is identical to the one used by Campos-Aguilar & Moles (1991) and more recently Noeske et al. (2001). A galaxy is considered to be non isolated when an objects is found within a projected separation $s_p \le 0.1$ Mpc and its difference in velocity is $\Delta v_{h} \le 500$ km s$^{-1}$. The angular search radius was done using the mean heliocentric redshift as found in *LEDA*. The fraction of galaxies in pair or group is presented in Table 4. The same tendency is found for the PDS starburst galaxies and Markarian SBNG. No significant relation is noted between the environment and the presence of a bar. The mean characteristics of the samples separated based on the isolation criteria are presented in Table 5. The mean values look more similar than when the samples are separated based on the presence or absence of a bar. The environment comparison is conclusive. It shows that whatever the differences are, they cannot be explained by different environmental effects. Discussion ========== ![Absolute magnitude as a function of the redshift a) for the PDS starburst galaxies; b) for the Markarian SBNG. The dashed and long dashed curves in the two graphics are linear regression on the PDS and Markarian respectively. They show the presence of a Malmquist bias. The regression for the Markarian is reported in the panel a) to show the difference between the sample. In a) the position occupied by the Markarian SBNG detected in IRAS are also shown.[]{data-label="fig3"}](fig3){width="200pt"} ![Diameter of the galaxies as a function of the redshift. a) for the PDS starburst galaxies; b) for the Markarian SBNG. The signification of the symbols is the same as in Figure 3.[]{data-label="fig4"}](fig4){width="210pt"} The criteria used to select the PDS seem to imply important differences in the characteristics of the galaxies sampled. It is important, however, to verify that these differences are not due to spurious observational biases. Observational biases (like Malmquist bias) are usually detected by looking for a relation between the parameter observed and the redshift. This method was already used in Figure 2 to show that an observational bias cannot explain why the Markarian galaxies are not detected at 12$\mu$ and 25$\mu$ in IRAS. A bias would have cause the detected fluxes to increase with the redshift. On the contrary, the decreasing slopes in Figure 2 indicate the real observational limits in these bands were not reached. Applying the same kind of test for the absolute magnitude in Figure 3, it can be seen that the luminosity of the galaxies is increasing with the redshift. This is a Malmquist bias. Although the bias seems less severe for the Markarian SBNG, the effect is insufficient to explain the difference in luminosity between the two samples. What Figure 3 shows also is that the PDS are more luminous than the Markarian SBNG at all redshift. For some reasons, therefore, the PDS sample seems to contain only the most luminous galaxies. ![Morphology as a function of the redshift: a) for the PDS starburst galaxies; b) for the Markarian SBNG. The signification of the symbols is the same as in Figure 3.[]{data-label="fig5"}](fig5){width="200pt"} The above result is consistent with the size difference observed between the galaxies in the two samples. Assuming the luminosity in B is correlated to the size of a galaxy, the PDS starburst galaxies being more luminous than the Markarian SBNG are naturally expected to have bigger size. This is verified in Figure 4. The similarity between Figure 3 and Figure 4 (Malmquist bias included) is obvious. It suggests that the large size of the PDS starburst galaxies is due to a bias in luminosity. But, how is this possible, since no condition on the B luminosity was applied? The B luminosity bias is not trivial. In fact, the only way one could obtain such a bias is through the FIR PDS selection criteria. By picking galaxies that emit in the four IRAS bands, only FIR luminous galaxies are selected. This was already noted in Coziol et al. (1998a). Now, assuming the FIR luminosity of a galaxy is correlated to its mass (which is consistent with the analysis in Coziol, Doyon & Demers 2001) and the mass is correlated to the B luminosity, then, FIR luminous galaxies are also expected to be luminous in B. Therefore, the B luminosity preference observed is not an observational bias, but a result of the FIR selection criteria used: by selecting IRAS galaxies that emit in the four IRAS bands, we pick only FIR luminous galaxies, which turn out to be massive (luminous in B) and large size galaxies. A strong indication that this interpretation is correct is the preference in morphology shown by the PDS. The morphology of the galaxies as a function of the redshift is examined in Figure 5. No significant relation (bias) is observed with the redshift. It can be seen that the PDS are mostly late-type spiral galaxies. Late-type spiral galaxies being richer in dust than early-type ones are obviously favoured by the FIR selection criteria. Implicit in the definition of an observational bias, there is also a notion of incompleteness (some objects are left aside). Obviously, such incompleteness does not apply to the PDS sample. The PDS is a sub-sample of the IRAS galaxy survey, which is an all sky survey (complete up to a certain magnitude). The PDS contains, therefore, all the nearby galaxies (non AGN) that emit in the four IRAS bands. The only galaxies that are missing in the present PDS sample are those emitting in the four IRAS bands, but that were not classified as starburst in Coziol et al. (1998a). The properties of the galaxies in this sample (the PDS normal galaxies) are presented in Table 6. The same preference towards massive (luminous) large size late-type spiral galaxies is observed. This is independent from the fact that these galaxies are less luminous in the FIR than the PDS starbursts ($\langle$L$_{IR}\rangle\ = 10$, compared to $\langle$L$_{IR}\rangle\ = 10.3$ for the PDS starburst). unbarred barred ------------------------------------------------ ---------- -------- N 74 83 $\langle$M$_B\rangle\ $ -20.2 -20.4 $\langle\mu_B\rangle\ (\frac{mag}{arsec^{2}})$ 23.1 23.1 $\langle$T$\rangle\ $ 3.7 4.2 $\langle$log D$_{25}\rangle\ (kpc)$ 25 26 $\langle$log V$_{max}\rangle\ (\frac{km}{s})$ 2.28 2.25 : Mean values and dispersion for the PDS normal galaxies[]{data-label="table6"} It is important to remember that the Markarian SBNG are also a sub-sample of the IRAS galaxies. In the present sample, 72% of the Markarian galaxies were detected at 100$\mu$m and/or 60$\mu$m in IRAS. These are shown with the PDS in Figure 3, 4 and 5. These IRAS starburst galaxies are not included in the PDS sample because they do not emit at 12$\mu$m and 25$\mu$m. If we relax this criterion the two samples merge together and we loose all distinction between the PDS and Markarian starburst galaxies. We then have a sample of IRAS starburst galaxies covering all ranges in luminosity, size and morphology. From the above analysis, it is concluded that there is a strong correlation between the physical characteristics of the galaxies sampled and the fact that they are weak or strong emitter at 12$\mu$m and 25$\mu$m in IRAS. It seems therefore secure to state that the only galaxies that emit in these bands in the nearby universe are massive (luminous) large size late-type spiral galaxies. Equivalently, we can also state that the reason the Markarian SBNG are not detected in these bands is because they are too small and have too early morphology. Note that the high number of early-type galaxies in the Markarian sample may already explain the large difference in barred galaxies observed in the different samples (see Table 3). In the Markarian sample, 111 of the unbarred galaxies are early-type ($T\leq0$). Since bars are less frequent (or more difficult to detect) in these galaxies, their high number in this sample leads naturally to a lower frequency of barred galaxies. Eliminating the 111 early-type unbarred galaxies, for example, increases the fraction of barred galaxies to 31%, which is still low, but more comparable to the 46% barred galaxies observed in the PDS sample. One important question is why are the Markarian SBNG so numerous in early-type galaxies? One possibility is to assume a special burst mechanisms. Merger of galaxies is usually recognized as a mechanism capable of producing early-type galaxies. This mechanism may also be fatal for a pre-existing spiral disk and a bar. If merger is the main burst mechanism of the Markarian SBNG, a high number of early type galaxies in this sample is thus understandable. In the case of the PDS galaxies, their different properties may be the result of gas accretion. In Coziol et al. (1998b), using a chemical evolution model, it was shown that a starburst galaxy that is accreting gas forms a late-type spiral with a metallicity that is high as compared to a merger case. Having a high metallicity, these galaxies are expected to be rich in dust and to possibly emit in the four IRAS bands. One difficulty with the above explanation is that nothing indicates what could trigger one mechanism instead of the other. The fact that the galaxies in the two samples were shown to share the same environment excludes an explanation in terms of external causes. Another possibility is to assume that we are observing a unique population of starburst galaxies where galaxies are observed at different level of evolution. As previously shown, this would be equivalent to relaxing the FIR criteria used to select the PDS. The problem now is to identify the process that corresponds to the evolutionary sequence observed. To explain the characteristics of the Markarian SBNG, it was proposed in Coziol et al. (2000) that they are presently forming a disk. Assuming this process is not instantaneous, we should then expect to find, in a single population, examples of galaxies at different stage of formation of their disk. According to this interpretation, the Markarian SBNG are simply at a less advanced stage than the PDS starbursts. One could easily test the above hypothesis. It would be sufficient to determine if the PDS have different metallicity and metallicity gradients (O/H and N/O) than the Markarian SBNG (Considère et al. 2000). Assuming the PDS are more evolved, a strong influence of the bar would also be expected in these galaxies. In Coziol et al. (2000), the proposition that the Markarian galaxies are in the process of forming a disk was said to be consistent with observations of galaxies in the HDF, suggesting that large size spiral galaxies become less frequent at higher redshift (van den Bergh et al. 2000). However, assuming the same process happens for galaxies in the HDF, we would then expect to also see a difference in size between the early-type and late-type galaxies in this sample. The fact that this difference was recently observed (Cohen et al. 2003) may be one new argument in support of the disk formation model. conclusion ========== The goal of this article was to verify if other nearby starburst galaxies share the characteristics noted for the Markarian SBNG in Coziol et al. (2000). Using the PDS starburst galaxies, which were selected based on FIR criteria, it was found that the type of galaxy detected differs quite significantly. These differences were then shown to be strongly correlated to the fact that the PDS are strong emitter at 12$\mu$m and 25$\mu$m in IRAS. In particular, it suggests that only massive and large size late-type spiral galaxies in the nearby universe have sufficiently hot dust to be observed in these two bands. The differences observed between the two samples are not in contradiction with the interpretation proposed in Coziol et al. (2000), which is that a large number of Markarian SBNG are in the process of forming a disk. If such process extends over a significant period of time, it is naturally expected to find, in a single population, examples of galaxies at different stages of the formation of their disk. The characteristics expected for galaxies at a more advanced stage are similar to those observed for the PDS. This interpretation is consistent with the observation that the PDS starburst and Markarian SBNG share the same environment. Similarities with what is observed in the HDF (van den Bergh et al. 2000; Cohen et al. 2003) suggests that we may see the same phenomenon at high and low redshift. This supports the view that nearby starburst in massive galaxies are examples of galaxy in formation. The author thanks the referee for important comments. This research was supported by CONACyT grant EX-000479. For this analysis, the Lyon-Meudon Extragalactic Database (*LEDA*), operated by the Lyon and Paris-Meudon Observatories (France) and the NASA/IPAC Extragalactic Database (*NED*) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration were used. Barns, J. E., Hernquist, L. 1996, ApJ, 471, 115 Bicay, M. D., Kojoian, G., Seal, J., Dickinson, D. F., Malkan, M. A. 1995, ApJS, 98, 369 Campos-Aguilar, A., Moles, M. 1991, A&A, 241, 358 Considère, S., Coziol, R., Contini, T., Davoust, E. 2000, A&A, 356, 89 Coziol, R. 1996, A&A, 309, 345 Coziol, R., Demers, S., Peña, M., Barnéoud, R. 1997, AJ, 113, 1548 Coziol, R., Torres, C. A. O., Quast, G. R.,Contini, T., Davoust, E. 1998a, ApJS, 119, 239 Coziol, R., Contini, T., Davoust, E., Considère, S. 1998b, in ASP conf. ser. Vol. 147, Abundance profiles: Diagnostic Tools for Galaxy History, ed. D. Friedli, M. G. Edmunds, C. Robert, L. Drissen (San Francisco: ASP), 219 Coziol, R., Considère, S., Davoust, E., Contini, T. 2000, A&A, 356, 102 Coziol, R., Doyon, R., Demers, S. 2001, MNRAS, 325, 1081 de Vaucouleurs G., Vaucouleurs A. de, Corwin H.G. Jr., Buta R.J., Paturel G., Fouqué, P., 1991, Third Reference Catalogue of Bright Galaxies, Springer-Verlag (RC3) Goldader, J. D., Joseph, R. D., Doyon, R., Sanders, D. B. 1997a, ApJ, 474, 104 Keel, W. C., van Soest, E. T. M. 1992, A&AS, 94, 553 Mihos, J. C., Hernquist, L. 1996, APJ, 464, 641 Noeske, K. G., Iglesias-Páramo, J., Vílchez, J. M., Papaderos, P., Fricke, K. J. 2001, A&A, 371, 806 Paturel G., Bottinelli L., Di Nella H., Durand N., Garnier R., Gouguenheim L., Lanoix P., Marthinet M.C., Petit C., Rousseau J., Theureau G., Vauglin I., 1997, Astronom. Astrophys. Suppl. Ser. 124, 109 Salzer, J. J., MacAlpine, G. M., Gordon, M. 1988, AJ, 96, 1192 Cohen, S. H., Windhorst, R. A., Odewahn, S. C., Chiarenza, C. A., Driver, S. P. 2003, AJ, 125, 1762 van den Bergh, S., Cohen, J. G., Hogg, D. W., Blandford, R. 2000, AJ, 120, 2190 One-way analysis of variance: description and results ===================================================== The one-way analysis of variance (ANOVA) test compares three or more unmatched groups of data, based on the assumption that the two populations are Gaussian. The P value answers this question: if the populations really have the same mean, what is the chance that random sampling would result in means as far apart as observed? The post tests (like Tukey’s and Dun’s post tests) are modifications of the t test. They account for multiple comparisons, as well as for the fact that the comparisons are interrelated. The unpaired t test computes the t ratio as the difference between two group means divided by the standard error of the difference (computed from the standard errors of the two group means, and the two sample sizes). The P value is then derived from t. The post tests work in a similar way. Instead of dividing by the standard error of the difference, they divide by a value computed from the residual mean square. Each test uses a different method to derive a P value from this ratio. For the difference between each pair of means, the tests report the P value as $>0.05$, $<0.05$, $<0.01$ or $<0.001$. If the null hypothesis is true (all the values are sampled from populations with the same mean), then there is only a 5% chance that any one or more comparisons will have a P value less than 0.05. For comparison the data for the PDS and Markarian starbursts were separated in 4 groups: group A, the PDS unbarred galaxies, group B, the PDS barred, group C, the Markarian unbarred and group D, the Markarian barred. Before performing the analysis, Kolmogorov-Smirnov tests were used to verify the normality distribution of the data. The only group of data that did not pass this test is the maximum rotation velocity for the Markarian unbarred galaxies (group C). A non-parametric test (Kruskall-Wallis) and Dunn’s post test were used for this parameter. Results of the one-way ANOVA and non-parametric tests are reported in Table A1. The results for the post tests, when the means were shown to varied significantly, are reported in Table A2.[^1] Parameters P Means signif. different ------------ ---------- ------------------------- M$_B$ $<0.001$ yes $\mu_B$ 0.4637 no T $<0.001$ yes v$_{h}$ $<0.001$ yes D$_{25}$ $<0.001$ yes V$_{max}$ $<0.001$ yes : Result for the one-way ANOVA and non parametric tests[]{data-label="TableA1"} groups M$_B$ T v$_{h}$ D$_{25}$ V$_{max}$ -------- ---------- ---------- ---------- ---------- ----------- A vs B $>0.05$ $>0.05$ $<0.05$ $>0.05$ $>0.05$ A vs C $<0.001$ $<0.01$ $<0.001$ $<0.001$ $<0.001$ A vs D $>0.05$ $>0.05$ $>0.05$ $>0.05$ $>0.05$ B vs C $<0.001$ $<0.001$ $<0.001$ $<0.001$ $<0.001$ B vs D $<0.05$ $>0.05$ $<0.001$ $<0.05$ $>0.05$ C vs D $<0.001$ $<0.001$ $<0.05$ $<0.05$ $>0.05$ : P values for the post tests[]{data-label="TableA2"} \[lastpage\] [^1]: For this analysis, the program Prism, version 3.00, for Windows was used. This program was made by GraphPad software, San Diego California USA (www.graphpad.com).

--- abstract: 'We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or ${{\ensuremath{\mathrm{MCSP}}}}$ for short), and of the variant denoted as ${{\ensuremath{\mathrm{MKTP}}}}$ where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$ hinged on the power of ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$ to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (${{\ensuremath{\mathrm{GI}}}}$). It yields a randomized reduction with zero-sided error from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$. We generalize the result and show that ${{\ensuremath{\mathrm{GI}}}}$ can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.' author: - 'Eric Allender [^1]' - 'Joshua A. Grochow [^2]' - 'Dieter van Melkebeek [^3]' - 'Cristopher Moore [^4]' - 'Andrew Morgan [^5]' bibliography: - 'refer.bib' title: | Minimum Circuit Size, Graph Isomorphism,\ and Related Problems[^6] --- Introduction {#sec:intro} ============ Finding a circuit of minimum size that computes a given Boolean function constitutes the overarching goal in nonuniform complexity theory. It defines an interesting computational problem in its own right, the complexity of which depends on the way the Boolean function is specified. A generic and natural, albeit verbose, way to specify a Boolean function is via its truth-table. The corresponding decision problem is known as the Minimum Circuit Size Problem ([[$\mathrm{MCSP}$]{}]{}): Given a truth-table and a threshold $\theta$, does there exist a Boolean circuit of size at most $\theta$ that computes the Boolean function specified by the truth-table? The interest in ${{\ensuremath{\mathrm{MCSP}}}}$ dates back to the dawn of theoretical computer science [@trakhtenbrot]. It continues today partly due to the fundamental nature of the problem, and partly because of the work on natural proofs and the connections between pseudorandomness and computational hardness. A closely related problem from Kolmogorov complexity theory is the Minimum KT Problem (${{\ensuremath{\mathrm{MKTP}}}}$), which deals with compression in the form of efficient programs instead of circuits. Rather than asking if the input has a small circuit when interpreted as the truth-table of a Boolean function, ${{\ensuremath{\mathrm{MKTP}}}}$ asks if the input has a small program that produces each individual bit of the input quickly. To be more specific, let us fix a universal Turing machine $U$. We consider descriptions of the input string $x$ in the form of a program $d$ such that, for every bit position $i$, $U$ on input $d$ and $i$ outputs the $i$-th bit of $x$ in $T$ steps. The ${{\ensuremath{\operatorname{KT}}}}$ cost of such a description is defined as $|d|+T$, , the bit-length of the program plus the running time. The ${{\ensuremath{\operatorname{KT}}}}$ complexity of $x$, denoted ${{\ensuremath{\operatorname{KT}}}}(x)$, is the minimum ${{\ensuremath{\operatorname{KT}}}}$ cost of a description of $x$. ${{\ensuremath{\operatorname{KT}}}}(x)$ is polynomially related to the circuit complexity of $x$ when viewed as a truth-table (see Section \[sec:prelim:complexity-measures\] for a more formal treatment). On input a string $x$ and an integer $\theta$, ${{\ensuremath{\mathrm{MKTP}}}}$ asks whether ${{\ensuremath{\operatorname{KT}}}}(x) \leq \theta$. Both ${{\ensuremath{\mathrm{MCSP}}}}$ and ${{\ensuremath{\mathrm{MKTP}}}}$ are in ${{\ensuremath{\mathsf{NP}}}}$ but are not known to be in ${\ensuremath{\mathsf{P}}}$ or ${{\ensuremath{\mathsf{NP}}}}$-complete. As such, they are two prominent candidates for ${{\ensuremath{\mathsf{NP}}}}$-intermediate status. Others include factoring integers, discrete log over prime fields, graph isomorphism ([[$\mathrm{GI}$]{}]{}), and a number of similar isomorphism problems. Whereas ${{\ensuremath{\mathsf{NP}}}}$-complete problems all reduce one to another, even under fairly simple reductions, less is known about the relative difficulty of presumed ${{\ensuremath{\mathsf{NP}}}}$-intermediate problems. Regarding ${{\ensuremath{\mathrm{MCSP}}}}$ and ${{\ensuremath{\mathrm{MKTP}}}}$, factoring integers and discrete log over prime fields are known to reduce to both under randomized reductions with zero-sided error [@powerk; @rudow]. Recently, Allender and Das [@adas] showed that ${{\ensuremath{\mathrm{GI}}}}$ and all of ${{\ensuremath{\mathsf{SZK}}}}$ (Statistical Zero Knowledge) reduce to both under randomized reductions with bounded error. Those reductions and, in fact, *all* prior reductions of supposedly-intractable problems to ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$ proceed along the same well-trodden path. Namely, ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$ is used as an efficient statistical test to distinguish random distributions from pseudorandom distributions, where the pseudorandom distribution arises from a hardness-based pseudorandom generator construction. In particular, [@kab.cai] employs the construction based on the hardness of factoring Blum integers, [@powerk; @adas; @pervasive; @rudow] use the construction from [@hill] based on the existence of one-way functions, and [@powerk; @carmosino] make use of the Nisan-Wigderson construction [@NisanW94]. The property that ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$ breaks the construction implies that the underlying hardness assumption fails relative to ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$, and thus that the supposedly hard problem reduces to ${{\ensuremath{\mathrm{MCSP}}}}$ / ${{\ensuremath{\mathrm{MKTP}}}}$. #### Contributions. The main conceptual contribution of our paper is a fundamentally different way of constructing reductions to ${{\ensuremath{\mathrm{MKTP}}}}$ based on a novel use of known interactive proof systems. Our approach applies to ${{\ensuremath{\mathrm{GI}}}}$ and a broad class of isomorphism problems. A common framework for those isomorphism problems is another conceptual contribution. In terms of results, our new approach allows us to eliminate the errors in the recent reductions from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$, and more generally to establish *zero-sided error* randomized reductions to ${{\ensuremath{\mathrm{MKTP}}}}$ from many isomorphism problems within our framework. These include Linear Code Equivalence, Matrix Subspace Conjugacy, and Permutation Group Conjugacy (see Section \[sec:iso:corollaries\] for the definitions). The technical contributions mainly consist of encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum. Before describing the underlying ideas, we note that our techniques remain of interest even in light of the recent quasi-polynomial-time algorithm for ${{\ensuremath{\mathrm{GI}}}}$ [@babai]. For one, ${{\ensuremath{\mathrm{GI}}}}$ is still not known to be in [$\mathsf{P}$]{}, and Group Isomorphism stands as a significant obstacle to this (as stated at the end of [@babai]). Moreover, our techniques also apply to the other isomorphism problems mentioned above, for which the current best algorithms are still exponential. Let us also provide some evidence that our approach for constructing reductions to ${{\ensuremath{\mathrm{MKTP}}}}$ differs in an important way from the existing ones. We claim that the existing approach can only yield zero-sided error reductions to ${{\ensuremath{\mathrm{MKTP}}}}$ from problems that are in ${{\ensuremath{\mathsf{NP}}}}\cap {{\ensuremath{\mathsf{coNP}}}}$, a class which ${{\ensuremath{\mathrm{GI}}}}$ and—*a fortiori*—none of the other isomorphism problems mentioned above are known to reside in. The reason for the claim is that the underlying hardness assumptions are fundamentally average-case,[^7] which implies that the reduction can have both false positives and false negatives. For example, in the papers employing the construction from [@hill], ${{\ensuremath{\mathrm{MKTP}}}}$ is used in a subroutine to invert a polynomial-time-computable function (see Lemma \[lemma:mktp-inverts-bbox\] in Section \[sec:prelim:complexity-measures\]), and the subroutine may fail to find an inverse. Given a reliable but imperfect subroutine, the traditional way to eliminate false positives is to use the subroutine for constructing an efficiently verifiable membership witness, and only accept after verifying its validity. As such, the existence of a traditional reduction without false positives from a language $L$ to ${{\ensuremath{\mathrm{MKTP}}}}$ implies that $L \in {{\ensuremath{\mathsf{NP}}}}$. Similarly, a traditional reduction from $L$ to ${{\ensuremath{\mathrm{MKTP}}}}$ without false negatives is only possible if $L \in {{\ensuremath{\mathsf{coNP}}}}$, and zero-sided error is only possible if $L \in {{\ensuremath{\mathsf{NP}}}}\cap {{\ensuremath{\mathsf{coNP}}}}$. #### Main Idea. Instead of using the oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ in the *construction* of a candidate witness and then verifying the validity of the candidate without the oracle, we use the power of the oracle in the *verification* process. This obviates the need for the language $L$ to be in ${{\ensuremath{\mathsf{NP}}}}\cap {{\ensuremath{\mathsf{coNP}}}}$ in the case of reductions with zero-sided error. Let us explain how to implement this idea for $L = {{\ensuremath{\mathrm{GI}}}}$. Recall that an instance of ${{\ensuremath{\mathrm{GI}}}}$ consists of a pair $(G_0,G_1)$ of graphs on the vertex set $[n]$, and the question is whether $G_0 \equiv G_1$, , whether there exists a permutation $\pi \in S_n$ such that $G_1=\pi(G_0)$, where $\pi(G_0)$ denotes the result of applying the permutation $\pi$ to the vertices of $G_0$. In order to develop a zero-sided error algorithm for ${{\ensuremath{\mathrm{GI}}}}$, it suffices to develop one without false negatives. This is because the false positives can subsequently be eliminated using the known search-to-decision reduction for ${{\ensuremath{\mathrm{GI}}}}$ [@kst]. The crux for obtaining a reduction without false negatives from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$ is a witness system for the complement $\overline{{{\ensuremath{\mathrm{GI}}}}}$ inspired by the well-known two-round interactive proof system for $\overline{{{\ensuremath{\mathrm{GI}}}}}$ [@GoldreichMW91]. Consider the distribution $R_G(\pi) \doteq \pi(G)$ where $\pi \in S_n$ is chosen uniformly at random. By the Orbit–Stabilizer Theorem, for any fixed $G$, $R_G$ is uniform over a set of size $N \doteq n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|$ and thus has entropy $s = \log(N)$, where ${{\ensuremath{\operatorname{Aut}}}}(G) \doteq \{ \pi \in S_n \, : \, \pi(G) = G \}$ denotes the set of automorphisms of $G$. For ease of exposition, let us assume that $|{{\ensuremath{\operatorname{Aut}}}}(G_0)|=|{{\ensuremath{\operatorname{Aut}}}}(G_1)|$ (which is actually the hardest case for ${{\ensuremath{\mathrm{GI}}}}$), so both $R_{G_0}$ and $R_{G_1}$ have the same entropy $s$. Consider picking $r \in {\{0,1\}}$ uniformly at random, and setting $G = G_r$. If $(G_0,G_1) \in {{\ensuremath{\mathrm{GI}}}}$, the distributions $R_{G_0}$, $R_{G_1}$, and $R_{G}$ are all identical, and therefore $R_G$ also has entropy $s$. On the other hand, if $(G_0,G_1) \not\in {{\ensuremath{\mathrm{GI}}}}$, the entropy of $R_G$ equals $s+1$. The extra bit of information corresponds to the fact that in the nonisomorphic case each sample of $R_G$ reveals the value of $r$ that was used, whereas that bit gets lost in the reduction in the isomorphic case. The difference in entropy suggests that a typical sample of $R_G$ can be compressed more in the isomorphic case than in the nonisomorphic case. If we can compute some threshold such that ${{\ensuremath{\operatorname{KT}}}}(R_G)$ *never* exceeds the threshold in the isomorphic case, and exceeds it with nonnegligible probability in the nonisomorphic case, we have the witness system for $\overline{{{\ensuremath{\mathrm{GI}}}}}$ that we aimed for: Take a sample from $R_G$, and use the oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ to check that it cannot be compressed at or below the threshold. The entropy difference of 1 may be too small to discern, but we can amplify the difference by taking multiple samples and concatenating them. Thus, we end up with a randomized mapping reduction of the following form, where $t$ denotes the number of samples and $\theta$ the threshold: $$\label{eq:reduction} \begin{array}{l} \text{Pick $r \doteq r_1\ldots r_t \in {\{0,1\}}^t$ and $\pi_i \in S_n$ for $i \in [t]$, each uniformly at random.} \\ \text{Output $(y,\theta)$ where $y \doteq y_1\ldots y_t$ and $y_i \doteq \pi_i(G_{r_i})$.} \end{array}$$ We need to analyze how to set the threshold $\theta$ and argue correctness for a value of $t$ that is polynomially bounded. In order to do so, let us first consider the case where the graphs $G_0$ and $G_1$ are *rigid*, , they have no nontrivial automorphisms, or equivalently, $s = \log(n!)$. - If $G_0 \not\equiv G_1$, the string $y$ contains all of the information about the random string $r$ and the $t$ random permutations $\pi_1, \ldots, \pi_t$, which amounts to $ts + t = t(s+1)$ bits of information. This implies that $y$ has ${{\ensuremath{\operatorname{KT}}}}$-complexity close to $t(s+1)$ with high probability. - If $G_0 \equiv G_1$, then we can efficiently produce each bit of $y$ from the adjacency matrix representation of $G_0$ ($n^2$ bits) and the function table of permutations $\tau_i \in S_n$ (for $i \in [t]$) such that $y_i \doteq \pi_i(G_{r_i}) = \tau_i(G_0)$. Moreover, the set of all permutations $S_n$ allows an efficiently decodable indexing, , there exists an efficient algorithm that takes an index $k \in [n!]$ and outputs the function table of the $k$-th permutation in $S_n$ according to some ordering. An example of such an indexing is the Lehmer code (see, , [@knuth3 pp. 12-13] for specifics). This shows that $$\label{eq:KT-bound} {{\ensuremath{\operatorname{KT}}}}(y) \leq t \lceil s \rceil+ (n + \log(t))^c$$ for some constant $c$, where the first term represents the cost of the $t$ indices of ${{\left\lceil{s}\right\rceil}}$ bits each, and the second term represents the cost of the $n^2$ bits for the adjacency matrix of $G_0$ and the polynomial running time of the decoding process. If we ignore the difference between $s$ and ${{\left\lceil{s}\right\rceil}}$, the right-hand side of becomes $ts+n^c$, which is closer to $ts$ than to $t(s+1)$ for $t$ any sufficiently large polynomial in $n$, say $t=n^{c+1}$. Thus, setting $\theta$ halfway between $ts$ and $t(s+1)$, i.e., $\theta \doteq t(s+\frac{1}{2})$, ensures that ${{\ensuremath{\operatorname{KT}}}}(y) > \theta$ holds with high probability if $G_0 \not\equiv G_1$, and never holds if $G_0 \equiv G_1$. This yields the desired randomized mapping reduction without false negatives, modulo the rounding issue of $s$ to ${{\left\lceil{s}\right\rceil}}$. The latter can be handled by aggregating the permutations $\tau_i$ into blocks so as to make the amortized cost of rounding negligible. The details are captured in the of Section \[sec:GI:rigid\]. What changes in the case of [non-rigid]{} graphs? For ease of exposition, let us again assume that $|{{\ensuremath{\operatorname{Aut}}}}(G_0)| = |{{\ensuremath{\operatorname{Aut}}}}(G_1)|$. There are two complications: - We no longer know how to efficiently compute the threshold $\theta \doteq t(s+\frac{1}{2})$ because $s \doteq \log(N)$ and $N \doteq \log(n!/|{{\ensuremath{\operatorname{Aut}}}}(G_0)|) = \log(n!/|{{\ensuremath{\operatorname{Aut}}}}(G_1)|)$ involves the size of the automorphism group. - The Lehmer code no longer provides sufficient compression in the isomorphic case as it requires $\log(n!)$ bits per permutation whereas we only have $s$ to spend, which could be considerably less than $\log(n!)$. In order to resolve (ii) we develop an efficiently decodable indexing of cosets for any subgroup of $S_n$ given by a list of generators (see Lemma \[lemma:graph-coding\] in Section \[sec:GI:quasirigid\]). In fact, our scheme even works for cosets of a subgroup within another subgroup of $S_n$, a generalization that may be of independent interest (see Lemma \[lemma:permutation-group-coding\] in the Appendix). Applying our scheme to ${{\ensuremath{\operatorname{Aut}}}}(G)$ and including a minimal list of generators for ${{\ensuremath{\operatorname{Aut}}}}(G)$ in the description of the program $p$ allows us to maintain . Regarding (i), we can deduce a good approximation to the threshold with high probability by taking, for both choices of $r \in {\{0,1\}}$, a polynomial number of samples of $R_{G_r}$ and using the oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ to compute the exact ${{\ensuremath{\operatorname{KT}}}}$-complexity of their concatenation. This leads to a randomized reduction from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$ with bounded error (from which one without false positives follows as mentioned before), reproving the earlier result of [@adas] using our new approach (see Remark \[remark:GI-BPP\] in Section \[sec:GI:quasirigid\] for more details). In order to avoid false negatives, we need to improve the above approximation algorithm such that it never produces a value that is too small, while maintaining efficiency and the property that it outputs a good approximation with high probability. In order to do so, it suffices to develop a *probably-correct overestimator* for the quantity $n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|$, i.e., a randomized algorithm that takes as input an $n$-vertex graph $G$, produces the correct quantity with high probability, and never produces a value that is too small; the algorithm should run in polynomial time with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. Equivalently, it suffices to develop a probably-correct *under*estimator of similar complexity for $|{{\ensuremath{\operatorname{Aut}}}}(G)|$. The latter can be obtained from the known search-to-decision procedures for ${{\ensuremath{\mathrm{GI}}}}$, and answering the oracle calls to ${{\ensuremath{\mathrm{GI}}}}$ using the above two-sided error reduction from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$. There are a number of ways to implement this strategy; here is one that generalizes to a number of other isomorphism problems including Linear Code Equivalence. 1. Find a list of permutations that generates a subgroup of ${{\ensuremath{\operatorname{Aut}}}}(G)$ such that the subgroup equals ${{\ensuremath{\operatorname{Aut}}}}(G)$ with high probability. Finding a list of generators for ${{\ensuremath{\operatorname{Aut}}}}(G)$ deterministically reduces to ${{\ensuremath{\mathrm{GI}}}}$. Substituting the oracle for ${{\ensuremath{\mathrm{GI}}}}$ by a two-sided error algorithm yields a list of permutations that generates ${{\ensuremath{\operatorname{Aut}}}}(G)$ with high probability, and always produces a subgroup of ${{\ensuremath{\operatorname{Aut}}}}(G)$. The latter property follows from the inner workings of the reduction, or can be imposed explicitly by checking every permutation produced and dropping it if it does not map $G$ to itself. We use the above randomized reduction from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$ as the two-sided error algorithm for ${{\ensuremath{\mathrm{GI}}}}$. 2. Compute the order of the subgroup generated by those permutations. There is a known deterministic polynomial-time algorithm to do this [@seress]. The resulting probably-correct underestimator for $|{{\ensuremath{\operatorname{Aut}}}}(G)|$ runs in polynomial time with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. Plugging it into our approach, we obtain a randomized reduction from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$ without false negatives. A reduction with zero-sided error follows as discussed earlier. Before applying our approach to other isomorphism problems, let us point out the important role that the Orbit–Stabilizer Theorem plays. A randomized algorithm for finding generators for a graph’s automorphism group yields a probably-correct underestimator for the size of the automorphism group, as well as a randomized algorithm for ${{\ensuremath{\mathrm{GI}}}}$ without false positives. The Orbit–Stabilizer Theorem allows us to turn a probably-correct underestimator for $|{{\ensuremath{\operatorname{Aut}}}}(G)|$ into a probably-correct overestimator for the size of the support of $R_G$, thereby switching the error from one side to the other, and allowing us to avoid false negatives instead of false positives. #### General Framework. Our approach extends to several other isomorphism problems. They can be cast in the following common framework, parameterized by an underlying family of group actions $(\Omega,H)$ where $H$ is a group that acts on the universe $\Omega$. We typically think of the elements of $\Omega$ as abstract objects, which need to be described in string format in order to be input to a computer; we let $\omega(z)$ denote the abstract object represented by the string $z$. \[def:iso\] An instance of an Isomorphism Problem consists of a pair $x=(x_0,x_1)$ that determines a universe $\Omega_x$ and a group $H_x$ that acts on $\Omega_x$ such that $\omega_0(x) \doteq \omega(x_0)$ and $\omega_1(x) \doteq \omega(x_1)$ belong to $\Omega_x$. Each $h \in H_x$ is identified with the permutation $h: \Omega_x \to \Omega_x$ induced by the action. The goal is to determine whether there exists $h \in H_x$ such that $h(\omega_0(x))=\omega_1(x)$. When it causes no confusion, we drop the argument $x$ and simply write $H$, $\Omega$, $\omega_0$, and $\omega_1$. We often blur the—sometimes pedantic—distinction between $z$ and $\omega(z)$. For example, in ${{\ensuremath{\mathrm{GI}}}}$, each $z$ is an $n\times n$ binary matrix (a string of length $n^2$), and represents the abstract object $\omega(z)$ of a graph with $n$ labeled vertices; thus, in this case the correspondence between $z$ and $\omega(z)$ is a bijection. The group $H$ is the symmetric group $S_n$, and the action is by permuting the labels. Table \[table:iso\] summarizes how the problems we mentioned earlier can be cast in the framework (see Section \[sec:iso:corollaries\] for details about the last three). Problem $H$ $\Omega$ ----------------------------- -------------------------------------------------------- ------------------------------------------------------------- Graph Isomorphism $S_n$ graphs with $n$ labeled vertices Linear Code Equivalence $S_n$ subspaces of dimension $d$ in ${\mathbb{F}}_q^n$ Permutation Group Conjugacy $S_n$ subgroups of $S_n$ Matrix Subspace Conjugacy ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ subspaces of dimension $d$ in ${\mathbb{F}}_q^{n \times n}$ : Instantiations of the Isomorphism Problem[]{data-label="table:iso"} We generalize our construction for ${{\ensuremath{\mathrm{GI}}}}$ to any Isomorphism Problem by replacing $R_G(\pi) \doteq \pi(G)$ where $\pi \in S_n$ is chosen uniformly at random, by $R_\omega(h) \doteq h(\omega)$ where $h \in H$ is chosen uniformly at random. The analysis that the construction yields a randomized reduction without false negatives from the Isomorphism Problem to ${{\ensuremath{\mathrm{MKTP}}}}$ carries over, provided that the Isomorphism Problem satisfies the following properties. 1. The underlying group $H$ is *efficiently samplable*, and the action $(\omega,h) \mapsto h(\omega)$ is efficiently computable. We need this property in order to make sure the reduction is efficient. 2. There is an efficiently computable *normal form* for representing elements of $\Omega$ as strings. This property trivially holds in the setting of ${{\ensuremath{\mathrm{GI}}}}$ as there is a unique adjacency matrix that represents any given graph on the vertex set $[n]$. However, uniqueness of representation need not hold in general. Consider, for example, Permutation Group Conjugacy. An instance of this problem abstractly consists of two permutation groups $(\Gamma_0,\Gamma_1)$, represented (as usual) by a sequence of elements of $S_n$ generating each group. In that case there are many strings representing the same abstract object, , a subgroup has many different sets of generators. For the correctness analysis in the isomorphic case it is important that $H$ acts on the abstract objects, and *not* on the binary strings that represent them. In particular, the output of the reduction should only depend on the abstract object $h(\omega)$, and not on the way $\omega$ was provided as input. This is because the latter may leak information about the value of the bit $r$ that was picked. The desired independence can be guaranteed by applying a normal form to the representation before outputting the result. In the case of Permutation Group Conjugacy, this means transforming a set of permutations that generate a subgroup $\Gamma$ into a canonical set of generators for $\Gamma$. In fact, it suffices to have an efficiently computable *complete invariant* for $\Omega$, , a mapping from representations of objects from $\Omega$ to strings such that the image only depends on the abstract object, and is different for different abstract objects. 3. There exists a probably-correct overestimator for $N \doteq |H|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|$ that is computable efficiently with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. We need this property to set the threshold $\theta \doteq t(s+\frac{1}{2})$ with $s \doteq \log(N)$ correctly. 4. There exists an encoding for cosets of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ in $H$ that achieves ${{\ensuremath{\operatorname{KT}}}}$-complexity close to the information-theoretic optimum (see Section \[sec:prelim:encodings\] for the definition of an encoding). This property ensures that in the isomorphic case the ${{\ensuremath{\operatorname{KT}}}}$-complexity is never much larger than the entropy. Properties 1 and 2 are fairly basic. Property 4 may seem to require an instantiation-dependent approach. However, in Section \[sec:flat-coding-lemma\] we develop a *generic* hashing-based encoding scheme that meets the requirements. In fact, we give a nearly-optimal encoding scheme for any samplable distribution that is almost flat, without reference to isomorphism. Unlike the indexings from Lemma \[lemma:permutation-group-coding\] for the special case where $H$ is the symmetric group, the generic construction does not achieve the information-theoretic optimum, but it comes sufficiently close for our purposes. The notion of a probably-correct overestimator in Property 3 can be further relaxed to that of a *probably-approximately-correct overestimator*, or *pac overestimator* for short. This is a randomized algorithm that with high probability outputs a value within an absolute deviation bound of $\Delta$ from the correct value, and never produces a value that is more than $\Delta$ below the correct value. More precisely, it suffices to efficiently compute with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ a pac overestimator for $s \doteq \log(|H|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|)$ with deviation $\Delta = 1/4$. The relaxation suffices because of the difference of about 1/2 between the threshold $\theta$ and the actual ${{\ensuremath{\operatorname{KT}}}}$-values in both the isomorphic and the non-isomorphic case. As $s = \log|H| - \log|{{\ensuremath{\operatorname{Aut}}}}(\omega)|$, it suffices to have a pac overestimator for $\log|H|$ and a pac *under*estimator for $\log|{{\ensuremath{\operatorname{Aut}}}}(\omega)|$, both to within deviation $\Delta/2 = 1/8$ and of the required efficiency. Generalizing our approach for ${{\ensuremath{\mathrm{GI}}}}$, one way to obtain the desired underestimator for $\log|{{\ensuremath{\operatorname{Aut}}}}(\omega)|$ is by showing how to efficiently compute with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$: - a list $L$ of elements of $H$ that generates a subgroup $\langle L \rangle$ of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ such that $\langle L \rangle = {{\ensuremath{\operatorname{Aut}}}}(\omega)$ with high probability, and - a pac underestimator for $\log|\langle L \rangle|$, the logarithm of the order of the subgroup generated by a given list $L$ of elements of $H$. Further mimicking our approach for ${{\ensuremath{\mathrm{GI}}}}$, we know how to achieve (a) when the Isomorphism Problem allows a search-to-decision reduction. Such a reduction is known for Linear Code Equivalence, but remains open for problems like Matrix Subspace Conjugacy and Permutation Group Conjugacy. However, we show that (a) holds for a *generic* isomorphism problem provided that products and inverses in $H$ can be computed efficiently (see Lemma \[lemma:sample-subgroups\] in Section \[sec:iso:conditions\]). The proof hinges on the ability of ${{\ensuremath{\mathrm{MKTP}}}}$ to break the pseudo-random generator construction of [@hill] based on a purported one-way function (Lemma \[lemma:mktp-inverts-bbox\] in Section \[sec:prelim:complexity-measures\]). As for (b), we know how to efficiently compute the order of the subgroup *exactly* in the case of permutation groups ($H=S_n$), even without an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$, and in the case of many matrix groups over finite fields ($H={{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$) with oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$, but some cases remain open (see footnote \[footnote:left-right\] in Section \[sec:iso:conditions\] for more details). Instead, we show how to *generically* construct a *pac underestimator* with small deviation given access to ${{\ensuremath{\mathrm{MKTP}}}}$ as long as products and inverses in $H$ can be computed efficiently, and $H$ allows an efficient complete invariant (see Lemma \[lemma:pacue-subgroup-order\] in Section \[sec:iso:conditions\]). The first two conditions are sufficient to efficiently generate a distribution $\widetilde{p}$ on $\langle L \rangle$ that is uniform to within a small relative deviation [@Bab1991]. The entropy $\widetilde{s}$ of that distribution equals $\log|\langle L \rangle|$ to within a small additive deviation. As $\widetilde{p}$ is almost flat, our encoding scheme from Section \[sec:flat-coding-lemma\] shows that $\widetilde{p}$ has an encoding whose length does not exceed $\widetilde{s}$ by much, and that can be decoded by small circuits. Given an efficient complete invariant for $H$, we can use an approach similar to the one we used to approximate the threshold $\theta$ to construct a pac underestimator for $\widetilde{s}$ with small additive deviation, namely the amortized ${{\ensuremath{\operatorname{KT}}}}$-complexity of the concatenation of a polynomial number of samples from $\widetilde{p}$. With access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ we can efficiently evaluate ${{\ensuremath{\operatorname{KT}}}}$. As a result, we obtain a pac underestimator for $\log|\langle L \rangle|$ with a small additive deviation that is efficiently computable with oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$. The above ingredients allow us to conclude that all of the isomorphism problems in Table \[table:iso\] reduce to ${{\ensuremath{\mathrm{MKTP}}}}$ under randomized reductions without false negatives. Moreover, we argue that Properties 1 and 2 are sufficient to generalize the construction of Allender and Das [@adas], which yields randomized reductions of the isomorphism problem to ${{\ensuremath{\mathrm{MKTP}}}}$ without false positives (irrespective of whether a search-to-decision reduction is known). By combining both reductions, we conclude that all of the isomorphism problems in Table \[table:iso\] reduce to ${{\ensuremath{\mathrm{MKTP}}}}$ under randomized reductions with zero-sided error. See Sections \[sec:iso\] and \[sec:iso:corollaries\] for more details. #### Open Problems. The difference in compressibility between the isomorphic and non-isomorphic case is relatively small. As such, our approach is fairly delicate. Although we believe it yields zero-sided error reductions to ${{\ensuremath{\mathrm{MCSP}}}}$ as well, we currently do not know whether that is the case. An open problem in the other direction is to develop zero-error reductions from all of ${{\ensuremath{\mathsf{SZK}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$. We refer to Section \[sec:conclusion\] for further discussion and other future research directions. #### Relationship with arXiv 1511.08189. This report subsumes and significantly strengthens the earlier report [@agm.arxiv]. - Whereas [@agm.arxiv] only proves the main result for ${{\ensuremath{\mathrm{GI}}}}$ on *rigid* graphs, and for Graph Automorphism (${{\ensuremath{\mathrm{GA}}}}$) on arbitrary graphs, this report proves it for ${{\ensuremath{\mathrm{GI}}}}$ on *arbitrary* graphs (which subsumes the result for ${{\ensuremath{\mathrm{GA}}}}$ on arbitrary graphs). - Whereas [@agm.arxiv] only contains the main result for ${{\ensuremath{\mathrm{GI}}}}$, this report presents a framework for a generic isomorphism problem, and generalizes the main result for ${{\ensuremath{\mathrm{GI}}}}$ to any problem within the framework that satisfies some elementary conditions. In particular, this report shows that the generalization applies to Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. The generalization involves the development of a generic efficient encoding scheme for samplable almost-flat distributions that is close to the information-theoretic optimum, and reductions to ${{\ensuremath{\mathrm{MKTP}}}}$ for the following two tasks: computing a generating set for the automorphism group, and approximating the size of the subgroup generated by a given list of elements. - The main technical contribution in [@agm.arxiv] (efficiently indexing the cosets of the automorphism group) was hard to follow. This report contains a clean proof using a different strategy, which also generalizes to indexing cosets of subgroups of any permutation group, answering a question that was raised during presentations of [@agm.arxiv]. - The exposition is drastically revised. Preliminaries {#sec:prelim} ============= We assume familiarity with standard complexity theory, including the bounded-error randomized polynomial-time complexity classes ${{\ensuremath{\mathsf{BPP}}}}$ (two-sided error), ${{\ensuremath{\mathsf{RP}}}}$ (one-sided error, , no false positives), and ${{\ensuremath{\mathsf{ZPP}}}}$ (zero-sided error, , no false positives, no false negatives, and bounded probability of no output). In the remainder of this section we provide more details about ${{\ensuremath{\operatorname{KT}}}}$-complexity, formally define the related notions of indexing and encoding, and review some background on graph isomorphism. KT Complexity {#sec:prelim:complexity-measures} ------------- The measure ${{\ensuremath{\operatorname{KT}}}}$ that we informally described in Section \[sec:intro\], was introduced and formally defined as follows in [@powerk]. We refer to that paper for more background and motivation for the particular definition. \[KTdef\] Let $U$ be a universal Turing machine. For each string $x$, define ${{\ensuremath{\operatorname{KT}}}}_U(x)$ to be $$\begin{aligned} \min \{\, |d| + T : \;\; (\forall \sigma \in \{0,1,*\}) \; (\forall i \leq |x|+1) \; U^d(i,\sigma) \mbox{ accepts in $T$ steps iff $x_i = \sigma$}\,\}. \end{aligned}$$ We define $x_i=*$ if $i > |x|$; thus, for $i=|x|+1$ the machine accepts iff $\sigma=*$. The notation $U^d$ indicates that the machine $U$ has random access to the description $d$. ${{\ensuremath{\operatorname{KT}}}}(x)$ is defined to be equal to ${{\ensuremath{\operatorname{KT}}}}_U(x)$ for a fixed choice of universal machine $U$ with logarithmic simulation time overhead [@powerk Proposition 5]. In particular, if $d$ consists of the description of a Turing machine $M$ that runs in time $t_M(n)$ and some auxiliary information $a$ such that $M^a(i) = x_i$ for $i \in [n]$, then ${{\ensuremath{\operatorname{KT}}}}(x) \leq |a| + c_M T_M(\log n) \log(T_M(\log n))$, where $n \doteq |x|$ and $c_M$ is a constant depending on $M$. It follows that $(\mu/\log n)^{\Omega(1)} \leq {{\ensuremath{\operatorname{KT}}}}(x) \leq (\mu \cdot \log n)^{O(1)}$ where $\mu$ represents the circuit complexity of the mapping $i \mapsto x_i$ [@powerk Theorem 11]. The Minimum KT Problem is defined as ${{\ensuremath{\mathrm{MKTP}}}}\doteq \{ (x,\theta) \mid {{\ensuremath{\operatorname{KT}}}}(x) \leq \theta\}$. [@powerk] showed that an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ is sufficient to invert on average any function that can be computed efficiently. We use the following formulation: \[lemma:mktp-inverts-bbox\] There exists a polynomial-time probabilistic Turing machine $M$ using oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$ so that the following holds. For any circuit $C$ on $n$ input bits, $$\Pr\left[ C(M(C, C(\sigma)) = C(\sigma) \right] \geq 1/{{\ensuremath{\operatorname{poly}}}}(n)$$ where the probability is over the uniform distribution of $\sigma \in {\{0,1\}}^n$ and the internal coin flips of $M$. Random Variables, Samplers, Indexings and Encodings {#sec:prelim:encodings} --------------------------------------------------- A finite probability space consists of a finite sample space $S$, and a probability distribution $p$ on $S$. Typical sample spaces include finite groups and finite sets of strings. The probability distributions underlying our probability spaces are always uniform. A *random variable* $R$ is a mapping from the sample space $S$ to a set $T$, which typically is the universe $\Omega$ of a group action or a set of strings. The random variable $R$ with the uniform distribution on $S$ induces a distribution $p$ on $T$. We sometimes use $R$ to denote the induced distribution $p$ as well. The support of a distribution $p$ on a set $T$ is the set of elements $\tau \in T$ with positive probability $p(\tau)$. A distribution is *flat* if it is uniform on its support. The *entropy* of a distribution $p$ is the expected value of $\log(1/p(\tau))$. The *min-entropy* of $p$ is the largest real $s$ such that $p(\tau) \leq 2^{-s}$ for every $\tau \in T$. The *max-entropy* of $p$ is the least real $s$ such that $p(\tau) \geq 2^{-s}$ for every $\tau \in T$. For a flat distribution, the min-, max-, and ordinary entropy coincide and equal the logarithm of the size of the support. For two distributions $p$ and $q$ on the same set $T$, we say that $q$ approximates $p$ within a factor $1+\delta$ if $q(\tau) / (1+\delta) \leq p(\tau) \leq (1+\delta) \cdot q(\tau)$ for all $\tau \in T$. In that case, $p$ and $q$ have the same support, and if $p$ has min-entropy $s$, then $q$ has min-entropy at least $s-\log(1+\delta)$, and if $p$ has max-entropy $s$, then $q$ has max-entropy at most $s+\log(1+\delta)$. A *sampler* within a factor $1+\delta$ for a distribution $p$ on a set $T$ is a random variable $R : {\{0,1\}}^\ell \to T$ that induces a distribution that approximates $p$ within a factor $1+\delta$. We say that $R$ *samples $T$ within a factor $1+\delta$ from length $\ell$*. If $\delta=0$ we call the sampler *exact*. The choice of ${\{0,1\}}^\ell$ reflects the fact that distributions need to be generated from a source of random bits. Factors $1+\delta$ with $\delta > 0$ are necessary in order to sample uniform distributions whose support is not a power of 2. We consider ensembles of distributions $\{p_x\}$ where $x$ ranges over ${\{0,1\}}^*$. We call the ensemble *samplable by polynomial-size circuits* if there exists an ensemble of random variables $\{R_{x,\delta}\}$ where $\delta$ ranges over the positive rationals such that $R_{x,\delta}$ samples $p_x$ within a factor $1+\delta$ from length $\ell_{x,\delta}$ and $R_{x,\delta}$ can be computed by a circuit of size ${{\ensuremath{\operatorname{poly}}}}(|x|/\delta)$. We stress that the circuits can depend on the string $x$, not just on $|x|$. If in addition the mappings $(x,\delta) \mapsto \ell_{x,\delta}$ and $(x,\delta,\sigma) \mapsto R_{x,\delta}(\sigma)$ can be computed in time ${{\ensuremath{\operatorname{poly}}}}(|x|/\delta)$, we call the ensemble *uniformly samplable in polynomial time*. One way to obtain strings with high ${{\ensuremath{\operatorname{KT}}}}$-complexity is as samples from distributions with high min-entropy. \[prop:complexity-at-least-entropy\] Let $y$ be sampled from a distribution with min-entropy $s$. For all $k$, we have ${{\ensuremath{\operatorname{KT}}}}(y) \geq s - k$ except with probability at most $2^{-k}$. One way to establish upper bounds on ${{\ensuremath{\operatorname{KT}}}}$-complexity is via efficiently decodable encodings into integers from a small range. Encodings with the minimum possible range are referred to as indexings. We use these notions in various settings. The following formal definition is for use with random variables and is general enough to capture all the settings we need. It defines an encoding via its decoder $D$; the range of the encoding corresponds to the domain of $D$. \[def:encoding\] Let $R: S \to T$ be a random variable. An *encoding* of $R$ is a mapping $D: [N] \to S$ such that for every $\tau \in T$ there exists $i \in [N]$ such that $R(D(i)) = \tau$. We refer to ${{\left\lceil{\log(N)}\right\rceil}}$ as the *length* of the encoding. An *indexing* is an encoding with $N = |T|$. Definition \[def:encoding\] applies to a set $S$ by identifying $S$ with the random variable that is the identity mapping on $S$. It applies to the cosets of a subgroup $\Gamma$ of a group $H$ by considering the random variable that maps $h \in H$ to the coset $h\Gamma$. It applies to a distribution induced by a random variable $R$ by considering the random variable $R$ itself. We say that an ensemble of encodings $\{D_x\}$ is *decodable by polynomial-size circuits* if for each $x$ there is a circuit of size ${{\ensuremath{\operatorname{poly}}}}(|x|)$ that computes $D_x(i)$ for every $i \in [N_x]$. If in addition the mapping $(x,i) \mapsto D_x(i)$ is computable in time ${{\ensuremath{\operatorname{poly}}}}(|x|)$, we call the ensemble *uniformly decodable in polynomial time*. Graph Isomorphism and the Orbit-Stabilizer Theorem {#sec:prelim:graphs} -------------------------------------------------- Graph Isomorphism (${{\ensuremath{\mathrm{GI}}}}$) is the computational problem of deciding whether two graphs, given as input, are isomorphic. A *graph* for us is a simple, undirected graph, that is, a vertex set $V(G)$, and a set $E(G)$ of unordered pairs of vertices. An *isomorphism* between two graphs $G_0, G_1$ is a bijection $\pi\colon V(G_0) \to V(G_1)$ that preserves both edges and non-edges: $(v,w) \in E(G_0)$ if and only if $(\pi(v), \pi(w)) \in E(G_1)$. An isomorphism from a graph to itself is an *automorphism*; the automorphisms of a given graph $G$ form a group under composition, denoted ${{\ensuremath{\operatorname{Aut}}}}(G)$. The Orbit–Stabilizer Theorem implies that the number of distinct graphs isomorphic to $G$ equals $n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|$. A graph $G$ is *rigid* if $|{{\ensuremath{\operatorname{Aut}}}}(G)|=1$, , the only automorphism is the identity, or equivalently, all $n!$ permutations of $G$ yield distinct graphs. More generally, let $H$ be a group acting on a universe $\Omega$. For $\omega \in \Omega$, each $h\in H$ is an isomorphism from $\omega$ to $h(\omega)$. ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ is the set of isomorphisms from $\omega$ to itself. By the Orbit–Stabilizer Theorem the number of distinct isomorphic copies of $\omega$ equals $|H|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|$. Graph Isomorphism {#sec:GI} ================= In this section we show: \[thm:GI\] ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. The crux is the randomized mapping reduction from deciding whether a given pair of $n$-vertex graphs $(G_0,G_1)$ is in ${{\ensuremath{\mathrm{GI}}}}$ to deciding whether $(y,\theta) \in {{\ensuremath{\mathrm{MKTP}}}}$, as prescribed by . Recall that involves picking a string $r \doteq r_1\ldots r_t \in {\{0,1\}}^t$ and permutations $\pi_i$ at random, and constructing the string $y = y_1\ldots y_t$, where $y_i = \pi_i(G_{r_i})$. We show how to determine $\theta$ such that a sufficiently large polynomial $t$ guarantees that the reduction has no false negatives. We follow the outline of Section \[sec:intro\], take the same four steps, and fill in the missing details. Rigid Graphs {#sec:GI:rigid} ------------ We first consider the simplest setting, in which both $G_0$ and $G_1$ are rigid. We argue that $\theta \doteq t(s+\frac{1}{2})$ works, where $s = \log(n!)$. [*Nonisomorphic Case.*]{} If $G_0 \not\equiv G_1$, then (by rigidity), each choice of $r$ and each distinct sequence of $t$ permutations results in a different string $y$, and thus the distribution on the strings $y$ has entropy $t(s+1)$ where $s \doteq \log(n!)$. Thus, by Proposition \[prop:complexity-at-least-entropy\], ${{\ensuremath{\operatorname{KT}}}}(y) > \theta = t(s+1) - \frac{t}{2}$ with all but exponentially small probability in $t$. Thus with high probability the algorithm declares $G_0$ and $G_1$ nonisomorphic. [*Isomorphic Case.*]{} If $G_0 \equiv G_1$, we need to show that ${{\ensuremath{\operatorname{KT}}}}(y) \leq \theta$ always holds. The key insight is that the information in $y$ is precisely captured by the $t$ permutations $\tau_1, \tau_2, \ldots, \tau_t$ such that $\tau_i(G_0) = y_i$. These permutations exist because $G_0 \equiv G_1$; they are unique by the rigidity assumption. Thus, $y$ contains at most $ts$ bits of information. We show that its ${{\ensuremath{\operatorname{KT}}}}$-complexity is not much larger that this. We rely on the following encoding, due to Lehmer (see, , [@knuth3 pp. 12–33]): \[prop:lehmer-coding\] The symmetric groups $S_n$ have indexings that are uniformly decodable in time ${{\ensuremath{\operatorname{poly}}}}(n)$. To bound ${{\ensuremath{\operatorname{KT}}}}(y)$, we consider a program $d$ that has the following information hard-wired into it: $n$, the adjacency matrix of $G_0$, and the $t$ integers $k_1, \ldots, k_t \in [n!]$ encoding $\tau_1, \ldots, \tau_t$. We use the decoder from Proposition \[prop:lehmer-coding\] to compute the $i$-th bit of $y$ on input $i$. This can be done in time ${{\ensuremath{\operatorname{poly}}}}(n,\log(t))$ given the hard-wired information. As mentioned in Section \[sec:intro\], a naïve method for encoding the indices $k_1, \ldots, k_t$ only gives the bound $t{{\left\lceil{s}\right\rceil}} + {{\ensuremath{\operatorname{poly}}}}(n,\log(t))$ on ${{\ensuremath{\operatorname{KT}}}}(y)$, which may exceed $t(s+1)$ and—*a fortiori*—the threshold $\theta$, no matter how large a polynomial $t$ is. We remedy this by aggregating multiple indices into blocks, and amortizing the encoding overhead across multiple samples. The following technical lemma captures the technique. For a set $T$ of strings and $b \in {\mathbb{N}}$, the statement uses the notation $T^b$ to denote the set of concatenations of $b$ strings from $T$; we refer to Section \[sec:prelim:encodings\] for the other terminology. \[lemma:blocking\] Let $\{T_x\}$ be an ensemble of sets of strings such that all strings in $T_x$ have the same length ${{\ensuremath{\operatorname{poly}}}}(|x|)$. Suppose that for each $x \in {\{0,1\}}^*$ and $b \in {\mathbb{N}}$, there is a random variable $R_{x,b}$ whose image contains $(T_x)^b$, and such that the $R_{x,b}$ is computable by a circuit of size ${{\ensuremath{\operatorname{poly}}}}(|x|,b)$ and has an encoding of length $s'(x,b)$ decodable by a circuit of size ${{\ensuremath{\operatorname{poly}}}}(|x|,b)$. Then there are constants $c_1$ and $c_2$ so that, for every constant $\alpha > 0$, every $t \in {\mathbb{N}}$, every sufficiently large $x$, and every $y \in (T_x)^t$ $${{\ensuremath{\operatorname{KT}}}}(y) \leq t^{1-\alpha}\cdot s'(x, {{\left\lceil{t^\alpha}\right\rceil}}) \;+\; t^{\alpha\cdot c_1} \cdot |x|^{c_2}.$$ We first show how to apply the and then prove it. For a given rigid graph $G$, we let $T_G$ be the image of the random variable $R_G$ that maps $\pi \in S_n$ to $\pi(G)$ (an adjacency matrix viewed as a string of $n^2$ bits). We let $R_{G,b}$ be the $b$-fold Cartesian product of $R_G$, , $R_{G,b}$ takes in $b$ permutations $\tau_1,\ldots,\tau_b \in S_n$, and maps them to $\tau_1(G)\tau_2(G)\cdots\tau_b(G)$. $R_{G,b}$ is computable by (uniform) circuits of size ${{\ensuremath{\operatorname{poly}}}}(n,b)$. To encode an outcome $\tau_1(G)\tau_2(G)\cdots\tau_b(G)$, we use as index the number whose base-$(n!)$ representation is written $k_1k_2{\cdots}k_b$, where $k_i$ is the index of $\tau_i$ from the Lehmer code. This indexing has length $s'(G,b) \doteq {{\left\lceil{\log(n!^b)}\right\rceil}} \leq bs+1$. Given an index, the list of permutations $\tau_1,\ldots,\tau_b$ can be decoded by (uniform) circuits of size ${{\ensuremath{\operatorname{poly}}}}(n,b)$. By the , we have that $$\label{eq:blocking:calculation} {{\ensuremath{\operatorname{KT}}}}(y) \leq t^{1-\alpha} ({{\left\lceil{t^\alpha}\right\rceil}} s +1) + t^{\alpha c_1} \cdot n^{c_2} \leq ts + t^{1-\alpha}\cdot n^{c_0} + t^{\alpha c_1} \cdot n^{c_2}$$ for some constants $c_0, c_1, c_2$, every constant $\alpha > 0$, and all sufficiently large $n$, where we use the fact that $s = \log n! \leq n^{c_0}$. Setting $\alpha = \alpha_0 \doteq 1/(c_1+1)$, this becomes ${{\ensuremath{\operatorname{KT}}}}(y) \leq ts + t^{1-\alpha_0} n^{(c_0+c_2)}$. Taking $t = n^{1+(c_0+c_2)/\alpha_0}$, we see that for all sufficiently large $n$, ${{\ensuremath{\operatorname{KT}}}}(y) \leq t(s+\frac{1}{2}) \doteq \theta$. Let $R_{x,b}$ and $D_{x,b}$ be the hypothesized random variables and corresponding decoders. Fix $x$ and $t$, let $m = {{\ensuremath{\operatorname{poly}}}}(|x|)$ denote the length of the strings in $T_x$, and let $b \in {\mathbb{N}}$ be a parameter to be set later. To bound ${{\ensuremath{\operatorname{KT}}}}(y)$, we first break $y$ into ${{\left\lceil{t/b}\right\rceil}}$ blocks $\widetilde{y}_1, \widetilde{y}_2, \ldots, \widetilde{y}_{{{\left\lceil{t/b}\right\rceil}}}$ where each $\widetilde{y}_i \in (T_x)^b$. As the image of $R_{x,b}$ contains $(T_x)^b$, $\widetilde{y}_i$ is encoded by some index $k_j$ of length $s'(x,b)$. Consider a program $d$ that has $x$, $t$, $m$, $b$, the circuit for computing $R_{x,b}$, the circuit for computing $D_{x,b}$, and the indices $k_1, k_2, \ldots, k_{{{\left\lceil{t/b}\right\rceil}}}$ hardwired, takes an input $i \in {\mathbb{N}}$, and determines the $i$-th bit of $y$ as follows. It first computes $j_0, j_1 \in {\mathbb{N}}$ so that $i$ points to the $j_1$-th bit position in $\widetilde{y}_{j_0}$. Then, using $D_{x,b}$, $k_{j_0}$, $\alpha_{x,b}$, and $j_1$, it finds $\sigma$ such that $R_{x,b}(\sigma)$ equals the $\widetilde{y}_{j_0}$. Finally, it computes $R_{x,b}(\sigma)$ and outputs the $j_1$-th bit, which is the $i$-th bit of $y$. The bit-length of $d$ is at most ${{\left\lceil{t/b}\right\rceil}} \cdot s'(x,b)$ for the indices, plus ${{\ensuremath{\operatorname{poly}}}}(|x|,b,\log t)$ for the rest. The time needed by $p$ is bounded by ${{\ensuremath{\operatorname{poly}}}}(|x|,b,\log t)$. Thus ${{\ensuremath{\operatorname{KT}}}}(y) \leq {{\left\lceil{t/b}\right\rceil}} \cdot s'(x,b) + {{\ensuremath{\operatorname{poly}}}}(|x|,b,\log t) \leq t/b \cdot s'(x,b) + {{\ensuremath{\operatorname{poly}}}}(|x|,b,\log t)$, where we used the fact that $s'(x,b) \leq {{\ensuremath{\operatorname{poly}}}}(|x|,b)$. The lemma follows by choosing $b = {{\left\lceil{t^\alpha}\right\rceil}}$. Known Number of Automorphisms {#sec:GI:quasirigid} ----------------------------- We generalize the case of rigid graphs to graphs for which we know the size of their automorphism groups. Specifically, in addition to the two input graphs $G_0$ and $G_1$, we are also given numbers $N_0, N_1$ where $N_i \doteq n!/|{{\ensuremath{\operatorname{Aut}}}}(G_i)|$. Note that if $N_0 \ne N_1$, we can right away conclude that $G_0 \not\equiv G_1$. Nevertheless, we do not assume that $N_0 = N_1$ as the analysis of the case $N_0 \ne N_1$ will be useful in Section \[sec:GI:assume-pcoe\]. The reduction is the same as in Section \[sec:GI:rigid\] with the correct interpretation of $s$. The main difference lies in the analysis, where we need to accommodate for the loss in entropy that comes from having multiple automorphisms. Let $s_i \doteq \log(N_i)$ be the entropy in a random permutation of $G_i$. Set $s \doteq \min(s_0, s_1)$, and $\theta \doteq t(s+\frac{1}{2})$. In the nonisomorphic case the min-entropy of $y$ is at least $t(s+1)$, so ${{\ensuremath{\operatorname{KT}}}}(y) > \theta$ with high probability. In the isomorphic case we upper bound ${{\ensuremath{\operatorname{KT}}}}(y)$ by about $ts$. Unlike the rigid case, we can no longer afford to encode an entire permutation for each permuted copy of $G_0$; we need a replacement for the Lehmer code. The following encoding, applied to $\Gamma = {{\ensuremath{\operatorname{Aut}}}}(G)$, suffices to complete the argument from Section \[sec:GI:rigid\]. \[lemma:graph-coding\] For every subgroup $\Gamma$ of $S_n$ there exists an indexing of the cosets[^8] of $\Gamma$ that is uniformly decodable in polynomial time when $\Gamma$ is given by a list of generators. We prove Lemma \[lemma:graph-coding\] in the Appendix as a corollary to a more general lemma that gives, for each $\Gamma \leq H \leq S_n$, an efficiently computable indexing for the cosets of $\Gamma$ in $H$. \[remark:GI-BPP\] Before we continue towards Theorem \[thm:GI\], we point out that the above ideas yield an alternate proof that ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ (and hence that ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$). This weaker result was already obtained in [@adas] along the well-trodden path discussed in Section \[sec:intro\]; this remark shows how to obtain it using our new approach. The key observation is that in both the isomorphic and the nonisomorphic case, with high probability ${{\ensuremath{\operatorname{KT}}}}(y)$ stays away from the threshold $\theta$ by a growing margin, Moreover, the above analysis allows us to efficiently obtain high-confidence approximations of $\theta$ to within any constant using sampling and queries to the ${{\ensuremath{\mathrm{MKTP}}}}$ oracle. More specifically, for $i \in {\{0,1\}}$, let $\widetilde{y}_i$ denote the concatenation of $\widetilde{t}$ independent samples from $R_{G_i}$. Our analysis shows that ${{\ensuremath{\operatorname{KT}}}}(\widetilde{y}_i) \leq \widetilde{t} s_i + \widetilde{t}^{1-\alpha_0} n^c$ always holds, and that ${{\ensuremath{\operatorname{KT}}}}(\widetilde{y}_i) \geq \widetilde{t} s_i - \widetilde{t}^{1-\alpha_0} n^c$ holds with high probability. Thus, $\widetilde{s}_i \doteq {{\ensuremath{\operatorname{KT}}}}(\widetilde{y}_i)/\widetilde{t}$ approximates $s_i$ with high confidence to within an additive deviation of $n^c/\widetilde{t}^{\alpha_0}$. Similarly, $\widetilde{s} \doteq \min(\widetilde{s}_0,\widetilde{s}_1)$ approximates $s$ to within the same deviation margin, and $\widetilde{\theta} \doteq t(\widetilde{s}+\frac{1}{2})$ approximates $\theta$ to within an additive deviation of $t n^c/\widetilde{t}^{\alpha_0}$. The latter bound can be made less than 1 by setting $\widetilde{t}$ to a sufficiently large polynomial in $n$ and $t$. Moreover, all these estimates can be computed in time ${{\ensuremath{\operatorname{poly}}}}(\widetilde{t},n)$ with access to ${{\ensuremath{\mathrm{MKTP}}}}$ as ${{\ensuremath{\mathrm{MKTP}}}}$ enables us to evaluate ${{\ensuremath{\operatorname{KT}}}}$ efficiently. Probably-Correct Underestimators for the Number of Automorphisms {#sec:GI:assume-pcoe} ---------------------------------------------------------------- The reason the ${{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$-algorithm in Remark \[remark:GI-BPP\] can have false negatives is that the approximation $\widetilde{\theta}$ to $\theta$ may be too small. Knowing the quantities $N_i \doteq n!/|{{\ensuremath{\operatorname{Aut}}}}(G_i)|$ exactly allows us to compute $\theta$ exactly and thereby obviates the possibility of false negatives. In fact, it suffices to compute overestimates for the quantities $N_i$ which are correct with non-negligible probability. We capture this notion formally as follows: \[def:estimator\] Let $g: \Omega \to {\mathbb{R}}$ be a function, and $M$ a randomized algorithm that, on input $\omega \in \Omega$, outputs a value $M(\omega) \in {\mathbb{R}}$. We say that $M$ is a *probably-correct overestimator* for $g$ if, for every $\omega \in \Omega$, $M(\omega) = g(\omega)$ holds with probability at least $1/{{\ensuremath{\operatorname{poly}}}}(|\omega|)$, and $M(\omega) > g(\omega)$ otherwise. A *probably-correct underestimator* for $g$ is defined similarly by reversing the inequality. We point out that, for any probably-correct over-/underestimator, taking the min/max among ${{\ensuremath{\operatorname{poly}}}}(|\omega|)$ independent runs yields the correct value with probability $1 - 2^{-{{\ensuremath{\operatorname{poly}}}}(|\omega|)}$. We are interested in the case where $g(G) = n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|$. Assuming this $g$ on a given class of graphs $\Omega$ has a probably-correct overestimator $M$ computable in randomized polynomial time with an ${{\ensuremath{\mathrm{MKTP}}}}$ oracle, we argue that ${{\ensuremath{\mathrm{GI}}}}$ on $\Omega$ reduces to ${{\ensuremath{\mathrm{MKTP}}}}$ in randomized polynomial time without false negatives. To see this, consider the algorithm that, on input a pair $(G_0,G_1)$ of $n$-vertex graphs, computes $\widetilde{N}_i = M(G_i)$ as estimates of the true values $N_i = \log(n!/|{{\ensuremath{\operatorname{Aut}}}}(G_i)|)$, and then runs the algorithm from Section \[sec:GI:quasirigid\] using the estimates $\widetilde{N}_i$. - In the case where $G_0$ and $G_1$ are not isomorphic, if both estimates $\widetilde{N}_i$ are correct, then the algorithm detects $G_0 \not\equiv G_1$ with high probability. - In the case where $G_0 \equiv G_1$, if $\widetilde{N}_i = N_i$ we showed in Section \[sec:GI:quasirigid\] that the algorithm always declares $G_0$ and $G_1$ to be isomorphic. Moreover, increasing $\theta$ can only decrease the probability of a false negative. As the computed threshold $\theta$ increases as a function of $\widetilde{N}_i$, and the estimate $\widetilde{N}_i$ is always at least as large as $N_i$, it follows that $G_0$ and $G_1$ are always declared isomorphic. Arbitrary Graphs {#sec:GI:pcoe-construction} ---------------- A probably-correct overestimator for the function $G \mapsto n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|$ on *any* graph $G$ can be computed in randomized polynomial time with access to ${{\ensuremath{\mathrm{MKTP}}}}$. The process is described in full detail in Section \[sec:intro\], based on a ${{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ algorithm for ${{\ensuremath{\mathrm{GI}}}}$ (taken from Remark \[remark:GI-BPP\] or from [@adas]). This means that the setting of Section \[sec:GI:assume-pcoe\] is actually the general one. The only difference is that we no longer obtain a mapping reduction from ${{\ensuremath{\mathrm{GI}}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$, but an oracle reduction: We still make use of , but we need more queries to ${{\ensuremath{\mathrm{MKTP}}}}$ in order to set the threshold $\theta$. This shows that ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{coRP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. As ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ follows from the known search-to-decision reduction for ${{\ensuremath{\mathrm{GI}}}}$, this concludes the proof of Theorem \[thm:GI\] that ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. Estimating the Entropy of Flat Samplable Distributions {#sec:flat-coding-lemma} ====================================================== In this section we develop a key ingredient in extending Theorem \[thm:GI\] from ${{\ensuremath{\mathrm{GI}}}}$ to other isomorphism problems that fall within the framework presented in Section \[sec:intro\], namely efficient near-optimal encodings of cosets of automorphism groups. More generally, our encoding scheme works well for any samplable distribution that is flat or almost flat. It allows us to probably-approximately-correctly underestimate the entropy of such distributions with the help of an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. We first develop our encoding, which only requires the existence of a sampler from strings of polynomial length. The length of the encoding is roughly the max-entropy of the distribution, which is the information-theoretic optimum for flat distributions. \[lemma:flat-coding\] Consider an ensemble $\{R_x\}$ of random variables that sample distributions with max-entropy $s(x)$ from length ${{\ensuremath{\operatorname{poly}}}}(|x|)$. Each $R_x$ has an encoding of length $s(x) + \log s(x) + O(1)$ that is decodable by polynomial-size circuits. To see how Lemma \[lemma:flat-coding\] performs, let us apply to the setting of ${{\ensuremath{\mathrm{GI}}}}$. Consider the random variable $R_G$ mapping a permutation $\pi \in S_n$ to $\pi(G)$. The induced distribution is flat and has entropy $s = \log(n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|)$, and each $\pi \in S_n$ can be sampled from strings of length $O(n\log n)$. The thus yields an encoding of length $s + \log s + O(1)$ that is efficiently decodable. The bound on the length is worse than Lemma \[lemma:graph-coding\]’s bound of ${{\left\lceil{s}\right\rceil}}$, but will still be sufficient for the generalization of Theorem \[thm:GI\] and yield the result for ${{\ensuremath{\mathrm{GI}}}}$. We prove the using hashing. Here is the idea. Consider a random hash function $h: {\{0,1\}}^\ell \to {\{0,1\}}^m$ where $\ell$ denotes the length of the strings in the domain of $R_x$ for a given $x$, and $m$ is set slightly below $\ell-s$. For any fixed outcome $y$ of $R_x$, there is a positive constant probability that no more than about $2^\ell / 2^m \approx 2^s$ of all samples $\sigma \in {\{0,1\}}^\ell$ have $h(\sigma)=0^m$, and at least one of these also satisfies $R_x(\sigma)=y$. Let us say that *$h$ works for $y$* when both those conditions hold. In that case—ignoring efficiency considerations—about $s$ bits of information are sufficient to recover a sample $\sigma_y$ satisfying $R_x(\sigma_y)=y$ from $h$. Now a standard probabilistic argument shows that there exists a sequence $h_1, h_2, \ldots$ of $O(s)$ hash functions such that for every possible outcome $y$, there is at least one $h_i$ that works for $y$. Given such a sequence, we can encode each outcome $y$ as the index $i$ of a hash function $h_i$ that works for $y$, and enough bits of information that allow us to *efficiently* recover $\sigma_y$ given $h_i$. We show that $s+O(1)$ bits suffice for the standard linear-algebraic family of hash functions. The resulting encoding has length $s+\log(s)+O(1)$ and is decodable by circuits of polynomial size. Recall that a family $\mathcal{H}_{\ell,m}$ of functions from ${\{0,1\}}^\ell$ to ${\{0,1\}}^m$ is *universal* if for any two distinct $\sigma_0, \sigma_1 \in {\{0,1\}}^\ell$, the distributions of $h(\sigma_0)$ and $h(\sigma_1)$ for a uniform choice of $h \in \mathcal{H}_{\ell,m}$ are independent and uniform over ${\{0,1\}}^m$. We make use of the specific universal family $\mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$ that consists of all functions of the form $\sigma \mapsto U\sigma+v$, where $U$ is a binary $(m \times \ell)$-matrix, $v$ is a binary column vector of dimension $\ell$, and $\sigma$ is also viewed as a binary column vector of dimension $\ell$ [@CarterW79]. Uniformly sampling from $\mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$ means picking $U$ and $v$ uniformly at random. \[claim:flat-coding-lemma-subclaim\] Let $\ell,m \in {\mathbb{N}}$ and $s \in {\mathbb{R}}$. 1. For every universal family $\mathcal{H}_{\ell,m}$ with $m=\ell-{{\left\lceil{s}\right\rceil}}-2$, and for every $S \subseteq {\{0,1\}}^\ell$ with $|S| \geq 2^{\ell-s}$, $$\Pr[ (\exists \sigma \in S) \, h(\sigma)=0^m \textrm{ and } |h^{-1}(0^m)| \leq 2^{{{\left\lceil{s}\right\rceil}}+3} ] \geq \frac{1}{4},$$ where the probability is over a uniformly random choice of $h \in \mathcal{H}_{\ell,m}$. 2. The sets $h^{-1}(0^m)$ have indexings that are uniformly decodable in time ${{\ensuremath{\operatorname{poly}}}}(\ell,m)$, where $h$ ranges over $\mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$. Assume for now that the claim holds, and let us continue with the proof of the lemma. Fix an input $x$, and let $\ell = \ell(x)$ and $s = s(x)$. Consider the family $\mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$ with $m = \ell - {{\left\lceil{s}\right\rceil}} - 2$. For each outcome $y$ of $R_x$, let $S_y$ consist of the strings $\sigma \in {\{0,1\}}^\ell$ for which $R_x(\sigma) = y$. Since the distribution induced by $R_x$ has max-entropy $s$, a fraction at least $1/2^s$ of the strings in the domain of $R_x$ map to $y$. It follows that $|S_y| \geq 2^{\ell-s}$. A hash function $h \in \mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$ *works* for $y$ if there is some $\sigma \in S_y$ with $h(\sigma) = 0^m$ and $|h^{-1}(0^m)| \leq 2^{{{\left\lceil{s}\right\rceil}}+3}$. By the first part of Claim \[claim:flat-coding-lemma-subclaim\], the probability that a random $h \in \mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$ works for a fixed $y$ is at least $1/4$. If we now pick $3{{\left\lceil{s}\right\rceil}}$ hash functions independently at random, the probability that none of them work for $y$ is at most $(3/4)^{3{{\left\lceil{s}\right\rceil}}} < 1/2^s$. Since there are at most $2^s$ distinct outcomes $y$, a union bound shows that there exists a sequence of hash functions $h_1, h_2, \ldots, h_{3{{\left\lceil{s}\right\rceil}}} \in \mathcal{H}_{\ell,m}^{\mathrm{(lin)}}$ such that for every outcome $y$ of $R_x$ there exists $i_y \in [3{{\left\lceil{s}\right\rceil}}]$ such that $h_{i_y}$ works for $y$. The encoding works as follows. Let $D^{\mathrm{(lin)}}$ denote the uniform decoding algorithm from part 2 of Claim \[claim:flat-coding-lemma-subclaim\] such that $D^{\mathrm{(lin)}}(h,\cdot)$ decodes the set $h^{-1}(0^m)$. For each outcome $y$ of $R_x$, let $j_y \in [2^{{{\left\lceil{s}\right\rceil}}+3}]$ be such that $D^{\mathrm{(lin)}}(h_{i_y},j_y) = \sigma_y \in S_y$. Such a $j_y$ exists since $h_{i_y}$ works for $y$. Let $k_y = 2^{{{\left\lceil{s}\right\rceil}}+3} i_y + j_y$. Given $h_1,h_2,\ldots, h_{3{{\left\lceil{s}\right\rceil}}}$ and $\ell$ and $m$ as auxiliary information, we can decode $\sigma_y$ from $k_y$ by parsing out $i_y$ and $j_y$, extracting $h_{i_y}$ from the auxiliary information, and running $D^{\mathrm{(lin)}}(h_{i_y},j_y)$. This gives an encoding for $R_x$ of length ${{\left\lceil{s}\right\rceil}}+3 + {{\left\lceil{\log(3{{\left\lceil{s}\right\rceil}})}\right\rceil}} = s + \log s + O(1)$ that can be decoded in time ${{\ensuremath{\operatorname{poly}}}}(|x|)$ with the hash functions as auxiliary information. As each hash function can be described using $(\ell+1)m$ bits and there are $3{{\left\lceil{s}\right\rceil}} \leq {{\ensuremath{\operatorname{poly}}}}(|x|)$ many of them, the auxiliary information consists of no more than ${{\ensuremath{\operatorname{poly}}}}(|x|)$ bits. Hard-wiring it yields a decoder circuit of size ${{\ensuremath{\operatorname{poly}}}}(|x|)$. For completeness we argue Claim \[claim:flat-coding-lemma-subclaim\]. For part 1, let $m=\ell-{{\left\lceil{s}\right\rceil}}-2$, and consider the random variables $X \doteq |h^{-1}(0^m) \cap S|$ and $Y \doteq |h^{-1}(0^m)|$. Because of universality we have that $\operatorname{\mathbb{V}}(X) \leq \operatorname{\mathbb{E}}(X) = |S|/2^m$, and by the choice of parameters $|S|/2^m \geq 4$. By Chebyshev’s inequality $$\Pr(X=0) \leq \Pr( |X-\operatorname{\mathbb{E}}(X)| \geq \operatorname{\mathbb{E}}(X) ) \leq \frac{\operatorname{\mathbb{V}}(X)}{(\operatorname{\mathbb{E}}(X))^2} \leq \frac{1}{\operatorname{\mathbb{E}}(X)} \leq \frac{1}{4}.$$ We have that $\operatorname{\mathbb{E}}(Y) = 2^\ell/2^m = 2^{{{\left\lceil{s}\right\rceil}}+2}$. By Markov’s inequality $$\Pr(Y \geq 2^{{{\left\lceil{s}\right\rceil}}+3}) = \Pr(Y \geq 2 \operatorname{\mathbb{E}}(Y)) \leq \frac{1}{2}.$$ A union bound shows that $$\Pr(X=0 \text{ or } Y \geq 2^{{{\left\lceil{s}\right\rceil}}+3}) \leq \frac{1}{4} + \frac{1}{2},$$ from which part 1 follows. For part 2, note that if $|h^{-1}(0^m)|>0$ then $|h^{-1}(0^m)| = 2^{\ell-r}$ where $r$ denotes the rank of $U$. In that case, given $U$ and $v$, we can use Gaussian elimination to find binary column vectors $\hat{\sigma}$ and $\sigma_1, \sigma_2, \ldots, \sigma_{\ell-r}$ such that $U\hat{\sigma}+v=0^m$ and the $\sigma_i$’s form a basis for the kernel of $U$. On input $j \in [2^{\ell-r}]$, the decoder outputs $\hat{\sigma} + \sum_{i=1}^{\ell-r}j_i \sigma_i$, where $\sum_{i=1}^{\ell-r} j_i 2^{i-1}$ is the binary expansion of $j-1$. The image of the decoder is exactly $h^{-1}(0^m)$. As the decoding process runs in time ${{\ensuremath{\operatorname{poly}}}}(\ell,m)$ when given $U$ and $u$, this gives the desired indexing. The proof of the shows a somewhat more general result: For any ensemble $\{R_x\}$ of random variables whose domains consist of strings of length ${{\ensuremath{\operatorname{poly}}}}(|x|)$, and for *any* bound $s(x)$, the set of outcomes of $R_x$ with probability at least $1/2^{s(x)}$ has an encoding of length $s(x) + \log s(x) + O(1)$ that is decodable by a circuit of size ${{\ensuremath{\operatorname{poly}}}}(|x|)$. In the case of flat distributions of entropy $s(x)$ that set contains all possible outcomes. We also point out that a similar construction (with a single hash function) was used in [@paturi-pudlak] to boost the success probability of randomized circuits that decide ${\ensuremath{\mathsf{CircuitSAT}}}$ as a function of the number of input variables.[^9] In combination with the , the yields upper bounds on ${{\ensuremath{\operatorname{KT}}}}$-complexity in the case of distributions $p$ that are samplable by polynomial-size circuits. More precisely, if $y$ is the concatenation of $t$ samples from $p$, we can essentially upper bound the amortized ${{\ensuremath{\operatorname{KT}}}}$-complexity ${{\ensuremath{\operatorname{KT}}}}(y)/t$ by the max-entropy of $p$. On the other hand, Proposition \[prop:complexity-at-least-entropy\] shows that if the samples are picked independently at random, with high probability ${{\ensuremath{\operatorname{KT}}}}(y)/t$ is not much less than the min-entropy of $p$. Thus, in the case of flat distributions, ${{\ensuremath{\operatorname{KT}}}}(y)/t$ is a good *probably-approximately-correct underestimator* for the entropy, a notion formally defined as follows. Let $g : \Omega \to {\mathbb{R}}$ be a function, and $M$ a randomized algorithm that, on input $\omega \in \Omega$, outputs a value $M(\omega) \in {\mathbb{R}}$. We say that $M$ is a *probably-approximately-correct underestimator* (or *pac underestimator*) for $g$ with deviation $\Delta$ if, for every $\omega \in \Omega$, $\left|M(\omega) - g(\omega)\right| \leq \Delta$ holds with probability at least $1/{{\ensuremath{\operatorname{poly}}}}(|\omega|)$, and $M(\omega) < g(\omega)$ otherwise. A *probably-approximately-correct overestimator* (or *pac overestimator*) for $g$ is defined similarly, by reversing the last inequality. Similar to the case of probably-correct under-/overestimators, we can boost the confidence level of a pac under-/overestimator from $1/{{\ensuremath{\operatorname{poly}}}}(|\omega|)$ to $1 - 2^{-{{\ensuremath{\operatorname{poly}}}}(|\omega|)}$ by taking the max/min of ${{\ensuremath{\operatorname{poly}}}}(|\omega|)$ independent runs. More generally, we argue that the amortized ${{\ensuremath{\operatorname{KT}}}}$-complexity of samples yields a good pac underestimator for the entropy when the distribution is *almost* flat, , the difference between the max- and min-entropy is small. As ${{\ensuremath{\operatorname{KT}}}}$ can be evaluated efficiently with oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$, pac underestimating the entropy of such distributions reduces to ${{\ensuremath{\mathrm{MKTP}}}}$. \[cor:flat-KT\] Let $\{p_x\}$ be an ensemble of distributions such that $p_x$ is supported on strings of the same length ${{\ensuremath{\operatorname{poly}}}}(|x|)$. Consider a randomized process that on input $x$ computes ${{\ensuremath{\operatorname{KT}}}}(y)/t$, where $y$ is the concatenation of $t$ independent samples from $p_x$. If $p_x$ is samplable by circuits of polynomial size, then for $t$ a sufficiently large polynomial in $|x|$, ${{\ensuremath{\operatorname{KT}}}}(y)/t$ is a pac underestimator for the entropy of $p_x$ with deviation $\Delta(x)+o(1)$, where $\Delta(x)$ is the difference between the min- and max-entropies of $p_x$. Since the entropy lies between the min- and max-entropies, it suffices to show that ${{\ensuremath{\operatorname{KT}}}}(y)/t$ is at least the min-entropy of $p_x$ with high probability, and is always at most the max-entropy of $p_x$ (both up to $o(1)$ terms) when $t$ is a sufficiently large polynomial. The lower bound follows from Proposition \[prop:complexity-at-least-entropy\]. It remains to establish the upper bound. Let $\{R_{x,\delta}\}$ be the ensemble of random variables witnessing the samplability of $\{p_x\}$ by circuits of polynomial size, and let $s(x)$ denote the max-entropy of $p_x$. The allows us to bound ${{\ensuremath{\operatorname{KT}}}}(y)$ by giving an encoding for random variables whose support contains the $b$-tuples of samples from $p_x$. Let $R'_{x,b}$ denote the $b$-fold Cartesian product of $R_{x,1/b}$. $R'_{x,b}$ induces a distribution that approximates to within a factor of $(1 + 1/b)^b = O(1)$ the distribution of the $b$-fold Cartesian product of $p_x$, which is a distribution of max-entropy $bs(x)$. It follows that the distribution induced by $R'_{x,b}$ has min-entropy at most $bs(x) + O(1)$. Its support is exactly the $b$-tuples of samples from $p_x$. Moreover, the ensemble $\{R'_{x,b}\}$ is computable by circuits of size ${{\ensuremath{\operatorname{poly}}}}(n,b)$. By the there exists an encoding of $R'_{x,b}$ of length $b s(x) + \log b + \log s(x) + O(1)$ that is decodable by circuits of polynomial-size. The then says that there exist constants $c_1$ and $c_2$ so that for all $\alpha > 0$ and all sufficiently large $n$ $$\begin{aligned} {{\ensuremath{\operatorname{KT}}}}(y) &\leq t^{1-\alpha}\cdot \left( {{\left\lceil{t^\alpha}\right\rceil}}\cdot s(x) + \log s(x) + \alpha \log t + O(1) \right) + t^{\alpha c_1} \cdot n^{c_2} \\ &\leq t s(x) + t^{1-\alpha}\cdot (n^{c_0} + c_0 \log n + \alpha \log t + O(1)) + t^{\alpha c_1} \cdot n^{c_2}, \end{aligned}$$ where we use the fact that there exists a constant $c_0$ such that $s(x) \leq n^{c_0}$. A similar calculation as the one following Equation  shows that ${{\ensuremath{\operatorname{KT}}}}(y) \leq ts(x) + t^{1-\alpha_0}n^{c_0+c_2)}$ for $t \geq n^c$ and $n$ sufficiently large, where $\alpha_0 = 1/(1+c_1)$ and $c = 1 + (1+c_1)(c_0+c_2)$. Dividing both sides by $t$ yields the claimed upper bound. Generic Isomorphism Problem {#sec:iso} =========================== In Section \[sec:intro\] we presented a common framework for isomorphism problems and listed some instantiations in Table \[table:iso\]. In this section we state and prove a generalization of Theorem \[thm:GI\] that applies to many problems in this framework, including the ones from Table \[table:iso\]. Generalization {#sec:iso:thm} -------------- The generalized reduction makes use of a complete invariant for the abstract universe $\Omega$. For future reference, we define the notion with respect to a representation for an arbitrary ensemble of sets. Let $\{ \Omega_x \}$ denote an ensemble of sets. A *representation* of the ensemble is a surjective mapping $\omega: {\{0,1\}}^* \to \cup_x \Omega_x$. A *complete invariant* for $\omega$ is a mapping $\nu: {\{0,1\}}^* \to {\{0,1\}}^*$ such that for all strings $x, z_0, z_1$ with $\omega(z_0), \omega(z_1) \in \Omega_x$ $$\omega(z_0) = \omega(z_1) \Leftrightarrow \nu(z_0) = \nu(z_1).$$ $\omega(z)$ denotes the set element represented by the string $z$. The surjective property of a representation guarantees that every set element has at least one string representing it. Note that for the function $\nu$ to represent a *normal form* (rather than just a complete invariant), it would need to be the case that $\omega(\nu(z)) = \omega(z)$. Although this additional property holds for all the instantiations we consider, it is not a requirement. In our setting, all that matters is that $\nu(z)$ only depends on the element $\omega(z)$ that $z$ represents, and is different for different elements.[^10] We are now ready to state the generalization of Theorem \[thm:GI\]. \[thm:iso\] Let ${\ensuremath{\mathrm{Iso}}}$ denote an Isomorphism Problem as in Definition \[def:iso\]. Consider the following conditions: 1. \[cond:uniform-sampling\] *\[action sampler\]* The uniform distribution on $H_x$ is uniformly samplable in polynomial time, and the mapping $(\omega,h) \mapsto h(\omega)$ underlying the action $(\Omega_x,H_x)$ is computable in ${{\ensuremath{\mathsf{ZPP}}}}$. 2. \[cond:normal-form\] *\[complete universe invariant\]* There exists a complete invariant $\nu$ for the representation $\omega$ that is computable in ${{\ensuremath{\mathsf{ZPP}}}}$. 3. \[cond:pcoe\] *\[entropy estimator\]* There exists a probably-approximately-correct overestimator for $(x,\omega) \mapsto \log\left(|H_x|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|\right)$ with deviation $\Delta = 1/4$ that is computable in randomized time ${{\ensuremath{\operatorname{poly}}}}(|x|)$ with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. With these definitions: 1. If conditions \[cond:uniform-sampling\] and \[cond:normal-form\] hold, then ${\ensuremath{\mathrm{Iso}}}\in{{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. 2. If conditions \[cond:uniform-sampling\], \[cond:normal-form\], and \[cond:pcoe\] hold, then ${\ensuremath{\mathrm{Iso}}}\in{{\ensuremath{\mathsf{coRP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. In the case of ${{\ensuremath{\mathrm{GI}}}}$, $\Omega$ denotes the universe of graphs on $n$ vertices (represented as adjacency matrices viewed as strings of length $n^2$), and $H$ the group of permutations on $[n]$ (represented as function tables). All conditions in the statement of Theorem \[thm:iso\] are met. The identity mapping can be used as the complete invariant $\nu$ in condition \[cond:normal-form\], and the probably-correct overestimator for $n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|$ that we argued in Sections \[sec:intro\] and \[sec:GI\] immediately yields the pac overestimator for $\log(n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|)$ required in condition \[cond:pcoe\]. Note that $\log(n!/|{{\ensuremath{\operatorname{Aut}}}}(G)|)$ equals the entropy of the distribution induced by the random variable $R_G$. In general, the quantity $\log(|H_x|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|)$ in condition \[cond:pcoe\] represents the entropy of $\nu(h(\omega))$ when $h \in H_x$ is picked uniformly at random. Let $x$ denote an instance of length $n \doteq |x|$, defining a universe $\Omega$, a group $H$ that acts on $\Omega$, and two elements $\omega_i = \omega_i(x)$ for $i \in {\{0,1\}}$. Both parts (a) and (b) make use of the random variables $R_i$ for $i \in {\{0,1\}}$ where $R_i: H \to {\{0,1\}}^*$ maps $h \in H$ to $\nu(h(\omega_i))$. #### Part (a). We follow the approach from [@adas]. Their argument uses Lemma \[lemma:mktp-inverts-bbox\], which states the existence of a randomized polynomial-time machine $M$ with access to an ${{\ensuremath{\mathrm{MKTP}}}}$ oracle which, given a random sample $y$ from the distribution induced by a circuit $C$, recovers with non-negligible probability of success an input $\sigma$ so that $C(\sigma) = y$. If we can model the $R_i$ as circuits of size ${{\ensuremath{\operatorname{poly}}}}(n)$ that take in an element $h$ from $H$ and output $R_i(h)$, this means that, with non-negligible probability over a random $h_0 \in H$, $M(R_0, R_0(h_0))$ outputs some $h_1$ so that $h_1(\omega_0) = h_0(\omega_0)$. The key observation is that when $\omega_0 \equiv \omega_1$, $R_0$ and $R_1$ induce the same distribution, and therefore, for a random element $h_0$, $M(R_1, R_0(h_0))$ outputs some $h_1$ so that $h_1(\omega_0) = h_0(\omega_0)$ with non-negligible probability probability of success. Thus ${{\ensuremath{\mathrm{Iso}}}}$ can be decided by trying the above a polynomial number of times, declaring $\omega_0 \equiv \omega_1$ if a trial succeeds, and declaring $\omega_0 \not\equiv \omega_1$ otherwise. We do not know how to model the $R_i$ exactly as circuits of size ${{\ensuremath{\operatorname{poly}}}}(n)$, but we can do so approximately. Condition \[cond:uniform-sampling\] implies that we can construct circuits $C_{i,\delta}$ in time ${{\ensuremath{\operatorname{poly}}}}(n/\delta)$ that sample $h(\omega_i)$ within a factor $1+\delta$. Combined with the ${{\ensuremath{\mathsf{ZPP}}}}$-computability of $\nu$ in condition \[cond:normal-form\] this means that we can construct a circuit $C_\nu$ in time ${{\ensuremath{\operatorname{poly}}}}(n)$ such that the composed circuit $C_\nu \circ C_{i,\delta}$ samples $R_i$ within a factor $1+\delta$ from strings $\sigma$ of length ${{\ensuremath{\operatorname{poly}}}}(n/\delta)$. We use the composed circuits in lieu of $R_i$ in the arguments for $M$ above. More precisely, we pick an input $\sigma_0$ for $C_{0,\delta}$ uniformly at random, and compute $\sigma_1 = M(C_{1,\delta},C_{0,\delta}(\sigma_0))$. Success means that $h_1(\omega_0) = h_0(\omega_0)$, where $h_i=C_{i,\delta}(\sigma_i)$. The probability of success for an approximation factor of $1+\delta$ is at least $1/(1+\delta)^2$ times the probability of success in the exact setting, which is $1/{{\ensuremath{\operatorname{poly}}}}(n/\delta)$ in the isomorphic case. Fixing $\delta$ to any positive constant, a single trial runs in time ${{\ensuremath{\operatorname{poly}}}}(n)$, success can be determined in ${{\ensuremath{\mathsf{ZPP}}}}$ (by the second part of condition \[cond:uniform-sampling\]), and the probability of success is at least $1/{{\ensuremath{\operatorname{poly}}}}(n)$ in the isomorphic case. Completing the argument as in the exact setting above, we conclude that ${\ensuremath{\mathrm{Iso}}}\in{{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. #### Part (b). We generalize the argument from Section \[sec:GI\]. Let $s_i \doteq \log\left( |H|/|{{\ensuremath{\operatorname{Aut}}}}(\omega_i)| \right)$ for $i \in {\{0,1\}}$, and let $M$ be the pac overestimator from condition \[cond:pcoe\]. We assume that $M$ has been amplified such that it outputs a good estimate with probability exponentially close to 1. Condition \[cond:uniform-sampling\] and the ${{\ensuremath{\mathsf{ZPP}}}}$-computability of $\nu$ imply that the distribution induced by $R_i$ is uniformly samplable in polynomial time, , for each $i \in {\{0,1\}}$ and $\delta>0$, there is a random variable $R_{i,\delta}$ that samples $R_i$ within a factor $1+\delta$ from length ${{\ensuremath{\operatorname{poly}}}}(|x|/\delta)$, and that is computable in time ${{\ensuremath{\operatorname{poly}}}}(|x|/\delta)$. Let $t \in {\mathbb{N}}$ and $\delta$ be parameters to be determined. On input $x$, the algorithm begins by computing the estimates $\widetilde{s}_i = M(x, \omega_i)$ for $i \in {\{0,1\}}$, and sets $\widetilde{s} \doteq \min(\widetilde{s}_0,\widetilde{s}_1)$ and $\widetilde{\theta} \doteq t(\widetilde{s}+\frac{1}{2})$. The algorithm then samples $r\in{\{0,1\}}^t$ uniformly, and constructs $y = (R_{r_i,\delta}(\sigma_i))_{i=1}^t$, where each $\sigma_i$ is drawn independently and uniformly from ${\{0,1\}}^{{{\ensuremath{\operatorname{poly}}}}(n,1/\delta)}$. If ${{\ensuremath{\operatorname{KT}}}}(y) > \widetilde{\theta}$, the algorithm declares $\omega_0 \not\equiv \omega_1$; otherwise, the algorithm declares $\omega_0 \equiv \omega_1$. [*Nonisomorphic Case.*]{} If $\omega_0 \not\equiv \omega_1$, we need to show ${{\ensuremath{\operatorname{KT}}}}(y) > \widetilde{\theta}$ with high probability. Since $R_{i,\delta}$ samples $R_i$ within a factor of $1+\delta$, and $R_i$ is flat with entropy $s_i$, it follows that $R_{i,\delta}$ has min-entropy at least $s_i-\log(1+\delta)$, and that $y$ is sampled from a distribution with min-entropy at least $$t(1+\min(s_0,s_1)-\log(1+\delta)).$$ Since $M$ is a pac overestimator with deviation $\Delta = 1/4$, $|\widetilde{s}_0 - s_0| \leq 1/4$ and $|\widetilde{s}_1 - s_1| \leq 1/4$ with high probability. When this happens, $\widetilde{s} \leq \min(s_0, s_1) + 1/4$, $$\widetilde{\theta} \leq t(\min(s_0,s_1) + 3/4),$$ and Proposition \[prop:complexity-at-least-entropy\] guarantees that ${{\ensuremath{\operatorname{KT}}}}(y) > \widetilde{\theta}$ except with probability exponentially small in $t$ as long as $\delta$ is a constant such that $1 - \log(1+\delta) > 3/4$. Such a positive constant $\delta$ exists. [*Isomorphic Case.*]{} If $\omega_0 \equiv \omega_1$, we need to show that ${{\ensuremath{\operatorname{KT}}}}(y) \leq \widetilde{\theta}$ always holds for $t$ a sufficiently large polynomial in $n$, and $n$ sufficiently large. Recall that, since $\omega_0 \equiv \omega_1$, $R_0$ and $R_1$ induce the same distribution, so we can view $y$ as the concatenation of $t$ samples from $R_0$. Each $R_0$ is flat, hence has min-entropy equal to its max-entropy, and the ensemble of all $R_0$ (across all inputs $x$) is samplable by (uniform) polynomial-size circuits. The with $\Delta(x) \equiv 0$ then implies that ${{\ensuremath{\operatorname{KT}}}}(y) \leq t(s_0 + o(1))$ holds whenever $t$ is a sufficiently large polynomial in $n$, and $n$ is sufficiently large. In that case, ${{\ensuremath{\operatorname{KT}}}}(y) \leq t(\widetilde{s}+\frac{1}{4}+o(1)) < \widetilde{\theta}$ holds because $s_0 \leq \widetilde{s} + 1/4$ follows from $M$ being a pac overestimator for $s_0$ with deviation $1/4$. \[remark:iso-efficiency\] The notion of efficiency in conditions \[cond:uniform-sampling\], and \[cond:normal-form\] can be relaxed to mean the underlying algorithm is implementable by a family of polynomial-size circuits which is constructible in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. It is important for our argument that the circuits themselves do not have oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$, but it is all right for them to be constructible in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ rather than ${\ensuremath{\mathsf{P}}}$ or ${{\ensuremath{\mathsf{ZPP}}}}$. For example, a sampling procedure that requires knowing the factorization of some number (dependent on the input $x$) is fine because the factorization can be computed in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ [@powerk] and then can be hard-wired into the circuit. In particular, this observation yields an alternate way to show that integer factorization being in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ implies that the discrete log over prime fields is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ [@rudow]. Recall that an instance of the discrete log problem consists of a triple $x=(g,z,p)$, where $g$ and $z$ are integers, and $p$ is a prime, and the goal is to find an integer $y$ such that $g^y \equiv z \bmod p$, or report that no such integer exists. The search version is known to reduce to the decision version in randomized polynomial time, and the above observation shows that the decision version is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. This is because computing the size of the subgroup of ${\mathbb{F}}_p^\times$ generated by $g$ or $z$ reduces to integer factorization, and can thus be computed in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. Construction of Probably-Correct Overestimators {#sec:iso:conditions} ----------------------------------------------- We now discuss some generic methods to satisfy condition \[cond:pcoe\] in Theorem \[thm:iso\], , how to construct a probably-approximately-correct overestimator for the quantity $\log(|H|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|)$ that is computable in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. Here is the generalization of the approach we used in Section \[sec:GI:pcoe-construction\] in the context of ${{\ensuremath{\mathrm{GI}}}}$: 1. Find a list $L$ of elements of $H$ that generates a subgroup $\langle L \rangle$ of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ such that $\langle L \rangle = {{\ensuremath{\operatorname{Aut}}}}(\omega)$ with high probability. 2. Pac underestimate $\log|\langle L \rangle|$ with deviation $1/8$. This yields a pac underestimator for $\log |{{\ensuremath{\operatorname{Aut}}}}(\omega)|$. 3. Pac overestimate $\log |H|$ with deviation $1/8$. 4. Return the result of step 3 minus the result of step 2. This gives a pac overestimator for $\log(|H|/|{{\ensuremath{\operatorname{Aut}}}}(\omega)|)$ with deviation $1/4$. Although in the setting of ${{\ensuremath{\mathrm{GI}}}}$ we used the oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ only in step 1, we could use it to facilitate steps 2 and 3 as well. The first step for ${{\ensuremath{\mathrm{GI}}}}$ follows from the known search-to-decision reduction. It relies on the fact that *Colored Graph Isomorphism* reduces to ${{\ensuremath{\mathrm{GI}}}}$, where Colored Graph Isomorphism allows one to assign colors to vertices with the understanding that the isomorphism needs to preserve the colors. For all of the isomorphism problems in Table \[table:iso\], finding a set of generators for the automorphism group reduces to a natural colored version of the Isomorphism Problem, but it is not clear whether the colored version always reduces to the regular version. The latter reduction is known for Linear Code Equivalence, but remains open for problems like Permutation Group Conjugacy and Matrix Subspace Conjugacy. However, there is a different, *generic* way to achieve step 1 above, namely based on Lemma \[lemma:mktp-inverts-bbox\], , the power of ${{\ensuremath{\mathrm{MKTP}}}}$ to efficiently invert on average any efficiently computable function. \[lemma:sample-subgroups\] Let ${\ensuremath{\mathrm{Iso}}}$ denote an Isomorphism Problem as in Definition \[def:iso\] that satisfies conditions \[cond:uniform-sampling\] and \[cond:normal-form\] of Theorem \[thm:iso\], and such that products and inverses in $H_x$ are computable in ${{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. There exists a randomized polynomial-time algorithm using oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$ with the following behavior: On input any instance $x$, and any $\omega\in\Omega_x$, the algorithm outputs a list of generators for a subgroup $\Gamma$ of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ such that $\Gamma = {{\ensuremath{\operatorname{Aut}}}}(\omega)$ with probability $1 - 2^{-|x|}$. Consider an instance $x$ of length $n \doteq |x|$, and $\omega \in \Omega_x$. We first argue that the uniform distribution on ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ is uniformly samplable in polynomial time with oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$. Let $R_\omega$ denote the random variable that maps $h \in H$ to $\nu(h(\omega))$. As in the proof of Part 1 of Theorem \[thm:iso\], we can sample $h$ from $H$ uniformly (to within a small constant factor) and use Lemma \[lemma:mktp-inverts-bbox\] to obtain some $h' \in H$ such that $h'(\omega)=h(\omega)$. In that case, $h^{-1}h'$ is an automorphism of $\omega$. The key observation is the following: if $h$ were sampled perfectly uniformly then, conditioned on success, the distribution of $h^{-1}h'$ is *uniform* over ${{\ensuremath{\operatorname{Aut}}}}(\omega)$. Instead, $h$ is sampled uniformly to within a factor $1+\delta$; in that case $h^{-1}h'$ is uniform on ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ to within a factor $1+\delta$ and, as argued in the proof of Theorem \[thm:iso\], the probability of success is $1/{{\ensuremath{\operatorname{poly}}}}(n/\delta)$. We run the process many times and retain the automorphism $h^{-1}h'$ from the first successful run (if any); ${{\ensuremath{\operatorname{poly}}}}(n/\delta)$ runs suffice to obtain, with probability $1-2^{-2n}$, an automorphism that is within a factor $1+\delta$ from uniform over ${{\ensuremath{\operatorname{Aut}}}}(\omega)$. By the computability parts of conditions \[cond:uniform-sampling\] and \[cond:normal-form\], and by the condition that products and inverses in $H$ can be computed in ${{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$, each trial runs in time ${{\ensuremath{\operatorname{poly}}}}(n/\delta)$. Success can be determined in ${{\ensuremath{\mathsf{ZPP}}}}$ as the group action is computable in ${{\ensuremath{\mathsf{ZPP}}}}$. It follows that the uniform distribution on ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ is uniformly samplable in polynomial time with oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$. Finally, we argue that a small number of independent samples $h_1, h_2, \ldots, h_k$ for some constant $\delta>0$ suffice to ensure that they generate all of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ with very high probability. Denote by $\Gamma_i$ the subgroup of $H_x$ generated by $h_1,\ldots,h_i$. Note that $\Gamma_i$ always is a subgroup of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$. For $i < k$, if $\Gamma_i$ is not all of ${{\ensuremath{\operatorname{Aut}}}}(\omega)$, then $|\Gamma_i| \leq |{{\ensuremath{\operatorname{Aut}}}}(\omega)|/2$. Thus, with probability at least $\frac{1}{2} \cdot \frac{1}{1+\delta}$, $h_{i+1} \not\in \Gamma_i$, in which case $|\Gamma_{i+1}| \geq 2 |\Gamma_i|$. For any constant $\delta>0$, if follows that $k \geq \Theta(n + \log|{{\ensuremath{\operatorname{Aut}}}}(\omega)|) = O({{\ensuremath{\operatorname{poly}}}}(n))$ suffices to guarantee that $\Gamma_k = {{\ensuremath{\operatorname{Aut}}}}(\omega)$ with probability at least $1-2^{-2n}$. The lemma follows. The second step for ${{\ensuremath{\mathrm{GI}}}}$ followed from the ability to efficiently compute the order of permutation groups exactly. Efficient exact algorithms (possibly with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$) are known for larger classes of groups, including most matrix groups over finite fields, but not for all.[^11] We show how to *generically* pac underestimate $\log|\langle L \rangle|$ with small deviation (step 2), namely under the prior conditions that only involve $H$, and the additional condition of a ${{\ensuremath{\mathsf{ZPP}}}}$-computable complete invariant $\zeta$ for $H$. The construction hinges on the and viewing $\log|\langle L \rangle|$ as the entropy of the uniform distribution $p_L$ on $\langle L \rangle$. 1. Provided that $p_L$ is samplable by circuits of polynomial size, the corollary allows us to pac underestimate $\log|\langle L \rangle|$ as ${{\ensuremath{\operatorname{KT}}}}(y)/t$, where $y$ is the concatenation of $t$ independent samples from $p_L$. 2. If we are able to uniformly sample $\{p_L\}$ *exactly* in polynomial time (possibly with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$), then we can evaluate the estimator ${{\ensuremath{\operatorname{KT}}}}(y)/t$ in polynomial time with access to ${{\ensuremath{\mathrm{MKTP}}}}$. This is because the oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ lets us evaluate ${{\ensuremath{\operatorname{KT}}}}$ in polynomial time. Thus, if we were able to uniformly sample $\{p_L\}$ *exactly* in polynomial time, we’d be done. We do not know how to do that, but we can do it *approximately*, which we argue is sufficient. The need for a ${{\ensuremath{\mathsf{ZPP}}}}$-computable complete invariant comes in when representing the abstract group elements as strings. In order to formally state the requirement, we make the underlying representation of group elements explicit; we denote it by $\eta$. \[lemma:pacue-subgroup-order\] Let $\{ H_x \}$ be an ensemble of groups. Suppose that the ensemble has a representation $\eta$ such that the uniform distribution on $H_x$ is uniformly samplable in polynomial-time, products and inverses in $H_x$ are computable in ${{\ensuremath{\mathsf{ZPP}}}}$, and there exists a ${{\ensuremath{\mathsf{ZPP}}}}$-computable complete invariant for $\eta$. Then for any list $L$ of elements of $H_x$, the logarithm of the order of the group generated by $L$, , $\log|\langle L \rangle|$, can be pac underestimated with any constant deviation $\Delta>0$ in randomized time ${{\ensuremath{\operatorname{poly}}}}(|x|, |L|)$ with oracle access to ${{\ensuremath{\mathrm{MKTP}}}}$. Let $\zeta$ be the ${{\ensuremath{\mathsf{ZPP}}}}$-computable complete invariant for $\eta$. For each list $L$ of elements of $H_x$, let $p_L$ denote the distribution of $\zeta(h)$ when $h$ is picked uniformly at random from $\langle L \rangle$. Note that $p_L$ is flat with entropy $s = \log|\langle L \rangle|$. \[claim:pacue-subgroup-order:subclaim\] The ensemble of distributions $\{p_L\}$ is uniformly samplable in polynomial time. For every constant $\delta>0$, the claim yields a family of random variables $\{R_{L,\delta}\}$ computable uniformly in polynomial time such that $R_{L,\delta}$ induces a distribution $p_{L,\delta}$ that approximates $p_L$ to within a factor $1+\delta$. Note that the min-entropy of $p_{L,\delta}$ is at least $s-\log(1+\delta)$, and the max-entropy of $p_{L,\delta}$ at most $s+\log(1+\delta)$, thus their difference is no more than $2\log(1+\delta)$. Let $M_\delta(L)$ denote ${{\ensuremath{\operatorname{KT}}}}(y)/t$, where $y$ is the concatenation of $t$ independent samples from $p_{L,\delta}$. 1. The guarantees that for any sufficiently large polynomial $t$, $M_\delta$ is a pac underestimator for the entropy of $p_{L,\delta}$ with deviation $2\log(1+\delta) + o(1)$, and thus a pac underestimator for $s = \log|\langle L \rangle|$ with deviation $3\log(1+\delta)+o(1)$. 2. For any polynomial $t$, we can compute $M_\delta$ in polynomial time with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. This is because $R_{L,\delta}$ enables us to generate $y$ in polynomial time. We then use the oracle for ${{\ensuremath{\mathrm{MKTP}}}}$ to compute ${{\ensuremath{\operatorname{KT}}}}(y)$ exactly, and divide by $t$. Thus, $M_\delta$ meets all the requirements for our estimator as long as $3\log(1+\delta) < \Delta$, which holds for some positive constant $\delta$. This completes the proof of Lemma \[lemma:pacue-subgroup-order\] modulo the proof of the claim. The proof of Claim \[claim:pacue-subgroup-order:subclaim\] relies on the notion of *Erdős–Rényi generators*. A list of generators $L = (h_1,\ldots,h_k)$ is said to be Erdős–Rényi with factor $1+\delta$ if a random subproduct of $L$ approximates the uniform distribution on $\langle L \rangle$ within a factor $1+\delta$, where a random subproduct is obtained by picking $r_i \in {\{0,1\}}$ for each $i \in [k]$ uniformly at random, and outputting $h_1^{r_1}h_2^{r_2}\cdots h_k^{r_k}$. By definition, if $L$ happens to be Erdős–Rényi with factor $1+\delta$, then $p_L$ can be sampled to within a factor $1+\delta$ with fewer than $|L|$ products in $H_x$. Erdős and Rényi [@erdos.renyi] showed that, for any finite group $\Gamma$, a list of ${{\ensuremath{\operatorname{poly}}}}(\log|\Gamma|,\log(1/\delta))$ random elements of $\Gamma$ form an Erdős–Rényi list of generators with factor $1+\delta$. For $\Gamma = \langle L \rangle$, this gives a list $L'$ for which we can sample $p_{L'} = p_L$. By hard-wiring the list $L'$ into the sampler for $p_{L'}$, it follows that $p_L$ is samplable by circuits of size ${{\ensuremath{\operatorname{poly}}}}(\log|\langle L\rangle|, \log(1/\delta)) \leq {{\ensuremath{\operatorname{poly}}}}(|L|/\delta)$. As for *uniformly* sampling $\{p_L\}$ in polynomial time, [@Bab1991 Theorem 1.1] gives a randomized algorithm that generates out of $L$ a list $L'$ of elements from $\langle L \rangle$ that, with probability $1-\varepsilon$, are Erdős–Rényi with factor $1+\delta$. The algorithm runs in time ${{\ensuremath{\operatorname{poly}}}}(|x|,|L|,\log(1/\delta),\log(1/\varepsilon))$ assuming products and inverses in $H_x$ can be computed in ${{\ensuremath{\mathsf{ZPP}}}}$. For $\varepsilon = \delta/|\langle L \rangle|$, the overall distribution of a random subproduct of $L'$ is within a factor $1+2\delta$ from $p_L$, and can be generated in time ${{\ensuremath{\operatorname{poly}}}}(|x|,|L|,\log(1/\delta)) \leq {{\ensuremath{\operatorname{poly}}}}(|x|,|L|,1/\delta)$. As $\delta$ can be an arbitrary positive constant, it follows that $p_L$ is uniformly samplable in polynomial time. Following the four steps listed at the beginning of this section, we can replace condition \[cond:pcoe\] in Theorem \[thm:iso\] by the conditions of Lemma \[lemma:sample-subgroups\] (for step 1), those of Lemma \[lemma:pacue-subgroup-order\] (for step 2), and the existence of an estimator for the size $|H|$ of the sample space as stated in step 3. This gives the following result: \[thm:iso-specific\] Let ${\ensuremath{\mathrm{Iso}}}$ denote an Isomorphism Problem as in Definition \[def:iso\]. Suppose that the ensemble $\{ H_x \}$ has a representation $\eta$ such that conditions \[cond:uniform-sampling\] and \[cond:normal-form\] of Theorem \[thm:iso\] hold as well as the following additional conditions: 4. \[cond:basic-operations\] *\[group operations\]* Products and inverses in $H_x$ are computable in ${{\ensuremath{\mathsf{ZPP}}}}$. 5. \[cond:cardinality\] *\[sample space estimator\]* The map $x \mapsto |H_x|$ has a pac overestimator with deviation $\Delta = 1/8$ computable in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. 6. \[cond:group-invariant\] *\[complete group invariant\]* There exists a complete invariant $\zeta$ for the representation $\eta$ that is computable in ${{\ensuremath{\mathsf{ZPP}}}}$. Then ${\ensuremath{\mathrm{Iso}}} \in {{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. As was the case for Theorem \[thm:iso\], the conditions of Theorem \[thm:iso-specific\] can be satisfied in a straightforward way for ${{\ensuremath{\mathrm{GI}}}}$. The representation $\eta$ of the symmetric groups $S_n$ meets all the requirements that only involve the underlying group: uniform samplability as in the first part of condition \[cond:uniform-sampling\], efficient group operations as in condition \[cond:basic-operations\], the sample space size $|H|=|S_n|=n!$ can be computed efficiently (condition \[cond:cardinality\]), and the identity mapping can be used as the complete group invariant $\zeta$ (condition \[cond:group-invariant\]). The efficiency of the action (the second part of condition \[cond:uniform-sampling\]) and condition \[cond:normal-form\] about a complete universe invariant are also met in the same way as before. We point out that Claim \[claim:pacue-subgroup-order:subclaim\] can be used to show that the uniform distribution on $H_x$ is uniformly samplable in polynomial time (the first part of condition \[cond:uniform-sampling\]), provided a set of generators for $H_x$ can be computed in ${{\ensuremath{\mathsf{ZPP}}}}$. This constitutes another use of [@Bab1991 Theorem 1.1]. On the other hand, the use of [@Bab1991 Theorem 1.1] in the proof of Theorem \[thm:iso-specific\] can be eliminated. Referring to parts ($\alpha$) and ($\beta$) in the intuition and proof of Lemma \[lemma:pacue-subgroup-order\], we note the following: 1. The first part of the proof of Claim \[claim:pacue-subgroup-order:subclaim\] relies on [@erdos.renyi] but not on [@Bab1991 Theorem 1.1]. It shows that $p_L$ is samplable by polynomial-size circuits, which is sufficient for the to apply and show that $M_\delta(L)={{\ensuremath{\operatorname{KT}}}}(y)/t$ is a pac underestimator for $\log|\langle L \rangle|$ with deviation $3\log(1+\delta)+o(1)$, where $y$ is the concatenation of $t$ independent samples from $p_{L,\delta}$ for a sufficiently large polynomial $t$. 2. In the special case where $\langle L \rangle = {{\ensuremath{\operatorname{Aut}}}}(\omega)$, the first part of the proof of Lemma \[lemma:sample-subgroups\] shows that, for any constant $\delta>0$, $p_{L,\delta}$ is *uniformly* samplable in polynomial time with access to an oracle for ${{\ensuremath{\mathrm{MKTP}}}}$. Once we have generated $y$ with the help of ${{\ensuremath{\mathrm{MKTP}}}}$, we use ${{\ensuremath{\mathrm{MKTP}}}}$ once more to evaluate ${{\ensuremath{\operatorname{KT}}}}(y)$ and output $M_\delta(L)={{\ensuremath{\operatorname{KT}}}}(y)/t$. This way, for any constant $\delta>0$ we obtain a pac underestimator $M_\delta$ for $\log|{{\ensuremath{\operatorname{Aut}}}}(\omega)|$ with deviation $3\log(1+\delta)+o(1)$ that is computable in polynomial time with access to ${{\ensuremath{\mathrm{MKTP}}}}$. This alternate construction replaces steps 1 and 2 in the outline from the beginning of this section. The resulting alternate proof of Theorem \[thm:iso-specific\] is more elementary (as it does not rely on [@Bab1991 Theorem 1.1]) but does not entirely follow the approach we used for ${{\ensuremath{\mathrm{GI}}}}$ of first finding a list $L$ of elements that likely generates ${{\ensuremath{\operatorname{Aut}}}}(\omega)$ (and never generates more) and then determining the size of the subgroup generated by $L$. \[remark:iso-specific-efficiency\] Remark \[remark:iso-efficiency\] on relaxing the efficiency requirement in conditions \[cond:uniform-sampling\] and \[cond:normal-form\] of Theorem \[thm:iso\] extends similarly to Theorem \[thm:iso-specific\]. For Theorem \[thm:iso-specific\], it suffices that all the computations mentioned in conditions \[cond:uniform-sampling\], \[cond:normal-form\], \[cond:basic-operations\], and \[cond:group-invariant\] be do-able by ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$-constructible ordinary circuits. Instantiations of the Isomorphism Problem {#sec:iso:corollaries} ========================================= In this section we argue that Theorem \[thm:iso-specific\] applies to the instantiations of the Isomorphism Problem listed in Table \[table:iso\] (other than ${{\ensuremath{\mathrm{GI}}}}$, which we covered in Section \[sec:GI\]). We describe each problem, provide some background, and show that the conditions of Theorem \[thm:iso-specific\] hold, thus proving that the problem is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. #### Linear code equivalence. A *linear code* over the finite field ${\mathbb{F}}_q$ is a $d$-dimensional linear subspace of ${\mathbb{F}}_q^n$ for some $n$. Two such codes are (permutationally) *equivalent* if there is a permutation of the $n$ coordinates that makes them equal as subspaces. *Linear Code Equivalence* is the problem of deciding whether two linear codes are equivalent, where the codes are specified as the row-span of a $d \times n$ matrix (of rank $d$), called a *generator matrix*. Note that two different inputs may represent the same code. There exists a mapping reduction from ${{\ensuremath{\mathrm{GI}}}}$ to Linear Code Equivalence over any field [@pet.roth; @grochow.lie]; Linear Code Equivalence is generally thought to be harder than ${{\ensuremath{\mathrm{GI}}}}$. In order to cast Code Equivalence in our framework, we consider the family of actions $(S_n,\Omega_{n,d,q})$ where $\Omega_{n,d,q}$ denotes the linear codes of length $n$ and dimension $d$ over ${\mathbb{F}}_q$, and $S_n$ acts by permuting the coordinates. To apply Theorem \[thm:iso-specific\], as the underlying group is $S_n$, we only need to check the efficiency of the action (second part of condition \[cond:uniform-sampling\]) and the complete universe invariant (condition \[cond:normal-form\]). The former holds because the action only involves swapping columns in the generator matrix. For condition \[cond:normal-form\] we can define $\nu(z)$ to be the reduced row echelon form of $z$. This choice works because two generator matrices define the same code iff they have the same reduced row echelon form, and it can be computed in polynomial time. \[cor:code\] Linear Code Equivalence is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. #### Permutation Group Conjugacy. Two permutation groups $\Gamma_0, \Gamma_1 \leq S_n$ are *conjugate* (or permutationally isomorphic) if there exists a permutation $\pi \in S_n$ such that $\Gamma_1 = \pi \Gamma_0 \pi^{-1}$; such a $\pi$ is called a conjugacy. The *Permutation Group Conjugacy* problem is to decide whether two subgroups of $S_n$ are conjugate, where the subgroups are specified by a list of generators. The problem is known to be in ${{\ensuremath{\mathsf{NP}}}}\cap {\ensuremath{\mathsf{coAM}}}$, and is at least as hard as Linear Code Equivalence. Currently the best known algorithm runs in time $2^{O(n)} {{\ensuremath{\operatorname{poly}}}}(|\Gamma_1|)$ [@BCQ]—that is, the runtime depends not only on the input size (which is polynomially related to $n$), but also on the size of the groups generated by the input permutations, which can be exponentially larger. Casting Permutation Group Conjugacy in the framework is similar to before: $S_n$ acts on the subgroup by conjugacy. The action is computable in polynomial time (second part of condition \[cond:uniform-sampling\]) as it only involves inverting and composing permutations. It remains to check condition \[cond:normal-form\]. Note that there are many different lists that generate the same subgroup. We make use of the normal form provided by the following lemma. \[lemma:permutation-group-normal-form\] There is a ${{\ensuremath{\operatorname{poly}}}}(n)$-time algorithm $\nu$ that takes as input a list $L$ of elements of $S_n$, and outputs a list of generators for the subgroup generated by the elements in $L$ such that for any two lists $L_0, L_1$ of elements of $S_n$ that generate the same subgroup, $\nu(L_0) = \nu(L_1)$. The normal form from Lemma \[lemma:permutation-group-normal-form\] was known to some experts (Babai, personal communication); for completeness we provide a proof in the Appendix. By Theorem \[thm:iso-specific\] we conclude: \[cor:perm\] Permutation Group Conjugacy is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. #### Matrix Subspace Conjugacy. A *linear matrix space* over ${\mathbb{F}}_q$ is a $d$-dimensional linear subspace of $n \times n$ matrices. Two such spaces $V_0$ and $V_1$ are *conjugate* if there is an invertible $n \times n$ matrix $X$ such that $V_1 = X V_0 X^{-1} \doteq \{ X \cdot M \cdot X^{-1} : M \in V_0\}$, where “$\cdot$” represents matrix multiplication. *Matrix Subspace Conjugacy* is the problem of deciding whether two linear matrix spaces are conjugate, where the spaces are specified as the linear span of $d$ linearly independent $n \times n$ matrices. There exist mapping reductions from ${{\ensuremath{\mathrm{GI}}}}$ and Linear Code Equivalence to Matrix Subspace Conjugacy [@grochow.lie]; Matrix Subspace Conjugacy is generally thought to be harder than Linear Code Equivalence. In order to cast Matrix Subspace Conjugacy in our framework, we consider the family of actions $({{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q),\Omega_{n,d,q})$ where ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ denotes the $n$-by-$n$ general linear group over ${\mathbb{F}}_q$ (consisting of all invertible $n$-by-$n$ matrices over ${\mathbb{F}}_q$ with multiplication as the group operation), $\Omega_{n,d,q}$ represents the set of $d$-dimensional subspaces of ${\mathbb{F}}_q^{n \times n}$, and the action is by conjugation. As was the case with Linear Code Equivalence, two inputs may represent the same linear matrix space, and we use the reduced row echelon form of $\omega$ when viewed as a matrix in ${\mathbb{F}}_q^{d \times n^2}$ as the complete universe invariant. This satisfies condition \[cond:normal-form\] of Theorem \[thm:iso-specific\]. The action is computable in polynomial time (second part of condition \[cond:uniform-sampling\]) as it only involves inverting and multiplying matrices in ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$. The remaining conditions only depend on the underlying group, which is different from before, namely ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ instead of $S_n$. Products and inverses in ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ can be computed in polynomial time (condition \[cond:basic-operations\]), and the identity mapping serves as the complete group invariant (condition \[cond:group-invariant\]). Thus, only the uniform sampler for ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ (first part of condition \[cond:uniform-sampling\]) and the pac overestimator for $|{{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)|$ (condition \[cond:cardinality\]) remain to be argued. The standard way of constructing the elements of ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ consists of $n$ steps, where the $i$-th step picks the $i$-th row as any row vector that is linearly independent of the $(i-1)$ prior ones. The number of choices in the $i$-th step is $q^n-q^{i-1}$. Thus, $|{{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)| = \prod_{i=1}^n (q^n-q^{i-1})$ which can be computed in time ${{\ensuremath{\operatorname{poly}}}}(|x|)$ (condition \[cond:cardinality\]). It also follows that the probability that a random $(n \times n)$-matrix over ${\mathbb{F}}_q$ is in ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ is at least some positive constant (independent of $n$ and $q$), which implies that $\{H_x\}$ can be uniformly sampled in time ${{\ensuremath{\operatorname{poly}}}}(|x|)$, satisfying the first part of condition \[cond:uniform-sampling\]. Matrix Subspace Conjugacy is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. Before closing, we note that there is an equivalent of the Lehmer code for ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$. We do not need it for our results, but it may be of interest in other contexts. In general, Lehmer’s approach works for indexing objects that consist of multiple components where the set of possible values for the $i$-th component may depend on the values of the prior components, but the *number* of possible values for the $i$-th component is independent of the values of the prior components. An efficiently decodable indexing follows provided one can efficiently index the possible values for the $i$-th component given the values of the prior components. The latter is possible for ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$. We include a proof for completeness. \[prop:pgl-coding\] For each $n$ and prime power $q$, ${{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$ has an indexing that is uniformly decodable in time ${{\ensuremath{\operatorname{poly}}}}(n,\log(q))$. Consider the above process. In the $i$-th step, we need to index the complement of the subspace spanned by the $i-1$ row vectors picked thus far, which are linearly independent. This can be done by extending those $i-1$ row vectors by $n-i+1$ new row vectors to a full basis, and considering all $q^{i-1}$ linear combinations of the $i-1$ row vectors already picked, and all $(q^{n-i+1}-1)$ *non-zero* linear combinations of the other basis vectors, and outputting the sum of the two components. More precisely, on input $k \in [q^n-q^{i-1}]$, write $k-1$ as $k_0 + k_1 q^{i-1}$ where $k_0$ and $k_1$ are nonnegative integers with $k_0 < q^{i-1}$, and output $v_0+v_1$ where $v_0$ is the combination of the $i-1$ row vectors already picked with coefficients given by the binary expansion of $k_0$, and $v_1$ is linear combination of the other basis vectors with coefficients given by the binary expansion of $k_1+1$. Using Gaussian elimination to construct the other basis vectors, the process runs in time ${{\ensuremath{\operatorname{poly}}}}(n,\log(q))$. Future Directions {#sec:conclusion} ================= We end with a few directions for further research. What about Minimum Circuit Size? {#sec:MCSP} -------------------------------- We suspect that our techniques also apply to ${{\ensuremath{\mathrm{MCSP}}}}$ in place of ${{\ensuremath{\mathrm{MKTP}}}}$, but we have been unsuccessful in extending them to ${{\ensuremath{\mathrm{MCSP}}}}$ so far. To show our result for the complexity measure $\mu={{\ensuremath{\operatorname{KT}}}}$, we showed the following property for polynomial-time samplable flat distributions $R$: There exists an efficiently computable bound $\theta(s,t)$ and a polynomial $t$ such that if $y$ is the concatenation of $t$ independent samples from $R$, then $$\begin{aligned} \mu(y) & > & \theta(s,t) \textrm{ holds with high probability if $R$ has entropy $s+1$, and} \label{eq:no} \\ \mu(y) & \leq & \theta(s,t) \textrm{ always holds if $R$ has entropy $s$.} \label{eq:yes}\end{aligned}$$ We set $\theta(s,t)$ slightly below $\kappa(s+1,t)$ where $\kappa(s,t) \doteq st$. followed from a counting argument, and by showing that $$\label{eq:mktp} \mu(y) \leq \kappa(s,t) \cdot \left(1+\frac{n^c}{t^{\alpha}}\right)$$ always holds for some positive constants $c$ and $\alpha$. We concluded by observing that for a sufficiently large polynomial $t$ the right-hand side of is significantly below $\kappa(s+1,t)$. Mimicking the approach with $\mu$ denoting circuit complexity, we set $$\kappa(s,t) = \frac{st}{\log(st)}\cdot\left(1 + (2-o(1))\cdot \frac{\log\log(st)}{\log(st)}\right).$$ Then follows from [@Yam2011]. As for , the best counterpart to we know of (see, e.g., [@FM2005]) is $$\mu(y) \leq \frac{st}{\log(st)}\cdot\left(1 + (3+o(1))\cdot \frac{\log\log(st)}{\log(st)} \right).$$ However, in order to make the right-hand side of smaller than $\kappa(s+1,t)$, $t$ needs to be exponential in $s$. One possible way around the issue is to boost the entropy gap between the two cases. This would not only show that all our results for ${{\ensuremath{\mathrm{MKTP}}}}$ apply to ${{\ensuremath{\mathrm{MCSP}}}}$ as well, but could also form the basis for reductions between different versions of ${{\ensuremath{\mathrm{MCSP}}}}$ (defined in terms of different circuit models, or in terms of different size parameters), and to clarify the relationship between ${{\ensuremath{\mathrm{MKTP}}}}$ and ${{\ensuremath{\mathrm{MCSP}}}}$. Until now, all of these problems have been viewed as morally equivalent to each other, although no efficient reduction is known between *any* two of these, in either direction. Given the central role that ${{\ensuremath{\mathrm{MCSP}}}}$ occupies, it would be desirable to have a theorem that indicates that ${{\ensuremath{\mathrm{MCSP}}}}$ is fairly robust to minor changes to its definition. Currently, this is lacking. On a related point, it would be good to know how the complexity of ${{\ensuremath{\mathrm{MKTP}}}}$ compares with the complexity of the ${{\ensuremath{\operatorname{KT}}}}$-random strings: ${{R_{\rm KT}}}= \{x : {{\ensuremath{\operatorname{KT}}}}(x) \geq |x|\}$. Until now, all prior reductions from natural problems to ${{\ensuremath{\mathrm{MCSP}}}}$ or ${{\ensuremath{\mathrm{MKTP}}}}$ carried over to ${{R_{\rm KT}}}$—but this would seem to require even stronger gap amplification theorems. The relationship between ${{\ensuremath{\mathrm{MKTP}}}}$ and ${{R_{\rm KT}}}$ is analogous to the relationship between ${{\ensuremath{\mathrm{MCSP}}}}$ and the special case of ${{\ensuremath{\mathrm{MCSP}}}}$ that is denoted ${{\ensuremath{\mathrm{MCSP}}}}'$ in [@murray.williams]: ${{\ensuremath{\mathrm{MCSP}}}}'$ consists of truth tables $f$ of $m$-ary Boolean functions that have circuits of size at most $2^{m/2}$. Statistical Zero Knowledge {#sec:other} -------------------------- Allender and Das [@adas] generalized their result that ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ to ${{\ensuremath{\mathsf{SZK}}}}\subseteq {{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ by applying their approach to a known ${{\ensuremath{\mathsf{SZK}}}}$-complete problem. Our proof that ${{\ensuremath{\mathrm{GI}}}}\in {{\ensuremath{\mathsf{coRP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ similarly generalizes to ${{\ensuremath{\mathsf{SZK}}}}\subseteq {{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. We use the ${{\ensuremath{\mathsf{SZK}}}}$-complete problem known as Entropy Difference: Given two circuits $C_0$ and $C_1$ that induce distributions whose entropy is at least one apart, decide which of the two has the higher entropy [@GoldreichV99]. By combining the Flattening Lemma [@GoldreichV99] with the , one can show that for any distribution of entropy $s$ sampled by a circuit $C$, the concatenation of $t$ random samples from $C$ has, with high probability, ${{\ensuremath{\operatorname{KT}}}}$ complexity between $ts - t^{1-\alpha_0}\cdot{{\ensuremath{\operatorname{poly}}}}(|C|)$ and $ts + t^{1-\alpha_0}\cdot{{\ensuremath{\operatorname{poly}}}}(|C|)$ for some positive constant $\alpha_0$. Along the lines of Remark \[remark:GI-BPP\], this allows us to determine which of $C_0$ or $C_1$ has the higher entropy in ${{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. A natural next question is whether this can be strengthened to show ${{\ensuremath{\mathsf{SZK}}}}\subseteq {{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. For this it suffices to prove that ${{\ensuremath{\mathsf{SZK}}}}$ is in ${{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ or in ${{\ensuremath{\mathsf{coRP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ as ${{\ensuremath{\mathsf{SZK}}}}$ is closed under complementation [@okamoto]. In fact, the above approach shows that the following variant is in ${{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$: Given a circuit $C$ and a threshold $\theta$ with the promise that $C$ induces a flat distribution of entropy either at least $\theta+1$ or else at most $\theta-1$, decide whether the former is the case. This is the problem Entropy Approximation [@GSV1999a] restricted to flat distributions. The general version is known to be complete for ${{\ensuremath{\mathsf{SZK}}}}$ under oracle reductions [@GSV1999a Lemma 5.1], and therefore is in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ if and only if all of ${{\ensuremath{\mathsf{SZK}}}}$ is. Thus, showing that Entropy Approximation is in ${{\ensuremath{\mathsf{RP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ is tantamount to reducing the two-sided error in the known result that ${{\ensuremath{\mathsf{SZK}}}}\subseteq {{\ensuremath{\mathsf{BPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$ to zero-sided error. This suggests that the difficulty lies in handling non-flat distributions. For example, it may be the case that the distribution sampled by $C$ is supported on every string, but the entropy $s$ is relatively small. In that case, there is no nontrivial worst-case bound on the ${{\ensuremath{\operatorname{KT}}}}$ complexity of samples from $C$; with positive probability, $t$ samples from $C$ may have ${{\ensuremath{\operatorname{KT}}}}$-complexity close to $t$ times the length of each sample, far above $t(s+1)$. Trying to go beyond ${{\ensuremath{\mathsf{SZK}}}}$, recall that except for the possible use of the ${{\ensuremath{\mathrm{MKTP}}}}$ oracle in the construction of the probably-correct overestimator from condition \[cond:pcoe\] in Theorem \[thm:iso\] (or as discussed in Remark \[remark:iso-efficiency\]), the reduction in Theorem \[thm:iso\] makes only one query to the oracle. It was observed in [@hirahara.watanabe] that the reduction also works for any relativized ${{\ensuremath{\operatorname{KT}}}}$ problem ${{\ensuremath{\mathrm{MKTP}}}}^A$ (where the universal machine for ${{\ensuremath{\operatorname{KT}}}}$ complexity has access to oracle $A$). More significantly, [@hirahara.watanabe] shows that any problem that is accepted with negligible error probability by a probabilistic reduction that makes only one query, relative to *every* set ${{\ensuremath{\mathrm{MKTP}}}}^A$, must lie in ${\ensuremath{\mathsf{AM}}} \cap {\ensuremath{\mathsf{coAM}}}$. Thus, without significant modification, our techniques cannot be used in order to reduce any class larger than ${\ensuremath{\mathsf{AM}}} \cap {\ensuremath{\mathsf{coAM}}}$ to ${{\ensuremath{\mathrm{MKTP}}}}$. The property that only one query is made to the oracle was subsequently used in order to show that ${{\ensuremath{\mathrm{MKTP}}}}$ is hard for the complexity class ${\ensuremath{\mathsf{DET}}}$ under mapping reductions computable in nonuniform ${\ensuremath{\mathsf{NC}}}^0$ [@allender.hirahara]. Similar hardness results (but for a more powerful class of reducibilities) hold also for ${{\ensuremath{\mathrm{MCSP}}}}$ [@oliveira.santhanam]. This has led to unconditional lower bounds on the circuit complexity of ${{\ensuremath{\mathrm{MKTP}}}}$ [@allender.hirahara; @hirahara.santhanam], showing that ${{\ensuremath{\mathrm{MKTP}}}}$ does not lie in the complexity class ${\ensuremath{\mathsf{AC}}}^0[p]$ for any prime $p$; it is still open whether similar circuit lower bounds hold for ${{\ensuremath{\mathrm{MCSP}}}}$. ##### Acknowledgments. {#acknowledgments. .unnumbered} EA acknowledges the support of National Science Foundation grant CCF-1555409. JAG was supported by an Omidyar Fellowship from the Santa Fe Institute and National Science Foundation grant DMS-1620484. DvM and AM acknowledge the support of National Science Foundation grant CCF-1319822. We thank V. Arvind for helpful comments about the graph automorphism problem and rigid graphs, Alex Russell and Yoav Kallus for helpful ideas on encoding and decoding graphs, Laci Babai and Peter Brooksbank for answering questions about computational group theory, and Oded Goldreich and Salil Vadhan for answering questions about ${{\ensuremath{\mathsf{SZK}}}}$. Appendix: Coset Indexings and Normal Forms for Permutation Groups {#sec:permutation-group-coding .unnumbered} ================================================================= In this appendix we develop the efficiently decodable indexings for cosets of permutation subgroups claimed in Lemma \[lemma:graph-coding\], and also use some of the underlying ideas to establish the normal form for permutation groups stated in Lemma \[lemma:permutation-group-normal-form\]. #### Indexing Cosets. The indexings are not strictly needed for our main results as the generic encoding from the can be used as a substitute. However, the information-theoretic optimality of the indexings may be useful in other contexts. In fact, we present a further generalization that may be of independent interest, namely an efficiently decodable indexing for cosets of permutation subgroups within another permutation subgroup. \[lemma:permutation-group-coding\] For all $\Gamma \leq H \leq S_n$, there exists an indexing of the cosets[^12] of $\Gamma$ within $H$ that is uniformly decodable in polynomial time when $\Gamma$ and $H$ are given by a list of generators. Lemma \[lemma:graph-coding\] is just the instantiation of Lemma \[lemma:permutation-group-coding\] with $H = S_n$. The proof of Lemma \[lemma:permutation-group-coding\] requires some elements of the theory of permutation groups. Given a list of permutations $\pi_1, \ldots, \pi_k \in S_n$, we write $\Gamma = \langle \pi_1, \ldots, \pi_k \rangle \leq S_n$ for the subgroup they generate. Given a permutation group $\Gamma \leq S_n$ and a point $i \in [n]$, the $\Gamma$-orbit of $i$ is the set $\{g(i) : g \in \Gamma\}$, and the $\Gamma$-stabilizer of $i$ is the subgroup $\{g \in \Gamma : g(i)=i\} \leq \Gamma$. We make use of the fact that (a) the number of cosets of a subgroup $\Gamma$ of a group $H$ equals $|H|/|\Gamma|$, and (b) the orbits of a subgroup $\Gamma$ of $H$ form a refinement of the orbits of $H$. We also need the following basic routines from computational group theory (see, for example, [@Holt05; @seress]). \[prop:cgt\] Given a set of permutations that generate a subgroup $\Gamma \leq S_n$, the following can be computed in time polynomial in $n$: - the cardinality $|\Gamma|$, - a permutation in $\Gamma$ that maps $u$ to $v$ for given $u,v \in [n]$, or report that no such permutation exists in $\Gamma$, and - a list of generators for the subgroup $\Gamma_v$ of $\Gamma$ that stabilizes a given element $v \in [n]$. The proof of Lemma \[lemma:permutation-group-coding\] makes implicit use of an efficient process for finding a *canonical representative* of $\pi \Gamma$ for a given permutation $\pi \in H$, where “canonical” means that the representative depends on the coset $\pi \Gamma$ only. The particular canonical representative the process produces can be specified as follows. \[def:canonical\] For a permutation $\pi \in S_n$ and a subgroup $\Gamma \leq S_n$, the *canonical representative* of $\pi$ modulo $\Gamma$, denoted $\pi \bmod \Gamma$, is the lexicographically least $\pi' \in \pi \Gamma$, where the lexicographic ordering is taken by viewing a permutation $\pi'$ as the sequence $(\pi'(1), \pi'(2), \dotsc, \pi'(n))$. We describe the process as it provides intuition for the proof of Lemma \[lemma:permutation-group-coding\]. \[lemma:canonical\] There exists a polynomial-time algorithm that takes as input a generating set for a subgroup $\Gamma \leq S_n$ and a permutation $\pi \in S_n$, and outputs the canonical representative $\pi \bmod \Gamma$. Consider the element 1 of $[n]$. Permutations in $\pi \Gamma$ map 1 to an element $v$ in the same $\Gamma$-orbit as $\pi(1)$, and for every element $v$ in the $\Gamma$-orbit of $\pi(1)$ there exists a permutation in $\pi \Gamma$ that maps 1 to $v$. We can canonize the behavior of $\pi$ on the element 1 by replacing $\pi$ with a permutation $\pi_1 \in \pi \Gamma$ that maps 1 to the minimum element $m$ in the $\Gamma$-orbit of $\pi(1)$. This can be achieved by multiplying $\pi$ to the right with a permutation in $\Gamma$ that maps $\pi(1)$ to $m$. Next we apply the same process to $\pi_1$ but consider the behavior on the element 2 of $[n]$. Since we are no longer allowed to change the value of $\pi_1(1)$, which equals $m$, the canonization of the behavior on 2 can only use multiplication on the right with permutations in $\Gamma_m$, i.e., permutations in $\Gamma$ that stabilize the element $m$. Doing so results in a permutation $\pi_2 \in \pi_1 \Gamma$. We repeat this process for all elements $k \in [n]$ in order. In the $k$-th step, we canonize the behavior on the element $k$ by multiplying on the right with permutations in $\Gamma_{\pi_{k-1}([k-1])}$, i.e., permutations in $\Gamma$ that pointwise stabilize all of the elements $\pi_{k-1}(\ell)$ for $\ell \in [k-1]$. The number of canonical representatives modulo $\Gamma$ in $H$ equals the number of distinct (left) cosets of $\Gamma$ in $H$, which is $|H|/|\Gamma|$. We construct an algorithm that takes as input a list of generators for $\Gamma$ and $H$, and an index $i \in [|H|/|\Gamma|]$, and outputs the permutation $\sigma$ that is the lexicographically $i$-th canonical representative modulo $\Gamma$ in $H$. The algorithm uses a prefix search to construct $\sigma$. In the $k$-th step, it knows the prefix $(\sigma(1),\sigma(2),\ldots,\sigma(k-1))$ of length $k-1$, and needs to figure out the correct value $v \in [n]$ to extend the prefix with. In order to do so, the algorithm needs to compute for each $v \in [n]$ the count $c_v$ of canonical representatives modulo $\Gamma$ in $H$ that agree with $\sigma$ on $[k-1]$ and take the value $v$ at $k$. The following claims allow us to do that efficiently when given a permutation $\sigma_{k-1} \in H$ that agrees with $\sigma$ on $[k-1]$. The claims use the notation $T_{k-1} \doteq \sigma_{k-1}([k-1])$, which also equals $\sigma([k-1])$. \[claim:1\] The canonical representatives modulo $\Gamma$ in $H$ that agree with $\sigma \in H$ on $[k-1]$ are exactly the canonical representatives modulo $\Gamma_{T_{k-1}}$ in $\sigma_{k-1}H_{T_{k-1}}$. The following two observations imply Claim \[claim:1\]. - A permutation $\pi \in H$ agrees with $\sigma \in H$ on $[k-1]$\ $\Leftrightarrow$ $\pi$ agrees with $\sigma_{k-1}$ on $[k-1]$\ $\Leftrightarrow$ $\sigma_{k-1}^{-1} \pi \in H_{T_{k-1}}$\ $\Leftrightarrow$ $\pi \in \sigma_{k-1} H_{T_{k-1}}$. - Two permutations in $\sigma_{k-1}H_{T_{k-1}}$, say $\pi \doteq \sigma_{k-1} g$ and $\pi' \doteq \sigma_{k-1} g'$ for $g, g' \in H_{T_{k-1}}$, belong to the same left coset of $\Gamma$ iff they belong to the same left coset of $\Gamma_{T_{k-1}}$. This follows because if $\sigma_{k-1} g' = \sigma_{k-1} g h$ for some $h \in \Gamma$, then $h$ equals $g^{-1} g' \in H_{T_{k-1}}$, so $h \in \Gamma \cap H_{T_{k-1}} = \Gamma_{T_{k-1}}$. \[claim:2\] The count $c_v$ for $v \in [n]$ is nonzero iff $v$ is the minimum of some $\Gamma_{T_{k-1}}$-orbit contained in the $H_{T_{k-1}}$-orbit of $\sigma_{k-1}(k)$. The set of values of $\pi(k)$ when $\pi$ ranges over $\sigma_{k-1} H_{T_{k-1}}$ is the $H_{T_{k-1}}$-orbit of $\sigma_{k-1}(k)$. Since $\Gamma_{T_{k-1}}$ is a subgroup of $H_{T_{k-1}}$, this orbit is the union of some $\Gamma_{T_{k-1}}$-orbits. Combined with Claim \[claim:1\] and the construction of the canonical representatives modulo $\Gamma_{T_{k-1}}$, this implies Claim \[claim:2\]. \[claim:3\] If a count $c_v$ is nonzero then it equals $|H_{T_{k-1}\cup\{v\}}|/|\Gamma_{T_{k-1}\cup\{v\}}|$. Since the count is nonzero, there exists a permutation $\sigma' \in H$ that is a canonical representative modulo $\Gamma$ that agrees with $\sigma_{k-1}$ on $[k-1]$ and satisfies $\sigma'(k)=v$. Applying Claim \[claim:1\] with $\sigma$ replaced by $\sigma'$, $k$ by $k' \doteq k+1$, $T_{k-1}$ by $T'_k \doteq T_{k-1} \cup \{v\}$, and $\sigma_{k-1}$ by any permutation $\sigma'_k \in H$ that agrees with $\sigma'$ on $[k]$, yields Claim \[claim:3\]. This is because the number of canonical representatives modulo $\Gamma_{T'_k}$ in $\sigma'_kH_{T'_k}$ equals the number of (left) cosets of $\Gamma_{T'_k}$ in $H_{T'_k}$, which is the quantity stated in Claim \[claim:3\]. The algorithm builds a sequence of permutations $\sigma_0, \sigma_1, \ldots, \sigma_n \in H$ such that $\sigma_k$ agrees with $\sigma$ on $[k]$. It starts with the identity permutation $\sigma_0 = id$, builds $\sigma_k$ out of $\sigma_{k-1}$ for increasing values of $k \in [n]$, and outputs the permutation $\sigma_n = \sigma$. Pseudocode for the algorithm is presented in Algorithm \[alg:T\]. Note that the pseudocode modifies the arguments $\Gamma$, $H$, and $i$ along the way. Whenever a group is referenced in the pseudocode, the actual reference is to a list of generators for that group. positive integer $n$, $\Gamma \leq H \leq S_n$, $i \in [|H|/|\Gamma|]$ lexicographically $i$-th canonical representative modulo $\Gamma$ in $H$ $\sigma_0 \gets id$ **return** $\sigma_n$ The correctness of the algorithm follows from Claims \[claim:2\] and \[claim:3\]. The fact that the algorithm runs in polynomial time follows from Proposition \[prop:cgt\]. #### Normal Form. Finally, we use the canonization captured in Definition \[def:canonical\] and Lemma \[lemma:canonical\] to establish the normal form for permutation groups given by Lemma \[lemma:permutation-group-normal-form\] (restated below): There is a polynomial-time algorithm $\nu$ that takes as input a list $L$ of elements of $S_n$, and outputs a list of generators for the subgroup generated by the elements in $L$ such that for any two lists $L_0, L_1$ of elements of $S_n$ that generate the same subgroup, $\nu(L_0) = \nu(L_1)$. Let $\Gamma$ denote the subgroup generated by $L$, and recall that $\Gamma_{[i]}$ denotes the subgroup of $\Gamma$ that stabilizes each element in $[i]$, for $i \in \{0,1,\ldots,n\}$. We have that $\Gamma_{[0]} = \Gamma$, and $\Gamma_{[n-1]}$ consists of the identity only. We define $\nu(L)$ as follows. Start with $\nu$ being the empty list. For $i \in [n-1]$, in the $i$-th step we consider each $j \in [n]$ that is in the $\Gamma_{[i-1]}$-orbit of $i$ in order. Note that for each such $j$, the permutations in $\Gamma_{[i-1]}$ that map $i$ to $j$ form a coset of $\Gamma_{[i-1]} \bmod \Gamma_{[i]}$. We append the canonical representative of this coset to $\nu$. $\nu(L)$ is the value of $\nu$ after step $n-1$. As we only include permutations from $\Gamma$, $\nu(L)$ generates a subgroup of $\Gamma$. By construction, for each $i \in [n-1]$, the permutations we add in the $i$-th step represent all cosets of $\Gamma_{[i-1]} \bmod \Gamma_{[i]}$. It follows by induction on $n-i$ that the permutations added to $\nu$ during and after the $i$-th step generate $\Gamma_{[i-1]}$ for $i \in [n]$. Thus, $\nu(L)$ generates $\Gamma_{[0]} = \Gamma$. That $\nu(L)$ only depends on the subgroup $\Gamma$ generated by $L$ follows from its definition, which only refers to the abstract groups $\Gamma_{[i]}$, their cosets, and their canonical representatives. That $\nu(L)$ can be computed in polynomial time follows by tracking a set of generators for the subgroups $\Gamma_{[i]}$ based on Proposition \[prop:cgt\]. More specifically, we use item 2 to check whether a given $j$ is in the $\Gamma_{[i-1]}$-orbit of $i$, and item 3 to obtain $\Gamma_{[i]}$ out of $\Gamma_{[i-1]}$ as $\Gamma_{[i]} = (\Gamma_{[i-1]})_i$. [^1]: Rutgers University, Piscataway, NJ, USA, [^2]: University of Colorado at Boulder, Boulder, CO, USA, [^3]: University of Wisconsin–Madison, Madison, WI, USA, [^4]: Santa Fe Institute, Santa Fe, NM, USA, [^5]: University of Wisconsin–Madison, Madison, WI, USA, [^6]: This work subsumes and significantly strengthens the earlier arXiv submission 1511.08189 [@agm.arxiv]. See the end of Section \[sec:intro\] for more details. [^7]: In some settings worst-case to average-case reductions are known, but these reductions are themselves randomized with two-sided error. [^8]: The choice of left ($\pi \Gamma$) vs right ($\Gamma \pi$) cosets is irrelevant for us; all our results hold for both, and one can usually switch from one statement to the other by taking inverses. Related to this, there is an ambiguity regarding the order of application in the composition $gh$ of two permutations: first apply $g$ and then $h$, or vice versa. Both interpretations are fine. For concreteness, we assume the former. \[footnote:left-right\] [^9]: More precisely, suppose there exists a randomized circuit family $A$ of size $f(n,m)$ that decides ${\ensuremath{\mathsf{CircuitSAT}}}$ without false positives on instances consisting of circuits $C$ with $n$ input variables and of description length $m$ such that the probability of success is at least $1/2^{\alpha n}$. Applying our encoding to the set of random bit sequences that make $A$ accept on a positive instance $C$, and hard-wiring the input $C$ into the circuit $A$, yields an equivalent instance $C'$ on $\alpha n$ variables of size $f(n,m)+\mu(D)$, where $\mu(D)$ denotes the circuit size of $D$. Applying $A$ to the description of this new circuit $C'$ yields a randomized circuit $A'$ to decide whether $C$ is satisfiable without false positives. For the linear-algebraic family of hash functions, $A'$ has size $O(f(n,m) {{\ensuremath{\operatorname{polylog}}}}(f(n,m)))$. Its success probability is at least $1/2^{\alpha^2 n}$, which is larger than $1/2^{\alpha n}$ when $\alpha<1$. [^10]: For complexity-theoretic investigations into the difference between complete invariants and normal forms, see, , [@blassGurevich1; @blassGurevich2; @FortnowGrochowPEq; @finkelsteinHescott]. [^11]: For many cases where $L \subseteq {{\ensuremath{\operatorname{GL}}}}_n({\mathbb{F}}_q)$, [@babai2009polynomial] shows how to compute the exact order of $\langle L \rangle$ in ${{\ensuremath{\mathsf{ZPP}}}}$ with oracles for integer factorization and the discrete log. Combined with follow-up results of [@kantorMagaard; @liebeckObrien; @kantorMagaard2], the only cases that remain open are those over a field of characteristic $2$ where $\langle L \rangle$ contains at least one of the Ree groups $^2 F_4(2^{2n+1})$ as a composition factor, and those over a field of characteristic $3$ where $\langle L \rangle$ contains at least one of the Ree groups $^2 G_2(3^{2n+1})$ as a composition factor. The claim follows as integer factorization and discrete log can be computed in ${{\ensuremath{\mathsf{ZPP}}}}^{{\ensuremath{\mathrm{MKTP}}}}$. [^12]: Recall footnote \[footnote:left-right\] on page .

--- abstract: | Infrared features of the ghost propagator of color diagonal and color antisymmetric ghost propagator of quenched SU(2) and quenched SU(3) are compared with those of unquenched Kogut-Susskind fermion SU(3) lattice Landau gauge. We compare 1) the fluctuation of the ghost propagator, 2) the ghost condensate parameter $v$ of the local composite operator (LCO) approach and 3) the Binder cumulant of color anti-symmetric ghost propagator between quenched and unquenched configurations. The color diagonal SU(3) ghost dressing function of unquenched configurations has weaker singularity than the quenched configurations. In both cases fluctuations become large in $q<0.5$GeV. The ghost condensate parameter $v$ in the ghost propagator of the unquenched MILC$_c$ configuration samples is $0.002\sim 0.04$GeV$^2$ while that of the SU(2) PT samples is consistent with 0. The Binder cumulant defined as $U(q)=1-\frac{1}{3}\frac{< \vec\phi^4 >}{(< \vec\phi^2 >)^2}$ where $\vec \phi(q)$ is the color anti-symmetric ghost propagator measured by the sample average of gauge fixed configurations via parallel tempering method becomes $\sim 4/9$ in all the momentum region. The Binder cumulant of the color antisymmetric ghost propagator of quenched SU(2) can be explained by the 3-d gaussian distribution, but that of the unquenched MILC$_c$ deviates slightly from that of the 8-dimensional gaussian distribution. The stronger singularity and large fluctuation in the quenched configuration could be the cause of the deviation of the Kugo-Ojima confinement parameter $c$ from 1, and the presence of ordering in the ghost propagator of unquenched configurations makes it closer to 1. author: - Sadataka Furui - Hideo Nakajima title: Effects of the quark field on the ghost propagator of Lattice Landau Gauge QCD --- Introduction ============ Infrared features of the ghost propagator are important in the analysis of color confinement mechanism and the running coupling. Kugo and Ojima[@KO] considered the two point function connected by the ghost propagator and expressed the confinement criterion as $$1+u(0)=1-c=\frac{Z_1}{Z_3}=\frac{\tilde Z_1}{\tilde Z_3}=0$$ at the renormalization point $\mu=0$[@kugo]. Here $Z_1$ and $\tilde Z_1$ are the vertex renormalization factor of the triple gluon vertex and the ghost anti-ghost gluon vertex, respectively and $Z_3$ and $\tilde Z_3$ are the wave function renormalization factor of the gluon and the ghost, respectively. If $\tilde Z_1$ is finite, divergence of $\tilde Z_3$ is a sufficient condition of the color confinement. The lattice data suggest that $\tilde Z_3$ is infrared divergent, but its singularity is not strong enough to hinder the running coupling measured as $$\label{alpha} \alpha_s(q)=\frac{g_0^2}{4\pi}\frac{Z(q^2){ G(q^2)}^2}{{\tilde Z_1}^2}\sim \alpha_s(\Lambda_{UV}) q^{-2(\alpha_D+2\alpha_G)},$$ approach zero in the infrared[@unquench]. Here $Z(q^2)$ and $G(q^2)$ are the gluon dressing function and the ghost dressing function, respectively. The same observation is reported in[@ilgen]. The ghost propagator in the infrared region was investigated by several authors. Common findings are that it is more singular than $q^{-2}$ and that in the infrared region its statistical fluctuation is large probably due to presence of Gribov copies[@FN04; @FN05; @muell; @bclm; @Orsay]. In the quenched $32^4, 48^4$ and $56^4$ SU(3) lattice simulation, the color diagonal ghost propagator showed singularity of $q^{-2-\alpha_G}$ with $\alpha_G\sim 0.25$. In the Dyson-Schwinger (DS) approach, the infrared power behavior of the ghost propagator and the gluon propagator $q^{-2-\alpha_D}$ have the relation $2\alpha_G+\alpha_D=0$ and the lattice data are consistent with this ansatz in $q>1$GeV region. As the magnitude of $\alpha_D$, Dyson-Schwinger approach[@vS] and Langevin approach[@Zw] predicts -0.59, while the lattice data and DS approaches[@Blo; @Kond] predicts -0.5. If $\alpha_D$ is smaller than -0.5 the gluon propagator in the infrared vanishes and the Gribov-Zwanziger’s conjecture on the color confinement of the gluon becomes satisfied. Recent detailed analysis of the finite size effect in the lattice confirms that infrared limit of $-\alpha_D$ in the DS approach $\kappa=0.5$ is compatible with the lattice data[@Orsay3; @OS]. The relation $2\alpha_G+\alpha_D=0$ suggests presence of an infrared fixed point[@vS]. The infrared finite quark wavefunction renormalization $Z_\psi$ of unquenched simulation[@FN06] also suggests that the running coupling is not infrared vanishing. We cannot measure the ghost propagator at zero momentum, since we evaluate it with the condition that it is zero-mode-less. Thus the infrared power fitted at finite lattice momentum $\alpha_G$ cannot predict the power behavior of the ghost propagator near momentum 0 i.e. the index $\kappa$. In [@unquench], we observed that the Kugo-Ojima confinement criterion is satisfied in the unquenched simulation but not in the quenched simulation of lattice sizes up to $56^4$. In order to study the role of fermion in the color confinement, we consider the BRST(Becchi-Rouet-Stora-Tyutin) quartet mechanism[@KO; @adfm]. In the BRST formulation[@KO], unphysical degrees of freedom are confined by the quartet mechanism. In the pure QCD in the Landau gauge, one can construct BRST quartet as $$\begin{array}{ccccccc} A_\mu & \to & D_\mu(A) c & \to & 0 & & \\ & &A_\mu \bar c &\to & D_\mu(A)c \bar c - A_\mu B & \to &0 . \end{array}$$ Here the arrow implies the BRST transformation $\delta_B$ and $B$ is the Nakanishi-Lautrup auxiliary field. The transverse gluon state $A_\mu$ is a BRST parent state of a daughter state $D_\mu(A)c$ and the state with opposite ghost number of the $D_\mu(A)c$ i.e. $A \bar c$ becomes a parent state, whose daughter and the above three states construct a quartet. Inclusion of the fermion field $\psi$ allows to construct another BRST quartet as $$\begin{array}{ccccccc} \psi & \to & -\psi c & \to & 0 & & \\ & & \psi \bar c & \to & -\psi c \bar c-\psi B & \to & 0. \end{array}$$ The Dirac fermion state $\psi$ is a BRST parent state of $\psi c$ and the state with opposite ghost number state of $\psi c$ is $\psi \bar c$, which becomes a parent state of the BRST partner that construct a quartet. Inclusion of fermion gives more restiction on the degrees of freedom of the ghost and it may change the fluctuation of the ghost propagator. Another current problem concerning the ghost propagator is the possibility of the ghost condensates. In the lattice Landau gauge QCD simulation, presence of $A^2$ condensates was suggested[@Orsay; @FN04; @FN05; @unquench; @lat05_a]. Since $A^2$ is not BRST invariant, a mixed condensate i.e. a combination with ghost condensates $$\int \langle tr_{G/H}[\frac{1}{2}{\mathcal A}_\mu{\mathcal A}^\mu-\xi i{{\mathcal C}}\bar{\mathcal C}]\rangle d^4 x$$ was proposed[@kondo; @grip] as on-shell BRST invariant, i.e. invariant for the $B$ field that satisfies $$B^a=-\frac{1}{\xi}\partial_\mu A^{a \mu}+i\frac{g}{2}f^{abc}c^b\bar c^c.$$ Here $G/H$ is the subset of gauge fixed configuration, and $\xi$ is the gauge fixing parameter. The Landau gauge $\xi=0$ is regarded as a specific limit of the Curci-Ferrari gauge. In recent studies, the space-time average of the vacuum expectation value $$\frac{1}{V}\int_V \langle\frac{1}{2} tr {\mathcal A}_\mu(x){\mathcal A}^\mu(x)\rangle d^4 x$$ is claimed to have gauge invariant meaning[@kondo; @slav]. In the Landau gauge QCD, the Faddeev-Popov(FP) gauge-fixing action is $$S_{FP}=B^a\partial_\mu A^a_\mu+i\bar {\mathcal C}^a{\partial_\mu D_\mu}^{ab}{\mathcal C}^b,$$ where the last term $\bar {\mathcal C}^a{\partial_\mu D_\mu}^{ab}{\mathcal C}^b$, where $D_\mu^{ab}=\delta^{ab}\partial_\mu +gf^{acb}A^c_\mu$. In analytical studies in the Curci-Ferrari gauge, presence of the ghost condensate $\langle f^{abc}c^b \bar c^c\rangle$ was discussed as the Overhauser effect in contrast to the $\langle f^{abc} c^b c^c\rangle$ or $\langle f^{abc}\bar c^b \bar c^c\rangle$ which are regarded as the BCS effect[@dvlsspvg]. Since the Landau gauge is a specific limit of the Curci-Ferrari gauge, it is of interest to study the ghost propagator. In [@FN04], we observed that in the SU(2) $\beta=2.1$ $16^4$ lattice, the expectation value of color off-diagonal ghost propagator $\langle \epsilon^{abc}\bar c^b c^c\rangle$ is consistent with 0 but the standard deviation of the color-diagonal ghost propagator has the momentum dependence of $\sigma(G^{aa}(q))\propto q^{-4}$. The investigation was extended by [@CMM] and this fluctuation was confirmed and although the expectation value of $\phi^a(q)=\epsilon^{abc}c^b \bar c^c$ is consistent with 0, the expectation value of its absolute value $|\phi^a(q)|$ was shown to behave as $q^{-4}$ and not zero. We extend this approach to unquenched MILC configurations. In [@CMM], the ghost condensate parameter $v$ and the Binder cumulant[@bind] of the color anti-symmetric ghost propagator was measured. In the Binder cumulant of an order parameter, renormalization factors cancel and one can extract the fixed point in the continuum limit by a suitable extrapolation. In the Zwanziger’s Lagrangian[@Zwa], the color anti-symmetric ghost field $\phi^{bc}_\mu(x)$ leads to the mass gap equation $$f^{abc}\langle A^{a\mu}(x)\phi^{bc}_\mu(x)\rangle=\frac{4(N_c^2-1) \gamma^2}{\sqrt 2 g^2}$$ where $\gamma^2$ is the mass dimension two Gribov mass parameter[@gra]. It is not evident that the Zwanziger’s Lagrangian expresses the effective theory of the lattice Landau gauge QCD, but analytical calculation of the ghost propagator in two loop[@gra], and the local composite operator approach[@dvlsspvg; @cdglssv] suggest hints for solving entanglements in the confinement problem. In this paper we study the ghost propagator of quenched SU(2) $\beta=2.2$ $16^4$ lattice gauge fixed to the Landau gauge via parallel tempering(PT) method[@NF] and investigate the Binder cumulant. We extend the study to unquenched SU(3), using the MILC$_c$ configurations[@milc]. Organization of the paper is as follows. In sect. II, we show definitions of the color diagonal and color anti-symmetric ghost propagator on the lattice. In sect.III, fluctuation of the ghost propagator of quenched and unquenched configurations are compared. In sect.IV, we compare the parameter $v$ of the ghost condensates from the color anti-symmetric ghost propagators of quenched SU(2) PT configurations and unquenched MILC$_c$ configurations. In sect.V, the Binder cumulant of the color anti-symmetric ghost propagator of the quenched SU(2) and unquenched SU(3) are compared. Summary and discussion are given in sect. VI. The ghost propagator ==================== The ghost propagator ${D_G}^{ab}(q^2)$ and the ghost dressing function $G^{ab}(q^2)$ is defined by the Fourier transform(FT) of the expectation value of the inverse Faddeev-Popov operator $\cal M=-\partial_\mu D_\mu$ $$\begin{aligned} FT[{D_G}^{ab}(x,y)]&=&FT\langle {\rm tr} ( \Lambda^{a \dagger} \{({\cal M}[U])^{-1}\}_{xy} \Lambda^b )\rangle,\nonumber\\ &=&{D_G}^{ab}(q^2)=\frac{G^{ab}(q^2)}{q^2}\end{aligned}$$ where antihermitian SU(3) generator $\Lambda^a$ is normalized as ${\rm tr} \Lambda^{a\dagger} \Lambda^b=\delta^{ab}$. We measure $${D_G}^{ab}(q^2)=\left\langle tr\langle \Lambda^a q|{\mathcal M}[U]^{-1}|\Lambda^b q\rangle \right\rangle$$ using the source vector $\displaystyle |\Lambda^a q\rangle= \frac{1}{\sqrt V}\Lambda^a e^{i q\cdot x}$. We select the momentum $q_\mu$ to be directed along the diagonal of the lattice momentum space. In the approach of calculating the fourier transform of ${\mathcal M}^{-1}S^a_0(x)$[@Orsay], compensation of hypercubic artefacts was necessary, but in our method the artifact-free momenta are selected and the translation invariance is fully utilized to improve the statistics. The Faddeev-Popov operator ${\mathcal M}[U]=-\partial D[U]$ is defined with use of the covariant derivative as $$D_\mu(U_{x,\mu})\phi=S(U_{x,\mu})\partial_\mu\phi+[A_{x,\mu},\bar\phi]$$ where $\partial_\mu \phi=\phi(x+\mu)-\phi(x)$, $\bar\phi=\frac{1}{2}(\phi(x+\mu)+\phi(x))$. In the $U$-linear version, ($A_{x,\mu}=\frac{1}{2}(U_{x,\mu}-U_{x,\mu}^\dagger)|_{trlp}$ where $|_{trlp}$ means the traceless part) $S(U_{x,\mu})B_{x,\mu}$ is defined as $$S(U_{x,\mu})B_{x,\mu}=\frac{1}{2}\{\frac{U_{x,\mu}+U_{x,\mu}^\dagger}{2},B_{x,\mu}\}|_{trlp}$$ and in the $\log-U$ version, ($U_{x,\mu}=e^{A_{x,\mu}}$) $$S(U_{x,\mu})B_{x,\mu}=\frac{A_{x,\mu}}{2\tanh(A_{x,\mu}/2)}B_{x,\mu},$$ where $A_{x,\mu}=adj A_{x,\mu}$[@NF]. In [@CMM], the Faddeev-Popov operator was parametrized as $${\mathcal M}^{bc}_U(x,y)=\delta^{bc}{\mathcal S}(x,y)-f^{bcd}{\mathcal A}^d(x,y)$$ The authors decomposed the inverse matrix ${D_G}^{bc}(x,y)=({\mathcal M}^{-1})^{bc}(x,y)$ into ${D_G}^{bc}_e(x,y)$ and ${D_G}^{bc}_o(x,y)$ i.e. the component containing even number of ${\mathcal A}$ odd number of ${\mathcal A}$, respectively. They derived the ghost propagator from $\displaystyle\frac{\delta^{bc}}{N_c^2-1}{D_G}^{bc}_e(x,y)$, and the color antisymmetric ghost propagator by multiplying ${\mathcal S}^{-1}{\mathcal A}$ to the color antisymmetric ghost propagator ${D_G}^{bc}_e(x,y)$ which contains perturbation series of even numbers of ${\mathcal A}$. We do not adopt this procedure, but derive directly the color anti-symmetric ghost propagators by the conjugate gradient method. The convergence condition on the series is set to less than a few % in the ${\it l}_2$ norm. We define ${\mathcal M}=-\partial_\mu D_\mu$ and solve $$-\partial_\mu D_\mu f_s^b({\bf x})=\frac{1}{\sqrt V}\Lambda^b \sin{\bf q}\cdot{\bf x}$$ and $$-\partial_\mu D_\mu f_c^b({\bf x})=\frac{1}{\sqrt V}\Lambda^b \cos{\bf q}\cdot{\bf x}.$$ Then we calculate the overlap to get the color diagonal ghost propagator $$\begin{aligned} &&D_G(q)=\frac{1}{N_c^2-1}\frac{1}{V}\nonumber\\ &&\times\delta^{ab}(\langle \Lambda^a\cos{\bf q}\cdot{\bf x}|f_c^b({\bf x})\rangle+\langle \Lambda^a\sin{\bf q}\cdot{\bf x}|f_s^b({\bf x})\rangle)\nonumber\\\end{aligned}$$ and color anti-symmetric ghost propagator $$\begin{aligned} &&\phi^c(q)=\frac{1}{\mathcal N}\frac{1}{V}\nonumber\\ &&\times f^{abc}(\langle \Lambda^a\cos{\bf q}\cdot{\bf x}|f_s^b({\bf x})\rangle-\langle \Lambda^a\sin{\bf q}\cdot{\bf x}|f_c^b({\bf x})\rangle)\nonumber\\\end{aligned}$$ where ${\mathcal N}=2$ for SU(2) and 6 for SU(3). Fluctuation of the ghost propagator =================================== We present in the following subsections square and the absolute value of the color antisymmetric ghost propagators of the quenched SU(2) $\beta=2.2$ $16^4$ lattice, and compare the corresponding values of SU(2) Landau gauge QCD of larger samples[@CMM]. We measure also ghost propagators of quenched SU(3) $\beta=6.45$ $56^4$ lattice and those of unquenched MILC$_c$ with lattice size $20^3\times 64$ and MILC$_f$ with lattice size $28^3\times 96$. We present square and the absolute value of color antisymmetric ghost propagator of MILC$_c$. Quenched SU(2) -------------- We select momenta $q$ following the cylinder cut, and in the case of unquenched SU(3) $20^3\times 64$ lattice calculation, it takes about 260 iterations in the $q=0.2$GeV region but several iterations in the $q=4$GeV region. The average of color anti-symmetric ghost propagator $\phi^c(q)$ is consistent with 0 but the average of its square $\phi^c(q)^2$ has a non-vanishing value. We define $$\vec \phi(q)^2=\frac{1}{N_c^2-1}\sum_{c}\phi^c(q)^2$$ ![Log of color anti-symmetric ghost propagator squared $\log_{10}[\phi(q)^2]$ as the function of $q$(GeV). $\beta=2.2$, $16^4$ PT gauge fixing. (Color online)[]{data-label="phi_ptn"}](logphi_ptn.eps){width="7.2cm"} The log of $\vec \phi(q)^2$ of $\beta=2.2$ $16^4$ lattice gauge fixed by the PT method (67 samples) is shown in Fig.\[phi\_ptn\]. The corresponding log-log plots (Fig.\[phi\_pt\]) is to be compared with that of the ghost propagator $D_G(q)$ (Fig.\[gh\_pt\]). ![Log of the color anti-symmetric ghost propagator squared $\log_{10}[\phi(q)^2]$ as the function of $\log_{10}[q$(GeV)\]. $\beta=2.2$, $16^4$ PT gauge fixing. (Color online)[]{data-label="phi_pt"}](logphi_pt.eps){width="7.2cm"} ![Log of the ghost propagator $\log_{10}[D_G(q)]$ as the function of $\log_{10}[q$(GeV)\]. $\beta=2.2$, $16^4$ PT gauge fixing. (Color online)[]{data-label="gh_pt"}](log_dg_pt.eps){width="7.2cm"} The infrared singularity of the standard deviation of the color anti-symmetric ghost dressing function and the color diagonal ghost dressing function are $q^{-4.4}$ and $q^{-4.5}$, respectively. Quenched SU(3) -------------- In FIG.\[gh645\_64\] we show the color diagonal ghost propagator of quenched SU(3) with $\beta=6.4$ ($1/a=3.66$GeV) and $\beta=6.45$ ($1/a=3.8697$GeV) on $56^4$ lattice. The corresponding ghost dressing function are in FIG.\[ghd645\_64\]. ![The ghost propagator as the function of the momentum $q$(GeV). $\beta=6.45$, $56^4$(stars) and $\beta=6.4, 56^4$(filled diamonds) in the $\log U$ definition. Solid line is the pQCD fit in $\widetilde{MOM}$ scheme[@FN05]. (Color online)[]{data-label="gh645_64"}](gh645_64.eps){width="7.2cm"} ![The ghost dressing function as the function of the momentum $q$(GeV). $\beta=6.45$, $56^4$(stars) and $\beta=6.4, 56^4$(filled diamonds) in the $\log-U$ definition. (Color online)[]{data-label="ghd645_64"}](ghd645_64a.eps){width="7.2cm"} The standard deviation of the color-diagonal ghost propagator of $\beta=6.45$ multiplied by $(qa)^4$ is almost constant in $q>1$GeV region but in the $q<0.5$GeV region it is enhanced as compared to the value at $q>1$GeV region. The $\log-\log$ plot of the standard deviation of the color-diagonal ghost dressing function of $\beta=6.45$ in the $q<1$GeV region behaves as $$\sigma(G(q))\propto q^{-2.8(1)}.$$ Unquenched SU(3) ---------------- In [@FN05] we showed lattice results of color diagonal ghost dressing function of unquenched JLQCD/CP-PACS and MILC. In these simulations the length of the time axis is longer than the spacial axes and the ghost propagator of low momentum region is extended. In FIG.\[ghdmilc\_645\] the log-log plot of the ghost dressing function of the MILC$_f$ $\beta_{imp}=7.09$ on $28^3\times 96$ lattice and that of quenched $\beta=6.45$ on $56^4$ lattice are shown. We observe suppression of the ghost propagator in the infrared region in the asymmetric lattice[@unquench; @lat05_a]. Systematic deviation of ghost propagator and gluon propagator of asymmetric lattice from those of symmetric lattice is recently confirmed in the large 3-dimensional SU(2) lattice[@CM]. The suppression in the infrared of the unquenched data may not be due to the presence of quarks but due to the geometry of the lattice. ![Log of the ghost dressing function $\log_{10} G(q)$ as a function of $\log_{10} q(\rm{GeV})$ of MILC$_f$ $\beta_{imp}={7.09}$ (diamonds) and that of quenched $\beta=6.45$ $56^4$ (stars). (Color online)[]{data-label="ghdmilc_645"}](ghdmilc709_645.eps){width="7.2cm"} There are difference in the momentum dependence of the standard deviation of the color diagonal ghost dressing function of unquenched MILC$_f$, $\beta_{imp}=7.09$ on $28^3\times 96$ lattice and the quenched $\beta=6.45$ on $56^4$ lattice in the infrared region as shown in FIG.\[sigma\_g\]. Since the sample size is different, the absolute value of the standard deviation is not meaningful, but the strength of the fluctuation defind by the slope influences the infrared behavior of the running coupling etc. ![Log of the standard deviation $\log_{10} \sigma(G(q))$ as a function of $\log_{10} q(\rm{GeV})$ of MILC$_f$ $\beta_{imp}={7.09}$ (upper points) and that of quenched $\beta=6.45$ $56^4$ (lower points). (Color online)[]{data-label="sigma_g"}](sigma_g.eps){width="7.2cm"} The momentum dependence of the standard deviation of the color diagonal ghost dressing function of MILC$_f$ is dramatically less singular than that of quenched configuration. We observed $$\sigma(G(q))\propto q^{-1.1(1)}.$$ The color anti-symmetric ghost propagator of MILC$_c$ (21 samples) is shown in Fig.\[phi\_milc\]. By comparing with FIG.\[phi\_ptn\], we observe decrease of the slope. ![Log of the color anti-symmetric ghost propagator squared $\log_{10}[\phi(q)^2]$ as the function of $q$(GeV). $\beta_{imp}=6.83$ and 6.76, $20^3\times 64$ MILC$_c$. (Color online)[]{data-label="phi_milc"}](phimilc_c.eps){width="7.2cm"} The ghost condensate ==================== In the case of SU(2), the ghost condensate appears in the color anti-symmetric ghost propagator ${D_G}^{bc}_o(q)$ related to $\phi^a(q)$ through $$\phi^a(q)=-i\frac{f^{abc}}{2}{D_G}^{bc}(q^2)=-i\frac{f^{abc}}{2}{D_G}^{bc}_o(q)$$ and $${D_G}^{bc}_o(q)=i\frac{r/L^2+v}{q^4+v^2}\epsilon^{bc},$$ where $\epsilon^{bc}$ is an anti-symmetric tensor, i.e. when $a=3$, $b$ and $c=1, 2$. In general, we parametrize the average of $|\phi^a(q)|$ as $$\frac{1}{N_c^2-1}\sum_a|\phi^a(q)|= \frac{r/L^2+v}{q^4+v^2}$$ Here $L$ is the lattice size and the parameter $r/L^2$ is the correction from the finite size effect. Quenched SU(2) -------------- In [@CMM], the fitting parameter $r$ of $|\vec\phi^a(q)|$ (color antisymmetric ghost propagator) on the lattice was derived from $$\frac{1}{3}\sum_a\frac{L^2}{\cos(\pi \bar q/L)}|\phi^a(q)|=\frac{r}{q^z},$$ in which $\bar q=0,1,\cdots L$. Our fit of $\displaystyle \frac{|\phi^a(q)|}{\cos(\pi\bar q a/L)}$ of PT samples using $r=10.13, z=4.215$ is shown in FIG.\[su2gha\_1\]. ![Log of $|\phi^a(q)|$ (color anti-symmetric ghost propagator) devided by $\cos(\pi\bar q a/L)$ as the function of $\log_{10}(q$(GeV)) of SU(2) PT samples. (Color online)[]{data-label="su2gha_1"}](su2gha_1.eps){width="7.2cm"} Using $r=10.13, L=16$, the fitting parameter $v$ of $|\phi(q)|$ is found to be -0.002GeV$^2$ and is consistent with 0. The fit is shown in FIG.\[su2gha\_2\]. ![Log of the absolute value $|\phi^a(q)|$ (color anti-symmetric ghost propagator) as the function of $\log_{10}(q$(GeV)) of SU(2) PT samples. (Color online)[]{data-label="su2gha_2"}](su2gha_2.eps){width="7.2cm"} Unquenched SU(3) ---------------- As in the SU(2) PT samples, we performed the fit of the parameter $v$ for the MILC$_c$ samples. We first fit the $\log$ of $|\phi^a(q)|$(color anti-symmetric ghost propagator) devided by $\cos(\pi\bar q a/L)$ using $L=\sqrt {20^3\times 64}$ and obtained $r=40.5, z=3.75$, as shown in FIG.\[milcgha\_1\]. The parameter $r$ and $z$ for the fit of $\vec \phi^2(q)/\cos^2(\pi \bar q/L)$ are $r=36.5, z=7.5$. ![Log of $|\vec\phi(q)|/\cos(\pi\bar q a/L)$ (color anti-symmetric ghost propagator) as the function of $\log_{10}(q$(GeV)) of MILC$_c$ samples. (Color online) []{data-label="milcgha_1"}](milcgha_1.eps){width="7.2cm"} Our fit of $|\phi(q)|$ ignoring two lowest momentum points and using $r=40.51$ gives $v=0.0020$GeV$^2$, which is small but positive. When the two lowest momentum points are included, $v$ decreases to -0.0005 but $\chi^2/d.o.f.$ increases. The former fit is shown in FIG.\[milcgha\_2\]. ![Log of the absolute value $|\vec\phi(q)|$ (color anti-symmetric ghost propagator) as the function of $\log_{10}(q$(GeV)) of MILC$_c$ samples. (Color online)[]{data-label="milcgha_2"}](milcgha_2.eps){width="7.2cm"} We fitted also $\log_{10}\vec \phi^2(q)$, where $$\vec \phi^2(q)=\frac{1}{N_c^2-1}\sum_a \phi^a(q)^2= (\frac{r/L^2+v}{q^4+v^2})^2$$ The fit with $r=40.5$, $v=0.035$GeV$^2$ is shown in FIG.\[milcgha\_3\]. The fit with $r=36.5, v=0.041$GeV$^2$ is not distinguished from this figure. ![Log of the $\vec\phi(q)^2$ (color anti-symmetric ghost propagator squared) as the function of $\log_{10}(q$(GeV)) of MILC$_c$ samples. (Color online)[]{data-label="milcgha_3"}](milcphi2fit.eps){width="7.2cm"} Binder cumulant =============== Two decades ago Binder[@bind] showed cumulants of the order parameter yields non-trivial fixed-point values. The theory was applied to the Ising model in which the magnetization $M$ is the order parameter[@pr; @bptr] and the cumulant was defined as $$\label{bind_ising} B=\frac{1}{2}(3-\frac{\langle M^4\rangle}{\langle M^2\rangle^2}).$$ When the distribution of $M$ is given by the 1 dimensional gaussian distribution, one finds $$\frac{\langle M^4\rangle}{\langle M^2\rangle^2}=3.$$ and $B$ becomes 0. In SU(2) and SU(3) lattice QCD, deconfinement phase transition was studied by measuring $$g=\frac{\langle P^4\rangle}{\langle P^2\rangle^2}-3$$ using the Polyakov line data $P$ as the order parameter[@abs; @karsch]. Since the color anti-symmetric ghost propagator could be an order parameter of the system, the authors of [@CMM] considered its Binder cumulant defined as $$U(q)=1-\frac{\langle\vec\phi(q)^4\rangle}{3\langle\vec\phi(q)^2\rangle^2}.\label{bind}$$ We measure $$\begin{aligned} &&\vec\phi(q)^2=\frac{1}{N_c^2-1}\sum_a [\frac{1}{\mathcal N}\nonumber\\ &&\times \frac{f^{abc}}{V}\left(\langle\Lambda^{b}\cos{\bf q}\cdot{\bf x}|{\mathcal M}^{-1}| \Lambda^{c}\sin{\bf q}\cdot{\bf x}\rangle\right.\nonumber\\ &&\left.-\langle\Lambda^{b}\sin{\bf q}\cdot{\bf x}|{\mathcal M}^{-1}| \Lambda^{c}\cos{\bf q}\cdot{\bf x}\rangle \right)]^2\end{aligned}$$ and $$\begin{aligned} \vec\phi(q)^4&=&(\frac{1}{N_c^2-1}\sum_a [\frac{1}{\mathcal N}\nonumber\\ &&\times \frac{f^{abc}}{V}\left(\langle\Lambda^{b}\cos{\bf q}\cdot{\bf x}|{\mathcal M}^{-1}| \Lambda^{c}\sin{\bf q}\cdot{\bf x}\rangle\right.\nonumber\\ &&\left.-\langle\Lambda^{b}\sin{\bf q}\cdot{\bf x}|{\mathcal M}^{-1}| \Lambda^{c}\cos{\bf q}\cdot{\bf x}\rangle\right)]^2)^2.\end{aligned}$$ In arbitrary d-dimensional space, corresponding expectation value for d-dimensional gaussian distribution becomes $$\frac{\langle \vec\phi^4\rangle}{\langle \vec\phi^2\rangle^2}=\frac{d+2}{d}.$$ Thus a natural extension to d-dimensional vector variable is $$\tilde U(q)=\frac{\langle \vec\phi^4\rangle}{\langle \vec\phi^2\rangle^2}-\frac{d+2}{d}$$ which becomes 0 in the system with gaussian distribution whose symmetry is not broken. When the symmetry of the system is broken, as in the Ising model at the 0 temperature, the ratio of $\langle\vec\phi(q)^2\rangle^2$ and $\langle\vec\phi(q)^4\rangle$ becomes 1 and $\tilde U(q)$ becomes $-\displaystyle\frac{2}{d}$. It corresponds to the 0 temperature fixed point. Quenched SU(2) -------------- We measure the Binder cumulant $U(q)$ of the quenched SU(2) $16^4$ $\beta=2.2$, $a=1.07$GeV$^{-1}$ configurations (67 samples) produced by the PT Landau gauge fixing and the corresponding first copy[@NF; @FN04]. The $q$ dependence of the $U(q)$ of PT gauge fixed samples and the first copies are shown in FIG. \[u\_pt\_ptr\]. The infrared fluctuation is large in the first copy but it is reduced in the PT gauge fixed samples. It implies a Gribov copy effect in the infrared region[@muell]. The average over $q>0.5$GeV becomes $U(q)=0.45(2)$. This value is comparable to that of [@CMM] obtained by 10000 samples using symmetric momentum $q_1=q_2=q_3=q_4\ne 0$. In [@CMM], the value $U(q)$ between 0 and 2/3 was interpreted as a system deviating from the gaussian distribution. However, since $\vec\phi(q)$ is a 3-dimensional vector, it would not be appropriate to treat it as an 1 dimensional object. The value 0.45 is very close to $$U(q)\sim 1-\frac{d+2}{3d}=\frac{4}{9},$$ or $$\label{u_tilde} \tilde U(q)\sim \frac{5}{3}-\frac{d+2}{d}= 0$$ corresponding to the 3-d gaussian distribution. ![The momentum dependence of Binder cumulant $U(q)$ of SU(2), $\beta=2.2$, $a=1.07$GeV$^{-1}$ of PT samples and first copy samples. (color online) []{data-label="u_pt_ptr"}](u_pt_ptrn.eps){width="7.2cm"} Unquenched SU(3) ---------------- We measured the Binder cumulant of the color anti-symmetric ghost propagator of MILC$_c$. We observed qualitatively different features from quenched SU(2). An average of 9 $\beta_{imp}=6.76$ samples and 12 $\beta_{imp}=6.83$ samples of MILC$_c$ is shown in Fig.\[u\_milc\]. When the $\vec \phi$ is distributed as a Gaussian vector in eight dimensional space, $U(q)=1-{10\over{3\cdot 8}}=0.58$. Data of Fig.\[u\_milc\] suggests that $U(q)$ is slightly larger than the 0.58, and that the shift from Gaussian distribution of $\beta_{imp}=6.76$ samples with light bare quark mass $m_0=11.5MeV$ is larger than that of $\beta_{imp}= 6.83$ samples with heavier bare quark mass $m_0=65.7MeV$. A qualitative difference of unquenched SU(3) (Fig.\[u\_milc\]) from quenched SU(2) (Fig.\[u\_pt\_ptr\]) is the smallness of the fluctuation at the lowest and next to the lowest momentum point \[$q=(0,0,0,1)$ and (0,0,0,2)\]. Relatively large fluctuation exists when one of the spacial components of $q$ is 1 and other components are 0. The difference from the quenched SU(2) could be due to the improvement in the Asqtad action used in the MILC$_c$ and/or the presence of dynamical fermions. ![The momentum dependence of Binder cumulant $U(q)$ of unquenched SU(3), $\beta_{imp}=6.83$ and 6.76, $a=1.64$GeV$^{-1}$ MILC$_c$. (color online) []{data-label="u_milc"}](milc_bindn.eps){width="7.2cm"} Summary and Discussion ====================== We presented color diagonal ghost propagator of quenched $\beta=6.45$ $56^4$ lattice and those of unquenched MILC$_c$ $20^3\times 64$ and MILC$_f$ $28^3\times 96$ configurations. The momentum dependence of standard deviation of the color diagonal ghost dressing function of the unquenched configurations is less singular than that of the quenched configurations. The standard deviation and the mean value of statistical distribution is important for determining the nature of the ensemble. The ghost pair creation operator in the BCS channel is expected to behave as the order parameter, and in the Landau gauge, where ghost pair creation is absent, the ghost anti-ghost pair creation in the Overhauser channel was speculated to become an order parameter. The parameter $v$ of LCO approach that characterize the ghost condensate was compatible with 0 in the SU(2) PT samples. In the unquenched SU(3) MILC$_c$ samples, we found small but positive value of $v$. Uncertainty on $v$ comes mainly from that of $r$, where finite size effect is crucial. We need to extend the analysis to larger lattices to get the definite conclusion. We showed that the Binder cumulant which measures the fluctuation of the ghost propagator differ between quenched and unquenched configurations. We confirmed that the Binder cumulant $U(q)$ of the color anti-symmetric ghost propagator of SU(2) obtained by 10000 samples[@CMM] $U(q)\sim \frac{4}{9}$ is consistent with that obtained by using PT gauge fixed samples. In 3-dimensional system, this data can be interpreted as $\tilde U(q)$ defined as eq.(\[u\_tilde\])$\sim 0$, i.e. the color symmetry is not broken and that the system is in the random phase. When the system is ordered, a certain direction in the color space will be selected and the Binder cumulant would deviate from the value expected by the gaussian distribution. The data of quenched SU(2) $\beta=2.2$ do not show this tendency but the unquenched SU(3) show deviation from the gaussian distribution. Whether it implies the precourser of the ghost condensation in the unquenched QCD is not evident. It would be interesting to extend the analysis to finite temperature and study qualitative differences. The ghost condensates and the $A^2$ condensates are expected to be related by the on-shell BRST symmetry. The observables of lattice Landau gauge in 1-3GeV region suggests presence of $A^2$ condensates. The larger standard deviation of the SU(3) quenched ghost propagator as compared to the unquenched ghost propagator may imply that the ghost propagator is more random in the quenched samples. The fluctuation of the ghost propagator could be the main cause of suppression of the running coupling in the infrared and saturation of the Kugo-Ojima parameter $c$ at about 0.8 in the quenched approximation. It is likely that the fermion field reduces the fluctuation of the color diagonal ghost propagator in the infrared, and renders the Kugo-Ojima parameter $c$ close to 1. We think infrared suppression of the running coupling of unquenched SU(3) measured by eq.(\[alpha\]) presented in [@unquench] is a finite size effect. In the process of measuring the ghost propagator for the running coupling, we observed large fluctuations of the norm and random orientation of the vector in adjoint color space i.e. weakening of the color diagonal structure of the ghost propagator in the infrared. Concerning the fixed points of the running coupling, Wilson[@Wilson] noted in 1971 that the renormalization group flow of the coupling could approach limit cycles which are more elaborate than simple isolated fixed points. A possibility of complicated fractal structure in fixed points was discussed in [@MN]. The Zamolodchikov’s c-theorem in two dimensional conformal field theory[@Zam], however, excludes the limit cycle structure of the infrared fixed points. In four dimensional QCD, the situation is obscure[@Card]. To clarify the nature of the infrared fixed points, it is necessary to investigate the continuum limit of the lattice Landau gauge QCD via systematic studies of finite size effects and the Gribov copy effects. This work was supported by the KEK supercomputing project 04-106 and 05-128. The numerical calculation was performed also at the Cyber Media Center of Osaka Universitry. H.N. is supported by the MEXT grant in aid of scientific research in priority area No.13135210. [99]{} T. Kugo and I. Ojima, [Suppl.]{} [**66**]{}, 1 (1979). T. Kugo, hep-th/9511033. S. Furui and H. Nakajima, hep-lat/0503029. E.-M. Ilgenfritz, M. Müller-Preussker and A. Sternbeck, hep-lat/0601027. Ph. Boucaud, J.P. Leroy, A. Le Yaouanc, A.Y. Lokhov, J. Michelli, O. Pène, J. Rodríguez-Quintero and C. Roiesnel, hep-lat/0506031. S. Furui and H. Nakajima, [**69**]{},[074505]{}[(2004)]{}, hep-lat/0305010. S. Furui and H. Nakajima, [**70**]{},[094504]{}[(2004)]{}, hep-lat/0403021. A. Sternbeck, E.-M. Ilgenfritz, M. Müller-Preussker and A. Schiller, [**72**]{} [014507]{} [(2005)]{}, hep-lat/0506007. J.C.R. Bloch, A. Cucchieri, K. Langfeld and T. Mendes, [**687**]{},[76]{}[(2004)]{}. C. Lerche and L. von Smekal,[**65**]{},[125006]{}[(2002)]{}, hep-ph/0202194 D. Zwanziger, [**69**]{},[016002]{}[(2004)]{}, hep-ph/0303028. J.C.R. Bloch, Few Body Syst [**33**]{},[111]{}[(2003)]{}. K-I. Kondo, hep-th/0303251. Ph. Boucaud et al., hep-lat/0602006. O. Oliveira and P.J. Silva, PoS [**LAT2005**]{}287(2005), hep-lat/0509037; hep-lat/0511043. S. Furui and H. Nakajima, [**73**]{},[074503]{}(2006), hep-lat/0511045. R. Alkofer, W. Detmold, C.S. Fischer and P. Maris, [**70**]{},[014014]{}[(2004)]{}. S. Furui and H. Nakajima, PoS [**LAT2005**]{} 291 (2005), hep-lat/0509035. K-I. Kondo, [**619**]{},[377]{}[(2005)]{}, hep-th/0504088. B.M. Gripaios, [**558**]{},[250]{}[(2003)]{}. A.A. Slavnov, [**608**]{},[171]{}[(2005)]{}. D. Dudal, H. Verschelde, V.E.R. Lemes, M.S. Sarandy, S.P. Sorella, M. Picariello, A. Vicini and J.A. Gracey, JHEP306 hep-th/0305020. A. Cucchieri, T. Mendes and A. Mihara, hep-lat/0508028. K. Binder, [**47**]{},[693]{}[(1981)]{}. D. Zwanziger, [**399**]{}[477]{}[(1993)]{}. J.A. Gracey, hep-ph/0510151. M.A.L. Capri, D. Dudal, J.A. Gracey, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella and H. Verschelde, hep-th/0508216. H. Nakajima and S. Furui, (Proc.Suppl)[**114**]{},[34]{}[(2005)]{}, hep-lat/0408001 C. Bernard et al., [**58**]{},[014503]{}[(1998)]{}. A. Cucchieri and T. Mendes, hep-lat/0602012. G. Parisi and J.J.Ruiz-Lorenzo, [**54**]{},[R3698]{}[(1996)]{}, cond-mat/9604166. M.N. Barber, R.B. Pearson, D. Toussaint and J.L. Richardson, [**32**]{},[1720]{}[(1985)]{}. N.A. Alves, B. A. Berg and S. Sanielevici, [**376**]{},[218]{}[(1992)]{}. J. Fingberg, U. Heller and F. Karsch, [**392**]{},[493]{}[(1993)]{}, hep-lat/9208012. A. Morozof and A.J. Niemi, [**666**]{},[311]{}[(2003)]{}, hep-th/0304178. K.G. Wilson, [**3**]{},[1818]{}[(1971)]{}. A.B. Zamolodchikov, Sov. J. Nucl. Phys.[**46**]{},[1090]{}[(1987)]{}. J.L. Cardy, [**215**]{},[749]{}[(1988)]{}.

--- abstract: 'The lines of lithium at 6708 Å and 6103 Å are analyzed in high resolution spectra of some sharp-lined and slowly rotating roAp stars. Three spectral synthesis codes - STARSP, ZEEMAN2 and SYNTHM were used. New lines of the rare earth elements from the DREAM database, and lines calculated on the basis of the NIST energy levels were included. Magnetic splitting and other line broadening processes were taken into account. Enhanced abundances of lithium in the atmospheres of the stars studied are obtained for both the lithium lines. High estimates of $^6$Li/$^7$Li ratio ($0.2\div0.5$) for the studied stars can be explained by Galactic Cosmic Ray (GCR) production by to spallation reactions and the preservation of the original $^6$Li and $^7$Li by the strong magnetic fields.' date: '?? and in revised form ??' --- Introduction ============ In the framework of the project “Lithium in CP stars", a significant series of observations was obtained at ESO and CrAO (R=100000 and 50000 respectively, 1996–2001) for 5 rapidly oscillating Ap (roAp) stars: 33 Lib (HD137947), $\gamma$ Equ (HD201601), HD134214, HD166473, HD101065, in the spectral region 6680–6730 Å. These series were supplemented by ESO (March 2004) and SAO-BTA (April 2004) spectra with R=100000 and 60000. The observations show very strong and non-variable resonance doublets of Li [i]{} at 6708Å. The spectra of these roAp stars are group II in the classification of lithium roAp stars in accordance with the Li [i]{} line 6708Å appearance over the phases ([@Polosukhina99 Polosukhina, Kurtz, Hack,  1999])). All these stars are characterized by sharp lines in their spectra, by the strong overabundance of rare earth elements, and by magnetic fields from 2 kG up to 6.8 kG. The sharp lines ($2\div3$ km s$^{-1}$) in the spectra of these stars result from small $v_e \sin{i}$. For the stars with short rotational periods the sharp lines appear to be due to the combination of equatorial velocity $v_e$ and a significant inclination angle $i$. For the stars with longer periods (of some years) – $\gamma$ Equ and 33 Lib – the width of the lines is attributed by slow rotation. (Note that the broadening of spectral lines due to rotation is not distinguished from the broadening due to the rapid oscillations). Some of the stars are therefore observed “pole-on", and an observer always sees only one hemisphere of these star. In this case the spectrum is essentially constant. ![Left: The estimation of field parameters from the lines Ce [ii]{} 6706.051Å and Pr [iii]{} 6706.705Å for HD101065, red solid line: $B_r=0$kG, $B_m=2.3$kG, $B_l=0$kG; blue solid line: $B_r=0$kG, $B_m=2.8$kG, $B_l=0$kG; green solid line: $B_r=2.3$kG, $B_m=0$kG, $B_l=0$kG; righ: for 33 Lib, blue line: $B_r=2$kG, $B_m=5$kG, $B_l=0$kG and red line: $B_r=5$kG, $B_m=2$kG, $B_l=0$kG.[]{data-label="fig1ab"}](fp14_fig1a.eps "fig:"){width="2.5in"} ![Left: The estimation of field parameters from the lines Ce [ii]{} 6706.051Å and Pr [iii]{} 6706.705Å for HD101065, red solid line: $B_r=0$kG, $B_m=2.3$kG, $B_l=0$kG; blue solid line: $B_r=0$kG, $B_m=2.8$kG, $B_l=0$kG; green solid line: $B_r=2.3$kG, $B_m=0$kG, $B_l=0$kG; righ: for 33 Lib, blue line: $B_r=2$kG, $B_m=5$kG, $B_l=0$kG and red line: $B_r=5$kG, $B_m=2$kG, $B_l=0$kG.[]{data-label="fig1ab"}](fp14_fig1b.eps "fig:"){width="2.5in"} Synthetic spectra ================= These stars with strong 6708Å lithium doublets are very poorly studied. We study their spectra in detail in a narrow range near 6708Å by the method of synthetic spectra, taking into account Zeeman magnetic splitting and blending by REE lines. The additional broadening, likely pulsational was described by the parameter $v \sin{i}$. Spectral calculations for HD166473, $\gamma$Equ and 33 Lib were carried out using the model atmospheres of [@Kurucz94] with parameters from the papers of [@Gelbman00], [@Ryabchikova97], [@Ryabchikova99]. For HD101065 Pavlenko’s model was used, as in the work of [@Shavrina03]. For synthetic spectra calculations we applied the magnetic spectrum synthesis code SYNTHM ([@Khan04 Khan 2004]), which is similar to Piskunov code SYNTHMAG and was tested in accordance with the paper of [@Wade01]. Also for initial calculations we used the code STARSP of [@Tsymbal96] and in some cases the code ZEEMAN2 [@Wade01]. ![Left: The fitting of observed and calculated spectra of HD101065 near 6708Å: black: observed spectrum; red line: calculated spectrum taking into account lines of the main isotope $^{7}$Li only; green line: spectrum with the ratio $^{6}$Li/$^{7}$Li = 0.4. The positions of those lines which are the main contributors in absorption are marked at the top of the figure; right: N(Li)=$-8.50 \pm 0.2$, $^{6}$Li/$^{7}$Li=0.4[]{data-label="fig2"}](fp14_fig2a.eps "fig:"){width="2.5in"} ![Left: The fitting of observed and calculated spectra of HD101065 near 6708Å: black: observed spectrum; red line: calculated spectrum taking into account lines of the main isotope $^{7}$Li only; green line: spectrum with the ratio $^{6}$Li/$^{7}$Li = 0.4. The positions of those lines which are the main contributors in absorption are marked at the top of the figure; right: N(Li)=$-8.50 \pm 0.2$, $^{6}$Li/$^{7}$Li=0.4[]{data-label="fig2"}](fp14_fig2b.eps "fig:"){width="2.5in"}\ The simplified model of the magnetic field is characterized by radial(along line of sight), meridional and longitudinal components of field $B_{\rm r}, B_{\rm m}, B_{\rm l}$ ($B_{\rm l}$ = 0 always, as it is justified for the plane-parallel model atmospheres), which were primarily determined from Fe [ii]{} lines 6147Å, 6149Å Ce [ii]{} 6706.05 and Pr [iii]{} 6706.70Å (see Table \[tab1\]). ![Fe [ii]{} lines 6147Å and 6149Å for 33Lib with magnetic field components. Left: $B_r=4.2$kG, $B_m=3.3$kG, $B_l=0$kG; right: $B_r=2$kG, $B_m=5$kG and $B_l=0$kG like for Pr [iii]{} 6708Å (see Fig \[fig1ab\]b).[]{data-label="fig2a"}](fp14_fig3a.eps "fig:"){width="2.5in"} ![Fe [ii]{} lines 6147Å and 6149Å for 33Lib with magnetic field components. Left: $B_r=4.2$kG, $B_m=3.3$kG, $B_l=0$kG; right: $B_r=2$kG, $B_m=5$kG and $B_l=0$kG like for Pr [iii]{} 6708Å (see Fig \[fig1ab\]b).[]{data-label="fig2a"}](fp14_fig3b.eps "fig:"){width="2.5in"}\   HD101065 HD134214 HD137949 HD137949 HD166473 HD201601 ------------------------------- ----------------- ------------- ------------ ------------ ------------ ------------- T$_{eff}$/$\log{g}$/\[m\] 6600/4.2/0 7500/4.0/0 7750/4.5/0 7250/4.5/0 7750/4.0/0 7750/4.0/0 N(FeI) 6103Å 6.95 (Fe[ii]{}) 7.60 8.00 7.80 - 7.80 N(Fe[ii]{}) 6149Å - 7.25 7.70 7.80 7.35 7.50 N(Li) 6708Å 3.1 3.9 4.1 3.6 3.7 3.8 N(Li) 6103Å 3.5 4.1 4.4 4.4 - 4.0 $^6$Li/$^7$Li 6708Å 0.4: 0.3: 0.2: 0.3: 0.5: 0.3: $B_r/ B_m/ B_l/$ (kG) Fe[ii]{} 6149Å - -2.9/-1.7/0 4.1/4.1/0 4.2/3.3/0 2.0/6.0/0 3.5/2.6/0.8 Pr [iii]{}I 6706.7Å 0/2.3/0 -2.3/-1.9/0 2.0/5.0/0 1.5/5.0/0 2.0/6.5/0 2.7/3.5/0 CaI 6102.7Å 0/2.4/0 -1.7/-2.8/0 3.0/4.0/0 3.5/4.0/0 - 0/4.0/0 $v \sin i ({\rm km\,s^{-1}})$ - 3.0 2.5 2.5 3.0 2.5 Fe [ii]{} - $v \sin i ({\rm km\,s^{-1}})$ 3.5 2.0 4.0 4.0 5.5 2.5 Pr [iii]{} : []{data-label="tab1"} REE lines with new atomic data ============================== We used the VALD ()) and DREAM [^1] databases for atomic spectral lines. These data do not in fact allow us to fit synthetic spectra to the observed ones for all stars studied. We therefore calculated additional REE [ii]{}-[iii]{} lines using NIST energy levels and estimated their “astrophysical” gf-values from the spectra of HD101065 using elemental abundances from [@Cowley00]. As well, the theoretical gf-values for important (under the lithium abundance determination) blending lines were especially computed by P. Quinet with Cowan’s code ([@Shavrina03 Shavrina, Polosukhina, Pavlenko,  2003]). HD101065 ======== We present a new version of the spectraa analysis of the star HD101065 in the lithium spectral ranges 6708Å and 6103Å using the new atomic data for REE lines and the new magnetic synthesis code SYNTHM ([@Khan04 Khan 2004]). The lithium abundance estimates from 6708Å and 6103Å are 3.1 dex and 3.4 dex respectively, in the scale of $\log{N(H)}$=12.0 dex, and its isotopic ratio $^6$Li/$^7$Li is about 0.4 (6708Å) and 0.3 (6103Å). ![For 33 Lib a) Li [i]{} 6708 Å, blue line: log N(Li) =-7.95, $^6$Li/$^7$Li =0.2; red line: log N(Li)= -7.88, only $^7$Li. b) Li [i]{} 6103 Å, green line: log N(Li)= -7.60 $\pm 0.3$, $^6$Li/$^7$Li =0.2 []{data-label="f1"}](fp14_fig4a.eps "fig:"){width="2.5in"} ![For 33 Lib a) Li [i]{} 6708 Å, blue line: log N(Li) =-7.95, $^6$Li/$^7$Li =0.2; red line: log N(Li)= -7.88, only $^7$Li. b) Li [i]{} 6103 Å, green line: log N(Li)= -7.60 $\pm 0.3$, $^6$Li/$^7$Li =0.2 []{data-label="f1"}](fp14_fig4b.eps "fig:"){width="2.5in"}\ Results ======= Results of the work are presented in the Table. I the first line, the HD numbers and in the second one - the parameters of used model atmospheres are given. The calculations for star HD 137949 were carried out for two model atmospheres in a possible effective temperature range - 7750/4.5/0 and 7250/4.5/0. In six column for each star(model) we give the abundances of Fe I and Fe II in the scale of log N(H)=12.0, derived from a group of the Fe I lines (6102-6103 Å) and Fe II 6149 Å(3,4 lines in the table). For HD 101065 with weak Fe lines we use the abundance of Fe II from the paper of Cowley et al. (2000) to take into account Fe II line 6103.496, which is near Li I lines (6103.538, 6103.649, 6103.664). The abundances of lithium determined from both 6708 A and 6103 A lines and isotopic ratio from 6708 A line are shown in 5-7 lines of table. Under the solid line we give the parameters of magnetic field and vsini found from the fitting of Fe II 6149 Å, Pr III 6706.7 and Ca I 6102.7 lines. Last value of vsini was used for spectra calculations in both lithium lines ranges. Magnetic field parameters from Ca I 6102.7 were used in the 6103 Årange and ones from Pr III 6707.6 Å- for 6708 Årange. Conclusions =========== - The lithium abundance for all stars determined from the Li [i]{}I 6103Å line is higher than the abundance determined from Li [i]{}I 6708Å. This may be evidence of vertical lithium stratification, an abnormal temperature distribution, or consistent unidentified blending with the 6103Å line. - Our work on two roAp stars, HD83368 and HD60435 ([@Shavrina01 Shavrina, Polosukhina, Zverko,  2001]) provides evidence of an enhanced lithium abundance near the magnetic field poles. We can expect similar effects in sharp-lined roAp stars. The high lithium abundance for all stars determined from the Li [i]{}I lines and the estimates of $^6$li/$^7$Li ratio ($0.2\div0.5$) can be explained by the Galactic Cosmic Ray (GCR) production due to spallation reactions with ISM in the areas of these stars formation and preservation of original both $^6$Li and $^7$Li by the strong magnetic fields of these stars. The values of the $^6$Li/$^7$Li ratio expected from GCR production are about $0.5\div0.8$ ([@Knauth03 Knauth et al. 2003], [@Webber02 Webber, Lukasiak, McDonald, 2002]). - The new laboratory and theoretical gf-values for REE lines are necessary in order to refine our estimates of lithium abundances and the isotopic ratio. The authors are grateful to Dr. J. Zverko and Dr. J. Žižňovský for their useful comments. A. Shavrina, N. Polosukhina, V. Khalack and V. Gopka would like also to express gratitude to the Local Organizing Committee and IAU for financial support. , D.R.E.A.M. Database on Rare Earth at Mons.Univ., *http://www.umh.ac.be/$\tilde{\ }$astro/dream.shtml* 2000, *MNRAS* 317, 299 , 2000, *A&A* 356, 200 2004, *JQSRT*, 88, N1-3, 71 2003, *ApJ* 586, 268 1999, *A&A* 138, 119 1994, CDR(1-23) 1999, *A&A* 351, 283 [^1]: \[*http://www.umh.ac.be/$\tilde{\ }$astro/dream.shtml*\]

--- abstract: 'In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus (“cat maps”). In Part II of the series, we construct quasimodes that are quantum ergodic but are not equidistributed at the logarithmical scales.' address: 'Department of Mathematics, California State University, Northridge, CA 91330, USA' author: - Xiaolong Han title: 'Small scale quantum ergodicity in cat maps. II. Quasimodes that are not equidistributed at the logarithmical scales' --- Introduction ============ We recall the setup of cat maps briefly and refer to Part I [@Han] of the series for more background. A classical cat map on the torus ${\mathbb{T}}^2$ is defined by a matrix $M\in{\mathrm{SL}}(2,{\mathbb{Z}})$ with $|{\mathrm{Tr}\,}M|>2$. Its iterations $M^t$ ($t\in{\mathbb{Z}}$) induce a discrete hyperbolic (i.e., chaotic) dynamical system. In the quantum cat system, the phase space is the $2$-$\dim$ torus ${\mathbb{T}}^2=\{(q,p):q,p\in{\mathbb{T}}^1\}$, in which $q$ and $p$ denote the position and momentum variables, respectively. Therefore, a quantum state can be represented by a distribution $\psi(q)$ on ${\mathbb{R}}^1$ such that $\psi$ and its Fourier transform are both periodic. One can then determine that the spaces of such quantum states are $N$-$\dim$ Hilbert spaces with $N\in{\mathbb N}$, by which we denote ${\mathcal H}_N$. Finally, the quantum cat map $\hat M$ is a unitary operator acting on ${\mathcal H}_N$ and $\hbar=1/(2\pi N)\to0$ plays the role of the Planck parameter. One of the main problems in *Quantum Chaos* for cat maps is concerned with the density distribution of eigenstates (or more generally, approximate eigenstates, i.e., quasimodes) of $\hat M$ as $\hbar\to0$. The density distribution of a quantum state $\psi$ in the physical space can be studied via $\int_\Omega|\psi(q)|^2\,dq$, in which $\Omega\subset{\mathbb{T}}^1$. In particular, we say that a sequence of normalized states $\{\psi_j\}_{j=1}^\infty$ with $\psi_j\in{\mathcal H}_{N_j}$ tend equidistributed at the macroscopic scale in the physical space ${\mathbb{T}}^1$ if for any open subset $\Omega\subset{\mathbb{T}}^1$, $$\int_\Omega|\psi_j(q)|^2\,dq\to{\mathrm{Vol}}(\Omega)\quad\text{as }j\to\infty.$$ We say that $\{\psi_j\}_{j=1}^\infty$ tend equidistributed at a small scale $r=r(N)\to0$ as $N\to\infty$ in the physical space if $$\int_{B_1(q_0,r_j)}|\psi_j(q)|^2\,dq={\mathrm{Vol}}(B_1(q_0,r_j))+o(r_j)\quad\text{as }j\to\infty,$$ uniformly for all $q_0\in{\mathbb{T}}^1$. Here, $r_j=r(N_j)$ and $B_d(x,r)$ is the geodesic ball in ${\mathbb{T}}^d$ with radius $r$ and center $x$. The density distribution of a quantum state $\psi$ in the phase space ${\mathbb{T}}^2$ can be studied via $\braket{\psi|\hat f|\psi}$, in which $f\in C^\infty({\mathbb{T}}^2)$ and $\hat f$ is its quantization. In particular, we say that a sequence of normalized states $\{\psi_j\}_{j=1}^\infty$ with $\psi_j\in{\mathcal H}_{N_j}$ is quantum ergodic (i.e., they tend equidistributed at the macroscopic scale in the phase space ${\mathbb{T}}^2$) if for all $f\in C^\infty({\mathbb{T}}^2)$, $$\braket{\psi_j|\hat f|\psi_j}\to\int_{{\mathbb{T}}^2}f(x)\,dx\quad\text{as }j\to\infty.$$ Here, $dx$ is the Lebesgue measure on ${\mathbb{T}}^2$. It is obvious that a quantum ergodic sequence is also equidistributed in the physical space. We say that $\{\psi_j\}_{j=1}^\infty$ is quantum ergodic at a small scale $r=r(N)\to0$ as $N\to\infty$ if $$\braket{\psi_j|\hat b_{x,r_j}^\pm|\psi_j}={\mathrm{Vol}}(B_1(x,r_j))+o\left(r_j^2\right)\quad\text{as }j\to\infty,$$ uniformly for all $x\in{\mathbb{T}}^2$. Here, $b_{x,r}^\pm$ are the appropriate smooth functions that approximate the indicator function of $B_2(x,r)\subset{\mathbb{T}}^2$. See Subsection \[sec:AW\] for more details. Quantum ergodicity at small scales characterize refined density distribution properties of the states than the one at the macroscopic scale. Establishing these small scale results in various dynamical systems, as well as proving optimal scales for such properties, have attracted a lot of attention. See Part I [@Han] of the series for the recent history. In the context of cat maps, the Quantum Ergodicity theorem [@BouDB; @Ze] asserts that a full density subsequence of any eigenbasis of the quantum cat map is quantum ergodic at the macroscopic scale. In Part I [@Han] of the series, we established quantum ergodicity of eigenstates at various small scales. In particular, we proved that a full density subsequence of any eigenbasis is quantum ergodic at logarithmical scales $(\log N)^{-\alpha}$ for some $\alpha>0$. In addition, the scale of quantum ergodicity can be greatly improved to polynomial scales $N^{-\beta}$ for some $\beta>0$, under certain conditions such as the Hecke symmetry requirement. In Part II, we investigate the potential failure of quantum ergodicity at certain small scales. In particular, we construct approximate eigenstates (i.e., quasimodes) which are quantum ergodic at the macroscopic scale but fail quantum ergodicity at some logarithmical scales. Here, we say that a sequence of states $\{\psi_j\}_{j=1}^\infty$ are quasimodes of order $R(N)\to0$ as $N\to\infty$ if $\psi_j\in{\mathcal H}_{N_j}$ and there are $\{\phi_j\}_{j=1}^\infty\subset{\mathbb{R}}$ such that $$\label{eq:qm} \left\|\left(\hat M-e^{i\phi_j}\right)\psi_j\right\|_{L^2({\mathbb{T}}^1)}=R(N_j)\|\psi_j\|_{L^2({\mathbb{T}}^1)}.$$ Note that since $\hat M$ is unitary, the eigenvalues (or energy) have module one. Therefore, $e^{i\phi_j}$ is the “quasi-energy” of the quasimode $\psi_j$. Setting the reminder $R=0$ reduces the quasimodes to exact eigenstates. Our main theorem states that \[thm:SSQE\] Let $M$ be a cat map on ${\mathbb{T}}^2$. Then there exist a quantum ergodic sequence of normalized quasimodes $\{\psi_j\}_{j=1}^\infty$ of order $O((\log N)^{-1/2})$ such that $\psi_j\in{\mathcal H}_{N_j}$ satisfies the following non-equidistribution conditions. For any ${\varepsilon}>0$, there are constants $c_0=c_0({\varepsilon},M)>0$ and $j_0=j_0({\varepsilon},M)\in{\mathbb N}$ such that for all $j\ge j_0$, 1. at the scale $r_j=c_0(\log N_j)^{-1}$ in the physical space, $$\sup_{q\in{\mathbb{T}}^1}\left\{\frac{\int_{B_1(q,r_j)}|\psi_j|^2}{{\mathrm{Vol}}(B_1(q,r_j))}\right\}\ge{\varepsilon}^{-1}\quad\text{and}\quad\inf_{q\in{\mathbb{T}}^1}\left\{\frac{\int_{B_1(q,r_j)}|\psi_j|^2}{{\mathrm{Vol}}(B_1(q,r_j))}\right\}\le{\varepsilon},$$ 2. at the scale $r_j=c_0(\log N_j)^{-1/2}$ in the phase space, $$\sup_{x\in{\mathbb{T}}^2}\left\{\frac{\braket{\psi_j|\hat b_{x,r}^\pm|\psi_j}}{{\mathrm{Vol}}(B_2(x,r_j))}\right\}\ge{\varepsilon}^{-1}\quad\text{and}\quad\inf_{x\in{\mathbb{T}}^2}\left\{\frac{\braket{\psi_j|\hat b_{x,r}^\pm|\psi_j}}{{\mathrm{Vol}}(B_2(x,r_j))}\right\}\le{\varepsilon}.$$ Theorem \[thm:SSQE\] does not apply to exact eigenstates and we leave it for the future study. In the case of quasimodes, we in fact prove a much more general result than Theorem \[thm:SSQE\]. To state the result, we say that a measure $\mu$ on ${\mathbb{T}}^2$ is a semiclassical measure induced by a sequence of states $\{\psi_j\}_{j=1}^\infty$ if for all $f\in C^\infty({\mathbb{T}}^2)$ $$\label{eq:scmeasures} \braket{\psi_j|\hat f|\psi_j}\to\int_{{\mathbb{T}}^2}f(x)\,d\mu\quad\text{as }j\to\infty.$$ So the semiclassical measure $\mu$ characterizes the density distribution of the states $\{\psi_j\}_{j=1}^\infty$ at the macroscopic scale. For example, if $\mu=\mu_\gamma$ as the delta measure on some closed orbit of the classical cat map $M$ on ${\mathbb{T}}^2$, then the states concentrate near the orbit asymptotically and are said to be “scarring”. (See for the delta measure $\mu_\gamma$ defined on a closed orbit $\gamma$.) On the other hand, the states are quantum ergodic at the macroscopic scale if and only if the corresponding semiclassical measure coincides with the Lebesgue measure. Our next theorem states that \[thm:sc\] Let $M$ be a cat map on ${\mathbb{T}}^2$ and $\mu$ be an invariant probability measure of $M$. Then there exists a sequence of normalized quasimodes $\{\psi_j\}_{j=1}^\infty$ of order $O((\log N)^{-1/2})$ with corresponding semiclassical measure $\mu$ such that the non-equidistribution conditions (i) and (ii) in Theorem \[thm:SSQE\] hold. Since the Lebesgue measure is an invariant probability measure of $M$, Theorem \[thm:SSQE\] follows directly from Theorem \[thm:sc\]. Suppose that $\{\psi_j\}_{j=1}^\infty$ are normalized quasimodes of order $O((\log N)^{-1/2})$ as considered in Theorems \[thm:SSQE\] and \[thm:sc\]. Then it is well known that the corresponding semiclassical measures are probability measures which are invariant under $M$. (In fact, the result remains valid for quasimodes of order $o(1)$. See Zworski [@Zw Chapter 5] and also Subsection \[sec:qtorus\] for a short proof in the context of cat maps.) From this point of view, Theorem \[thm:sc\] provides a reverse statement that any invariant probability measure of $M$ must arise as a semiclassical measure that is induced by quasimodes of order $O((\log N)^{-1/2})$, moreover, these quasimodes are not equidistributed at the same logarithmical scales as in Theorem \[thm:SSQE\]. We shall also point out that Theorem \[thm:sc\] is invalid for exact eigenstates. Indeed, the set of semiclassical measures corresponding to eigenstates is smaller than the one of invariant probability measures, see [@B1; @BonDB; @FN; @R]. For example, the delta measure on a closed prime orbit can not be the semiclassical measure induced by eigenstates. Therefore, Theorem \[thm:sc\] demonstrates the sharp difference between the exact eigenstates and quasimodes of logarithmical order. Outline of the plan {#outline-of-the-plan .unnumbered} ------------------- The construction of the quasimodes in Theorems \[thm:SSQE\] and \[thm:sc\] is inspired by Faure-Nonnenmacher-De Bièvre [@FNDB]. Let $\phi\in{\mathbb{R}}$ and $\gamma=\{x_t\}_{t=0}^{T-1}$ be a closed prime orbit of the cat map $M$ on ${\mathbb{T}}^2$. Construct the quantum state $\Psi^\gamma_\phi\in{\mathcal H}_N$ by $$\ket{\Psi^\gamma_\phi}=\sum_{t=0}^{T-1}e^{-i\phi t}\hat M^t\ket{x_0,{{\tilde c_0}},\theta}.$$ Here, $\ket{x_0,{{\tilde c_0}},\theta}\in{\mathcal H}_N$ is a coherent state centered at $x_0\in{\mathbb{T}}^2$ and localized in a region with width $\sim\hbar^\frac12$. (The precise localization of a quantum state is analyzed via its Husimi function. See Subsection \[sec:qmHusimi\].) Under the quantum evolution with $t>0$, $\hat M^t\ket{x_0,{{\tilde c_0}},\theta}$ becomes less localized. Assume that $M$ has Lyapunov exponent $\lambda>0$ (so the eigenvalues of $M$ are $e^{\pm\lambda}$). Then $\hat M^t\ket{x_0,{{\tilde c_0}},\theta}$ has center at $M^tx_0=x_t$ and localization with width $\sim\hbar^\frac12e^{\lambda t}$. We therefore introduce the Ehrenfest time $$\label{eq:TEhrenfest} T_E=\frac{|\log\hbar|}{\lambda}.$$ Thus, $\hat M^t\ket{x_0,{{\tilde c_0}},\theta}$ remains well localized near $x_t$ if $t\le\delta T_E$ for $\delta<1/2$. From the basic property of cat maps, the points on a closed prime orbit $\{x_t\}_{t=0}^{T-1}$ are separated by distance at least $e^{-\lambda T}$ in ${\mathbb{T}}^2$. Combining these two estimates, $$\hbar^\frac12e^{\lambda T}\ll e^{-\lambda T}\quad\text{if }T\le\delta T_E\text{ for }\delta<\frac14.$$ That is, within this time frame, $\hat M^t\ket{x_0,{{\tilde c_0}},\theta}$ ($t=0,...,T-1$) are well localized in disjoint regions so the state $\Psi^\gamma_\phi$ is approximately a direct sum of them. Suppose that $\mu$ is an invariant probability measure of $M$ on ${\mathbb{T}}^2$. Then by Sigmund [@S], there is a sequence of closed prime orbits $\{\gamma_j\}_{j=1}^\infty$ such that the delta measures $\mu_{\gamma_j}\to\mu$ weakly. If the lengths $|\gamma_j|$ of $\gamma_j$ are bounded, then $\mu$ must be itself a delta measure on some closed prime orbit. This is because the orbits of $M$ are enumerable by length. In this case, the construction of quasimodes with corresponding semiclassical measure $\mu$ was done by Faure-Nonnenmacher-De Bièvre [@FNDB]. (Indeed, the quasimodes are the localized parts of the ones constructed in [@FNDB]. See related liturature below for the difference between the approach in [@FNDB] and the one in this paper.) These quasimodes are not quantum ergodic at the macroscopic scale therefore satisfy the non-equidistribution conditions as in (i) and (ii) of Theorem \[thm:SSQE\] at any small scale. In this paper, we discuss the case when the lengths $|\gamma_j|\to\infty$. Fix $0<\delta<1/4$. Assign $\hbar_j\sim e^{\lambda|\gamma_j|/\delta}$ so $|\gamma_j|\sim\delta|\log\hbar_j|/\lambda$. Construct the quantum state $\Psi_{\phi_j}^{\gamma_j}\in{\mathcal H}_{N_j}$ as above. For notational simplicity, we drop the subscription for now. We have seen that $\Psi^{\gamma}_{\phi}$ localizes near $\gamma=\{x_t\}_{t=0}^{T-1}$, on which the points are also well separated. It then follows immediately that $\|\Psi^{\gamma}_{\phi}\|_{L^2}\sim\sqrt{T}$. Observe that $$\label{eq:hatMphi} \left\|\left(\hat M-e^{i\phi}\hat I\right)\sum_{t=0}^{T-1}e^{-i\phi t}\hat M^t\right\|=\left\|e^{-i\phi T}\hat M^T-\hat I\right\|\le2.$$ The states $\Psi^{\gamma}_{\phi}$ therefore are quasimodes of order $O(1/\sqrt{T})=O(|\log\hbar|^{-1/2})$. In addition, $\Psi^{\gamma}_{\phi}$ localizes near the classical orbit $\gamma$, on which the delta measure tends to $\mu$ as $T\to\infty$. Then the semiclassical measure induced by $\Psi^{\gamma}_{\phi}$ is also $\mu$. The rigorous analysis requires the detailed study of their Husimi functions. To observe the non-equidistribution phenomenon, we first note that a closed prime orbit $\gamma$ of length $T$ in ${\mathbb{T}}^2$ can not be equidistributed at any scale $r$ if $r\ll T^{-1/2}\sim|\log\hbar|^{-1/2}$. Indeed, one can find $\sim r^{-2}\gg T$ disjoint balls in ${\mathbb{T}}^2$. From the pigeon-hole principle, there are balls that do not intersect $\gamma$. This readily shows the non-equidistribution of $\gamma$ in ${\mathbb{T}}^2$. Since the quasimode $\Psi^{\gamma}_{\phi}$ is localized near the orbit $\gamma$, it must also display non-equidistribution at the same scale in ${\mathbb{T}}^2$. The analysis here again needs the study of their Husimi functions. The non-equidistribution of $\Psi^{\gamma}_{\phi}$ at scales $r\ll T^{-1}\sim|\log\hbar|^{-1}$ in the physical space ${\mathbb{T}}^1$ can be argued similarly. Related literature {#related-literature .unnumbered} ------------------ The cat maps are the simplest examples of hyperbolic dynamical systems. We expect that the *Quantum Chaos* study in this series would motivate a more general approach for other hyperbolic systems, such as the geodesic flow on compact manifolds with negative curvature. The eigenstates in the corresponding quantum system can be described by Laplacian eigenfunctions on the manifold. The density equidistribution of eigenfunctions as well as quasimodes have been extensively studied. See Part I [@Han] for the discussion on these results. The study of non-equidistribution of quasimodes of logarithmical order have been studied by Brooks [@B2] on surfaces of constant curvature, Eswarathasan-Nonnenmacher [@EN] on surfaces of variable curvature, and Eswarathasan-Silberman [@ES] on higher dimensional manifolds of constant curvature. In these various settings, they construct quasimodes that concentrate near a fixed closed orbit and therefore fail equidistribution at the macroscopic scale. These arguments are similar in spirit to Faure-Nonnenmacher-De Bièvre [@FNDB] for cat maps. In our construction however, there are a family of closed orbits with lengths tending infinity. The quasimodes are associated with this family, instead of one fixed orbit. When the delta measures on the orbits of this family tend to a measure, the quasimodes are then designed to recover the same measure in the semiclassical limit. It is interesting to see if such construction can also be carried out on manifolds. Organization of the paper {#organization-of-the-paper .unnumbered} ------------------------- In Sections \[sec:plane\] and \[sec:torus\], we recall the necessary tools of classical and quantum dynamics to construct the quasimodes . In Section \[sec:qm\], we describe the Husimi functions of these quasimodes and prove the properties in Theorem \[thm:sc\]. Classical dynamics and quantum dynamics on the plane {#sec:plane} ==================================================== In this section, we introduce the classical linear hyperbolic systems on the phase space ${\mathbb{R}}^2$ and their quantum systems. We follow the setup in Faure-Nonnenmacher-De Bièvre [@FNDB]. In particular, we mention several interpretations of the classical system, define the corresponding quantum system, and analyze the quantum evolution of coherent states. Classical dynamics on the plane ------------------------------- Consider the quadratic Hamiltonian on the phase space ${\mathbb{R}}^2$ that is given by $$\label{eq:Ham} H(q,p)=\frac12\alpha q^2+\frac12\beta p^2+\gamma qp.$$ It generates the Hamiltonian flow $M(t):x(0)=(q(0),p(0))\to x(t)=(q(t),p(t))$, in which $$M(t)=\exp\left\{t\begin{pmatrix} \gamma & \beta\\ -\alpha & -\gamma \end{pmatrix}\right\}.$$ Define $$\label{eq:M} M=M(1)=\exp\left\{\begin{pmatrix} \gamma & \beta\\ -\alpha & -\gamma \end{pmatrix}\right\}=\begin{pmatrix} A & B\\ C & D \end{pmatrix}\in{\mathrm{SL}}(2,{\mathbb{R}}).$$ Set $\lambda=\sqrt{\gamma^2-\alpha\beta}$. Then $$\begin{cases} A=\cosh\lambda+\frac{\gamma}{\lambda}\sinh\lambda,\quad B=\frac\beta\lambda\sinh\lambda,\\ C=-\frac\alpha\lambda\sinh\lambda,\quad D=\cosh\lambda-\frac\gamma\lambda\sinh\lambda. \end{cases}$$ - If $\gamma^2>\alpha\beta$, then $M(t)$ is a hyperbolic flow with Lyapunov exponent $\lambda=\sqrt{\gamma^2-\alpha\beta}$ and $M$ is a hyperbolic map with eigenvalues $e^{\pm\lambda}$ and two eigenaxes that correspond to the unstable and stable directions for the dynamics. They have slopes $s_+=\tan\psi_+$ and $s_-=\tan\psi_-$. - If $\gamma^2<\alpha\beta$, then $M$ is an elliptic map. Any hyperbolic map $M\in{\mathrm{SL}}(2,{\mathbb{R}})$ with ${\mathrm{Tr}\,}M>2$ is of the above form. (If ${\mathrm{Tr}\,}M<-2$, then consider the map $-M$.) Throughout the paper, we use $M$ to denote both the map and the matrix that defines it. We remark that $M\in{\mathrm{SL}}(2,{\mathbb{R}})$ preserves the symplectic product on ${\mathbb{R}}^2$: $$Mu\wedge Mv=u\wedge v.$$ Here, $$u\wedge v=u_2v_1-u_1v_2\quad\text{for }u=(u_1,u_2)\text{ and }v=(v_1,v_2)\in{\mathbb{R}}^2.$$ We now rewrite the hyperbolic flow $M(t)$ and the hyperbolic map $M$ in complex coordinates. Let $z=(q+ip)/\sqrt2$. Then the Hamiltonian in is $$H(z,{\overline}z)=\frac c2z^2+\frac{{\overline}c}{2}{\overline}z^2+bz{\overline}z,\quad\text{in which }b=\frac{\alpha+\beta}2\in{\mathbb{R}}\text{ and }c=\frac{\alpha-\beta}{2}-i\gamma\in{\mathbb C}.$$ Since $\gamma^2-\alpha\beta=|c|^2-b^2$, $M=M_{(c,b)}$ is hyperbolic if $|c|^2>b^2$ and is elliptic if $|c|^2<b^2$. Let $\mu\in(0,\infty)$. - Define $$D(\mu)=M_{(c=-i\mu,b=0)}=\begin{pmatrix} e^{\mu} & 0\\ 0 & e^{-\mu} \end{pmatrix},$$ which is hyperbolic with $q$-axis and $p$-axis as the unstable and stable axes, respectively. - Define $$B(\mu)=M_{(c=-\mu,b=0)}=\begin{pmatrix} \cosh\mu & \sinh\mu\\ \sinh\mu & \cosh\mu \end{pmatrix},$$ which is hyperbolic with the unstable and stable axes forming $\psi_+=\pi/4$ and $\psi_-=-\pi/4$ with the $q$-axis. - Define $$R(\mu)=M_{(c=0,b=-\mu)}=\begin{pmatrix} \cos\mu & -\sin\mu\\ \sin\mu & \cos\mu \end{pmatrix},$$ which is a rotation of angle $\mu$ and is therefore elliptic. Then any hyperbolic map $M_{(c,b)}$ can be decomposed as follows: There are $b_1\in[\frac\pi2,\frac\pi2]$ and $b_2\in{\mathbb{R}}$ such that $$\label{eq:Mdecomp} M_{(c,b)}=QD(\lambda)Q^{-1},\quad\text{in which }Q=R(b_1)B(b_2).$$ That is, $M_{(c,b)}$ is obtained from the special case $D(\lambda)$ ($\lambda=\sqrt{|c|^2-b^2}>0$) by a change of coordinates. Notice that $D(\lambda)$ has unstable and stable axes given by vectors $e_q$ and $e_p$ in the $q$-axis and $p$-axis, respectively. The map $Q$ transforms from this $(q,p)$-frame to the unstable/stable-frame given by vectors $v_+=Qe_q$ and $v_-=Qe_p$. Quantum dynamics on the plane ----------------------------- Let $h$ be the Planck constant and we are interested in the semiclassical limit as $h\to0$ in this paper. Denote $\hbar=h/(2\pi)$. The states in the quantum system are represented by functions in $L^2({\mathbb{R}})$; the quantum observables are operators acting on $L^2({\mathbb{R}})$ which are quantization of the classical observables in $C^\infty({\mathbb{R}}^2)$. We first define the quantization of the position and momentum observables as the self-adjoint operators $$\hat q\psi(q)=q\psi(q)\quad\text{and}\quad\hat p\psi=\frac\hbar i\frac{d\psi(q)}{dq}\quad\text{for }\psi\in C^\infty_0({\mathbb{R}}).$$ So we have that $$[\hat q,\hat p]=\hat q\hat p-\hat p\hat q=i\hbar\hat I.$$ Here, $\hat I$ is the identity map that $\hat I\psi=\psi$. The Weyl quantization of the Hamiltonian in is the self-adjoint operator $$\hat H=\frac12\alpha\hat q^2+\frac12\beta\hat p^2+\frac\gamma2(\hat q\hat p+\hat p\hat q).$$ It generates the Schrödinger flow $\psi(0)\to\psi(t)$ such that $$\psi(t)=e^{-it\hat H/\hbar}\psi(0)$$ solves the Schrödinger equation $$i\hbar\frac{\partial\psi(t)}{\partial t}=\hat H\psi(t).$$ The quantum map (or quantum evolution operator) corresponding to a hyperbolic map $M$ is defined as $\hat M=e^{-i\hat H/\hbar}$. Let $v=(v_1,v_2)\in{\mathbb{R}}^2$ and the translation $T_v(x)=x+v$ for $x\in{\mathbb{R}}^2$. Define the quantum translation operator as $$\hat T_v=\exp\left(-\frac{i}{\hbar}(v_1\hat p-v_2\hat q)\right).$$ \[prop:TR\] Let $u=(u_1,u_2)$ and $v=(v_1,v_2)$ in ${\mathbb{R}}^2$. Then 1. the adjoint operator $$\hat T_v^\star=\hat T_{-v},$$ 2. the conjugation $$\label{eq:MTM} \hat M\hat T_v\hat M^{-1}=\hat T_{Mv},$$ 3. the composition $$\label{eq:TuTv} \hat T_u\hat T_v=e^{\frac{iu\wedge v}{2\hbar}}\hat T_{u+v}.$$ Coherent states and their evolution on the plane {#sec:coR} ------------------------------------------------ The standard coherent state $\ket0$ at the origin is the ground state of the quantum harmonic oscillator $\hat q^2+\hat p^2$. The standard coherent state at $x=(q,p)\in{\mathbb{R}}^2$ is then $\ket x=\hat T_x\ket0$. In the $L^2({\mathbb{R}})$ representation, $\ket x$ is a Gaussian wave packet that is given by $$\ket x(q')=\frac{1}{(\pi\hbar)^\frac14}e^{\frac i\hbar pq'}e^{-\frac{1}{2\hbar}|q'-q|^2},$$ which is localized at $x=(q,p)$ with width $\sim\sqrt\hbar$. Let $M$ be a hyperbolic map on ${\mathbb{R}}^2$ and $\hat M$ be its quantum map. The evolution of the standard coherent state $\ket x$ under $\hat M$ is not straightforward. This is partly due to the fact that the unstable/stable-frame of $M$ is in general different from the $(q,p)$-frame. To remedy this issue, we introduce the squeezed coherent states. Let ${{\tilde c}}\in{\mathbb C}$. Define the squeezed coherent states $$\ket{{\tilde c}}=\hat M_{({{\tilde c}},0)}\ket0\quad\text{and}\quad\ket{x,{{\tilde c}}}=\hat T_x\ket{{\tilde c}}.$$ We use the notations $\ket{{\tilde c}}$ and $\ket{x,{{\tilde c}}}$ with tildes to indicate the squeezed coherent states. In particular, choosing $\tilde c=0$ reduces $\ket{{\tilde c}}$ and $\ket{x,{{\tilde c}}}$ to the standard coherent states $\ket0$ and $\ket x$. The properties of the coherent states $\ket{{\tilde c}}$ are analyzed via the Bargmann and Husimi functions. Let $\tilde w\in{\mathbb C}$ and $\psi\in L^2({\mathbb{R}})$. Define - the Bargmann function of $\psi$ as $${\mathcal{B}}_{\tilde w,\psi}(x)=\braket{x,\tilde w|\psi},$$ - the Husimi function of $\psi$ as $$\label{eq:Husimi} {\mathcal H}_{\tilde w,\psi}(x)=\frac{\left|\braket{x,\tilde w|\psi}\right|^2}{2\pi\hbar},\quad\text{which satisfies }\frac{1}{2\pi\hbar}\int_{{\mathbb{R}}^2}\ket{x,\tilde w}\bra{x,\tilde w}\,dx=\hat I.$$ That is, taking the inner product of $\psi$ with the coherent state $\ket{x,\tilde w}$, ${\mathcal{B}}_{\tilde w,\psi}(x)$ and ${\mathcal H}_{\tilde w,\psi}(x)$ measure the localization of the state $\psi$ at the point $x$ in the phase space ${\mathbb{R}}^2$. We use the squeezed coherent states $\ket{x,\tilde w}=\hat T_x\ket{\tilde w}$ here to allow full generality so ${\mathcal{B}}_{\tilde w,\psi}(x)$ and ${\mathcal H}_{{{\tilde c}},\psi}(x)$ depend on $\tilde w$. The Bargmann and Husimi functions of various states $\psi$ can be simplified by making appropriate choices of $\tilde w$ and the frame in ${\mathbb{R}}^2$. Let $\ket{{\tilde c}}=\hat M_{({{\tilde c}},0)}\ket0$ be a squeezed coherent state. Choose $\tilde w=0$ and the unstable/stable-frame $x=(\tilde q,\tilde p)$ of $M_{({{\tilde c}},0)}$. Then we have that $${\mathcal{B}}_{0,{{\tilde c}}}(x)=\braket{x,0|{{\tilde c}}}=\frac{1}{\sqrt{\cosh|{{\tilde c}}|}}\exp\left(-\frac{i\tanh|{{\tilde c}}|}{2\hbar}\tilde q\tilde p\right)\exp\left(-\frac12\left(\frac{\tilde q^2}{\Delta\tilde q^2}+\frac{\tilde p^2}{\Delta\tilde p^2}\right)\right),$$ in which $$\Delta\tilde q^2=\frac{2\hbar}{1-\tanh|{{\tilde c}}|}\quad\text{and}\quad\Delta\tilde p^2=\frac{2\hbar}{1+\tanh|{{\tilde c}}|}.$$ In particular, if ${{\tilde c}}=-i\mu$ for $\mu>0$, then $M_{({{\tilde c}},0)}=D(\mu)$ and the unstable/stable-frame coincides with the $(q,p)$-frame. In this case, we have that $$\label{eq:cBargmannmu} \braket{x,0|{{\tilde c}}}=\braket{x|\hat D(\mu)|0}=\frac{1}{\sqrt{\cosh\mu}}\exp\left(-\frac{i\tanh\mu}{2\hbar}qp\right)\exp\left(-\frac12\left(\frac{q^2}{\Delta q^2}+\frac{p^2}{\Delta p^2}\right)\right).$$ We next describe the quantum evolution of coherent states under $\hat M=\hat M_{(c,b)}$. Based on the decomposition of $M=M_{(c,b)}$ in , the evolution can be described explicitly for properly chosen squeezed coherent states. Recall that $M_{(c,b)}=QD(\lambda)Q^{-1}$, in which $Q=R(b_1)B(b_2)$ with $b_1\in[\frac\pi2,\frac\pi2]$ and $b_2\in{\mathbb{R}}$. Put $$\label{eq:co} {{\tilde c_0}}=-b_2e^{-2ib_1}.$$ Since $M_{({{\tilde c_0}},0)}=R(b_1)B(b_2)R(-b_1)$, $\hat M_{({{\tilde c_0}},0)}=\hat R(b_1)\hat B(b_2)\hat R(-b_1)=\hat Q\hat R(-b_1)$, which implies that $$\ket{{\tilde c_0}}=M_{({{\tilde c_0}},0)}\ket0=\hat Q\hat R(-b_1)\ket0=e^{-\frac{ib_1}{2}}\hat Q\ket0.$$ It thus follows from that $$\hat M^t\ket{{\tilde c_0}}=e^{-\frac{ib_1}{2}}\hat Q\hat D(\lambda t)\ket0\quad\text{and}\quad\braket{{{\tilde c_0}}|\hat M^t|{{\tilde c_0}}}=\braket{0|\hat D(\lambda t)|0}=\frac{1}{\sqrt{\cosh(\lambda t)}}.$$ Denote the quantum evolution of the coherent state $\ket{{\tilde c_0}}$ in (1) by $$\label{eq:tco} \ket{t;{{\tilde c_0}}}=\hat M^t\ket{{\tilde c_0}}.$$ Then the Husimi function of $\ket{t;{{\tilde c_0}}}$ is explicit by choosing $\tilde w={{\tilde c_0}}$ and the unstable/stable frame of $M$. \[prop:Husimiplane\] Let $x=(q',p')=Q^{-1}(q,p)$. Then for $t\ge0$, $${\mathcal H}_{{{\tilde c_0}},t}(x)=\frac{\left|\braket{x,{{\tilde c_0}}|t;{{\tilde c_0}}}\right|^2}{2\pi\hbar}=\frac{1}{2\pi\hbar\cosh(\lambda t)}\exp\left(-\left(\frac{q'^2}{\Delta q'^2}+\frac{p'^2}{\Delta p'^2}\right)\right),$$ in which $$\begin{cases} \Delta q'^2=\frac{2\hbar}{1-\tanh(\lambda t)}\sim\hbar e^{2\lambda t} & \text{as }t\to\infty,\\ \Delta p'^2=\frac{2\hbar}{1+\tanh(\lambda t)}=e^{-2\lambda t}\Delta q'^2\to\hbar & \text{as }t\to\infty. \end{cases}$$ - The Husimi function ${\mathcal H}_{{{\tilde c_0}},t}(x)$ of $\ket{t;{{\tilde c_0}}}$ spreads in the unstable direction of the map $M$ by a rate of $\sqrt\hbar e^{\lambda t}$ while stays in the $\sqrt\hbar$ neighborhood of the stable direction. - ${\mathcal H}_{{{\tilde c_0}},t}(x)$ is concentrated in the elliptic region around the origin with two axes $\Delta q'$ and $\Delta p'$ (in the unstable/stable-frame of $M$). - The concentration region of ${\mathcal H}_{{{\tilde c_0}},t}(x)$ has area $\sim\Delta q'\Delta p'\sim\hbar e^{\lambda t}$. Hence, ${\mathcal H}_{{{\tilde c_0}},t}(x)\sim\sqrt\hbar e^{\lambda t/2}$ in this region due to conservation of the $L^2$ norm in . - The Husimi function of the evolution $\ket{t;{{\tilde c_0}}}$ in negative times $t<0$ can be described similarly as above. In this case, the concentration region of ${\mathcal H}_{{{\tilde c_0}},t}(x)$ spreads in the stable direction of the map $M$ by a rate of $\sqrt\hbar e^{\lambda t}$ while stays in the $\sqrt\hbar$ neighborhood of the unstable direction. - Because of the explicit Husimi function of the evolution $\ket{t;{{\tilde c_0}}}$, we exclusively use the squeezed coherent states $\ket{{\tilde c_0}}$ (which depends on $M$) in our construction of quasimodes. See Subsection \[sec:coT\]. Classical dynamics and quantum dynamics on the torus {#sec:torus} ==================================================== In this section, we introduce the classical linear hyperbolic systems on the phase space ${\mathbb{T}}^2={\mathbb{R}}^2/{\mathbb{Z}}^2$ and their quantum systems, which are referred as classical and quantum cat maps, respectively. We again follow the setup in Faure-Nonnenmacher-De Bièvre [@FNDB]. Classical dynamics on the torus ------------------------------- Let $M\in{\mathrm{SL}}(2,{\mathbb{R}}):{\mathbb{R}}^2\to{\mathbb{R}}^2$ be a hyperbolic map. Suppose further that $M\in{\mathrm{SL}}(2,{\mathbb{Z}})$, i.e., $A,B,C,D\in{\mathbb{Z}}$ in . Since $$M(x+n)=Mx+Mn=Mx\mod1\quad\text{for }x\in{\mathbb{R}}^2\text{ and }n\in{\mathbb{Z}}^2,$$ $M$ induces a map on ${\mathbb{T}}^2$ that is hyperbolic, by which we refer as a classical cat map. The Arnold cat map is defined by $$M_{\mathrm{Arnold}}=\begin{pmatrix} 2 & 1\\ 1 & 1 \end{pmatrix}.$$ The eigenvalues of $M_{\mathrm{Arnold}}$ are $(3\pm\sqrt5)/2$ with the Lyapunov exponent $\log((3+\sqrt5)/2)$. Consider the classical cat map $M$ with Lyapunov exponent $\lambda>0$ so the eigenvalues are $e^{\pm\lambda}$. We recall some standard facts about the discrete hyperbolic dynamical system $M^t:{\mathbb{T}}^2\to{\mathbb{T}}^2$ ($t\in{\mathbb{Z}}$) that are useful later. See Katok-Hasselblatt [@KH] for more details. - We say that $x\in{\mathbb{T}}^2$ is periodic if $x$ is a fixed point of $M^t$ for some $t\ge1$, i.e., $M^tx=x$. Then $\gamma=\{M^sx\}_{s=0}^{t-1}$ form a closed orbit of $M$ and we denote the length of $\gamma$ by $|\gamma|=t$. - The period of a periodic point $x\in{\mathbb{T}}^2$ is defined as $$p(x)=\min\left\{T:T\ge1\text{ and }M^Tx=x\right\}.$$ Then $\gamma=\{M^tx\}_{t=0}^{p(x)-1}$ is called a prime closed orbit of $M$. In this case, $p(x)=|\gamma|$ for all $x\in\gamma$. Denote ${\mathcal F}=[-\frac12,\frac12)\times[-\frac12,\frac12)$ as a fundamental domain of ${\mathbb{T}}^2$. Then each periodic point $x=(q,p)\in{\mathcal F}$ satisfies that $$M^tx=M^t\begin{pmatrix} q\\ p \end{pmatrix}=\begin{pmatrix} q\\ p \end{pmatrix}+\begin{pmatrix} j\\ k \end{pmatrix}\quad\text{for some }j,k\in{\mathbb{Z}}.$$ Since $M^t\in{\mathrm{SL}}(2,{\mathbb{Z}})$, $$\begin{pmatrix} q\\ p \end{pmatrix}=(M^t-{\mathrm{Id}})^{-1}\begin{pmatrix} j\\ k \end{pmatrix}$$ has rational coordinates. Here, ${\mathrm{Id}}$ is the identity matrix. Moreover, if $x\in{\mathcal F}$ is periodic with period $p(x)$, then its coordinates can be written as rational numbers with denominator that is not larger than $\det(M^{p(x)}-{\mathrm{Id}})$. To summarize, \[prop:periodicpts\] 1. Let $x\in{\mathcal F}$ be a periodic point with period $p(x)$. Then $$x\in{\mathcal{L}}_l,\quad\text{in which }l=\det\left(M^{p(x)}-{\mathrm{Id}}\right)=e^{\lambda p(x)}+e^{-\lambda p(x)}-2.$$ Here, $${\mathcal{L}}_l=\left\{\left(\frac jl,\frac kl\right):-\frac l2\le j,k<\frac l2\right\}\subset{\mathcal F}$$ is the lattice of rational points with denominator $l\in{\mathbb N}$ (which are not necessarily in the simplest form). 2. Let $x\in{\mathcal{L}}_l$. Then $x$ is periodic with period $p(x)\le l^2$. Given a closed orbit $\gamma=\{x_t\}_{t=0}^{T-1}$, define the delta measure on $\gamma$ $$\label{eq:delta} \mu_\gamma=\frac1T\sum_{t=0}^{T-1}\delta_{x_t},$$ in which $\delta_x$ is the delta measure at $x$. That is, for any $f\in C({\mathbb{T}}^2)$, $$\int_{{\mathbb{T}}^2}f\,d\mu_\gamma=\frac1T\sum_{t=0}^{T-1}f(x_t).$$ It is clear that $\mu_\gamma$ is an invariant probability measure of the cat map $M$ on ${\mathbb{T}}^2$. Moreover, by Sigmund [@S], \[thm:Sigmund\] For any invariant probability measure $\mu$ of a cat map $M$ on the torus ${\mathbb{T}}^2$, there is a sequence of closed prime orbits $\{\gamma_j\}_{j=1}^\infty$ such that the delta measures $\mu_{\gamma_j}\to\mu$ weakly, that is, for any $f\in C({\mathbb{T}}^2)$, $$\int_{{\mathbb{T}}^2}f\,d\mu_{\gamma_j}\to\int_{{\mathbb{T}}^2}f\,d\mu,\quad\text{as }j\to\infty.$$ - In particular, since the Lebesgue measure $dx$ is invariant on ${\mathbb{T}}^2$, there is a sequence of closed prime orbits $\{\gamma_j\}_{j=1}^\infty$ such that the delta measures $\mu_{\gamma_j}$ converge to the Lebesgue measure weakly. - Let $\{\gamma_j\}_{j=1}^\infty$ be a sequence of closed prime orbits such that $\mu_{\gamma_j}\to\mu$. Suppose that $|\gamma_j|\le C$ for some uniform constant $C>0$. Since the closed prime orbits are enumerable by their lengths, there are only finitely many orbits with length bounded by $C$. Therefore, $\mu$ is itself a delta measure on some closed prime orbit. Quantum dynamics on the torus {#sec:qtorus} ----------------------------- We first need to describe the space of states in the quantum system of a cat map with phase space ${\mathbb{T}}^2$. Each state is represented by $\psi\in L^2({\mathbb{R}})$ that is periodic in position and in momentum. This means that $\psi$ is invariant under the phase translations $\hat T_n$ for $n=(n_1,n_2)\in{\mathbb{Z}}^2$. In particular, $$\label{eq:theta} \hat T_{(1,0)}\ket\psi=e^{i\theta_1}\ket\psi\quad\text{and}\quad\hat T_{(0,1)}\ket\psi=e^{i\theta_2}\ket\psi.$$ Here, we allow the phase shifts $e^{i\theta_1}$ and $e^{i\theta_2}$ for some angle $\theta=(\theta_1,\theta_2)\in[0,2\pi)\times[0,2\pi)$, because under such phase shifts the function defines the same quantum state. It then follows from such periodicity that $$\hat T_{(1,0)}\hat T_{(0,1)}=\hat T_{(0,1)}\hat T_{(1,0)}$$ restricted to the space of quantum states. But in the view of , since $(1,0)\wedge(0,1)=-1$, it requires that $e^{i/\hbar}=1$. Hence, $$\label{eq:N} N=\frac{1}{2\pi\hbar}\in{\mathbb N}.$$ Under the conditions and , the space of quantum states ${\mathcal H}_{N,\theta}$ is an $N$-$\dim$ Hilbert space. Moreover, $L^2({\mathbb{R}})$ can then be decomposed as $$L^2({\mathbb{R}})=\frac{1}{(2\pi)^2}\int^\oplus{\mathcal H}_{N,\theta}\,d\theta.$$ The projector ${{\hat P_\theta}}:{\mathcal{S}}'({\mathbb{R}})\to{\mathcal H}_{N,\theta}$ is defined as $$\label{eq:Ptheta} {{\hat P_\theta}}=\sum_{n=(n_1,n_2)\in{\mathbb{Z}}^2}e^{-in_1\theta_1-in_2\theta_2}\hat T_{(1,0)}^{n_1}\hat T_{(0,1)}^{n_2}=\sum_{n=(n_1,n_2)\in{\mathbb{Z}}^2}e^{-i\theta\cdot n+i\delta_n}\hat T_n,$$ in which $\delta_n=-n_1n_2N\pi$ by . Let $M\in{\mathrm{SL}}(2,{\mathbb{Z}})$ be a hyperbolic map on ${\mathbb{R}}^2$. Then by Section \[sec:plane\], $\hat M$ defines a quantum map that acts on $L^2({\mathbb{R}})$. From , we have that $$\hat M{{\hat P_\theta}}\hat M^{-1}=\sum_{n=(n_1,n_2)\in{\mathbb{Z}}^2}e^{-i\theta\cdot n+i\delta_n}\hat T_{Mn}=\sum_{n=(n_1,n_2)\in{\mathbb{Z}}^2}e^{-i\theta\cdot M^{-1}n+i\delta_{M^{-1}n}}\hat T_n.$$ Hence, $$\hat M{{\hat P_\theta}}=\hat P_{\theta'}\hat M,\quad\text{in which }\theta'=M^{-1}\theta+N\pi\begin{pmatrix}CD\\AB\end{pmatrix}.$$ If $\theta'=\theta\mod(2\pi)$, then $\hat M$ commutes with ${{\hat P_\theta}}$ and therefore defines an endomorphism on ${\mathcal H}_{N,\theta}$. For each $N\in{\mathbb N}$, such choices of $\theta$ are always possible, for example, $\theta=(0,0)$ if $N$ is even and $\theta=(\pi,\pi)$ if $N$ is odd. Let $M\in{\mathrm{SL}}(2,{\mathbb{Z}})$ be a classical cat map. Then for any $N\in{\mathbb N}$, there exists $\theta\in[0,2\pi)\times[0,2\pi)$ such that $\hat M:{\mathcal H}_{N,\theta}\to{\mathcal H}_{N,\theta}$. We fix such choice of $\theta$ that depends on $M$ and $N$. The operator $\hat M$ restricted on ${\mathcal H}_{N,\theta}$ is called the quantum cat map. (If there is no confusion, then we simply write ${\mathcal H}_N$.) Any quantum translation operator $\hat T_v$ acts on ${\mathcal H}_N$ only if $\hat T_v$ commutes with $\hat T_n$ for all $n\in{\mathbb{Z}}^2$. Applying again, $e^{i(v\wedge n)/\hbar}=1$ for all $n\in{\mathbb{Z}}^2$. So $v\in{\mathbb{Z}}^2/N$. For notational convenience, we write $$\hat T_N(n)=\hat T_{n/N}.$$ Let $a\in C^\infty({\mathbb{T}}^2)$. Define its Weyl quantization as an operator on ${\mathcal H}_N$: $$\label{eq:OpW} \hat a^{\mathrm{w}}=\sum_{n\in{\mathbb{Z}}^2}\tilde a(n)\hat T_N(n).$$ Here, $a$ is called the symbol and $\tilde a(n)$ is the Fourier coefficients of $a$ that $$a(x)=\sum_{n\in{\mathbb{Z}}^2}\tilde a(n)e^{2\pi i(n\wedge x)}.$$ We have that $\hat a^{\mathrm{w}}$ is bounded on $L^2({\mathbb{T}}^1)$, that is, $$\braket{\psi|\hat a^{\mathrm{w}}|\psi}\le C\braket{\psi|\psi},$$ in which $C$ depends on finite number of derivatives of $a$. See Zworski [@Zw Setion 4.5]. Since the cat maps are linear, we have the following exact Egorov’s theorem. See Part I [@Han] for a shoot proof. \[thm:Egorov\] Let $a\in C^\infty({\mathbb{T}}^2)$. Then $$\hat M^{-t}\hat a^{\mathrm{w}}\hat M^t={\widehat}{a\circ M^t}^{\mathrm{w}}\quad\text{for all }t\in{\mathbb{Z}}.$$ As an immediate consequence, we have that Let $\{\psi_j\}_{j=1}^\infty$ be a sequence of quasimodes of order $o(1)$ in . Suppose that $\mu$ is the corresponding semiclassical measure. Then $\mu$ is invariant under $M$. It suffices to prove that for any $f\in C^\infty({\mathbb{T}}^2)$, $$\int_{{\mathbb{T}}^2}f\,d\mu=\int_{{\mathbb{T}}^2}f\circ M\,d\mu.$$ Since $\{\psi_j\}_{j=1}^\infty$ is a sequence of quasimodes of order $o(1)$, we have that $$\hat M\ket{\psi_j}=e^{i\phi_j}\ket{\psi_j}+o_{L^2}(1)\quad\text{for some }\phi_j\in{\mathbb{R}}.$$ Then by the $L^2$ boundedness of $\hat f^{\mathrm{w}}$ and the Egorov’s theorem above, $$\begin{aligned} \braket{\psi_j|{\widehat}{f\circ M}^{\mathrm{w}}|\psi_j}&=&\braket{\psi_j|\hat M^{-1}\hat f^{\mathrm{w}}\hat M|\psi_j}\\ &=&\braket{\psi_j|\hat f^{\mathrm{w}}|\psi_j}+o_f(1).\end{aligned}$$ The proposition follows by taking limits of both sides as $j\to\infty$. Coherent states and their evolution on the torus {#sec:coT} ------------------------------------------------ Let $M$ be a classical cat map on the torus ${\mathbb{T}}^2$ and $\hat M$ be its quantization on ${\mathcal H}_{N,\theta}$ with $N=1/(2\pi\hbar)\in{\mathbb N}$. In this section, we investigate the coherent states on the torus and their evolution under the quantum cat map $\hat M$. For technical convenience, we begin from the squeezed coherent state $\ket{{\tilde c_0}}$ on the plane . Its evolution $\ket{t;{{\tilde c_0}}}=\hat M^t\ket{{\tilde c_0}}$ on the plane has an explicit Husimi function which is given in Proposition \[prop:Husimiplane\]. The squeezed coherent states $\ket{{{\tilde c_0}},\theta}$ and $\ket{x,{{\tilde c_0}},\theta}$ on the torus are defined via the projector ${{\hat P_\theta}}$ in : $$\ket{{{\tilde c_0}},\theta}={{\hat P_\theta}}\ket{{\tilde c_0}}\quad\text{and}\quad\ket{x,{{\tilde c_0}},\theta}={{\hat P_\theta}}\ket{x,{{\tilde c_0}}}.$$ Write the evolution of $\ket{{{\tilde c_0}},\theta}$ under $\hat M$ as $\ket{t;{{\tilde c_0}},\theta}=\hat M^t\ket{{{\tilde c_0}},\theta}\in{\mathcal H}_{N,\theta}$ for $t\in{\mathbb{Z}}$. We use the Husimi function on the torus to analytize $\ket{t;{{\tilde c_0}},\theta}$. For any quantum state $\psi$ on the torus, define the Husimi function of $\psi$ as $$\label{eq:Husimitorus} {\mathcal H}_{{{\tilde c_0}},\psi,\theta}(x)=N\left|\braket{x,{{\tilde c_0}},\theta|\psi}\right|^2,\quad\text{which satisfies }\int_{{\mathbb{T}}^2}N\ket{x,{{\tilde c_0}},\theta}\bra{x,{{\tilde c_0}},\theta}\,dx=\hat I.$$ Then the Husimi function of $\ket{t;{{\tilde c_0}},\theta}$ $$\begin{aligned} {\mathcal H}_{{{\tilde c_0}},t,\theta}(x)&=&N\left|\braket{x,{{\tilde c_0}},\theta|t;{{\tilde c_0}},\theta}\right|^2\\ &=&N\left|\braket{x,{{\tilde c_0}}|{{\hat P_\theta}}|t;{{\tilde c_0}}}\right|^2\\ &=&N\left|\sum_{n\in{\mathbb{Z}}^2}e^{-i\theta\cdot n+i\delta_n}\braket{x,{{\tilde c_0}}|\hat T_n|t;{{\tilde c_0}}}\right|^2\\ &=&N\left|\sum_{n\in{\mathbb{Z}}^2}e^{-i\theta\cdot n+i\delta_n}\braket{x+n,{{\tilde c_0}}|t;{{\tilde c_0}}}\right|^2.\end{aligned}$$ That is, $\braket{x,{{\tilde c_0}}|{{\hat P_\theta}}|t;{{\tilde c_0}}}$ is the sum (up to some phases) of the translates of $\braket{x,{{\tilde c_0}}|t;{{\tilde c_0}}}$ in different phase space cells of size $1$. Use $x=(q',p')$ in the unstable/stable-frame of $M$ and recall that ${\mathcal F}$ is a fundamental cell of ${\mathbb{T}}^2$. By Proposition \[prop:Husimiplane\], the Husimi function $${\mathcal H}_{{{\tilde c_0}},t}(x)=\frac{\left|\braket{x,{{\tilde c_0}}|t;{{\tilde c_0}}}\right|^2}{2\pi\hbar}=\frac{N}{\cosh(\lambda t)}\exp\left(-\left(\frac{q'^2}{\Delta q'^2}+\frac{p'^2}{\Delta p'^2}\right)\right),$$ in which $$\Delta q'^2=\frac{2\hbar}{1-\tanh(\lambda t)}\sim\hbar e^{2\lambda t}\quad\text{and}\quad\Delta p'=\frac{2\hbar}{1+\tanh(\lambda t)}\sim\hbar.$$ Recall that the Ehrenfect time $T_E=|\log\hbar|/\lambda$ in . Suppose that $0\le t\le\delta T_E$ for some $0\le\delta<1/2$. Then ${\mathcal H}_{{{\tilde c_0}},t}$ is concentrated in the region $$\left\{|q'|\lesssim\sqrt\hbar e^{\lambda t},|p'|\lesssim\sqrt\hbar\right\}\subset B_2\left(o,C\hbar^\frac12e^{\lambda t}\right)\subset{\mathcal F},$$ in which $C>0$ is an absolute constant. Here, $B_2(o,r)\subset{\mathbb{T}}^2$ is the geodesic ball centered at the origin $o$ and with radius $r$. Therefore, within such a time frame, all the terms but the one when $n=(0,0)$ are negligible in the sum of ${\mathcal H}_{{{\tilde c_0}},t,\theta}$. In particular, $$\label{eq:tcothetapt} {\mathcal H}_{{{\tilde c_0}},t,\theta}(x)=\begin{cases} \frac{N}{\cosh(\lambda t)}\exp\left(-\left(\frac{q'^2}{\Delta q'^2}+\frac{p'^2}{\Delta p'^2}\right)\right)+O\left(e^{-\frac{1}{C\hbar}}\right) & \text{if }x\in B_2\left(o,C\hbar^\frac12e^{\lambda t}\right),\\ O\left(e^{-\frac{1}{C\hbar}}\right) & \text{if }x\in{\mathcal F}\setminus B_2\left(o,C\hbar^\frac12e^{\lambda t}\right). \end{cases}$$ Taking $x=o$ and $t=0$, we have that ${\mathcal H}_{{{\tilde c_0}},0,\theta}(o)=N+O(e^{-\frac{1}{C\hbar}})$. This means that the squeezed coherent state $\ket{{{\tilde c_0}},\theta}$ is asymptotically normalized in $L^2({\mathbb{T}}^2)$: $$\braket{{{\tilde c_0}},\theta|{{\tilde c_0}},\theta}=\frac{{\mathcal H}_{{{\tilde c_0}},0,\theta}(o)}{N}=1+O\left(e^{-\frac{1}{C\hbar}}\right).$$ Since $\hat M$ preserves the $L^2$ norm, the above estimate reminds valid for $\ket{t;{{\tilde c_0}},\theta}$: $$\label{eq:tcothetaL2} \braket{t;{{\tilde c_0}},\theta|t;{{\tilde c_0}},\theta}=1+O\left(e^{-\frac{1}{C\hbar}}\right).$$ Moreover, by and , the $L^2$ norm of $\ket{t;{{\tilde c_0}},\theta}$ can be recovered by the integral of the Husimi function in its concentration region modulo an exponential error. \[cor:tcothetaL2\] Let $0\le t\le\delta T_E$ for some $0\le\delta<1/2$. Then $$\braket{t;{{\tilde c_0}},\theta|t;{{\tilde c_0}},\theta}=\int_{B_2\left(o,C\hbar^\frac12e^{\lambda t}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta}(x)\,dx+O\left(e^{-\frac{1}{C\hbar}}\right).$$ If $t\ge T_E/2$, then $\Delta q'\gtrsim\hbar e^{\lambda t}$ reaches the size of the fundamental cell ${\mathcal F}$. That is, the concentration region of ${\mathcal H}_{{{\tilde c_0}},t}$ spreads from ${\mathcal F}$ to other cells. Hence, the terms when $n\ne(0,0)$ may contribute in the sum of ${\mathcal H}_{{{\tilde c_0}},0,\theta}$. This phenomenon of “interference effects” has been extensively studied in Faure-Nonnenmacher-De Bièvre [@FNDB]. In this paper, we restrict the evolution below $T_E/2$ so the interference effects are negligible. Anti-Wick quantization {#sec:AW} ---------------------- In the previous subsection, we discussed the coherent states and their evolution under the quantum cat map. The localization of these quantum states are described by their Husimi functions . It is therefore most convenient to use the anti-Wick quantization to study the density distribution of the states. Let $a\in L^\infty({\mathbb{T}}^2)$. Define its anti-Wick quantization as an operator on ${\mathcal H}_N$: $$\label{eq:OpaW} \hat a^{\mathrm{aw}}=N\int_{{\mathbb{T}}^2}a(x)\ket{x,{{\tilde c_0}},\theta}\bra{x,{{\tilde c_0}},\theta}\,dx.$$ Then we immediately have that $$\label{eq:AWT2} \braket{\psi|\hat a^{\mathrm{aw}}|\psi}=N\int_{{\mathbb{T}}^2}a(x)\left|\braket{x,{{\tilde c_0}},\theta|\psi}\right|^2\,dx=\int_{{\mathbb{T}}^2}a(x){\mathcal H}_{{{\tilde c_0}},\psi,\theta}(x)\,dx.$$ In particular, for any $\Omega\subset{\mathbb{T}}^1$, $$\label{eq:AWT1} \int_\Omega|\psi(q)|^2\,dq=\braket{\psi|\hat\chi_\Omega^{\mathrm{aw}}|\psi}=\int_{\Omega}\int_{{\mathbb{T}}^1}{\mathcal H}_{{{\tilde c_0}},\psi,\theta}(q,p)\,dpdq.$$ To accommodate the discussion of density distribution of states at small scales in the phase space ${\mathbb{T}}^2$ in Theorems \[thm:SSQE\] and \[thm:sc\], we allow the classical symbols in quantizations and to depend on the semiclassical parameter $\hbar$: Let $\rho\in[0,1/2)$ and $\hbar_0\in(0,1)$. We say that $a(x;\hbar)\in S_\rho({\mathbb{T}}^2)$ if $a\in C^\infty({\mathbb{T}}^2\times(0,\hbar_0))$ and for each multiindex $\alpha$, there is a constant $C_\alpha>0$ such that $$\left|\partial_x^\alpha a(x;\hbar)\right|\le C_\alpha\hbar^{-\rho|\alpha|},$$ for all $x\in{\mathbb{T}}^2$ and $\hbar\in(0,\hbar_0)$. The Weyl and anti-Wick quantizations are asymptotically equivalent for symbols in $S_\rho({\mathbb{T}}^2)$ with $\rho\in[0,1/2)$. See for example Bouzouina-De Bièvre [@BouDB] for a proof. \[lemma:WAW\] Let $\rho\in[0,1/2)$ and $a(x;\hbar)\in S_\rho({\mathbb{T}}^2)$. Then $$\hat a^{\mathrm{w}}-\hat a^{\mathrm{aw}}=O_{L^2\to L^2}\left(\hbar^{1-2\rho}\right).$$ Hence, $$\braket{\psi|\hat a^{\mathrm{w}}|\psi}-\braket{\psi|\hat a^{\mathrm{aw}}|\psi}=O\left(\hbar^{1-2\rho}\right),$$ for all normalized states $\psi$. A direct consequence is that the semiclassical measure defined in are independent of the quantization. In Theorems \[thm:SSQE\] and \[thm:sc\], we are concerned with the density distribution at small scales in the physical space and in the phase space. 1. Let $B_1(q_0,r)\subset{\mathbb{T}}^1$ with $r=r(\hbar)\ge\hbar^\rho$ for $\rho\in[0,1/2)$. From , $$\label{eq:bqr} \int_{B_1(q_0,r)}|\psi(q)|^2\,dq=\braket{\psi|\hat\chi_{B_1(q_0,r)}^{\mathrm{aw}}|\psi}=\int_{\Omega}\int_{B_1(q_0,r)}{\mathcal H}_{{{\tilde c_0}},\psi,\theta}(q,p)\,dpdq.$$ 2. Let $B_2(x,r)\subset{\mathbb{T}}^2$ with $r=r(\hbar)\ge\hbar^\rho$ for $\rho\in[0,1/2)$. Then as in Part I [@Han Lemma 3.1], there are $b^\pm_{x_0,r}\in C^\infty({\mathbb{T}}^1\times(0,\hbar))$ such that $$\label{eq:bxr} b^-_{x_0,r}\le\chi_{B_2(x_0,r)}\le b^+_{x_0,r}\quad\text{and}\quad\int_{{\mathbb{T}}^2}b^\pm_{x_0,r}(x)\,dx={\mathrm{Vol}}(B(x_0,r))+o\left(r^2\right).$$ In addition, $b^\pm_{x_0,r}\in S_\rho({\mathbb{T}}^2)$. For later use in Subsection \[sec:nonequi\], we also require that $b^-_{x_0,r}=1$ in $B_2(x_0,2r/3)$ and $b^-_{x_0,r}=1$ in $B_2(x_0,3r/2)$. The symbols $b^\pm_{x_0,r}$ are therefore the appropriate approximation of the indicator function $\chi_{B_2(x_0,r)}$. By Lemma \[lemma:WAW\], we have that for any normalized state $\psi$, $$\braket{\psi|\hat b_{x_0,r}^{\mathrm{w}}|\psi}-\braket{\psi|\hat b_{x_0,r}^{\mathrm{aw}}|\psi}=O\left(\hbar^{1-\rho}\right).$$ With the choice of $r(\hbar)=O(|\log\hbar|^{-1/2})$ in Theorems \[thm:SSQE\] and \[thm:sc\], the symbols $b^\pm_{x_0,r}\in S_\rho({\mathbb{T}}^2)$ for all $\rho\in(0,1/2)$. Hence, Statement (ii) there applies to both Weyl and anti-Wick quantizations. Construction of the quasimodes {#sec:qm} ============================== Throughout this section, we fix $M:{\mathbb{T}}^2\to{\mathbb{T}}^2$ as a classical cat map with Lyapunov $\lambda$. Denote $\hat M:{\mathcal H}_N\to{\mathcal H}_N$ its quantum cat map. Recall that $N=1/(2\pi\hbar)\in{\mathbb N}$. In This section, we construct the quasimodes in ${\mathcal H}_N$ that satisfy the conditions in Theorem \[thm:sc\]. Their localization properties are described by the Husimi functions , while the density distribution properties are studied via the anti-Wick quantization . Let $\gamma=\{x_t\}_{t=0}^{T-1}\subset{\mathbb{T}}^2$ be a closed prime orbit of $M$ with length $|\gamma|=T$. Here, we allow $T$ to depend on $\hbar$ and require that $$\label{eq:T} T\le\delta T_E=\frac{\delta|\log\hbar|}{\lambda},$$ in which $T_E$ is the Ehrenfest time and $\delta>0$ is to be determined later. Suppose that $\phi\in{\mathbb{R}}$. Construct the quantum state $\Psi^\gamma_\phi$ associated with $\gamma$ by $$\label{eq:qmT} \ket{\Psi^\gamma_\phi}=\sum_{t=0}^{T-1}e^{-i\phi t}\hat M^t\ket{x_0,{{\tilde c_0}},\theta}=\sum_{t=0}^{T-1}e^{-i\phi t}\hat M^t{{\hat P_\theta}}\hat T_{x_0}\ket{{\tilde c_0}}\in{\mathcal H}_N.$$ We first describe the Husimi function of $\Psi^\gamma_\phi$ in Subsection \[sec:qmHusimi\]. Then we use this description to establish the distribution of $\Psi^\gamma_\phi$ at various scales in Subsections \[sec:nonequi\] and \[sec:scmeasures\]. Description of the Husimi function {#sec:qmHusimi} ---------------------------------- For notational simplicity, we omit the scripts in $\Psi^\gamma_\phi$ and write $\Psi$. Compute the Husimi function of $\Psi$: $${\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)=N\left|\braket{x,{{\tilde c_0}},\theta|\Psi}\right|^2=N\left|\sum_{t=0}^{T-1}e^{i\phi t}\braket{{{\tilde c_0}}|\hat T_{-x}{{\hat P_\theta}}\hat M^t\hat T_{x_0}|{{\tilde c_0}}}\right|^2$$ We know from the Egorov’s theorem in Theorem \[thm:Egorov\] that $\hat M^t\hat T_{x_0}=\hat T_{M^tx_0}\hat M^t=\hat T_{x_t}\hat M^t$. Thus, $$\begin{aligned} N\left|\braket{{{\tilde c_0}}|\hat T_{-x}{{\hat P_\theta}}\hat M^t\hat T_{x_0}|{{\tilde c_0}}}\right|^2&=&N\left|\braket{{{\tilde c_0}}|\hat T_{-x}\hat T_{x_t}{{\hat P_\theta}}\hat M^t|{{\tilde c_0}}}\right|^2\\ &=&N\left|\braket{x-x_t,{{\tilde c_0}}|t;{{\tilde c_0}},\theta}\right|^2\\ &=&{\mathcal H}_{{{\tilde c_0}},t,\theta}(x-x_t).\end{aligned}$$ Now if $\delta<1/2$ in , then by , ${\mathcal H}_{{{\tilde c_0}},t,\theta}(x-x_t)$ is exponentially small unless $x-x_t\in B_2(o,C\hbar^\frac12e^{\lambda t})$, that is, $x\in B_2(x_t,C\hbar^\frac12e^{\lambda t})\subset B_2(x_t,C\hbar^\frac12e^{\lambda T})$. Next we are concerned about the separation of the balls $B_2(x_t,C\hbar^\frac12e^{\lambda t})\subset{\mathbb{T}}^2$ for $t=0,...,T-1$. By Proposition \[prop:periodicpts\], the prime closed orbit $\gamma$ of length $T$ lives on the lattice ${\mathcal{L}}_l$ of rational points with denominator $$l=e^{\lambda T}+e^{-\lambda T}-2\le e^{\lambda T}.$$ Since $\gamma=\{x_t\}_{t=0}^{T-1}$ is prime, $$\left|x_t-x_s\right|\ge\frac1l\ge e^{-\lambda T},\quad\text{if }t\ne s.$$ Thus, if $\delta<1/4$ in , then for all $T\le\delta T_E$, $$\left|x_t-x_s\right|\ge e^{-\lambda T}\ge2C\hbar^\frac12e^{\lambda T}.$$ This means that $$B_2\left(x_t,C\hbar^\frac12e^{\lambda T}\right)\cap B_2\left(x_s,C\hbar^\frac12e^{\lambda T}\right)=\emptyset,\quad\text{if }t\ne s.$$ Combining with and Corollary \[cor:tcothetaL2\], we summarize the description of the Husimi function of $\Psi$ as follows. \[thm:PsiHusimi\] Let $0<\delta<1/4$. Then there is a constant $C_0>0$ depending only on $M$ and $\delta$ such that the following statement holds. Suppose that $\phi\in{\mathbb{R}}$ and $\gamma=\{x_t\}_{t=0}^{T-1}$ is a closed prime prime orbit with length $|\gamma|=T\le\delta T_E$. Construct $\Psi=\Psi_\phi^\gamma$ as in . Then $$U_\Psi=\bigcup_{t=0}^{T-1}B_2\left(x_t,C_0\hbar^\frac12e^{\lambda T}\right)\quad\text{is a disjoint union.}$$ Moreover, 1. if $x\in U_\Psi$, then there is a unique $t\in\{0,...,T-1\}$ such that $${\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)={\mathcal H}_{{{\tilde c_0}},t,\theta}(x-x_t)+O\left(e^{-\frac{1}{C_0\hbar}}\right),$$ in which $$\int_{{\mathbb{T}}^2}{\mathcal H}_{{{\tilde c_0}},t,\theta}(x-x_t)\,dx=\int_{B_2\left(o,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta}(x)\,dx+O\left(e^{-\frac{1}{C_0\hbar}}\right)=1+O\left(e^{-\frac{1}{C_0\hbar}}\right),$$ 2. if $x\not\in U_\Psi$, then $${\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)=O\left(e^{-\frac{1}{C_0\hbar}}\right).$$ The reminder estimates in (i) and (ii) are uniform for all $\phi\in{\mathbb{R}}$, closed prime orbits $\gamma$ with $|\gamma|\le\delta T_E$, and $x\in{\mathbb{T}}^2$. The proposition asserts that for all $T\le\delta T_E$ such that $\delta<1/4$, the Husimi function ${\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)$ of $\Psi$ is a direct sum of ${\mathcal H}_{{{\tilde c_0}},t,\theta}(x-x_t)$ ($t=0,...,T-1$) with disjoint essential supports in $B_2(x_t,C_0\hbar^\frac12e^{\lambda T})$. The analysis in the rest of this section are based upon this assertion. For example, we immediately have the $L^2$ norm estimate of $\Psi$. \[prop:PsiL2\] Let $0<\delta<1/4$ and $T\le\delta T_E$. Then there is a constant $C>0$ depending only on $M$ and $\delta$ such that $$\|\Psi\|^2_{L^2({\mathbb{T}}^1)}=\braket{\Psi|\Psi}=T+O\left(e^{-\frac{1}{C\hbar}}\right).$$ By and Theorem \[thm:PsiHusimi\], we have that $$\begin{aligned} \braket{\Psi|\Psi}&=&\int_{{\mathbb{T}}^2}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)\,dx\\ &=&\sum_{t=0}^{T-1}\int_{B_2\left(x_t,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)\,dx+O\left(e^{-\frac{1}{C\hbar}}\right)\\ &=&\sum_{t=0}^{T-1}\int_{B_2\left(x_t,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta}(x-x_t)\,dx+O\left(e^{-\frac{1}{C\hbar}}\right)\\ &=&T\int_{B_2\left(o,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta}(x)\,dx+O\left(e^{-\frac{1}{C\hbar}}\right)\\ &=&T+O\left(e^{-\frac{1}{C\hbar}}\right).\end{aligned}$$ From the above $L^2$ norm estimate, we now see that \[cor:qm\] Let $0\le\delta<1/4$ and $T\le\delta T_E$. Then the normalized state $\ket\Psi_n=\ket\Psi/\sqrt{\braket{\Psi,\Psi}}$ are quasimodes of order $O(1/\sqrt T)$ with quasi-energy $e^{i\phi}$. In the view of , we have that $$\left\|\left(\hat M-e^{i\phi}\right)\ket\Psi\right\|_{L^2({\mathbb{T}}^2)}\le2\|\Psi\|_{L^2({\mathbb{T}}^2)},$$ that is, $$\left\|\left(\hat M-e^{i\phi}\right)\ket\Psi_n\right\|\le\frac{2}{\sqrt{T+O\left(e^{-\frac{1}{C\hbar}}\right)}}=O\left(\frac{1}{\sqrt T}\right),$$ by the $L^2$ norm estimate of $\Psi$ in Proposition \[prop:PsiL2\]. Non-equidistribution of the quasimodes at small scales {#sec:nonequi} ------------------------------------------------------ Let $\Psi=\Psi^\gamma_\phi$ be the quasimode that is associated with a closed prime orbit $\gamma=\{x_t\}_{t=0}^{T-1}$ of length $T=|\gamma|$. In this section, we show that $\Psi$ must display non-equidistribution at certain scales depending on $T$. The next lemma is a direct consequence of the description of the Husimi function in Proposition \[thm:PsiHusimi\]. Let $0<\delta<1/4$ and $T\le\delta T_E$. Suppose that $r=r(\hbar)\ge2C_0\hbar^\frac12e^{\lambda T}$ in Theorem \[thm:PsiHusimi\]. Then there is a constant $C>0$ depending only on $M$ and $\delta$ such that 1. $$\begin{cases} \braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q,3r)}|\Psi}\ge1+O\left(e^{-\frac{1}{C\hbar}}\right) & \text{if }B_1(q,r)\cap P_q(\gamma)\ne\emptyset,\\ \braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q,r/3)}|\Psi}=O\left(e^{-\frac{1}{C\hbar}}\right) & \text{if }B_1(q,r)\cap P_q(\gamma)=\emptyset, \end{cases}$$ 2. $$\begin{cases} \braket{\Psi|\hat b^{-,{\mathrm{aw}}}_{x,3r}|\Psi}\ge1+O\left(e^{-\frac{1}{C\hbar}}\right) & \text{if }B_2(x,r)\cap\gamma\ne\emptyset,\\ \braket{\Psi|\hat b^{+,{\mathrm{aw}}}_{x,r/3}|\Psi}=O\left(e^{-\frac{1}{C\hbar}}\right) & \text{if }B_2(x,r)\cap\gamma=\emptyset. \end{cases}$$ Here, $b^\pm_{x,r}$ are given in and $P_q(x)=q$ for $x=(q,p)\in{\mathbb{T}}^2$ is the projection onto space of the position variable $q$. (i). If $B_1(q,r)\cap P_q(\gamma)\ne\emptyset$, then there is $x_t=(q_t,p_t)\in\gamma$ such that $q_t\in B_1(q,r)$. Hence, $B_2(x_t,C_0\hbar^\frac12e^{\lambda T})\subset P_q^{-1}(B_1(q,3r))$ since $r\ge2C_0\hbar^\frac12e^{\lambda T}$. Therefore, by and Theorem \[thm:PsiHusimi\], $$\begin{aligned} \braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q,3r)}|\Psi}&=&\int_{B_1(q,3r)}\int_{{\mathbb{T}}^1}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(q',p')\,dp'dq'\\ &\ge&\int_{B_2\left(x_t,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(q',p')\,dp'dq'\\ &=&1+O\left(e^{-\frac{1}{C\hbar}}\right).\end{aligned}$$ On the other hand, if $B_1(q,r)\cap P_q(\gamma)=\emptyset$, then $B_2(x_t,C_0\hbar^\frac12e^{\lambda T})\cap P_q^{-1}(B_1(q,r/3))=\emptyset$ for all $t=0,...,T-1$. Hence, $U_\Psi\cap P_q^{-1}(B_1(q,r/3))=\emptyset$. By and Theorem \[thm:PsiHusimi\] again, $$\braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q,r/3)}|\Psi}=O\left(e^{-\frac{1}{C\hbar}}\right).$$ (ii). If $B_2(x,r)\cap\gamma\ne\emptyset$, then there is $x_t\in\gamma$ such that $x_t\in B_2(x,r)$ since $r\ge2C_0\hbar^\frac12e^{\lambda T}$. Hence, $B_2(x_t,C_0\hbar^\frac12e^{\lambda T})\subset B_2(x,2r)$. Therefore, by and Theorem \[thm:PsiHusimi\], $$\begin{aligned} \braket{\Psi|\hat b^{-,{\mathrm{aw}}}_{x,3r}|\Psi}&\ge&\int_{B_2(x,2r)}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x')\,dx'\\ &\ge&\int_{B_2\left(x_t,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x')\,dx'\\ &=&1+O\left(e^{-\frac{1}{C\hbar}}\right).\end{aligned}$$ On the other hand, if $B_1(q,r)\cap\gamma=\emptyset$, then $B_2(x_t,C_0\hbar^\frac12e^{\lambda T})\cap B_1(q,r/2)=\emptyset$ for all $t=0,...,T-1$. Hence, $U_\Psi\cap B_1(x,r/2)=\emptyset$. By and Theorem \[thm:PsiHusimi\] again, $$\braket{\Psi|\hat b^{+,{\mathrm{aw}}}_{x,r/3}|\Psi}\le\int_{B_2(x,r/2)}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x')\,dx'=O\left(e^{-\frac{1}{C\hbar}}\right).$$ Next we establish the non-equidistribution of $\Psi$ at certain scales. \[thm:nonequi\] Let $0<\delta<1/4$ and $T\le\delta T_E=\delta|\log\hbar|/\lambda$. Then there are constant $c_1,C>0$ depending only on $M$ and $\delta$ such that for any closed prime orbit $\gamma$ with length $T$, the associated quasimode $\Psi$ in satisfies the following non-equidistribution conditions. 1. For any $r\in[2C_0\hbar^\frac12e^{\lambda T},c_1T^{-1}]$, there are $q_1,q_2\in{\mathbb{T}}^1$ such that $$\braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q_1,r)}|\Psi}\ge1+O\left(e^{-\frac{1}{C\hbar}}\right)\quad\text{and}\quad\braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q_2,r)}|\Psi}=O\left(e^{-\frac{1}{C\hbar}}\right).$$ 2. For any $r\in[2C_0\hbar^\frac12e^{\lambda T},c_1T^{-\frac12}]$, then there are $x_1,x_2\in{\mathbb{T}}^2$ such that $$\braket{\Psi|\hat b_{x_1,r}^{\mathrm{aw}}|\Psi}\ge1+O\left(e^{-\frac{1}{C\hbar}}\right)\quad\text{and}\quad\braket{\Psi|\hat b_{x_2,r}^{\mathrm{aw}}|\Psi}=O\left(e^{-\frac{1}{C\hbar}}\right).$$ (i). Given any closed prime orbit $\gamma$ of length $T$ and $x_1=(q_1,p_1)\in\gamma$, $$\braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q,r)}|\Psi}\ge\int_{B_2\left(x_1,C_0\hbar^\frac12e^{\lambda T}\right)}{\mathcal H}_{{{\tilde c_0}},\Psi,\theta}(x)\ge1+O\left(e^{-\frac{1}{C\hbar}}\right),$$ using Theorem \[thm:PsiHusimi\]. On the other hand, select a maximal family of disjoint balls $\{B_1(q_k,3r)\}_{k=1}^K\subset{\mathbb{T}}^1$. Then we have that $K\ge cr^{-1}$ for some absolute constant $c>0$. Suppose that $r\le c/(2T)$ so $K\ge cr^{-1}\ge2T>T$. However, there are only $T$ points on the orbit $\gamma$. By the pigeon-hole principle, there is $1\le k\le K$ such that $B_1(q_k,3r)\cap P_q(\gamma)=\emptyset$. In this case, by the previous lemma, $$\braket{\Psi|\hat\chi^{\mathrm{aw}}_{B_1(q_k,r)}|\Psi}=O\left(e^{-\frac{1}{C\hbar}}\right).$$ That is, (i) is proved by choosing $c_1=c/2$. \(ii) can be proved in the same fashion so we omit the details. Semiclassical measures of the quasimodes {#sec:scmeasures} ---------------------------------------- Let $\mu$ be any invariant probability measure of the cat map $M$ on the torus ${\mathbb{T}}^2$. In this subsection, we construct a sequence of quasimodes for which the corresponding semiclassical measure is $\mu$, and in addition, they satisfy the non-equidistribution conditions as in Theorem \[thm:sc\]. By Theorem \[thm:Sigmund\], there is a sequence of closed prime orbits $\{\gamma_j\}_{j=1}^\infty$ such that the delta measures $\mu_{\gamma_j}\to\mu$ weakly. Denote $T_j=|\gamma_j|$. Then for all $f\in C({\mathbb{T}}^2)$, $$\label{eq:mujmu} \int_{{\mathbb{T}}^2}f\,d\mu_{\gamma_j}=\frac{1}{T_j}\sum_{t=0}^{T_j-1}f(x^j_t)\to\int_{{\mathbb{T}}^2}f\,d\mu\quad\text{as }j\to\infty,$$ in which the closed prime orbit $\gamma_j=\{x^j_t\}_{t=0}^{T_j-1}$. If $\{T_j\}_{j=1}^\infty$ is bounded, then $\mu=\mu_\gamma$ is itself a delta measure on some closed prime orbit $\gamma$. As mentioned in the introduction, this case has been treated in Faure-Nonnenmacher-De Bièvre [@FNDB]. We therefore assume that $\{T_j\}_{j=1}^\infty$ is unbounded. Without loss of generality, we assume that $T_j\to\infty$ is increasing. (If not, then choose a subsequence of $\{\gamma_j\}_{j=1}^\infty$ such that the lengths are increasing.) Fix $0<\delta<1/4$. Let $$N_j=\frac{1}{2\pi\hbar_j}=\left\lceil\frac{e^{\frac{\lambda T_j}{\delta}}}{2\pi}\right\rceil.$$ Then $$\label{eq:Tj} T_j\le\frac{\delta|\log(2\pi N_j)|}{\lambda}=\frac{\delta|\log\hbar_j|}{\lambda}\quad\text{and}\quad T_j=\frac{\delta|\log\hbar_j|}{\lambda}+O(\hbar_j).$$ For any $\phi_j\in{\mathbb{R}}$, construct the quantum states $\Psi^{\gamma_j}_{\phi_j}\in{\mathcal H}_{N_j}$ associated with the prime closed orbit $\gamma_j$ as in . In Subsection \[sec:qmHusimi\], we know that the normalized states $\ket{\Psi^{\gamma_j}_{\phi_j}}_n$ are quasimodes of order $O(T_j^{-1/2})=O(|\log\hbar_j|^{-1/2})$. Next, we show that the semiclassical measure induced by the normalized quasimodes $$\ket{\psi_j}=\ket{\Psi^{\gamma_j}_{\phi_j}}_n=\frac{\ket{\Psi^{\gamma_j}_{\phi_j}}}{\sqrt{\braket{\Psi^{\gamma_j}_{\phi_j}|\Psi^{\gamma_j}_{\phi_j}}}}$$ coincides with the probability measure $\mu$ on ${\mathbb{T}}^2$. By Theorem \[thm:PsiHusimi\], the Husimi function ${\mathcal H}_{{{\tilde c_0}},\Psi_{\psi_j}^{\gamma_j},\theta_j}(x)$ of $\Psi^{\gamma_j}_{\phi_j}$ is a direct sum of ${\mathcal H}_{{{\tilde c_0}},t,\theta_j}(x-x^j_t)$ ($t=0,...,T-1$) with disjoint essential support in $B_2(x_t^j,C_0\hbar^\frac12e^{\lambda T_j})$, module exponential errors. Together with the $L^2$ norm estimate of $\Psi_{\phi_j}^{\gamma_j}$ in Proposition \[prop:PsiL2\], we have that $$\begin{aligned} &&\braket{\psi_j|\hat f^{\mathrm{aw}}|\psi_j}\\ &=&\frac{1}{\braket{\Psi_{\phi_j}^{\gamma_j}|\Psi_{\phi_j}^{\gamma_j}}}\braket{\Psi_{\phi_j}^{\gamma_j}|\hat f^{\mathrm{aw}}|\Psi_{\phi_j}^{\gamma_j}}\\ &=&\frac{1}{T_j+O\left(e^{-\frac{1}{C\hbar_j}}\right)}\left(\int_{{\mathbb{T}}^2}{\mathcal H}_{{{\tilde c_0}},\Psi_{\phi_j}^{\gamma_j},\theta_j}(x)f(x)\,dx+O_f\left(\hbar_j^\frac12\right)\right)\\ &=&\frac{1}{T_j}\sum_{t=0}^{T_j-1}\int_{{\mathbb{T}}^2}{\mathcal H}_{{{\tilde c_0}},t,\theta_j}\left(x-x^j_t\right)f(x)\,dx+O_f\left(\hbar_j^\frac12\right)\\ &=&\frac{1}{T_j}\sum_{t=0}^{T_j-1}\int_{B_2\left(x_t^j,C_0\hbar^\frac12e^{\lambda T_j}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta_j}\left(x-x^j_t\right)f(x)\,dx+O_f\left(\hbar_j^\frac12\right)\\ &=&\frac{1}{T_j}\sum_{t=0}^{T_j-1}\left(f(x^j_t)\int_{B_2\left(x_t^j,C_0\hbar^\frac12e^{\lambda T_j}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta_j}\left(x-x^j_t\right)\,dx+O_f\left(\hbar^\frac12e^{\lambda T_j}\right)\right)+O_f\left(\hbar_j^\frac12\right)\\ &=&\frac{1}{T_j}\sum_{t=0}^{T_j-1}f(x^j_t)\int_{B_2\left(o,C\hbar^\frac12e^{\lambda T_j}\right)}{\mathcal H}_{{{\tilde c_0}},t,\theta_j}\left(x\right)\,dx+O_f\left(\hbar_j^{\frac12-\delta}\right)\\ &=&\frac{1}{T_j}\sum_{t=0}^{T_j-1}f(x^j_t)+O_f\left(\hbar_j^{\frac12-\delta}\right)\\ &\to&\int_{{\mathbb{T}}^2}f\,d\mu\quad\text{as }j\to\infty,\end{aligned}$$ in which the last step follows from . Finally, we use Theorem \[thm:nonequi\] to establish the non-equidistribution conditions in Theorem \[thm:sc\] for $\{\psi_j\}_{j=1}^\infty$. Let ${\varepsilon}>0$. For the non-equidistribution in the physical space, observe that if $r_j=c_0(\log N_j)^{-1}$ for $c_0$ small enough, then $r_j\in[2C_0\hbar^\frac12e^{\lambda T_j},c_1T_j^{-1}]$ so Theorem \[thm:nonequi\] applies. Note that $$\braket{\Psi_{\phi_j}^{\gamma_j}|\Psi_{\phi_j}^{\gamma_j}}=T_j+O\left(e^{-\frac{1}{C\hbar_j}}\right)$$ by the $L^2$ norm estimate of $\Psi_{\phi_j}^{\gamma_j}$ in Proposition \[prop:PsiL2\]. Then by Theorem \[thm:nonequi\], there are $q_1,q_2\in{\mathbb{T}}^1$ such that $$\int_{B_1(q_1,r_j)}|\psi_j|^2=\braket{\psi_j|\hat\chi^{\mathrm{aw}}_{B_1(q_1,r_j)}|\psi_j}\ge\frac{1}{T_j}+O\left(e^{-\frac{1}{C\hbar_j}}\right)\ge\frac{c}{\log N_j},$$ where $c>0$ depends only on $M$ and $\delta$ in the view of , and $$\int_{B_1(q_2,r_j)}|\psi_j|^2=\braket{\psi_j|\hat\chi^{\mathrm{aw}}_{B_1(q_2,r)}|\psi_j}=O\left(e^{-\frac{1}{C\hbar_j}}\right).$$ However, $${\mathrm{Vol}}(B_1(q,r_j))=2r_j=\frac{2c_0}{\log N_j}.$$ Hence, there are $c_0>0$ and $j_0\in{\mathbb N}$ such that $$\frac{\int_{B_1(q_1,r_j)}|\psi_j|^2}{{\mathrm{Vol}}(B_1(q,r_j))}\ge\frac{c}{2c_0}\ge{\varepsilon}^{-1}\quad\text{and}\quad\frac{\int_{B_1(q_2,r_j)}|\psi_j|^2}{{\mathrm{Vol}}(B_1(q,r_j))}=O\left(e^{-\frac{1}{C\hbar_j}}\right)\le{\varepsilon}.$$ The non-equidistribution of $\{\psi_j\}_{j=1}^\infty$ at small scale $c_0(\log N_j)^{-1/2}$ in the phase space ${\mathbb{T}}^2$ can be argued similarly so we omit the details. [99]{} S. Brooks, *On the entropy of quantum limits for $2$-dimensional cat maps*. Comm. Math. Phys. 293 (2010), no. 1, 231–255. S. Brooks, *Logarithmic-scale quasimodes that do not equidistribute*. Int. Math. Res. Not. IMRN 2015, no. 22, 11934–11960. A. Bouzouina and S. De Bièvre, *Equipartition of the eigenfunctions of quantized ergodic maps on the torus*, Comm. Math. Phys. 178 (1996), no. 1, 83–105. F. Bonechi and S. De Bièvre, *Controlling strong scarring for quantized ergodic toral automorphisms*. Duke Math. J. 117 (2003), no. 3, 571–587. F. Faure and S. Nonnenmacher, *On the maximal scarring for quantum cat map eigenstates*. Comm. Math. Phys. 245 (2004), no. 1, 201–214. S. Eswarathasan and S. Nonnenmacher, *Strong scarring of logarithmic quasimodes*. Ann. Inst. Fourier (Grenoble) 67 (2017), no. 6, 2307–2347. S. Eswarathasan and L. Silberman, *Scarring of quasimodes on hyperbolic manifolds*. Nonlinearity 31 (2018), no. 1, 1–29. F. Faure, S. Nonnenmacher, and S. De Bièvre, *Scarred eigenstates for quantum cat maps of minimal periods*, Comm. Math. Phys. 239 (2003), no. 3, 449–492. X. Han, *Small scale quantum ergodicity in cat maps*. I. [arXiv:1810.11949](https://arxiv.org/abs/1810.11949). A. Katok and B. Hasselblatt, *Introduction to the modern theory of dynamical systems*. Cambridge University Press, Cambridge, 1995. G. Riviére, *Entropy of semiclassical measures for symplectic linear maps of the multidimensional torus*. Int. Math. Res. Not. IMRN 2011, no. 11, 2396–2443. K. Sigmund, *Generic properties of invariant measures for Axiom A diffeomorphisms*. Invent. Math. 11 (1970), 99–109. S. Zelditch, *Index and dynamics of quantized contact transformations*. Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 305–363. M. Zworski, *Semiclassical analysis*. American Mathematical Society, Providence, 2012.

--- abstract: 'Implicit discourse relation classification is one of the most difficult steps in discourse parsing. The difficulty stems from the fact that the coherence relation must be inferred based on the content of the discourse relational arguments. Therefore, an effective encoding of the relational arguments is of crucial importance. We here propose a new model for implicit discourse relation classification, which consists of a classifier, and a sequence-to-sequence model which is trained to generate a representation of the discourse relational arguments by trying to predict the relational arguments including a suitable implicit connective. Training is possible because such implicit connectives have been annotated as part of the PDTB corpus. Along with a memory network, our model could generate more refined representations for the task. And on the now standard 11-way classification, our method outperforms the previous state of the art systems on the PDTB benchmark on multiple settings including cross validation.' author: - | Wei Shi$^\dag$ and Vera Demberg$^{\dag,\ddag}$\ $^\dag$Dept. of Language Science and Technology\ $^\ddag$Dept. of Mathematics and Computer Science, Saarland University\ Saarland Informatics Campus, 66123 Saarbrücken, Germany\ [{w.shi, vera}@coli.uni-saarland.de]{} bibliography: - 'iwcs2019.bib' title: | Learning to Explicitate Connectives with Seq2Seq Network\ for Implicit Discourse Relation Classification --- Introduction ============ Discourse relations describe the logical relation between two sentences/clauses. When understanding a text, humans infer discourse relation between text segmentations. They reveal the structural organization of text, and allow for additional inferences. Many natural language processing tasks, such as machine translation, question-answering, automatic summarization, sentiment analysis, and sentence embedding learning, can also profit from having access to discourse relation information. Recent years have seen more and more works on this topic, including two CoNNL shared tasks [@xue2015conll; @xue2016conll]. Penn Discourse Tree Bank [@Prasad08thepenn PDTB] provides lexically-grounded annotations of discourse relations and their two discourse relational arguments (i.e., two text spans). Discourse relations are sometimes signaled by explicit discourse markers (e.g., *because, but*). Example \[eg\_1\] shows an explicit discourse relation marked by “because"; the presence of the connective makes it possible to classify the discourse relation with high reliability: [@miltsakaki2005experiments] reported an accuracy of 93.09% for 4-way classification of explicits. Discourse relations are however not always marked by an explicit connective. In fact, implicit discourse relations (i.e. relations not marked by an explicit discourse cue) outnumber explicit discourse relations in naturally occurring text. Readers can still infer these implicit relations, but automatic classification becomes a lot more difficult in these cases, and represents the main bottleneck in discourse parsing today. Example \[eg\_2\] shows an implicit contrastive relation which can be inferred from the two text spans that have been marked *Arg1* and *Arg2*. When annotating implicit relations in the PDTB, annotators were asked to first insert a connective which expresses the relation, and then annotate the relation label. This procedure was introduced to achieve higher inter-annotator agreement for implicit relations between human annotators. In the approach taken in this paper, our model mimics this procedure by being trained to explicitate the discouse relation, i.e. to insert a connective as a secondary task. 1. [\[eg\_1\] ]{} 2. [\[eg\_2\] ]{} The key in implicit discourse relation classification lies in extracting relevant information for the relation label from (the combination of) the discourse relational arguments. Informative signals can consist of surface cues, as well as the semantics of the relational arguments. Statistical approaches have typically relied on linguistically informed features which capture both of these aspects, like temporal markers, polarity tags, Levin verb classes and sentiment lexicons, as well as the Cartesian products of the word tokens in the two arguments [@lin2009recognizing]. More recent efforts use distributed representations with neural network architectures [@qin2016implicit]. The main question in designing neural networks for discourse relation classification is how to get the neural networks to effectively encode the discourse relational arguments such that all of the aspects relevant to the classification of the relation are represented, in particular in the face of very limited amounts of annotated training data, see e.g. @rutherford2017systematic. The crucial intuition in the present paper is to make use of the annotated implicit connectives in the PDTB: in addition to the typical relation label classification task, we also train the model to encode and decode the discourse relational arguments, and at the same time predict the implicit connective. This novel secondary task forces the internal representation to more completely encode the semantics of the relational arguments (in order to allow the model to decode later), and to make a more fine-grained classification (predicting the implicit connective) than is necessary for the overall task. This more fine-grained task thus aims to force the model to represent the discourse relational arguments in a way that allows the model to also predict a suitable connective. Our overall discourse relation classifier combines representations from the relational arguments as well as the hidden representations generated as part of the encoder-decoder architecture to predict relation labels. What’s more, with an explicit memory network, the network also has access to history representations and acquire more explicit context knowledge. We show that our method outperforms previous approaches on the 11-way classification on the PDTB 2.0 benchmark. The remaining of the paper is organized as follows: Section 2 discusses related work; Section 3 describes our proposed method; Section 4 gives the training details and experimental results, which is followed by conclusion and future work in section 5. Related Work ============ Implicit Discourse Relation Classification ------------------------------------------ Implicit discourse relation recognition is one of the most important components in discourse parsing. With the release of PDTB [@Prasad08thepenn], the largest available corpus which annotates implicit examples with discourse relation labels and implicit connectives, a lot of previous works focused on typical statistical machine learning solutions with manually crafted sparse features [@rutherford2014discovering]. Recently, neural networks have shown an advantage of dealing with data sparsity problem, and many deep learning methods have been proposed for discourse parsing, including convolutional [@zhang2015shallow], recurrent [@ji2016latent], character-based [@qin2016implicit], adversarial [@qin2017adversarial] neural networks, and pair-aware neural sentence modeling [@cai2017pair]. Multi-task learning has also been shown to be beneficial on this task [@lan2017multi]. However, most neural based methods suffer from insufficient annotated data.[@wu2016bilingually] extracted bilingual-constrained synthetic implicit data from a sentence-aligned English-Chinese corpus. [@shi2017using; @shi2018acquiring] proposed to acquire additional training data by exploiting *explicitation* of connectives during translation. Explicitation refers to the fact that translators sometimes add connectives into the text in the target language which were not originally present in the source language. They used explicitated connectives as a source of weak supervision to obtain additional labeled instances, and showed that this extension of the training data leads to substantial performance improvements. The huge gap between explicit and implicit relation recognition (namely, 50% vs. 90% in 4-way classification) also motivates to incorporate connective information to guide the reasoning process. @zhou2010predicting used a language model to automatically insert discourse connectives and leverage the information of these predicted connectives. The approach which is most similar in spirit to ours, @qin2017adversarial, proposed a neural method that incorporates implicit connectives in an adversarial framework to make the representation as similar as connective-augmented one and showed that the inclusion of implicit connectives could help to improve classifier performance. Sequence-to-sequence Neural Networks ------------------------------------ Sequence to sequence model is a general end-to-end approach to sequence learning that makes minimal assumptions on the sequence structure, and firstly proposed by @sutskever2014sequence. It uses multi-layered Long Short-Term Memory (LSTM) or Gated Recurrent Units (GRU) to map the input sequence to a vector with a fixed dimensionality, and then decode the target sequence from the vector with another LSTM / GRU layer. Sequence to sequence models allow for flexible input/output dynamics and have enjoyed great success in machine translation and have been broadly used in variety of sequence related tasks such as Question Answering, named entity recognition (NER) / part of speech (POS) tagging and so on. If the source and target of a sequence-to-sequence model are exactly the same, it is also called Auto-encoder, @dai2015semi used a sequence auto-encoder to better represent sentence in an unsupervised way and showed impressive performances on different tasks. The main difference between our model and this one is that we have different input and output (the output contains a connective while the input doesn’t). In this way, the model is forced to explicitate implicit relation and try to learn the latent pattern and discourse relation between implicit arguments and connectives and then generate more discriminative representations. Methodology =========== Our model is based on the sequence-to-sequence model used for machine translation [@luong2015effective], an adaptation of an LSTM [@hochreiter1997long] that encodes a variable length input as a fix-length vector, then decodes it into a variable length of outputs. As illustrated in Figure \[model\], our model consists of three components: Encoder, Decoder and Discourse Relation Classifier. We here use different LSTMs for the encoding and decoding tasks to help keep the independence between those two parts. The task of implicit discourse relation recognition is to recognize the senses of the implicit relations, given the two arguments. For each discourse relation instance, The Penn Discourse Tree Bank (PDTB) provides two arguments (*$Arg_1$, $ Arg_2$*) along with the discourse relation (*Rel*) and manually inserted implicit discourse connective (*$Conn_i$*). Here is an implicit example from section 0 in PDTB: 1. [\[eg\_3\] $\mathbf{Arg_1}$:   This is an old story.\ $\mathbf{Arg_2}$:   We’re talking about years ago before anyone heard of asbestos having any questionable properties.\ $\mathbf{Conn_i}$: in fact\ $\mathbf{Rel}$:      Expansion.Restatement ]{} During training, the input and target sentences for sequence-to-sequence neural network are $\left[\textit{$Arg_1$}; \textit{$Arg_2$} \right]$ and $\left[\textit{$Arg_1$}; \textit{$Conn_i$}; \textit{$Arg_2$} \right]$ respectively, where “;” denotes concatenation. Model Architecture ------------------ ### Encoder Given a sequence of words, an encoder computes a joint representation of the whole sequence. After mapping tokens to Word2Vec embedding vectors [@mikolov2013distributed], a LSTM recurrent neural network processes a variable-length sequence $x=(x_1, x_2, ..., x_n)$. At time step $t$, the state of memory cell $c_t$ and hidden $h_t$ are calculated with the Equations \[lstm\]: $$\label{lstm} \small \begin{gathered} \left[\begin{array}{c} i_t \\ f_t \\ o_t \\ \hat{c_t} \end{array} \right] = \left[\begin{array}{c} \sigma \\ \sigma \\ \sigma \\ \tanh \end{array} \right] W \cdot [h_{t-1}, x_t] \\ c_t = f_t \odot c_{t-1} + i_t \odot \hat{c_t} \\ h_t = o_t \odot \tanh(c_t)\\ \end{gathered}$$ where $x_t$ is the input at time step $t$, $i$, $f$ and $o$ are the input, forget and output gate activation respectively. $\hat{c_t}$ denotes the current cell state, $\sigma$ is the logistic sigmoid function and $\odot$ denotes element-wise multiplication. The LSTM separates the memory $c$ from the hidden state $h$, which allows for more flexibility incombining new inputs and previous context. For the sequence modeling tasks, it is beneficial to have access to the past context as well as the future context. Therefore, we chose a bidirectional LSTM as the encoder and the output of the word at time-step $t$ is shown in the Equation \[bi\_h\]. Here, element-wise sum is used to combine the forward and backward pass outputs. $$\label{bi_h} \small h_t = \left[\overrightarrow{h_t} \oplus \overleftarrow{h_t}\right]$$ Thus we get the output of encoder: $$\small h_{e} = \left[h^e_1, h^e_2, ..., h^e_n\right]$$ ### Decoder With the representation from the encoder, the decoder tries to map it back to the targets space and predicts the next words. Here we used a separate LSTM recurrent network to predict the target words. During training, target words are fed into the LSTM incrementally and we get the outputs from decoder LSTM: $$\small h_{d} = \left[h^d_1, h^d_2, ..., h^d_n\right]$$ ### Global Attention {#global-attention .unnumbered} In each time-step in decoding, it’s better to consider all the hidden states of the encoder to give the decoder a full view of the source context. So we adopted the global attention mechanism proposed in @luong2015effective. For time step $t$ in decoding, context vector $c_t$ is the weighted average of $h_{e}$, the weights for each time-step are calculated with $h_t^d$ and $h_{e}$ as illustrated below: $$\small \alpha_t = \frac{\exp({h_t^d}^\top \mathbf{W_{\alpha}} h_{e})}{\sum\limits_{t=1}^n \exp({h_t^d}^\top \mathbf{W_{\alpha}} h_{e})}$$ $$\small c_t = \alpha h_{e}$$ ### Word Prediction {#word-prediction .unnumbered} Context vector $c_t$ captured the relevant source side information to help predict the current target word $y_t$. We employ a concatenate layer with activation function $\tanh$ to combine context vector $c_t$ and hidden state of decoder $h^d_t$ at time-step t as follows: $$\small \hat{h^d_t} = \tanh(\mathbf{W_c}\left[c_t; h^d_t \right])$$ Then the predictive vector is fed into the softmax layer to get the predicted distribution $\hat{p}(y_t|s)$ of the current target word. $$\small \begin{gathered} \hat{p}(y_t|s) = softmax(\mathbf{W}_s \hat{h_d} + \mathbf{b}_s)\\ \hat{y_t} = \arg\max_y\hat{p}(y_t|s) \end{gathered}$$ After decoding, we obtain the predictive vectors for the whole target sequence $\hat{h_d}=\left[h^d_1, h^d_2, ..., h^d_n \right]$. Ideally, it contains the information of exposed implicit connectives. ### Gated Interaction {#gated-interaction .unnumbered} In order to predict the coherent discourse relation of the input sequence, we take both the $h_{encoder}$ and the predictive word vectors $h_d$ into account. K-max pooling can “draw together” features that are most discriminative and among many positions apart in the sentences, especially on both the two relational arguments in our task here; this method has been proved to be effective in choosing active features in sentence modeling [@blunsom2014convolutional]. We employ an average k-max pooling layer which takes average of the top k-max values among the whole time-steps as in Equation \[k\_max\_1\] and \[k\_max\_2\]: $$\label{k_max_1} \small \bar{h}_e=\frac{1}{k}\sum\limits^{k}_{i=1}topk(h_{e})$$ $$\label{k_max_2} \small \bar{h}_d=\frac{1}{k}\sum\limits^{k}_{i=1}topk(\hat{h^d})$$ $\bar{h}_e$ and $\bar{h}_d$ are then combined using a linear layer [@lan2017multi]. As illustrated in Equation \[equ\_interaction\], the linear layer acts as a gate to determine how much information from the sequence-to-sequence network should be mixed into the original sentence’s representations from the encoder. Compared with bilinear layer, it also has less parameters and allows us to use high dimensional word vectors. $$\small \label{equ_interaction} h^* = \bar{h}_e \oplus \sigma(\mathbf{W}_i \bar{h}_d + \mathbf{b}_i)$$ ### Explicit Context Knowledge {#explicit-context-knowledge .unnumbered} To further capture common knowledge in contexts, we here employ a memory network proposed in @liu2018learning, to get explicit context representations of contexts training examples. We use a memory matrix $M \in R^{K \times N}$, where $K, N$ denote hidden size and number of training instances respectively. During training, the memory matrix remembers the information of training examples and then retrieves them when predicting labels. Given a representation $h^*$ from the interaction layer, we generate a **knowledge vector** by weighted memory reading: $$\small k = M softmax(M^Th^*)$$ We here use dot product attention, which is faster and space-efficient than additive attention, to calculate the scores for each training instances. The scores are normalized with a softmax layer and the final knowledge vector is a weighted sum of the columns in memory matrix $M$. Afterwards, the model predicts the discourse relation using a softmax layer. $$\small \begin{gathered} \hat{p}(r|s) = softmax(\mathbf{W}_r [k; h^*] + \mathbf{b}_r)\\ \hat{r} = \arg\max_y\hat{p}(r|s) \end{gathered}$$ Multi-objectives ---------------- In our model, the decoder and the discourse relation classifier have different objectives. For the decoder, the objective consists of predicting the target word at each time-step. The loss function is calculated with masked cross entropy with $\mathtt{L2}$ regularization, as follows: $$\small \mathit{Loss_{de}} = -\frac{1}{n}\sum\limits^n_{t=1}y_t\log(\hat{p_y}) + \frac{\lambda}{2}\parallel\theta_{de}\parallel^2_2 \label{decoder_loss}$$ where $y_t$ is one-hot represented ground truth of target words, $\hat{p_y}$ is the estimated probabilities for each words in vocabulary by softmax layer, $n$ denotes the length of target sentence. $\lambda$ is a hyper-parameter of $L2$ regularization and $\theta$ is the parameter set. The objective of the discourse relation classifier consists of predicting the discourse relations. A reasonable training objective for multiple classes is the categorical cross-entropy loss. The loss is formulated as: $$\small \mathit{Loss_{cl}} = -\frac{1}{m}\sum\limits^m_{i=1}r_i\log(\hat{p_r}) + \frac{\lambda}{2}\parallel\theta_{cl}\parallel^2_2$$ where $r_i$ is one-hot represented ground truth of discourse relation labels, $\hat{p_r}$ denotes the predicted probabilities for each relation class by softmax layer, $m$ is the number of target classes. Just like above, $\lambda$ is a hyper-parameter of $L2$ regularization. For the overall loss of the whole model, we set another hyper-parameter $w$ to give these two objective functions different weights. Larger $w$ means that more importance is placed on the decoder task. $$\small \mathit{Loss} = \mathit{w} \cdot \mathit{Loss_{de}} + \mathit{(1-w)} \cdot \mathit{Loss_{cl}} \label{weighted_losses}$$ Model Training -------------- To train our model, the training objective is defined by the loss function we introduced above. We use Adaptive Moment Estimation (Adam) [@kingma2014adam] with different learning rate for different parts of the model as our optimizer. Dropout layers are applied after the embedding layer and also on the top feature vector before the softmax layer in the classifier. We also employ $L_2$ regularization with small $\lambda$ in our objective functions for preventing over-fitting. The values of the hyper-parameters, are provided in Table \[table:hyper\]. The model is trained firstly to minimize the loss in Equation \[decoder\_loss\] until convergence, we use scheduled sampling [@bengio2015scheduled] during training to avoid “teacher-forcing problem". And then to minimize the joint loss in Equation \[weighted\_losses\] to train the implicit discourse relation classifier. Experiments and Results ======================= Experimental Setup {#exp_setup} ------------------ We evaluate our model on the PDTB. While early work only evaluated classification performance between the four main PDTB relation classes, more recent work including the CoNLL 2015 and 2016 shared tasks on Shallow Discourse Parsing [@xue2015conll; @xue2016conll] have set the standard to second-level classification. The second-level classification is more useful for most downstream tasks. Following other works we directly compare to in our evaluation, we here use the setting where AltLex, EntRel and NoRel tags are ignored. About 2.2% of the implicit relation instances in PDTB have been annotated with two relations, these are considered as two training instances. To allow for full comparability to earlier work, we here report results for three different settings. The first one is denoted as PDTB-Lin [@lin2009recognizing]; it uses sections 2-21 for training, 22 as dev and section 23 as test set. The second one is labeled PDTB-Ji [@ji2015one], and uses sections 2-20 for training, 0-1 as dev and evaluates on sections 21-22. Our third setting follows the recommendations of @shi2017need, and performs 10-fold cross validation on the whole corpus (sections 0-23). Table \[table:train\_num\] shows the number of instances in train, development and test set in different settings. Settings Train Dev Test ---------------------------- ------- ------ ------ PDTB-Lin 13351 515 766 PDTB-Ji 12826 1165 1039 Cross valid. per fold avg. 12085 1486 1486 : Numbers of train, development and test set on different settings for 11-way classification task. Instances annotated with two labels are double-counted and some relations with few instances have been removed.[]{data-label="table:train_num"} The advantage of the cross validation approach is that it addresses problems related to the small corpus size, as it reports model performance across all folds. This is important, because the most frequently used test set (PDTB-Lin) contains less than 800 instances; taken together with a lack in the community to report mean and standard deviations from multiple runs of neural networks [@reimers2018comparing], the small size of the test set makes reported results potentially unreliable. ### Preprocessing {#preprocessing .unnumbered} We first convert tokens in PDTB to lowercase and normalize strings, which removes special characters. The word embeddings used for initializing the word representations are trained with the CBOW architecture in *Word2Vec*[^1] [@mikolov2013distributed] on PDTB training set. All the weights in the model are initialized with uniform random. To better locate the connective positions in the target side, we use two position indicators ($\langle conn \rangle$, $\langle /conn \rangle$) which specify the starting and ending of the connectives [@zhou2016attention], which also indicate the spans of discourse arguments. Since our main task here is not generating arguments, it is better to have representations generated by correct words rather than by wrongly predicted ones. So at test time, instead of using the predicted word from previous time step as current input, we use the source sentence as the decoder’s input and target. As the implicit connective is not available at test time, we use a random vector, which we used as “impl\_conn” in Figure \[fig:att\_weights\], as a placeholder to inform the sequence that the upcoming word should be a connective. ### Hyper-parameters {#hyper-parameters .unnumbered} There are several hyper-parameters in our model, including dimension of word vectors $d$, two dropout rates after embedding layer $q_1$ and before softmax layer $q_2$, two learning rates for encoder-decoder $lr_1$ and for classifier $lr_2$, top $k$ for k-max pooling layer, different weights $w$ for losses in Equation and $\lambda$ denotes the coefficient of regularizer, which controls the importance of the regularization term, as shown in Table \[table:hyper\]. $d$ $q_1$ $q_2$ ${lr}_1$ ${lr}_2$ $k$ $w$ $\lambda$ ----- ------- ------- ------------- ----------- ----- ----- ----------- 100 0.5 0.2 $2.5e^{-3}$ $5e^{-3}$ 5 0.2 $5e^{-4}$ : Hyper-parameter settings.[]{data-label="table:hyper"} Experimental Results -------------------- We compare our models with six previous methods, as shown in Table \[table:performance\_level2\]. The baselines contain feature-based methods [@lin2009recognizing], state-of-the-art neural networks [@qin2016implicit; @cai2017pair], including the adversarial neural network that also exploits the annotated implicit connectives [@qin2017adversarial], as well as the data extension method based on using explicitated connectives from translation to other languages [@shi2017using]. Additionally, we ablate our model by taking out the prediction of the implicit connective in the sequence to sequence model. The resulting model is labeled Auto-Encoder in Table \[table:performance\_level2\]. And seq2seq network without knowledge memory, which means we use the output of gated interaction layer to predict the label directly, as denoted as Seq2Seq w/o Mem Net. Methods PDTB-Lin PDTB-Ji Cross Validation ---------------------------------------- ----------- ----------- ------------------ Majority class 26.11 26.18 25.59 @lin2009recognizing 40.20 - - @qin2016implicit 43.81 45.04 - @cai2017pair - 45.81 - @qin2017adversarial 44.65 **46.23** - @shi2017using (with extra data) **45.50** - **37.84** Encoder only (Bi-LSTM) [@shi2017using] 34.32 - 30.01 Auto-Encoder 43.86 45.43 39.50 Seq2Seq w/o Mem Net 45.75 47.05 40.29 Proposed Method **45.82** **47.83** **41.29** Our proposed model outperforms the other models in each of the settings. Compared with performances in @qin2017adversarial, although we share the similar idea of extracting highly discriminative features by generating connective-augmented representations for implicit discourse relations, our method improves about 1.2% on setting PDTB-Lin and 1.6% on the PDTB-Ji setting. The importance of the implicit connective is also illustrated by the fact that the “Auto-Encoder” model, which is identical to our model except it does not predict the implicit connective, performs worse than the model which does. This confirms our initial hypothesis that training with implicit connectives helps to expose the latent discriminative features in the relational arguments, and generates more refined semantic representation. It also means that, to some extent, purely increasing the size of tunable parameters is not always helpful in this task and trying to predict implicit connectives in the decoder does indeed help the model extract more discriminative features for this task. What’s more, we can also see that without the memory network, the performances are also worse, it shows that with the concatenation of knowledge vector, the training instance may be capable of finding related instances to get common knowledge for predicting implicit relations. As @shi2017need argued that it is risky to conclude with testing on such small test set, we also run cross-validation on the whole PDTB. From Table \[table:performance\_level2\], we have the same conclusion with the effectiveness of our method, which outperformed the baseline (Bi-LSTM) with more than 11% points and 3% compared with @shi2017using even though they have used a very large extra corpus. For the sake of obtaining a better intuition on how the global attention works in our model, Figure \[fig:att\_weights\] demonstrates the weights of different time-steps in attention layer from the decoder. The weights show how much importance the word attached to the source words while predicting target words. We can see that without the connective in the target side of test, the word filler still works as a connective to help predict the upcoming words. For instance, the true discourse relation for the right-hand example is *Expansion.Alternative*, at the word filler’s time-step, it attached more importance on the negation “don’t” and “tastefully appointed". It means the current representation could grasp the key information and try to focus on the important words to help with the task. Here we see plenty room for adapting this model to discourse connective prediction task, we would like to leave this to the future work. We also try to figure out which instances’ representations have been chosen from the memory matrix while predicting. Table \[table:mem\_att\_examples\] shows two examples and their context instances with top 2 memory attentions among the whole training set. We can see that both examples show that the memory attention attached more importance on the same relations. This means that with the Context Memory, the model could facilitate the discourse relation prediction by choosing examples that share similar semantic representation and discourse relation during prediction. [.33]{} {width="\linewidth"} [.33]{} {width="\linewidth"} [.33]{} {width="\linewidth"} [|p[15.5cm]{}|]{} *In recent years, U.S. steelmakers have supplied about 80% of the 100 million tons of steel used annually by the nation. (**in addition,**) Of the remaining 20% needed, the steel-quota negotiations allocate about 15% to foreign suppliers.*\ — Expansion.Conjunction\ \ 1\. The average debt of medical school graduates who borrowed to pay for their education jumped 10% to \$42,374 this year from \$38,489 in 1988, says the Association of American Medical Colleges. (**furthermore**) that’s 115% more than in 1981\ — Expansion.Conjunction\ 2. ... he rigged up an alarm system, including a portable beeper, to alert him when Sventek came on the line. (**and**) Some nights he slept under his desk.\ — Expansion.Conjunction\ *Prices for capital equipment rose a hefty 1.1% in September, while prices for home electronic equipment fell 1.1%. (**Meanwhile,**) food prices declined 0.6%, after climbing 0.3% in August.*\ — Comparison.Contrast\ \ 1\. Lloyd’s overblown bureaucracy also hampers efforts to update marketing strategies. (**Although**) some underwriters have been pressing for years to tap the low-margin business by selling some policies directly to consumers.\ — Comparison.Contrast 2\. Valley National "isn’t out of the woods yet. (**Specifically**), the key will be whether Arizona real estate turns around or at least stabilizes\ — Expansion.Restatement\ Relation Train Dev Test ------------- ------- ------ ------ Comparison 1855 189 145 Contingency 3235 281 273 Expansion 6673 638 538 Temporal 582 48 55 Total 12345 1156 1011 : Distribution of top-level implicit discourse relations in the PDTB.[]{data-label="table:relation_num_top"} ---------------------------- ----------- ----------- ----------- ----------- ----------- ----------- $F_1$ Acc. Comp. Cont. Expa. Temp. @rutherford2014discovering 38.40 55.50 39.70 54.42 70.23 28.69 @qin2016stacking - - **41.55** 57.32 71.50 35.43 @liu2016implicit 44.98 57.27 37.91 55.88 69.97 37.17 @ji2016latent 42.30 **59.50** - - - - @liu2016recognizing 46.29 57.17 36.70 54.48 70.43 **38.84** @qin2017adversarial - - 40.87 54.46 72.38 36.20 @lan2017multi **47.80** 57.39 40.73 **58.96** **72.47** 38.50 Our method 46.40 **61.42** **41.83** **62.07** 69.58 35.72 ---------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ### Top-level Binary and 4-way Classification A lot of the recent works in PDTB relation recognition have focused on first level relations, both on binary and 4-ways classification. We also report the performance on level-one relation classification for more comparison to prior works. As described above, we followed the conventional experimental settings [@rutherford2015improving; @liu2016recognizing] as closely as possible. Table \[table:relation\_num\_top\] shows the distribution of top-level implicit discourse relation in PDTB, it’s worth noticing that there are only 55 instances for Temporal Relation in the test set. To make the results comparable with previous work, we report the $F_1$ score for four binary classifications and both $F_1$ and Accuracy for 4-way classification, which can be found in Table \[performance\_level1\]. We can see that our method outperforms all alternatives on <span style="font-variant:small-caps;">Comparison</span> and <span style="font-variant:small-caps;">Contingency</span>, and obtain comparable scores with the state-of-the-art in others. For 4-way classification, we got the best accuracy and second-best $F_1$ with around 2% better than in @ji2016latent. Conclusion and Future Work ========================== We present in this paper a novel neural method trying to integrate implicit connectives into the representation of implicit discourse relations with a joint learning framework of sequence-to-sequence network. We conduct experiments with different settings on PDTB benchmark, the results show that our proposed method can achieve state-of-the-art performance on recognizing the implicit discourse relations and the improvements are not only brought by the increasing number of parameters. The model also has great potential abilities in implicit connective prediction in the future. Our proposed method shares similar spirit with previous work in @zhou2010predicting, who also tried to leverage implicit connectives to help extract discriminative features from implicit discourse instances. Comparing with the adversarial method proposed by @qin2017adversarial, our proposed model more closely mimics humans’ annotation process of implicit discourse relations and is trained to directly explicitate the implicit relations before classification. With the representation of the original implicit sentence and the explicitated one from decoder, and the help of the explicit knowledge vector from memory network, the implicit relation could be classified with higher accuracy. Although our method has not been trained as a generative model in our experiments, we can see potential for applying it to generative tasks. With more annotated data, minor modification and fine-tuned training, we believe our proposed method could also be applied to tasks like implicit discourse connective prediction, or argument generation in the future. Acknowledgments =============== This work was supported by German Research Foundation (DFG) as part of SFB 1102 “Information Density and Linguistic Encoding”. We would like to thank the anonymous reviewers for their careful reading and insightful comments. [^1]: <https://code.google.com/archive/p/word2vec/>

--- abstract: 'We study a generalization of the multi-armed bandit problem with multiple plays where there is a cost associated with pulling each arm and the agent has a budget at each time that dictates how much she can expect to spend. We derive an asymptotic regret lower bound for any uniformly efficient algorithm in our setting. We then study a variant of Thompson sampling for Bernoulli rewards and a variant of KL-UCB for both single-parameter exponential families and bounded, finitely supported rewards. We show these algorithms are asymptotically optimal, both in rate and leading problem-dependent constants, including in the thick margin setting where multiple arms fall on the decision boundary.' author: - 'Alexander R. Luedtke' - Emilie Kaufmann - Antoine Chambaz bibliography: - 'persrule.bib' title: '**Asymptotically Optimal Algorithms for Budgeted Multiple Play Bandits**' --- \#1 Introduction ============ In the classical multi-armed bandit problem, an agent is repeatedly confronted with a set of $K$ probability distributions $\nu_1,\dots,\nu_K$ called *arms* and must at each round select one of the available arms to pull based on their knowledge from previous rounds of the game. Each arm presents the agent with a reward drawn from the corresponding distribution, and the agent’s objective is to maximize the expected sum of their rewards over time or, equivalently, to minimize the total regret (the expected reward of pulling the optimal arm at every time step minus the expected sum of the rewards corresponding to their selected actions). To play the game well, the agent must balance the need to gather new information about the reward distribution of each arm (exploration) with the need to take advantage of the information that they already have by pulling the arm for which they believe the reward will be the highest (exploitation). While the general setup of the bandit problem is timeless, the problem first started receiving rigorous mathematical attention slightly under a century ago [@Thompson1933]. This early work focused on Bernoulli rewards, that are relevant in the simplest modeling of a sequential clinical trial, and presented a Bayesian algorithm now known as Thompson sampling. Since that time, many authors have contributed to a deeper understanding of the multi-armed bandit problem, both with Bernoulli and other reward distributions and either from a Bayesian [@Gittins1979] or frequentist [@Robbins1952] perspective. established a lower bound on the (frequentist) regret of any algorithm that satisfies a general uniform efficiency condition. This lower bound provides a concise definition of asymptotic (regret) optimality for an algorithm: an algorithm is asymptotically optimal when it achieves this lower bound. also introduced what are known as upper confidence bound (UCB) procedures for deciding which arm to pull at a given time step. In short, these procedures compute a UCB for the expected reward of each arm at each time and pull the arm with the highest UCB. Many variants of UCB algorithms have been proposed since then (see the Introduction of [@Cappeetal2013] for a thorough review), with more explicit indices and/or finite-time regret guarantees. Among them the KL-UCB algorithm [@Cappeetal2013] is proved to be asymptotically optimal for rewards that belong to a one-parameter exponential family and finitely-supported rewards. Meanwhile, there has been a recent interest in the theoretical understanding of the previously discussed Thompson sampling algorithm, whose first regret bound was obtained by . Since then, Thompson Sampling has been proved to be asymptotically optimal for Bernoulli rewards and for reward distributions belonging to univariate exponential families [@Kordaetal2013]. There has recently been a surge of interest in the multi-armed bandit problem, due to its applications to (online) sequential content recommendation. In this context each arm models the feedback of an agent to a specific item that can be displayed (e.g. an advertisement). In this framework, it might be relevant to display *several* items at a time, and some variants of the classical bandit problems that have been proposed in the literature may be considered. In the *multi-armed bandit with multiple plays*, $m \geq 1$ out of $K$ arms are sampled at each round and all the associated rewards are observed by the agent, who receives their sum. [@Anantharametal1987] present a regret lower bound for this problem, together with a (non-explicit) matching strategy. More explicit strategies can be obtained when viewing this problem as a particular instance of a *combinatorial bandit problem with semi-bandit feedback*. Combinatorial bandits, originally introduced by in a non-stochastic setting, present the agent with possibly structured subsets of arms at each round: once a subset is chosen, the agent receives the sum of their rewards. The semi-bandit feedback corresponds to the case where the agent is able to see the reward of each of the sampled arms [@Audibertetal2011]. Several extensions of UCB procedures have been proposed for the combinatorial setting (see e.g. [@Chenetal2013; @Combesetal2015b]), with logarithmic regret garantees. However, existing regret upper bounds do not match the lower bound of [@Anantharametal1987]. In particular, despite the strong practical performance of KL-UCB-based algorithms in some combinatorial settings (including multiple-plays), their asymptotic optimality has never been established. Extending the optimality result from the single-play setting has proven challenging, especially in settings where the optimal set of $m$ arms in non-unique. Recently, [@Komiyamaetal2015] proved the asymptotic optimality of Thompson sampling for multiple-play bandits with Bernoulli rewards in the case where the arm with the $m^\textnormal{th}$ largest mean is unique. An important consequence of the uniqueness of the $m^\textnormal{th}$ largest mean is that the optimal set of $m$ arms is necessarily unique, which may not be plausible in practice. In this paper, we extend the multiple plays model in two directions, incorporating a *budget constraint* and an *indifference point*. Given a known cost $c_a$ associated with pulling each arm $a$, at each round a subset of arms $\hat{\cA}(t)$ is selected, so that the expected cost of pulling the chosen arms is at most the budget $B$. More formally, letting $C(t)\equiv\sum_{a \in \hat{\cA}(t)} c_a$, one requires $\operatorname{\mathbb{E}}[C(t)] \leq B$, where the expectation over the random selection of the subset $\hat{\cA}(t)$ is taken conditionally on past observations. The agent observes the reward associated to the selected arms and receives a total reward $R(t) = \sum_{a=1}^K Y_{a}(t) \ind_{(a \in \hat{\cA}(t))}$, where $Y_{a}(t)$ is drawn from $\nu_a$. This reward is then compared to what she could have obtained, had she spent the same budget on some other activity, for which the expect reward per cost unit is $\rho \geq 0$ (that is, the agent may prefer to use that money for some purpose that has reward to cost ratio greater than $\rho$ and is external to the bandit problem). The agent’s gain at round $t$ is thus defined as $$G(t) = R(t) - \rho C(t) = \sum_{a \in \hat{\cA}(t)}\left(Y_{a}(t) - c_a \rho\right).$$ The goal of the agent is to devise a sequential subset selection strategy that maximizes the expected sum of her gains, up to some horizon $T$ and for which the budget constraint $\operatorname{\mathbb{E}}[C(t)] \leq B$ is satisfied at each round $t \leq T$. In particular, arm $a$ is “worth” drawing (in the sense that it increases the expected gain) only if its average reward per cost unit, $\mu_a/{c}_a$ (where $\mu_a$ is the expectation of $\nu_a$), is at least the indifference point $\rho$. This new framework no longer requires the number of arm draws to be fixed. Rather, the number of arm draws is selected to exhaust the budget, which makes sense in several online marketing scenarios. One can imagine for example a company targeting a new market on which it is willing to spend a budget $B$ per week. Each week, the company has to decide which products to advertise for, and the cost of the advertising campaign may vary. After each week, the income associated to each campaign $a$ is measured and compared to the minimal income of $\rho c_a$ that can be obtained when targeting other (known) markets or investing the money in some other well-understood venture. One can also think of a scenario in which the same item can be displayed on several marketplaces for different costs, and the seller has to sequentially choose the different places it wants to display the items on. Our first contribution is to characterize the best attainable performance in terms of regret (with respect to the gain $G(t)$, not the total reward $R(t)$) in this multiple-play bandit scenario with cost constraints, thanks to a lower bound that generalizes that of [@Anantharametal1987]. We then study natural extensions of two existing bandit algorithms (KL-UCB and Thompson sampling) to our setting. We prove both rate and problem-dependent leading constant optimality for KL-UCB and Thompson sampling. The most difficult part of the proof is to show that the optimal arms away from the margin are pulled in almost every round (specifically, they are pulled in all but a sub-logarithmic number of rounds). [@Komiyamaetal2015] studied this problem for Thompson sampling in multiple-play bandits using an argument different than that used in this paper. We provide a novel proof technique that leverages the asymptotic lower bound on the number of draws of any suboptimal arm. While this lower bound on suboptimal arm draws is typically used to prove an asymptotic lower bound on the regret of any reasonable algorithm, we use it as a key ingredient for our proof of an asymptotically optimal *upper bound* on the regret of KL-UCB and Thompson sampling, i.e. to prove the asymptotic optimality of these two algorithms. Also, throughout the manuscript, we do not assume that the set of optimal arms is unique, unlike most of the existing work on (standard) multiple-play bandits. The rest of the article is organized as follows. Section \[sec:methintro\] outlines our problem of interest. Section \[sec:lb\] provides an asymptotic lower bound on the number of suboptimal arm draws and on the regret. Section \[sec:algs\] presents the two sampling algorithms we consider in this paper and theorems establishing their asymptotic optimality: KL-UCB (Section \[sec:kl\]) and Thompson sampling (Section \[sec:thom\]). Section \[sec:exp\] presents numerical experiments supporting our theoretical findings. Section \[sec:proofoutlines\] presents the proofs of our asymptotic optimality (rate and leading constant) results for KL-UCB and Thompson Sampling. Section \[sec:conc\] gives concluding remarks. Technical proofs are postponed to the appendix. Appendix \[proofs:Oracle\] focuses on oracle strategy and regret decomposition. Appendix \[app:lbproof\] contains proofs establishing the asymptotic lower bound on the number of suboptimal arm draws. Appendices \[app:klucbproof\] and \[app:thomproof\] contain technical proofs for KL-UCB and the Thompson sampling, respectively. Multiple plays bandit with cost constraint {#sec:methintro} ========================================== We consider a finite collection of arms $a\in\{1,\ldots,K\}$, where each arm has real-valued marginal reward distribution $\nu_a$ whose mean we denote by both $\mu_a$ and $E(\nu_a)$. Each arm belongs to a (possibly nonparametric) class of distributions $\cD$. We use $\mathcal{V}$ to denote $(\nu_1,\ldots,\nu_{K})$, where $\mathcal{V}$ belongs to any model $\cD_K$ that is variation-independent in the sense that, for each $a\in\{1,\ldots,K\}$, knowing the joint distribution of the rewards $a'\not=a$ places no restrictions on the collection of possible marginal distributions of $\nu_a$, i.e. $\nu_a$ could be equal to any element in $\cD$. An example of a statistical model satisfying this variation-independence assumption is the distribution in which the rewards of all of the arms are independent and the marginal distributions $\nu_a$ fall in $\cD$ for all $a$, though this assumption also allows for high levels of dependence between the rewards of the arms, i.e. is not to be confused with the *much stronger* model assumption of independence between the different arms. The sequential decision problem ------------------------------- Let $\{(Y_1(t),\ldots,Y_{K+1}(t))\}_{t=1}^\infty$ be an independent and identically distributed (i.i.d.) sample from the distribution $\cV$. In the multiple-play bandit with cost constraint, each arm $a$ is associated with a known *cost* $c_a>0$. The model also depends on a known *budget per round* $B$ and *indifference parameter* $\rho \ge 0$. At round $t$, the agent selects a subset $\operatorname{\mathcal{A}}(t)$ of arms and subsequently observes the action-reward pairs $\{(a,Y_a(t)) : a\in \operatorname{\mathcal{A}}(t)\}$. We emphasize that the agent is aware that reward $Y_a(t)$ corresponds to the action $a\in\operatorname{\mathcal{A}}(t)$. This subset $\operatorname{\mathcal{A}}(t)$ is drawn from a distribution $Q(t-1)$ over $\cS_K$, the set of all subsets of $\{1,\dots,K\}$, that depends on the observations gathered at the $(t-1)$ previous rounds. More precisely, $Q(t)$ is $\cF(t)$-measurable, where $\cF(t)$ is the $\sigma$-field generated by all action-reward pairs seen at times $1,\ldots,t$, and possibly also some exogenous stochastic mechanism. We use $q_a(t)$ to denote the probability that arm $a$ falls in $\operatorname{\mathcal{A}}(t+1)\sim Q(t)$. Given the budget $B$ and the indifference parameter $\rho$, at each round $(t+1)$ the distribution $Q(t)$ must respect the budget constraint $$\begin{aligned} \operatorname{\mathbb{E}}_{\cA\sim Q(t)}\left[\sum_{a\in \cA} {c}_a \right] \le B, \ \ \ \text{or, equivalently,} \ \ \ \sum_{a=1}^{K} {c}_a q_a(t) \leq B.\label{SoftBC}\end{aligned}$$ Upon selecting the arms, the agent receives a reward $R(t+1) = \sum_{a \in \operatorname{\mathcal{A}}(t+1)} Y_a(t+1)$ and incurs a gain $G(t+1) =\sum_{a \in \operatorname{\mathcal{A}}(t+1)} (Y_a(t+1) - c_a\rho)$. Given a (possibly unknown) horizon $T$, the goal of the agent is to adopt a strategy for sequentially selecting the distributions $Q(t)$ that maximizes $${\mathbb{E}}\left[\sum_{t=1}^T G(t)\right],$$ while satisfying, at each round $t=0,\dots,T-1$ the budget constraint . This constraint may be viewed as a ‘soft’ budget constraint, as it allows the agent to (slightly) exceed the budget at some rounds, as long as the expected cost remains below $B$ at each round. We shall see below that considering a ‘hard’ budget constraint, that is selecting at each round a deterministic subset $\hat{\mathcal{A}}(t)$ that satisfies $\sum_{a=1}^K {c}_a \ind_{(a \in \hat{\cA}(t))} \leq B$, is a much harder problem. Besides, in the marketing examples described in the introduction, it makes sense to consider a large time horizon and to allow for minor budget crossings. Under the soft budget constraint , if we knew the vector of expected mean rewards $\bm\mu \equiv (\mu_1,\dots,\mu_K)$, at each round $t$ we would draw a subset from a distribution $$Q^\star \in \underset{Q}{\text{argmax}} \ {\mathbb{E}}_{S \sim Q}\left[\sum_{a \in S} (\mu_a - c_a \rho)\right] \ \ \ \text{such that} \ \ \ {\mathbb{E}}_{S \sim Q} \left[\sum_{a \in S} c_a\right] \leq B.\label{OriginalPb}$$ Above, the argmax is over distributions $Q$ with support on the power set of $\{1,\ldots,K\}$. Noting that the two expectations only depend on the marginal probability of inclusions $q_a = {\mathbb{P}}_{S \sim Q}\left( a \in S\right)$, it boils down to finding a vector $\bm q^\star = (q_a)_{a =1}^K$ that satisfies $$\label{FractionalKnapsack}\bm q^\star \in \underset{\bm q \in [0,1]^K}{\text{argmax}} \ \sum_{a \in S} q_a(\mu_a - c_a \rho) \ \ \ \text{such that} \ \ \ \sum_{a \in S} q_a c_a \leq B.$$ An oracle strategy would then draw $S$ from a distribution $Q^\star$ with marginal probabilities of inclusions given by $\bm q^\star$ (e.g. including independently each arm $a$ with probability $q^\star_a$). The optimization problem is known as a fractional knapsack problem [@Dantzig1957], and its solution is a greedy strategy, that is described below. It is expressed in terms of the reward to cost ratio of each arm $a$, defined as $\rho_a \equiv \mu_a/c_a$. \[prop:Oracle\]Introduce $$\rho^\star = \left\{ \begin{array}{cl} &\rho \ \text{ if } \ \sum_{a : \rho_a > \rho} c_a < B, \\ &\sup \{ r \geq 0 : \sum_{a : \rho_a > r} c_a \geq B\} \ge \rho\ \text{ otherwise},\\ \end{array} \right.$$ and define the three sets $$\begin{aligned} \mbox{optimal arms away from the margin: }\ \ &\cL\equiv \{a : \rho_a>\rho^{\star}\}, \\ \mbox{arms on the margin: }\ \ &\cM\equiv \{a : \rho_a=\rho^{\star}\}, \\ \mbox{suboptimal arms away from the margin: }\ \ &\cN\equiv \{a : \rho_a<\rho^{\star}\}.\end{aligned}$$ Then $\bm q^\star$ is solution to if and only if $q_a^\star = 1$ for all $a \in \cL$, $q^\star_b =0$ for all $b \in \cN$ and $\sum_{a \in \cM} c_aq_a = B - \sum_{a \in \cL} c_a$ if $\rho^\star > \rho$. As proved in Appendix \[proofs:Oracle\], the optimal strategy sorts the items by decreasing order of $\rho_a$, and include them one by one ($q^\star_a=1$), as long as the value increases and the budget is not exceeded. Hereafter we will write $\rho^\star(\bm\mu)> \rho$ for $\rho^{\star}$ to emphasize the dependence of $\rho^{\star}$ on $\bm\mu$. Then we can identify two situations: if $\rho^\star(\bm\mu) = \rho$, there are not enough interesting items (i.e. such that $\rho_a > \rho$) to saturate the budget, and the optimal strategy is to include all the interesting items. If $\rho^\star(\bm\mu)> \rho$, some probability of inclusion is further given to the items on the margin in order to saturate the budget constraint. In that case, the margin is always non-empty: there exist items $a$ such that $\rho^\star(\bm\mu) = \rho_a$. #### Recovering the multiple-play bandit model. By choosing $c_a = 1$ for all arm $a$, $B=m$ and $\rho=0$, we recover the classical multiple-play bandit model. In that case $\rho^\star(\bm\mu) = \mu_{[m]}$, where $[m]$ is the arm with $m$ largest mean and $Q^\star = \delta_{\{[1],\dots,[m]\}}$ is a solution to : the corresponding oracle strategy always plays the $m$ arms with largest means. #### Hard and soft constraints. Under hard budget constraints, if we knew the vector of expected mean rewards $\bm\mu$, at each round $t$ we would pick the subset $$S^\star \in \underset{S \in \cS_K}{\text{argmax}} \ \sum_{a \in S} (\mu_a - c_a \rho) \ \ \ \text{such that} \ \ \ \sum_{a \in S} c_a \leq B.\label{Knapsack}$$ This is a $0/1$ knapsack problem, that is much harder to solve than the above fractional knapsack problem. In fact, $0/1$ knapsack problems are NP-hard, though they are, admittedly, some of the easiest problems in this class, and reasonable approximation schemes exist [@Karp1972]. Nonetheless, the greedy strategy (including arms by decreasing order of $\rho_a$ while the budget is not exceeded, with ties broken arbitrarily) is not generally a solution to . However, using Proposition \[prop:Oracle\], one can identify some examples where there exist deterministic solutions to , i.e. solutions such that $q_a^\star \in \{0,1\}$ that are therefore solutions to : if $\rho^\star(\bm\mu)=\rho$ or if there exists $m \in \cM$ such that $\sum_{a \in \cL\cup \{m\}} c_a = B$. Hence the multiple-play bandit model can be viewed as a particular instance of the multiple plays model under both hard or soft budget constraint. In the rest of the article, we only consider soft budget constraints, as there is generally no tractable oracle under hard budget constraints. Regret decompositions {#sec:regretdecomp} --------------------- The best achievable (oracle) performance consists in choosing, at every round $t$, $Q(t)$ to be the optimal distribution $Q^\star$ whose probabilities of inclusions are described in Proposition \[prop:Oracle\]. Such a strategy ensures an expected gain at each round of $$G^\star \equiv \sum_{a \in \cL} \mu_a + \rho^\star\left(B - \sum_{a \in \cL} c_a\right) - B\rho,$$ with the definitions introduced in Proposition \[prop:Oracle\]. Maximizing the expected total gain is equivalent to minimizing the regret, that is the difference in performance compared to the oracle strategy: $$\begin{aligned} \Reg(T,\cV,\texttt{Alg})&\equiv T G^\star - \operatorname{\mathbb{E}}_\cV\left[\sum_{t=1}^T G(t)\right],\end{aligned}$$ where the sequence of gains $G(t)$ is obtained under algorithm `Alg`. The following statement, proved in Appendix \[proofs:Oracle\], provides an interesting decomposition of the regret, as a function of the number of selections of each arm, denoted by $N_a(T)\equiv \sum_{t=1}^T \operatorname{\mathds{1}}\{a\in\operatorname{\mathcal{A}}(t)\}$. \[prop:RegretDec\] With $\rho^\star = \rho(\bm\mu)$, $\cL,\cN$ defined as in Proposition \[prop:Oracle\], for any algorithm `Alg` $$\begin{aligned} \Reg(T,\cV,\texttt{Alg}) & = & \sum_{a^\star\in\cL} {c}_{a^\star}(\rho_{a^\star}-\rho^{\star} )\left(T- \operatorname{\mathbb{E}}_\cV[N_{a^\star}(T)]\right) + \sum_{a\in\cN} {c}_{a} [\rho^{\star}-\rho_a] \operatorname{\mathbb{E}}_\cV[N_a(T)] \nonumber \\& & \ \ + (\rho^\star - \rho)\left(BT - \sum_{a=1}^K c_a{\mathbb{E}}_\cV[N_a(T)]\right). \label{eq:regretDecomp2} \end{aligned}$$ This decomposition writes the regret as a sum of three non-negative terms. In order for the regret to be small, each optimal arm $a^\star \in \cL$ should be drawn very often (of order $T$ times, to make the first term small) and each suboptimal arm $a^\star \in \cN$ should be drawn seldomly (to make the second term small). Finally if $\rho^\star>\rho$, that is if there are sufficiently many ‘worthwhile’ arms to exceed the budget, then the third term appears as a penalty for not using the whole budget at every round. It means that arms on the margin $\cM$ have to be drawn sufficiently often so as to saturate the budget constraint. #### An extended bandit interpretation. Here we propose another view on this regret decomposition, by means of an extended bandit game with an extra arm, which we term a pseudo-arm, that represents the choice not to pull arms. Whenever an algorithm does not saturate the budget constraint , one can view this algorithm as putting weight on a pseudo-arm in the bandit, that yields zero gain but permits saturation of the budget. Letting $\mu_{K+1}=B\rho$ and $c_{K+1}=B$, the gain associated with drawing arm $(K+1)$ (whose distribution is a point mass at $B\rho$) is indeed zero (as $\mu_{K+1} - \rho c_{K+1} = 0$) and, for any $\bm q(t)$ such that $\sum_{a=1}^{K}q_a(t) c_a \leq B$, there exists $q_{K+1}(t)$ such that $\sum_{a=1}^{K+1}q_a(t) c_a = B$, as $c_{K+1}=B$. Any algorithm for the original bandit problem selecting $\hat{S}(t) \in \cS_{K}$ at time $t$ can thus be viewed as an algorithm selecting $\tilde{S}(t) \in \cS_{K+1}$, that additionnaly includes arm $(K+1)$ with probability $q_{K+1}(t)$. As the pseudo-arm is associated with a null gain, the cumulated gain and regret are similar in both settings. Moreover, as $q_{K+1}(t) = (B - \sum_{a=1}^Kc_aq_a(t))/B$, one easily sees that the number of (artificial) selections of the pseudo-arm is such that $$B{\mathbb{E}}[N_{K+1}(T)] = BT - \sum_{a=1}^K c_a{\mathbb{E}}[N_a(T)],$$ which equals the third term in the regret decomposition, up to the factor $(\rho^{\star} - \rho)$. In this extended bandit model, the three sets of arms introduced in Proposition \[prop:Oracle\] remain unchanged, with $\cL\equiv \{a \in \{1,\dots,K+1\}: \rho_a>\rho^{\star}\}$, $\cM\equiv \{a \in \{1,\dots,K+1\}: \rho_a=\rho^{\star}\}$ and $\cN\equiv \{a \in \{1,\dots,K+1\} : \rho_a<\rho^{\star}\}$. As $\rho_{K+1}=\rho \leq \rho^\star$, the pseudo-arm may only belong to $\cM$ or $\cN$, and the margin $\cM$ is always non-empty. Considering the extended bandit model, the regret decomposition can be rewritten in a more compact way: $$\Reg(T,\cV, \texttt{Alg}) = \sum_{a^\star\in\cL} {c}_{a^\star}(\rho_{a^\star}-\rho^{\star} )\left(T- \operatorname{\mathbb{E}}[N_{a^\star}(T)]\right) + \sum_{a\in\cN} {c}_{a} [\rho^{\star}-\rho_a] \operatorname{\mathbb{E}}[N_a(T)].$$ Our proofs make use of this extended bandit model, since many of the results we present apply to both the “actual” arms $a=1,\ldots,K$ and the pseudo-arm $(K+1)$. Our proofs also make use of a set $\cS$, which, in the extended bandit model, refers to all arms in $(\cL\cup\cM)\backslash\{K+1\}$ whereas, in the unextended bandit model, it refers simply to all optimal arms both on and away from the margin. Related work ------------ There has been considerable work on various forms of “budgeted” or “knapsack” bandit problems . The main difference between our work and these works is that we consider a round-wise budget constrain, and allow for several arms to be selected at each round, possibly in a randomized way in order to satisfy the budget constraint in expectation. In contrast, in most existing works, one arm is (deterministically) selected at each round, and the game ends when a global budget is exhausted. The recent work of [@Xiaetal2016b] appears to be the most closely related to ours: in their setup the agent may play multiple arms at each round, though the number of arms pulled at each round is fixed and the cost of pulling each arm is random and observed upon pulling each arm. Compared to all these mentioned budgeted bandit problems, the focus of our analysis differs substantially, in that our primary objective is to not only prove rate optimality, but also leading constant optimality of our regret bounds. Proving constant optimality is especially challenging in situations where the set of optimal arms is non-unique, but we give careful arguments that overcome this challenge. Several other extensions of the multiple-play bandit model have been studied in the literature. UCB algorithms have been widely used in the combinatorial semi-bandit setting, in which at each time step a subset of arms has to be select among a given class of subsets, and the rewards of every individual arms in the subset are observed. The most natural use of UCBs and the “optimism in face of uncertainty principle” is to choose at every time step the subset that would be the best if the unknown means were equal to the corresponding UCBs. This was studied by [@Chenetal2013; @Kvetonetal2014; @Kvetonetal2015c], who exhibit good empirical performance and logarithmic regret bounds. [@Combesetal2015b] further study instance-dependent optimality for combinatorial semi bandits, and propose an algorithm based on confidence bounds on the value of each subset, rather than on confidence bounds on the arms’ means. Their ESCB algorithm is proved to be order-optimal for several combinatorial problems. As a by product of our results, we will see that in the multiple-play setting, using KL-based confidence bounds on the arms’ means is sufficient to achieve asymptotic optimality. Another interesting direction of extension is the possibility to have only partial feedback over the $m$ proposed item. Variants of KL-UCB and Thompson Sampling were proposed for the Cascading bandit model [@Kvetonetal2015; @Kvetonetal2015b], Learning to Rank [@Combesetal2015] or the Position-Based model [@Lagreeetal2016]. It would be interesting to try to extend the results presented in this work to these partial feedback settings. Regret Lower Bound {#sec:lb} ================== We first give in Lemma \[lem:lb\] asymptotic lower bounds on the number of draws of suboptimal arms, either in high-probability or in expectation, in the spirit of those obtained by . Compared to these works, the lower bounds obtained here hold under our more general assumptions on the arm distributions, which is reminiscent of the work of . While could also easily be obtained using the recent change-of-distribution tools introduced by [@Garivieretal2016], note that we need to go back to Lai and Robbins’ technique to prove the high-probability result , which will be crucial in the sequel. Indeed, we will use it to prove optimal regret of our algorithms: in essence we need to ensure that we have enough information about arms in $\cM\cup\cN$ to ensure that we pull the optimal arms in $\cL$ sufficiently often. To be able to state our regret lower bound, we now introduce the following notation. We let $\operatorname{KL}(\nu,\nu')$ denote the KL-divergence between distributions $\nu$ and $\nu'$. If $\nu$ and $\nu'$ are uniquely parameterized by their respective means $\mu$ and $\mu'$ as in a single parameter exponential family (e.g. Bernoulli distributions), then we abuse notation and let $\operatorname{KL}(\mu,\mu')\equiv \operatorname{KL}(\nu,\nu')$. For a distribution $\nu\in\cD$ and a real $\mu$, we define $$\begin{aligned} \operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu,\mu)&\equiv \inf\left\{\operatorname{KL}(\nu,\nu') : \nu'\in\cD\textnormal{ and }\mu<E(\nu')\textnormal{ and }\nu\ll\nu'\right\}, \label{eq:Kinfdef}\end{aligned}$$ with the convention that $\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu,\mu)=\infty$ if there does not exist a $\nu\ll\nu'$ with $\mu<E(\nu')$. We will also use the convention that, for finite constants $d_1,d_2$, $d_1/(d_2 + \operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu,\mu))=0$ when $\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu,\mu)=\infty$. We make one final assumption, and introduce two disjoint sets $\underline{\cN}$ and $\overline{\cN}$, whose union is $\cN$. The assumption is that, for each arm $a\in\{1,\ldots,K\}$, $\mu_a$ falls below the upper bound of the expected reward parameter space, i.e. $\mu_a < \mu_{+}\equiv \sup\{E(\nu) : \nu\in\cD\}$. We define the sets $\underline{\cN}$ and $\overline{\cN}$ respectively as the subsets of $\cN$ for which optimality is and is not feasible given our parameter space, namely $$\begin{aligned} &\underline{\cN}\equiv \left[\cN\cap\left\{a : {c}_a \rho^\star< \mu_{+}\right\}\right]\backslash\{K+1\} \\ &\overline{\cN}\equiv \left[\cN\cap\left\{a : {c}_a \rho^\star\ge \mu_{+}\right\}\right]\backslash\{K+1\}.\end{aligned}$$ By defining $\underline{\cN}$ and $\overline{\cN}$ in this way, these sets agree in the extended and unextended bandit models. The lower bounds presented in this section will also agree in these two models. We now define a uniformly efficient algorithm, that generalizes the class of algorithms considered in . An algorithm `Alg` is uniformly efficient if, for all $\cV \in \cD_K$, $\Reg(T,\cV,\texttt{Alg}) = o(T^\alpha)$ as $T$ goes to infinity (from now on, the limits in $T$ will be for $T \to \infty$). From the regret decomposition , this is equivalent to 1. $T-\operatorname{\mathbb{E}}_{{\mathcal{V}}}\left[N_{a^\star}(T)\right] = o\left(T^\alpha\right)$ for all arms $a^\star$ such that $\rho_{a^\star}>\rho^\star(\bm\mu)$; 2. \[it:cond2\] $\operatorname{\mathbb{E}}_{\mathcal{V}}[N_a(T)]=o(T^{\alpha})$ for all arms $a$ such that $\rho_a < \rho^\star(\bm\mu)$; 3. if $\rho^\star(\bm\mu) > \rho,$ $BT - \sum_{a=1}^K c_a{\mathbb{E}}_{\cV}[N_a(T)] = o(T^\alpha)$. \[lem:lb\] If an algorithm is uniformly efficient, then, for any arm $a\in(\cM\cup\underline{\cN})\backslash\{K+1\}$ and any $\delta\in(0,1)$ and $\epsilon>0$, $$\begin{aligned} &\lim_T \operatorname{\mathbb{P}}\left\{N_a(T)< (1-\delta)\frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^{\star}) + \epsilon}\right\}=0. \label{eq:problb} \\ \intertext{One can take $\epsilon=0$ if $a\in\underline{\cN}$. Furthermore, for any suboptimal arm $a\in\underline{\cN}$,} &\liminf_T \frac{\operatorname{\mathbb{E}}[N_a(T)]}{\log T}\ge \frac{1}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^{\star})}. \label{eq:explb}\end{aligned}$$ We defer the proof of this result to Appendix \[app:lbproof\]. We now present a corollary to this result which provides a regret lower bound, as well as sufficient conditions for an algorithm to asymptotically match it. As already noted by [@Komiyamaetal2015] in the Bernoulli case for the bandit with multiple-play problems, an algorithm achieving the asymptotic lower bound on the expected number of draws of arms in $\underline{\cN}$ does not necessarily achieve optimal regret, unlike in classic bandit problems. Thus, we emphasize that the upcoming condition alone is not sufficient to prove asymptotic optimality. The conditions of this proof can be easily obtained from the regret decomposition , and so the proof is omitted. \[thm:reglb\] If an algorithm `Alg` is uniformly efficient, then $$\begin{aligned} \liminf_T \frac{\Reg(T,\cV,\texttt{Alg})}{\log T}&\ge \sum_{a\in\underline{\cN}} \frac{{c}_a(\rho^{\star}-\rho_a)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho^{\star})}. \label{eq:reglb}\end{aligned}$$ Moreover, any algorithm `Alg` satisfying $$\begin{aligned} \mbox{for arms $a\in \underline{\cN}$: }\ \ &\operatorname{\mathbb{E}}_\cV[N_a(T)]= \frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^{\star})} + o(\log T), \label{eq:subopt} \\ \mbox{for arms $a\in \overline{\cN}$}\ \ &\operatorname{\mathbb{E}}_\cV[N_a(T)]= o(\log T), \label{eq:suboptindifference} \\ \mbox{for arms $a^\star\in \cL$: }\ \ &\operatorname{\mathbb{E}}_\cV[N_{a^\star}(t)]= T-o(\log T), \label{eq:nonmargin} \end{aligned}$$ and, if $\rho^\star(\bm\mu) > \rho$, $$BT - \sum_{a=1}^K c_a{\mathbb{E}}_\cV[N_a(T)] = o(\log(T)), \label{eq:exhaustbudget}$$ is asymptotically optimal, in the sense that it satisfies $$\begin{aligned} \limsup_T \frac{\Reg(T,\cV,\texttt{Alg})}{\log T}\leq \sum_{a\in\underline{\cN}} \frac{{c}_a(\rho^{\star}-\rho_a)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho^{\star})}. \label{eq:regub}\end{aligned}$$ Algorithms {#sec:algs} ========== Algorithms rely on estimates of the arm distributions and their means, that we formally introduce below. For each arm $a$ and natural number $n$, define $\tau_{a,n}=\min\{t\ge 1 : N_a(t)=n\}$ to be the (stopping) time at which the $n^{\textnormal{th}}$ draw of arm $a$ occurs. Let $X_{a,n}\equiv Y_a(\tau_{a,n})$ denote the $n^{\textnormal{th}}$ draw from $\nu_a$. One can show that $\{X_{a,n}\}_{n=1}^\infty$ is an i.i.d. sequence of draws from $\nu_a$ for each $a$, though we note that our variation independence assumption is too weak to ensure that these sequences are independent for two arms $a\not=a'$ (this is not problematic – most of our arguments end up focusing on arm-specific sequences $\{X_{a,n}\}_{n=1}^\infty$)[^1]. We denote the empirical distribution function of observations drawn from arm $a$ by any time $T$ by $$\begin{aligned} \hat{\nu}_a(T)&\equiv \frac{1}{N_a(T)}\sum_{t=1}^T \delta_{Y_a(t)} \operatorname{\mathds{1}}\{a\in\operatorname{\mathcal{A}}(t)\} = \frac{1}{N_a(T)}\sum_{n=1}^{N_a(T)} \delta_{X_{a,n}}.\end{aligned}$$ We similarly define $\hat{\nu}_{a,n}$ to be the empirical distribution function of the obervations $X_{a,1}$, $\ldots$, $X_{a,n}$. Thus, $\hat{\nu}_a(t)=\hat{\nu}_{a,N_a(t)}$. We further define $\hat{\mu}_a(t)$ to be the empirical mean of observations drawn from arm $a$ by time $t$ and $\hat{\mu}_{a,N_a(t)}=\hat{\mu}_a(t)$. KL-UCB {#sec:kl} ------ At time $t$, UCB algorithms leverage high probability upper bound $U_a(t)$ on $\mu_a$ for each $a$. The methods used to build these confidence bounds vary, as does the way the algorithm uses these confidence bounds. In our setting, we derive these bounds using the same technique as for KL-UCB in [@Cappeetal2013]. A round $(t+1)$, the KL-UCB algorithm computes an optimistic oracle strategy $(q_a(t))_{a=1,\dots,K}$, that is an oracle strategy assuming the unknown mean of each arm $a$ is equal to its best possible value, $U_a(t)$. From Proposition \[prop:Oracle\], this optimistic oracle depends on $\hat{\rho^\star}(t) = \rho^\star\left(U_a(t) : a=1,\dots,K\right)$, where $\rho^\star(\bm\mu)$ is the function defined in Proposition \[prop:Oracle\]. Then each arm is included in $\hat{\cA}(t+1)$ independently with probability $q_a(t)$. Due to the structure of an oracle strategy, KL-UCB can be rephrased as successively drawing the arms by decreasing order of the ratio $U_a(t)/c_a$ until the point that the budget is exhausted, with some probability to include the arms on the margin. We choose to keep the name KL-UCB for this straightforward generalization of the original KL-UCB algorithm. The definition of the upper bound $U_a(t)$ is closely related to that of $\operatorname{\mathcal{K}_{\textnormal{inf}}}$ given in . Let $\Pi_{\cD}$ be a problem-specific operator mapping each empirical distribution function $\hat{\nu}_a(t)$ to an element of the model $\cD$. Furthermore, let $f : \mathbb{N}\rightarrow\mathbb{R}$ be a non-decreasing function, where this function is usually chosen so that $f(t)\approx \log t$. The UCB is then defined as $$\begin{aligned} U_a(t)&\equiv \sup\left\{E(\nu) : \nu\in\cD\textnormal{ and }\operatorname{KL}\left(\Pi_{\cD}\left(\hat{\nu}_a(t)\right),\nu\right)\le \frac{f(t)}{N_a(t)}\right\}\textnormal{, $a=1,\ldots,K$}. \label{eq:Uadef}\end{aligned}$$ As we will see, the closed form expression for $U_a(t)$ can be made slightly more explicit for exponential family models, though the expression still has the same general flavor. If a number $\mu$ satisfies $\mu\ge U_a(t)$, then this implies that, for every $\nu\in\cD$ for which $E(\nu)>\mu$, $\operatorname{KL}\left(\Pi_{\cD}(\hat{\nu}_a(t)),\nu\right)>\frac{f(t)}{N_a(t)}$. Consequently, $\operatorname{\mathcal{K}_{\textnormal{inf}}}(\Pi_{\cD}(\hat{\nu}_a(t)),\mu)\ge \frac{f(t)}{N_a(t)}$. We now describe two settings in which the algorithm that we have described achieves the optimal asymptotic regret bound. These two settings and the presentation thereof follows [@Cappeetal2013]. The first family of distributions we consider for $\cD$ is a canonical one-dimensional exponential family $\mathcal{E}$. For some dominating measure $\lambda$ (not necessarily Lebesgue), open set $H\subseteq\mathbb{R}$, and twice-differentiable strictly convex function $b : H\rightarrow\mathbb{R}$, $\mathcal{E}$ is a set of distributions $\nu_\eta$ such that $$\begin{aligned} \frac{d\nu_\eta}{d\lambda}(x)&= \exp\left[x\eta-b(\eta)\right].\end{aligned}$$ We assume that the open set $H$ is the natural parameter space, i.e. the set of all $\eta\in\mathbb{R}$ such that $\int \exp(x\eta) d\lambda(x)<\infty$. We define the corresponding (open) set of expectations by $I\equiv \{E(\nu_\eta) : \eta\in H\}\equiv (\mu_{-},\mu_{+})$ and its closure by $\bar{I}=[\mu_{-},\mu_{+}]$. We have omitted the dependence of $\mathcal{E}$ on $\lambda$ and $b$ in the notation. It is easily verified that $\operatorname{\mathcal{K}_{\textnormal{inf}}}(\mu_a,{c}_a \rho^\star)= \operatorname{KL}(\nu_a,{c}_a \rho^\star)$. For the moment suppose that $\hat{\nu}_a(t)$ is such that $\hat{\mu}_a(t)\in I$. In this case we let $\Pi_{\cD}$ denote the maximum likelihood operator so that $\Pi_{\cD}\left(\hat{\nu}_a(t)\right)$ returns the unique distribution in $\cD$ indexed by the $\eta$ satisfying $b'(\eta)=\hat{\mu}_a(t)$. Thus, in this setting where $\hat{\mu}_a(t)\in I$, the UCB $U_a(t)$ then takes the form of the expression in . More generally, we must deal with the case that $\hat{\mu}_a(t)$ equals $\mu_{+}$ or $\mu_{-}$. For $\mu\in I$, define by convention $\operatorname{KL}(\mu_{-},\mu) = \lim_{\mu'\rightarrow\mu_{-}} \operatorname{KL}(\mu_{-},\mu)$, $\operatorname{KL}(\mu_{+},\mu) = \lim_{\mu'\rightarrow\mu_{+}} \operatorname{KL}(\mu',\mu)$, and analogously for $\operatorname{KL}(\mu,\mu_{-})$ and $\operatorname{KL}(\mu,\mu_{+})$. Finally, define $\operatorname{KL}(\mu_{-},\mu_{-})$ and $\operatorname{KL}(\mu_{+},\mu_{+})$ to be zero. This then gives the following general expression for $U_a(t)$ that we use to replace in the KL-UCB Algorithm: $$\begin{aligned} U_a(t)&\equiv \sup\left\{\mu\in\bar{I} : \operatorname{KL}\left(\hat{\mu}_a(t),\mu\right)\le \frac{f(t)}{N_a(t)}\right\}\textnormal{, $a=1,\ldots,K$}\label{indexKLUCB}.\end{aligned}$$ Note that this definition of $U_a(t)$ does not explicitly include a mapping $\Pi_{\cD}$ mapping any empirical distribution function to an element of the model $\cD$. Thus we have avoided any problems that could arise in defining such a mapping when $\hat{\mu}_a(t)$ falls on the boundary of $\bar{I}$. The KL-UCB variant that we have presented achieves the asymptotic regret bound in the setting where $\cD=\mathcal{E}$. \[thm:expfam\] Suppose that $\cD = \mathcal{E}$. Further let $f(t)=\log t + 3\log\log t$ for $t\ge 3$ and $f(1)=f(2)=f(3)$. This variant of KL-UCB satisfies , , and . Thus, KL-UCB achieves the asymptotic regret lower bound for uniformly efficient algorithms. Another interesting family of distributions for $\cD$ is a set $\mathcal{B}$ of distributions on $[0,1]$ with finite support. If the support of $\cD$ is instead bounded in some $[-M,M]$, then the observations can be rescaled to $[0,1]$ when selecting which arm to pull using the linear transformation $x\mapsto (x + M)/(2M)$. If $\cD$ is equal to $\mathcal{B}$, then [@Cappeetal2013] observe that rewrites as $$\begin{aligned} U_a(t)&= \sup\left\{E(\nu) : \operatorname{Support}[\nu]\subseteq \operatorname{Support}\left[\hat{\nu}_a(t)\right]\cup\{1\}\textnormal{ and }\operatorname{KL}\left(\hat{\nu}_a(t),\nu\right)\le\frac{f(t)}{N_a(t)}\right\}\end{aligned}$$ where, for a measure $\nu'$, we use $\operatorname{Support}[\nu']$ to denote the support of $\nu'$. They furthermore observe that this expression admits an explicit solution via the method of Lagrange multipliers. \[thm:finsup\] Suppose that $\cD=\mathcal{B}$. Let $\Pi_{\cD}$ denote the identity map and $f(t)=\log t + \log\log t$ for $t\ge 2$ and $f(1)=f(2)$. Suppose that $\mu_a\in(0,1)$ for all $a=1,\ldots,K$. The variant of KL-UCB satisfies , , and . Thus, KL-UCB achieves the asymptotic regret lower bound for uniformly efficient algorithms. In both theorems, the little-oh notation hides the problem-dependent but $T$-independent quantities. In the proofs of Theorems \[thm:expfam\] and \[thm:finsup\] we refer to equations in [@Cappeetal2013b] where the reader can find explicit finite-sample, problem-dependent expressions for the $o(\log T)$ term in for the settings of Theorems \[thm:expfam\] and \[thm:finsup\]. The argument used to establish considers similar $o(\log T)$ terms to those that appear in the proof of , though the simplest argument for establishing (which, for brevity, is the one that we have elected to present here) invokes asymptotics. The argument used to establish in these settings, on the other hand, seems to be fundamentally asymptotic and does not appear to easily yield finite sample constants. Nonetheless, this is to our knowledge the first handling of thick margins in the multiple-play bandit literature, and so we believe that it is of interest despite its asymptotic nature. Thompson Sampling {#sec:thom} ----------------- \[alg:tom\] *Parameters* For each arm $a=1,\ldots,K$, let $\Pi_a(0)$ be a prior distribution on $\mu_a$. For each arm $a=1,\ldots,K$, draw $\theta_a(t)\sim \Pi_a(t)$. Let $\hat{\rho}^\star(t)\equiv \rho^\star\left(\left(\theta_a(t) : a=1,\ldots,K\right)\right)$. For $a\in\{1,\ldots,K\}$, let $q_a(t) = \operatorname{\mathds{1}}\{U_a(t)>{c}_a \hat{\rho}^\star(t)\}$. For $a\in\widehat{\cM}(t)$, let $q_a(t)\propto 1$, where $\sum_{a\in\widehat{\cM}(t)}{c}_a q_a(t) = B-\sum_{a : U_a(t)>{c}_a\hat{\rho}^\star(t)} {c}_a$. Draw $\operatorname{\mathcal{A}}(t+1)$ from any distribution $Q(t)$ with marginal probabilities $q_a(t)$. Draw the corresponding rewards $Y_a(t+1)$, $a\in\operatorname{\mathcal{A}}(t+1)$. For each $a\in\operatorname{\mathcal{A}}(t+1)$, obtain a new posterior $\Pi_a(t+1)$ by updating $\Pi_a(t)$ with the observation $Y_a(t+1)$. For each $a\not\in\operatorname{\mathcal{A}}(t+1)$, let $\Pi_a(t+1)=\Pi_a(t)$. Thompson sampling uses Bayesian ideas to account for the uncertainty in the estimated reward distributions. In a classical bandit setting, one first posits a (typically non-informative) prior over the means of the reward distributions, and then at each time updates the posterior and takes a random draw of the $K$ means from the posterior and pulls the arm whose posterior draw is the largest. In our setting, this corresponds to drawing the subset of arms for which the posterior draw to cost ratio is largest (up until the budget constraint is met), which generalizes the idea initially proposed by [@Thompson1933]. In the above algorithm we focus on independent priors so that the only posteriors updated at time $(t+1)$ are those of arms in $\operatorname{\mathcal{A}}(t+1)$. At time $(t+1)$, Thompson Sampling first draws one sample $\theta_a(t)$ from the posterior distribution on the mean of each arm $a$, and then selects a subset according an oracle strategy assuming $(\theta_a(t))_{a=1,\dots,K}$ are the true parameters. We prove the optimality of Thompson sampling for Bernoulli rewards, for the particular choice of a uniform prior distribution on the mean of each arm. Note that the algorithm is easy to implement in that case, since $\Pi_a(t)$ is a Beta distribution with parameters $N_a(t)\hat{\mu}_a(t) +1$ and $N_a(t)(1-\hat{\mu}_a(t))+1$. Our proof relies on the same techniques as those used to prove the optimality of Thompson sampling in the standard bandit setting for Bernoulli rewards by . We note that [@Komiyamaetal2015] also made use of some of the techniques in to prove the optimality of Thompson sampling for Bernoulli rewards in the multiple-play bandit setting. \[thm:thom\] If the reward distributions are Bernoulli and $\Pi_a(0)$ is a standard uniform distribution for each $a$, then Thompson sampling satisfies , , and . Thus, Thompson sampling achieves the asymptotic regret lower bound for uniformly efficient algorithms. For any $\epsilon>0$ and $a\in\underline{\cN}$, the proof shows that Thompson sampling satisfies $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]\le (1+\epsilon)^2\frac{f(T)}{\operatorname{KL}(\mu_a,{c}_a \rho^\star)} + o(\log T).\end{aligned}$$ The proof gives an explicit bound on the $o(\log T)$ term that depends on both the problem and the choice of $\epsilon$. Numerical Experiments {#sec:exp} ===================== We now run four simulations to evaluate our theoretical results in practice, all with Bernoulli reward distributions, a horizon of $T=100\,000$, and $K=5$. The simulation settings are displayed in Table \[tab:simsettings\]. Simulations 1-3 are run using $5\,000$ Monte Carlo repetitions, and Simulation 4 was run using $50\,000$ repetitions to reduce Monte Carlo uncertainty. The `R` [@R2014] code for running one repetition of our simulation is available in the Supplementary Materials. $\mu$ ${c}$ $B$ $\rho$ $\cL$ $\cM$ $\overline{\cN}$ ------- --------------------------- --------------------- ----- -------- ----------- ------------- ------------------ Sim 1 $(0.5,0.45,0.45,0.4,0.3)$ $(1,1,1,1,1)$ $2$ $0$ $\{1\}$ $\{2,3\}$ $\emptyset$ Sim 2 $(0.7,0.6,0.5,0.3,0.2)$ $(1,1,1,1,1)$ $3$ $0$ $\{1,2\}$ $\{3\}$ $\emptyset$ Sim 3 $(0.5,0.45,0.45,0.4,0.3)$ $(0.8,1,1,0.8,0.6)$ $2$ $0.5$ $\{1\}$ $\{4,5,6\}$ $\emptyset$ Sim 4 $(0.7,0.6,0.5,0.3,0.2)$ $(1.5,1,1,1,2.5)$ $3$ $0.4$ $\{2,3\}$ $\{1\}$ $\{5\}$ : Simulation settings considered. Simulations 1 and 3 have non-unique margins so that $q_a$ must be less than one for at least one arm $a\in\cM$ for the budget constraint to be satisfied. In Simulation 3, the pseudo-arm $(K+1)=6$ is in $\cM$, and in Simulation 4 arm $5$ is in $\overline{\cN}$.[]{data-label="tab:simsettings"} For $d \in \mathbb{R}$, we define the KL-UCB $d$ algorithm as the instance of KL-UCB using the function $f(t)=\log t + d\log\log t$. Note that the use of both KL-UCB 3 and KL-UCB 1 are theoretically justified by the results of Theorems \[thm:expfam\] and \[thm:finsup\], as Bernoulli distributions satisfy the conditions of both theorems. In the settings of Simulations 1 and 2, which represent multiple-play bandit instances as $B$ is an integer in $[1,K]$ and the cost of pulling each arm is one, we compare Thompson sampling and KL-UCB to the ESCB algorithm of [@Combesetal2015b]. As quickly explained earlier, ESCB is a generalization of the KL-UCB algorithm, designed for the combinatorial semi-bandit setting (that includes multiple-play). This algorithm computes an upper confidence bound for the sum of the arm means for each of the ${K\choose B}$ candidate sets $\mathcal{S}$, defined by the optimal value to $$\begin{aligned} \sup_{(\mu_1,\ldots,\mu_K)\in[0,1]^K} \sum_{a\in\mathcal{S}} \mu_K\,\textnormal{ subject to }\,\sum_{a\in\mathcal{S}} N_a(t) \operatorname{KL}\left(\hat{\mu}_a(t),\mu_a\right)\le f(t) \label{indexESCB}\end{aligned}$$ and draws the arms in the set $\mathcal{S}$ with the maximal index. Just like KL-UCB, ESCB uses confidence bounds whose level rely on a function $f$ such that $f(t)\approx \log t$. Because the optimization problem solved to compute the indices and are different, the $f$ functions used by KL-UCB and ESCB are not directly comparable. Nonetheless, a side-by-side comparison of the two algorithms seems to indicate that $f(t)=\log t + cB\log\log t$ for ESCB is comparable to $f(t)=\log t + c\log\log t$ for KL-UCB. @Combesetal2015b prove an $O(\log T)$ regret bound (with a sub-optimal constant) for the version of ESCB corresponding to the constant $c=4$, that we refer to as ESCB 4$B$. ![Regret of the four algorithms with theoretical guarantees. ESCB only run for Simulations 1 and 2 for which the cost is identically one for all arms.[]{data-label="fig:reg"}](Regret_cost.pdf){width="\linewidth"} Figure \[fig:reg\] displays the regret of the four algorithms with theoretical guarantees. All but ESCB 4$B$ have been proven to be asymptotically optimal, and thus are guaranteed to achieve the theoretical lower bound asymptotically. In our finite sample simulation, Thompson sampling performs better than this theoretical guarantee may suggest (the regret lower bounds at time $T=100\,000$ are approximately equal to $150$ and $45$ in Simulations 1 and 2, respectively). Indeed, Thompson sampling outperforms the KL-UCB algorithms in all but Simulation 4, while KL-UCB 1 outperforms KL-UCB 3 and KL-UCB 3 outperforms ESCB 4$B$ in Simulations 1 and 2. To give the reader intuition on the relative performance of KL-UCB variants, note that in the proofs of Theorems \[thm:expfam\] and \[thm:finsup\] we prove that the number of pulls on each suboptimal arm $a$ is upper bounded by $f(T)/\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho^\star) + o(\log T)$, with an explicit finite sample constant for the $o(\log T)$ term. While $f(T) = \log T + o(\log T)$ for KL-UCB 1 and KL-UCB 3, for finite $T$ the quantities $\log T$ and $\log T + c\log \log T$, $c=1,3$, are quite different. At $T=10^5$, $\log T + \log\log T$ is 20% larger than $\log T$, and $\log T + 3\log\log T$ is 60% larger. This difference does not decay quickly with sample size: at $T=10^{15}$, these two quantities are still respectively 10% and 30% larger than $\log T$. This makes clear the practical benefit to choosing $f(t)$ as close to $\log t$ as is theoretically justifiable: for Bernoullis, the choice of $f(t)$ in Theorem \[thm:finsup\] yields much better results than the choice of $f(t)$ in Theorem \[thm:expfam\]. We also compared the performance of KL-UCB 0 and ESCB 0 in Simulations 1 and 2 (details omitted here, but the exact results of this simulation are given in Figure 2 of the earlier technical report ). Though not theoretically justified, this choice of $f(t)=\log t$ has been used quite a lot in practice. The ordering of the three algorithms is the same in Simulations 1 and 2: Thompson Sampling performs best while ESCB 0 slightly outperforms KL-UCB 0. This should however be mitigated by the gap of numerical complexity between the two algorithms, especially when $B$ and $K$ are large and $B/K$ is not close to $0$ or $1$: while KL-UCB only requires running $K$ univariate root-finding procedures regardless of $B$, the current proposed ESCB algorithm requires running ${K\choose B}$ univariate root-finding procedures. For $K=100$ and $B=10$, this is a difference of running $100$ root-finding procedures versus more than $10^{13}$ of them. ![Time minus the number of optimal arm draws (top) and number of suboptimal arm draws (bottom) in Simulation 4.[]{data-label="fig:N"}](N_model4_cost.pdf){width="0.9\linewidth"} Figure \[fig:N\] displays the number of optimal and suboptimal arm draws in Simulation 4. None of the algorithms pulled the arm in $\overline{\cN}$ (arm 5) often. Thompson Sampling pulled the indifference point pseudo-arm surprisingly often in the first $10^3$ draws, and as a result arm 3 (above the margin) was also not pulled as often as would be expected in these early draws. By time $10^4$, the regret of Thompson sampling appears to have stabilized, and soon outperforms that of the two KL-UCB algorithms. We also checked what would happen if the indifference point were increased from $0.4$ to $0.45$ (details not shown). In this case, it takes even longer for the algorithm to differentiate between arm 3 (with $\rho_3=0.5$) and the pseudo-arm, though by time $10^5$ the algorithm again appears to have succeeded in learning that pulling arm 3 is to be preferred over pulling the peudo-arm. Proofs of Optimality of KL-UCB and Thompson Sampling Schemes {#sec:proofoutlines} ============================================================ We now outline our proofs of optimality for the KL-UCB and Thompson sampling schemes. We break this section into three subsections. Section \[sec:suboptrare\] establishes that the arms in $\cN$, i.e. the suboptimal arms, are not pulled often (satisfy Equations \[eq:subopt\] and \[eq:suboptindifference\]). Due to the differences in proof methods, we consider the KL-UCB and Thompson sampling schemes separately in this subsection. Section \[sec:budgetsat\] justifies that when $\rho^*>\rho$, the budget constraint is most often satured, that is the third term in the regret is negligible. Finally Section \[sec:optcommon\] establishes that the arms in $\cL$, i.e. the optimal arms away from the margin, are pulled often (satisfy Equation \[eq:nonmargin\]). We give the outline of the proofs for the KL-UCB and Thompson sampling schemes simultaneously, though we provide the detailed arguments separately in Appendices \[app:klucbproof\] and \[app:thomproof\], respectively. We note that the order of presentation of the two subsections is important: the arguments used in Section \[sec:optcommon\] rely on the validity of and , which is established in Section \[sec:suboptrare\]. To ease the presentation, we find it convenient to consider the extended bandit model presented in Section \[sec:regretdecomp\], in which a pseudo-arm $K+1$ of cost $B$ is added to the bandit instance, with a positive probability of pulling arm $K+1$ representing the decision not to spend the entire budget on pulling arms $1,\ldots,K$. Though both the KL-UCB and Thompson Sampling algorithms were presented without this extra arm, we already noted thatfor each $t$, $q_{K+1}(t)= 1-\frac{1}{B}\sum_{a=1}^K {c}_a q_a(t)$. The UCB index $U_{K+1}(t)$ and posterior draw $\theta_{K+1}(t)$ for arm $K+1$ are both equal to $B\rho$ for all $t$. For the sake of condensing notation in our study of (expected) regret, it will be convenient to consider a hypothetical scenario in which arm $K+1$ is pulled with probability $q_{K+1}(t)$ at each time point, even though the outcome of these pulls has no effect on the behavior of the algorithms. Suboptimal arms not pulled often {#sec:suboptrare} -------------------------------- In this section, we establish and for KL-UCB and Thompson Sampling. ### KL-UCB {#kl-ucb .unnumbered} #### *Preliminary: a general analysis.* We start by giving a general analysis of KL-UCB in our setting, and then use it to prove Theorems \[thm:expfam\] and \[thm:finsup\]. Fix $a\in \cN\backslash\{K+1\}$. The arguments in this section generalize those given in [@Cappeetal2013; @Cappeetal2013b] for the case where one arm is drawn at each time point and there is no budget constraint. Let $\mu^\dagger\in(\mu_a,\mu_{+})$ be some real number. If $a\in\underline{\cN}$, then we will choose $\mu^\dagger={c}_a \rho^\star$. If, on the other hand, $a\in\overline{\cN}$, then we will choose $\mu^\dagger$ to be slightly less than $\mu_{+}$. Let $\rho^\dagger$ be a constant that is either equal to or slightly less than $\mu^\dagger/{c}_a$. Below we take minimums over $a^\star\in\cS\equiv (\cL\cup \cM)\backslash\{K+1\}$: if $\cS=\emptyset$, then we take these minimums to be equal to negative infinity. When we later take sums over $a^\star\in\cS$, we let empty sums equal zero. We now establish that, for all $t\ge K$, $$\begin{aligned} \left\{a\in\operatorname{\mathcal{A}}(t+1)\right\}&\subseteq \left[\cup_{a^\star\in\cS}\left\{{c}_{a^\star} \rho^\dagger\ge U_{a^\star}(t)\right\}\right]\cup \left\{a\in\operatorname{\mathcal{A}}(t+1), {c}_a \rho^\dagger< U_a(t)\right\}.\label{eq:Atp1a}\end{aligned}$$ We separately handle the cases that $\rho^\star>\rho$ and $\rho^\star\not=\rho$. If $\rho^\star > \rho$, playing all of the arms in $\cS$ would spend at least the allotted budget $B$. Hence, on the event $\left\{\forall a^\star\in \cS, U_{a^\star}(t)/c_{a^\star} > \rho^\dagger\right\}$, it holds that $\hat{\rho}^\star(t) > \rho^\dagger$. If moreover $a\in\operatorname{\mathcal{A}}(t+1)$, one has $U_a(t) \geq c_a\hat{\rho}^\star(t) > c_a\rho^\dagger$. If $\rho = \rho^\star$, it holds that $\{a \in \hat{\cA}(t+1) \} \subseteq \{a \in \hat{\cA}(t+1) , c_a \rho^\dagger < U_a(t)\}$. Indeed, if $\hat{\rho}^\star(t) > \rho$ the algorithm only pulls arms $a$ if $U_a(t) \geq \hat{\rho}^\star(t) c_a > \rho c_a$ and if $\hat{\rho}^\star(t) = \rho$, then the algorithm only pulls arm $a$ if $U_a(t) > c_a\rho$, see Footnote \[foot:extraif\]. As $\rho^\dagger$ is smaller or equal to $\rho^\star=\rho$, it follows that $U_a(t) > c_a\rho^\dagger$ in both cases. For each $\zeta>0$ and $\tilde{\mu}< \mu_{+}$, we now introduce the set $\mathcal{C}_{\tilde{\mu},\zeta}$. In the setting of Theorem \[thm:expfam\], $$\begin{aligned} \mathcal{C}_{\tilde{\mu},\zeta}&\equiv \left\{\nu' : \operatorname{Support}[\nu']\subseteq\bar{I}\right\} \cap \left\{\nu' : \exists\,\mu\in(\tilde{\mu},\mu_{+}]\textnormal{ with }\operatorname{KL}(E(\nu'),\mu)\le \zeta\right\},\end{aligned}$$ where above $\operatorname{KL}(E(\nu'),\mu)$ is the KL-divergence in the canonical exponential family $\mathcal{E}$. In the setting of Theorem \[thm:finsup\], $$\begin{aligned} \mathcal{C}_{\tilde{\mu},\zeta}&\equiv \left\{\nu' : \operatorname{Support}[\nu']\subseteq[0,1]\right\} \cap \left\{\nu' : \exists\,\nu\in\mathcal{B}\textnormal{ with }\tilde{\mu}<E(\nu)\textnormal{ and }\operatorname{KL}(\Pi_{\cD}(\nu'),\nu)\le \zeta\right\}.\end{aligned}$$ In both settings, we will invoke this set at $\tilde{\mu}={c}_a \rho^\dagger<\mu_{+}$. The set $\mathcal{C}_{\tilde{\mu},\zeta}$ is defined in both settings so that $\tilde{\mu}< U_a(t)$ if and only if $\hat{\nu}_a(t)\in \mathcal{C}_{\tilde{\mu},f(t)/N_a(t)}$. Recalling that $\operatorname{\mathbb{E}}[N_a(T)]=\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\{a\in\mathcal{A}(t+1)\}$, a union bound gives $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]\le&\, 1 + \sum_{a^\star\in\cS} \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{{c}_{a^\star} \rho^\dagger\ge U_{a^\star}(t)\right\} + \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \hat{\nu}_{a,N_a(t)}\in \mathcal{C}_{{c}_a \rho^\dagger,f(t)/N_a(t)}\right\}.\end{aligned}$$ In analogue to Equation 8 in [@Cappeetal2013], the above rightmost term satisfies $$\begin{aligned} \sum_{t=K}^{T-1} &\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \hat{\nu}_{a,N_a(t)}\in \mathcal{C}_{{c}_a \rho^\dagger,f(t)/N_a(t)}\right\} \nonumber \\ \le& \sum_{t=K}^{T-1}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \hat{\nu}_{a,N_a(t)}\in \mathcal{C}_{{c}_a \rho^\dagger,f(T)/N_a(t)}\right\} \nonumber \\ =& \sum_{t=K}^{T-1} \sum_{n=2}^{T-K+1}\operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n-1}\in \mathcal{C}_{{c}_a \rho^\dagger,f(T)/(n-1)},\tau_{a,n}=t+1\right\} \label{eq:taunKL} \\ \le& \sum_{n=1}^{T-K}\operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in \mathcal{C}_{{c}_a \rho^\dagger,f(T)/n}\right\}, \nonumber\end{aligned}$$ where the final inequality holds because, for each $n$, $\tau_{a,n}=t+1$ for at most one $t$ in $\{K,\ldots,T-1\}$. We will upper bound the terms with $n=1,\ldots,b_a^{\star}(T)$ in the sum on the right by $1$, where $$\begin{aligned} b_a^{\star}(T)&\equiv \left\lceil\frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)}\right\rceil\le \frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)}+1.\end{aligned}$$ This gives the bound $$\begin{aligned} \sum_{n=1}^{T-K} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{{c}_a \rho^\dagger,f(T)/n}\right\}&\le \frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)} + 1 + \sum_{n=b_a^{\star}(T) + 1}^{\infty} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{{c}_a \rho^\dagger,f(T)/n}\right\}.\end{aligned}$$ Hence, $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]&\le \frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)} + \underbrace{\sum_{n=b_a^{\star}(T) + 1}^{\infty} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{{c}_a \rho^\dagger,f(T)/n}\right\}}_{\textnormal{Term 1}} + \sum_{a^\star\in\cS} \underbrace{\sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{{c}_{a^\star} \rho^\dagger\ge U_{a^\star}(t)\right\}}_{\textnormal{Term 2}a^\star} + 2. \label{eq:T1T2}\end{aligned}$$ Up until this point we have not committed to any particular choice of $\mu^\dagger$, $\rho^\dagger$, or $f$. We now give proofs of and in the settings of Theorems \[thm:expfam\] and \[thm:finsup\]. For each proof we use the choice of $f$ from the theorem statement and make particular choices of $\mu^\dagger$ and $\rho^\dagger$. #### Proof of . Fix $a\in\cN\backslash\{K+1\}$. If $a\in\underline{\cN}$, then let $\mu^\dagger={c}_a \rho^\star$ and, if $a\in\overline{\cN}$, then let $\mu^\dagger\in(\mu_a,\mu_{+})$. In the setting of Theorem \[thm:expfam\] let $\rho^\dagger=\mu^\dagger/{c}_a$ and in the setting of Theorem \[thm:finsup\] let $\rho^\dagger=\left[1-\log(T)^{-1/5}\right]\mu^\dagger/{c}_a$. Lemma \[lem:term1\] shows that Term 1 is $o(\log T)$ and includes references on where to find an explicit finite sample upper bound, where this upper bound will rely on the choice of $\mu^\dagger<\mu_{+}$ if $a\in\overline{\cN}$. Fix $a^\star\in\cS$. Noting that $\rho^\dagger\le \left[1-\log(T)^{-1/5}\right]\rho_{a^\star}$ (Theorem \[thm:expfam\]) and $\rho^\dagger\le \rho_{a^\star}$ (Theorem \[thm:finsup\]), Term 2$a^\star$ is $o(\log T)$ in both settings by Lemma \[lem:term2astar\], with an exact finite sample upper bound given in the proof thereof. Thus, $\sum_{a^\star\in\cS}\textnormal{Term 2}a^\star = o(\log T)$. This completes the proof of . #### Proof of . For $a\in\overline{\cN}$, so far we have established that, for arbitrary $\mu^\dagger\in(\mu_a,\mu_{+})$, $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]&\le \frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)} + r(T,\mu^\dagger),\end{aligned}$$ where $r(T,\mu^\dagger)/\log T\rightarrow 0$ for fixed $\mu^\dagger$. As this holds for every $\mu^\dagger$, there exists a sequence $\mu^\dagger(T)\rightarrow\mu_{+}$ such that $r(T,\mu^\dagger(T))/\log T\rightarrow 0$. In both settings $\liminf_{\mu^\dagger\rightarrow\mu_{+}}\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)=+\infty$, and so using this $\mu^\dagger(T)$ sequence shows that $\operatorname{\mathbb{E}}[N_a(T)]=o(\log T)$. ### Thompson Sampling {#thompson-sampling .unnumbered} This proof is inspired by the analysis of Thompson sampling proposed by . We work with a suboptimal arm $a\in\cN\backslash\{K+1\}$ in most of this section, though we state one of the results (Lemma \[lem:suboptasopt\]) for general arms $a\in\{1,\ldots,K+1\}$ since it will prove useful later. We will let $\rho^\dagger$ and $\rho^\ddagger$ be numbers (to be specified later) satisfying $\rho_a<\rho^\dagger<\rho^\ddagger<1/{c}_a$. Observe that $\left\{a\in\operatorname{\mathcal{A}}(t+1)\right\}$ equals $$\begin{aligned} &\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)\le {c}_a\rho^\ddagger\right\}\cup \left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a\rho^\ddagger\right\} \\ &\subseteq \left[\cup_{a^\star\in\cL\cup\cM}\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)\le{c}_a\rho^\ddagger,\theta_{a^\star}(t)\le {c}_{a^\star}\hat{\rho}^\star\right\}\right]\cup \left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a\rho^\ddagger\right\}.\end{aligned}$$ By the absolute continuity of the beta distribution, with probability one at most one $a'\in\{1,\ldots,K+1\}$ satisfies $\theta_{a'}(t)= {c}_{a'}\hat{\rho}^\star$, and hence, conditional on $\mathcal{F}(t)$, the leading event above is almost surely equivalent to the event $$\begin{aligned} \cup_{a^\star\in\cL\cup\cM}&\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)\le{c}_a\rho^\ddagger,\theta_{a^\star}(t)< {c}_{a^\star}\hat{\rho}^\star\right\}.\end{aligned}$$ If $K+1 \in\cM$, then the fact that $a\in\operatorname{\mathcal{A}}(t+1)$ implies that $\theta_a(t)/{c}_a(t)\ge \hat{\rho}^\star(t)$ shows that the event in the union above at $a^\star=K+1$ never occurs, since on this event $\rho_{K+1}=\theta_{K+1}(t)/{c}_{K+1}<\rho^\ddagger$, which contradicts our choice that $\rho^\ddagger<\rho^\star=\rho_{K+1}$. Hence, the union above can be taken over $\cS$ regardless of whether or not $K+1\in\cM$. Furthermore, $$\begin{aligned} &\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a\rho^\ddagger\right\} \\ &\subseteq \left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a\rho^\ddagger,\hat{\mu}_a(t)\le{c}_a\rho^\dagger\right\}\cup \left\{a\in\operatorname{\mathcal{A}}(t+1),\hat{\mu}_a(t)>{c}_a\rho^\dagger\right\}.\end{aligned}$$ Recalling that $\operatorname{\mathbb{E}}[N_a(T)] = \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1)\right\}$, $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]\le&\, \sum_{a^\star\in\cS} \underbrace{\sum_{t=0}^{T-1}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)\le{c}_a\rho^\ddagger,\theta_{a^\star}(t)< {c}_{a^\star}\hat{\rho}^\star\right\}}_{\textnormal{Term I}a^\star} \nonumber \\ &+ \underbrace{\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a\rho^\ddagger,\hat{\mu}_a(t)\le{c}_a\rho^\dagger\right\}}_{\textnormal{Term II}} \nonumber \\ &+ \underbrace{\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\hat{\mu}_a(t)>{c}_a\rho^\dagger\right\}}_{\textnormal{Term III}}. \label{eq:termsitoiii}\end{aligned}$$ The above decomposition does not depend on the algorithm. Bounding Terms I$a^\star$, $a^\star\in\cS$, and Term II will rely on arguments that are specific to Thompson Sampling. Fix $a^\star\in \cS$ and let $p_{a^\star}^{\rho^\ddagger}(t)\equiv \operatorname{\mathbb{P}}(\left.\theta_{a^\star}(t)>{c}_{a^\star}\rho^\ddagger\right|\mathcal{F}(t))$. Note that $p_{a^\star}^{\rho^\ddagger}(t)\not= p_{a^\star}^{\rho^\ddagger}(t+1)$ implies $a^\star\in\operatorname{\mathcal{A}}(t+1)$. Thus $p_{a^\star}^{\rho^\ddagger}(t)$ is equal to $p_{a^\star,n}^{\rho^\ddagger}\equiv p_{a^\star}^{\rho^\ddagger}(\tau_{a^\star,n})$ for all $t$ such that $N_{a^\star}(t)=n$. We now state Lemma \[lem:suboptasopt\], that generalizes Lemma 1 in . \[lem:suboptasopt\] If $a\in\{1,\ldots,K+1\}$, $a^\star\in\cS$, and $\rho^\ddagger$ satisfies ${c}_{a^\star}\rho^\ddagger<1$, then, for all $t\ge 0$, $$\begin{aligned} \operatorname{\mathbb{P}}\left(\left.a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)\le{c}_a \rho^{\ddagger},\theta_{a^\star}(t)< {c}_{a^\star}\hat{\rho}^\star\right|\mathcal{F}(t)\right)&\le \frac{1-p_{a^\star}^{\rho^\ddagger}(t)}{p_{a^\star}^{\rho^\ddagger}(t)}\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^{\star}(t)\right|\mathcal{F}(t)\right).\end{aligned}$$ The proof can be found in Appendix \[app:thomproof\]. Observe that the upper bound in the above lemma does not rely on $a$. We have another lemma, that relies on a lower bound on the probability $\mathring{q}_{a^\star}$, to be defined shortly, that is possible for $q_{a^\star}(t)$ given that $\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^\star(t)$. By the absolute continuity of the beta distribution, we also have that $$\begin{aligned} \operatorname{\mathbb{P}}\left(a^\star\in\operatorname{\mathcal{A}}(t+1)\middle|\mathcal{F}(t)\right)&= \operatorname{\mathbb{P}}\left(a^\star\in\operatorname{\mathcal{A}}(t+1),\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}\ge \hat{\rho}^{\star}(t)\middle|\mathcal{F}(t)\right) \\ &= \operatorname{\mathbb{P}}\left(a^\star\in\operatorname{\mathcal{A}}(t+1)\middle|\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}\ge \hat{\rho}^{\star}(t),\mathcal{F}(t)\right)\operatorname{\mathbb{P}}\left(\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}\ge \hat{\rho}^{\star}(t)\middle|\mathcal{F}(t)\right). $$ We lower bound the leading term in the product on the right by $$\begin{aligned} \mathring{q}_{a^\star}&\equiv \min\left\{1,\min_{\mathcal{H}\subseteq \{1,\ldots,K\}\backslash\{a^\star\} : \sum_{\tilde{a}\in\mathcal{H}}{c}_{\tilde{a}} < B} \frac{B-\sum_{\tilde{a}\in\mathcal{H}}{c}_{\tilde{a}}}{c_{a^\star}}\right\}.\end{aligned}$$ Because ${c}_{K+1}=B$, one could equivalently take the minimum over $\mathcal{H}\subseteq \{1,\ldots,K+1\}\backslash\{a^\star\}$. To see that this is a lower bound, consider two cases. If $\theta_{a^\star}(t)/{c}_{a^\star}> \hat{\rho}^{\star}(t)$, then $a\in\operatorname{\mathcal{A}}(t+1)$ with probability one, and so the above is a lower bound. If $\theta_{a^\star}(t)/{c}_{a^\star}= \hat{\rho}^{\star}(t)$, then the numerator $B - \sum_{\tilde{a}\in\mathcal{H}}{c}_{\tilde{a}}$ of the inner minimum (over $\mathcal{H}$) above represents the minimum possible amount of remaining budget when arm $a^\star$ is the unique arm on the estimated margin. The estimated margin is almost surely (over the draws of $\theta(t)$) singleton. Clearly, $\mathring{q}_{a^\star}>0$. As a consequence,, $$\begin{aligned} \operatorname{\mathbb{P}}\left(\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}\ge \hat{\rho}^{\star}(t)\middle|\mathcal{F}(t)\right)&\le \mathring{q}_{a^\star}^{-1} \operatorname{\mathbb{P}}\left(a^\star\in\operatorname{\mathcal{A}}(t+1)\middle|\mathcal{F}(t)\right). \label{eq:calSvscalA}\end{aligned}$$ We have the following lemma, whose proof can be found in Appendix \[app:thomproof\]. \[lem:thomtransferTtoN\] If $a^\star\in\cS$ and ${c}_{a^\star}\rho^\ddagger<1$, then, for all $t\ge 0$, $$\begin{aligned} \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \frac{1-p_{a^\star}^{\rho^\ddagger}(t)}{p_{a^\star}^{\rho^\ddagger}(t)}\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^{\star}(t)\right|\mathcal{F}(t)\right)\right]&\le \mathring{q}_{a^\star}^{-1}\operatorname{\mathbb{E}}\left[\sum_{n=0}^{T-1} \frac{1-p_{a^\star,n}^{\rho^\ddagger}}{p_{a^\star,n}^{\rho^\ddagger}}\right].\end{aligned}$$ Combining the two preceding lemmas yield the inequality $$\begin{aligned} \textnormal{Term I}a^\star&\le \mathring{q}_{a^\star}^{-1} \operatorname{\mathbb{E}}\left[\sum_{n=0}^{T-1} \frac{1-p_{a^\star,n}^{\rho^\ddagger}}{p_{a^\star,n}^{\rho^\ddagger}}\right]. \label{eq:suboptasoptsum}\end{aligned}$$ Note crucially that we have upper bounded the sum over time on the left-hand side by a sum over the number of pulls of arm $a^\star$ on the right-hand side. There appears to be a steep price to pay for this transfer from a sum over time to a sum over counts: the right-hand side inverse weights by a conditional probability, which may be small for certain realizations of the data. Lemma 2 in , that we restate below using our modified notation, establishes that this inverse weighting does not cause a problem for Thompson sampling with Bernoulli rewards and independent beta priors. If $\rho^\ddagger<\rho^\star$, then the proceeding lemma implies that, for each $a^\star\in\cS$, Term I$a^\star$ is $O(1)$, i.e. is $o(\log T)$ with much to spare. Obviously, this implies that $\sum_{a^\star\in\cS}\textnormal{Term I}a^\star = o(\log T)$ as well. \[lem:carefulbinomialbound\] If $a^\star\in\cS$ and $\rho^\ddagger<\rho_{a^\star}$, then, with $\Delta\equiv \mu_{a^\star}-{c}_{a^\star} \rho^\ddagger$, $$\begin{aligned} \operatorname{\mathbb{E}}\left[\frac{1-p_{a^\star,n}^{\rho^\ddagger}}{p_{a^\star,n}^{\rho^\ddagger}}\right]&=\begin{cases} \frac{3}{\Delta},&\mbox{ for }n<\frac{8}{\Delta} \\ \Theta\left(e^{-\Delta^2 n/2} + \frac{1}{(n+1)\Delta^2}e^{-\operatorname{KL}({c}_{a^\star}\rho^\ddagger,\mu_{a^\star})n} + \frac{1}{\exp(\Delta^2 n/4)-1}\right),&\mbox{ for }n\ge \frac{8}{\Delta}. \end{cases}\end{aligned}$$ Above $\Theta(\cdot)$ is used to represent big-Theta notation. We now turn to Term II. The following result mimics Lemma 4 in , and is a consequence of the close link between beta and binomial distributions and the Chernoff-Hoeffding bound. We provide a proof of this result in Appendix \[app:thomproof\]. \[lem:termiii\] If $a\in(\cM\cup\cN)\backslash\{K+1\}$ and $\rho_a<\rho^\dagger<\rho^\ddagger$, where ${c}_a\rho^\ddagger< 1$, then $$\begin{aligned} \textnormal{Term II}\equiv\sum_{t=0}^{T-1}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a\rho^\ddagger,\hat{\mu}_a(t)\le{c}_a\rho^\dagger\right\}&\le \frac{\log T}{\operatorname{KL}({c}_a\rho^\dagger,{c}_a\rho^\ddagger)}.\end{aligned}$$ We now turn to Term III. Note that $$\begin{aligned} \textnormal{Term III}&= \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\hat{\mu}_{a,N_{a}(t)}>{c}_a \rho^\dagger\right\}\right] \nonumber \\ &= \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \sum_{n=0}^{T-1} \operatorname{\mathds{1}}\left\{\tau_{a,n+1}=t+1,\hat{\mu}_{a,n}>{c}_a \rho^\dagger\right\}\right] \nonumber \\ &\le \sum_{n=0}^{T-1} \operatorname{\mathbb{P}}\left\{\hat{\mu}_{a,n}>{c}_a \rho^\dagger\right\}, \label{eq:termii}\end{aligned}$$ where the latter inequality holds because $\tau_{a,n+1}=t+1$ for at most one $t$ in $\{0,\ldots,T-1\}$. The following lemma controls the right-hand side of the above. \[lem:termii\] Fix an arm $a\in\{1,\ldots,K\}$. If $\rho^\dagger>\rho_a$ and ${c}_a \rho^\dagger<1$, then $$\begin{aligned} \sum_{n=0}^{T-1} \operatorname{\mathbb{P}}\left\{\hat{\mu}_{a,n}>{c}_a \rho^\dagger\right\}&\le 1 + \frac{1}{\operatorname{KL}({c}_a \rho^\dagger,\mu_a)}.\end{aligned}$$ The proof is omitted, but is an immediate consequence of the Chernoff-Hoeffding bound and the additional bounding from the proof of Lemma 3 in . Thus we have shown that Term III is $o(\log T)$, with much to spare as well. The proof of and in the setting of Theorem \[thm:thom\] is now straightforward. #### Proof of . Fix $a\in\cN\backslash\{K+1\}$. Let $\mu^\dagger= {c}_a \rho^\star$ if $a\in\underline{\cN}$, and let $\mu^\dagger$ be slightly less than $\mu_{+}$ if $a\in\overline{\cN}$. Fix $\rho^\dagger<\rho^\ddagger$ and $\rho^\ddagger$ (to be specified shortly) so that $\rho_a<\rho^\dagger<\rho^\ddagger<\mu^\dagger/{c}_a$ and $\epsilon\in(0,1]$ a constant. Plugging our results on each Term I$a^\star$ and on Terms II and III into then yields that $$\begin{aligned} \operatorname{\mathbb{E}}\left[N_a(T)\right]&\le \frac{\log T}{\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger)} + 1 + \frac{1}{\operatorname{KL}({c}_a \rho^\dagger,\mu_a)} + O(1).\end{aligned}$$ Select $\rho^\dagger$ so that $\operatorname{KL}({c}_a \rho^\dagger,\mu^\dagger)=\frac{\operatorname{KL}(\mu_a,\mu^\dagger)}{1+\epsilon}$ and $\rho^\ddagger$ so that $\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger)=\frac{\operatorname{KL}({c}_a \rho^\dagger,\mu^\dagger)}{1+\epsilon}$, since this gives $\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger) = \frac{\operatorname{KL}(\mu_a,\mu^\dagger)}{(1+\epsilon)^2}$. Hence, $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]\le (1+\epsilon)^2\frac{f(T)}{\operatorname{KL}(\mu_a,\mu^\dagger)} + r(T,\mu^\dagger),\end{aligned}$$ where $r(T,\mu^\dagger)/\log T\rightarrow 0$ for fixed $\mu^\dagger$. #### Proof of . If $a\in\underline{\cN}$, then dividing both sides by $\log T$, and then taking $T\rightarrow\infty$ followed by $\epsilon\rightarrow 0$ gives . If, on the other hand, $a\in\overline{\cN}$, then we use that there exists a sequence $\mu^\dagger(T)$ such that $r(T,\mu^\dagger(T))/\log T\rightarrow 0$. Because $\liminf_{\mu^\dagger\rightarrow\mu_{+}}=+\infty$, then dividing both sides by $\log T$, taking the limit as $T\rightarrow\infty$, followed by $\epsilon\rightarrow 0$, gives in the case where $a\in\overline{\cN}$. Budget saturation when $\rho^\star > \rho$ {#sec:budgetsat} ------------------------------------------ Assuming $\rho^\star > \rho$, we prove for KL-UCB and Thompson Sampling in the setting of Theorems \[thm:expfam\] and \[thm:finsup\] and Theorem \[thm:thom\] respectively. Recall that the third term in the regret decomposition can be expressed in terms of the number of draws of the supplementary arm $K+1$ in the extended bandit model: $$BT - \sum_{a=1}^Kc_a {\mathbb{E}}_{\nu}[N_a(T)] = B {\mathbb{E}}[N_{K+1}(T)].$$ We prove below for each algorithm that ${\mathbb{E}}[N_{K+1}(T)] = o(\log(T))$, as a by product from specific elements already established when controlling the number of suboptimal draws. ### KL-UCB {#kl-ucb-1 .unnumbered} For any $\rho^\dagger\in(\rho,\rho^\star]$ and any $t\ge K$, it holds that, for $T$ large enough, $$\left\{K+1\in\operatorname{\mathcal{A}}(t+1)\right\}\subseteq \bigcup_{a^\star\in\cS}\left\{{c}_{a^\star}\rho\ge U_{a^\star}(t)\right\}\subseteq \bigcup_{a^\star\in\cS}\left\{{c}_{a^\star}\rho^\dagger\ge U_{a^\star}(t)\right\}.$$ The first inclusion must hold because if all the arms in $\cS$ had satisfied $U_{a^\star}/c_{a^\star} \geq \rho$, then including all of those arms in $\operatorname{\mathcal{A}}(t+1)$ would have been enough to saturate the budget and $K+1$ would not have been selected. The second inclusion holds because $\rho^\dagger>\rho$. Hence, $\operatorname{\mathbb{E}}[N_{K+1}(T)]\le \sum_{a^\star\in\cS} \textnormal{Term 2}a^\star$ (see Equation \[eq:T1T2\] for its definition). This condition is always satisfied by the choice $\rho^\dagger=\rho^\star$ that we have used in the setting of Theorem \[thm:expfam\], and it holds for all $T$ sufficiently large for the choice $\rho^\dagger=\left[1-\log(T)^{-1/5}\right]\rho^\star$ that we have used in the setting of Theorem \[thm:finsup\]. Lemma \[lem:term2astar\] shows that each Term 2$a^\star$ is again $o(\log T)$. ### Thompson Sampling {#thompson-sampling-1 .unnumbered} We have that $$\{K+1\in\operatorname{\mathcal{A}}(t+1)\}\subseteq\bigcup_{a^\star\in\cS}\{K+1\in\operatorname{\mathcal{A}}(t+1),\theta_{a^\star}(t)\le {c}_{a^\star} \hat{\rho}^\star\}.$$ As $\rho < \rho^\star$ and $\theta_{K+1}(t)={c}_{K+1}\rho$ with probability one, $\operatorname{\mathbb{E}}[N_a(T)]\le \sum_{a^\star\in \cS} \textnormal{Term I}a^\star$ provided $\rho^\dagger\in (\rho,\rho^\star)$ (see Equation \[eq:termsitoiii\] for its definition). Thus, we can invoke Lemma \[lem:suboptasopt\] (that holds for $a=K+1$), followed by Lemmas \[lem:thomtransferTtoN\] and \[lem:carefulbinomialbound\], to show that $\operatorname{\mathbb{E}}[N_{K+1}(T)]=O(1)$, and therefore is $o(\log T)$ with much to spare. Optimal arms away from margin pulled $T-o(\log T)$ times {#sec:optcommon} -------------------------------------------------------- We now show that the optimal arms away from the margin ($a^\star\in\cL$) are pulled often. We start by giving an analysis that applies to any algorithm that, to decide which arms to draw at time $t+1$, based on $\mathcal{F}(t)$ and possibly some external stochastic mechanism, defines indices $I_a(t)$, $a=1,\ldots,K+1$, and then defines the threshold $\hat{\rho}^\star(t)\equiv\rho^\star(c_aI_a(t) : a=1,\ldots,K+1)$, and, for all arms $a$ with $I_a(t)\not=\hat{\rho}^\star(t)$, assigns mass $q_a(t)=\operatorname{\mathds{1}}\{I_a(t) > \hat{\rho}^\star(t)\}$. The arms with $I_a(t)=\hat{\rho}^\star(t)$ are assumed to be drawn so that $\sum_{a=1}^{K+1} {c}_a q_a(t)=B$. We then specialize the discussion to KL-UCB and Thompson sampling, where $I_a(t)$ is respectively equal to $U_a(t)/{c}_a$ and $\theta_a(t)/{c}_a$. For the remainder of this section, we fix an optimal arm $a^\star\in\cL$. Observe that, for $t\ge K$ (KL-UCB) or $t\ge 0$ (Thompson sampling), $$\begin{aligned} \left\{I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}=\,& \cup_{a\in\cM\cup\cN}\left\{I_a(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\} \\ = &\left[\cup_{a\in(\cM\cup\cN)\backslash\{K+1\}}\left\{I_a(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t),I_{K+1}(t)<\hat{\rho}^\star(t)\right\}\right] \\ &\,\cup\left\{I_{K+1}(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}.\end{aligned}$$ Recalling , we see that, for Thompson sampling, $$\begin{aligned} T - \operatorname{\mathbb{E}}[N_{a^\star}(T)]=\,& T - \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a^\star\in\operatorname{\mathcal{A}}(t+1)\middle|\mathcal{F}(t)\right\}\right] \nonumber \\ =\,& \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a^\star\not\in\operatorname{\mathcal{A}}(t+1)\middle|\mathcal{F}(t)\right\}\right] \nonumber \\ \le\,& \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_{a^\star}(t)< \hat{\rho}^\star(t)\middle|\mathcal{F}(t)\right\}\right] \nonumber \\ \le\,& \sum_{a\in(\cM\cup\cN)\backslash\{K+1\}} \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_a(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t),I_{K+1}(t)<\hat{\rho}^\star(t)\right\} \nonumber \\ &+ \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_{K+1}(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}, \label{eq:bdthom}\end{aligned}$$ where the first inequality holds because $\{a\not\in\operatorname{\mathcal{A}}(t+1)\}\subseteq\{I_{a^\star}(t)< \hat{\rho}^\star(t)\}$ and the second inequality holds by the preceding display. We have a similar identity for KL-UCB, though the identity is slightly different due to the initiation of each of the $K$ arms. Specifically, $$\begin{aligned} T - K + 1 - \operatorname{\mathbb{E}}[N_{a^\star}(T)]\le\,& \sum_{a\in(\cM\cup\cN)\backslash\{K+1\}} \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_a(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t),I_{K+1}(t)<\hat{\rho}^\star(t)\right\} \nonumber \\ &+ \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_{K+1}(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}. \label{eq:bdKL}\end{aligned}$$ For $a\in\cM\cup\cN$, let $\mathscr{H}$ denote the collection of all subsets $\mathcal{H}$ of $\{1,\ldots,K\}\backslash\{a,a^\star\}$ for which $\sum_{\tilde{a}\in\mathcal{H}}{c}_{\tilde{a}} < B$. For $a\in\cM\cup\cN$, we then define $$\begin{aligned} \check{q}_{a}^{a^\star}\equiv \begin{cases} \min\left\{1,\min_{\mathcal{H}\in \mathscr{H}} \frac{B-\sum_{\tilde{a}\in\mathcal{H}}{c}_{\tilde{a}}}{{c}_a}\right\},&\mbox{ if }a=K+1\textnormal{ or Thompson Sampling,} \\ \min\left\{1,\min_{\mathcal{H}\in \mathscr{H}} \frac{B-\sum_{\tilde{a}\in\mathcal{H}}{c}_{\tilde{a}}}{\sum_{\tilde{a}\in\{1,\ldots,K\}\backslash[\mathcal{H}\cup\{a^\star\}]}{c}_{\tilde{a}}}\right\}&\mbox{ if }a\not=K+1\textnormal{ and KL-UCB.} \end{cases}\end{aligned}$$ Above “Thompson Sampling” and “KL-UCB” in the conditioning statements refers to which of the two algorithms is under consideration. The latter condition represents the extreme scenario where the arms in $\tilde{a}\in\mathcal{H}$ have $I_{\tilde{a}}(t)>\hat{\rho}^\star(t)$, whereas the arms $\tilde{a}$ outside of $\mathcal{H}\cup\{a^\star,K+1\}$ have $I_{\tilde{a}}(t)=\hat{\rho}^\star(t)$. One can verify that $\check{q}_{a}^{a^\star}>0$. Similarly to , for each $a\in(\cM\cup\cN)\backslash\{K+1\}$ and $t\ge K$ (KL-UCB) or $t\ge 0$ (Thompson sampling), $$\begin{aligned} \operatorname{\mathbb{P}}&\left\{I_a(t)\ge \hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t),I_{K+1}(t)<\hat{\rho}^\star(t)\middle| \mathcal{F}(t)\right\} \\ &\le \frac{1}{\check{q}_{a}^{a^\star}} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),I_{a^\star}(t)< \hat{\rho}^\star(t)\middle| \mathcal{F}(t)\right\},\end{aligned}$$ and thus $$\begin{aligned} \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_a(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t),I_{K+1}(t)<\hat{\rho}^\star(t)\right\}&\le \frac{1}{\check{q}_{a}^{a^\star}} \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}.\end{aligned}$$ For $a=K+1$, we similarly have $$\begin{aligned} \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{I_{K+1}(t)\ge\hat{\rho}^\star(t),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}&\le \frac{1}{\check{q}_{a}^{a^\star}} \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),I_{a^\star}(t)< \hat{\rho}^\star(t)\right\}.\end{aligned}$$ For each $a\in\cM\cup\cN$, let $$\begin{aligned} M_a^{a^\star}(T)\equiv\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\{a\in\operatorname{\mathcal{A}}(t+1),I_{a^\star}(t)< \hat{\rho}^\star(t)\}.\end{aligned}$$ The bounds and yield the key observation that we use in this section: $$\begin{aligned} \mbox{for KL-UCB:}&\ \ T - K + 1 - \operatorname{\mathbb{E}}[N_{a^\star}(T)]\le \sum_{a\in\cM\cup\cN} \frac{1}{\check{q}_a^{a^\star}}\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]; \nonumber \\ \mbox{for Thompson sampling:}&\ \ T - \operatorname{\mathbb{E}}[N_{a^\star}(T)]\le \sum_{a\in\cM\cup\cN} \frac{1}{\check{q}_a^{a^\star}}\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]. \label{eq:keyobs}\end{aligned}$$ The remainder of the analysis involves controlling $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]$ for arms $a\in\cM\cup\cN$. Let $G$ be some integer in $[0,+\infty($ and $\delta\in(0,1)$ be a constant to be specified shortly. For convenience, we let $T^{(g)}\equiv\lfloor T^{(1-\delta)^g}\rfloor$ for $g\in\mathbb{N}$. We also define $$\begin{aligned} &\overline{\cU}\equiv\left\{a\in (\cM\cup\cN)\backslash\{K+1\} : {c}_a\rho_{a^\star}\ge\mu_{+}\right\}, \\ &\underline{\cU}\equiv\left\{a\in (\cM\cup\cN)\backslash\{K+1\} : {c}_a\rho_{a^\star}<\mu_{+}\right\},\end{aligned}$$ where we note that $\overline{\cU}\cup\underline{\cU}=(\cM\cup\cN)\backslash\{K+1\}$. Our analysis relies on the following bound (for which we provide the arguments below): $$\begin{aligned} \sum_{a\in\cM\cup\cN} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le\,& \operatorname{\mathbb{E}}[N_{K+1}(T)] + \sum_{a\in\overline{\cU}} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)] + \sum_{a\in\underline{\cU}} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)] \nonumber \\ =\,& o(\log T) + \underbrace{\sum_{a\in\overline{\cU}} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)] + \sum_{a\in\underline{\cU}} \operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(G)})]}_{\textnormal{Term A}} \nonumber \\ &+ \underbrace{\sum_{g=1}^G \sum_{a\in\underline{\cU}} \operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(g-1)})-M_a^{a^\star}(T^{(g)})]}_{\textnormal{Term B}}. \label{eq:margindecomp}\end{aligned}$$ The inequality uses that $\operatorname{\mathbb{E}}[M_{K+1}^{a^\star}(T)]\le \operatorname{\mathbb{E}}[N_{K+1}(T)]$, and the equality holds using (i) a telescoping series and (ii) the fact that the algorithm achieves : indeed, this was proven for both KL-UCB and Thompson sampling in Section \[sec:budgetsat\]. We now present the key ingredients to bound Term A and B. Each lemma stated below holds for both KL-UCB in the settings of Theorems \[thm:expfam\] and \[thm:finsup\] and for Thompson sampling in the setting of Theorem \[thm:thom\]. Though these lemmas hold for both algorithms, the methods of proof for KL-UCB and for Thompson sampling are quite different. Thus we give the proofs of the lemmas in the settings of Theorems \[thm:expfam\] and \[thm:finsup\] in Appendix \[app:klucbproof\] and the proofs in the setting of Theorem \[thm:thom\] in Appendix \[app:thomproof\]. \[lem:TermA\] In the settings of Theorem \[thm:expfam\], \[thm:finsup\], and \[thm:thom\], $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]=o(\log T)$ for $a\in \overline{\cU}$ and, for fixed $G\ge 0$, $$\begin{aligned} \operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(G)})]\le (1-\delta)^G\frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})}\end{aligned}$$ for $a\in\underline{\cU}$. As a consequence, $$\begin{aligned} \textnormal{Term A}&\le (1-\delta)^G \sum_{a\in\underline{\cU}}\frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})} + o(\log T).\end{aligned}$$ The proof of Lemma \[lem:TermA\] borrows a lot from the proofs of and for each algorithm. Controlling Term B relies on a careful choice of $\delta>0$, which is specified in Lemma \[lem:TermB\] below. The proof of this lemma is highly original: indeed we first prove that the considered algorithm is uniformly efficient, which allows to exploit the lower bound given in Theorem \[thm:reglb\]. Its proof is provided in the appendix for both KL-UCB and Thompson Sampling, and we sketch it below. \[lem:TermB\] Let $d\in (0,1)$ and $\delta$ chosen such $$\begin{aligned} \delta = d\left[1-\left(\max_{a\in\cN\cap\underline{\cU}}\frac{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho^\star)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho_{a^\star})}\right)^{1/2}\right], \label{def:ChoiceDelta}\end{aligned}$$ and $\delta=d$ if $\cN\cap\underline{\cU} = \emptyset$. Then in the setting of Theorems \[thm:expfam\], \[thm:finsup\], and \[thm:thom\], Term B is $o(\log T)$. We first show that the algorithms are uniformly efficient in the sense defined in Section \[sec:lb\]. This result is an immediate consequence of the results in Section \[sec:suboptrare\], which show that the arms in $\cN\backslash\{K+1\}$ are not pulled too often, plus the preliminary results in this section, which show that arms in $\cL$ are pulled often. \[lem:cons\] KL-UCB is uniformly efficient in the settings of Theorems \[thm:expfam\] and \[thm:finsup\] and Thompson sampling is uniformly efficient in the setting of Theorem \[thm:thom\]. Fix an arbitrary reward distribution $\mathcal{V}$. By the already proven and in the settings of Theorems \[thm:expfam\], \[thm:finsup\], and \[thm:thom\] and by Lemma \[lem:TermA\], both of which hold for $\mathcal{V}$, $$\begin{aligned} T-\operatorname{\mathbb{E}}_{\mathcal{V}}[N_{a^\star}(T)]&\le \sum_{a\in\cM\cup\cN}\frac{1}{\check{q}_a^{a^\star}}\operatorname{\mathbb{E}}_{\mathcal{V}}[M_a^{a^\star}(T)] + O(1)\\ &\le o(\log T) + \sum_{a\in\overline{\cU}}\frac{1}{\check{q}_a^{a^\star}}\operatorname{\mathbb{E}}_{\mathcal{V}}[M_a^{a^\star}(T)] + \sum_{a\in\underline{\cU}}\frac{1}{\check{q}_a^{a^\star}}\operatorname{\mathbb{E}}_{\mathcal{V}}[M_a^{a^\star}(T)] + O(1) $$ for any $a^\star\in\cL$, where the $O(1)$ term is equal to zero for Thompson sampling and, by , is $K-1$ for KL-UCB. The right-hand side is $O(\log T)$ by applying the results of Lemma \[lem:TermA\] to control the sums over $\overline{\cU}$ and $\underline{\cU}$. Section \[sec:suboptrare\] showed that arms in $\cN$ are not pulled often (at most $O(\log T)$ times). By , it follows that $R(T)=O(\log T)$, which is $o(T^{\alpha})$ for any $\alpha>0$. Fix $g\in\mathbb{N}$ and an arm $a\in\cN\cap\underline{\cU}$. By the uniform efficiency of the algorithm established in Lemma \[lem:cons\], we will be able to apply from Lemma \[lem:lb\] to show that $N_a(T^{(g)})\ge (1-\delta)\frac{\log T^{(g)}}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_{a},{c}_a \rho^\star)}$ with probability approaching 1. For now suppose this holds almost surely (in the proofs we deal with the fact that this happens with probability approaching rather than exactly 1). Our objective will be to show that this lower bound on $N_a(T^{(g)})$ suffices to ensure that $M_a^{a^\star}(T^{(g-1)})-M_a^{a^\star}(T^{(g)})$ is $o(\log T)$, in words that arm $a$ is pulled while arm $a^\star$ is pulled with probability zero ($I_{a^\star}(t)<\hat{\rho}^\star(t)$) at most $o(\log T)$ times from time $t=T^{(g)},\ldots,T^{(g-1)}$. We will see that $\frac{\log T^{(g-1)}}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho_{a^\star})}$ pulls of arm $a$ by time $T^{(g)}$ suffices to ensure this in both settings. Using that $(1-\delta)\log T^{(g)}\approx (1-\delta)^2\log T^{(g-1)}$, it will follow that we can control the sum in Term B for each $a\in\cN$ provided we choose $\delta\in(0,1)$ so that $$(1-\delta)^2\frac{1}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho^\star)}> \frac{1}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho_{a^\star})}\textnormal{ for all $a\in\cN$.}\label{eq:deltalogic}$$ It is easy to check to for any $d\in (0,1)$, $\delta$ as defined in Lemma \[lem:TermB\] satisfies this inequality. Note that $\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho_{a^\star})\ge\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho^\star)$, and thus $\delta\in(0,1)$. So far we have only considered suboptimal arms $a\in\cN\cap\underline{\cU}$, but the fact that, for any $a\in\cM\cap\underline{\cU}$, Lemma \[lem:lb\] ensures that $N_a(T^{(g)})> \log T^{(g)}/\epsilon$ with probability approaching 1 for *any* $\epsilon>0$ shows that $a\in\cN\cap\underline{\cU}$ is indeed the harder case. Indeed, this is what we see in our proofs controlling Term B for the two algorithms. We now conclude the analysis. Combining Equations and with the bounds on Term A and B obtained in Lemma \[lem:TermA\] and Lemma \[lem:TermB\] yield, for any finite $G$ and for the particular choice of $\delta \in (0,1)$ given in $$\begin{aligned} \limsup_T \frac{T-\operatorname{\mathbb{E}}[N_{a^\star}(T)]}{\log T}&\le (1-\delta)^G \sum_{a\in \underline{\cU}} \frac{1}{\check{q}_a^{a^\star} \operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a\rho_{a^\star})}. \label{eq:fixedG}\end{aligned}$$ Taking $G$ to infinity yields the result. Conclusion {#sec:conc} ========== We have established the asymptotic efficiency of KL-UCB and Thompson sampling for budgeted multiple-play bandit problem in which the cost of pulling each arm is known and, in each round, the agent may use any strategy for which the expected cost is no more than their budget. We have also introduced a pseudo-arm so that the agent has the option of reserving the remainder of their budget if the remaining arms have reward-to-cost ratios that fall below a prespecified indifference point. Thompson sampling outperforms KL-UCB in three of our four simulations scenarios. Despite the strong performance of Thompson sampling for Bernoulli rewards, we have been able to prove stronger results about KL-UCB in this work, dealing with more general distributions. Understanding for which distributions one of these algorithms is preferable to the other is an interesting area for future work. In future work, it would be interesting to consider an extension of our setting where the budget ($B_t$), indifference points ($\rho_t$), and costs (${c}_t$) are random over time according to some exogeneous source of randomness. If only the budget is random over time, then, under some regularity conditions, the regret lower bound and regret of our algorithms would seem to be driven by the behavior of our algorithm for the fixed budget representing the upper edge of the support for the random budget, since this is the setting in which the most information is learned about the arm distributions (arms that are otherwise suboptimal can be optimal in this setting). If only the indifference point is variable over time, then the behavior of our algorithm will similarly be driven by the lowest indifference point, since the most information is available in this case. Combinations of variable budgets and indifference points will result in a similar analysis. Variable but known costs are more complex, because they have the potential to change the order and indices of the optimal arms. For sufficiently variable costs, we in fact expect that all arms will be pulled more than order $\log T$ times, since all arms will be optimal for certain cost realizations. Therefore, a careful study of a variable cost budgeted bandit problem may require very different techniques than those used in this work. #### Acknowledgments. The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-13-BS01-0005 (project SPADRO) and ANR-16-CE40-0002 (project BADASS). Alex Luedtke gratefully acknowledges the support of a Berkeley Fellowship. Appendix {#appendix .unnumbered} ======== We begin with an outline of the results proven in this appendix and how they are related to one another. Lemma \[lem:lb\] gives a lower bound on the number of draws of each suboptimal arm for a uniformly efficient algorithm. Deduced from Lemma \[lem:lb\], Theorem \[thm:reglb\] gives an asymptotic regret lower bound for a uniformly efficient algorithm. The asymptotic lower bound is achieved whenever the expected number of draws of each suboptimal arm satisfies the appropriate asymptotic condition, either or depending on the arm, and the expected number of draws of each optimal arm away from the margin satisfies the asymptotic condition . Theorems \[thm:expfam\] and \[thm:finsup\] state that the variants of KL-UCB are uniformly efficient and achieve for rewards sampled either from a single parameter exponential family or from bounded and finitely supported distributions. Theorem \[thm:thom\] states that Thompson sampling is uniformly efficient and achieves for Bernoulli distributed rewards. The first step of the proof of Theorems \[thm:expfam\], \[thm:finsup\], and \[thm:thom\] consists in showing that KL-UCB and Thompson sampling achieve the asymptotically optimal expected number of suboptimal arm draws, i.e. that and hold in their contexts. For KL-UCB, this is a consequence of a preliminary analysis given in Lemmas \[lem:term1\] and \[lem:term2astar\]. For Thompson sampling, this is a consequence of another preliminary analysis given in Lemmas \[lem:suboptasopt\] through \[lem:termiii\]. The proof of Lemma \[lem:termiii\] relies on a link between the beta and binomial distributions given in Lemma \[lem:betabinomial\]. The second step of the proof of Theorems \[thm:expfam\], \[thm:finsup\], and \[thm:thom\] consists in showing that KL-UCB and Thompson sampling are uniformly efficient in their respective contexts. This is a consequence of yet another preliminary analysis, , , and Lemma \[lem:TermA\]. The third step of the proof of Theorems \[thm:expfam\], \[thm:finsup\], and \[thm:thom\] consists in showing that KL-UCB and Thompson sampling achieve the asymptotically optimal expected number of optimal draws away from the margin, i.e. that holds in their contexts. This is a consequence of the preliminary analysis undertaken in step two and of Lemmas \[lem:TermA\] and \[lem:TermB\]. The proofs of Lemmas \[lem:TermA\] and \[lem:TermB\] hinge on Lemmas \[lem:suboptasopt\] through \[lem:termiii\]. The proof of Lemma \[lem:TermB\] also relies on Lemma \[lem:betabinomial\]. The fourth and final step of the proof of Theorems \[thm:expfam\], \[thm:finsup\], and \[thm:thom\] boils down to applying Theorem \[thm:reglb\]. Oracle strategy and regret decomposition {#proofs:Oracle} ======================================== Proof of Proposition \[prop:Oracle\] ------------------------------------ Recall that $$\bm q^\star \in \underset{{\bm q \in [0,1]^K}}{\operatorname{argmax}} \sum_{a=1}^K q_a (\mu_a - c_a \rho) \ \ \ \text{such that} \ \ \ \sum_{a=1}^K q_a c_a \leq B.$$ Introducing $c_{K+1} = B$ and $\mu_{K+1} = B\rho$, one can prove that $\bm q^\star$ coincides with the first $K$ components of $\bm q^\star_{K+1} \in [0,1]^{K+1}$, that is defined as the solution to $$\bm q^\star_{K+1} \in \underset{\bm q \in [0,1]^{K+1}}{\operatorname{argmax}} \sum_{a=1}^{K+1} q_a (\mu_a - c_a \rho) \ \ \ \text{such that} \ \ \ \sum_{a=1}^{K+1} q_a c_a = B\label{opt:2}$$ and that the two optimization problems have the same value. This is because as $\mu_{K+1} - c_{K+1}\rho = 0$, the two objective functions coincide: $$f_K(\bm q) \equiv \sum_{a=1}^{K} q_a (\mu_a - c_a \rho) =\sum_{a=1}^{K+1} q_a (\mu_a - c_a \rho) \equiv f_{K+1}(\bm q_{K+1})$$ and if $\bm q$ satisfies the first constraint, there exists $q_{K+1}$ such that $\bm q_{K+1} = (\bm q,q_{K+1})$ satisfies the second constraint: $\sum_{a=1}^{K+1} q_a c_a = B$ (as $c_{K+1}=B$). Conversely, if $\bm q_{K+1}$ satisfies the second constraint, its first $K$ marginals clearly satisfy the first constraint. The common value $M^\star$ of these two optimization problem, that is the maximal achievable reward, can be rearranged a bit, using that $\sum_{a=1}^{K+1} q_a c_a \rho = \rho B$: $$M^\star = \sum_{a=1}^{K+1} q_a^\star \mu_a - \rho B,$$ where $\bm q^\star_{K+1} \in [0,1]^{K+1}$ is the solution to $$\bm q^\star_{K+1} \in \underset{\bm q \in [0,1]^{K+1}}{\operatorname{argmax}} \sum_{a=1}^{K+1} q_a \mu_a \ \ \ \text{such that} \ \ \ \sum_{a=1}^{K+1} q_a c_a = B.\label{opt:3}$$ Now introduce $$L^\star \equiv \sum_{a \in \cL} \mu_{a} + \rho^\star\left(B - \sum_{a \in \cL} c_{a}\right).$$ The optimal weights are also defined by $$\bm q^\star_{K+1} \in \operatorname{argmin}{\bm q \in [0,1]^{K+1}} \left[L^\star - \sum_{a=1}^{K+1} q_a \mu_a \right] \ \ \ \text{such that} \ \ \ \sum_{a=1}^{K+1} q_a c_a = B.$$ The new objective can be rewritten as follows, where the ‘virtual’ arm $K+1$ that has characteristics $\mu_{K+1} = B\rho$ and $c_{K+1}=B$ is added to either the set $\cM$ (if $\rho_{K+1}=\rho=\rho^\star$) or $\cN$ (if $\rho_{K+1}=\rho < \rho^\star$). $$\begin{aligned} &L^\star - \sum_{a=1}^{K+1} q_a \mu_a = \sum_{a \in \cL} \mu_{a} + \rho^\star\left(B - \sum_{a \in \cL} c_{a}\right) - \sum_{a \in \cL} c_a q_a \rho_a - \sum_{a \in \cM} c_a q_a \rho^\star - \sum_{b \in \cN} c_b q_b \rho_b\\ & = \sum_{a \in \cL} c_a\rho_{a} + \rho^\star\left(B - \sum_{a \in \cL} c_{a}\right) - \sum_{a \in \cL} c_a q_a \rho_a - \rho^\star \left( B - \sum_{a \in \cL}c_aq_a - \sum_{b \in \cN}c_b q_b\right) - \sum_{b \in \cN} c_b q_b \rho_b\\ & = \sum_{a \in \cL} c_a\underbrace{(\rho_a - \rho^\star)}_{> 0}(1-q_a) + \sum_{b \in \cN}c_b\underbrace{(\rho^\star - \rho_b)}_{> 0} q_b.\end{aligned}$$ This shows that the objective function is always non negative, and that it can actually be set to the zero by choosing weights that satisfy $q_a = 1$ for all $a\in \cL$ and $q_b = 0$ for all $b \in \cL$. It remains to justify that such a choice is indeed feasible for some choices of weights on the arms in the margin $\cM$. This margin is never empty, as in the case $\rho^\star = \rho$, it does contain the ‘pseudo-arm’ mentioned above. By definition of the sets $\cL$ and $\cM$, $$\sum_{a \in \cL} c_a < B \ \ \ \text{and} \ \ \ \sum_{a \in \cL \cup \cM} c_a \geq B$$ hence, the solution can be “completed” by putting weight on the margin such that $\sum_{a \in \cL}c_a + \sum_{a \in \cN}q_ac_a = B$. If $\rho < \rho^\star$, then the arm $K+1$ belongs to $\cN$ and as such $q_{K+1} = 0$ and the first $K$ marginals indeed satisfy the statement of Proposition \[prop:Oracle\], with a non-empty margin. If $\rho = \rho^\star$, our ‘extended’ margin only contains arm $K+1$, while the original margin is empty. As such the only arms with non-zero weights among the first $K$ marginals are the arms in $\cL$, for which the weight is one. Proof of Proposition \[prop:RegretDec\] --------------------------------------- $$\Reg(T,\cV) = {\mathbb{E}}\left[\sum_{t=1}^T (G^\star - G(t))\right] = {\mathbb{E}}\left[\sum_{t=1}^T (G^\star - \sum_{a=1}^Kq_a(t)(\mu_a - c_a \rho))\right]$$ The proof follows from a rewriting of $$\begin{aligned} G^\star - \sum_{a=1}^Kq_a(t)(\mu_a - c_a \rho) = &\sum_{a\in \cL}c_a \rho_a + \rho^\star\left(B - \sum_{a \in \cL} c_a\right) - B\rho - \sum_{a=1}^Kq_a(t)(\mu_a -c_a\rho) \\ = & \sum_{a\in \cL}c_a \rho_a + \rho^\star\left(B - \sum_{a \in \cL} c_a\right) - B\rho - \sum_{a=1}^{K+1}q_a(t)(\mu_a -c_a\rho),\end{aligned}$$ where we define $\mu_{K+1}=\rho B$, $c_K = B$ and let $q_{K+1}(t)$ be such that $\sum_{a=1}^{K+1}q_a(t) c_a = B$. This is possible as $\sum_{a=1}^{K}q_a(t) c_a \leq B$ due to the soft budget constraints and $c_{K+1}=B$ and $$q_{K+1}(t) = \frac{B - \sum_{a=1}^K c_aq_a(t)}{B}.$$ Thus one can further write $$\begin{aligned} &G^\star - \sum_{a=1}^Kq_a(t)(\mu_a - c_a \rho) = \sum_{a\in \cL}c_a \rho_a + \rho^\star\left(B - \sum_{a \in \cL} c_a\right) - \sum_{a=1}^{K+1}q_a(t)\mu_a \\ & \ \ \ = \sum_{a\in \cL}c_a \rho_a + \rho^\star\left(B - \sum_{a \in \cL} c_a\right) - \sum_{a\in \cL}q_a(t)c_a\rho_a - \rho^\star\sum_{a\in \cM}q_a(t)c_a- \sum_{a\in \cN}q_a(t)c_a\rho_a - q_{K+1}(t)\rho B\end{aligned}$$ Using that $$\sum_{a\in \cM}q_a(t)c_a = B - \sum_{a\in \cL}q_a(t)c_a - \sum_{a\in \cM}q_a(t)c_a - q_{K+1}(t)B,$$ one obtains $$\begin{aligned} & G^\star - \sum_{a=1}^Kq_a(t)(\mu_a - c_a \rho) = \sum_{a\in \cL}c_a (\rho_a - \rho^\star)(1-q_a(t)) + \sum_{a \in \cN}c_a(\rho^\star - \rho_a) q_a(t) + B(\rho^\star - \rho) q_{K+1}(t)\\ & \ \ \ = \sum_{a\in \cL}c_a (\rho_a - \rho^\star)(1-q_a(t)) + \sum_{a \in \cN}c_a(\rho^\star - \rho_a) q_a(t) + (\rho^\star - \rho)\left(B - \sum_{a=1}^K c_a q_a(t)\right)\end{aligned}$$ Summing over $t$, the regret can be decomposed as $$\sum_{a\in \cL}c_a (\rho_a - \rho^\star)\left(T- {\mathbb{E}}\left[\sum_{t=1}^Tq_a(t)\right]\right) + \sum_{a \in \cN}c_a(\rho^\star - \rho_a) {\mathbb{E}}\left[ \sum_{t=1}^T q_a(t)\right] + (\rho^\star - \rho)\left(B - \sum_{a=1}^K c_a {\mathbb{E}}\left[\sum_{t=1}^T q_a(t)\right]\right)$$ and the conclusion follows by noting that $N_a(T) = {\mathbb{E}}\left[\sum_{a=1}^T q_a(t)\right]$. Proof of Lower Bound on Suboptimal Arm Draws {#app:lbproof} ============================================ Fix some arm $a\in(\cM\cup\underline{\cN})\backslash\{K+1\}$, natural number $T$, and $\delta\in(0,1)$. By definition, ${c}_a \rho^{\star}<\mu_{+}$ for all $a\in\underline{\cN}$, and, for $a\in\cM$ the same property holds by our assumption that ${c}_a\rho^\star=\mu_a<\mu_{+}$. Hence, the set $\{\tilde{\nu}_a\in\cD : E(\tilde{\nu}_a)>{c}_a \rho^{\star}\}$ is non-empty. If the intersection of this set with the set of distributions $\{\tilde{\nu}_a\in\cD : \nu_a \ll \tilde{\nu}_a\}$ is empty, then the bounds are trivial by our convention that $d/\infty=0$ for finite $d$. Otherwise, let $\mathcal{V}'$ be some distribution that is equal to $\mathcal{V}$ except in the $a^{\textnormal{th}}$ component, where its $a^{\textnormal{th}}$ component $\nu_a'\in\cD$ is such that $\mu_a'\equiv E(\nu_a')>{c}_a \rho^{\star}$ and $\nu_a \ll \nu_a'$. Furthermore, one can select $\mathcal{V}'$ to fall in the statistical model for the joint distribution of the arm-specific rewards by our variation-independence assumption. For each $b$, let $\rho_{b}'=\rho_{b}$, $b\not=a$, and let $\rho_a'=\mu_a'/{c}_a$. Observe that $\mu_a'>{c}_a\rho^{\star}\ge \mu_a$ implies that $\operatorname{KL}(\nu_a,\nu_a')>0$. Define the log-likelihood ratio random variable $L_a(T)\equiv L_{a,N_a(T)}\equiv \sum_{n=1}^{N_a(T)} \log\frac{d\nu_a}{d\nu_a'}(X_{a,n})$. Let $b_a(T)\equiv (1-\delta)\frac{\log T}{\operatorname{KL}(\nu_a,\nu_a')}$ and $d(T)\equiv (1-\delta/2)\log T$. We have that $$\begin{aligned} &\operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)<b_a(T)\right\} \nonumber \\ &\le \operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)<b_a(T),L_a(T)\le d(T)\right\} + \operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)<b_a(T),L_a(T)> d(T)\right\} \nonumber \\ &\le e^{d(T)}\operatorname{\mathbb{P}}_{\mathcal{V}'}\left\{N_a(T)<b_a(T)\right\} + \operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)<b_a(T),L_a(T)> d(T)\right\}, \label{eq:lbdecomp}\end{aligned}$$ where the final inequality holds because, for any event $D\subseteq\{N_a(T)=b,L_a(T)\le d(T)\}$, a change of measure shows that $\operatorname{\mathbb{P}}_{\mathcal{V}}\{D\}={\mathbb{E}}_{\mathcal{V}'}\left[e^{L_{a,b}}\ind_{\{D\}}\right]\le e^{d(T)}\operatorname{\mathbb{P}}_{\mathcal{V}'}\{D\}$ . Let $\tilde{\rho}^\star\equiv \rho^\star({c}_a \rho_{a}' : a=1,\ldots,K+1)$. Observe that arm $a$ under the reward distribution involving $\nu_a'$ satisfies either (i) $\rho_{a'}'>\tilde{\rho}^\star$ or (ii) $\rho_a'=\tilde{\rho}^\star$ and $g_a\equiv B - \sum_{\tilde{a}\not=a : \rho_{\tilde{a}}\ge \tilde{\rho}^\star} {c}_{\tilde{a}}>0$, where the sum over the empty set is zero. Under (i), we note that the uniform efficiency of the algorithm and Markov’s inequality yield that $$\begin{aligned} \operatorname{\mathbb{P}}_{\mathcal{V}'}\left\{N_a(T)<b_a(T)\right\}&= \operatorname{\mathbb{P}}_{\mathcal{V}'}\left\{T-N_a(T)>T-b_a(T)\right\} = o\left(T^{\delta/2-1}\right).\end{aligned}$$ Thus, the first term in converges to zero as $T\rightarrow\infty$ when (i) holds. We now show the same result when (ii) holds. We first note that $$\begin{aligned} g_a T - {c}_a \operatorname{\mathbb{E}}[N_a(T)]&\ge BT-\sum_{\tilde{a}\not=a : \rho_{\tilde{a}}\ge \tilde{\rho}^\star} {c}_{\tilde{a}}\operatorname{\mathbb{E}}[N_{\tilde{a}}(T)] - {c}_a \operatorname{\mathbb{E}}[N_a(T)]= BT-\sum_{\tilde{a} : \rho_{\tilde{a}}\ge \tilde{\rho}^\star} {c}_{\tilde{a}}\operatorname{\mathbb{E}}[N_{\tilde{a}}(T)].\end{aligned}$$ The right-hand side is $o(T^{\delta/2})$ by the uniform efficiency of the algorithm. Hence, Markov’s inequality yields that, $$\begin{aligned} \operatorname{\mathbb{P}}_{\mathcal{V}'}\left\{N_a(T)<b_a(T)\right\}&= \operatorname{\mathbb{P}}_{\mathcal{V}'}\left\{g_aT-{c}_a N_a(T)>g_aT-{c}_a b_a(T)\right\} = o\left(T^{\delta/2-1}\right).\end{aligned}$$ Thus, the first term in also converges to zero as $T\rightarrow\infty$ when (ii) holds. For the second term, observe that $$\begin{aligned} \left\{N_a(T)<b_a(T),L_a(T)> d(T)\right\}&\subseteq \left\{\max_{n\le b_a(T)} \frac{L_{a,n}}{b_a(T)}> \frac{d(T)}{b_a(T)}\right\} \\ &= \left\{\max_{n\le b_a(T)} \frac{L_{a,n}}{b_a(T)}> \frac{1-\delta/2}{1-\delta}\operatorname{KL}(\nu_a,\nu_a')>\operatorname{KL}(\nu_a,\nu_a')\right\}.\end{aligned}$$ By the strong law of large numbers, $b_a(T)^{-1}L_{a,\lfloor b_a(T)\rfloor}\rightarrow \operatorname{KL}(\nu_a,\nu_a')$ almost surely under $\nu_a$. Further, $\max_{n\le b_a(T)} b_a(T)^{-1}L_{a,n}\rightarrow \operatorname{KL}(\nu_a,\nu_a')$ almost surely as $T\rightarrow\infty$. It follows that the second term in converges to zero as $T\rightarrow\infty$ so that $$\begin{aligned} \operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)<(1-\delta) \frac{\log(T)}{\operatorname{KL}(\nu_a,\nu_a')}\right\}\rightarrow 0. \label{eq:problbnup}\end{aligned}$$ For convenience, we let $\mathcal{K}\equiv \operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^{\star})$ in what follows. By the definition of the infimum, for every $\epsilon>0$ there exists some $\nu_a'$ such that $\mathcal{K}+\epsilon>\operatorname{KL}(\nu_a,\nu_a')$. This proves . If $a\in\underline{\cN}$ so that $\mathcal{K}>0$, then take $\epsilon = \left[(1-\delta)^{-1/2}-1\right]\mathcal{K}$ and write $$\begin{aligned} \operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)<(1-\delta)^{3/2} \frac{\log(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)}\right\}\rightarrow 0. \end{aligned}$$ Applying the above to $\delta' = 1 - (1-\delta)^{2/3}$ (such that $(1-\delta')^{3/2}=(1-\delta)$) yield the result for $a\in \underline{\cN}$. For $a\in \underline{\cN}$, it also follows that for all $\delta \in (0,1)$ one has $$\begin{aligned} \operatorname{\mathbb{E}}[N_a(T)]\ge \frac{(1-\delta)\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)}\operatorname{\mathbb{P}}_{\mathcal{V}}\left\{N_a(T)\ge (1-\delta) \frac{\log T }{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)}\right\} \underset{T \rightarrow \infty}{\sim} \frac{(1-\delta)\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)},\end{aligned}$$ which yields , letting $\delta$ go to zero. Supplementary Proofs for KL-UCB {#app:klucbproof} =============================== \[lem:term1\] Fix an $a\in\{1,\ldots,K\}$ and a fixed $\mu^\dagger$ (not relying on $T$) with $\mu_a<\mu^\dagger$. In the setting of Theorem \[thm:expfam\] with $\rho^\dagger=\mu^\dagger/{c}_a$ or in the setting of Theorem \[thm:finsup\] with $\rho^\dagger=\left[1-\log(T)^{-1/5}\right]\mu^\dagger/{c}_a$, it holds that $$\begin{aligned} \sum_{n=b(T) + 1}^{\infty} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{{c}_a \rho^\dagger,f(T)/n}\right\}&= o(\log T),\end{aligned}$$ where $b(T)$ is any number satisfying $$\begin{aligned} b(T)&\ge \left\lceil\frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)}\right\rceil.\end{aligned}$$ An explicit finite sample bound on the $o(\log T)$ term can be found in [@Cappeetal2013b]. In the setting of Theorem \[thm:expfam\], Equation 25 in [@Cappeetal2013b] gives the result for $\rho^\dagger=\mu^\dagger/{c}_a$. We refer the readers to that equation for the explicit finite sample bound that we are summarizing with little-oh notation. In the setting of Theorem \[thm:finsup\], Equation 33 combined with the unnumbered equation preceding Equation 36 in Section B.4 of [@Cappeetal2013b] gives the result for $\rho^\dagger=\left[1-\log(T)^{-1/5}\right]\mu^\dagger/{c}_a$. An explicit finite sample upper bound on this quantity can be found in Section B.4 of [@Cappeetal2013b]. \[lem:term2astar\] Fix an arm $a^\star\in\cS$. In the setting of Theorem \[thm:expfam\] with $\rho^\dagger\le\rho_{a^\star}$ or in the setting of Theorem \[thm:finsup\] with $\rho^\dagger\le\left[1-\log(T)^{-1/5}\right]\rho_{a^\star}$, it holds that $$\begin{aligned} \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{{c}_{a^\star}\rho^\dagger\ge U_{a^\star}(t)\right\}&= o(\log T).\end{aligned}$$ Explicit finite sample constants can be found in the proof. In the setting of Theorem \[thm:expfam\], it holds that $\{{c}_{a^\star}\rho^\dagger\ge U_{a^\star}(t)\}\subseteq \{\mu_{a^\star}\ge U_{a^\star}(t)\}$. Hence, $$\begin{aligned} \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{{c}_{a^\star}\rho^\dagger\ge U_{a^\star}(t)\right\}&\le \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\{\mu_{a^\star}\ge U_{a^\star}(t)\}.\end{aligned}$$ Furthermore, $$\begin{aligned} \{\mu_{a^\star}\ge U_{a^\star}(t)\}&\subseteq \bigcup_{n=1}^{t-K+1}\left\{\mu_{a^\star}\ge \hat{\mu}_{a^\star,n},\operatorname{KL}(\hat{\mu}_{a^\star,n},\mu_{a^\star})\ge\frac{f(t)}{n}\right\}.\end{aligned}$$ Using the above, Equations 17 and 18 in [@Cappeetal2013b] show that $\sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\{\mu_{a^\star}\ge U_{a^\star}(t)\}$ is upper bounded by $3 + 4e\log\log T = o(\log T)$ provided $T\ge 3$. In the setting of Theorem \[thm:finsup\], it holds that $\{{c}_{a^\star}\rho^\dagger\ge U_{a^\star}(t)\}\subseteq \{\left[1-\log(T)^{-1/5}\right]\mu_{a^\star}\ge U_{a^\star}(t)\}$. Hence, $$\begin{aligned} \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{{c}_{a^\star}\rho^\dagger\ge U_{a^\star}(t)\right\}&\le \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{\left[1-\log(T)^{-1/5}\right]\mu_{a^\star}\ge U_{a^\star}(t)\right\}. \label{eq:PmudaggtUastar}\end{aligned}$$ Let $\epsilon\equiv \log(T)^{-1/5} \mu_{a^\star}>0$. Arguments given in Section B.2 of [@Cappeetal2013b] show that $$\begin{aligned} \left\{\mu_{a^\star}-\epsilon\ge U_{a^\star}(t)\right\}&\subseteq \left\{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\hat{\nu}_{a^\star}(t),\mu_{a^\star}-\epsilon)\ge \frac{f(t)}{N_{a^\star}(t)}\right\} \\ &\subseteq \left\{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\hat{\nu}_{a^\star}(t),\mu_{a^\star})\ge \frac{f(t)}{N_{a^\star}(t)} + \frac{\epsilon^2}{2}\right\} \\ &\subseteq \cup_{n=1}^{t-K+1} \left\{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\hat{\nu}_{a^\star,n},\mu_{a^\star})\ge \frac{f(t)}{n}+\frac{\epsilon^2}{2}\right\}.\end{aligned}$$ The remainder of the proof is now the same as in [@Cappeetal2013b]. In particular, their Equation 26 combined with the bounds given after their Equation 35 shows that the right-hand side of is upper bounded by $36\mu_{a^\star}^{-4}\left(2 + \log\log T\right)\left(\log T\right)^{4/5}=o(\log T)$. Fix $a\in\underline{\cU}\cup\overline{\cU}$. For ease of notation, we analyze $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]$ rather than $\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(G)})]$, but for fixed $G<\infty$ there is no loss of generality in doing so. If $a\in\underline{\cU}$, then let $\mu^{\dagger}= {c}_a\rho_{a^\star}$, and otherwise, fix $\mu^\dagger\in(\mu_a,\mu_{+})$. Let $\rho^\dagger\equiv\mu^\dagger/{c}_a$ (setting of Theorem \[thm:expfam\]) or $\rho^\dagger\equiv\left[1-\log(T)^{-1/5}\right]\mu^\dagger/{c}_a$ (setting of Theorem \[thm:finsup\]). Note that $\rho^\dagger<\mu_{+}/{c}_a$. Analogous arguments to those used for show that $$\begin{aligned} &\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{U_{a^\star}(t)}{{c}_{a^\star}}<\hat{\rho}^\star(t)\right\} \nonumber \\ &\subseteq\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{U_{a^\star}(t)}{{c}_{a^\star}}<\hat{\rho}^\star(t),\rho^\dagger\ge \frac{U_{a}(t)}{{c}_a}\right\}\cup\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{U_{a^\star}(t)}{{c}_{a^\star}}<\hat{\rho}^\star(t), \rho^\dagger<\frac{U_{a}(t)}{{c}_a}\right\} \nonumber \\ &\subseteq \left\{\rho^\dagger\ge \frac{U_{a^\star}(t)}{{c}_{a^\star}}\right\}\cup \left\{a\in\operatorname{\mathcal{A}}(t+1), \rho^\dagger<\frac{U_a(t)}{{c}_a}\right\}. \label{eq:Atp1atilde}\end{aligned}$$ Let $$\begin{aligned} b_a^{a^\star}(T)&\equiv \left\lceil\frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)}\right\rceil.\end{aligned}$$ Similarly to , we have that $$\begin{aligned} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)]&\le \frac{f(T)}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu_{a^\star})} + \sum_{n=b_a^{a^\star}(T) + 1}^{\infty} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{{c}_a\rho^\dagger,f(T)/n}\right\} + \sum_{t=K}^{T-1} \operatorname{\mathbb{P}}\left\{\rho^\dagger\ge \frac{U_{a^\star}(t)}{{c}_{a^\star}}\right\} + 2.\end{aligned}$$ By Lemmas \[lem:term1\] and \[lem:term2astar\], $$\begin{aligned} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le \frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)} + o(\log T). \label{eq:TnotTG}\end{aligned}$$ In what follows we refer to this $o(\log T)$ term as $r(T,\mu^\dagger)$, where we note that $r(T,\mu^\dagger)/\log T\rightarrow 0$ for each fixed $\mu^\dagger\in(\mu_a,\mu_{+})$. If $a\in\overline{\cU}$, we will obtain our result by letting $\mu^\dagger\rightarrow \mu_{+}$. Thus, there exists a sequence $\mu^\dagger(T)\rightarrow\mu_{+}$ such that $r(T,\mu^\dagger(T))/\log T\rightarrow 0$. Noting $\liminf_{\mu^\dagger\rightarrow\mu_{+}} \operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger) = +\infty$ in the setting of both theorems, we see that $$\begin{aligned} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le \frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger(T))} + r(T,\mu^\dagger(T)) = o(\log T).\end{aligned}$$ This is the desired result when $a\in\overline{\cU}$. If, instead, $a\in\underline{\cU}$, then replacing $T$ by $T^{(G)}$ in (for $T$ large enough so that $T^{(G)}>1$), and recalling that $\mu^\dagger={c}_a \rho_{a^\star}$ when $a\in\underline{\cU}$, gives the desired result. Fix $g\in\mathbb{N}$, $a\in\underline{\cU}\subset\cM\cup\cN$, and $T^{(g)}$ such that $T^{(g)}>1$. In the setting of Theorem \[thm:expfam\] let $\rho^\dagger=\rho_{a^\star}$, and in the setting of Theorem \[thm:finsup\] let $\rho^\dagger=[1-\log(T)^{-1/5}]\rho_{a^\star}$. By and the fact that $\{\rho^\dagger<U_a(t)/{c}_a\} = \left\{\hat{\nu}_{a,N_a(t)}\in\mathcal{C}_{{c}_a \rho^\dagger,f(t)/N_a(t)}\right\}$, $$\begin{aligned} &\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(g-1)})-M_a^{a^\star}(T^{(g)})] \\ &\le \sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathbb{P}}\left\{\rho^\dagger\ge \frac{U_{a^\star}(t)}{{c}_{a^\star}}\right\} + \sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \hat{\nu}_{a,N_a(t)}\in\mathcal{C}_{{c}_a \rho^\dagger,f(t)/N_a(t)}\right\}.\end{aligned}$$ The first term in the right hand side is upper bounded by the same sum from $t=K$ to $T-1$, and is thus $o(\log T)$ by Lemma \[lem:term2astar\]. For the second term, let $b_a'(T,g)\equiv \lceil(1-\delta)\frac{f(T^{(g)})}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)}\rceil$ if $a\in\cN$ and let $b_a'(T,g)\equiv \lceil\frac{f(T^{(g)})}{(1-\delta)\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})}\rceil$ if $a\in\cM$. Similar arguments to those used to derive in Section \[sec:suboptrare\] show that, for $T$ large enough so that $T^{(g)}\ge K$, $$\begin{aligned} \sum_{t=T^{(g)}}^{T^{(g-1)}-1} &\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \hat{\nu}_{a,N_a(t)}\in\mathcal{C}_{{c}_a \rho^\dagger,f(t)/N_a(t)}\right\} \\ \le&\, \sum_{n=1}^{T^{(g-1)}-K}\;\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{{c}_a \rho^\dagger,f(T^{(g-1)})/n}, \tau_{a,n+1}=t+1\right\}.\end{aligned}$$ We split the sum over $n$ into a sum $S_1$ from $n=1$ to $b_a'(T,g)$ and a sum $S_2$ from $n=b_a'(T,g)+1$ to $T^{(g-1)}-K$. For the latter sum, the fact that, for each $n$, $\tau_{a,n+1}=t+1$ for at most one $t$ in a given interval, yields that $$\begin{aligned} S_2&\le \sum_{n= b_a'(T,g)+1}^{T^{(g-1)}-K} \operatorname{\mathbb{P}}\left\{\hat{\nu}_{a,n}\in\mathcal{C}_{\mu^\dagger,f(T^{(g-1)})/n}\right\}.\end{aligned}$$ If $a\in\cN$, then $\delta$ satisfying yields that $b_a'(T,g)> \frac{f(T^{(g-1)})}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})}$, and so the above sum is $o(\log T)$ by Lemma \[lem:term1\]. If $a\in\cM$, then $b_a'(T,g)=\lceil\frac{f(T^{(g-1)})}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})}\rceil$, and so again the above sum is $o(\log T)$. We now bound $S_1$. Note that if $N_a(T^{(g)}-1)>b_a'(T,g)$, then, for every $n\le b_a'(T,g)$, $\tau_{a,n+1}< T^{(g)}$ and $S_1=0$ (the sum over $t$ is void). Therefore, $$\begin{aligned} S_1&\le \sum_{n=1}^{b_a'(T,g)} \operatorname{\mathbb{P}}\left\{N_a(T^{(g)}-1)\le b_a'(T,g)\right\} = b_a'(T,g)\operatorname{\mathbb{P}}\left\{N_a(T^{(g)}-1)\le b_a'(T,g)\right\}.\end{aligned}$$ From Lemma \[lem:cons\], KL-UCB is uniformly efficient. Thus, by , for any $a\in\cM$ one has $$\lim_{T\rightarrow \infty} \operatorname{\mathbb{P}}\left(N_a(T^{(g)}-1) \leq \frac{2\log(T^{(g)})}{(1-\delta)\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})}\right) = 0,$$ where we use the fact that $\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)=0$ and choose $\epsilon =(1-\delta)\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho_{a^\star})/2 >0$. This yields that $\operatorname{\mathbb{P}}\left\{N_a\left(T^{(g)}-1\right)< b_a'(T,g)\right\}\rightarrow 0$ as $T\rightarrow\infty$ and $S_1=o(\log T)$. If $a\in\cN$, then Lemma \[lem:cons\] and from Lemma \[lem:lb\] yield that $$\begin{aligned} \operatorname{\mathbb{P}}\left\{N_a\left(T^{(g)}-1\right)< (1-\delta)\frac{\log T^{(g)}}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)}\right\}\rightarrow 0\textnormal{ as $T\rightarrow 0$.}\end{aligned}$$ The fact that $\lim_T f(T)/\log T= 1$ shows that $b_a'(T,g)=(1-\delta)\frac{\log T^{(g)}}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,{c}_a \rho^\star)} + o(\log T)$. Plugging this into from Lemma \[lem:lb\] (which holds for *every* $\delta$ between $0$ and $1$) yields that $\operatorname{\mathbb{P}}\left\{N_a\left(T^{(g)}-1\right)< b_a'(T,g)\right\}\rightarrow 0$ as $T\rightarrow\infty$. It follows that $S_1=o(\log T)$. We have then shown that $\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(g-1)})-M_a^{a^\star}(T^{(g)})]= o(\log T)$ for each $a\in\underline{\cU}\subset\cM\cup\cN$ and each $g\le G$. As Term B is a sum of finitely many such terms, Term B is $o(\log T)$. Supplementary Proofs for Thompson Sampling {#app:thomproof} ========================================== We begin with a lemma. \[lem:betabinomial\] For any fixed real number $L$, arm $a$, $\mu_a<\mu^\dagger<\theta^\dagger$, and $t\ge 1$, $$\begin{aligned} I\left\{\hat{\mu}_a(t)\le \mu^\dagger,N_a(t)\ge L\right\}\operatorname{\mathbb{P}}\left\{\theta_a(t)>\theta^\dagger\middle|\mathcal{F}(t)\right\}\le e^{-(L+1)\operatorname{KL}(\mu^\dagger,\theta^\dagger)}.\end{aligned}$$ From Fact 3 in , $$\begin{aligned} \operatorname{\mathbb{P}}\left(\left.\theta_a(t)>\theta^\dagger\right|\mathcal{F}(t)\right)&= \operatorname{\mathbb{P}}\left(\left.\sum_{n=1}^{N_a(T)+1} Z_n\le \sum_{n=1}^{N_a(T)} \operatorname{\mathds{1}}\{X_{a,n}=0\} \right|\mathcal{F}(t)\right),\end{aligned}$$ where $\{Z_n\}$ is an i.i.d. sequence (independent of all other quantities under consideration) of Bernoulli random variables with mean $\theta^\dagger$. Upper bounding the right-hand side yields $$\begin{aligned} \operatorname{\mathbb{P}}\left(\left.\theta_a(t)>\theta^\dagger\right|\mathcal{F}(t)\right)&\le \operatorname{\mathbb{P}}\left(\left.\frac{1}{N_a(T)+1}\sum_{n=1}^{N_a(T)+1} Z_n\le \hat{\mu}_a(T) \right|\mathcal{F}(t)\right).\end{aligned}$$ Using that $\mu^\dagger<\theta^\dagger$, the Chernoff-Hoeffding bound gives that $\operatorname{\mathbb{P}}\left(\left.\theta_a(t)>\theta^\dagger\right|\mathcal{F}(t)\right)$ is no larger than $e^{-[N_a(t)+1]\operatorname{KL}(\hat{\mu}_a(t),\theta^\dagger)}$. Multiplying the left-hand side by $I\left\{\hat{\mu}_a(t)\le \mu^\dagger,N_a(t)\ge L\right\}$, this yields the upper bound $e^{-(L+1)\operatorname{KL}(\mu^\dagger,\theta^\dagger)}$. Let $\tilde{\theta}_{\tilde{a}}(t)=\theta_{\tilde{a}}(t)$ for all $\tilde{a}\not=a^\star$ and let $\tilde{\theta}_{a^\star}(t)=-\infty$. Define the event $B\equiv \left\{\rho^\star(\tilde{\theta}_a(t) : a=1,\ldots,K+1)<\theta_a(t)/{c}_a\le \rho^\ddagger\right\}$. Observe that $$\begin{aligned} \operatorname{\mathbb{P}}&\left\{\hat{\rho}^\star(t)\le \frac{\theta_a(t)}{{c}_a}\le \rho^\ddagger,\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}< \hat{\rho}^\star(t)\middle| \mathcal{F}(t)\right\} \nonumber \\ &= \operatorname{\mathbb{P}}\left(\left\{\hat{\rho}^\star(t)\le \frac{\theta_a(t)}{{c}_a},\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}< \hat{\rho}^\star(t)\right\}\cap B\middle| \mathcal{F}(t)\right) \nonumber \\ &\le \operatorname{\mathbb{P}}\left(\left\{\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}\le\rho^\ddagger\right\}\cap B\middle| \mathcal{F}(t)\right). \label{eq:firstordering}\end{aligned}$$ The event $\{\theta_{a^\star}(t)/{c}_{a^\star}> \rho^\ddagger\}$ is independent of the event $B$ conditional on $\mathcal{F}(t)$, and so the fact that $\left\{\theta_{a^\star}(t)/{c}_{a^\star}> \rho^\ddagger\right\}\cap B\subseteq\left\{\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^{\star}(t)\right\}$ yields $$\begin{aligned} \operatorname{\mathbb{P}}(B|\mathcal{F}(t))\le \frac{\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^{\star}(t)\right|\mathcal{F}(t)\right)}{\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)/{c}_{a^\star}> \rho^\ddagger\right|\mathcal{F}(t)\right)}.\end{aligned}$$ We note that $\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)>{c}_{a^\star}\rho^\ddagger\right|\mathcal{F}(t)\right)$ is positive (a beta distribution with at least one success is larger than ${c}_{a^\star}\rho^\ddagger<1$ with positive probability). Finally, since $a\in\operatorname{\mathcal{A}}(t+1)$ implies that $\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^{\star}(t)$, yields $$\begin{aligned} \operatorname{\mathbb{P}}&\left(\left.a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)/{c}_a\le \rho^\ddagger,\theta_{a^\star}(t)/{c}_{a^\star}< \hat{\rho}^\star(t)\right|\mathcal{F}(t)\right) \\ &\le \operatorname{\mathbb{P}}\left(\left\{\theta_{a^\star}(t)/{c}_{a^\star}\le\rho^\ddagger\right\}\cap B\middle| \mathcal{F}(t)\right) \\ &= \operatorname{\mathbb{P}}\left(\theta_{a^\star}(t)/{c}_{a^\star}\le\rho^\ddagger\middle| \mathcal{F}(t)\right)\operatorname{\mathbb{P}}\left(\left.B\right|\mathcal{F}(t)\right) \\ &\le \operatorname{\mathbb{P}}\left(\theta_{a^\star}(t)/{c}_{a^\star}\le\rho^\ddagger\middle| \mathcal{F}(t)\right)\frac{\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)/{c}_{a^\star}\ge \hat{\rho}^{\star}(t)\right|\mathcal{F}(t)\right)}{\operatorname{\mathbb{P}}\left(\left.\theta_{a^\star}(t)/{c}_{a^\star}> \rho^\ddagger\right|\mathcal{F}(t)\right)}.\end{aligned}$$ Using , one can write $$\begin{aligned} \operatorname{\mathbb{E}}&\left[\sum_{t=0}^{T-1} \frac{1-p_{a^\star}^{\rho^\ddagger}(t)}{p_{a^\star}^{\rho^\ddagger}(t)}\operatorname{\mathbb{P}}\left(\frac{\theta_{a^\star}(t)}{{c}_{a^\star}}\ge \hat{\rho}^{\star}(t)\middle|\mathcal{F}(t)\right)\right] \\ &\le \mathring{q}_{a^\star}^{-1} \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \frac{1-p_{a^\star}^{\rho^\ddagger}(t)}{p_{a^\star}^{\rho^\ddagger}(t)}\operatorname{\mathbb{P}}\left(a^\star\in\operatorname{\mathcal{A}}(t+1)\middle|\mathcal{F}(t)\right)\right] \\ &= \mathring{q}_{a^\star}^{-1} \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \frac{1-p_{a^\star,N_{a^\star}(t)}}{p_{a^\star,N_{a^\star}(t)}}\operatorname{\mathds{1}}\left\{a^\star\in\operatorname{\mathcal{A}}(t+1)\right\}\right] \\ &= \mathring{q}_{a^\star}^{-1} \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \sum_{n=0}^{T-1} \frac{1-p_{a^\star,n}^{\rho^\ddagger}}{p_{a^\star,n}^{\rho^\ddagger}}\operatorname{\mathds{1}}\left\{\tau_{a^\star,n+1}=t+1\right\}\right] \\ &\le \mathring{q}_{a^\star}^{-1} \operatorname{\mathbb{E}}\left[\sum_{n=0}^{T-1} \frac{1-p_{a^\star,n}^{\rho^\ddagger}}{p_{a^\star,n}^{\rho^\ddagger}}\right],\end{aligned}$$ where the latter inequality holds because $\tau_{a^\star,n+1}=t+1$ for at most one $t$ in $\{0,\ldots,T-1\}$. Let $L^\dagger(T)\equiv \frac{\log T}{\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger)}$. We have that $$\begin{aligned} \sum_{t=0}^{T-1}& \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a \rho^\ddagger,\hat{\mu}_a(t)\le{c}_a \rho^\dagger\right\} \nonumber \\ =&\, \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\left\{N_a(t)< L^\dagger(T)-1,\hat{\mu}_a(t)\le{c}_a \rho^\dagger\right\}\operatorname{\mathbb{P}}\left(\left.a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a \rho^\ddagger\right|\mathcal{F}(t)\right)\right] \nonumber \\ &+ \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\left\{N_a(t)\ge L^\dagger(T)-1,\hat{\mu}_a(t)\le{c}_a \rho^\dagger\right\}\operatorname{\mathbb{P}}\left(\left.a\in\operatorname{\mathcal{A}}(t+1),\theta_a(t)>{c}_a \rho^\ddagger\right|\mathcal{F}(t)\right)\right] \nonumber \\ \le&\, \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\left\{N_a(t)< L^\dagger(T)-1\right\}\operatorname{\mathbb{P}}\left(\left.a\in\operatorname{\mathcal{A}}(t+1)\right|\mathcal{F}(t)\right)\right] \nonumber \\ &+ \operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\left\{N_a(t)\ge L^\dagger(T)-1,\hat{\mu}_a(t)\le{c}_a \rho^\dagger\right\}\operatorname{\mathbb{P}}\left(\left.\theta_a(t)>{c}_a \rho^\ddagger\right|\mathcal{F}(t)\right)\right]. \label{eq:thompsconstbd}\end{aligned}$$ The first term in the right hand side equals $\operatorname{\mathbb{E}}\left[\sum_{t=0}^{T-1} \operatorname{\mathds{1}}\left\{N_a(t)< L^\dagger(T)-1,a\in\operatorname{\mathcal{A}}(t+1)\right\}\right]$. Hence it is no larger than $L^\dagger(T)-1$ (the sum has at most $L^\dagger(T)-1$ nonzero terms). For the second term, Lemma \[lem:betabinomial\] yields $$\begin{aligned} &\operatorname{\mathds{1}}\left\{\hat{\mu}_a(t)<{c}_a \rho^\dagger,N_a(T)\ge L^\dagger(T)-1\right\}\operatorname{\mathbb{P}}\left(\left.\theta_a(t)>{c}_a \rho^\ddagger\right|\mathcal{F}(t)\right)\le e^{-L^\dagger(T)\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger)}= T^{-1}.\end{aligned}$$ It follows that the second term on the right of is upper bounded by $\sum_{t=0}^{T-1} T^{-1} = 1$. This completes the proof. Fix $a\in\underline{\cU}\cup\overline{\cU}$ and $\epsilon\in(0,1)$. For ease of notation, we analyze $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]$ rather than $\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(G)})]$, but for fixed $G<\infty$ there is no loss of generality in doing so. If $a\in\underline{\cU}$, then let $\mu^\dagger= {c}_a\rho_{a^\star}$, and otherwise, fix $\mu^\dagger\in(\mu_a,\mu_{+})$. Let $\rho^\dagger$ and $\rho^\ddagger$ satisfy $\rho_a<\rho^\dagger<\rho^\ddagger<\mu^\dagger/{c}_a$ (exact quantities to be specified at the end of the proof). Note that $$\begin{aligned} &\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_{a^\star}(t)}{{c}_{a^\star}}<\hat{\rho}^\star(t)\right\} \\ &\subseteq \left\{a\in\operatorname{\mathcal{A}}(t+1),\frac{\theta_a(t)}{{c}_a}\le\rho^\ddagger, \frac{\theta_{a^\star}(t)}{{c}_{a^\star}}<\hat{\rho}^\star(t)\right\}\cup \left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger\right\}.\end{aligned}$$ Recalling that $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]$ is equal to $\sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \theta_{a^\star}(t)/{c}_{a^\star}<\hat{\rho}^\star(t)\right\}$, the above yields $$\begin{aligned} \operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le&\, \sum_{t=0}^{T-1}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\frac{\theta_a(t)}{{c}_a}\le\rho^\ddagger, \frac{\theta_{a^\star}(t)}{{c}_{a^\star}}<\hat{\rho}^\star(t)\right\} \nonumber \\ &+ \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\hat{\mu}_a(t)}{{c}_a}>\rho^\dagger\right\} \nonumber \\ &+ \sum_{t=0}^{T-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger,\frac{\hat{\mu}_a(t)}{{c}_a}\le\rho^\dagger\right\}. \label{eq:termsitoiiiagain}\end{aligned}$$ Note that the right-hand side of the above is almost identical to . Note that all of the results used to control the three terms on the right-hand side of hold for any $a$ with $\rho_a\le\rho^\star$ provided $\rho_a<\rho^\dagger<\rho^\ddagger<\mu^\dagger/{c}_a$. In particular, we are referring to Lemma \[lem:suboptasopt\], , Lemma \[lem:carefulbinomialbound\], , Lemma \[lem:termii\], and Lemma \[lem:termiii\]. Hence, $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le \frac{\log T}{\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger)} + o(\log T)$. Selecting $\rho^\dagger$ and $\rho^\ddagger$ as in the proof of and from Theorem \[thm:thom\] yields $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le (1+\epsilon)^2\frac{\log T}{\operatorname{KL}(\mu_a,\mu^\dagger)} + o(\log T)$. As $\epsilon$ was arbitrary, dividing both sides by $\log T$ and taking $T\rightarrow\infty$ followed by $\epsilon\rightarrow 0$ yields that $\operatorname{\mathbb{E}}[M_a^{a^\star}(T)]\le \frac{\log T}{\operatorname{KL}(\mu_a,\mu^\dagger)} + o(\log T)$. If $a\in\underline{\cU}$, then replacing $T$ by $T^{(G)}$ (for $T$ large enough so that $T^{(G)}>1$) gives the desired $\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(G)})]\le (1-\delta)^G\frac{\log T}{\operatorname{\mathcal{K}_{\textnormal{inf}}}(\nu_a,\mu^\dagger)} + o(\log T)$ in light of the fact that $\mu^\dagger={c}_a\rho_{a^\star}$. If, on the other hand, $a\in\overline{\cU}$, then the same arguments used to conclude the $a\in\overline{\cU}$ result in the proof of Lemma \[lem:TermA\] for KL-UCB, namely selecting an appropriate sequence $\mu^\dagger(T)\rightarrow\mu_{+}$, can be used to show that $\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(G)})]=o(\log T)$. Fix $g\in\mathbb{N}$, an arm $a\in\underline{\cU}\subset\cM\cup\cN$, and $T^{(g)}$ such that $T^{(g)}>1$. Let $\rho^\dagger$ and $\rho^\ddagger$ satisfy $\rho_a<\rho^\dagger<\rho^\ddagger<\rho_{a^\star}$ and $\operatorname{KL}({c}_a \rho^\dagger,{c}_a \rho^\ddagger)\ge (1-\delta)\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})$. By the same arguments used for , $$\begin{aligned} \operatorname{\mathbb{E}}&[M_a^{a^\star}(T^{(g-1)})-M_a^{a^\star}(T^{(g)})] \nonumber \\ \le&\, \sum_{t=T^{(g)}}^{T^{(g-1)}-1}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1),\frac{\theta_a(t)}{{c}_a}\le\rho^\ddagger,\frac{\theta_{a^\star}(t)}{{c}_{a^\star}} < \hat{\rho}^\star(t)\right\} \nonumber \\ &+ \sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\hat{\mu}_a(t)}{{c}_a}>\rho^\dagger\right\} \nonumber \\ &+ \sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger,\frac{\hat{\mu}_a(t)}{{c}_a}\le\rho^\dagger\right\}. \label{eq:thomNdiff}\end{aligned}$$ The first two sums are trivially upper bounded by the sums from $t=0$ to $T-1$, and thus are $o(\log T)$ by Lemma \[lem:suboptasopt\], , Lemma \[lem:carefulbinomialbound\], , and Lemma \[lem:termii\]. If $a\in\cN$, then let $b_a(T,g)\equiv(1-\delta)\frac{\log T^{(g)}}{\operatorname{KL}(\mu_a,{c}_a \rho^\star)}$, and if $a\in\cM$ then let $b_a(T,g)\equiv\frac{\log T^{(g)}}{(1-\delta)\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})}$. We have that $$\begin{aligned} &\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger,\frac{\hat{\mu}_a(t)}{{c}_a}\le\rho^\dagger\right\} \nonumber \\ =&\, \operatorname{\mathbb{E}}\left[\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathds{1}}\left\{\frac{\hat{\mu}_a(t)}{{c}_a}\le\rho^\dagger,N_a(t)\ge b_a(T,g)\right\}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger\middle|\mathcal{F}(t)\right\}\right] \nonumber \\ &+ \operatorname{\mathbb{E}}\left[\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathds{1}}\left\{\frac{\hat{\mu}_a(t)}{{c}_a}\le\rho^\dagger,N_a(t)< b_a(T,g)\right\}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger\middle|\mathcal{F}(t)\right\}\right]. \label{eq:thomNdecomp}\end{aligned}$$ If $a\in\cN$, then Lemma \[lem:betabinomial\] and $\operatorname{KL}({c}_a\rho^\dagger,{c}_a\rho^\ddagger)\ge (1-\delta)\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})$ yield that the first term on the right is upper bounded by $$\begin{aligned} \sum_{t=T^{(g)}}^{T^{(g-1)}-1} &\exp\left[-(1-\delta)^2\frac{\log T^{(g-1)}}{\operatorname{KL}(\mu_a,{c}_a \rho^\star)}\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})\right] \\ &\le T^{(g-1)}\exp\left[-(1-\delta)^2\frac{\log T^{(g-1)}}{\operatorname{KL}(\mu_a,{c}_a \rho^\star)}\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})\right]\le 1,\end{aligned}$$ where the second inequality holds because $\delta$ satisfies . If $a\in\cM$, then we instead have that this term is no larger than $$\begin{aligned} \sum_{t=T^{(g)}}^{T^{(g-1)}-1} \exp\left[-\frac{\log T^{(g-1)}}{\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})}\operatorname{KL}(\mu_a,{c}_a \rho_{a^\star})\right]\le 1.\end{aligned}$$ For the second term in , note that $$\begin{aligned} \operatorname{\mathbb{E}}&\left[\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathds{1}}\left\{\frac{\hat{\mu}_a(t)}{{c}_a}\le\rho^\dagger,N_a(t)< b_a(T,g)\right\}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1), \frac{\theta_a(t)}{{c}_a}>\rho^\ddagger\middle|\mathcal{F}(t)\right\}\right] \\ &\le \operatorname{\mathbb{E}}\left[\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathds{1}}\left\{N_a(t)< b_a(T,g)\right\}\operatorname{\mathbb{P}}\left\{a\in\operatorname{\mathcal{A}}(t+1)\middle|\mathcal{F}(t)\right\}\right] \\ &= \operatorname{\mathbb{E}}\left[\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathds{1}}\left\{N_a(t)< b_a(T,g),a\in\operatorname{\mathcal{A}}(t+1)\right\}\right] \\ &= \operatorname{\mathbb{E}}\left[\operatorname{\mathds{1}}\left\{N_a\left(T^{(g)}\right)< b_a(T,g)\right\}\sum_{t=T^{(g)}}^{T^{(g-1)}-1} \operatorname{\mathds{1}}\left\{N_a(t)< b_a(T,g),a\in\operatorname{\mathcal{A}}(t+1)\right\}\right] \\ &\le b_a(T,g)\operatorname{\mathbb{P}}\left\{N_a\left(T^{(g)}\right)< b_a(T,g)\right\},\end{aligned}$$ where the final inequality uses that the sum inside the expectation is at most $b_a(T,g)$. By the uniform efficiency of the algorithm established in Lemma \[lem:cons\] and from Lemma \[lem:lb\], the probability in the final inequality is $o(1)$, and thus the above is $o(b_a(T,g))=o(\log T)$. Thus is $o(\log T)$. Plugging this into yields that $\operatorname{\mathbb{E}}[M_a^{a^\star}(T^{(g-1)})-M_a^{a^\star}(T^{(g)})]=o(\log T)$ for each $a\in\underline{\cU}\subset\cM\cup\cN$ and each $g\le G$. As Term B is a sum of finitely many such terms, Term B is $o(\log T)$. 0.2in [^1]: It is *a priori* possible that $\tau_{a,n}=\infty$ for all $n$ large enough (though, as we showed in Section \[sec:lb\], not possible for any reasonable algorithm). To deal with this case, let $X_{a,n}\equiv Y_a(\tau_{a,n})$ denote the $n^{\textnormal{th}}$ draws from $\nu_a$ for all $\tau_{a,n}<\infty$ and let $\{X_{a,n}\}_{n : \tau_{a,n}=\infty}$ denote an i.i.d. sequence independent of $\{X_{a,n}\}_{n : \tau_{a,n}<\infty}$.

--- abstract: 'The ALICE experiment at the Large Hadron Collider LHC is presently being commissioned. ALICE consists of a central barrel, a muon spectrometer and neutron calorimeters at $0^0$. Additional detectors for event classification and trigger purposes are located on both sides of the central barrel. The geometry of the ALICE detector allows the implementation of a diffractive double gap trigger by requiring two or more tracks in the central barrel but no activity in the event classification detectors. Some selected diffractive physics channels are discussed which become accessible by a double gap trigger. The interest of such diffractive measurements in proton-proton as well as in lead-lead collisions is outlined.' author: - 'R. Schicker' bibliography: - 'sample.bib' title: 'ALICE diffractive physics in p-p and Pb-Pb collisions at the LHC' --- [ address=[Phys. Inst., Philosophenweg 12, 69120 Heidelberg]{} ]{} The ALICE Experiment ==================== The ALICE experiment is designed for taking data in the high multiplicity environment of lead-lead collisions at the Large Hadron Collider (LHC)[@Alice1; @Alice2]. The ALICE experiment consists of a central barrel covering the pseudorapidity range $-0.9 < \eta < 0.9$ and a muon spectrometer in the range $-4.0<\eta<-2.4$. Additional detectors for trigger purposes and for event classification exist in the range $ -4.0 < \eta < 5.0 $. The ALICE Central Barrel ------------------------ The detectors in the ALICE central barrel track and identify hadrons, electrons and photons in the pseudorapidity range $ -0.9 < \eta < 0.9$. The magnetic field strength of allows the measurement of tracks from very low transverse momenta of about to fairly high values of about 100 GeV/c. The tracking detectors are designed to reconstruct secondary vertices resulting from decays of hyperons, D and B mesons. The main detector systems for these tasks are the Inner Tracking System, the Time Projection Chamber, the Transition Radiation Detector and the Time of Flight array. These systems cover the full azimuthal angle within the pseudorapidity range $ -0.9 < \eta < 0.9$. Additional detectors with partial coverage of the central barrel are a PHOton Spectrometer (PHOS), an electromagnetic calorimeter (EMCAL) and a High-Momentum Particle Identification Detector (HMPID). The ALICE Zero Degree Neutron Calorimeter ----------------------------------------- The Zero Degree Neutron Calorimeters (ZDC) are placed on both sides of the interaction point at a distance of 116 m[@ZDC]. The ZDC information can be used to select different diffractive topologies. Events of the types $pp \rightarrow ppX, pN^{*}X, N^{*}N^{*}X$ will have no signal, signal in one or in both of the ZDC calorimeters, respectively. Here, X denotes a centrally produced diffractive state from which the diffractive L0 trigger is derived. The ALICE diffractive gap trigger ================================= Additional detectors for event classification and trigger purposes are located on both sides of the ALICE central barrel. First, an array of scintillator detectors (V0) is placed on both sides of the central barrel. These arrays are labeled V0A and V0C on the two sides, respectively. Each of these arrays covers a pseudorapidity interval of about two units with a fourfold segmentation of half a unit. The azimuthal coverage is divided into eight segments of 45$^{0}$ degrees hence each array is composed of 32 individual counters. Second, a Forward Multiplicity Detector (FMD) is located on both sides of the central barrel. The pseudorapidity coverage of this detector is $-3.4 < \eta < -1.7$ and $1.7 < \eta < 5.1$, respectively. ![Pseudorapidity coverage of trigger detectors and of detectors in central barrel[]{data-label="fig:acc_eta"}](acc_eta.eps){height=".28\textheight"} Fig.\[fig:acc\_eta\] shows the pseudorapidity coverage of the detector systems described above. The geometry of the central barrel in conjunction with the additional detectors V0 and FMD is well suited for the definition of a rapidity gap trigger. Such a gap trigger can be defined by the requirement of signals coming from the central barrel detectors while V0 and FMD not showing any activity. This scheme requires a trigger signal from within the central barrel for L0 decision. The pixel detector of the Inner Tracking System or the Time Of Flight array can deliver such a signal[@pixel]. The high level trigger HLT has access to the information of all the detectors shown in Fig.\[fig:acc\_eta\] and will hence be able to select events with rapidity gaps in the range $-4 < \eta < -1$ and $1 < \eta < 5$. These gaps extend over seven units of pseudorapidity and are hence expected to suppress minimum bias inelastic events by many orders of magnitude. In addition to the scheme described above, the ALICE diffractive L0 trigger signal can be generated from the Neutron ZDC if no central state is present in the reaction. A L0 signal from ZDC does not arrive at the central trigger processor within the standard L0 time window. A L0 trigger from ZDC is, however, possible during special data taking runs for which the standard L0 time limit is extended. ALICE diffractive physics ========================= The tracking capabilities at very low transverse momenta in conjunction with the excellent particle identification make ALICE an unique facility at LHC to pursue a long term physics program of diffractive physics. The low luminosity of ALICE as compared to the other LHC experiments restricts the ALICE physics program to reactions with cross section at a level of a few nb per unit of rapidity. ![Rapidity and transverse momentum acceptance of the LHC experiments[]{data-label="fig:acc_all"}](acc_all.eps){height="0.25\textheight"} Fig.\[fig:acc\_all\] shows the transverse momentum acceptance of the four main LHC experiments. Not shown in this figure is the TOTEM experiment. The acceptance of the TOTEM telescopes is in the range of $ 3.1 < | \eta | < 4.7$ and $5.3 < | \eta | <6.5$. The CMS transverse momentum acceptance of about 1 GeV/c shown in Fig.\[fig:acc\_all\] represents a nominal value. The CMS analysis framework foresees the reconstruction of a few selected data samples to values as low as 0.2 GeV/c[@CMS]. Pomeron signatures in proton-proton collisions ============================================== The ALICE experiment will take data in proton-proton mode at a luminosity of Double pomeron events in proton-proton collisions as shown in Fig.\[fig:ex\_pp\] are expected to possess a few interesting properties. ![Pomeron-pomeron fusion in proton-proton[]{data-label="fig:ex_pp"}](ex_pp.eps){height="0.16\textheight"} - The production cross section of glueball states is expected to be enhanced in Pomeron fusion events as compared to minimum bias inelastic events. It will therefore be interesting to study the resonances produced in the central region when two rapidity gaps are required[@close]. - The slope $\alpha'$ of the Pomeron trajectory is rather small: $\alpha' \sim$ 0.25 GeV$^{-2}$ in DL fit and $\alpha' \sim$ 0.1 GeV$^{-2}$ in vector meson production at HERA[@DL]. These values of $\alpha'$ in conjunction with the small $t$-slope ($<$ 1 GeV$^{-2}$ ) of the triple Pomeron vertex indicate that the mean transverse momentum $k_t$ in the Pomeron wave function is relatively large $\alpha' \sim$ 1/$k_t^2$, most probably $k_t >$ 1 GeV. The transverse momenta of secondaries produced in Pomeron-Pomeron interactions are of the order of this $k_t$. Thus the mean transverse momenta of secondaries produced in Pomeron-Pomeron fusion is expected to be larger as compared to inelastic minimum bias events. - The large $k_t$ described above corresponds to a large effective temperature. A suppression of strange quark production is not expected. Hence the K/$\pi$ ratio is expected to be enhanced in Pomeron-Pomeron fusion as compared to inelastic minimum bias events[@akesson]. Similarly, the $\eta$/$\pi$ and $\eta'$/$\pi$ ratios are expected to be enhanced due to the hidden strangeness content and due to the gluon components in the Fock states of $\eta,\eta'$. Pomeron signatures in lead-lead collisions ========================================== ALICE will take data in lead-lead mode at a luminosity of $\cal{L}$ = $5 x 10^{27}$cm$^{-2}$s$^{-1}$. ![Pomeron-pomeron fusion in lead-lead[]{data-label="fig:ex_AA"}](ex_AA.eps){height="0.16\textheight"} The cross section of double pomeron induced reaction channels will be modified as compared to the proton-proton case due to absorption and screening as illustrated in The A-dependence of the cross section for specific pomeron induced channels hence reflects the contribution of these multi-pomeron diagrams. The study of multi-pomeron couplings is an important ingredient in the analysis of soft diffraction data and in the evaluation of the total pp cross section at LHC energies[@amartin]. Odderon signatures ================== Odderon signatures can be looked for in exclusive reactions where the Odderon (besides the Photon) is the only possible exchange. Diffractively produced C-odd states such as vector mesons $\phi, J/\psi, \Upsilon$ can result from Photon-Pomeron or Odderon-Pomeron exchange. Any excess beyond the Photon contribution is indication of Odderon exchange. Cross section estimates for diffractively produced $J/\psi$ in pp collisions at LHC energies were first given by Schäfer[@schaefer]. More refined calculations result in a $t$-integrated photon and Odderon contribution of $\frac{d\sigma}{dy}\mid_{y=0} \;\sim$ 15 nb and 1 nb, respectively[@bzdak]. If the diffractively produced final state is not an eigenstate of C-parity, then interference effects between photon-Pomeron and photon-Odderon amplitudes can be analyzed. ![photon-Pomeron and photon-Odderon amplitudes[]{data-label="fig:odderon_inter"}](odderon_inter.eps){height="0.146\textheight"} Fig.\[fig:odderon\_inter\] shows the photon-Pomeron and the photon-Odderon amplitudes for $q\bar{q}$ production. A study of open charm diffractive photoproduction estimates the asymmetry in fractional energy to be on the order of 15%[@brodsky]. The forward-backward charge asymmetry in diffractive production of pion pairs is calculated to be on the order of 10% for pair masses in the range I thank Otto Nachtmann and Carlo Ewerz for inspiring conversations.This work is supported in part by German BMBF under project 06HD197D. [9]{} F. Carminati et al, ALICE Collaboration, J. Phys. G [**30**]{} (2004) 1517 B. Alessandro et al, ALICE Collaboration, J. Phys. G [**32**]{} (2006) 1295 R. Arnaldi et al, Nucl. Instr. and Meth. A [**564**]{} (2006) 235 D. d’Enterria et al, Addendum CMS technical design report, J. Phys. G [**34**]{} (2007) 2307 F. Close et al, Phys. Lett. B [**477**]{} (2000) 13 A. Donnachie et al, Phys. Lett. B [**595**]{} (2004) 393 T. Åkesson et al, Nucl. Phys. B [**264**]{} (1986) 154 A. Martin, these proceedings A. Schäfer et al, Phys. Lett. B [**272**]{} (1991) 419 A. Bzdak et al, Phys. Rev. D [**75**]{} (2007) 094023 S.J. Brodsky et al, Phys. Lett. B [**461**]{} (1999) 114 P. Hägler et al, Phys. Lett. B [**535**]{} (2002) 117 I.F. Ginzburg et al, Eur. Phys. J. C[**5**]{} (2003) 02

--- abstract: 'With the increasing popularity of user equipments (UEs), the corresponding UEs’ generating big data (UGBD) is also growing substantially, which makes both UEs and current network structures struggling in processing those data and applications. This paper proposes a Near-Far Computing Enhanced C-RAN (NFC-RAN) architecture, which can better process big data and its corresponding applications. NFC-RAN is composed of near edge computing (NEC) and far edge computing (FEC) units. NEC is located in remote radio head (RRH), which can fast respond to delay sensitive tasks from the UEs, while FEC sits next to baseband unit (BBU) pool which can do other computational intensive tasks. The task allocation between NEC or FEC is introduced in this paper. Also WiFi indoor positioning is illustrated as a case study of the proposed architecture. Moreover, simulation and experiment results are provided to show the effectiveness of the proposed task allocation and architecture.' author: - 'Lianming Zhang, Kezhi Wang, *IEEE Member*, Du Xuan and Kun Yang, *IEEE Senior Member*. [^1]' bibliography: - 'bare\_jrnl.bib' title: | \ <span style="font-variant:small-caps;">Optimal Task Allocation in Near-Far Computing Enhanced C-RAN for Wireless Big Data Processing</span> --- NFC-RAN, Task Allocation, Near Far Computing, Wireless Big Data Introduction ============ With the increasing popularity of user equipments (UEs) such as smartphones and hand-held devices, more and more resource-hungry applications like high definition video gaming and virtual reality applications are developing and coming into play in our mobile devices. Due to limited resources in UEs, it is very difficult for them to compute the resource intensive applications. Moreover, UEs’ generating big data (UGBD) is growing accordingly, which poses big challenges to the existing mobile devices and wireless networks [@7295483]. Mobile edge computing (MEC) [@MEC] and offloading techniques [@6195845] have been proposed to enable UEs to send its tasks to their corresponding virtual machines. In this case, UEs’ experience will be increased substantially and their battery life will be prolonged largely. By taking advantage of the cloud technology, wireless networks have also undergone revolution recently [@7444125; @7143336]. Cloud radio access network (C-RAN) has been proposed [@China] by moving most of the signal processing tasks, which previous were done in special hardware, now to the cloud, i.e., baseband unit (BBU) pool. In this architecture, remote radio heads (RRHs) can be distributed easily according to the requirement. In C-RAN, we can dynamically and easily adjust and allocate computing resource to the wireless communications. Previous studies [@7511044; @7393804] have proposed to have mobile edge computing resource next to BBU pool to better provide service to the UEs. However, due to the transmission latency and limited bandwidth in fronthaul, the above architecture may be not beneficial to the delay sensitive tasks and big data applications. Moreover, the studies in [@7562344] have investigated the big data applications in wireless communications and shown that it is not easy to process those data in wireless networks due to the required large amount of computation resource. Reference [@7909159] has proposed a big data computing architecture for smart grid analytics. Some key technologies to enable big-data-aware wireless communication for smart grid were investigated in this paper. Reference [@7864795] has shown that most of the applications require very short response time which is typically composed of two parts: transmission (i.e. communication) and processing (i.e., computation). To better process big data applications, we propose a near-far computing enhanced C-RAN (NFC-RAN) architecture, by extending the C-RAN enhanced with mobile cloud [@7393804] with another layer of cloud computing, called near edge computing (NEC). In comparison with the mobile cloud in [@7393804], which is referred to as far edge computing (FEC), NEC is deployed in the RRHs, namely much closer to UEs. Also, in this paper, we introduce how we allocate different tasks between NEC and FEC, as proper allocation affects the performance of the whole networks and the experiences of the user significantly [@6863135]. Furthermore, in this paper, we will use the indoor positioning [@7442075; @7841583] as an example to showcase the benefits of the proposed network architecture. To fulfil indoor positioning effectively, a large amount of wireless data concerning signal strength of positioning beacons needs to be collected, transmitted and processed. Also, unlike the outdoor positioning where the distance is usually measured in terms of tens of meters, indoor positioning techniques require to capture movement at a level of no more than 2-3 meters. This requires a very short response time when processing above mentioned large amount of information and the corresponding processing has to be conducted promptly in order to show a walker’s current position in a real-time manner. The remainder of the paper is organized as follows. Section II describes the proposed NFC-RAN architecture on top of the popular C-RAN and presents various tasks involved in indoor positioning. Then the big data nature of indoor positioning problem is also discussed. Section III generalizes the task allocation issue in NFC-RAN into an optimization problem covering both the computation and communication aspects. Also, simulation result is given in this section. Section IV introduces the indoor positioning as a case study to illustrate our proposed architecture. Also, experimental results are reported in this section. Finally, conclusion remarks are given in Section V. NFC-RAN Architecture and its Application to Indoor Positioning ============================================================== NFC-RAN Architecture -------------------- Our proposed NFC-RAN is shown in Fig. \[c-ran\]. NFC-RAN is composed of the BBU pool, which is responsible of doing most of the signal processing tasks, and RRHs, which is in charge of sending and receiving data to and from UEs. RRHs can serve as the access points, which can be distributed closer to UEs as required. Also, RRHs is connected to the BBU pool through high speed fronthaul link. To support wireless big data processing, similar to [@7393804], we propose to have FEC located next to BBU pool. If FEC decides to execute the task for UEs, UEs will send all the task data to RRHs through wireless channel first, and then RRHs will forward the data to FEC via fronthaul link. This may not be beneficial to the delay sensitive tasks or the tasks involving big transmission data. Thus, in addition to FEC, we propose to have the NEC located in each RRH as well. NEC can respond to UEs’ requests much faster, due to its closer geographic location. In this architecture, UE does not have to send the data all the way to the central cloud, i.e., FEC. This can not only save the bandwidth for fronthaul, but also reduce the response time for the tasks. However, NEC may not have enough computational resource to process requests from all the UEs from its serving premises. Some delay tolerant tasks which require more computation can be forwarded to FEC instead. Thus, it is important to identify whether the tasks from UEs are delay tolerant or computation intensive or both and then allocate them to FEC and NEC accordingly. Next, we will analyze the big data tasks involved in indoor positioning. ![Near-Far Edge Computing Enhanced C-RAN Architecture to Support Big Data Processing.[]{data-label="c-ran"}](c-ran.eps){width="5.5in"} Tasks Involved in Indoor Positioning ------------------------------------ Outdoor positioning has been widely used in real life thanks to the Global Positioning System (GPS) technology [@7731599]. However, GPS does not work indoor whereas demands on finding indoor locations are high, as people spend most of their time indoor rather outdoor. Much effort has been made for indoor positioning, especially regarding techniques that do not involve much effort for initial deployment of positioning beacons. Due to the wide spread of WiFi hot spots (technically access points or APs), WiFi becomes a cost-effective option for indoor positioning. The WiFi assisted indoor positioning works in the steps as follows. Firstly, WiFi AP’s signal fingerprints which are typically composed of received signal strength (RSS) are collected against a particular physical indoor location (also known as reference point). Then, after certain signal processing in cloud, those (RSS, location) pairs are stored in a database, which can be referred to as Signal Fingerprints Database (SFD). Secondly, one can repeat the first step until all the designated locations are traversed. This procedure can be called as the site survey, which is conducted in an offline manner. Then during the online positioning stage, the UE which requires to show its current location collects the current RSS of the WiFi signal at the surrounding area and then compare the RSS with the data in SFD. To this end, the location with the best-matched fingerprint can be considered as the UE’s current location. The software architecture and the corresponding tasks of this widely used fingerprint-based indoor positioning is depicted in Fig. \[fig2\]. ![The Software Architecture and the Corresponding Tasks in WiFi assisted Indoor Positioning.[]{data-label="fig2"}](fig2.eps){width="4.2in"} To make positioning more accurate and efficient, extra procedures are introduced, such as signal pattern processing to clean out and correct the wrong fingerprints. Also, as RSS may change along with the change of indoor environment, to reflect in the signal pattern database on the positioning server, feedback is also applied in Fig. \[fig2\]. In summary there are two types of tasks. The first one is non-realtime or delay tolerant tasks that make long-term effect to the system such as signal pattern collection, processing and map generation, etc. The other type is realtime or delay sensitive tasks that are more concerned with end user’s positioning or display, such as instantaneous signal fingerprint collection, location display and signal pattern matching, etc. The non-realtime tasks can run remotely on FEC whereas the realtime tasks have to be executed on NEC to ensure fast response action. The experimental results in Section IV show the effectiveness of NEC in improving the accuracy of location. Wireless Big Data Involved in Indoor Positioning ------------------------------------------------ To have an estimate of how many WiFi signals are needed for indoor positioning, certain tests are carried out in our experimental site, i.e., the 5th Floor of the Network Centre Building on the Colchester main campus of the University of Essex. The pie chart in Fig. \[fig3\] shows the percentage of observed APs with different appearance frequency (denoted by $N$) when a UE is moving along a corridor for 25 seconds. Each AP is uniquely identified by a Basic Service Set Identifier (BSSID). Around 200 APs were detected during the 25 seconds’ movement. Around one third of APs are observed only once and about a quarter of APs are observed for more than 3 times. The AP appearing less times normally means its signal strength are weak and thus can be observed only within a certain range. The bar chart in Fig. \[fig3\] illustrates the distribution of observed APs in 2.4 GHz and 5 GHz frequency band respectively. APs of 5 GHz dominates the observations with smallest and largest appearance frequency, which reveals that 5 GHz channels may be less crowded and weak signals in 5 GHz are more likely to be observed than the signals in 2.4 GHz. ![Illustration of Wireless Big Data Involved in Indoor Positioning.[]{data-label="fig3"}](fig3.eps){width="6.2in"} Fig. \[fig3\] also shows that approximately 600 observations are collected within 25 seconds, which means that about 25 APs are observed every second on average. For each AP, a large amount of information has to be collected, such as MAC address, RSS, working frequency and time stamp, etc, amounting to at least 8 bytes. Thus, a stream of data of more than 200 bytes per second is generated. When the number of UEs or APs increases, the amount of data just for the purpose of indoor positioning will increase significantly. Note that this is a stream of data that is generated constantly and continually. Thus, it is nearly impossible for UEs to process those tasks with huge amount of data. The only way is to offload the corresponding tasks to the cloud, i.e., NEC and FEC. Next, we will introduce how we allocate the tasks to NEC and FEC in our proposed architecture. Task Allocation in NFC-RAN for Wireless Big Data ================================================ System Model ------------ To better describe the task allocation algorithms, we assume that there are $M$ RRHs and each of which $j=1,2,...,M$ forms a small cell, which can support $N_j$ UEs, as shown in Fig. \[c-ran\]. Also, assume each UE employs the orthogonal channel to transmit its data and there is no interference between each other. We assume each UE is only served by its nearest RRH which is predefined by its geographical position and signal strength. We denote UE $i=1,2,...,N_j$ in the coverage of $j$-th RRH as $ij$-th UE, which has a task $U_{ij}=(F_{ij}, D_{ij}, T_{ij}), \forall i\in N_j, \forall j\in M$, where $F_{ij}$ (in cycles) describes the computation requirement of this task, $D_{ij}$ (in bits) denotes the data required to be transmitted to NEC, or FEC and $T_{ij}$ (in seconds) is the delay requirement in order to satisfy the UE’s quality of service. We assume there is one FEC, next to BBU pool, with huge computation capacity $F^{FE}$ (in cycles/seconds) and it can be allocated to any UE in any cell. We also assume each RRH $j$ has a small NEC with limited computation capacity $F_j^{NE}$ (in cycles/seconds). Also, we assume that the tasks cannot be executed in UEs, as UEs may not have enough processing capacity and thus UEs can either offload the tasks to FEC or NEC. We define the indication parameters, i.e., $a_{ij}$, $b_{ij}$, $\forall i\in N_j, \forall j\in M$ to indicate where the task should be executed ($a_{ij}=1$ denotes the task is executed by FEC whereas $b_{ij}=1$ denotes the task is executed by $j$-th NEC). On the other hand, if $a_{ij}=b_{ij}=0$, it means this task can neither be executed by the NEC nor FEC and thus it has to be delayed to the next time slot. If NEC decides to execute the task for $ij$-th UE, then it will allocate the CPU capacity $f_{ij}^{NE}$ to the UE, which needs to send its data through wireless channel to $j$-th RRH with data rate $r_{ij}^W$. In this case, the task with a large amount of data $D_{ij}$ (big data application) does not have to send all the data to the central cloud through the fronthaul link. On the other hand, if the task requires a huge amount of calculation (computation intensive application), NEC may not be able to complete this task, due to its limited computation capacity, i.e. $F_j^{NE}$. Then, UE has to send all the data to the central cloud, i.e., FEC. If FEC decides to execute the task for UE, then it will allocate the CPU capacity $f_{ij}^{FE}$ to the $ij $-th UE. In this case, UE should first send its data to $j$-th RRH with wireless data rate $r_{ij}^W$, and then RRH will forward the data to FEC with fronthual transmission data rate as $r_{ij}^F$. Also, as the computation capacity of FEC is not infinite, i.e. constrained by the capacity of the physical machine, i.e., $F^{FE}$, some UEs may still not be able to complete the tasks. Moreover, if UEs decide to send the task to FEC, the capacity of fronthaul has to be taken into account. Thus, we assume the capability of the $j$-th frontaul as $R_j $. We model the task allocation problem as follows. $$\label{en25} \begin{aligned} \mathcal{P}: \;\;\;& \underset{a_{ij}, b_{ij} }{\text{max}}\;\;\; \sum_{i\in N_j}\sum_{j\in M}( a_{ij} + b_{ij}) \\& \text{subject to }: C1: a_{ij} \left(\frac{D_{ij}}{r_{ij}^W} +\frac{D_{ij}}{r_{ij}^F} +\frac{F_{ij}}{f_{ij}^{FE}}\right )\leq T_{ij} \\& C2: b_{ij} \left(\frac{D_{ij}}{r_{ij}^W} +\frac{F_{ij}}{f_{ij}^{NE}}\right )\leq T_{ij} \\& C3: \sum_{i\in N_j}\sum_{j\in M} a_{ij} f_{ij}^{FE} \leq F^{FE} \\& C4: \sum_{i\in N_j} b_{ij} f_{ij}^{NE} \leq F_j^{NE} \\& C5: \sum_{i\in N_j} a_{ij} r_{ij}^{F} \leq R_j \\& C6: a_{ij} + b_{ij} \leq 1 \\& C7: a_{ij}, b_{ij} =\{ 0,1\}, \forall i\in N_j, \forall j\in M \end{aligned}$$ where we aim to maximize the successful rate of all the offloading tasks, by deciding where the tasks should be executed. In another words, we try to accommodate as many tasks in cloud as possible. In above problem, $C1$ and $C2$ denote that the task has to be completed in certain amount of time by FEC or NEC, respectively, $C3$ and $C4$ denotes that the computation resources are limited in FFC and $j$-th NEC respectively, $C5$ is the constraint for the $j$-th fronthaul, $C6$ and $C7$ can not only show where each task should be executed, but also make the problem feasible. The above problem may be modified as the multi-dimension multi-choice 0-1 knapsack problem (MMKP), which can be solved effectively by using heuristic algorithm. Simulation Result ----------------- We assume that there are five $M=5$ RRHs, each of which forms a small cell. In each cell, there are $N$ UEs, each of which has a task to be completed. For each task, we assume the latency requirement is 3 seconds. We assume that other parameters are randomly assigned from the sets indicated in left hand side of Fig. \[figtable\]. ![Simulation Parameters Setting (Left) and Simulation Result (Right).[]{data-label="figtable"}](figtable.eps){width="6.6in"} In the right hand side of Fig. \[figtable\], we show that the relation between task successful rate versus the number of offloading tasks. The task successful rate or the task completion rate is defined as the ratio of the number of completion to the overall offloaded tasks. We compare the new NFC-RAN and the traditional C-RAN architecture only with FEC. The number of UEs is set from 10 to 50 in each of the 5 cells. Also, to compare fairly, $F_{FE}$ is set to $10^7$ GHz for traditional C-RAN. One can see that with the increase of the number of the offloading tasks, the successful rate decreases. This is because the cloud has limited computation resource and some tasks may be dropped or delayed to the next time interval. Our proposed NFC-RAN outperforms the traditional C-RAN with FEC, as NFC-RAN not only supports FEC, but also NEC which is much closer to UEs. This structure is beneficial to the delay sensitive tasks and therefore can increase the overall tasks’ successful rate. In the next section, we will use indoor positioning as a case study to show the benefit of NFC-RAN. Case Study: Indoor Positioning ============================== This section use indoor positioning as a case study and we assume that the task allocation is predetermined, namely, all the offline tasks and the feedback tasks (refer to Fig. \[fig2\]) are executed at FEC whereas the online computation tasks are executed at NEC. The experiments are carried out to show the performance improvement of positioning by using NFC-RAN architecture. Experimental Setup ------------------ ![Illustration of Indoor Map of the Experimental Site (Left) and Positioning Result (Right) Shown on Android Smartphone.[]{data-label="fig4"}](fig4.eps){width="5in"} We setup our testbed on the 5th Floor of the Network Centre Building of the University of Essex, where the experiments were conducted. The map of the environment is depicted in left hand side of Fig. \[fig4\]. The site is covered by about 60 wireless APs of Aruba mounted on the ceiling. They are part of the campus WiFi infrastructure. Each physical AP may generate multiple SSIDs, which means much more APs are observed by UEs. The circle in right hand side of Fig. \[fig4\] indicates the location of the UE. Experimental Results -------------------- In this section, two experiments were conducted. In the first one, most of the online tasks such as fingerprint matching are offloaded to and conducted by NFC-RAN, whereas in the second experiment, UE itself executes most of the task. We compare these two cases in Fig. \[wired\] where in Fig. \[wired\] (a), we show the cumulative distribution function (CDF) of location errors for both cases. One can see that our proposed NFC-RAN assisted system shows better location accuracy than UE-based one. The percentage of accuracy within 1 meter is approximately 70% in the NFC-RAN assisted system, which doubles the percentage of UE-based approach (35%). In 90% of positioning results, the NFC-RAN assisted and UE-based systems provide accuracy of 3 meters and 5 meters respectively. Through the comparison of the location errors when the CDF is 100%, it is apparent that the maximum error distance of UE-based system is almost 7.5 meters whereas the NFC-RAN assisted system can decrease it to 4.5 meters. The major reasons why NFC-RAN architecture outperforms UE based one are two-fold. Firstly, the processing capacity of UE is in no comparison with that of NFC-RAN system which is composed of much more powerful computation resource. The high computation capacity in NFC-RAN can process more patterns (big data applications) and to execute more comprehensive tasks. Secondly, the proposal of NEC, which brings computation closer to the UE, can fast calculate and execute tasks for the UE, such as signal pattern matching tasks. However, UE based positioning approach may lead to a situation where the produced location result by UE is out of date sometimes, thus resulting in a worse location accuracy. In Fig. \[wired\] (b), we show that the overall energy consumption on UE in the situation where NFC-RAN dealing with computation or UE itself conducting computation. From Fig. \[wired\] (b), we can see that UE will save a lot of energy on the NFC-RAN assisted situation, as most of the tasks are offloaded from UE to NFC-RAN. This is particularly important for the practical indoor positioning system as the positioning process may incur a large amount of data and therefore drain the UE’s battery quickly. Conclusions =========== In this paper, we have proposed NFC-RAN architecture, which can facilitate wireless big data processing. NFC-RAN is composed of NEC and FEC, where FEC is located next to BBU, which can provide large amount of computational resource to UEs, while NEC, located in RRH, can fast respond to the delay sensitive applications. Also, task allocation in NFC-RAN for wireless big data is illustrated and indoor positioning, as a case study, is exemplified to show the benefit of the proposed architecture. Future work will focus on how to allocate and execute the tasks dynamically, including in the NEC, FEC and UE itself. Also, more efficient task allocation algorithms for wireless big data will be investigated. Acknowledgements ================ This work was supported in part by Natural Science Foundation of China (Grant No. 61620106011, 61572389 and 61572191), UK EPSRC NIRVANA project (EP/L026031/1), EU Horizon 2020 iCIRRUS project (GA-644526) and EU FP7 Project CROWN (GA-2013-610524). We also would like to thank Dr Yuansheng Luo for the very useful discussion. [^1]: Lianming Zhang (zlm@hunnu.edu.cn) is with College of Physics and Information Science, Hunan Normal University, Changsha, China, Kezhi Wang (kezhi.wang@northumbria.ac.uk) is with Department of Computer and Information Sciences, Northumbria University, NE2 1XE, Newcastle upon Tyne, U.K, Xuan Du (xdua@essex.ac.uk) and Kun Yang (kunyang@essex.ac.uk) are with the School of Computer Sciences and Electrical Engineering, University of Essex, CO4 3SQ, Colchester, UK, Kun Yang is also with University of Electronic Science and Technology of China, Chengdu, China (Corresponding Author: Kun Yang).

--- abstract: 'The helicity amplitudes for the transitions $N-S_{11}$ and $N-S_{31}$ are presented. The amplitudes have been obtained within our front-form CQM model, based on hadron eigenstates of a relativistic mass operator and CQ current with Dirac and Pauli form factors.' --- -0.3in -0.50cm 6.5in 8.5in 1.2cm epsfig.sty [**Investigation of N-N\* Electromagnetic Form Factors within a Front-Form CQM**]{} 1em E. Pace$^a$, G. Salmè$^b$ and S. Simula$~^{c}$ [$^a$*Dipartimento di Fisica, Università di Roma “Tor Vergata”, and Istituto Nazionale di Fisica Nucleare, Sezione Tor Vergata, Via della Ricerca Scientifica 1, I-00133, Rome, Italy*]{} [$^b$*Istituto Nazionale di Fisica Nucleare, Sezione di Roma I, P.le A. Moro 2, I-00185 Rome, Italy*]{} [$^c$*Istituto Nazionale di Fisica Nucleare, Sezione di Roma III, Via della Vasca Navale 84, I-00146 Rome, Italy*]{} ------------------------------------------------------------------------ width5cm Hadron electromagnetic (em) form factors have been recently investigated within the front-form constituent quark (CQ) model of [@nuc95] for space-like values of the four-momentum transfer. The main features of the model are: i) the use of hadron eigenfunctions of a relativistic mass operator, that includes an effective $q-q$ interaction and reproduces the hadron spectra for a large set of quantum numbers [@CI]; ii) the use of a one-body em current operator containing phenomenological Dirac and Pauli form factors for CQ’s, which are determined by the request of reproducing the existing experimental data for the pion and nucleon elastic form factors (cf. [@nuc95]). Such a model has been already applied for obtaining a parameter-free prediction of the em form factors for the transitions to $N^*(1440)$ and $\Delta(1232)$, including the possible effects due to the $D$-wave components in the $\Delta$ wave function, [@nuc95]. In this contribution, we will present an analysis of transition form factors for $N \rightarrow S_{11}(1535)$, $N \rightarrow S_{11}(1650)$ and $N \rightarrow S_{31}(1620)$. The current for negative-parity transition with $J_f=1/2$ is given in terms of Dirac ($F^{f\tau}_1$) and Pauli-like ($F^{f\tau}_2$) form factors by (cf. [@Weber]) $$\begin{aligned} \bar{\Psi}_f~J^{\mu}~\Psi_{\tau}=\bar{\Psi}_f\gamma^5 \left [ {p_f^{\mu}+ p_i^{\mu} \over M_f-M_i} F^{f\tau}_2 -{M_f+M_i \over M_f-M_i} q^{\mu} F^{f\tau}_1 + \gamma^{\mu} (F^{f\tau}_1+F^{f\tau}_2 ) \right ]\Psi_{\tau} \label {cur}\end{aligned}$$ where $\tau =p,n$. By using such a current, the helicities for negative-parity transition can be written as follows $$\begin{aligned} &S_{1/2}^{\tau}\left (Q^2 \right )= \zeta~\sqrt{{ 2 \pi \alpha \over k^*}} \sqrt{{ Q^+ \over 2 M_i M_f}}\sqrt{{ Q^+ Q^-\over 4 M_f}} {M_f- M_i \over Q^2 \sqrt{2}} \left [F^{f\tau}_1- {Q^2 \over (M_f- M_i)^2} F^{f\tau}_2 \right ] \nonumber \\ &A_{1/2}^{\tau}\left (Q^2 \right )= -\zeta~\sqrt{{ 2 \pi \alpha \over k^*}} \sqrt{{ Q^+ \over 2 M_i M_f}} \left (F^{f\tau}_1+ F^{f\tau}_2 \right ) \label{hel}\end{aligned}$$ where $\zeta $ is the sign of the $\pi N$ decay amplitude, $k^*= (M^2_f -M^2_i)/2M_f$, $Q^{\pm}= (M_f\pm M_i)^2 +Q^2$. The invariant form factors in Eq. (\[hel\]) can be obtained within the front-form CQ model following standard procedures (see, e.g., [@nuc95]), namely approximating the plus component of the transition current, $\cal{I}^+$, in terms of the sum of one-body CQ currents, containing CQ Dirac and Pauli form factors. In particular $$\begin{aligned} F_1^{f\tau}= -{1 \over 2} Tr \left (\sigma_z {\cal I}^+(\tau) \right ) ~~~~~~~~F_2^{f\tau}= -{M_f- M_i \over 2Q} Tr \left (\sigma_x {\cal I}^+(\tau) \right ). \label{ff}\end{aligned}$$ where ${\cal{I}}^{+}_{\nu_f \nu_i}(\tau)=\bar{u}^f_{LF}(\nu_f)\sum_{j=1}^3 ~ \left ( e_j \gamma^+ f_1^j(Q^2) ~ + ~ i \kappa_j {\sigma^{+ \rho} q_{\rho} \over 2 m_j}f_2^j(Q^2) \right ) u^{\tau}_{LF}( \nu_i)$. In Figs. 1-5, our [*parameter-free*]{} evaluation of the helicity amplitudes, $A_{1/2}$ and $S_{1/2}$ are shown for $N \rightarrow S_{11}(1535)$, $S_{11}(1650)$ and $S_{31}(1620)$, respectively. In the case of $S_{31}(1620)$ the results for $p$ and $n$ coincides (as in the case of $P_{33}(1232)$), since only the isovector part of the CQ current is effective, given the isospin of the resonance. [**Figure 1.**]{} - (a) The transverse helicity $A_{1/2}$ for the transition $p \rightarrow S_{11}(1535)$ vs. $Q^2$. Solid line: $A_{1/2}$ from the hadron wave functions corresponding to the interaction of [@CI] and the nucleon em current with CQ form factors of [@nuc95]; dashed line: a non relativistic CQM calculation [@Giannini]. Solid dot: PDG ’96 [@PDG]; triangles: data analysis from [@Volker]. - (b) The same as in Fig. 1(a), but for $n \rightarrow S_{11}(1535)$. [**Figure 2.**]{} The same as in Fig. 1, but for the longitudinal helicity $S_{1/2}$. [**Figure 3.**]{} - (a) The transverse helicity $A_{1/2}$ for the transition $p \rightarrow S_{11}(1650)$ vs. $Q^2$. Solid line: $A_{1/2}$ from the hadron wave functions corresponding to the interaction of [@CI] and the nucleon em current with CQ form factors of [@nuc95]; dashed line: a non relativistic CQM calculation [@Giannini]. Solid dot: PDG ’96 [@PDG]; triangles: data analysis from [@Volker]. - (b) The same as in Fig. 3(a), but for $n \rightarrow S_{11}(1650)$. [**Figure 4.**]{} The same for Fig. 3, but for the longitudinal helicity $S_{1/2}$. [**Figure 5.**]{} - (a) The transverse helicity $A_{1/2}$ for the transition $p \rightarrow S_{31}(1620)$ vs. $Q^2$. Solid line: $A_{1/2}$ from the hadron wave functions corresponding to the interaction of [@CI] and the nucleon em current with CQ form factors of [@nuc95]; dashed line: a non relativistic CQM calculation [@Giannini]. - (b) The same as in Fig. 5a, but for $S_{1/2}$. The overall agreement between our predictions and the data is encouraging, though a most accurate set of data is necessary in order to reliably discriminate between different models. However, the sensitivity to relativistic effects for the P-wave resonances seems sizable. [9]{} F. Cardarelli, E. Pace, G. Salmè and S. Simula: Phys. Lett. [**B 357**]{}, 267 (1995); Few Body Systems Suppl. [**8**]{}, 345 (1995); Phys. Lett. [**B 371**]{}, 7 (1996); Phys. Lett. [**B 397**]{}, 13 (1997); Nucl. Phys.[**A 623**]{}, 361c (1997). S. Capstick and N. Isgur: Phys. Rev. [**D 34**]{}, 2809 (1986). R.H. Stanley and H. J. Weber: Phys. Rev. [**C 52**]{}, 435 (1995). M. Aiello, M. M. Giannini and E. Santopinto: Jou. of Phys. [**G24**]{}, 753 (1998). Particle Data Group: Phys. Rev. [**D 54**]{}, 1 (1996). V. D. Burkert: Czech. Jou. Phys. [**46**]{}, 627 (1996) and private communications.

--- abstract: 'We consider a space $\mathcal{U}$ of 3-dimensional diffeomorphisms $f$ with hyperbolic fixed points $p$ the stable and unstable manifolds of which have quadratic tangencies and satisfying some open conditions and such that $Df(p)$ has non-real expanding eigenvalues and a real contracting eigenvalue. The aim of this paper is to study moduli of diffeomorphisms in $\mathcal{U}$. We show that, for a generic element $f$ of $\mathcal{U}$, all the eigenvalues of $Df(p)$ are moduli and the restriction of a conjugacy homeomorphism to a local unstable manifold is a uniquely determined linear conformal map.' address: - | Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan.\ [E-mail address: hashimoto-shinobu@ed.tmu.ac.jp]{} - | Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratuka Kanagawa, 259-1292, Japan.\ [E-mail address: kiriki@tokai-u.jp]{} - | Department of Mathematical Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan.\ [E-mail address: tsoma@tmu.ac.jp]{} author: - Shinobu Hashimoto - Shin Kiriki - Teruhiko Soma title: 'Moduli of 3-dimensional diffeomorphisms with saddle-foci' --- [^1] The topological classification of structurally unstable diffeomorphisms or vector fields on a manifold $M$ is an important subject in the study of dynamical systems. Palis [@pa] suggested that moduli play important roles in such a classification. For a subspace $\mathcal{N}$ of the diffeomorphism space $\mathrm{Diff}^r(M)$ with $r\geq 1$, we say that a value $m(f)$ determined by $f\in \mathcal{N}$ is a *modulus* in $\mathcal{N}$ if $m(g)=m(f)$ holds for any $g\in \mathcal{N}$ topologically conjugate to $f$, that is, there exists a homeomorphism $h:M\to M$ with $g=h\circ f\circ h^{-1}$. A modulus for a certain class of vector fields is defined similarly. We say that a set $\mu_{\mathcal{N}}$ of moduli is *complete* if any $f$, $g\in \mathcal{N}$ with $m(f)=m(g)$ for all $m\in \mu_{\mathcal{N}}$ are topologically conjugate. For given vector fields $X$, $Y$ on $M$, a candidate for a conjugacy homeomorphism between $X$ and $Y$ is found in a usual manner. In many cases, such a map is well defined in a most part of $M$. So it remains to show that the map is extended to a homeomorphism on $M$ by using the condition that $X$ and $Y$ have the same value for any moduli in $\mu_{\mathcal{N}}$. On the other hand, in the diffeomorphism case, it would be difficult to find a complete set of moduli except for very restricted classes $\mathcal{N}$ in $\mathrm{Diff}^r(M)$. First we consider the case that $\dim M=2$ and $f_j$ $(j=0,1)$ are elements of $\mathrm{Diff}^r(M)$ $(r\geq 2)$ with two saddle fixed points $p_j$, $q_j$. Suppose moreover that $W^u(p_j)$ and $W^s(q_j)$ have a quadratic heteroclinic tangency $r_j$ and there exists a conjugacy homeomorphism $h$ between $f_1$ and $f_2$ with $h(p_0)=p_1$, $h(q_0)=q_1$ and $h(r_0)=r_1$. Then, Palis [@pa] proved that $\dfrac{\log |\lambda_0|}{\log |\mu_0|}= \dfrac{\log |\lambda_1|}{\log |\mu_1|}$ holds under ordinary conditions, where $\lambda_j$ is the contracting eigenvalue of $Df(p_j)$ and $\mu_i$ is the expanding eigenvalue of $Df(q_j)$. In [@po], Posthumus proved that the homoclinic version of Palis’ results. In fact, he proved that, if $f_j$ $(j=0,1)$ has a saddle fixed point $p_j$ with a homoclinic quadratic tangency, then $\dfrac{\log |\lambda_0|}{\log |\mu_0|}= \dfrac{\log |\lambda_1|}{\log |\mu_1|}$ holds, where $\lambda_j,\mu_j$ are the contracting and expanding eigenvalues of $Df(p_i)$. Moreover, he showed that, by using some results of de Melo [@dm], $\lambda_0=\lambda_1$ and $\mu_0=\mu_1$ hold if $\dfrac{\log |\lambda_0|}{\log |\mu_0|}$ is irrational. We refer to [@dmp; @dmvs; @pt; @mp1; @gpvs; @ha] and references therein for more results on moduli of 2-dimensional diffeomorphisms. Moduli for 2-dimensional flows with saddle-connections are studied by Palis [@pa] and Takens [@ta] and so on. In those papers, they present finite sets of moduli which are complete in a neighborhood of the saddle connection in $M$. In this paper, we consider 3-dimensional diffeomorphisms $f$ with a hyperbolic fixed point $p$ such that $W^u(p)$ and $W^s(p)$ have a quadratic tangency and $Df(p)$ has non-real expanding eigenvalues $re^{\pm\sqrt{-1}\theta}$ with $r>1$ and a contracting eigenvalue $0<\lambda<1$. Moduli for diffeomorphisms of dimension more than two have been already studied by [@npt; @du2; @mp2] and so on. First we will prove the following theorem. \[thm\_A\] Let $M$ be a $3$-manifold and $f_j$ $(j=0,1)$ elements of $\mathrm{Diff}^r(M)$ for some $r\geq 3$ which have hyperbolic fixed points $p_j$ and homoclinic quadratic tangencies $q_j$ positively associated with $p_j$ and satisfy the following conditions. - For $j=0,1$, there exists a neighborhood $U(p_j)$ of $p_j$ in $M$ such that $f_j|_{U(p_j)}$ is linear and $Df_j(p_j)$ has non-real eigenvalues $r_je^{\pm \sqrt{-1}\theta_j}$ and a real eigenvalue $\lambda_j$ with $r_j>1$, $\theta_j\neq 0\mod \pi$ and $0<\lambda_j<1$. - $f_0$ is topologically conjugate to $f_1$ by a homeomorphism $h:M\to M$ with $h(p_0)=p_1$ and $h(q_0)=q_1$. Then the following and hold. 1. \[A1\] $\dfrac{\log \lambda_0}{\log r_0}=\dfrac{\log \lambda_1}{\log r_1}$. 2. \[A2\] Either $\theta_0=\theta_1$ or $\theta_0=-\theta_1\mod 2\pi$. Here we say that a homoclinic quadratic tangency $q_0$ is *positively associated* with $p_0$ if both $f_0^n(q_0)$ and $f_0^{-n}(\alpha)$ lie in the same component of $U(p_0)\setminus W_{\mathrm{loc}}^u(p_0)$ for a sufficiently large $n\in\mathbb{N}$ and any small curve $\alpha$ in $W^s(p_0)$ containing $q_0$. Theorem \[thm\_A\] holds also in the case when $\theta_0=0\mod \pi$ or $-1<\lambda_j<0$ except for some rare case, see Remark \[r\_exception\] for details. Assertion of Theorem \[thm\_A\] is implied in the case (D) of Theorem 1.1 in [@npt Chapter III]. Assertion is also proved by Dufraine [@du2] under weaker assumptions. The author used non-spiral curves in $W_{\mathrm{loc}}^u(p)$ emanating from $p$. On the other hand, we employ unstable bent disks defined in Section \[S\_front\_folding\] which are originally introduced by Nishizawa [@ni]. By using such disks, we construct a convergent sequence of mutually parallel straight segments in $W_{\mathrm{loc}}^u(p)$ which are mapped to straight segments in $W_{\mathrm{loc}}^u(h(p))$ by $h$, see Figure \[fig\_parallel\]. An *advantage* of our proof is that these sequences are applicable to prove our main theorem, Theorem \[thm\_B\] below. Results corresponding to Theorem \[thm\_A\] for 3-dimensional flows with Shilnikov cycles are obtained by Togawa [@to], Carvalho-Rodrigues [@cr] and for those with connections of saddle-foci by Bonatti-Dufraine [@bd], Dufraine [@du], Rodrigues [@ro] and so on. See the Section 2 in [@ro] for details. Moreover Carvalho and Rodrigues [@cr] present results on moduli of 3-dimensional flows with Bykov cycles. \[thm\_B\] Under the assumptions in Theorem \[thm\_A\], suppose moreover that $\theta_0/2\pi$ is irrational. Then the following conditions hold. 1. \[B1\] $\lambda_0=\lambda_1$ and $r_0=r_1$. 2. \[B2\] The restriction $h|_{W_{\mathrm{loc}}^u(p_0)}:W_{\mathrm{loc}}^u(p_0)\to W_{\mathrm{loc}}^u(p_1)$ is a uniquely determined linear conformal map. In contrast to Posthumus’ results for 2-dimensional diffeomorphisms, the eigenvalues $\lambda_0$ and $r_0$ are proved to be moduli without the assumption that $\dfrac{\log \lambda_0}{\log r_0}$ is irrational. The restriction $h|_{W_{\mathrm{loc}}^u(p_0)}$ is said to be a *linear conformal map* if $h|_{W_{\mathrm{loc}}^u(p_0)}$ is represented as $h|_{W_{\mathrm{loc}}^u(p_0)}(z)=\rho e^{\sqrt{-1}\omega}z$ $(z\in W_{\mathrm{loc}}^u(p_0))$ for some $\rho\in\mathbb{R}\setminus \{0\}$ and $\omega\in\mathbb{R}$ under the natural identification of $W_{\mathrm{loc}}^u(p_0)$, $W_{\mathrm{loc}}^u(p_1)$ with neighborhoods of the origin in $\mathbb{C}$ via their linearizing coordinates. For any $r_j>1$ and $\theta_j\in \mathbb{R}$ $(j=0,1)$, let $\varphi_j:\mathbb{C}\to \mathbb{C}$ be the map defined by $\varphi_j(z)=r_je^{\sqrt{-1}\theta_j}z$. Then there are many choices of conjugacy homeomorphisms on $\mathbb{C}$ for $\varphi_0$ and $\varphi_1$. For example, we take two-sided Jordan curves $\Gamma_j$ in $\mathbb{C}$ with $\varphi_j(\Gamma_j)\cap \Gamma_j=\emptyset$ and bounding disks in $\mathbb{C}$ containing the origin arbitrarily. Then there exists a conjugacy homeomorphism $h:\mathbb{C}\to \mathbb{C}$ for $\varphi_0$ and $\varphi_1$ with $h(\Gamma_0)=\Gamma_1$. On the other hand, Theorem \[thm\_B\] implies that we have severe constraints in the choice of conjugacy homeomorphisms for 3-dimensional diffeomorphisms as above. Intuitively, it says that only a homeomorphism $h$ with $h|_{W^u_{\mathrm{loc}}}(p)$ linear and conformal can be a candidate for a conjugacy between $f_0$ and $f_1$. As an application of the linearity and conformality of $h|_{W^u_{\mathrm{loc}}}(p)$, we will present a new modulus for $f_0$ other than $\theta_0$, $\lambda_0$, $r_0$, see Corollary \[c\_C\] in Section \[S\_PB\]. Front curves and folding curves {#S_front_folding} =============================== For $j=0,1$, let $f_j$ be a diffeomorphism and $q_j$ a quadratic tangency associated with a hyperbolic fixed point $p_j$ satisfying the conditions of Theorem \[thm\_A\]. We will define in this section front curves in $W^u(p_j)$ and folding curves in $W_{\mathrm{loc}}^u(p_j)$ and show in the next section that these curves converge to straight segments which are preserved by any conjugacy homeomorphism between $f_0$ and $f_1$. We set $f_0=f$, $p_0=p$, $q_0=q$, $r_0=r$, $\theta_0=\theta$ and $\lambda_0=\lambda$ for short. Similarly, let $f_1=f'$, $p_1=p'$, $q_1=q'$, $r_1=r'$, $\theta_1=\theta'$ and $\lambda_1=\lambda'$. Suppose that $(z,t)=(x,y,t)$ with $z=x+\sqrt{-1}y$ is a coordinate around $p$ with respect to which $f$ is linear. For a small $a>0$, let $D_a(p)$ be the disk $\{z\in\mathbb{C}\,; |z|\leq a\}$. We may assume that $q$ is contained in the interior of $D_a(p)\times \{0\}\subset W_{\mathrm{loc}}^u(p)$ and $\widehat q=f^N(q)$ is in the interior of the upper half $W_{\mathrm{loc}}^{s+}(p)=\{0\}\times [0,a]$ of $W_{\mathrm{loc}}^s(p)$ for some $N\in\mathbb{N}$. See Figure \[fig\_1\]. Let $U_a(p)$ be the circular column in the coordinate neighborhood defined by $U_a(p)=D_a(p)\times [0,a]$ and $V_{\widehat q}$ a small neighborhood of $\widehat q$ in $U_a(p)$. Suppose that $U_a(p)$ has the Euclidean metric induced from the linearizing coordinate on $U_a(p)$. By choosing the coordinate suitably and replacing $\theta$ by $-\theta$ if necessary, we may assume that the restriction $f|_{D_a(p)}$ is represented as $re^{\sqrt{-1}\theta}z$ for $z\in \mathbb{C}$ with $|z|<a$. Similarly, one can suppose that $f'|_{D_{a'}(p')}$ is represented as $r'e^{\sqrt{-1}\theta'}z$ for some $a'>0$. The orthogonal projection $\mathrm{pr}:U_a(p)\to D_a(p)$ is defined by $\mathrm{pr}(x,y,t)=(x,y)$. In this section, we construct an unstable bent disk $\widetilde H_0$ in $W^u(p)\cap U_a(p)$, the front curve $\widetilde\gamma_0$ in $\widetilde H_0$ and the folding curves $\gamma_0$ in $U_a(p)$. We also define the sequence of unstable bent disks $\widetilde H_m$ in $W^u(p)\cap U_a(p)$ converging to $\widetilde H_0$, which will be used in the next section to construct the sequence of front curves converging to $\widetilde\gamma_0$. Construction of unstable bent disks, front curves and folding curves {#ss_bent_disk} -------------------------------------------------------------------- We set $\widehat q=(0,t_0)$. Let $\widetilde H$ be the component of $W^u(p)\cap V_{\widehat q}$ containing $\widehat q$. One can retake the linearizing coordinate on $\mathbb{C}$ if necessary so that the line in $V_{\widehat q}$ passing through $\widehat q$ and parallel to the $x$-axis in $U_a(p)$ meets $\widetilde H$ transversely. Then $\widetilde H$ is represented as the graph of a $C^r$-function $x=\varphi(y,t)$ with $$\label{eqn_vp} \varphi(0,t_0)=0,\quad\frac{\partial \varphi}{\partial t}(0,t_0)=0\quad\text{and}\quad\frac{\partial^2 \varphi}{\partial t^2}(0,t_0)\neq 0.$$ By the implicit function theorem, there exists a $C^{r-1}$-function $t=\eta(y)$ defined in a small neighborhood $V$ of $0$ in the $y$-axis and satisfying $\eta(0)=t_0$ and $\partial \varphi(y,\eta(y))/\partial t=0$. Then the curve $\widetilde \gamma$ in $V_{\widehat q}$ parametrized by $\bigl(\varphi(y,\eta(y)),y,\eta(y)\bigr)$ divides $\widetilde H$ into two components and $\gamma=\mathrm{pr}(\widetilde\gamma)$ is a $C^{r-1}$-curve embedded in $D_a(p)$. Let $\widetilde H^+$ (resp. $\widetilde H^-$) be the closure of the upper (resp. lower) component of $\widetilde H\setminus \widetilde\gamma$. For a sufficiently large $n_0\in\mathbb{N}$, the component $\widetilde H_0$ of $f^{n_0}(\widetilde H)\cap U_a(p)$ containing $q_0=f^{n_0}(\widehat q)$ is an *unstable bent disk* in $U_a(p)$ such that $\partial \widetilde H_0$ is a simple closed $C^r$-curve in $\partial_{\mathrm{side}}U_a(p)$, where $$\partial_{\mathrm{side}}U_a(p)=\{(x,t)\in\mathbb{C}\times \mathbb{R}\,; |z|= a, 0\leq t< a\}\subset \partial U_a(p).$$ See Figure \[fig\_H\_0\]. We set $\widetilde\gamma_0=f^{n_0}(\widetilde\gamma)\cap \widetilde H_0$, $\widetilde H_0^+=f^{n_0}(\widetilde H^+)\cap \widetilde H_0$, $\widetilde H_0^-=f^{n_0}(\widetilde H^-)\cap \widetilde H_0$, $H_0=\mathrm{pr}(\widetilde H_0^+)=\mathrm{pr}(\widetilde H_0^-)$ and $\gamma_0=\mathrm{pr}(\widetilde \gamma_0)$. Then $\widetilde \gamma_0$ is called the *front curve* of $\widetilde H_0$ and $\gamma_0$ is the *folding curve* of $H_0$. We note that Nishizawa [@ni] has studied unstable bent disks similar to $\widetilde H_0$ as above in a different situation. In fact, he considered a 3-dimensional diffeomorphism $g$ which has a saddle fixed point $s$ such that all the eigenvalues of $Dg(s)$ are real and has a homoclinic quadratic tangency associated with $s$. Here we consider the component $\widetilde H_{0;u}^-$ of $f^u(\widetilde H_0^-)\cap U_a(p)$ containing $f^u(q_0)$ for $u\in\mathbb{N}$. Since the homoclinic tangency $q$ is positively associated with $p$, one can show that there exists $\widetilde H_{0;u}^-$ which meets $W^s(p)$ transversely at a point $\widehat z$ near $q$ by using an argument similar to that in [@ni Lemma 4.4]. See Figure \[fig\_H\_u\]. To show the claim, the assumption of $\theta_0\neq 0\mod \pi$ in Theorem \[thm\_A\] is crucial. In fact, the condition implies that the following property: (A) There exists an arbitrarily large $u$ such that the interior of $H_{0;u}=\mathrm{pr}(\widetilde H_{0;u}^-)$ in $D_a(p)$ contains $q$. \[r\_exception\] (1) We here suppose $\theta=0\mod \pi$. Even in this case, if $f$ has the property (P), then the component of $W^s(p)$ containing $q$ and $W^u(p)$ have a homoclinic transverse intersection point. Then Theorems \[thm\_A\] and \[thm\_B\] will be proved quite similarly. Since $\theta=0\mod \pi$, all $f^u(\gamma_0)$ are tangent to a unique straight segment $\gamma_\infty$ in $D_a(p)$ at $p$. Thus the property (P) is satisfied if $\gamma_\infty$ does not pass through $q$. \(2) Even in the case of $-1<\lambda<0$, one can show that $f$ has the property (P) similarly by using $f^2$ instead of $f$ if $2\theta\neq 0\mod \pi$. Moreover, since either $q$ or $f(q)$ is a homoclinic tangency positively associated with $p$, Theorems \[thm\_A\] and \[thm\_B\] hold without the assumption that $q$ is positively associated with $p$. Construction of convergent sequence of unstable bent disks ---------------------------------------------------------- Take $v\in\mathbb{N}$ such that $\widehat z_0=f^v(\widehat z)$ is a point $(0,\widehat t\,)$ contained in $U_a(p)$, where $\widehat z$ is the transverse intersection point of $\widetilde H_{0;u}^-$ and $W^s(p)$ given in the previous subsection. Let $D$ be a small disk in $W^u(p)\cap U_a(p)$ whose interior contains $\widehat z_0$. The *absolute slope* $\sigma(\boldsymbol{v})$ of a vector $\boldsymbol{v}=(v_1,v_2,v_3)$ in $U_a(p)$ with $(v_1,v_2)\neq (0,0)$ is given as $$\sigma(\boldsymbol{v})=\frac{|v_3|}{\sqrt{v_1^2+v_2^2}}.$$ The *maximum absolute slope* $\sigma(D)$ of $D$ is defined by $$\sigma(D)=\max\{\sigma(\boldsymbol{v})\,;\, \text{unit vectors $\boldsymbol{v}$ in $U_a(p)$ tangent to $D$}\}.$$ Fix $m_0\in \mathbb{N}$ such that, for any $m\in \mathbb{N}\cup\{0\}$, the component $D_m$ of $f^{m_0+m}(D)\cap U(p)$ containing $f^{m_0+m}(\widehat z_0)$ is a properly embedded disk in $U_a(p)$ with $\partial D_m\subset \partial_{\mathrm{side}}U_a(p)$. Note that $D_m$ intersects $W_{\mathrm{loc}}^s(p)$ transversely at $(0,\lambda^mt_0)$, where $t_0=\lambda^{m_0}\widehat t$. See Figure \[fig\_Dm\]. The maximum absolute slope of $D_m$ satisfies $$\label{eqn_sigma_D} \sigma(D_m)\leq \sigma_0\lambda^mr^{-m},$$ where $\sigma_0=\sigma(D)\lambda^{m_0}r^{-m_0}$. Consider a short straight segment $\rho$ in $U_a(p)$ meeting $\widetilde H_0$ orthogonally at $q_0$. Then $\widetilde\rho=f^{-(N+n_0)}(\rho)$ is a $C^r$-curve meeting $D_a(p)$ transversely at $q$, where $N$, $n_0$ are the positive integers given as above. One can choose $m_0\in\mathbb{N}$ so that, for any $m\in\mathbb{N}\cup\{0\}$, $\widetilde\rho$ meets $D_m$ transversely at a single point $\boldsymbol{w}_m=(z_m,s_m)$. Then implies that $|t_0\lambda^m-s_m|\leq \widetilde a\sigma_0\lambda^mr^{-m}$, where $\widetilde a=\sup_{m\geq 0}\{|z_m|\}<\infty$. It follows that $s_m=t_0\lambda^m+O(\lambda^mr^{-m})$. Since $\widetilde\rho$ has a tangency of order at least two with a straight segment at $q$, $$\label{eqn_s_m} \mathrm{dist}(\boldsymbol{w}_m,q)=\widetilde t_0\lambda^m+O(\lambda^mr^{-m})+O(\lambda^{2m})=\widetilde t_0\lambda^m +o(\lambda^m)$$ for some constant $\widetilde t_0>0$. By the inclination lemma, $D_m$ uniformly $C^r$-converges to $D_a(p)$. A short curve in $W^s(p)$ containing $q$ as an interior point meets $D_m$ transversely in two points for all sufficiently large $m$. Let $\widetilde H_m$ be the component of $f^{N+n_0}(D_m)\cap U_a(p)$ containing $f^{N+n_0}(\boldsymbol{w}_m)$. Then $\widetilde H_m$ $C^r$-converges to $\widetilde H_0$ as $m\to \infty$. By , there exist $C^r$-functions $\varphi_m(y,t)$ $C^r$-converging to $\varphi$ and representing $\widetilde H_m$ as the graph of $x=\varphi_m(y,t)$. Then the front curve $\widetilde \gamma_m$ in $\widetilde H_m$ is defined as the front curve $\widetilde\gamma_0$ in $\widetilde H_0$. Since $\partial\varphi_m(y,t)/\partial t$ $C^{r-1}$-converges to $\partial \varphi(y,t)/\partial t$, $\widetilde\gamma_m$ also $C^{r-1}$-converges to $\widetilde\gamma_0$. Note that $\widetilde \gamma_m$ divides $\widetilde H_m$ into the upper surface $\widetilde H_m^+$ and the lower surface $\widetilde H_m^-$ with $\widetilde \gamma_m=\widetilde H_m^+\cap \widetilde H_m^-$ and $H_m=\mathrm{pr}(\widetilde H_m)=\mathrm{pr}(\widetilde H_m^+)=\mathrm{pr}(\widetilde H_m^-)$. The image $\gamma_m=\mathrm{pr}(\widetilde \gamma_m)$ is called the folding curve of $H_m$. Limit straight segments {#S_limit} ======================= A curve $\gamma$ in $D_a(p)$ is called a *straight segment* if $\gamma$ is a segment with respect to the Euclidean metric on $D_a(p)$. In this section, we will construct a proper straight segment $\gamma_0^\natural$ in $D_a(p)$ with $p\not\in\gamma_0^\natural$ which is mapped to a straight segment in $U_{a'}(p')$ by $h$. Sequences of folding curves converging to straight segments ----------------------------------------------------------- Let $\alpha$ be an oriented $C^{r-1}$-curve in $D_a(p)$ of bounded length. Since $r-1\geq 2$, there exists the maximum absolute curvature $\kappa(\alpha)$ of $\alpha$. If $\alpha$ passes near the center $0$ of $D_a(p)$ and satisfies $\kappa(\alpha)<1/a$, then $\alpha$ has a unique point $z(\alpha)$ with $\mathrm{dist}(0,z(\alpha))=\mathrm{dist}(0,\alpha)$. In fact, if $\alpha$ had two points $z_i$ $(i=1,2)$ with $\mathrm{dist}(0,z_i)=\mathrm{dist}(0,\alpha)$, then for a point $z_3$ in $\alpha$ with the maximum $\mathrm{dist}(0,z_3)$ between $z_1$ and $z_2$, the curvature of $\alpha$ at $z_3$ is not less than $1/\mathrm{dist}(0,z_3)\geq 1/a$, a contradiction. We denote by $\vartheta(\alpha)\mod 2\pi$ the angle between $\widehat\alpha$ and the positive direction of the $x$-axis at $0$, where $\widehat\alpha$ is the oriented curve in $D_a(p)$ obtained from $\alpha$ by the parallel translation taking $z(\alpha)$ to $0$. By , there exists a constant $\widetilde d_0>0$ such that $$\label{eqn_dcm} \mathrm{dist}(\widetilde \gamma_m,\text{the $t$-axis})=\widetilde d_0(\widetilde t_0\lambda^m+o(\lambda^m))+o(\lambda^m) =\widetilde d_0\widetilde t_0\lambda^m+o(\lambda^m).$$ Since $\gamma_m$ $C^{r-1}$-converges to $\gamma_0$, $\kappa(\gamma_m)$ also converges to $\kappa(\gamma_0)$ as $m\to\infty$. This shows that $$\label{eqn_kappa} \sup_m\{\kappa(\gamma_m)\}=\kappa_0<\infty.$$ It follows that, for all sufficiently large $m$, there exists a unique point $c_m$ of $\gamma_m$ with $$\mathrm{dist}(c_m,0)=\mathrm{dist}(\gamma_m,0)=\mathrm{dist}(\widetilde c_m,\text{the $t$-axis}) =\mathrm{dist}(\widetilde \gamma_m,\text{the $t$-axis}),$$ where $\widetilde c_m$ is the point of $\widetilde\gamma_m$ with $\mathrm{pr}(\widetilde c_m)=c_m$. Fix $w$ with $0<w<a/2$ arbitrarily. For any $n\in\mathbb{N}$, let $m(n)$ be the minimum positive integer such that $f^n(c_m)$ is contained in $D_w(p)$ for any $m\geq m(n)$. Then $\lim_{n\to \infty}m(n)=\infty$ holds. For any $m\geq m(n)$, the component $\widetilde H_{m,n}$ of $f^n(\widetilde H_m)\cap U_a(p)$ containing $\widetilde c_{m,n}=f^n(\widetilde c_m)$ is a proper disk in $U_a(p)$ with $\partial \widetilde H_{m,n}\subset \partial_{\mathrm{side}}U_a(p)$. Then $\widetilde \gamma_{m,n}=f^n(\widetilde \gamma_m)\cap \widetilde H_{m,n}$ is the front curve of $\widetilde H_{m,n}$ and $\gamma_{m,n}=\mathrm{pr}(\widetilde \gamma_{m,n})$ is the folding curve of $H_{m,n}=\mathrm{pr}(\widetilde H_{m,n})$. Then $c_{m,n}=\mathrm{pr}(\widetilde c_{m,n})$ is a unique point of $\gamma_{m,n}$ closest to $0$. Here we orient $\widetilde\gamma_m=\widetilde \gamma_{m,0}$ so that $\widetilde \gamma_{m,0}$ $C^{r-1}$-converges as oriented curves to $\widetilde \gamma_0$ as $m\to\infty$. Suppose that $\gamma_{m,n}$ has the orientation induced from that on $\widetilde\gamma_{m,0}$ via $\mathrm{pr}\circ f^n$. In particular, it follows that $$\label{eqn_theta} \lim_{m\to \infty}\vartheta(\gamma_{m,0})=\vartheta(\gamma_0).$$ We set $d_{m,n}=\mathrm{dist}(c_{m,n},0)$. By , $$\label{eqn_dmn} d_{m,n}=r^n(\widetilde d_0\widetilde t_0\lambda^m+o(\lambda^m)).$$ There exist subsequences $\{m_j\}$, $\{n_j\}$ of $\mathbb{N}$ and $w\lambda/2\leq w_0\leq w$ such that $$\label{eqn_limit_dt} \lim_{j\to\infty}\widetilde d_0\widetilde t_0\lambda^{m_j}r^{n_j}=w_0.$$ If necessary taking subsequences of $\{m_j\}$ and $\{n_j\}$ simultaneously, we may also assume that $\vartheta(\gamma_{m_j,n_j})$ has a limit $\theta^\natural$. Since $f(z)=re^{\sqrt{-1}\theta}z$ on $D_a(p)$, by we have $$\kappa(\gamma_{m_j,n_j})\leq r^{-n_j}\kappa(\gamma_{m_j,0})\leq r^{-n_j}\kappa_0 \to 0\quad\text{as}\quad j\to \infty.$$ Thus the following lemma is obtained immediately. \[l\_gamma\_star\] The sequence $\gamma_{m_j,n_j}$ uniformly converges as oriented curves to an oriented straight segment $\gamma_0^\natural$ in $D_a(p)$ with $\vartheta(\gamma_0^\natural)=\theta^\natural$ and $\mathrm{dist}(\gamma_0^\natural,0)=w_0$. We say that $\gamma_0^\natural$ is the *limit straight segment* of $\gamma_{m_j,n_j}$. Limit straight segments preserved by the conjugacy -------------------------------------------------- Let $U_{a'}(p')$, $U_{b'}(p')$ be the circular columns defined as $U_a(p)$ for some $0<a'<b'$ which are contained in a coordinate neighborhood around $p'$ with respect to which $f'$ is linear. One can retake $a>0$ and choose such $a'$, $b'$ so that $U_{a'}(p')\subset h(U_a(p))\subset U_{b'}(p')$. Let $\widetilde H_{m,n}'$ be the component of $h(\widetilde H_{m,n})\cap U_{a'}(p')$ defined as $\widetilde H_{m,n}$ and $\mathrm{pr}(\widetilde H_{m,n}')=H_{m,n}'$. One can define the front and folding curves $\widetilde \gamma_{m,n}'$, $\gamma_{m,n}'$ in $\widetilde H_{m,n}'$ and $H_{m,n}'$ as $\widetilde\gamma_{m,n}$, $\gamma_{m,n}$ in $\widetilde H_{m,n}$ and $H_{m,n}$ respectively. See Figure \[fig\_UaUb\]. Since $h$ is only supposed to be a homeomorphism, $h(\widetilde\gamma_{m,n})\cap U_{a'}(p')$ would not be equal to $\widetilde\gamma_{m,n}'$. We will show that this equality holds in the limit. For the sequences $\{m_j\}$, $\{n_j\}$ given in Section \[S\_limit\], we set $\widetilde H_{m_j,n_j}=\widetilde H_{(j)}$, $H_{m_j,n_j}=H_{(j)}$, $\widetilde H_{m_j,n_j}'=\widetilde H_{(j)}'$ and $H_{m_j,n_j}'=H_{(j)}'$ for simplicity. Similarly, suppose that $\widehat H_{(j)}'$ is the component of $W^u(p')\cap U_{b'}(p')$ containing $\widetilde H_{(j)}'$ and $\widehat\gamma_{m_j,n_1}'$ is the front curve of $\widehat H_{(j)}'$. The distance between $\boldsymbol{x}$, $\boldsymbol{y}$ in $U_a(p)$ is denoted by $d(\boldsymbol{x},\boldsymbol{y})$ and that between $\boldsymbol{x}'$, $\boldsymbol{y}'$ in $U_{a'}(p')$ by $d'(\boldsymbol{x}',\boldsymbol{y}')$. The *path metric* on $\widetilde H_{(j)}$ is denoted by $d_{\widetilde H_{(j)}}$. That is, for any $\boldsymbol{x}$, $\boldsymbol{y}\in \widetilde H_{(j)}$, $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})$ is the length of a shortest piecewise smooth curve in $\widetilde H_{(j)}$ connecting $\boldsymbol{x}$ with $\boldsymbol{y}$. The path metrics $d_{\widetilde H_{(j)}'}$ on $\widetilde H_{(j)}'$ and $d_{\widehat H_{(j)}'}$ on $\widehat H_{(j)}'$ are defined similarly. \[l\_d\_H\] 1. For any $\varepsilon>0$, there exists a constant $\eta(\varepsilon)>0$ independent of $j\in\mathbb{N}$ and satisfying the following conditions. - $\lim_{\varepsilon\to 0}\eta(\varepsilon)=0$. - Let $\boldsymbol{x}$, $\boldsymbol{y}$ be any points of $\widetilde H_{(j)}$ both of which are contained in one of $\widetilde H_{(j)}^+$ and $\widetilde H_{(j)}^-$. If $d(\boldsymbol{x},\boldsymbol{y})<\eta(\varepsilon)$, then $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})<\varepsilon$. 2. For any $\varepsilon>0$, there exists a constant $\delta(\varepsilon)>0$ independent of $j\in\mathbb{N}$ and satisfying the following conditions. - $\lim_{\varepsilon\to 0}\delta(\varepsilon)=0$. - Let $\boldsymbol{x}$, $\boldsymbol{y}$ be any points of $\widetilde H_{(j)}$ both of which are contained in one of $\widetilde H_{(j)}^+$ and $\widetilde H_{(j)}^-$. If $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})<\delta(\varepsilon)$ and $\boldsymbol{x}'=h(\boldsymbol{x})$ and $\boldsymbol{y}'=h(\boldsymbol{y})$ are contained in $\widetilde H_{(j)}'$, then $d_{\widetilde H_{(j)}'}(\boldsymbol{x}',\boldsymbol{y}')<\varepsilon$. One can take these constants $\eta(\varepsilon)$, $\delta(\varepsilon)$ so that they work also for $d_{\widetilde H_{(j)}'}$ and $d_{\widehat H_{(j)}'}$. \(i) The assertion is proved immediately from the fact that $\widetilde H_{(j)}^\pm$ uniformly converges to a disk $H^\natural$ in $D_a(p)$ such that $d(\boldsymbol{x},\boldsymbol{y})=d_{H^\natural}(\boldsymbol{x},\boldsymbol{y})$ for any $\boldsymbol{x},\boldsymbol{y}\in H^\natural$. \(ii) Suppose that $\boldsymbol{x},\boldsymbol{y}\in \widetilde H_{(j)}^+$. First we consider the case that both $\boldsymbol{x}'$ and $\boldsymbol{y}'$ are contained in one of $\widetilde H_{(j)}'^+$ and $\widetilde H_{(j)}'^-$, say $\widetilde H_{(j)}'^+$. If $d_{\widetilde H_{(j)}'}(\boldsymbol{x}',\boldsymbol{y}')\geq \varepsilon$, then it follows from the assertion (i) that $d'(\boldsymbol{x}',\boldsymbol{y}')\geq \eta(\varepsilon)$. Since $h$ is uniformly continuous on $U_a(p)$, there exists a constant $\delta_1(\varepsilon)>0$ with $\lim_{\varepsilon\to 0}\delta_1(\varepsilon)=0$ and $d(\boldsymbol{x},\boldsymbol{y})\geq \delta_1(\varepsilon)$. Hence, in particular, $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})\geq \delta_1(\varepsilon)$. Thus $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})< \delta_1(\varepsilon)$ implies $d_{\widetilde H_{(j)}'}(\boldsymbol{x}',\boldsymbol{y}')<\varepsilon$. Next we suppose that $\boldsymbol{x}'\in \widetilde H_{(j)}'^+$ and $\boldsymbol{y}'\in \widetilde H_{(j)}'^-$. Consider a shortest curve $\alpha$ in $\widetilde H_{(j)}$ connecting $\boldsymbol{x}$ and $\boldsymbol{y}$. Since $\alpha'=h(\alpha)$ is contained in $\widehat H_{(j)}'$, $\alpha'$ intersects $\widehat\gamma_{m_j,n_j}'$ non-trivially. Let $\boldsymbol{z}$ be one of the intersection points of $\alpha$ with $h^{-1}(\widehat\gamma_{m_j,n_j}')$. See Figure \[fig\_d\_H\]. Suppose that $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})<\delta_1(\varepsilon/2)$. Since $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})=d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{z})+d_{\widetilde H_{(j)}}(\boldsymbol{z},\boldsymbol{y})$, $$d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{z})<\delta_1(\varepsilon/2)\quad\text{and}\quad d_{\widetilde H_{(j)}}(\boldsymbol{z},\boldsymbol{y})<\delta_1(\varepsilon/2).$$ Since $\boldsymbol{x}',\boldsymbol{z}'\in \widehat H_{(j)}'^+$ and $\boldsymbol{z}',\boldsymbol{y}'\in \widehat H_{(j)}'^-$, by the result in the previous case we have $d_{\widehat H_{(j)}'}(\boldsymbol{x}',\boldsymbol{z}')<\varepsilon/2$ and $d_{\widehat H_{(j)}'}(\boldsymbol{z}',\boldsymbol{y}')<\varepsilon/2$, and hence $$d_{\widetilde H_{(j)}'}(\boldsymbol{x}',\boldsymbol{y}')=d_{\widehat H_{(j)}'}(\boldsymbol{x}',\boldsymbol{y}')<\varepsilon.$$ Thus $\delta(\varepsilon):=\delta_1(\varepsilon/2)$ satisfies the conditions of (ii). The following result is a key of this paper. \[l\_HH\] For any $\varepsilon>0$, there exists $j_0\in\mathbb{N}$ such that, for any $j\geq j_0$, $$h(\widetilde\gamma_{m_j,n_j})\cap \widetilde H_{(j)}'\subset \mathcal{N}_\varepsilon(\widetilde\gamma_{m_j,n_j}',\widetilde H_{(j)}'),$$ where $\mathcal{N}_\varepsilon(\widetilde\gamma_{m_j,n_j}',\widetilde H_{(j)}')$ is the $\varepsilon$-neighborhood of $\widetilde\gamma_{m_j,n_j}'$ in $\widetilde H_{(j)}'$. Figure \[fig\_HH\] illustrates the situation of Lemma \[l\_HH\]. For $\sigma=\pm$, we will show that $h^{-1}(\widetilde H_{(j)}'^{\sigma}\setminus \mathcal{N}_\varepsilon(\gamma_{m_j,n_j}',\widetilde H_{(j)}'))\subset \widetilde H_{(j)}^{\sigma}$ for all sufficiently large $j$. Since $h^{-1}|_{U_{a'}(p')}$ is uniformly continuous, there exists $\nu(\varepsilon)>0$ such that, for any $\boldsymbol{x}',\boldsymbol{y}'\in U_{a'}(p')$ with $d'(\boldsymbol{x}',\boldsymbol{y}')<\nu(\varepsilon)$, the inequality $d(\boldsymbol{x},\boldsymbol{y})<\eta(\delta(\varepsilon))$ holds, where $\boldsymbol{x}=h^{-1}(\boldsymbol{x}')$, $\boldsymbol{y}=h^{-1}(\boldsymbol{y}')$. Since both $\widetilde H_{(j)}'^+$ and $\widetilde H_{(j)}'^{-}$ uniformly converge to the same half disk $H'^\natural$ in $D_{a'}(p')$, there exists $j_0\in \mathbb{N}$ such that, for any $j\geq j_0$ and any $\boldsymbol{x}' \in \widetilde H_{(j)}'^\sigma\setminus \mathcal{N}_\varepsilon(\widetilde\gamma_{(j)}',\widetilde H_{(j)}')$, $d'(\boldsymbol{x}',\boldsymbol{y}')$ is less than $\nu(\varepsilon)$, where $\boldsymbol{y}'$ is the element of $\widetilde H_{(j)}'^{-\sigma}$ with $\mathrm{pr}(\boldsymbol{x}')=\mathrm{pr}(\boldsymbol{y}')$. Then we have $d(\boldsymbol{x},\boldsymbol{y})<\eta(\delta(\varepsilon))$. If both $\boldsymbol{x}$ and $\boldsymbol{y}$ were contained in one of $\widetilde H_{(j)}^\sigma$ and $\widetilde H_{(j)}^{-\sigma}$, then by Lemma \[l\_d\_H\](i) $d_{\widetilde H_{(j)}}(\boldsymbol{x},\boldsymbol{y})<\delta(\varepsilon)$. Then, by Lemma \[l\_d\_H\](ii), $d_{\widetilde H_{(j)}'}(\boldsymbol{x}',\boldsymbol{y}')$ would be less than $\varepsilon$. This contradicts that $\boldsymbol{x}' \in \widetilde H_{(j)}'^\sigma\setminus \mathcal{N}_\varepsilon(\widetilde\gamma_{m_j,n_j}',\widetilde H_{(j)}')$ and $\boldsymbol{y}'\in \widetilde H_{(j)}'^{-\sigma}$. See Figure \[fig\_HH2\]. Thus, if $\boldsymbol{y}$ is contained in $\widetilde H_{(j)}^\sigma$, then $\boldsymbol{x}$ is not in $\widetilde H_{(j)}^\sigma$. In particular, $\boldsymbol{x}$ is not contained in $\widetilde\gamma_{m_j,n_j}=\widetilde H_{(j)}^+\cap \widetilde H_{(j)}^-$, and so $\widetilde\gamma_{m_j,n_j}\cap h^{-1}(\widetilde H_{(j)}'^\sigma\setminus \mathcal{N}_\varepsilon(\widetilde\gamma_{m,n}',\widetilde H_{(j)}'))=\emptyset$. Since $h^{-1}(\widetilde H_{(j)}'^\sigma\setminus \mathcal{N}_\varepsilon(\widetilde\gamma_{m,n}',\widetilde H_{(j)}'))$ is connected, it follows that $h^{-1}(\widetilde H_{(j)}'^\sigma\setminus \mathcal{N}_\varepsilon(\widetilde\gamma_{m,n}',\widetilde H_{(j)}'))\subset \widetilde H_{(j)}^\sigma$ for $\sigma=\pm$, and hence $h^{-1}(\mathcal{N}_\varepsilon(\widetilde\gamma_{m_j,n_j}',\widetilde H_{(j)}'))\supset \widetilde\gamma_{m_j,n_j}\cap h^{-1}(\widetilde H_{(j)}')$. This completes the proof. From the proof of Lemma \[l\_HH\], we know that there exists a simple curve in $h(\widetilde\gamma_{m_j,n_j})\cap \widetilde H_{(j)}'$ connecting the two components of $\partial \widetilde H_{(j)}'\cap \partial \mathcal{N}_\varepsilon(\widetilde\gamma_{m_j,n_j}',\widetilde H_{(j)}')$. The following corollary says that the images of certain straight segments in $D_a(p)$ by the homeomorphism $h$ are naturally straight segments in $D_{a'}(p')$. \[c\_HH\] For the limit straight segment $\gamma_0^\natural$ of $\gamma_{m_j,n_j}$, $h(\gamma_0^\natural)\cap D_{a'}(p')$ is the limit straight segment of $\gamma_{m_j,n_j}'$, i.e. $h(\gamma_0^\natural)\cap D_{a'}(p')=\gamma_0'^{\,\natural}$. Since $\gamma_0^\natural$ is the limit straight segment of $\widetilde \gamma_{m_j,n_j}$ and $h$ is uniformity continuous, $h(\gamma^\natural_0)\cap D_{a'}(p')$ is the limit of $h(\widetilde\gamma_{m_j,n_j})\cap \widetilde H_{(j)}'$. It follows from Lemma \[l\_HH\] that $h(\gamma_0^\natural)\cap D_{a'}(p')$ is also the limit of $\mathrm{pr}(\widetilde\gamma_{m_j,n_j}')=\gamma_{m_j,n_j}'$, that is, $h(\gamma_0^\natural)\cap D_{a'}(p')$ is equal to the limit straight segment of $\gamma_{m_j,n_j}'$. For any straight segment $l$ in $D_a(p)$ such that $h(l)$ is also a straight segment in $D_{b'}(p')$, we denote $h(l)\cap D_{a'}(p')$ simply by $h(l)$. In particular, Corollary \[c\_HH\] implies that $h(\gamma_0^\natural)=\gamma_0'^{\,\natural}$. Proof of Theorem \[thm\_A\] {#S_pA} =========================== Suppose that $\mathrm{St}_a(p)$ is the set of oriented proper straight segments in $D_a(p)$ passing through $0$, that is, each element of $\mathrm{St}_a(p)$ is an oriented diameter of the disk $D_a(p)$. For any $l\in \mathrm{St}_a(p)$ and $n\in \mathbb{N}$, the component of $f^n(l)\cap U_a(p)$ containing $0$ is also an element of $\mathrm{St}_a(p)$. We denote the element simply by $f^n(l)$. Since $f^n|_{D_a(p)}$ preserves angles on $D_a(p)$, by , for any $k,n\in\mathbb{N}$, $$\vartheta(\gamma_{m,n})-\vartheta(\gamma_{m+k,n})=\vartheta(\gamma_{m,0})-\vartheta(\gamma_{m+k,0}) \to \vartheta(\gamma_0)-\vartheta(\gamma_0)=0$$ as $m\to \infty$. Moreover it follows from that $\lim_{j\to\infty}d_{m_j+k,n_j}=w_0\lambda^k$. By these facts together with Lemma \[l\_gamma\_star\], one can show that $\gamma_{m_j+k,n_j}$ uniformly converges as $m\to \infty$ to a straight segment $\gamma_k^\natural$ in $U_a(p)$ with $$\label{eqn_zeta_k} \vartheta(\gamma_k^\natural)=\theta^\natural\quad\text{and}\quad d(0,\gamma_k^\natural)=w_0\lambda^k.$$ Thus we have obtained the parallel family $\{\gamma_k^\natural\}$ of oriented straight segments in $D_a(p)$. See Figure \[fig\_parallel\]. By Corollary \[c\_HH\], $\{\gamma_k'^{\,\natural}\}$ with $\gamma_k'^{\,\natural}=h(\gamma_k^\natural)$ is also a parallel family of oriented straight segments in $D_{a'}(p')$. Since $\gamma_k'^\natural$ is the limit of $\gamma_{m_j+k,n_j}'$ as $j\to\infty$, we have the equations $$\label{eqn_zeta_k'} \vartheta(\gamma_k'^{\,\natural})=\theta'^{\,\natural}\quad\text{and}\quad d(0,\gamma_k'^{\,\natural})=w_0'\lambda'^k.$$ corresponding to for some $\theta'^{\,\natural}$ and $w_0'>0$. Let $\gamma_\infty^\natural\in \mathrm{St}_a(p)$ (resp. $\gamma_\infty'^{\,\natural}\in \mathrm{St}_{a'}(p')$) be the limit of $\gamma_k^\natural$ (resp. $\gamma_k'^{\,\natural}$). By Lemma \[l\_gamma\_star\] and , $w_0=\lim_{j\to \infty}\widetilde d_0\widetilde t_0\lambda^{m_j}r^{n_j}$. This implies that $$\lim_{j\to \infty}\left(\frac{m_j}{n_j}\log \lambda+\log r\right)=\lim_{j\to\infty}\frac1{n_j}\log \frac{w_0}{\widetilde d_0\widetilde t_0}=0$$ and hence $\lim_{j\to\infty}\dfrac{m_j}{n_j}=-\dfrac{\log r}{\log \lambda}$. Applying the same argument to $\gamma_{m_j,n_j}'^{\,\natural}$, we also have $\lim_{j\to\infty}\dfrac{m_j}{n_j}=-\dfrac{\log r'}{\log \lambda'}$. This shows the part of Theorem \[thm\_A\]. Now we will prove the part . For any $n\in \mathbb{N}\cup \{0\}$, we set $f^n(\gamma_\infty^\natural)=\gamma_{\infty,n}^\natural$ and $f'^n(\gamma_\infty'^{\,\natural})=\gamma_{\infty,n}'^{\,\natural}$. By Corollary \[c\_HH\], $$\label{eqn_h_gamma} h(\gamma_{\infty,n}^\natural)=h(f^n(\gamma_\infty^\natural))=f'^n(h(\gamma_\infty^\natural))=f'^n(\gamma_\infty'^{\,\natural})= \gamma_{\infty,n}'^{\,\natural}.$$ We identify $\mathrm{St}_a(p)$ with the unit circle $S^1=\{z\in\mathbb{C}\,;\,|z|=1\}$ by corresponding $l\in \mathrm{St}_a(p)$ to $e^{\sqrt{-1}\vartheta(l)}$. Then the action of $f$ on $\mathrm{St}_a(p)$ is equal to the $\theta$-rotation $R_\theta$ on $S^1$ defined by $R_\theta(z)=e^{\sqrt{-1}\theta}z$. If $\theta/2\pi=v/u$ for coprime positive integers $u$, $v$ with $0\leq v<u$. Since $h(\gamma_\infty^\natural)=\gamma_\infty'^{\,\natural}$, we have $f'^k(\gamma_\infty'^{\,\natural})\neq \gamma_\infty'^{\,\natural}$ for $k=1,\dots,u-1$ and $f'^u(\gamma_\infty'^{\,\natural})=\gamma_\infty'^{\,\natural}$. This implies that $\theta'/2\pi=v'/u$ for some $v'\in\mathbb{N}$ with $0\leq v'<u$. Since $h|_{D_a(p)}:D_a(p)\to D_{a'}(p')$ is a homeomorphism with the correspondence $h(R_\theta^k(\gamma_\infty^\natural))=R_{\theta'}^k(\gamma_\infty'^\natural)$ $(k=0,1,\dots,u-1)$, there exists an orientation-preserving homeomorphism $\eta_0:S^1\to S^1$ with $\eta_0(e^{\sqrt{-1}(\theta^\natural+k\theta)})=e^{\sqrt{-1}(\theta'^\natural+k\theta')}$ for $k=0,1,\dots,u-1$. We set $\Gamma=\bigl\{e^{\sqrt{-1}(\theta^\natural+k\theta)};\,k=0,1,\dots,u-1\bigr\}$ and $\Gamma'=\bigl\{e^{\sqrt{-1}(\theta'^\natural+k\theta')};\,k=0,1,\dots,u-1\bigr\}$. Then $\bigl[e^{\sqrt{-1}\theta^\natural},e^{\sqrt{-1}(\theta^\natural+\theta)}\bigr)\cap \Gamma$ consists of $v$ points, where $[a,b)$ denotes the positively oriented half-open interval in $S^1$ for $a,b\in S^1$ with $a\neq b$. Since moreover $\eta_0\bigl(\bigl[e^{\sqrt{-1}\theta^\natural},e^{\sqrt{-1}(\theta^\natural+\theta)}\bigr)\cap \Gamma\bigr) =\bigl[e^{\sqrt{-1}\theta'^\natural},e^{\sqrt{-1}(\theta'^\natural+\theta')}\bigr)\cap \Gamma'$ consists of $v'$ points, it follows that $v=v'$, and hence $\theta=\theta'$. Next we suppose that $\theta/2\pi$ is irrational. Then, for any $l\in \mathrm{St}_a(p)$, there exists a subsequence $\{n_k\}$ of $\mathbb{N}$ such that the sequence $\gamma_{\infty,n_k}^\natural$ uniformly converges to $l$ as $k\to \infty$. By , $\gamma_{\infty,n_k}'^{\,\natural}$ uniformly converges to $l'=h(l)$. Since $\gamma_{\infty,n_k}'^{\,\natural}\in \mathrm{St}_{a'}(p')$, $l'$ is also an element of $\mathrm{St}_{a'}(p')$. Thus we have a homeomorphism $\eta:S^1\to S^1$ with respect to which $R_\theta$ and $R_{\theta'}$ are conjugate. Since the rotation number is invariant under topological conjugations, $\theta/2\pi=\theta'/2\pi\mod 1$ holds. This completes the proof of the part . Proof of Theorem \[thm\_B\] {#S_PB} =========================== In this section, we will prove Theorem \[thm\_B\]. Suppose that $f,f'$ are elements of $\mathrm{Diff}^r(M)$ satisfying the conditions of Theorems \[thm\_A\] and $\theta/2\pi$ is irrational. Since $\theta=\theta'\mod 2\pi$, for any $k,j\in\mathbb{N}$, $$\label{eqn_theta_gamma} \vartheta(\gamma_{\infty,k}^\natural)-\vartheta(\gamma_{\infty,j}^\natural)=\vartheta(\gamma_{\infty,k}'^{\,\natural})-\vartheta(\gamma_{\infty,j}'^{\,\natural})=(k-j)\theta\mod 2\pi.$$ Let $l_j$ $(j=1,2)$ be any elements of $\mathrm{St}_a(p)$. As in the proof of Theorem \[thm\_A\], there exist subsequences $\{n_k\}$, $\{n_j\}$ of $\mathbb{N}$ such that the sequencers $\{\gamma_{\infty,n_k}^\natural\}$, $\{\gamma_{\infty,n_j}^\natural\}$ uniformly converge to $l_1$ and $l_2$ respectively. Then, $\{\gamma_{\infty,n_k}'^{\,\natural}\}$, $\{\gamma_{\infty,n_j}'^{\,\natural}\}$ also uniformly converge to the elements $l_1'=h(l_1)$ and $l_2'=h(l_2)$ of $\mathrm{St}_{a'}(p')$ respectively. Then, by , $$\label{eqn_theta_l} \vartheta(l_2)-\vartheta(l_1)=\vartheta(l_2')-\vartheta(l_1')\mod 2\pi.$$ For the proof of Theorem \[thm\_B\], we need another family of straight segments in $D_a(p)$. Fix an integer $a_0$ with $$a_0>\max\left\{\frac{\log(2r)}{\log (\lambda^{-1})},\frac{\log(2r')}{\log (\lambda'^{-1})}\right\}.$$ For any $k\geq 0$, we consider the straight segment $\xi_k^\natural=f^k(\gamma_{a_0k}^\natural)\cap D_a(p)$. By , $$\label{eqn_theta_xi} \vartheta(\xi_k^\natural)-\vartheta(\xi_0^\natural)=k\theta\mod 2\pi \quad\text{and}\quad d(0,\xi_k^\natural)=w_0\lambda^{a_0k}r^k<2^{-k}w_0.$$ Similarly, by , $\xi_k'^{\,\natural}=h(\xi_k^\natural)$ is a straight segment in $D_{a'}(p')$ with $$\label{eqn_theta_xi'} \vartheta(\xi_k'^{\,\natural})-\vartheta(\xi_0'^{\,\natural})=k\theta\mod 2\pi \quad\text{and}\quad d(0,\xi_k'^{\,\natural})=w_0'\lambda'^{a_0k}r'^k<2^{-k}w_0'.$$ Let $\alpha$ be the element of $\mathrm{St}_a(p)$ with $\vartheta(\xi_0^\natural)-\vartheta(\alpha)=\pi/2$ and $\alpha'=h(\alpha)\in \mathrm{St}_{a'}(p')$. We will show that $\theta_{\alpha'}:=\vartheta(\xi_0'^{\,\natural})-\vartheta(\alpha')$ is also equal to $\pi/2\mod 2\pi$. See Figure \[fig\_beta\]. In fact, since $\theta/2\pi$ is irrational, by there exists a subsequence $\xi_{k_j}^\natural$ uniformly converges to $\alpha$. Since $h|_{D_a(p)}$ is uniformly continuous, $\xi_{k_j}'^{\,\natural}$ also uniformly converges to $\alpha'$. On the other hand, since $\vartheta(\xi_{k_j}^\natural)-\vartheta(\alpha)=k_j\theta+\pi/2\mod 2\pi$ and $\vartheta(\xi_{k_j}'^{\,\natural})-\vartheta(\alpha')=k_j\theta+\theta_{\alpha'}\mod 2\pi$, $$\theta_{\alpha'}-\frac{\pi}2=\bigl(\vartheta(\xi_{k_j}'^{\,\natural})-\vartheta(\alpha')\bigr) -\bigl(\vartheta(\xi_{k_j}^\natural)-\vartheta(\alpha)\bigr)\to 0\mod 2\pi$$ as $j\to\infty$. Thus we have $\theta_{\alpha'}=\pi/2\mod 2\pi$. We denote by $z(\boldsymbol{x})\in \mathbb{C}$ the entry of $\boldsymbol{x}\in D_{a}(p)$ with respect to the linearizing coordinate on $D_a(p)$. Similarly, the entry of $\boldsymbol{x}'\in D_{a'}(p')$ is denoted by $z'(\boldsymbol{x}')$. Let $\boldsymbol{x}_0$ be the intersection point of $\alpha$ and $\xi_0^\natural$, and let $\boldsymbol{x}_0'=h(\boldsymbol{x}_0)$. One can set $z(\boldsymbol{x}_0)=\rho_0e^{\sqrt{-1}\omega_0}$ and $z'(\boldsymbol{x}_0')=\rho_0'e^{\sqrt{-1}\omega_0'}$ for some $\rho_0>0$, $\rho_0'>0$ and $\omega_0$, $\omega_0'\in \mathbb{R}$. We define the new linearizing coordinate on $D_{a'}(p')$ by using the linear conformal map such that, for any $\boldsymbol{x}'\in D_{a'}(p')$, $z'^{\,\mathrm{new}}(\boldsymbol{x}')=\rho_0\rho_0'^{-1}e^{\sqrt{-1}(\omega_0-\omega_0')}z'(\boldsymbol{x}')$. Then $z(\boldsymbol{x}_0)=z'^{\,\mathrm{new}}(\boldsymbol{x}_0')$ holds. For any $\boldsymbol{x}\in \xi_0^\natural$, there exists $l\in \mathrm{St}_a(p)$ with $\{\boldsymbol{x}\}=\xi_0^\natural \cap l$. Then $\boldsymbol{x}'=h(\boldsymbol{x})$ is the intersection of $\xi_0'^{\,\natural}$ and $l'=h(l)$. By , $\vartheta(l)-\vartheta(\alpha)=\vartheta(l')-\vartheta(\alpha')\mod 2\pi$ and hence $z(\boldsymbol{x})=z'^{\,\mathrm{new}}(\boldsymbol{x}')$. We say the property that $h$ is *identical* on $\xi_0^\natural$. Since $\theta/2\pi$ is irrational, there exists $k_*\in\mathbb{N}$ satisfying $$\frac{\pi}3\leq \vartheta(\xi_{k_*}^\natural)-\vartheta(\xi_0^\natural)\leq \frac{\pi}2\mod 2\pi.$$ Then $\xi_{k_*}^\natural$ meets $\xi_0^\natural$ at a single point $\boldsymbol{x}_{k_*}$ in $D_a(p)$. For $\alpha_{k_*}=f^{k_*}(\alpha)$ and $\alpha_{k_*}'=h(\alpha_{k_*})$, we have $\vartheta(\xi_{k_*}^\natural)-\vartheta(\alpha_{k_*})=\vartheta(\xi_{k_*}'^{\,\natural})-\vartheta(\alpha_{k_*}')=\pi/2$. Since $h$ is identical at $\boldsymbol{x}_{k_*}$, $h$ is proved to be identical on $\xi_{k_*}^\natural$ by an argument as above. Then one can show inductively that, for any $n\in\mathbb{N}$, $h$ is identical on $\xi_{nk_*}^\natural$. See Figure \[fig\_beta\_2\]. By , $\lim_{n\to\infty}d(0,\xi_{nk_*}^\natural)=0$. Since moreover $k_*\theta/2\pi$ is irrational, $\overline{\bigcup_{n=1}^\infty \xi_{nk_*}^\natural}$ is equal to $D_a(p)$. This shows that $h$ is identical on $D_a(p)$. In particular, this implies that $h|_{D_a(p)}$ is a linear conformal map with respect to the original coordinates. We write $z(q)=\rho_1e^{\sqrt{-1}\omega_1}$ and $z'(q')=\rho_1'e^{\sqrt{-1}\omega_1'}$. It follows from the assumption of $h(q)=q'$ in our theorems that $h(z)=\rho_1'\rho_1^{-1}e^{\sqrt{-1}(\omega_1'-\omega_1)}z$ for any $z\in\mathbb{C}$ with $|z|\leq a$. In particular, this implies that $h|_{W_{\mathrm{loc}}^u(p)}$ is a linear conformal map. Let $\widetilde h$ be any other conjugacy homeomorphism between $f$ and $f'$ satisfying the conditions in Theorems \[thm\_A\] and \[thm\_B\]. In particular, $\widetilde h(p)=p'$ and $\widetilde h(q)=q'$ hold. Since $z(q)=\rho_1e^{\sqrt{-1}\omega_1}$ and $z'(q')=\rho_1'e^{\sqrt{-1}\omega_1'}$, one can show as above that $\widetilde h(z)=\rho_1'\rho_1^{-1}e^{\sqrt{-1}(\omega_1'-\omega_1)}z$ for any $z\in\mathbb{C}$ with $|z|\leq a$ and hence $\widetilde h|_{D_a(p)}=h|_{D_a(p)}$. This shows the assertion of Theorem \[thm\_B\] and $r=r'$. Then, by the assertion of Theorem \[thm\_A\], we also have $\lambda=\lambda'$. This completes the proof. Let $\widehat z$ be the homoclinic transverse point of $W^u(p)$ and $W^s(p)$ given in Subsection \[ss\_bent\_disk\]. Fix a sufficiently large $n\in \mathbb{N}$ with $s=f^{-n}(\widehat z)\in D_p(a)$. Then $s'=h(s)$ is contained in $D_{b'}(p')$. The following corollary shows that $z(s)/z(q)$ is a modulus for $f$. Recall that $z(\boldsymbol{x})\in\mathbb{C}$ is the entry of $\boldsymbol{x}$ with respect to the complex linearizing coordinate on $D_a(a)$. The complex number $z'(\boldsymbol{x}')$ is defined similarly for $\boldsymbol{x}'\in D_{a'}(p')$. \[c\_C\] Let $f$, $f'$ be elements of $\mathrm{Diff}^r(M)$ satisfying the conditions of Theorems \[thm\_A\] and \[thm\_B\], and let $h$ be a conjugacy homeomorphism between $f$ and $f'$ with $h(p)=p'$ and $h(q)=q'$. If $h|_{W_{\mathrm{loc}}^u(p)}$ is orientation-preserving, then $z(s)/z(q)=z'(s')/z'(q')$. Otherwise, $z(s)/z(q)=\overline{z'(s')/z'(q')}$. Here we only consider the case that $h$ is orientation-preserving. Since $h|_{D_a(p)}$ is a linear conformal map, the triangle with vertices $0,z(q),z(s)$ is similar to that with vertices $0,z'(q'),z'(s')$ with respect to Euclidean geometry. This shows $z(s)/z(q)=z'(s')/z'(q')$. Acknowledgement {#acknowledgement .unnumbered} --------------- The authors would like to thank the referee and the editors for helpful comments and suggestions. [99]{} C. Bonatti and E. Dufraine, Équivalence topologique de connexions de selles en dimension 3, Ergodic Theory Dynam. Sys. **23** (2003), no. 5, 1347–1381. M. Carvalho and A. Rodrigues, Complete set of invariants for a Bykov attractor, Regular Chaotic Dynam. **23** (2018), no. 3, 227–247. W. de Melo, Moduli of stability of two-dimensional diffeomorphisms, Topology **19** (1980), no. 1, 9–21. W. de Melo and J. Palis, Moduli of stability for diffeomorphisms, Global theory of dynamical systems, Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979, pp. 318–339, Lecture Notes in Math., 819, Springer, Berlin, 1980. W. de Melo and S.J. van Strien, Diffeomorphisms on surfaces with a finite number of moduli, Ergodic Theory Dynam. Sys. **7** (1987), no. 3, 415–462. E. Dufraine, Some topological invariants for three-dimensional flows, Chaos **11** (2001), 443–448. E. Dufraine, Un critère d’existence d’invariant pour la conjugaison de difféomorphismes et de champs de vecteurs, C. R. Math. Acad. Sci. Paris **334** (2002), no. 1, 53–58. V.Z. Grines, O.V. Pochinka and S.J. van Strien, On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J. **16** (2016), no. 4, 727–749. S. Hashimoto, Moduli of surface diffeomorphisms with cubic tangencies, Tokyo J. Math. (to appear). T.M Mitryakova and O.V. Pochinka, On necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency, Proc. Steklov Inst. Math. [**270**]{} (2010), no. 1, 194–215. T.M Mitryakova and O.V. Pochinka, Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies, Trans. Moscow Math. Soc. (2016), 69–86. S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. [**57**]{} (1983), 5–71. Y. Nishizawa, Existence of horseshoe sets with nondegenerate one-sided homoclinic tangencies in $\mathbb{R}^3$, Hokkaido Math. J. **37** (2008), no. 1, 133–145. J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, *Dynamical systems*, Vol. III–Warsaw, pp. 335–346, Astérisque, No. 51, Soc. Math.  France, Paris, 1978. R.A. Posthumus, Homoclinic points and moduli, Ergodic Theory Dynam. Sys. **9** (1989), no. 2, 389–398. R.A. Posthumus and F. Takens, Homoclinic tangencies: moduli and topology of separatrices, Ergodic Theory Dynam. Sys. **13** (1993), no. 2, 369–385. A. Rodrigues, Moduli for heteroclinic connections involving saddle-foci and periodic solutions, Discrete Cont. Dyn. Sys. **35** (2015), no. 7, 3155–3182. F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil.  Mat. **25** (1994), 107–120. Y. Togawa, A modulus of $3$-dimensional vector fields, Ergodic Theory Dynam. Sys. **7** (1987), no. 2, 295–301. [^1]: This work was partially supported by JSPS KAKENHI Grant Numbers 17K05283 and 26400093.

--- abstract: 'For a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic $L$-functions. In this paper we explicitly calculate archimedean zeta integrals of this type for $U(n,1)$, assuming that the corresponding archimedean component of the automorphic representation is a holomorphic discrete series.' address: - 'School of Mathematics, Sichuan University, Chengdu 610065, P.R. China' - 'School of Mathematical Science, Zhejiang University, Hangzhou 310027, P.R. China' author: - Bingchen Lin - Dongwen Liu title: 'Archimedean zeta integrals on $U(n,1)$' --- Introduction ============ In order to obtain deep arithmetic applications in the theory of automorphic forms, it is often necessary to have explicit computable results at each place of a number field. This paper is concerned with certain archimedean zeta integrals on unitary groups and central $L$-values, which arise from theta correspondence of cuspidal automorphic representations. We shall briefly explain the motivation and background of this paper, following [@H; @HLS; @L2]. Let $F^+$ be a totally real number field, $F$ a totally imaginary quadratic extension of $F^+$, ${\bf A}={\bf A}_{F^+}$ the adele ring of $F^+$. Let $V$ (resp. $V'$) be a hermitian (resp. skew-hermitian) vector space of dimension $n+1$ over $F$, and $W=V\otimes_{F}V'$, a symplectic space over $F^+$. Fixing an additive character $\psi$ and a complete polarization $W=X\oplus Y$, we have the Schrödinger model of the oscillator representation $\omega_\psi$ of $\widetilde{Sp}(W)({\bf A})$, realized on the space ${\mathcal S}(X({\bf A}))$ of Schwartz-Bruhat functions on $X({\bf A})$. Let $G=U(V),$ $G'=U(V')$. By choosing a global splitting character $\chi$ of ${\bf A}_{F}^\times/F^\times$ as in [@HLS], $\omega_\psi$ then defines an oscillator representation $\omega_{V,V',\chi}$ of $G({\bf A}) \times G'({\bf A})$ on ${\mathcal S}(X({\bf A}))$. As usual, for $\phi\in {\mathcal S}(X({\bf A}))$ we have the theta lifting $f\mapsto \theta_\phi(f)$ for a cusp form $f$ on $G(F^+)\backslash G({\bf A})$. Let $\pi$ be a cuspidal automorphic representations of $G$, $f\in\pi$, $\tilde{f}\in\pi^\vee$. Let $H=U(V\oplus (-V))$, $i_V: G\times G\hookrightarrow H$ be the natural inclusion, following the doubling method. The Piatetski-Shapiro-Rallis zeta integral is then defined by $$Z(s, f, \tilde{f},\varphi,\chi)=\int_{(G\times G)(F^+)\backslash (G\times G)({\bf A})}E(i_V(g,\tilde{g}), s, \varphi, \chi)f(g)\tilde{f}(\tilde{g})\chi^{-1}\big(\det(\tilde{g})\big)dg d\tilde{g},$$ where $E(\cdot, s,\varphi, \chi)$ is the Eisenstein series on $H({\bf A})$ as in [@H §1], and $\varphi=\varphi(s)$ is a section of a degenerate principal series $I_{n+1}(s,\chi)$ varying in $s$. This integral converges absolutely for $\textrm{Re }s\gg0$ and admits an Euler expansion if $\varphi$, $f$ and $\tilde{f}$ are factorizable. In this case, for $S$ a sufficiently large finite set of places of $F^+$ including archimedean ones, one has $$Z(s, f, \tilde{f}, \varphi, \chi)=\prod_{v\in S}Z(s, f_v, \tilde{f}_v, \varphi_v, \chi_v)d_{n+1}^S(s)^{-1}L^S(s+\frac{1}{2},\pi, St,\chi),$$ where $L^S(s+\frac{1}{2},\pi, St,\chi)$ is the partial $L$-function of $\pi$ twisted by $\chi$, attached to $2(n+1)$-dimensional standard representation of the $L$-group, and $d_{n+1}^S(s)$ is a product of certain partial $L$-functions attached to the extension $F/F^+$ as in [@H]. Take $\phi=\otimes_v\phi_v\in {\mathcal S}(X({\bf A}))$ and $\varphi=\delta(\phi\otimes\bar{\phi})$ in the notation of [@L2 p.182]. Then after proper normalization the [*Rallis inner product formula*]{} can be written as $$\langle\theta_\phi(f),\theta_{\bar{\phi}}(\bar{f})\rangle=\prod_{v\in S}Z(0, f_v,\bar{f}_v,\varphi_v,\chi_v)d_n^S(0)^{-1}L^S(\frac{1}{2},\pi, St,\chi),$$ The central $L$-value $L(\frac{1}{2},\pi, St,\chi)$ is of great arithmetic interest and it is quite useful to have explicit local value at each place. In [@HLS] under certain assumptions it is shown that $L^S(\frac{1}{2},\pi, St,\chi)\geq 0$ for any finite set $S$ of places of $F^+$. As explained in [@L2 §2], one has $$Z(0,f_v, \bar{f}_v, \varphi_v,\chi_v)=\int_{G(F_v^+)}(\omega_{\chi_v} (g)\phi_v, \phi_v) (\pi_v(g)f_v, f_v)dg,$$ which integrates matrix coefficient of the oscillator representation against that of $\pi_v$. From now on we assume that $v$ is real, $\pi_v$ is in the discrete series, $\phi_v$ is in the space of joint harmonics, and we replace $(\pi_v(g)f_v, f_v)$ by a canonical matrix coefficient $\psi_{\pi_v}(g)$ of $\pi_v$ (see Section \[s4\]). The aim of this paper is to explicitly compute the archimedean zeta integral $$\label{zeta'} \int_{G(F_v^+)}(\omega_{\chi_v}(g)\phi_v,\phi_v)\cdot \psi_{\pi_v}(g)dg$$ in the case that $G(F_v^+)$ is the real unitary group $U(n,1)$ and $\pi_v$ is a holomorphic discrete series. We mention that the cases $U(1,1)$ and $U(2,1)$ were solved completely in [@Lin] and [@Liu] respectively. However for $U(n,1)$ when $\pi_v$ is a general discrete series, this problem seems to out of reach at the moment. The main results of this paper can be formulated as follows. Fix an additive character $\psi$ of ${\bf R}$. Let $V$ be an $(n+1)$-dimensional complex Hermitian space, and let $G$ be the unitary group attached to $V$. For each complex skew-Hermitian space $V'$, the group $G$ is a subgroup of the real symplectic group $Sp(V\otimes_{\bf C} V')$ as usual. Define the metaplectic double cover $\widetilde{G}$ of $G$ to be the double cover of $G$ induced by the metaplectic double cover $\widetilde{Sp}(V\otimes_{\bf C}V')\to Sp(V \otimes_{\bf C}V')$. This is independent of $V'$. Let $\pi_\lambda$ be the genuine discrete series representation of $\widetilde{G}$ with Harish-Chandra parameter $\lambda:=(\lambda_1,\ldots, \lambda_{n+1})$. By theta dichotomy for real unitary groups [@P] and a result in [@L1] on discrete spectrum of local theta correspondence, up to isometry there exists a unique $(n+1)$-dimensional skew-Hermtian space $V'$ such that $\pi^\vee_\lambda$ occurs as a subrepresentation of $\omega_{V,V',\chi}$. Let $P_\lambda:\omega_{V,V',\chi}\to \omega_{V,V',\chi}$ be the orthogonal projection to the $\pi^\vee_\lambda$-isotypic subspace. Fix a maximal compact subgroup $K$ of $G$, which induces a maximal compact subgroup $\widetilde{K}$ of $\widetilde{G}$. Denote by $\tau^\vee_\lambda$ the lowest $\widetilde{K}$-type of $\pi^\vee_\lambda$. Then there is a positive number $c_{\psi, V, \lambda}$ such that $$\| P_\lambda(\phi)\| = c_{\psi, V, \lambda} \|\phi\|$$ for all $\phi$ in the $\tau^\vee_\lambda$-isotypic subspace of the space of joint harmonics (with respect to $K$ and an arbitrary maximal compact subgroup of the unitary group attached to $V'$). The constant $c_{\psi,V,\lambda}$ is 1 when either $V$ or $V'$ is anisotropic. The main result of this paper is equivalent to an explicit calculation of $c_{\psi, V, \lambda}$ when $V$ is of signature $(n,1)$, $\psi$ is chosen to be $\psi_a: t\mapsto e^{2\pi i a t}$ for some $a>0$, and $\pi_\lambda$ is holomorphic. In this case we list the explicit values of $c_{\psi,V,\lambda}$ below (Corollary \[7.3\]). Follow above notations, assume that $V$ has signature $(n,1)$, $\psi=\psi_a$ for some $a>0$ and $\pi_\lambda$ is holomorphic. Let $$\Lambda=\lambda+(-\frac{n}{2}+1,-\frac{n}{2}+2,\ldots, \frac{n}{2}, -\frac{n}{2}).$$ Let $\alpha$’s, $\beta$’s, $\gamma$, $p, q$ below stand for non-negative integers with $p+q=n+1$. Then\ (i) if $\Lambda=[(\alpha_1,\ldots,\alpha_n)+\det^{-(1-n)/2}]\otimes[\gamma+\det^{(1-n)/2}]$ with $\alpha_1\geq\cdots\geq \alpha_n\geq\gamma+2$, then $$c^2_{\psi,V,\lambda}=\prod^n_{i=1}\frac{\alpha_i-i+n-1-\gamma}{\alpha_i-i+n};$$ (ii) if $\Lambda=[(\alpha_1,\ldots,\alpha_{q-1},-\beta_p,\ldots,-\beta_1)+\det^{-(p-q)/2}]\otimes[-\gamma+\det^{(p-q)/2}]$ with $\alpha_1\geq\cdots\geq\alpha_{q-1}\geq -\beta_p\geq\cdots\geq -\beta_1\geq -\gamma+2p$, then $$c^2_{\psi,V,\lambda}=\prod^n_{i=1}\frac{\gamma+i-\delta_i-2p}{\gamma+i-p},$$ where $(\delta_1,\ldots,\delta_n):=(\beta_1,\ldots,\beta_p,-\alpha_{q-1},\ldots,-\alpha_1)$. The organization of the paper is as follows. In section \[s2\] we give the pair of weights appearing in the local theta correspondence. Section \[s3\] describes the structure and measure of the real Lie group $U(n,1)$. Section \[s4\] deals with the canonical matrix coefficient of a holomorphic discrete series following [@G]. Sections \[s5\] and \[s6\] are concerned with the matrix coefficient of oscillator representation, which is calculated using joint harmonics. In section \[s7\] we combine previous results and apply the technique of [@G] to evaluate the zeta integral. We remark that the method of this paper should be applicable to general $U(p,q)$, at least when one of the components of the lowest $\widetilde{K}$-type is one-dimensional. Furthermore, it also brings us some enlightenment to study certain period integrals for unitary groups. [**Notations.**]{} Let $1_n$ and $0_n$ be the $n\times n$ identity matrix and zero matrix respectively. Let $1_{p,q}$ stand for the square matrix $$\begin{pmatrix} 1_p & 0 \\ 0 & -1_q\end{pmatrix}.$$ In this paper, $U(p,q)$ is the real unitary group of the hermitian or skew-hermitian form represented by the matrix $1_{p,q}$ or $i1_{p,q}$, where $i=\sqrt{-1}$, and $Sp_{2N}({\bf R})$ is the isometry group of the real symplectic form represented by the matrix $$\begin{pmatrix} 0 & 1_N \\ -1_N & 0\end{pmatrix}.$$ For a complex matrix $g$, let ${}^tg$ be its transpose, and $g^*={}^t\bar{g}$ be the complex conjugate transpose. For a field $k$, $M_n(k)$ is the set of $n\times n$ matrices with entries in $k$. We usually regard vectors in $k^n$ as column vectors, unless otherwise specified. For $u, v\in k^n$, as usual $u\cdot v$ stands for their dot product, and $|u|^2=u\cdot \bar{u}$ if $k={\bf R}$ or ${\bf C}$. [**Acknowledgement.**]{} Both authors would like to thank Professor J.-S. Li for suggesting this problem. This joint work was started during the workshop “Automorphic Forms, Geometry and Representation Theory" held at Zhejiang University in July 2015. Both authors thank the hospitality of the organizers. Pair of weights {#s2} =============== Let $G=U(n,1)$ be the unitary group of a complex hermitian space of signature $(n,1)$. The absolute root system of $G_{\bf{C}}=GL(n+1,\bf{C})$ is of type $A_n$. Fix the maximal compact subgroup $K=U(n)\times U(1)$, the set of compact positive roots $\Delta^+_c=\{e_i-e_j: 1\leq i<j\leq n\}$, and the set of positive roots $\Delta^+=\{e_i-e_j: 1\leq i<j\leq n+1\}$ that contains $\Delta_c^+$. We consider $\Delta_c^+$-dominant Harish-Chandra parameters of genuine discrete series of $\widetilde{G}$. Those of holomorphic discrete series are in fact $\Delta^+$-dominant, i.e. strictly decreasing $(n+1)$-tuples $\lambda=(\lambda_1,\ldots,\lambda_{n+1})$ of half-integers. The corresponding lowest $\widetilde{K}$-type is given by the Blattner parameter $$\label{hc} \Lambda=\lambda+\rho-2\rho_c=\lambda+(-\frac{n}{2}+1,-\frac{n}{2}+2,\ldots, \frac{n}{2}, -\frac{n}{2}),$$ where $\rho$ (resp. $\rho_c$) is the half sum of all positive (resp. compact positive) roots. Consider the dual pair $(G,G')=(U(n,1),U(p,q))\hookrightarrow Sp_{2N}(\bf{R})$, where $p+q=n+1$ and $N=(n+1)^2$. Fix the additive character $\psi: t\mapsto e^{2\pi it}$ of ${\bf R}$, and consider the oscillator representation $\omega_\psi$ of $\widetilde{Sp}_{2N}({\bf R})$. Take an irreducible $\widetilde{K}\times \widetilde{K}'$-module $$\mathcal{H}_{\Lambda^\vee,\Lambda'}\cong \sigma_{\Lambda^\vee}\otimes \sigma_{\Lambda'}$$ that occurs in the space of joint harmonics of $\omega_\psi$, where $\Lambda^\vee$ and $\Lambda'$ are the highest weights of $\sigma_{\Lambda^\vee}$ and $\sigma_{\Lambda'}$ respectively. It is well-known that $\mathcal{H}_{\Lambda^\vee, \Lambda'}$ occurs with multiplicity one, and moreover $\Lambda^\vee$ and $\Lambda'$ determine each other. Let $\sigma_\Lambda$ be the contragradient of $\sigma_{\Lambda^\vee}$, which has highest weight $\Lambda$. Assume that $\sigma_\Lambda$ is the lowest $\widetilde{K}$-type of the holomorphic discrete series $\pi_\lambda$ of $\widetilde{G}$ so that $\lambda$ and $\Lambda$ are related by (\[hc\]), and that the theta lifting $\pi'=\theta(\pi_{\lambda}^\vee)$ of $\pi_\lambda^\vee$ is a non-zero discrete series of $\widetilde{G}'$. The Harish-Chandra parameter of the anti-holomorphic discrete series $\pi_\lambda^\vee$ is $\lambda^\vee=(-\lambda_n,\ldots, -\lambda_1, -\lambda_{n+1})$, and one has $$\Lambda^\vee=\lambda^\vee+(-\frac{n}{2},-\frac{n}{2}+1,\ldots, \frac{n}{2}-1, \frac{n}{2}).$$ Let $a$ and $b$ be the number of non-negative entries in $(-\lambda_n,\ldots, -\lambda_1)$ and $(-\lambda_{n+1})$ respectively. Then by [@L1], above assumption requires that $$\lambda_n> \lambda_{n+1}\quad\textrm{and}\quad p=a-b+1.$$ Let us write $$\label{lamv} \Lambda^\vee=[(\beta_1,\ldots,\beta_l,0,\ldots,0,-\alpha_k,\ldots,-\alpha_1)+\textrm{det}^{(p-q)/2}] \otimes [m+\textrm{det}^{-(p-q)/2}],$$ where $\alpha_1\geq\cdots\geq \alpha_k>0$, $\beta_1\geq\cdots\geq \beta_l>0$. Then $$\Lambda=[(\alpha_1,\ldots,\alpha_k,0,\ldots,0,-\beta_l,\ldots,-\beta_1)+\textrm{det}^{-(p-q)/2}] \otimes[-m+\textrm{det}^{(p-q)/2}].$$ We have two cases. Case (i): $b=0$. Then $\lambda_n>\lambda_{n+1}>0$, which implies that $a=p-1=0$ hence $p=1$, $q=n$, i.e. $G'=U(1,n)$. The first entry of $\Lambda^\vee$ is $$-\lambda_n-\frac{n}{2}<\frac{p-q}{2}=\frac{1-n}{2},$$ which by (\[lamv\]) implies that $l=0$, $k=n$. Let $$\gamma:=-m=\lambda_{n+1}-\frac{1}{2}\geq 0.$$ By the formulas for the pair of weights $\Lambda^\vee$, $\Lambda'$ in [@L1], we see that $$\label{w1} \left\{\begin{array}{ll} \Lambda^\vee=[(-\alpha_n,\ldots,-\alpha_1)+\textrm{det}^{(1-n)/2}] \otimes [-\gamma+\textrm{det}^{-(1-n)/2}], \\ \Lambda'=[-\gamma+\textrm{det}^{(n-1)/2}]\otimes[(-\alpha_n,\ldots,-\alpha_1)+\textrm{det}^{-(n-1)/2}].\end{array}\right.$$ The condition $\lambda_n>\lambda_{n+1}$ reads $$\alpha_n\geq\gamma+2.$$ Case (ii): $b=1$. Then $\lambda_{n+1}\leq 0$, $a=p$. Let $$\gamma:=m=-\lambda_{n+1}+\frac{n}{2}+\frac{p-q}{2}>0.$$ Again by [@L1] we have $$\Lambda'=[(\beta_1,\ldots,\beta_l,0,\ldots,0)+\textrm{det}^{(n-1)/2}]\otimes[(\gamma,0,\ldots,0, -\alpha_k,\ldots,-\alpha_1)+\textrm{det}^{(1-n)/2}].$$ Note that the obvious constraints $l\leq p$, $k+1\leq q$ apply. For convenience let us define $\beta_i=0$, $\alpha_j=0$ for $l<i\leq p$ and $k<j\leq q-1$, so that we may write $$\label{w2} \left\{\begin{array}{ll} \Lambda^\vee=[(\beta_1,\ldots,\beta_p, -\alpha_{q-1},\ldots,-\alpha_1)+\textrm{det}^{(p-q)/2}] \otimes [\gamma+\textrm{det}^{-(p-q)/2}], \\ \Lambda'=[(\beta_1,\ldots,\beta_p)+\textrm{det}^{(n-1)/2}]\otimes[(\gamma, -\alpha_{q-1},\ldots,-\alpha_1)+\textrm{det}^{-(n-1)/2}]. \end{array}\right.$$ The condition $\lambda_n> \lambda_{n+1}$ reads $$-\beta_1\geq-\gamma+2p.$$ Structure of $G$ {#s3} ================ Let $\frak{g}=\frak{u}(n,1)$ be the Lie algebra of $G$, and $\frak{g}=\frak{k}\oplus\frak{p}$ be the Cartan decomposition with respect to the Cartan involution $\theta(X)=-X^*$. Let $\frak{a}$ be the maximal abelian subalgebra of $\frak{p}$, which is one-dimensional and spanned by, say, $$H=E_{1,n+1}+E_{n+1,1},$$ where $E_{ij}$ is the elementary matrix with $1$ on the $(i,j)$-entry and $0$ everywhere else. Let $$a_t=\exp(tH)=\begin{pmatrix} \cosh t & 0 & \sinh t \\ 0 & 1_{n-1} & 0\\ \sinh t & 0 & \cosh t \end{pmatrix}.$$ The Cartan decomposition of $G$ is $G=C\cdot K \cong C\times K$, where $$C=\{g\in G: g=g^* \textrm{ is positive-definite hermitian}\}.$$ We normalize the measure on $K=U(n)\times U(1)$ so that the masses of $U(n)$ and $U(1)$ are both equal to 1. The set $C$ can be parametrized by $$\label{c} D_{n,1}\to C,\quad z\mapsto h_z=\begin{pmatrix} (1_n-zz^*)^{-1/2} & z(1-z^*z)^{-1/2}\\ (1-z^*z)^{-1/2}z^* & (1-z^*z)^{-1/2}\end{pmatrix}$$ where $D_{n,1}$ is the classical domain $$D_{n,1}=\{z\in {\bf C}^n: 1_n-zz^*\textrm{ is positive definite}\}.$$ $G$ acts on $D_{n,1}$ by generalized fractional linear transformations, and we fix the invariant measure on $D_{n,1}$ to be $$d^*z=\frac{dz}{(1-z^*z)^{n+1}}=\frac{dz}{\det(1_n-zz^*)^{n+1}},$$ where $dz$ is the product of the usual additive Haar measures. One may further parametrize $D_{n,1}$ by $z=x\underline{r}y$, where $x\in U(n)$, $y\in U(1)$, and ${\underline}{r}={}^t(r, 0, \ldots, 0)$ with $-1<r<1$. If we write $r=\tanh t$, $t\in{\bf R}$, then substituting this parametrization into (\[c\]) yields $$\label{mea} h_z=k_z a_t k_z^{-1},\quad k_z=\begin{pmatrix} x & 0 \\ 0 & y^*\end{pmatrix}\in K.$$ Holomorphic discrete series {#s4} =========================== We shall briefly review the treatment in [@G]. Recall that $\frak{g}$ is the Lie algebra of $G$, and let $\frak{g}_{\bf C}$ be its complexification. Let $$\frak{p}_+=\left\{\begin{pmatrix} 0_n & * \\ 0 & 0\end{pmatrix}\in\frak{g}_{\bf C}\right\},\quad \frak{p}_-=\left\{\begin{pmatrix} 0_n & 0 \\ * & 0\end{pmatrix}\in\frak{g}_{\bf C}\right\}$$ and $N_\pm = \exp\frak{p}_\pm$. Then one has the Harish-Chandra decomposition $$G\subset N_+\cdot K_{\bf C}\cdot N_-\subset G_{\bf C}.$$ Let $\pi=\pi_\lambda$ be a holomorphic discrete series with lowest $K$-type $\sigma=\sigma_\Lambda$. In [@G] it is shown that the canonical $K$-conjugation invariant matrix coefficient of $\pi$ is given by $$\psi_\pi(g)=\psi_\pi (n_+\theta n_-)=\textrm{tr } \sigma(\theta) \in \sigma \otimes \sigma\subset \pi\otimes \pi^\vee \subset L^2(G)$$ if $g=n_+\theta n_-$ under the Harish-Chandra decomposition. Here we use the holomorphic extension of $\sigma$ to $K_{\bf C}$. We remark that $\psi_\pi$ is equivalent to the canonical matrix coefficient considered in [@F-J; @HLS; @L1; @Liu]. Recall the Cartan decomposition $g=h_z k$. The Harish-Chandra decomposition $h_z=n_z^+\theta_z n_z^-$ is $$h_z=\begin{pmatrix} 1_n & z \\ 0 & 1\end{pmatrix}\begin{pmatrix} (1_n-zz^*)^{1/2} & 0 \\ 0 & (1-z^*z)^{-1/2}\end{pmatrix}\begin{pmatrix} 1_n & 0 \\ z^* & 1 \end{pmatrix}.$$ In particular $$\label{thetaz} \theta_z=\begin{pmatrix} (1_n-zz^*)^{1/2} & 0 \\ 0 & (1-z^*z)^{-1/2}\end{pmatrix}.$$ Then we have $$\label{dismc} \psi_\pi(g)=\psi_\pi (n^+_z\theta_z k k^{-1}n^-_z k)=\textrm{tr }\sigma(\theta_z k),$$ noting that $K$ normalizes $N_\pm$. Parametrizing $z\in D_{n,1}$ as in Section \[s3\], we may write $$\theta_z= k_z \theta_t k_z^{-1},$$ where $k_z$ is as in (\[mea\]) and $\theta_t$ is the $K_{\bf C}$-component of $a_t$ under Harish-Chandra decomposition, i.e. $$\label{thetat} \theta_t= \begin{pmatrix} (\cosh t)^{-1} & 0 & 0 \\ 0 & 1_{n-1} & 0 \\ 0 & 0 & \cosh t\end{pmatrix}.$$ Therefore one may further write $$\label{dismc2} \psi_\pi(g)=\textrm{tr }\sigma(k_z\theta_t k_z^{-1}k )=\textrm{tr }\sigma(\theta_t k_z^{-1}k k_z).$$ Finally we remark that by Corollary of [@H-C Lemma 23.1], the formal degree $d_\pi$ of a general discrete series $\pi=\pi_\lambda$ is given by $$\label{fd} d_\pi=C\prod_{1\leq i<j\leq n+1}|\lambda_i-\lambda_j|$$ where $C$ is a constant depending on the choice of the Haar measure of $G$. Fock model {#s5} ========== The smooth model $\omega^\infty_\psi$ of the oscillator representation of $\widetilde{Sp}_{2N}({\bf R})$ can be realized on the Fock space $\mathscr{F}_N$ of entire functions on ${\bf C}^N$ which are square integrable with respect to the hermitian inner product $$\langle f, g\rangle_\omega=\int_{{\bf C}^N} f(z)\overline{g(z)} e^{-\pi|z|^2}dz.$$ The monomials $\left\{\sqrt{\dfrac{\pi^{|\alpha|}}{\alpha!}}z^\alpha, |\alpha|\geq 0\right\}$ forms an orthonormal basis of ${\mathscr F}_N$. The Harish-Chandra module $\omega^{HC}_\psi$ can be realized as the subspace $\mathscr{P}_N$ of polynomials on ${\bf C}^N$. Following [@F], introduce the linear map $$\label{gc} M_{2N}({\bf R})\to M_{2N}({\bf C}),\quad g=\begin{pmatrix} A & B\\ C & D\end{pmatrix}\mapsto g^c=\frac{1}{2}\begin{pmatrix} A+D+i(C-B) & A-D+i(C+B)\\ A-D-i(C+B) & A+D-i(C-B)\end{pmatrix}.$$ Denote by $Sp^c_{2N}$ the image of $Sp_{2N}({\bf R})$. Let $\nu$ be the Fock projective representation of $Sp^c_{2N}$ on ${\mathscr F}_N$. Then for $g^c= \begin{pmatrix} P & Q\\ \overline{Q} & \overline{P}\end{pmatrix} \in Sp^c_{2N}$, up to a factor of $\pm 1$ the operator $\nu(g^c)$ is given by $$\label{fockaction} \left\{ \begin{aligned} &\nu(g^c)f(z) = \int_{{\bf C}^N}K_{g^c}(z, \bar{w})f(w)e^{-\pi |w|^2} dw,\\ &K_{g^c}(z,\bar{w})= (\det P)^{-\frac{1}{2}} \exp\bigg[\frac{\pi}{2}\big({}^tz\overline{Q}P^{-1}z+2{}^t\bar{w} P^{-1}z -{}^t\bar{w}P^{-1} Q \bar{w}\big)\bigg]. \end{aligned}\right.$$ Let $J=1_{n,1}\otimes 1_{p,q}$. We have the embedding $$\label{gembed} i_G: G\hookrightarrow Sp_{2N}({\bf R}),\quad X+iY\mapsto \begin{pmatrix} X\otimes 1_{n+1} & (Y\otimes 1_{n+1})J\\ -J(Y\otimes 1_{n+1}) & J(X\otimes 1_{n+1})J\end{pmatrix}.$$ In spirit of ${\bf C}^N\cong {\bf C}^{n+1}\otimes {\bf C}^{n+1}\cong M_{n+1}({\bf C})$, it is more convenient to label the variables $z_1, \ldots, z_N$ as $z_{11},\ldots, z_{1,n+1}, \ldots, z_{n+1,1},\ldots, z_{n+1,n+1}$. For instance, if we write the matrix of variables in the block form $$\label{block} z=(z_{ij})_{i,j=1,\ldots,n+1}=\begin{pmatrix} A_{n\times p} & B_{n\times q} \\ C_{1\times p} & D_{1\times q}\end{pmatrix},$$ then from (\[gc\]-\[gembed\]), for $k=(x, y)\in K=U(n)\times U(1)$ one has up to $\pm1$ $$\label{fockk} \omega(k) f(z)=(\det x)^{(p-q)/2}(\det y)^{-(p-q)/2}f\begin{pmatrix} {}^tx A & x^{-1}B \\ y^{-1}C & {}^t y D\end{pmatrix}.$$ We need to know the action of $\omega(a_t)$. We use the notation ${\underline}{z}_i= (z_{i,1},\ldots, z_{i,n+1})$, $i=1,\ldots, n+1$, so that we can write $f(z)=f({\underline}{z}_1,\ldots, {\underline}{z}_{n+1})\in \mathscr{F}_N$. As a preliminary step we have \[4.1\] For $f(z)\in\mathscr{P}_N$, $$\begin{aligned} &\omega(a_t)f(z)=(\cosh t)^{-n-1}\exp\left(\pi(\tanh t) {\underline}{z}_1\cdot {\underline}{z}_{n+1}\right) \\ &~ \times \int_{{\bf C}^{n+1}}f({\underline}{w}_1, {\underline}{z}_2,\ldots, {\underline}{z}_n, (\cosh t)^{-1}{\underline}{z}_{n+1}- (\tanh t) \bar{{\underline}{w}}_1)\exp\left(\pi(\cosh t)^{-1}{\underline}{z}_1\cdot \bar{{\underline}{w}}_1-\pi |{\underline}{w}_1|^2\right)d{\underline}{w}_1.\end{aligned}$$ One can show that $$i_G(a_t)=\begin{pmatrix} a_t\otimes 1_{n+1} & 0 \\ 0 & a_{-t}\otimes 1_{n+1}\end{pmatrix},\quad i_G(a_t)^c=\begin{pmatrix} P_t & Q_t \\ Q_t & P_t\end{pmatrix}$$ where $$P_t=\begin{pmatrix} \cosh t & 0 & 0 \\ 0 & 1_{n-1} & 0 \\ 0 & 0 & \cosh t\end{pmatrix}\otimes 1_{n+1},\quad Q_t=\begin{pmatrix} 0 & 0 & \sinh t \\ 0 & 0_{n-1} & 0 \\ \sinh t & 0 & 0\end{pmatrix}\otimes 1_{n+1}.$$ We calculate that $$\begin{aligned} &{}^t zQ_t P_t^{-1}z=2(\tanh t) {\underline}{z}_1\cdot{\underline}{z}_{n+1},\quad {}^t\bar{w}P_t^{-1}Q_t\bar{w}=2(\tanh t)\bar{{\underline}{w}}_1\cdot \bar{{\underline}{w}}_{n+1}\\ &{}^t\bar{w}P_t^{-1}z=(\cosh t)^{-1}({\underline}{z}_1\cdot\bar{{\underline}{w}}_1+{\underline}{z}_{n+1}\cdot\bar{{\underline}{w}}_{n+1})+\sum^n_{i=2} {\underline}{z}_i\cdot \bar{{\underline}{w}}_i.\end{aligned}$$ The lemma follows from integrating over ${\underline}{w}_2,\ldots, {\underline}{w}_{n+1}$ in (\[fockaction\]) and applying the following formula, which will be used later as well. $$\label{formula} \int_{\bf C} z^i\bar{z}^j e^{\pi c \bar{z}-\pi|z|^2}dz=\left\{\begin{array}{ll} \dfrac{i!}{(i-j)!}\dfrac{c^{i-j}}{\pi^j}& \textrm{if }i\geq j,\\ 0 & \textrm{if } i<j, \end{array}\right.$$ where $i, j\geq 0$ are integers and $c$ is a constant. Joint harmonics {#s6} =============== The notion of joint harmonics was introduced in [@Ho]. It is the subspace $\mathcal{H}\subset \mathscr{P}_N$ annihilated by certain second order differential operators from the the centralizers of $\frak{k}$ and $\frak{k}'$ in $\frak{sp}$, under the action of oscillator representation. We refer the readers to [@Ho] for the precise definition. It is known that $\mathcal{H}$ admits a multiplicity free decomposition $$\mathcal{H}\cong \bigoplus \sigma\otimes \sigma'$$ into irreducible $\widetilde{K}\times \widetilde{K}'$-modules such that $\sigma$ and $\sigma'$ determine each other. Moreover, the lowest $\widetilde{K}$- and $\widetilde{K}'$-type of discrete series correspond to each other under this decomposition. We consider the subspace of joint harmonics $\mathcal{H}_{\Lambda^\vee,\Lambda'}\cong \sigma_{\Lambda^\vee}\otimes \sigma_{\Lambda'}$ as in Section \[s2\]. The joint highest weight vector of $\mathcal{H}_{\Lambda^\vee,\Lambda'}$ can be expressed in terms of principal minors. For $i=1,\ldots, n$, let $$\Delta_i=\det\begin{pmatrix} z_{11} & \cdots & z_{1i} \\ &\cdots & \\ z_{i1} & \cdots & z_{ii}\end{pmatrix},\quad \Delta_i'=\det\begin{pmatrix} z_{n-i+1, n-i+2} & \cdots & z_{n-i+1, n+1} \\ & \cdots & \\ z_{n, n-i+2} & \cdots & z_{n,n+1}\end{pmatrix},$$ which are determinants of $i\times i$ minors, hence homogeneous polynomials. Then in the two cases of Section \[s2\], we have the following harmonic polynomials of joint highest weight, which are unique up to scalar. Case (i): We take $$\phi(z)=\Delta_1'^{\alpha_1-\alpha_2}\Delta_2'^{\alpha_2-\alpha_3}\cdots \Delta_n'^{\alpha_n}z_{n+1,1}^\gamma.$$ For any $k\in K$, from (\[fockk\]) we see that the block $C$ in (\[block\]), i.e. $z_{n+1,1}$, is the only variable of ${\underline}{z}_{n+1}$ that appears in $\omega(k)\phi$, but the block $A$, in particular $z_{11}$, does not show up in $\omega(k)\phi$. This observation together with Lemma \[4.1\] and (\[formula\]) gives us $$\begin{aligned} \omega(a_t k)\phi(z)&=(\cosh t)^{-n-1}\exp\left(\pi(\tanh t) {\underline}{z}_1\cdot {\underline}{z}_{n+1}\right) \omega(k)\phi\left((\cosh t)^{-1}{\underline}{z}_1,{\underline}{z}_2,\ldots, {\underline}{z}_n, (\cosh t)^{-1}{\underline}{z}_{n+1}\right)\\ &=(\cosh t)^{-n-1}\exp\left(\pi(\tanh t) {\underline}{z}_1\cdot {\underline}{z}_{n+1}\right) \sigma_{\Lambda^\vee}(b_t k)\phi,\end{aligned}$$ where we use the extension of $\sigma_{\Lambda^\vee}$ to $K_{\bf C}$, and $$\label{bt} b_t=\begin{pmatrix} \cosh t & 0 & 0 \\ 0 & 1_{n-1} & 0 \\ 0 & 0 & \cosh t\end{pmatrix}.$$ Since $\sigma_{\Lambda^\vee}$-action preserves the degree, and monomials are orthogonal basis, we may use Taylor expansion to drop the factor $\exp\left(\pi(\tanh t) {\underline}{z}_1\cdot {\underline}{z}_{n+1}\right)$ and obtain $$\label{wmc} \langle\omega(k'a_tk)\phi, \phi\rangle=(\cosh t)^{-n-1}\langle \sigma_{\Lambda^\vee}(k' b_t k)\phi, \phi\rangle$$ for any $k, k'\in K$. Case (ii): We take $$\phi(z)=\Delta_1^{\beta_1-\beta_2}\Delta_2^{\beta_2-\beta_3}\cdots \Delta_p^{\beta_p}\Delta_1'^{\alpha_1-\alpha_2}\Delta_2'^{\alpha_2-\alpha_3}\cdots\Delta_{q-1}'^{\alpha_{q-1}}z_{n+1,p+1}^\gamma.$$ The argument is similar to above. We note that $z_{n+1,p+1}$ is the only variable of ${\underline}{z}_{n+1}$ that appears in $\omega(k)\phi$, while the first $p$ rows of the block $B$ in (\[block\]), in particular ${\underline}{z}_{1,p+1}$, do not show up. The same argument as above gives us $$\label{wmc2} \langle\omega(k'a_tk)\phi, \phi\rangle=(\cosh t)^{-n-1}\langle \sigma_{\Lambda^\vee}(k' b_t^{-1} k)\phi, \phi\rangle$$ We may summarize our results as \[6.1\] Under the assumptions of Section \[s2\], for a vector $\phi\in\mathcal{H}_{\Lambda^\vee,\Lambda'}$ of joint highest weight, $k, k'\in K$, one has $$\langle\omega(k'a_tk)\phi, \phi\rangle=(\cosh t)^{-n-1}\langle \sigma_{\Lambda^\vee}(k' b_t^{\pm 1} k)\phi, \phi\rangle,$$ where the $\pm$ sign depends on whether the first (or equivalently, the last) component of $\Lambda^\vee$ is negative or positive. In particular, by the Harish-Chandra decomposition $g=h_zk=k_z a_t k_z^{-1}k$, one has $$\langle\omega(g)\phi,\phi\rangle=(\cosh t)^{-n-1}\langle\sigma_{\Lambda^\vee} (k_zb_t^{\pm1} k_z^{-1}k)\phi,\phi\rangle.$$ Define $$b_z= k_z b_t k_z^{-1}=\begin{pmatrix} (1_n-zz^*)^{-1/2} & 0 \\ 0 & (1-z^*z)^{-1/2}\end{pmatrix}$$ so that $$\label{wmc3} \langle\omega(g)\phi,\phi\rangle=(\cosh t)^{-n-1}\langle \sigma_{\Lambda^\vee}(b_z^{\pm 1} k)\phi,\phi\rangle.$$ Zeta integrals {#s7} ============== We are ready to compute the archimedean zeta integrals on $U(n,1)$ that involves oscillator representations and holomorphic discrete series, combining the results in the previous sections. In terms of the Harish-Chandra decomposition, by (\[dismc\]) and (\[wmc3\]) we have $$\begin{aligned} \int_G \langle \omega(g)\phi,\phi\rangle \cdot \psi_\pi(g)dg&=\int_C \int_K \langle \omega(h_zk)\phi,\phi\rangle\cdot\psi_\pi(h_zk)dk d^*z\\ &= \int_C\int_K \langle \sigma_{\Lambda^\vee}(b_z^{\pm1}k)\phi, \phi\rangle\cdot \textrm{tr }\sigma_\Lambda(\theta_z k) \det(1_n-zz^*)^{(n+1)/2}dk d^*z,\end{aligned}$$ noting that $\det(1_n-zz^*)=(\cosh t)^{-2}$. We shall follow the strategy in [@G] to evaluate above integral, or more generally the integral $$\label{is} I^\pm_s=\int_C\int_K \langle \sigma_{\Lambda^\vee}(b_z^{\pm1}k)\phi, \phi\rangle\cdot \textrm{tr }\sigma_\Lambda(\theta_z k) \det(1_n-zz^*)^s dk d^*z$$ which converges absolutely for $\textrm{Re }s\gg0$. Here the $\pm$ sign is determined by $\Lambda^\vee$ as in Proposition \[6.1\], i.e. depends on whether we have Case (i) or (ii). Our main result is the following \[main\] Under the assumptions and notations of Sections \[s2\] and \[s3\], for $\phi\in \mathcal{H}_{\Lambda^\vee, \Lambda'}$ and $\pi=\pi_\lambda$ one has the zeta integral Case (i): $$\int_G \langle \omega(g)\phi,\phi\rangle \cdot \psi_\pi(g)dg=\frac{\pi^n}{\dim\sigma_\Lambda}\prod^n_{i=1}\frac{1}{\alpha_i-i+n}\|\phi\|^2;$$ Case (ii): $$\int_G \langle \omega(g)\phi,\phi\rangle \cdot \psi_\pi(g)dg=\frac{\pi^n}{\dim \sigma_\Lambda}\prod^n_{i=1}\frac{1}{\gamma+i-p}\|\phi\|^2,$$ where $\dim\sigma_\Lambda$ is given by the well-known Weyl formula (\[dim\]). Since $\sigma_{\Lambda^\vee}(b_z)^*=\sigma_{\Lambda^\vee}(b_z^*)=\sigma_{\Lambda^\vee}(b_z)$, $\sigma_\Lambda(\theta_z)^*=\sigma_\Lambda(\theta_z^*)=\sigma_\Lambda(\theta_z)$, we have $$\begin{aligned} \langle \sigma_{\Lambda^\vee}(b_z^{\pm1}k)\phi, \phi\rangle\cdot \textrm{tr }\sigma_\Lambda(\theta_z k) &= \sum_i \langle \sigma_{\Lambda^\vee}(k)\phi, \sigma_{\Lambda^\vee}(b_z^{\pm 1})\phi\rangle \cdot \langle \sigma_{\Lambda}(k)x_i, \sigma_\Lambda(\theta_z)x_i\rangle\end{aligned}$$ where $\{x_i\}$ is an orthonormal basis of $\sigma_\Lambda$. By Schur orthogonality relation, the integration over $K$ leaves us $$\begin{aligned} I^\pm_s&=\frac{1}{\dim\sigma_\Lambda}\int_C\sum_i \langle\phi, x_i\rangle \cdot \overline{\langle \sigma_{\Lambda^\vee}( b_z^{\pm 1})\phi, \sigma_\Lambda(\theta_z)x_i\rangle} \det(1_n-zz^*)^s d^*z\\ &=\frac{1}{\dim\sigma_\Lambda}\int_C\sum_i \langle\phi, x_i\rangle \cdot \overline{\langle \sigma_{\Lambda^\vee}( \theta_z^{-1}b_z^{\pm 1})\phi, x_i\rangle} \det(1_n-zz^*)^s d^*z\\ &=\frac{1}{\dim\sigma_\Lambda}\int_C\langle\phi, \sigma_{\Lambda^\vee}(\theta_z^{-1} b_z^{\pm 1})\phi\rangle \det(1_n-zz^*)^s d^*z\\ &=\frac{1}{\dim\sigma_\Lambda}\langle\phi, (\int_C\sigma_{\Lambda^\vee}(\theta_z^{-1} b_z^{\pm 1}) \det(1_n-zz^*)^s d^*z)\phi\rangle.\end{aligned}$$ Hence we need to compute the endomorphism $$T^\pm_s=\int_C\sigma_{\Lambda^\vee}(\theta_z^{-1} b_z^{\pm 1}) \det(1_n-zz^*)^s d^*z\in\textrm{End}_{\bf C}(\sigma_{\Lambda^\vee}).$$ We find that $$\theta_z^{-1} b_z=\begin{pmatrix} (1_n-zz^*)^{-1} & 0 \\ 0 & 1\end{pmatrix},\quad \theta_z^{-1}b_z^{-1}=\begin{pmatrix} 1 & 0 \\ 0 & 1-z^*z\end{pmatrix}.$$ If we decompose $\sigma_{\Lambda^\vee}\cong \sigma_1\otimes \sigma_2$ as the outer tensor product of irreducibles $\sigma_1$ of $\widetilde{U}(n)$ and $\sigma_2$ of $\widetilde{U}(1)$, then $$\left\{ \begin{aligned} &T^+_s=\int_C \sigma^{-1}_1(1_n-zz^*)\det(1_n-zz^*)^s d^*z\in \textrm{End}_{\bf C}(\sigma_1), \\ & T^-_s=\int_C \sigma_2(1-z^*z)\det(1_n-zz^*)^s d^*z\in \textrm{End}_{\bf C}(\sigma_2). \end{aligned} \right.$$ By the parametrization of $D_{n,1}$ in Section \[s3\], a change of variables in the defining integral shows that $T_s^+$ commutes with $\sigma_1(k)$ for any $k\in \widetilde{U}(n)$, hence must be a scalar thanks to Schur’s lemma. Similarly $T_s^-$ is a scalar as well. In other words, we have $$I_s^\pm=\frac{\overline{T_s^\pm}}{\dim\sigma_\Lambda}\|\phi\|^2=\frac{T_s^\pm}{\dim\sigma_\Lambda}\|\phi\|^2,$$ noting that $T_s^\pm$ is real since the integrand is real. Recall the Weyl dimension formula $$\begin{aligned} \label{dim} \dim \sigma_\Lambda&=\dim \sigma_1 = \prod_{\alpha\in \Delta_c^+} \frac{\langle\Lambda+\rho_c, \alpha\rangle}{\langle\rho_c,\alpha\rangle}=\prod_{\alpha\in\Delta^+_c}\frac{\langle \lambda+\rho-\rho_c,\alpha\rangle}{\langle\rho_c,\alpha\rangle}\\ &=\prod_{\alpha\in\Delta^+_c}\frac{\langle\lambda,\alpha\rangle}{\langle\rho_c,\alpha\rangle}=\prod_{1\leq i<j\leq n} \frac{\lambda_i-\lambda_j}{j-i}.\nonumber\end{aligned}$$ It remains to find the scalar $T_s^\pm$, which is essentially a special case of the computation in [@G]. However the second proposition in [@G §3] was not stated correctly, which caused a mistake in the formulas of the main theorem therein. For reader’s convenience a correct variant form is given in the Lemma \[gar\] below. Applying this lemma for the representations $\sigma_1\otimes 1$ and $1\otimes \sigma_2$ of $(U(n)\times U(1))^\sim$ respectively, we obtain Case (i): $\Lambda^\vee=[(-\alpha_n,\ldots,-\alpha_1)+\textrm{det}^{(1-n)/2}] \otimes [-\gamma+\textrm{det}^{-(1-n)/2}]$, $$T_s^+=S_{\sigma_1\otimes 1, s}=\pi^n\prod^n_{i=1}\frac{1}{\alpha_i-i+s-(1-n)/2};$$ Case (ii): $\Lambda^\vee=[(\beta_1,\ldots,\beta_p, -\alpha_{q-1},\ldots,-\alpha_1)+\textrm{det}^{(p-q)/2}] \otimes [\gamma+\textrm{det}^{-(p-q)/2}]$, $$T_s^-=S_{1\otimes \sigma_2, s}=\pi^n\prod^n_{i=1}\frac{1}{\gamma-i+s-(p-q)/2}.$$ The theorem follows from specializing $s=(n+1)/2$. \[gar\] [@G] Let $\sigma=\sigma_1\otimes\sigma_2$ be an irreducible representation of $(U(p)\times U(q))^\sim$, where $\sigma_1$ and $\sigma_2$ have highest weights $(\kappa_1,\ldots, \kappa_p)$ and $(\iota_{1},\ldots, \iota_{q})$ respectively. Define $$S_{\sigma,s}=\int_{D_{p,q}}\sigma_1^{-1}(1_p-zz^*)\otimes \sigma_2(1_q-z^*z)\det(1_p-zz^*)^s d^*z\in\textrm{End}_{\bf C}(\sigma)$$ for $\textrm{Re }s\gg 0$, where $$D_{p,q}=\{z\in{\bf C}^{p\times q}: 1_p-zz^*\textrm{ is positive definite}\},\quad d^*z=\frac{dz}{\det(1_p-zz^*)^{p+q}}.$$ Then $S_{\sigma,s}$ is a scalar and one has \(i) if $\sigma_2=\det^\iota$ is one-dimensional, then $$\begin{aligned} S_{\sigma,s}&= \pi^{pq}\prod^p_{i=1}\frac{\Gamma(\iota-\kappa_i-(p+q-i)+s)}{\Gamma(\iota-\kappa_i-(p-i)+s)}\\ &=\pi^{pq}\prod^p_{i=1}\frac{1}{(\iota-\kappa_i-(p+q-i)+s)\cdots(\iota-\kappa_i-(p+1-i)+s)};\end{aligned}$$ \(ii) if $\sigma_1=\det^\kappa$ is one-dimensional, then $$\begin{aligned} S_{\sigma,s}&= \pi^{pq}\prod^q_{i=1}\frac{\Gamma(\iota_i-\kappa-(p+i-1)+s)}{\Gamma(\iota_i-\kappa-(i-1)+s)}\\ &=\pi^{pq}\prod^q_{i=1}\frac{1}{(\iota_i-\kappa-(p+i-1)+s)\cdots(\iota_i-\kappa-i+s)};\end{aligned}$$ \(iii) if $\sigma$ is the lowest $\widetilde{K}$-type of an anti-holomorphic discrete series $\pi$ of $\widetilde{U}(p,q)$, then under above measure the formal degree of $\pi$ is given by $$\frac{1}{d_\pi} =\frac{S_{\sigma,0}}{\dim\sigma}.$$ We remark that the formulation of [@G] is in terms of holomorphic discrete series, and our reformulation here about anti-holomorphic case is just for convenience. Recall from [@HLS] that $$\int_G \langle \omega(g)\phi,\phi\rangle \cdot \psi_\pi(g)dg=\frac{c^2_{\psi,\pi}}{d_\pi}\|\phi\|^2,$$ where $c_{\psi,\pi}$ is the positive number such that $$\| P_{\psi,\pi}(\phi)\|=c_{\psi,\pi}\|\phi\|$$ for $\phi\in \mathcal{H}_{\Lambda^\vee,\Lambda'}$ and $P_{\psi,\pi}$ the orthogonal projection from $\omega_\psi$ onto the closed subspace $\sigma_{\Lambda^\vee}\otimes \pi'$. We are interested in the explicit value of $c_{\psi, \pi}$. The formal degree $d_\pi$ is given by (\[fd\]), depending on the measure of $G$. Instead of specifying the explicit dependence, we may compare our zeta integral with the formal degree given by Lemma \[gar\] (iii). This will enable us to find out $c_{\psi, \pi}$. \[7.3\]The explicit value of $c_{\psi,\pi}$ is given by Case (i): $$c^2_{\psi,\pi}=\prod^n_{i=1}\frac{\alpha_i-i+n-1-\gamma}{\alpha_i-i+n};$$ Case (ii): $$c^2_{\psi,\pi}=\prod^n_{i=1}\frac{\gamma+i-\delta_i-2p}{\gamma+i-p},$$ where $(\delta_1,\ldots,\delta_n):=(\beta_1,\ldots,\beta_p,-\alpha_{q-1},\ldots,-\alpha_1)$. The proof of Theorem \[main\] shows that $$\frac{c^2_{\psi,\pi}}{d_\pi}=\frac{T^\pm_{(n+1)/2}}{\dim \sigma_\Lambda}.$$ On the other hand, by Lemma \[gar\] we have $$\frac{1}{d_\pi}=\frac{S_{\sigma_{\Lambda^\vee,0}}}{\dim \sigma_\Lambda}.$$ Comparison of the last two equations yields $$c^2_{\psi,\pi}=\frac{T^\pm_{(n+1)/2}}{S_{\sigma_{\Lambda^\vee,0}}}=\frac{S_{\sigma_1\otimes 1, (n+1)/2}}{S_{\sigma_1\otimes\sigma_2,0}}\quad \textrm{or} \quad \frac{S_{1\otimes \sigma_2, (n+1)/2}}{S_{\sigma_1\otimes\sigma_2,0}}$$ in Case (i) or (ii) respectively. Plugging in the parameter $\Lambda^\vee$ gives the corollary. [CERP]{} M. Flensted-Jensen, [*Discrete series for semisimple symmetric spaces*]{}, Ann. Math. [**111**]{} (1980), 253–311. G.B. Folland, [*Harmonic Analysis in Phase Space*]{}, Princeton University Press, 1989. P. Garrett, [*Values of Archimedean zeta integrals for unitary groups,*]{} Eisenstein series and applications, 125–148, Progr. Math., [**258**]{}, Birkhäuser Boston, Boston, MA, 2008. Harish-Chandra, [*Harmonic analysis on real reductive groups, III. The Maass-Selberg relations and the Plancherel formula*]{}, Ann. of Math. [**104**]{} (1976), 117-201. M. Harris, [*Cohomological automorphic forms on unitary groups, II: period relations and values of $L$-functions*]{}, in Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory, Lect. Notes Series Vol. [**12**]{}, Institute of Math. Sci., National Univ. of Singapore (2007), 89-150. M. Harris, S. Kudla and W.J. Sweet, [*Theta dichotomy for unitary groups*]{}, J. Amer. Math. Soc. [**9**]{} (1996), 941-1004. M. Harris, J.-S. Li and B. Sun, [*Theta correspondence for close unitary groups*]{}, Arithmetic geometry and automorphic forms, 265-307, Adv. Lect. Math. [**19**]{}, Int. Press, Somerville, MA, 2011. R. Howe, [*Transcending classical invariant theory*]{}, J. Amer. Math. Soc. [**2**]{} (1989), no. 3, 535–552. J.-S. Li, [*Theta lifting for unitary representations with nonzero cohomology*]{}, Duke. Math. J. [**61**]{} (1990), no. 3, 913-937. J.-S. Li, [*Nonvanishing theorems for the cohomology of certain arithmetic quotients*]{}, J. Reine Angew. Math. [**428**]{} (1992), 177-217. B. Lin, [*On the archimedean local zeta integrals and $L$-factors*]{}, HKUST Thesis (2011). D. Liu, [*Archimedean zeta integrals on $U(2,1)$*]{}, J. Funct. Anal. [**269**]{} (2015), 229–270. A. Paul, [*Howe correspondence for real unitary groups*]{}, J. Funct. Anal. [**159**]{} (1998), 384-431.

--- abstract: | This paper has two parts, on Baumslag–Solitar groups and on general $G$–trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag–Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces. In the second part we establish a universal lower bound of $1/12$ for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group $BS(2,3)$ show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions. Returning to Baumslag–Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval $(0, 1/12)$. address: - 'Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA' - 'Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA' - 'Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA' author: - Matt Clay - Max Forester - Joel Louwsma bibliography: - 'bib.bib' title: 'Stable commutator length in Baumslag–Solitar groups and quasimorphisms for tree actions' --- Introduction ============ Stable commutator length has been the subject of a significant amount of recent work, especially by Danny Calegari and his collaborators. See [@Calegari:scl] for an introduction to stable commutator length and a desciption of much of this work. A major breakthrough in this area was Calegari’s algorithm [@Calegari:free] for computing stable commutator length in free groups. This algorithm can also be used to compute stable commutator length in certain classes of groups that are built from free groups in simple ways. However, there are few other instances in which stable commutator length can be computed explicitly, with the exception of certain elements and classes of groups for which it is known to vanish. Other work involves studying the *spectrum* of values taken by stable commutator length on a given group. In certain cases, this spectrum has been shown to have a *gap*, i.e. there is a range of values that are the stable commutator length of no element of the group. For example, results of this type have been shown for free groups [@DH], for word-hyperbolic groups [@CF], and recently for mapping class groups [@BBF]. Such results have often involved constructing quasimorphisms with certain properties, thus relying on a dual interpretation of stable commutator length in terms of quasimorphisms. The primary goal of this paper is to understand stable commutator length in Baumslag–Solitar groups. We obtain both quantitative and qualitative results. On the way to establishing the *gap theorem* below, we digress in Section \[sec:qm\] to construct efficient quasimorphisms in the completely general setting of groups acting on trees, and derive some consequences. These results may be of independent interest to some readers. Stable commutator length in Baumslag–Solitar groups {#stable-commutator-length-in-baumslagsolitar-groups .unnumbered} --------------------------------------------------- We use the presentation $\langle \,a, t \mid t a^m t^{-1} = a^{\ell} \,\rangle$ for the Baumslag–Solitar group $BS(m,\ell)$, and we generally assume that $m\not= \ell$. Then, stable commutator length is defined exactly on the elements of $t$–exponent zero. We build on the approach taken in [@BCF] and attempt to encode the computation of stable commutator length as the output of a linear programming problem. This approach used the notions of the *turn graph* and *turn circuits* to encode the geometric data of an admissible surface. In the present setting, encoding this geometric data requires the use of a *weighted* turn graph instead, to account for winding numbers not present in the case of free groups. Even so, there is further winding data, and the natural encoding leads to an infinite-dimensional linear programming problem. By restricting to words of alternating $t$–shape, we are able to reduce to a finite-dimensional problem. \[mainthm\] Suppose $g \in BS(m,\ell)$, $m\not= \ell$, has alternating $t$–shape. Then there is a finite-dimensional, rational linear programming problem whose solution yields the stable commutator length of $g$. In particular, $\operatorname{scl}(g)$ is computable and is a rational number. More generally, the linear programming problem constructed in the proof of Theorem \[mainthm\] is defined for any element $g$ of $t$–exponent zero, and its solution provides a *lower bound* for $\operatorname{scl}(g)$ (see Theorem \[scl-lowerbound-lp\]). What is difficult is to convert the solution into an admissible surface to obtain a matching upper bound; the encoding procedure from surfaces to vectors *loses* information, and not every vector can be realized by a surface. In some cases the solution to the linear programming problem in Theorem \[mainthm\] can be expressed in a closed formula. We show in Proposition \[prop:length2\] that if $m \nmid i$ and $\ell \nmid j$ then $$\label{eq:formula} \operatorname{scl}\bigl(ta^i t^{-1}a^j\bigr) \ = \ \frac{1}{2} \left( 1 - \frac{\gcd(i,m)}{{\left\lvert {m} \right\rvert}} - \frac{\gcd(j,\ell)}{{\left\lvert {\ell} \right\rvert}}\right).$$ Next we characterize the elements of alternating $t$–shape for which there is a surface, known as an *extremal surface*, that realizes the infimum in the definition of stable commutator length. Such surfaces are important in applications of stable commutator length to problems in topology. It turns out that many elements have extremal surfaces, and many do not. [\[extremal-characterization\]]{} Let $g =\prod_{k=1}^r ta^{i_k} t^{-1}a^{j_k}\in BS(m,\ell)$, $m\neq\ell$. There is an extremal surface for $g$ if and only if $$\ell\sum_{k=1}^r i_k =-m\sum_{k=1}^r j_k.$$ This allows us to find many examples of elements with rational stable commutator length for which no extremal surface exists. Previous examples of this phenomenon were found in free products of abelian groups of higher rank (see [@Calegari:sails]). Our last main result for Baumslag–Solitar groups is more qualitative in nature and concerns the scl spectrum. [\[th:gap\]]{}\[Gap theorem\] For every element $g \in BS(m, \ell)$, either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/12$. Thus, similar to hyperbolic groups, the spectrum has a gap above zero. This theorem is proved in Section \[sec:gap\], and it depends heavily on results in Section \[sec:qm\] to be discussed shortly. Nevertheless, these latter results do not apply to every element of $BS(m,\ell)$ (namely, those that are not *well-aligned*). To study stable commutator length of these left-over elements, we take advantage of special properties of the Bass–Serre trees for these groups. It is interesting to note that, in contrast with Theorem \[cor:acylindrical\] below, it is the *failure* of acylindricity of these trees that is used in establishing the scl gap. Stable commutator length in groups acting on trees {#stable-commutator-length-in-groups-acting-on-trees .unnumbered} -------------------------------------------------- In order to prove the gap theorem we turn to the dual viewpoint of quasimorphisms on groups. According to Bavard Duality [@Bavard:duality], a lower bound for $\operatorname{scl}(g)$ can be obtained by finding a homogeneous quasimorphism $f$ on $G$ with $f(g) = 1$ and of small defect. Indeed, if the defect of $f$ is $D$ then $\operatorname{scl}(g) {\geqslant}1/2D$. Many authors have constructed quasimorphisms on groups in settings involving negative curvature. For the most part these constructions are variants and generalizations of the Brooks counting quasimorphisms on free groups [@Rhemtulla; @Brooks:qm; @Grigorchuk:qm]. These settings include hyperbolic groups [@EF], groups acting on Gromov-hyperbolic spaces [@Fujiwara:gromov; @CF], amalgamated free products and HNN extensions [@Fujiwara:amalgam], and mapping class groups [@BF; @BBF]. One such result is Theorem D of [@CF], due to Calegari and Fujiwara. They showed that for any amalgamated product $G = A \ast_C B$ and any appropriately chosen hyperbolic element $g\in G$, there is a homogeneous quasimorphism $f$ on $G$ with $f(g) = 1$ and of defect at most $312$. This bound is of interest since it is universal, independent of the group. In Theorem \[defect\] we construct efficient quasimorphisms, of defect at most $6$, for any group acting on a tree. These are similar to the “small” counting quasimorphisms introduced by Epstein–Fujiwara [@EF], except that they are specifically tailored to the geometry of tree actions; the counting takes place in the tree rather than a Cayley graph. Moreover, by working directly with the homogenization of the counting quasimorphism, we obtain a further improvement in the defect. Using the calculation in the group $BS(2,3)$ (or alternatively, a different calculation in $\operatorname{PSL}(2,{{\mathbb Z}})$) we determine that $6$ is the smallest possible defect that can be achieved in this generality, thus answering Question 8.4 of [@CF]. Expressed in terms of stable commutator length, the result can be stated as follows. [\[th:well-aligned\]]{} Suppose $G$ acts on a simplicial tree $T$. If $g\in G$ is well-aligned then $\operatorname{scl}(g) {\geqslant}1/12$. The same result holds for groups acting on ${{\mathbb R}}$–trees as well (Remark \[rem:rtree\]). Again, the bound of $1/12$ is the best possible. The condition of being *well-aligned* is necessary, and agrees with the double coset condition in [@CF] in the case of the Bass–Serre tree of an amalgam. Not every hyperbolic element is well-aligned. Indeed, there are examples of $3$–manifold groups that split as amalgams containing hyperbolic elements with very small stable commutator length; see [@CF]. If we consider trees that are *acylindrical* (see Section \[sec:qm\]) then we can obtain an additional lower bound that applies to all hyperbolic elements. This bound is almost universal, depending only on the acylindricity constant. Alternatively, there is a genuinely uniform bound if one considers only elements of translation length greater than or equal to the acylindricity constant. [\[cor:acylindrical\]]{} Suppose $G$ acts $K$–acylindrically on a tree $T$ and let $N$ be the smallest integer greater than or equal to $\frac{K}{2} + 1$. 1. If $g \in G$ is hyperbolic then either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/12N$. 2. If $g \in G$ is hyperbolic and ${\left\lvert {g} \right\rvert} {\geqslant}K$ then either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/24$. In both cases, $\operatorname{scl}(g) = 0$ if and only if $g$ is conjugate to $g^{-1}$. Matt Clay is partially supported by NSF grant DMS-1006898. Max Forester is partially supported by NSF grant DMS-1105765. Preliminaries ============= Stable commutator length {#stable-commutator-length .unnumbered} ------------------------ Stable commutator length may be defined as follows, according to Proposition 2.10 of [@Calegari:scl]. \[def1\] Let $G = \pi_1(X)$ and suppose $\gamma {\colon \thinspace}S^1 \to X$ represents the conjugacy class of $g\in G$. The *stable commutator length* of $g$ is given by $$\label{scldef} \operatorname{scl}(g) \ = \ \inf_S \frac{-\chi(S)}{2n(S)},$$ where $S$ ranges over all singular surfaces $S \to X$ such that - $S$ is oriented and compact with $\partial S \not= \emptyset$ - $S$ has no $S^2$ or $D^2$ components - the restriction $\partial S \to X$ factors through $\gamma$; that is, there is a commutative diagram: $$\begin{CD} \partial S @>>> S \\ @VVV @VVV\\ S^1 @>{\gamma}>> X \end{CD}$$ - the total degree, $n(S)$, of the map $\partial S \to S^1$ (considered as a map of oriented $1$–manifolds) is non-zero. A surface $S$ satisfying the conditions above is called an *admissible surface*. If, in addition, each component of $\partial S$ maps to $S^1$ with *positive* degree, we call $S$ a *positive admissible surface*. It is shown in Proposition 2.13 of [@Calegari:scl] that the infimum in the definition of scl may be taken over positive admissible surfaces. Such surfaces (admissible or positive admissible) exist if and only if $g^k \in [G,G]$ for some nonzero integer $k$. If this does not occur then by convention $\operatorname{scl}(g) = \infty$ (the infimum of the empty set). A surface $S \to X$ is said to be *extremal* if it realizes the infimum in . Notice that if this occurs, then $\operatorname{scl}(g)$ is a rational number. In order to bound scl from above, one needs to construct an admissible surface realizing a given value of $\frac{-\chi(S)}{2n(S)}$. Sometimes a procedure for building a surface cannot be completed, leaving a surface with portions missing. The following result can be used in this situation. \[fill-with-zero\] Let $S$ be a compact oriented surface with no $S^2$ or $D^2$ components, and whose boundary is expressed as two non-empty families of curves $\partial_1S$ and $\partial_2 S$. Suppose $S\to X$ is a map taking the components of $\partial_1 S$ to group elements $a_1, \ldots, a_k \in \pi_1(X)$ and all components of $\partial_2 S$ to powers of the single element $g \in \pi_1(X)$, with total degree $n\not= 0$. Then there is an inequality $$\operatorname{scl}(g) \ {\leqslant}\ \frac{-\chi(S)}{2n} + \frac{1}{n}\Bigl(\sum_i \operatorname{scl}(a_i)\Bigr).$$ More generally, if one has defined scl for chains, the sum on the right hand side may be replaced by $\operatorname{scl}(\sum_i a_i)$, which may be finite even when the original sum was not. We first show how to construct a cover of $S$ that unwraps the curves in $\partial_1 S$ to give a collection of curves each of which is trivial in $H_1(X)$. Let $b$ be the number of boundary components of $S$. Let $c_i$ be the order of the conjugacy class of $a_i$ in the abelianization of $\pi_1(X)$. If the conjugacy class of some $a_i$ has infinite order in the abelianization of $\pi_1(X)$, then $\operatorname{scl}(a_i)=\infty$ and the lemma is tautological. Therefore we assume each $c_i$ is finite. Let $M=\operatorname{lcm}(c_1,\dotsc,c_k)$, and consider the prime factorization $M=p_1^{d_1}\dotsm p_q^{d_q}$. We construct a tower of covers $S_q\to S_{q-1}\to\dotsb\to S_1\to S_0=S$ as follows. For all $i$, the boundary $\partial S_i$ will be partitioned into two families of curves $\partial_1 S_i$ and $\partial_2 S_i$, where the induced map $S_i\to X$ takes the curves in $\partial_1 S_i$ to powers of the elements $a_1,\dotsc,a_k$ and the curves in $\partial_2 S_i$ to powers of the element $g$. For all $i$, $\partial_1 S_i$ will consist of exactly $k$ curves and $\partial_2 S_i$ will consist of at least $b-k$ curves. Suppose $S_{i-1}$ has been constructed. Since $b-k{\geqslant}1$, there is some integer $e_i$ satisfying $k{\leqslant}e_i{\leqslant}b$ such that $e_i-1$ is relatively prime to $p_i$, and hence to $p_i^{q_i}$. Therefore Lemma 1.12 of [@Calegari:scl] shows that, for any $e_i$ boundary components of $S_{i-1}$, there is a $p_i^{d_i}$–sheeted covering $S_{i}\to S_{i-1}$ that unwraps these $e_i$ boundary components. We choose these $e_i$ boundary components to be the $k$ curves in $\partial_1 S_{i-1}$ and any $e_i-k$ curves in $\partial_2 S_{i-1}$. Then $\partial S_i$ is also partitioned into two collections of curves: those in the preimage of $\partial_1 S_{i-1}$ are said to be in $\partial_1 S_i$, and those in the preimage of $\partial_2 S_{i-1}$ are said to be in $\partial_2 S_i$. By construction, $\partial_1 S_i$ consists of exactly $k$ curves and $\partial_2 S_i$ consists of at least $b-k$ curves. Iterating this procedure, we obtain a surface $S_q$ that is a degree $M$ cover of $S$. The induced map $S_q\to X$ takes the curves in $\partial_1 S_{q}$ to $a_1^M,\dotsc,a_k^M$ and the curves in $\partial_2 S_q$ to powers of $g$ with total degree $nM$. Note that, for each $i$, $a_i^M$ is trivial in the abelianization of $\pi_1(X)$. Fix $\epsilon > 0$. For all $N$ relatively prime to $k-1$, we can construct a further cover $S_{q,N}\to S_q$ such that the curves in $\partial S_{q,N}$ are again partitioned into classes $\partial_1 S_{q,N}$ and $\partial_2 S_{q,N}$, where the curves in $\partial_1 S_{q,N}$ map to $a_1^{MN},\dotsc,a_k^{MN}$ in $X$. Choose $N$ sufficiently large that, for all $i$, the element $a_i^{MN}$ bounds an admissible surface $S'_i$ that approximates $\operatorname{scl}\bigl(a_i^{MN}\bigr)$ to within $\epsilon /k$. Since $\operatorname{scl}\bigl(a_i^{MN}\bigr)=MN\operatorname{scl}(a_i)$, we can also regard $S'_i$ as an admissible surface for $a_i$ that approximates $\operatorname{scl}(a_i)$ within $\epsilon/kMN$. More precisely, $$\frac{-\chi(S'_i)}{2MN} \ {\leqslant}\ \operatorname{scl}(a_i) + \frac{\epsilon}{kMN}$$ for each $i$. Now join the surfaces $S'_i$ along their boundaries to the corresponding curves in $\partial_1 S_{q,N}$. We thus obtain an admissible surface $S''$ for $g$, with $n(S'') =nMN$. We have $$\begin{aligned} \frac{-\chi(S'')}{2n(S'')} \ &= \ \frac{-\chi(S_{q,N}) + \sum_i - \chi(S'_i)}{2nMN} \\ &= \ \frac{-MN \chi(S) + \sum_i -\chi(S'_i)}{2nMN} \\ &{\leqslant}\ \frac{-\chi(S)}{2n} + \frac{1}{n}\sum_i\Bigl(\operatorname{scl}(a_i) + \frac{\epsilon}{kNM}\Bigr) \\ &= \ \frac{-\chi(S)}{2n} + \frac{1}{n}\Bigl(\sum_i\operatorname{scl}(a_i)\Bigr) + \frac{\epsilon}{nMN}.\end{aligned}$$ Hence $\operatorname{scl}(g) \ {\leqslant}\ \frac{-\chi(S)}{2n} + \frac{1}{n}\bigl(\sum_i \operatorname{scl}(a_i)\bigr)$. Baumslag–Solitar groups {#baumslagsolitar-groups .unnumbered} ----------------------- Before discussing Baumslag–Solitar groups per se, we make a general observation: \[scl-a\] In any group $G$, if $t$ and $a$ are elements satisfying the Baumslag–Solitar relation $t a^m t^{-1} = a^{\ell}$ with $m \not= \ell$ then $\operatorname{scl}(a) = 0$. For any space $X$ with fundamental group $G$ there is a singular annulus $S \to X$, whose oriented boundary components represent $a^m$ and $a^{-\ell}$ respectively (since $a^m$ and $a^{\ell}$ are conjugate in $G$). This surface can be made admissible with $\chi(S) = 0$ and $n(S) = m -\ell \not= 0$, so $\operatorname{scl}(a) = 0$. The *Baumslag–Solitar group* $BS(m,\ell)$ is defined by the presentation $$\label{bs-pres} \langle \, a, t \mid t a^m t^{-1} = a^{\ell} \, \rangle.$$ The corresponding presentation $2$–complex will be denoted $X_{m, \ell}$, or simply $X$, in this paper. One thinks of $X$ as being constructed by attaching both ends of an annulus to a circle, by covering maps of degrees $m$ and $\ell$ respectively; see Section \[sec:turn\]. Clearly, $BS(1,1)$ is ${{\mathbb Z}}\times {{\mathbb Z}}$ and $BS(1,-1)$ is the Klein bottle group. The cases $BS(m, \pm m)$ are also of special interest. By constructing a suitable covering space of $X$, one finds that this group contains a subgroup of index $2m$ isomorphic to $F_{2m-1} \times {{\mathbb Z}}$. In particular, stable commutator length can be computed in $BS(m, \pm m)$ and is always rational, using the rationality theorem for free groups [@Calegari:free] and results from [@Calegari:scl] (such as Proposition 2.80) on subgroups of finite index. In this paper we will study stable commutator length in $BS(m, \ell)$ under the standing assumption that $m \not= \ell$. \[t-exp\] The abelianization of $BS(m, \ell)$ is ${{\mathbb Z}}\times {{\mathbb Z}}_{{\left\lvert {m-\ell} \right\rvert}}$ with generators $t$ and $a$ respectively. Since we are assuming that $m\not= \ell$, an element of $BS(m, \ell)$ has finite order in the abelianization if and only if it has $t$–exponent zero. Thus scl is finite on exactly these elements. \[t-length\] Given a word $w$ in the letters $a^{\pm 1}$ and $t^{\pm 1}$ we denote by $|w|_t$ the *$t$–length* of $w$. That is, $|w|_t$ is the number of occurrences of $t$ and $t^{-1}$ in $w$. Given an element $g \in BS(m,\ell)$ we denote by $|g|_t$ the *$t$–length* of the conjugacy class of $g$. That is, $|g|_t$ is the minimum value of $|w|_t$ over all words $w$ that represent a conjugate of $g$. \[cyc-red\] Any element $g \in BS(m,\ell)$ has a conjugate that can be expressed as $$\label{eq:cyc-red} w = t^{\epsilon_1}a^{k_1}t^{\epsilon_2} \cdots t^{\epsilon_n}a^{k_n},$$ where: - $\epsilon_i \in \{1,-1\}$ for $i = 1,\ldots,n$, - $m \nmid k_i$ if $\epsilon_i = 1$ and $\epsilon_{i+1} = -1$, - $\ell \nmid k_i$ if $\epsilon_i = -1$ and $\epsilon_{i+1} = 1$, and - $|g|_t = |w|_t = n$. The subscripts in the second and third bullet are read modulo $n$. We refer to such a representative word of the conjugacy class of $g$ as *cyclically reduced*. Up to cyclic permutation, the cyclically reduced word representing a conjugacy class is not unique. Two other modifications to the word can be made, resulting in cyclically reduced words representing the same element: $$a^i t a^j \ \leftrightarrow \ a^{i-\ell} t a^{j+m} \ \text{ and } \ a^i t^{-1} a^j \ \leftrightarrow \ a^{i+m} t^{-1} a^{j-\ell}.$$ Collins’ Lemma [@Collins; @Lyndon-Schupp] characterizes precisely when two cyclically reduced words represent the same conjugacy class. It implies easily that modulo the two moves above and cyclic permutation, the expression is unique. Surfaces in $X_{m,\ell}$ {#sec:turn} ======================== Transversality {#transversality .unnumbered} -------------- Transversality will be used to convert a singular admissible surface $S \to X$ into a more combinatorial object. We will follow the approach from [@BCF], which treated the case of surfaces mapping into graphs. Recall that $X = X_{m, \ell}$ is the presentation $2$–complex for the presentation . We can build $X$ in the following way. Let $A$ be the annulus $S^1 \times [-1, 1]$, and let $C$ be a space homeomorphic to the circle. Fix orientations of $S^1$ and $C$ and attach the boundary circles $S^1 \times \{\pm 1\}$ to $C$ via covering maps of degrees $m$ and $\ell$ respectively, to form $X$. Note that the natural map $\phi{\colon \thinspace}A \to X$ is surjective, and maps the interior of $A$ homeomorphically onto $X - C$. Thus we have an identification of $X-C$ with $S^1 \times (-1,1)$. The space $X$ is also a cell complex with $C$ as a subcomplex. The $1$–skeleton of $X$ may be taken to be $C$ (having one $0$–cell and one $1$–cell, labeled $a$) along with an additional $1$–cell labeled $t$, which is a fiber in $A$ whose endpoints are attached to the $0$–cell of $C$. Let $C' = S^1 \times \{0\} \subset X-C$. This is a codimension-one submanifold. For any compact surface $S$ and continuous map $f {\colon \thinspace}S \to X$, we may perturb $f$ by a small homotopy to make it *transverse* to $C'$. Then, $f^{-1}(C')$ is a properly embedded codimension-one submanifold $N \subset S$. By a further homotopy, we can arrange that $N$ has an embedded $I$–bundle neighborhood $N \times [-1,1] \subset S$ (with $N = N \times \{0\}$) such that $f^{-1}(X-C) = N \times (-1, 1)$ and $$f\vert_{N \times (-1,1)} {\colon \thinspace}N \times (-1,1) \to S^1 \times (-1,1)$$ is a map of the form $f_0 \times \operatorname{id}$. Let $N_b \subset N$ be the union of the components that are intervals (rather than circles). Let $S_b \subset S$ be the subset $N_b \times [-1,1]$, each component of which is a *band* $I \times [-1,1]$ with $(I \times [-1,1]) \cap \partial S = \partial I \times [-1,1]$. By a further homotopy of $f$ in a neighborhood of $\partial S$, and using transversality for the map $S - (N \times (-1,1)) \to C$, we can arrange that in addition to the structure given so far, there is a collar neighborhood $S_{\partial} \subset S$ on which $f$ has a simple description. This map takes $S_{\partial}$ into the $1$–skeleton of $X$ by a retraction onto $\partial S$ followed by the restriction $\partial S \to X$. Each annulus component of $S_{\partial}$ decomposes into squares that retract into $\partial S$ and then map to $X$ by the characteristic maps of $1$–cells. These squares are labeled $a$– or $t$–squares depending on the $1$–cell. The $t$–squares are exactly the components of $S_{\partial} \cap S_b$. In particular, each band ends in two $t$-squares, representing one instance each of $t$ and $t^{-1}$ along the boundary. See Figure \[fig:surface\]. \[l\] at 249 59 \[r\] at 343 59 \[t\] at 297 66 \[t\] at 298 16 \[b\] at 58 -5 \[b\] at 139 -5 \[b\] at 31 -5 \[b\] at 87 -5 \[b\] at 113 -5 \[b\] at 167 -5 ![An admissible surface after the transversality procedure. The gray regions map into $C$.[]{data-label="fig:surface"}](surface "fig:") Finally, we define $S_0 = S_{\partial} \cup S_b$ and $S_1 = S - \operatorname{int}(S_0)$. Observe that $f$ maps $\partial S_1$ into $C$. The boundary $\partial S_0$ decomposes into two subsets: $\partial S$, called the *outer boundary*, and components in the interior of $S$, called the *inner boundary*, denoted $\partial^- S_0$. Note that $\partial^- S_0 = S_0 \cap S_1 = \partial S_1$. In particular, components of the inner boundary map by $f$ to loops in $X$ representing conjugacy classes of powers of $a$. \[boundarymap\] Call a loop $f {\colon \thinspace}S^1 \to X$ *regular* if $S^1$ can be decomposed into vertices and edges such that the restriction of $f$ to each edge factors through the characteristic map of a $1$–cell of $X$. Note that a regular map is completely described (up to reparametrization) by a cyclic word in the generators $a^{\pm 1}$, $t^{\pm 1}$ representing the conjugacy class of $f$ in $\pi_1(X)$. If a singular surface $S \to X$ has the property that its restriction to each boundary component is regular, then the transversality procedure described above can be performed rel boundary, so that the cyclic orderings of oriented $a$– and $t$–squares in $S_{\partial}$ agree with the cyclic boundary words one started with. Recall that $\operatorname{scl}(g)$ is the infimum of $\frac{-\chi(S)}{2n(S)}$ over all positive admissible surfaces. We will show how to compute $\operatorname{scl}(g)$ using the decomposition described above. Choose a cyclically reduced word $w$ representing the conjugacy class of $g$. For any positive admissible surface $S$, each boundary component maps by a loop representing a positive power of $g$ in $\pi_1(X)$. Modify $f$ by a homotopy to arrange that its boundary maps are regular, with corresponding cyclic words equal to positive powers of $w$. Then perform the transversality procedure given above, keeping the boundary map fixed (cf. Remark \[boundarymap\]). At this point, the subsurfaces $S_0$, $S_1$ are defined. Each boundary component is labeled by a positive power of $w$ and these powers add to $n(S)$. Note that $\chi(S) = \chi(S_0) + \chi(S_1)$ since $S_0$ and $S_1$ meet along circles. Also, $$\chi(S_0) \ = \ \frac{-n(S) {\left\lvert {g} \right\rvert}_t}{2},$$ as this is exactly the number of bands in $S_b$, each band connecting two instances of $t^{\pm 1}$ in $w^{n(S)}$ and contributing $-1$ to $\chi(S_0)$. (Note that $\chi(S_{\partial}) = 0$.) Let ${\chi^+}(S_1)$ denote the number of disk components in $S_1$. We have $$\chi(S) \ = \ \frac{-n(S){\left\lvert {g} \right\rvert}_t}{2} + \chi(S_1) \ {\leqslant}\ \frac{-n(S) {\left\lvert {g} \right\rvert}_t}{2} + {\chi^+}(S_1),$$ and therefore $$\label{chi-lowerbound} \frac{-\chi(S)}{2n(S)} \ {\geqslant}\ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ - \ \frac{{\chi^+}(S_1)}{2 n(S)} .$$ From this, we conclude that $$\label{scl-lowerbound} \operatorname{scl}(g) \ {\geqslant}\ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ + \ \inf_S \frac{-{\chi^+}(S_1)}{2n(S)},$$ where the infimum is taken over all positive admissible surfaces. In fact, the reverse of inequality holds as well: \[scl-g\] There is an equality $$\label{eq:scl-g} \operatorname{scl}(g) \ = \ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ + \ \inf_S \frac{-{\chi^+}(S_1)}{2n(S)}.$$ Given an admissible surface $S \to X$ decomposed as above, let $S'$ be the union of $S_0$ and the disk components of $S_1$. Recall that the components of $\partial S'$ in $\partial^- S_0$ map to loops in $X$ representing conjugacy classes of powers of $a$. Thus Lemma \[fill-with-zero\] and Lemma \[scl-a\] imply $$\operatorname{scl}(g) \ {\leqslant}\ \frac{-\chi(S')}{2n(S)} + \frac{1}{n(S)}\sum \operatorname{scl}(a^{p_i}) \ = \ \frac{-\chi(S')}{2n(S)}.$$ Since $$\frac{-\chi(S')}{2n(S)} \ = \ \frac{-\chi(S_0)}{2n(S)} - \frac{{\chi^+}(S_1)}{2n(S)} \ = \ \frac{{\left\lvert {g} \right\rvert}_t}{4} - \frac{{\chi^+}(S_1)}{2n(S)}$$ and $S$ was arbitrary, the reverse of inequality holds, as desired. \[extremal-disks-annuli\] If $S$ is an extremal surface for $g$, then $S_1$ consists only of disks and annuli. Let $S_2$ be the union of the components of $S_1$ that have nonnegative Euler characteristic, and let $S_3$ be the union of the components of $S_1$ that have negative Euler characteristic. Then $S_2$ consists only of disks and annuli and $\chi(S_2)={\chi^+}(S_1)$. If $S$ is extremal, we must have $$\operatorname{scl}(g) \ =\ \frac{-\chi(S)}{2n(S)} \ = \ \frac{{\left\lvert {g} \right\rvert}_t}{4}-\frac{\chi(S_2)}{2n(S)}-\frac{\chi(S_3)}{2n(S)} \ = \ \frac{{\left\lvert {g} \right\rvert}_t}{4}-\frac{{\chi^+}(S_1)}{2n(S)}-\frac{\chi(S_3)}{2n(S)}.$$ Comparing with Lemma \[scl-g\], this means $\chi(S_3){\geqslant}0$, meaning that $S_3$ must be empty. Thus $S_1$ consists only of disks and annuli. The weighted turn graph {#the-weighted-turn-graph .unnumbered} ----------------------- As in [@BCF], we use a graph to keep track of the combinatorics of the inner boundary $\partial^- S_0$. Consider a cyclically reduced word $w$ as in . A *turn* in $w$ is a subword of the form $a^k$ between two occurrences of $t^{\pm 1}$ considered as a cyclic word. The turns are indexed by the numbers $i = 1,\ldots,n$; the $i^{\rm th}$ turn is labeled by the subword $t^{\epsilon_i}a^{k_i}t^{\epsilon_{i+1}}$. A turn labeled $ta^kt^{-1}$ is of *type m*; a turn labeled $t^{-1}a^kt$ is of *type $\ell$*; all other turns are of *mixed* type. The *weighted turn graph* $\Gamma(w)$ is a directed graph with integer weights assigned to each vertex. The vertices correspond to the turns of $w$ and the weight associated to the $i^{\rm th}$ turn is $k_i$. There is a directed edge from turn $i$ to turn $j$ whenever $-\epsilon_i = \epsilon_{j+1}$. In other words, if the label of a turn begins with $t^{\pm 1}$, then there is a directed edge from this turn to every other turn whose label ends with $t^{\mp 1}$. The vertices of $\Gamma(w)$ are partitioned into four subsets where the presence of a directed edge between two vertices depends only on which subsets the vertices lie in. Figure \[fig:turngraph\] shows the turn graph in schematic form. at (-1,0) (tT) [$t \cdot t^{-1}$]{}; at (1,0) (Tt) [$t^{-1} \cdot t$]{}; at (0,-1) (TT) [$t^{-1} \cdot t^{-1}$]{}; at (0,1) (tt) [$t \cdot t$]{}; (tT) .. controls (-2.2,0.7) and (-2.2,-0.7) .. (tT); (Tt) .. controls (2.2,0.7) and (2.2,-0.7) .. (Tt); (TT) .. controls (-0.3,0) .. (tt); (tt) .. controls (0.3,0) .. (TT); (tt) – (tT); (tT) – (TT); (TT) – (Tt); (Tt) – (tt); The edges of the turn graph come in *dual pairs*: if $e \in \Gamma(w)$ is an edge from turn $i$ to turn $j$, then one verifies easily that there is also an edge $\bar{e}$ from turn $j+1$ to turn $i-1$, and moreover $\bar{\bar{e}} = e$. See Figure \[fig:dual-edge\]. at (-1,0) (i) [$t^{\epsilon_{i-1}}a^{k_{i-1}}t^{\epsilon_{i}}a^{k_{i}}t^{\epsilon_{i+1}}$]{}; at (1,0) (j) [$t^{\epsilon_{j}}a^{k_{j}}t^{-\epsilon_{i}}a^{k_{j+1}}t^{\epsilon_{j+1}}$]{}; (-0.75,0.1) .. controls (-0.5,0.5) and (0.4,0.5) .. (0.65,0.1); (1.1,0.1) .. controls (0.7,0.85) and (-0.7,0.85) .. (-1.21,0.1); A directed circuit in $\Gamma(w)$ is of *type m* or *type $\ell$* if every vertex it visits corresponds to a turn of type $m$ or of type $\ell$, respectively. Otherwise, the circuit is of *mixed* type. The *weight* $\omega(\gamma)$ of a directed circuit $\gamma$ is the sum of the weights of the vertices it visits (counted with multiplicity). Given a directed circuit $\gamma$, define $$\mu(\gamma) \ = \ \left\{ \begin{array}{ll} m & \text{ if $\gamma$ is of type $m$} \\ \ell & \text{ if $\gamma$ is of type $\ell$} \\ \gcd(m,\ell) & \text{ otherwise.} \end{array}\right.$$ A directed circuit $\gamma$ is a *potential disk* if $\omega(\gamma) \equiv 0 \mod \mu(\gamma)$. Turn circuits {#turn-circuits .unnumbered} ------------- Let $S \to X$ be a positive admissible surface whose boundary map is regular and labeled by $w^{n(S)}$. Decomposing $S$ as $S_0 \cup S_1$, each inner boundary component of $S_0$ can be described as follows. Traversing the curve in the positively oriented direction, one alternately follows the boundary arcs (or *sides*) of bands in $S_b$ and visits turns of $w$ along $S_{\partial}$; such a visit consists in traversing the inner edges of some $a$–squares before proceeding up along another side of a band (cf. Figure \[fig:surface\]). If the side of the band leads from turn $i$ to turn $j$, then $(t^{\epsilon_i})^{-1} = t^{\epsilon_j}$ and therefore there is an edge in $\Gamma(w)$ from turn $i$ to turn $j$. In this way, $\partial^- S_0$ gives rise to a finite collection (possibly with repetitions) of directed circuits in $\Gamma(w)$, called the *turn circuits* for $S_0$. Since $\partial S$ is labeled by $w^{n(S)}$, there are $n(S)$ occurrences of each turn on $\partial S$. The turn circuits do not contain the information of which particular instances of turns are joined bands, nor do they record how many times the band corresponding to a given edge in the circuit wraps around the annulus $X - C$. \[turngraph\] Given two cyclically reduced words $w,w'$ representing the same conjugacy class in $BS(m,\ell)$, there is an isomorphism $\Gamma(w) \to \Gamma(w')$ of the underlying directed graph structure that respects vertex type and edge duality but not necessarily the vertex weights. However, a directed circuit is a potential disk with one sets of weights if and only if it is a potential disk with the other set. The difference in weights of a type $m$ vertex is a multiple of $m$, the difference in weights of a type $\ell$ vertex is a multiple of $\ell$, and the difference in weights of mixed type vertex is a multiple of $\gcd(m,\ell)$. See Remark \[cyc-red\]. In what follows, only the property of being a potential disk is used and therefore this ambiguity in the weighed turn graph associated to a conjugacy class is not an issue. \[potential-disk\] Suppose $\gamma$ is a turn circuit for $S_0$ that corresponds to an inner boundary component in $\partial^- S_0$ that bounds a disk in $S_1$. Then $\gamma$ is a potential disk. For any band in $S_b$, the core arc (a component of $N_b$) maps to $C'$ as a loop of some degree $d$. The two sides then map to $C$ as loops of degrees $dm$ and $d\ell$ respectively. If $\iota$ is the side of a band that leads from a turn labeled $ta^kt^*$ to a turn labeled $t^*a^{k'}t^{-1}$ then the map $\iota \to C$ has degree a multiple of $m$. Likewise if $\iota$ leads from a turn labeled $t^{-1}a^kt^*$ to a turn labeled $t^*a^{k'}t$ then $\iota$ maps to $C$ with degree a multiple of $\ell$. Therefore the total degree of an inner boundary component corresponding to a turn circuit $\gamma$ is $\omega(\gamma) + dm + d'\ell$ for some integers $d,d'$. If $\gamma$ is of type $m$ then $d' = 0$. Likewise, if $\gamma$ is of type $\ell$ then $d = 0$. If the boundary component actually bounds a disk in $S_1$ then this total degree is $0$. Hence $\omega(\gamma) \equiv 0 \mod \mu(\gamma)$ and therefore $\gamma$ is a potential disk. Linear optimization {#sec:lp} =================== We would like to convert the optimization problem in Lemma \[scl-g\] to a problem of optimizing a certain linear functional on a vector space whose coordinates correspond to possible potential disks, subject to certain linear constraints. Here the functional would count the number of potential disks, and the constraints would arise from the pairing of edges in the turn graph. The objective would then be to compute stable commutator length using classical linear programming. The main difficulty in such an approach is arranging that the optimization takes place over a *finite dimensional* object. In this section, we show how to convert an admissible surface to a vector in a finite dimensional vector space in such a way that the number of disk components of $S_1$ is less than the value of an appropriate linear functional. We thus obtain computable, rational lower bounds for the stable commutator length of elements of Baumslag–Solitar groups (Theorem \[scl-lowerbound-lp\]). In Section \[sec:alt\], we will show that these bounds are sharp for a certain class of elements. We construct the finite dimensional vector space as follows. Let $w$ be a conjugate of $g$ of the form given in Remark \[cyc-red\]. Let $M=\max\{{\left\lvert {m} \right\rvert},{\left\lvert {\ell} \right\rvert}\}$. We consider two sets of directed circuits in $\Gamma(w)$: - ${\mathfrak{X}}$: the set of potential disks that are a sum of not more than $M$ embedded circuits, and - ${\mathfrak{Y}}$: the set of all embedded circuits. Note that both ${\mathfrak{X}}$ and ${\mathfrak{Y}}$ are finite sets and that they may have some circuits in common. Enumerate these sets as ${\mathfrak{X}}= \{ \alpha_1,\ldots,\alpha_p \}$ and ${\mathfrak{Y}}= \{\beta_1, \ldots, \beta_q\}$. Let ${{\mathbb X}}$ be a $p$–dimensional real vector space with basis $\{{\mathbf{x}}_1,\ldots,{\mathbf{x}}_p\}$, and let ${{\mathbb Y}}$ be a $q$–dimensional real vector space with basis $\{{\mathbf{y}}_1,\ldots,{\mathbf{y}}_q\}$. Equip both ${{\mathbb X}}$ and ${{\mathbb Y}}$ with an inner product that makes the respective bases orthonormal. By Remark \[turngraph\], the vector spaces ${{\mathbb X}}$ and ${{\mathbb Y}}$ depend only on the conjugacy class in $BS(m,\ell)$ represented by $w$. Abusing notation, we let $\{{\mathbf{x}}_1,\ldots,{\mathbf{x}}_p,{\mathbf{y}}_1,\ldots,{\mathbf{y}}_q\}$ denote the corresponding orthonormal basis of ${{\mathbb X}}\oplus{{\mathbb Y}}$. This is the vector space with which we will work. The linear functional on the vector space ${{\mathbb X}}\oplus{{\mathbb Y}}$ whose values will be compared with the number of disk components of $S_1$ is the functional that is the sum of the coordinates corresponding to ${{\mathbb X}}$, i.e. the functional that takes in ${\mathbf{u}}\in {{\mathbb X}}\oplus {{\mathbb Y}}$ and gives out ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}:= \sum_{i = 1}^p {\mathbf{u}}\cdot{\mathbf{x}}_i$. One thinks of this functional as counting the number of potential disks. There are additional linear functionals on ${{\mathbb X}}\oplus{{\mathbb Y}}$ that count the number of times turn circuits in a given collection visit a specific vertex or edge. For each vertex $v \in \Gamma(w)$, define $F_v{\colon \thinspace}{{\mathbb X}}\oplus {{\mathbb Y}}\to {{\mathbb R}}$ by letting $F_v({\mathbf{x}}_i)$ be the number of times $\alpha_i$ visits $v$, letting $F_v({\mathbf{y}}_i)$ be the number of times $\beta_i$ visits $v$, and extending by linearity. For each edge $e \subset \Gamma(w)$, define $F_e{\colon \thinspace}{{\mathbb X}}\oplus {{\mathbb Y}}\to {{\mathbb R}}$ by letting $F_e({\mathbf{x}}_i)$ be the number of times $\alpha_i$ traverses $e$, letting $F_v({\mathbf{y}}_i)$ be the number of times $\beta_i$ traverses $e$, and extending by linearity. One thinks of the next lemma as saying that, if a collection of turn circuits traverses each edge the same number of times as its dual edge, then this collection of turn circuits visits each vertex the same number of times. \[eqaul-v\] If $F_e({\mathbf{u}}) = F_{\bar{e}}({\mathbf{u}})$ for each dual edge pair $e,\bar{e}$ of $\Gamma(w)$, then $F_v({\mathbf{u}}) = F_{v'}({\mathbf{u}})$ for any vertices $v,v' \in \Gamma(w)$. For a vertex $v \in \Gamma(w)$, let $E^+(v)$ be the set of directed edges that are outgoing from $v$ and let $E^-(v)$ be the set directed edges that are incoming to $v$. Then $$F_v({\mathbf{u}}) = \sum_{e \in E^+(v)} F_e({\mathbf{u}}) = \sum_{e \in E^-(v)} F_e({\mathbf{u}}).$$ First suppose that $v$ corresponds to turn $i$ and $v'$ corresponds to turn $i-1$. Then edge duality gives a pairing between edges in $E^+(v)$ and edges in $E^-(v')$. Since $F_e({\mathbf{u}}) = F_{\bar{e}}({\mathbf{u}})$ for every dual edge pair $e,\bar{e}$, we have that $$F_v({\mathbf{u}}) = \sum_{e \in E^+(v)} F_e({\mathbf{u}}) = \sum_{e \in E^+(v)} F_{\bar{e}}({\mathbf{u}}) =\sum_{e \in E^-(v')} F_{e}({\mathbf{u}}) = F_{v'}({\mathbf{u}}).$$ Letting $i$ vary, we obtain a similar statement for all pairs of vertices corresponding to adjacent turns. It follows that $F_v({\mathbf{u}}) = F_{v'}({\mathbf{u}})$ for any vertices $v,v' \in \Gamma(w)$. Let $C \subset {{\mathbb X}}\oplus {{\mathbb Y}}$ be the cone of non-negative vectors ${\mathbf{u}}$ such that $F_e({\mathbf{u}}) = F_{\bar{e}}({\mathbf{u}})$ for every dual edge pair $e,\bar{e}$ of $\Gamma(w)$. In light of the lemma, we denote by $F {\colon \thinspace}C \to {{\mathbb R}}$ the function $F_v\big|_C$ for any vertex $v \in \Gamma(w)$. The following proposition shows how to convert an admissible surface into a vector ${\mathbf{u}}\in C$ in such a way that ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}$ is at least the number of disk components of $S_1$. \[better-vector\] Given an admissible surface $S \to X$, there is a vector ${\mathbf{u}}\in C$ such that $$\label{eq:better-vector} \frac{{\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}}{F({\mathbf{u}})} {\geqslant}\frac{{\chi^+}(S_1)}{n(S)}.$$ Suppose the surface $S$ has been decomposed as $S_0\cup S_1$ as described in Section \[sec:turn\], and consider the collection of turn circuits for $S_0$. Let $\gamma$ be one of the turn circuits in this collection. As a cycle we can decompose $\gamma$ as a sum of embedded circuits, i.e.  $\gamma = \beta_{i_1} + \cdots + \beta_{i_k}$. This decomposition may not be unique, but we only need its existence. For each turn circuit $\gamma$, we will construct a corresponding vector ${\mathbf{u}}(\gamma)$, depending on this decomposition and on whether the corresponding boundary component of $\partial^- S_0$ bounds a disk in $S_1$. If the corresponding inner boundary component of $\partial^- S_0$ does not bound a disk in $S_1$, we define $${\mathbf{u}}(\gamma) = \sum_{j = 1}^k {\mathbf{y}}_{i_j}.$$ Otherwise, the corresponding inner boundary component of $\partial^- S_0$ does bound a disk in $S_1$, in which case Lemma \[potential-disk\] implies that $\gamma$ is a potential disk. If $k {\leqslant}M$, then $\gamma \in{\mathfrak{X}}$; say $\gamma= \alpha_i$. In this case we define ${\mathbf{u}}(\gamma) = {\mathbf{x}}_i$. Otherwise, if $k>M$, we proceed as follows. For each $\beta_{i_j}$, let $\mu(\beta_{i_j})\beta_{i_j}$ denote the sum of $\mu(\beta_{i_j})$ copies of $\beta_{i_j}$. Notice that $\mu(\beta_{i_j})\beta_{i_j}$ is a potential disk that is not the sum of more than $M$ embedded circuits. Hence $\mu(\beta_{i_j})\beta_{i_j}\in{\mathfrak{X}}$, so $\mu(\beta_{i_j})\beta_{i_j}= \alpha_{i'_j}$ for some $i'_j \in \{1,\ldots,p\}$. In this case we define $${\mathbf{u}}(\gamma) = \sum_{j = 1}^k \frac{1}{\mu(\beta_{i_j})} {\mathbf{x}}_{i'_j}.$$ The vector we will consider is ${\mathbf{u}}= \sum_\gamma {\mathbf{u}}(\gamma)$, where this sum is taken over all $\gamma$ in the collection of turn circuits for $S_0$ (with multiplicity). Establishing the following three claims will complete the proof of the proposition. 1. ${\mathbf{u}}\in C$,\[u-in-C\] 2. $F({\mathbf{u}}) = n(S)$,\[F-is-n\] and 3. $|{\mathbf{u}}|_{{\mathbb X}}{\geqslant}{\chi^+}(S_1)$.\[geq-disk\] : The vector ${\mathbf{u}}(\gamma)$ was constructed so that $F_e({\mathbf{u}}(\gamma))$ counts the number of times the turn circuit $\gamma$ traverses the edge $e$. Thus $F_e({\mathbf{u}})$ records the number of times turn circuits for $S_0$ traverse $e$. Every time an edge $e$ is traversed by a turn circuit for $S_0$, there is a band in $S_b$ one side of which represents $e$. The other side of this band represents $\bar{e}$, so therefore we have that $F_e({\mathbf{u}}) = F_{\bar{e}}({\mathbf{u}})$ for all edges $e$. Thus ${\mathbf{u}}\in C$. : The vector ${\mathbf{u}}(\gamma)$ was also constructed so that $F_v({\mathbf{u}}(\gamma))$ counts the number of times the turn circuit $\gamma$ visits the vertex $v$. Therefore $F({\mathbf{u}})$ records the number of times turn circuits for $S_0$ visit any given vertex. As each turn occurs once in $w$, each vertex must be visited exactly $n(S)$ times. Thus $F({\mathbf{u}})=n(S)$. : Let $\gamma$ be a turn circuit for $S_0$, and suppose the corresponding inner boundary component of $S_0$ bounds a disk in $S_1$. Decompose $\gamma$ as a sum $\beta_{i_1} + \cdots + \beta_{i_k}$ of embedded circuits as above. If $k {\leqslant}M$, then ${\mathbf{u}}(\gamma) = {\mathbf{x}}_i$ for some $i \in \{1,\ldots,p\}$, and thus ${\left\lvert {{\mathbf{u}}(\gamma)} \right\rvert}_{{\mathbb X}}= 1$. Otherwise, we have $${\left\lvert {{\mathbf{u}}(\gamma)} \right\rvert}_{{\mathbb X}}= \sum_{j = 1}^k \frac{1}{\mu(\beta_{i_j})} {\geqslant}\sum_{j = 1}^k \frac{1}{M} = \frac{k}{M} {\geqslant}1.$$ In either case, ${\left\lvert {{\mathbf{u}}(\gamma)} \right\rvert}_{{\mathbb X}}{\geqslant}1$. As ${\left\lvert {{{\mathchoice{\mkern1mu\mbox{\raise2.2pt\hbox{$ \centerdot$}} \mkern1mu}{\mkern1mu\mbox{\raise2.2pt\hbox{$\centerdot$}}\mkern1mu}{ \mkern1.5mu\centerdot\mkern1.5mu}{\mkern1.5mu\centerdot\mkern1.5mu}}}} \right\rvert}_{{\mathbb X}}$ is a linear functional, we thus have that ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}{\geqslant}{\chi^+}(S_1)$. \[scl-lowerbound-lp\] Let $g \in BS(m,\ell)$, $m \neq \ell$, be of $t$–exponent zero. Then there is a computable, finite sided, rational polyhedron $P \subset {{\mathbb X}}\oplus {{\mathbb Y}}$ such that $$\label{eq:scl-lowerbound-lp} \operatorname{scl}(g) {\geqslant}\frac{|g|_t}{4} - \frac{1}{2}\max\left\{ {\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}\mid {\mathbf{u}}\text{ is a vertex of } P \right\}.$$ Let $P = F^{-1}(1)$. If $V$ is the number of vertices of $\Gamma(w)$, we can extend $F$ to a linear functional $\widetilde{F}{\colon \thinspace}{{\mathbb X}}\oplus {{\mathbb Y}}\to {{\mathbb R}}$ by setting $\widetilde{F}({\mathbf{u}}) = \frac{1}{V}\sum_v F_v({\mathbf{u}})$, where the sum is taken over all vertices of $\Gamma(w)$. The linear functional $\widetilde{F}$ is positive on all basis vectors of ${{\mathbb X}}\oplus {{\mathbb Y}}$, and hence a level set of $\widetilde{F}$ intersects the positive cone in a compact set. Clearly $P = C \cap \widetilde{F}^{-1}(1)$. Thus $P$ is a finite sided, rational, compact polyhedron. By Lemma \[scl-g\], Proposition \[better-vector\], and the linearity of ${\left\lvert {{{\mathchoice{\mkern1mu\mbox{\raise2.2pt\hbox{$ \centerdot$}} \mkern1mu}{\mkern1mu\mbox{\raise2.2pt\hbox{$\centerdot$}}\mkern1mu}{ \mkern1.5mu\centerdot\mkern1.5mu}{\mkern1.5mu\centerdot\mkern1.5mu}}}} \right\rvert}_{{\mathbb X}}$ and $F$, we have that $$\begin{aligned} \operatorname{scl}(g) & = \ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ + \ \frac{1}{2}\inf_S \frac{-{\chi^+}(S_1)}{n(S)} \\ &= \ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ - \ \frac{1}{2}\sup_S \frac{{\chi^+}(S_1)}{n(S)} \\ & {\geqslant}\ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ - \ \frac{1}{2}\sup_{{\mathbf{u}}\in C} \frac{{\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}}{F({\mathbf{u}})} \\ & {\geqslant}\ \frac{{\left\lvert {g} \right\rvert}_t}{4} \ - \ \frac{1}{2}\sup_{{\mathbf{u}}\in P} {\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}. \end{aligned}$$ As $P$ is a finite sided, compact polyhedron and ${\left\lvert {{{\mathchoice{\mkern1mu\mbox{\raise2.2pt\hbox{$ \centerdot$}} \mkern1mu}{\mkern1mu\mbox{\raise2.2pt\hbox{$\centerdot$}}\mkern1mu}{ \mkern1.5mu\centerdot\mkern1.5mu}{\mkern1.5mu\centerdot\mkern1.5mu}}}} \right\rvert}_{{\mathbb X}}$ is a linear functional, the supremum is realized at one of the vertices of $P$. This gives . \[only-X\] If ${\mathbf{u}}$ is a vertex of $P$ that maximizes ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}$ in $P$, then ${\mathbf{u}}\cdot{\mathbf{y}}_i = 0$ for all $i \in \{1,\ldots,q\}$. Indeed, suppose not and let $\beta_i \in {\mathfrak{Y}}$ be such that ${\mathbf{u}}\cdot{\mathbf{y}}_i = c > 0$. Then there is some $\alpha_{i'} \in {\mathfrak{X}}$ such that $\mu(\beta_i)\beta_i = \alpha_{i'}$. One then observes that ${\mathbf{u}}' = {\mathbf{u}}- c{\mathbf{y}}_i + \frac{c}{\mu(\beta_i)}{\mathbf{x}}_{i'} \in P$ and ${\left\lvert {{\mathbf{u}}'} \right\rvert}_{{\mathbb X}}> {\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}$. The linear programming problem described in this section has been implemented using `Sage` [@sage] and is available from the first author’s webpage. The number of embedded circuits in $\Gamma(w)$ is on the order of ${\left\lvert {w} \right\rvert}_{t}!$ and so the algorithm is only practical for elements with small $t$–length. Elements of alternating $t$–shape {#sec:alt} ================================= The bounds given in Theorem \[scl-lowerbound-lp\] are not always sharp, as we will point out in Remark \[strict inequality\]. However, we show in Theorem \[scl-equality-alternating\] that these bounds are sharp for a class of elements that have what we call *alternating $t$–shape*. We thus show that stable commutator length is computable and rational for such elements. We also characterize which elements of alternating $t$–shape admit extremal surfaces (Theorem \[extremal-characterization\]). We say that an element $g\in BS(m,\ell)$ has *alternating $t$–shape* if it has a conjugate of the form given in Remark \[cyc-red\] where $n$ is even and $\epsilon_i = (-1)^{i-1}$. In this section, we restrict attention to elements of alternating $t$–shape and express this conjugate as $$\label{eq:alternating-form} w=\prod_{k=1}^r ta^{i_k} t^{-1}a^{j_k}.$$ Note that if $g$ has alternating $t$–shape then it has $t$–exponent zero. Hence stable commutator length is finite for elements of alternating $t$–shape. Constructing surfaces {#constructing-surfaces .unnumbered} --------------------- Let $P$ be as in the proof of Theorem \[scl-lowerbound-lp\], and let $$L(g)=\frac{|g|_t}{4} - \frac{1}{2}\max\left\{ {\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}\mid {\mathbf{u}}\text{ is a vertex of } P \right\}.$$ To show that the lower bound $L(g)$ on stable commutator length is sharp, we would like to find a surface $S$ that gives the same upper bound on stable commutator length. Specifically, given a vertex ${\mathbf{u}}\in P$, we want to construct a corresponding surface $S=S_0\cup S_1$ of the type discussed in Section \[sec:turn\], where $\partial S$ maps to loops representing conjugacy classes of powers of $g$ and $\partial S_1$ maps to loops representing conjugacy classes of powers of $a$. Such a surface $S_0$ can be built (in fact, many such surfaces can be built); the construction is given in the proof of Theorem \[scl-equality-alternating\]. The difficulty is arranging $S_0$ so that its inner boundary components can be efficiently capped off by $S_1$. If the degree of each inner boundary component of $S_0$ were zero, we could take each component of $S_1$ to be a disk. In this case, we would have ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}= {\chi^+}(S_1) = \chi(S_1)$ and $$\operatorname{scl}(g) {\leqslant}\frac{-\chi(S)}{2n(S)} = \frac{{\left\lvert {g} \right\rvert}_t}{4} - \frac{\chi(S_1)}{2n(S)} = L(g) {\leqslant}\operatorname{scl}(g).$$ This would mean the bound in Theorem \[scl-lowerbound-lp\] is sharp and the surface $S$ is extremal. It may not be the case that all inner boundary components of $S_0$ can be made to have degree zero. Nevertheless, when $g$ has alternating $t$–shape, we can control the number of inner boundary components of $S_0$ that have nonzero degree in such a way as to show that there are surfaces $S$ for which $\frac{-\chi(S)}{2n(S)}$ is arbitrarily close to $L(g)$. The details are given in the proof of Theorem \[scl-equality-alternating\]. In this way, we establish that the lower bound given in Theorem \[scl-lowerbound-lp\] is sharp for elements of alternating $t$–shape. \[scl-equality-alternating\] Let $g \in BS(m,\ell)$, $m\neq\ell$, have alternating $t$–shape. Then $$\operatorname{scl}(g)=L(g).$$ We will show that $\operatorname{scl}(g)<L(g)+\epsilon$ for all $\epsilon>0$. Note that, since $g$ is of alternating $t$–shape, all circuits in the turn graph are either of type $m$ or of type $\ell$, not of mixed type. Let ${\mathbf{u}}$ be a vertex of $P$ on which ${\left\lvert {{{\mathchoice{\mkern1mu\mbox{\raise2.2pt\hbox{$ \centerdot$}} \mkern1mu}{\mkern1mu\mbox{\raise2.2pt\hbox{$\centerdot$}}\mkern1mu}{ \mkern1.5mu\centerdot\mkern1.5mu}{\mkern1.5mu\centerdot\mkern1.5mu}}}} \right\rvert}_{{\mathbb X}}$ is maximal. Since $P$ is a rational polyhedron on which all coordinates are nonnegative, all coordinates of ${\mathbf{u}}$ are nonnegative rational numbers. By Remark \[only-X\], ${\mathbf{u}}$ has nonzero entries only in coordinates corresponding to ${{\mathbb X}}$. Let $K$ denote the number of edges in the turn graph. Let $N$ be an integer such that each coordinate of $N{\mathbf{u}}$ is a nonnegative integer and such that $N>K/2\epsilon$ (so that $K/2N<\epsilon$). Each coordinate ${\mathbf{x}}_i$ of ${{\mathbb X}}$ represents a directed circuit $\gamma$ in the turn graph. For each such directed circuit $\gamma$ of length $n$, we consider a $2n$–gon with alternate sides labeled by the powers of $a$ corresponding to the vertices of the turn graph through which $\gamma$ passes and alternate sides labeled by the intervening edges of the turn graph traversed by $\gamma$. See Figure \[fig:polygon\]. (0,0)–(1,0)–(1,1)–(0,1)–cycle; (0,0)–(0.6,0); (1,0)–(1,0.6); (1,1)–(0.4,1); (0,1)–(0,0.4); at (0.5,0) [$a^{i_{k_1}}$]{}; at (0.5,1) [$a^{i_{k_2}}$]{}; at (0,0.5) [$e_1$]{}; at (1,0.5) [$e_2$]{}; For each $i$ we take $N{\mathbf{u}}\cdot{\mathbf{x}}_i$ copies of the polygon corresponding to ${\mathbf{x}}_i$, thus obtaining a collection $Q_1,\dotsc,Q_s$ of polygons. Since $F_e(N{\mathbf{u}})=F_{\bar{e}}(N{\mathbf{u}})$, there exists a pairing of the edges of these polygons corresponding to edges of the turn graph such that each edge labeled by $e$ on a polygon $Q_i$ is paired with an edge labeled by $\bar{e}$ on a polygon $Q_j$. Let $\Delta$ be the graph dual to this pairing, i.e. the graph with a vertex for each polygon $Q_i$ and an edge between the vertex corresponding to $Q_i$ and the vertex corresponding to $Q_j$ for each edge of $Q_i$ that is paired with an edge from $Q_j$. The graph $\Delta$ may have many components. However, we can adjust the pairings of edges of polygons to obtain some control over the number of components of $\Delta$. Suppose $Q_{i_1}$ and $Q_{i_2}$ are polygons where an edge labeled $e$ of $Q_{i_1}$ has been paired with an edge labeled $\bar{e}$ of $Q_{i_2}$, and suppose $Q_{j_1}$ and $Q_{j_2}$ are polygons in another component of $\Delta$ where an edge labeled $e$ of $Q_{j_1}$ has been paired with an edge labeled $\bar{e}$ of $Q_{j_2}$. Then we can modify the pairing of edges to instead pair the edge labeled $e$ of $Q_{i_1}$ with the edge labeled $\bar{e}$ of $Q_{j_2}$ and the edge labeled $e$ of $Q_{j_1}$ with the edge labeled $\bar{e}$ of $Q_{i_2}$. The graph $\Delta$ corresponding to this pairing will have one fewer component than the graph corresponding to the original pairing. See Figure \[fig:edge-swap\]. Such a modification can be done any time there are two components of $\Delta$ on which edges with the same labels have been paired. Therefore, we can arrange that the number of components of $\Delta$ is no more than $K$, the number of edges in the turn graph. Note that $\Delta$ is naturally a bipartite graph, with vertices partitioned into those corresponding to turn circuits of type $m$ (“type $m$ vertices”) and those corresponding to turn circuits of type $\ell$ (“type $\ell$ vertices”). (-1,1) circle \[radius=0.04cm\]; (-2,0) circle \[radius=0.04cm\]; (-1,0) circle \[radius=0.04cm\]; (-2,1) circle \[radius=0.04cm\]; (1,0) circle \[radius=0.04cm\]; (1,1) circle \[radius=0.04cm\]; (2,0) circle \[radius=0.04cm\]; (2,1) circle \[radius=0.04cm\]; (-2,0) – (-2,1); (-1,0) – (-1,1); (1,0) – (2,1); (2,0) – (1,1); at (-2,-0.3) [$Q_{i_{1}}$]{}; at (-1,-0.3) [$Q_{i_{2}}$]{}; at (1,-0.3) [$Q_{i_{1}}$]{}; at (2,-0.3) [$Q_{i_{2}}$]{}; at (-2,1.3) [$Q_{j_{1}}$]{}; at (-1,1.3) [$Q_{j_{2}}$]{}; at (1,1.3) [$Q_{j_{1}}$]{}; at (2,1.3) [$Q_{j_{2}}$]{}; in [1,...,3]{} [ (-2,0) – ([-2+0.3\*cos(120+15\*)]{},[0.3\*sin(120+15\*)]{}); (-2,1) – ([-2+0.3\*cos(120+15\*)]{},[1-0.3\*sin(120+15\*)]{}); (-1,0) – ([-1+0.3\*cos(15\*)]{},[0.3\*sin(15\*)]{}); (-1,1) – ([-1+0.3\*cos(15\*)]{},[1-0.3\*sin(15\*)]{}); (2,0) – ([2-0.3\*cos(120+15\*)]{},[0.3\*sin(120+15\*)]{}); (2,1) – ([2-0.3\*cos(120+15\*)]{},[1-0.3\*sin(120+15\*)]{}); (1,0) – ([1-0.3\*cos(15\*)]{},[0.3\*sin(15\*)]{}); (1,1) – ([1-0.3\*cos(15\*)]{},[1-0.3\*sin(15\*)]{}); ]{} If a vertex $v\in\Delta$ corresponds to a turn circuit $\gamma$, we define the weight of $v$ to be $\omega(v):=\omega(\gamma)$. If $v$ is a type $m$ vertex we have that $m\mid\omega(v)$, and if $v$ is a type $\ell$ vertex we have that $\ell\mid\omega(v)$. We wish to assign an integer $\omega(e)$ to each edge $e\in\Delta$ such that, whenever $v$ is of type $m$, we have $$\label{eq:type-m-weight} \omega(v)-m\sum_{e\ni v}\omega(e)=0,$$ and, whenever $v$ is of type $\ell$, we have $$\label{eq:type-l-weight} \omega(v)+\ell\sum_{e\ni v}\omega(e)=0.$$ On each connected component of $\Delta$, proceed as follows. For each type $m$ vertex $v$, choose a preferred edge $e_v$ emanating from $v$. For each $v$, set $\omega(e_v)=\omega(v)/m$, and let $\omega(e)=0$ for all other edges $e$. This makes hold for all vertices of type $m$. Now choose a preferred type $\ell$ vertex $v_0$. For another type $\ell$ vertex $v_1$, let $$\Omega=\frac{\omega(v_1)}{\ell}+\sum_{e\ni v_1}\omega(e).$$ Choose a path $e_1,\dotsc,e_k$ connecting $v_1$ to $v_0$, and modify the weights $\omega(e_i)$ by decreasing $\omega(e_i)$ by $\Omega$ whenever $i$ is odd and increasing $\omega(e_i)$ by $\Omega$ whenever $i$ is even. For all vertices other than $v_1$ and $v_0$, this does not change the quantities in and . Moreover, this causes to now be true for $v_1$. Fixing $v_0$ and letting $v_1$ vary over all type $\ell$ vertices other than $v_0$, we obtain edge weights $\omega(e)$ such that and are true for all vertices on this component of $\Delta$ except for $v_0$. Thus we obtain edge weights $\omega(e)$ such that and are true for all vertices except for one vertex in each component of $\Delta$. We now proceed to build a surface. Rather than building a surface from the polygons $Q_i$, we use them to build a band surface $S_0$, then attempt to fill various components of $\partial S_0$ with disks. For each pairing of an edge of $Q_i$ with an edge of $Q_j$, insert a rectangle with sides labeled by $t$, $a^{m\omega(e)}$, $t^{-1}$, and $a^{-\ell \omega(e)}$, where $\omega(e)$ is the weight assigned to the corresponding edge of $\Delta$. See Figure \[fig:building-surface\]. (0.5,1.5)–(1,1)–(2,1)–(2,3)–(1.5,3.5)–(3.5,3.5)–(3,3)–(3,1)–(4,1)–(4.5,1.5)–(4.5,0)–(0.5,0)–cycle; (0,0)–(5,0); (0,2)–(1,1)–(2,1)–(2,3)–(1,4); (4,4)–(3,3)–(3,1)–(4,1)–(5,2); (2,1)–(3,1); (2,3)–(3,3); (2,1)–(2.6,1); (3,1)–(3,2.1); (3,3)–(2.4,3); (2,3)–(2,1.9); (1,0)–(1.6,0); (2,0)–(2.6,0); (3,0)–(3.6,0); (1,1)–(1.6,1); (3,1)–(3.6,1); (1,0.1)–(1,-0.1); (2,0.1)–(2,-0.1); (3,0.1)–(3,-0.1); (4,0.1)–(4,-0.1); at (0.9,2.45) [$Q_i$]{}; at (4.1,2.45) [$Q_j$]{}; at (1.5,0) [$a^{j_{k-1}}$]{}; at (2.5,0) [$t$]{}; at (3.5,0) [$a^{i_k}$]{}; at (1.5,1) [$a^{j_{k-1}}$]{}; at (3.5,1) [$a^{i_k}$]{}; at (2,2) [$a^{-\ell \omega(e)}$]{}; at (3,2) [$a^{m \omega(e)}$]{}; at (2.5,1) [$t$]{}; at (2.5,3) [$t^{-1}$]{}; Note that the edges of these rectangles labeled $t$ and $t^{-1}$, together with the edges of the polygons labeled by powers of $a$, form paths that would map to the $1$–skeleton of $X$. By construction, these paths correspond exactly to powers of $w$. To each of these paths, attach an annulus $S^1\times[0,1]$ labeled on both sides by this power of $w$. The rectangles and annuli together form $S_0$, shown in Figure \[fig:building-surface\]. Note that each rectangle maps naturally to the $2$–cell of $X$ with degree $\omega(e)$. The annuli map to the $1$–skeleton of $X$, as indicated by the labels, with the map factoring through the projection $S^1\times[0,1]\to S^1$. As in Section \[sec:turn\], we refer to the boundary components of $S_0$ that map to a power of $w$ as outer boundary components and to those corresponding to a polygon $Q_i$ as inner boundary components. Each of the inner boundary components of $S_0$ maps to a power of $a$; let $d$ be the number of components of the inner boundary for which this power is zero. The powers of $a$ on the inner boundary components are exactly the quantities on the left-hand sides of and . The weights $\omega(e)$ have been chosen so that these quantities are zero for all but $K$ components of the inner boundary, so therefore $d{\geqslant}s-K$. Fill these $d$ components of the inner boundary with disks, and call the resulting surface $S$. Each inner boundary component of $S$ maps to a power of $a$, and Lemma \[scl-a\] says that $\operatorname{scl}(a)=0$. Therefore, applying Lemma \[fill-with-zero\], we have that $$\begin{aligned} \operatorname{scl}(g)&{\leqslant}\frac{-\chi(S)}{2n(S)}\\ &=\frac{{\left\lvert {g} \right\rvert}_t}{4}-\frac{d}{2N}\\ &{\leqslant}\frac{{\left\lvert {g} \right\rvert}_t}{4}-\frac{s-K}{2N}\\ &= \frac{{\left\lvert {g} \right\rvert}_t}{4}-\frac{s}{2N}+\frac{K}{2N}\\ &< \frac{{\left\lvert {g} \right\rvert}_t}{4}- \frac{1}{2}\max\left\{ {\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}\mid {\mathbf{u}}\text{ is a vertex of } P \right\}+\epsilon\\ &= L(g)+\epsilon.\end{aligned}$$ Thus $\operatorname{scl}(g)=L(g)$, as desired. \[scl-alternating-rational\] If $g \in BS(m,\ell)$, $m\neq\ell$, has alternating $t$–shape, then $\operatorname{scl}(g)$ is rational. Since each vertex ${\mathbf{u}}$ of $P$ has rational coordinates and ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}$ is the sum of certain of these coordinates, we know that $L(g)$ is rational. Therefore it follows from Theorem \[scl-equality-alternating\] that $\operatorname{scl}(g)$ is rational. \[strict inequality\] In general one suspects the inequality in Theorem \[scl-lowerbound-lp\] is strict. For example, the function ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}$ has a unique maximum on the polyhedron $P$ in Theorem \[scl-lowerbound-lp\] for the element $a^2t^2at^{-1}at^{-1} \in BS(2,3)$. When attempting to build a surface from this unique optimal vertex ${\mathbf{u}}$, it turns out that every component of the dual graph $\Delta$ has a constant proportion of vertices that cannot be filled, regardless of the edge weights. Therefore, in contrast to Theorem \[scl-equality-alternating\], where all but at most a constant number of vertices can be filled, there is no sequence of surfaces associated to ${\mathbf{u}}$ to for which $\frac{{\chi^+}(S)}{n(S)}$ approaches ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}$. An explicit formula {#an-explicit-formula .unnumbered} ------------------- When ${\left\lvert {g} \right\rvert}_t = 2$ the turn graph consists of two vertices, each adjacent to a one-edge loop. In this case the vector space ${{\mathbb X}}$ is essentially two dimensional and the linear optimization problem can be solved easily by hand, resulting in a formula for stable commutator length for such elements. This calculation is interesting for two reasons. First, it is rare that one can derive a formula for $\operatorname{scl}$ in non-trivial cases. Second, the minimal value for $\operatorname{scl}$ among all “well-aligned” elements (see Definition \[def:well-aligned\] and Theorem \[th:well-aligned\]) is realized by an element of this type. \[prop:length2\] In the group $BS(m,\ell)$ with $m\not= \ell$, if $m\nmid i$ and $\ell \nmid j$ then $$\operatorname{scl}(ta^i t^{-1}a^j) \ = \ \frac{1}{2} \left( 1 - \frac{\gcd(i,m)}{{\left\lvert {m} \right\rvert}} - \frac{\gcd(j,\ell)}{{\left\lvert {\ell} \right\rvert}}\right).$$ The divisibility hypotheses simply mean that the word $ta^i t^{-1}a^j$ is cyclically reduced (cf. Remark \[cyc-red\]). The turn graph for the word $ta^it^{-1}a^j$ is as shown in Figure \[fig:turn-example\]. (-0.8,-0.28) rectangle (1.8,0.28); at (0,0) (left) [$a^i$]{}; at (1,0) (right) [$a^j$]{}; (left) .. controls (-1,0.7) and (-1,-0.7) .. (left); (right) .. controls (2,0.7) and (2,-0.7) .. (right); There are two types of potential disks: 1. Circuits of type $m$ that traverse the left loop of the turn graph $p$ times, where $m\mid pi$. 2. Circuits of type $\ell$ that traverse the right loop of the turn graph $q$ times, where $\ell\mid qj$. Note that the condition $m\mid pi$ is equivalent to $\frac{{\left\lvert {m} \right\rvert}}{\gcd(i,m)}\mid p$, and the condition $\ell\mid qj$ is equivalent to $\frac{{\left\lvert {\ell} \right\rvert}}{\gcd(j,\ell)}\mid q$. Suppose $p=\frac{k{\left\lvert {m} \right\rvert}}{\gcd(i,m)}$, where $p{\leqslant}\max\{{\left\lvert {m} \right\rvert},{\left\lvert {\ell} \right\rvert}\}$, for some positive integer $k$, and let ${\mathbf{x}}_{i_k}$ be the corresponding basis vector of ${{\mathbb X}}$. We claim that, if $k>1$ and ${\mathbf{u}}$ is a vertex of $P$ that maximizes ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}$, then ${\mathbf{u}}\cdot{\mathbf{x}}_{i_k}=0$. Indeed, suppose not, and consider the vector ${\mathbf{u}}'={\mathbf{u}}-({\mathbf{u}}\cdot{\mathbf{x}}_{i_k}){\mathbf{x}}_{i_k}+k({\mathbf{u}}\cdot{\mathbf{x}}_{i_k}){\mathbf{x}}_{i_1}$. Then $F_e({\mathbf{u}}')=F_e({\mathbf{u}})$ for all $e$ and $F({\mathbf{u}}')=F({\mathbf{u}})=1$, but $${\left\lvert {{\mathbf{u}}'} \right\rvert}_{{{\mathbb X}}}={\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}-{\mathbf{u}}\cdot{\mathbf{x}}_{i_k}+k({\mathbf{u}}\cdot{\mathbf{x}}_{i_k})={\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}+(k-1)({\mathbf{u}}\cdot{\mathbf{x}}_{i_k})>{\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}.$$ A similar argument applies to coordinates of ${{\mathbb X}}$ corresponding to potential disks of type $\ell$. Thus, if ${\mathbf{u}}$ is a vertex of $P$ that maximizes ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}$, only two coordinates of ${\mathbf{u}}$ are nonzero, one corresponding to a potential disk of type $m$ where $p=\frac{{\left\lvert {m} \right\rvert}}{\gcd(i,m)}$ and the other corresponding to a potential disk of type $\ell$ where $q=\frac{{\left\lvert {\ell} \right\rvert}}{\gcd(j,\ell)}$. Let $c$ be the value of the coordinate corresponding to this potential disk of type $m$, and let $d$ be the value of the coordinate corresponding to this potential disk of type $\ell$. Then the conditions $F_e({\mathbf{u}})=F_{\bar{e}}({\mathbf{u}})$ and $F({\mathbf{u}})=1$ become $$\frac{{\left\lvert {m} \right\rvert}}{\gcd(i,m)}c=\frac{{\left\lvert {\ell} \right\rvert}}{\gcd(j,\ell)}d=1.$$ Therefore we have that $c=\frac{\gcd(i,m)}{{\left\lvert {m} \right\rvert}}$ and $d=\frac{\gcd(j,\ell)}{{\left\lvert {\ell} \right\rvert}}$. This means that $$\begin{aligned} \operatorname{scl}(ta^it^{-1}a^j)&=\frac{|ta^it^{-1}a^j|_t}{4} - \frac{1}{2}\max\left\{ {\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}\mid {\mathbf{u}}\text{ is a vertex of } P \right\}\\ &=\frac12-\frac12\left(\frac{\gcd(i,m)}{{\left\lvert {m} \right\rvert}}+\frac{\gcd(j,\ell)}{{\left\lvert {\ell} \right\rvert}}\right), \end{aligned}$$ as desired. Extremal surfaces {#extremal-surfaces .unnumbered} ----------------- We now characterize the elements $g\in BS(m,\ell)$ of alternating $t$–shape for which an extremal surface exists. \[extremal-alternating-disks\] Suppose $S$ is an admissible surface for some $g\in BS(m,\ell)$, $m\neq\ell$, of alternating $t$–shape that has been decomposed as described in Section \[sec:turn\]. If $S$ is extremal, then $S_1$ consists only of disks. If $S$ is extremal, we know by Lemma \[extremal-disks-annuli\] that $S_1$ consists of only disks and annuli. Suppose that some component of $S_1$ is an annulus. This means that some component of the inner boundary of $S_0$ does not bound a disk in $S$. Using the construction from the proof of Proposition \[better-vector\], there is a ${\mathbf{u}}\in P$ such that ${\left\lvert {{\mathbf{u}}} \right\rvert}_{{\mathbb X}}{\geqslant}{\chi^+}(S_1)$ and ${\mathbf{u}}\cdot{\mathbf{y}}_i>0$ for some $i$. Remark \[only-X\] shows how to find ${\mathbf{u}}'\in P$ such that ${\left\lvert {{\mathbf{u}}'} \right\rvert}_{{{\mathbb X}}}>{\left\lvert {{\mathbf{u}}} \right\rvert}_{{{\mathbb X}}}$, so we have ${\left\lvert {{\mathbf{u}}'} \right\rvert}_{{{\mathbb X}}}>{\chi^+}(S_1)$. But then Theorem \[scl-equality-alternating\] shows that $S$ is not extremal. Thus $S_1$ cannot have an annular component, meaning it consists only of disks. \[extremal-characterization\] Let $g =\prod_{k=1}^r ta^{i_k} t^{-1}a^{j_k}\in BS(m,\ell)$, $m\neq\ell$. There is an extremal surface for $g$ if and only if $$\label{eq:balance} \ell\sum_{k=1}^r i_k =-m\sum_{k=1}^r j_k.$$ The status of equation does not change under cancellation of $t^{\epsilon} t^{-\epsilon}$ pairs in $g$, nor under applications of the defining relator in $BS(m,\ell)$; hence we may assume without loss of generality that $g$ is cyclically reduced. First, suppose $g$ has an extremal surface $S$. Decompose $S$ as described in Section \[sec:turn\]. By Lemma \[extremal-alternating-disks\], $S_1$ consists only of disks. Let $\Delta$ be the graph that has a vertex for each component of $S_1$ and an edge for each band of $S_b$ that connects the vertices corresponding to the two disks it adjoins. There is a weight function on the vertices of $\Delta$, where $\omega(v)$ is the total degree of $a$ at all vertices of the circuit in the turn graph corresponding to $v$. There is also a natural weight function $w$ on the edges of $\Delta$, where $\omega(e)$ is the signed degree of the map from the band corresponding to $e$ to the $2$–cell of $X$. Since all vertices of $\Delta$ bound disks, we know that whenever $v$ corresponds to a circuit of type $m$, we have $$\omega(v)-m\sum_{e\ni v}\omega(e)=0,$$ and, whenever $v$ corresponds to a circuit of type $\ell$, we have $$\omega(v)+\ell\sum_{e\ni v}\omega(e)=0.$$ Summing over all vertices of type $m$, we obtain $$\label{eq:type-m-sum} m\sum_{e\in\Delta}\omega(e)=\sum_{\substack{v\in\Delta\\\text{of type $m$}}}\omega(v)=n(S)\sum_{k=1}^r i_k.$$ Summing over all vertices of type $\ell$, we obtain $$\label{eq:type-l-sum} -\ell\sum_{e\in\Delta}\omega(e)=\sum_{\substack{v\in\Delta\\\text{of type $\ell$}}}\omega(v)=n(S)\sum_{k=1}^r j_k.$$ Multiplying by $\ell$ and by $-m$ and combining gives . Conversely, suppose the element $g$ satisfies . Let $S_0$ and $\Delta$ be as in the proof of Theorem \[scl-equality-alternating\]. Restrict to one connected component of $S_0$, and let $\Delta_0$ be the corresponding connected component of $\Delta$. Then holds for all vertices of type $m$ in $\Delta_0$. Let $N_0$ be the power of $w$ corresponding to the image of the map on the outer boundary of this component of $S_0$. Summing over all vertices of type $m$ in $\Delta_0$, we have that $$\label{eq:type-m-sum0} m\sum_{e\in\Delta_0}\omega(e)=\sum_{\substack{v\in\Delta_0\\\text{of type $m$}}}\omega(v)=N_0\sum_{k=1}^r i_k$$ also holds. Multiplying by $\ell$ and combining with shows that $$\label{eq:type-l-sum0} -\ell\sum_{e\in\Delta_0}\omega(e)=\sum_{\substack{v\in\Delta_0\\\text{of type $\ell$}}}\omega(v)=N_0\sum_{k=1}^r j_k.$$ Since the procedure in the proof of Theorem \[scl-equality-alternating\] ensures that holds for all but one $v\in\Delta_0$ of type $\ell$, implies that in fact holds for all $v\in\Delta_0$. The same argument applies to each component of $\Delta$, so hence and hold for all $v\in\Delta$. Thus all inner boundary components of $S_0$ can be filled with disks, meaning the resulting surface achieves the lower bound on $\operatorname{scl}(g)$ given by linear programming. Hence this surface is extremal. Corollary \[scl-alternating-rational\] and Theorem \[extremal-characterization\] combine to give many examples of elements for which stable commutator length is rational but for which no extremal surface exists. Previous examples of this phenomenon were found in free products of abelian groups of higher rank. See [@Calegari:sails]. Quasimorphisms on groups acting on trees {#sec:qm} ======================================== We now turn our attention from analyzing $\operatorname{scl}$ for a single element in $BS(m,\ell)$ to analyzing properties of the $\operatorname{scl}$ spectrum $\operatorname{scl}(BS(m,\ell)) \subset {{\mathbb R}}$. Our main theorem about the spectrum (Theorem \[th:gap\]) shows that either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/12$. In other words, there is a *gap* in the spectrum. The proof has two parts. In this section we will provide a general condition (*well-aligned*) for an element $g$ in a group $G$ acting on a tree that implies $\operatorname{scl}(g) {\geqslant}1/12$ (Theorem \[th:well-aligned\]). This is not quite enough for the Gap Theorem for Baumslag–Solitar groups. In Section \[sec:gap\] we use the specific structure of $BS(m,\ell)$ as an HNN extension and show that if a stronger form of the well-aligned property does not hold, then $\operatorname{scl}(g) = 0$ (Theorem \[infiniteoverlap\] and Proposition \[sclzero\]). The key to the argument in Theorem \[th:well-aligned\] is the construction of a certain function $f {\colon \thinspace}BS(m,\ell) \to {{\mathbb R}}$ for each hyperbolic element $g \in BS(m,\ell)$, satisfying certain properties described below, that provides a lower bound on $\operatorname{scl}(g)$. The material in this section applies to any group $G$. Quasimorphisms and stable commutator length {#quasimorphisms-and-stable-commutator-length .unnumbered} ------------------------------------------- The functions we will construct are homogeneous quasimorphisms. \[def:qm\] A function $f {\colon \thinspace}G \to {{\mathbb R}}$ is called a *quasimorphism* if there is a number $D$ such that $$\label{qm} {\left\lvert {f(gh) - f(g) - f(h)} \right\rvert} \ {\leqslant}\ D$$ for all $g, h \in G$. The smallest such $D$ is called the *defect* of $f$. A quasimorphism $f$ is *homogeneous* if $f(g^n) = n f(g)$ for all $g\in G$, $n \in {{\mathbb Z}}$. Bavard Duality [@Bavard:duality] provides the link between homogeneous quasimorphisms and stable commutator length. We only need one direction of this link. \[bavard\] Given $g\in G$, suppose there is a homogeneous quasimorphism $f$ with defect at most $D$ such that $f(g) = 1$. Then $\operatorname{scl}(g) {\geqslant}1/2D$. One checks easily using that if $g^n$ is a product of $m$ commutators then ${\left\lvert {f(g^n)} \right\rvert} {\leqslant}2mD$. Hence $1 = {\left\lvert {f(g)} \right\rvert} {\leqslant}2 \operatorname{cl}(g^n)D/n$, and taking the limit as $n \to \infty$ gives the desired result. Hence to derive a large lower bound for $\operatorname{scl}(g)$ one tries to construct a homogenous quasimorphism $f{\colon \thinspace}G \to {{\mathbb R}}$ such that $f(g) = 1$ and with defect as small as possible. $G$–trees {#gtrees .unnumbered} --------- We consider simplicial trees not just as combinatorial objects, but also as metric spaces with each edge being isometric to an interval of length one. A *segment* in a tree $T$ is a subset $\alpha \subset T$ that is isometric to a closed segment in ${{\mathbb R}}$. Suppose we are given an action of $G$ on a simplicial tree $T$, always assumed to be without inversions. Every element $g \in G$ has a *characteristic subtree* $T_g$, consisting of those points $x \in T$ where the displacement function $x \mapsto d(x, gx)$ achieves its minimum. This minimum is denoted ${\left\lvert {g} \right\rvert}$, and called the *translation length* of $g$, or simply the *length* of $g$. If the length is zero then $T_g$ is the set of fixed points of $g$ and we call $g$ *elliptic*. Otherwise, $T_g$ is a linear subtree on which $g$ acts by a shift of amplitude ${\left\lvert {g} \right\rvert}$. In this case $T_g$ is called the *axis* of $g$ and $g$ is *hyperbolic*. Note that $T_g$ has a natural orientation, given by the direction of the shift by $g$. A *fundamental domain* for $g$ is a segment (of length ${\left\lvert {g} \right\rvert}$) contained in the axis, of the form $[x,gx]$. We specifically allow $x$ to be a point in the interior of an edge. If $k\not=0$ then $g^k$ has the same type (elliptic or hyperbolic) as $g$. If $g$ is hyperbolic then $T_{g^k} = T_g$ and ${\left\lvert {g^k} \right\rvert} = {\left\lvert {k} \right\rvert} {\left\lvert {g} \right\rvert}$. Also, ${\left\lvert {hgh^{-1}} \right\rvert} = {\left\lvert {g} \right\rvert}$ for all $g, h$. \[easyway\] There is an easy way to identify the axis of a hyperbolic element $g\in G$. Namely, if $\alpha$ is an oriented segment or edge in $T$, then $\alpha$ is on the axis if and only if $\alpha$ and $g\alpha$ are *coherently oriented* in $T$, i.e., there is an oriented segment that contains both $\alpha$ and $g\alpha$ as oriented subsegments. When this occurs, if $x$ is any point in $\alpha$, then the segment $[x,gx]$ is a fundamental domain for $g$. Let $\gamma$ be an oriented segment in $T$. The *reverse* of $\gamma$ is the same segment with the opposite orientation, denoted $\overline{\gamma}$. A *copy of $\gamma$* is a segment of the form $g \gamma$ for some $g\in G$. If $g$ is hyperbolic then the quotient of $T_g$ by the action of $\langle g \rangle$ is a circuit of length ${\left\lvert {g} \right\rvert}$. A *copy of $\gamma$* in $T_g / \langle g \rangle$ is the image of a copy of $\gamma$ in $T_g$, provided that ${\left\lvert {\gamma} \right\rvert} {\leqslant}{\left\lvert {g} \right\rvert}$. (If ${\left\lvert {\gamma} \right\rvert} > {\left\lvert {g} \right\rvert}$ then there are no copies of $\gamma$ in $T_g/\langle g \rangle$.) We say that two segments *overlap* if their intersection is a non-trivial segment. Let $\gamma$ be an oriented segment in $T$. For an oriented segment $\alpha$, let $c_{\gamma}(\alpha)$ be the maximal number of non-overlapping positively oriented copies of $\gamma$ in $\alpha$. Note that $c_{\gamma}(\overline{\alpha}) = c_{\overline{\gamma}}(\alpha)$. Also define $$f_{\gamma}(\alpha) \ = \ c_{\gamma}(\alpha) - c_{\overline{\gamma}}(\alpha).$$ If $g\in G$ is hyperbolic, let $c_{\gamma}(g)$ be the maximal number of non-overlapping positively oriented copies of $\gamma$ in $T_g/\langle g \rangle$. If $g$ is elliptic, let $c_{\gamma}(g) = 0$. In either case, define $$\label{f-def} f_{\gamma}(g) \ = \ c_{\gamma}(g) - c_{\overline{\gamma}}(g)$$ and $$\label{h-def} h_{\gamma}(g) \ = \ \lim_{n \to \infty} \frac{f_{\gamma}(g^n)}{n}.$$ We will see shortly that $f_{\gamma}$ is a quasimorphism. Therefore, by [@Calegari:scl Lemma 2.21], the limit defining $h_{\gamma}$ exists and $h_{\gamma}$ is a homogeneous quasimorphism. \[junctures\] Let $g\in G$ be hyperbolic and suppose that a fundamental domain for $g$ is expressed as a concatenation of non-overlapping segments $\alpha_1, \ldots, \alpha_k$, each given the same orientation as $T_g$. Then for any $\gamma$ there is an estimate $$\sum_i c_{\gamma}(\alpha_i) \ {\leqslant}\ c_{\gamma}(g) \ {\leqslant}\ k + \sum_i c_{\gamma}(\alpha_i).$$ In the situation of the lemma, we will refer to the images in $T_g/\langle g \rangle$ of the endpoints of the segments $\alpha_i$ as *junctures*. There are $k$ junctures in $T_g/\langle g \rangle$. Start with maximal collections of non-overlapping copies of $\gamma$ in the segments $\alpha_i$. The union of these sets of copies projects to a non-overlapping collection in $T_g/\langle g \rangle$, yielding the first inequality. For the second inequality, start with a maximal collection of non-overlapping copies of $\gamma$ in $T_g / \langle g \rangle$. At most $k$ of these copies contain junctures in their interiors. Each remaining copy lifts to a copy of $\gamma$ in one of the segments $\alpha_i$, and no two of these lifts overlap. Hence $\sum_i c_{\gamma}(\alpha_i) {\geqslant}c_{\gamma}(g) - k$. The main technical result of this section is the following theorem. \[defect\] Suppose $G$ acts on a simplicial tree $T$. Let $\gamma$ be an oriented segment in $T$ (with endpoints possibly not at vertices). Then the functions $f_{\gamma}$ and $h_{\gamma}$ defined in and are quasimorphisms on $G$ with defect at most $6$. We will prove the result for $h_{\gamma}$ directly. Replacing “$n$” throughout by “$1$” yields a proof of the result for $f_{\gamma}$. Fix elements $g, h \in G$. We wish to show that ${\left\lvert {h_{\gamma}(gh) - h_{\gamma}(g) - h_{\gamma}(h)} \right\rvert} {\leqslant}6$. There are several cases, corresponding to different configurations of the characteristic subtrees $T_g$, $T_h$, and $T_{gh}$. *Case I: $T_g$ and $T_h$ are disjoint.* Let $\rho$ be the segment joining $T_h$ to $T_g$, oriented from $T_h$ and towards $T_g$. Let $\rho' = g\overline{\rho}$ (which is a copy of $\overline{\rho}$). If $g$ and $h$ are both hyperbolic, let $\alpha$ and $\beta$ be fundamental domains for $h$ and $g$ respectively, as indicated in Figure \[fig:I\]. \[b\] at 146 32 \[t\] at 199 12 \[r\] at 171 21 \[r\] at 226 23 \[l\] at 225 43 \[l\] at 278 0 \[l\] at 337 27 ![Case I, $g$ and $h$ hyperbolic. $T_h$ is green, $T_g$ is purple, $T_{gh}$ is black. The red edges are of the form $e$, $he$, and $ghe$.[]{data-label="fig:I"}](I "fig:") Note that $gh$ has a fundamental domain given by the concatenation $\alpha \cdot \rho \cdot \beta \cdot \rho'$, by Remark \[easyway\]. More generally, $(gh)^n$ has a fundamental domain made of $n$ copies each of $\alpha$, $\rho$, $\beta$, and $\overline{\rho}$. Lemma \[junctures\] yields $$\begin{aligned} \label{e1} n(c_{\gamma}(\alpha) + c_{\gamma}(\rho) + c_{\gamma}(\beta) + c_{\gamma}(\overline{\rho})) \ &{\leqslant}\ c_{\gamma}((gh)^n) \ \notag \\ &{\leqslant}\ n(c_{\gamma}(\alpha) + c_{\gamma}(\rho) + c_{\gamma}(\beta) + c_{\gamma}(\overline{\rho})) + 4n\end{aligned}$$ and $$\begin{aligned} \label{e2} n(c_{\overline{\gamma}}(\alpha) + c_{\overline{\gamma}}(\rho) + c_{\overline{\gamma}}(\beta) + c_{\overline{\gamma}}(\overline{\rho})) \ &{\leqslant}\ c_{\overline{\gamma}}((gh)^n) \ \notag \\ &{\leqslant}\ n(c_{\overline{\gamma}}(\alpha) + c_{\overline{\gamma}}(\rho) + c_{\overline{\gamma}}(\beta) + c_{\overline{\gamma}}(\overline{\rho})) + 4n. \end{aligned}$$ Subtracting from yields $$\label{e3} n(f_{\gamma}(\alpha) + f_{\gamma}(\beta)) - 4n \ {\leqslant}\ f_{\gamma}((gh)^n) \ {\leqslant}\ n(f_{\gamma}(\alpha) + f_{\gamma}(\beta)) + 4n.$$ Since $h^n$ and $g^n$ have fundamental domains made of $n$ copies of $\alpha$ and $\beta$ respectively, Lemma \[junctures\] also yields, in a similar way, $$\label{e4} nf_{\gamma}(\alpha) - n \ {\leqslant}\ f_{\gamma}(h^n) \ {\leqslant}\ nf_{\gamma}(\alpha) + n$$ and $$\label{e5} nf_{\gamma}(\beta) - n \ {\leqslant}\ f_{\gamma}(g^n) \ {\leqslant}\ nf_{\gamma}(\beta) + n.$$ Subtracting and from yields $$-6n \ {\leqslant}\ f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n) \ {\leqslant}\ 6n.$$ Dividing by $n$ and taking a limit, we obtain ${\left\lvert {h_{\gamma}(gh) - h_{\gamma}(g) - h_{\gamma}(h)} \right\rvert} \ {\leqslant}\ 6$, as desired. *Remark.* In the argument just given, the fundamental domains for $h$, $g$, and $gh$ respectively were decomposed into one, one, and four segments; hence $T_h/\langle h\rangle$, $T_g/\langle g\rangle$, and $T_{gh}/\langle gh \rangle$ contained a total of six junctures. These junctures were the only source of defect, since the individual segments always contributed zero to ${\left\lvert {f_{\gamma}(gh) - f_{\gamma}(g) - f_{\gamma}(h)} \right\rvert}$. Every case below follows the same pattern: the defect will be bounded above by the total number of junctures appearing in the quotient circuits. In what follows, we will describe the structure of $T_h/\langle h\rangle$, $T_g/\langle g\rangle$, and $T_{gh}/\langle gh \rangle$ in each case and leave some of the details of the estimates to the reader. Returning to Case I, suppose $g$ and $h$ are both elliptic. Then $g^n$ and $h^n$ are also elliptic, and $(gh)^n$ has a fundamental domain made of $n$ copies of $\rho$ and $n$ copies of $\overline{\rho}$. See Figure \[fig:I-e\]. Using Lemma \[junctures\] one obtains $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ {\left\lvert {f_{\gamma}((gh)^n)} \right\rvert} \ {\leqslant}\ 2n,$$ for a defect of at most $2$. \[tr\] at 111 24 \[c\] at 92 42 \[c\] at 129 7 \[l\] at 293 26 ![Case I, $g$ and $h$ elliptic. $T_h$ is green, $T_g$ is purple, $T_{gh}$ is black. The red edges are of the form $e$, $he$, and $ghe$.[]{data-label="fig:I-e"}](I-e "fig:") If one of $g$ and $h$ is elliptic, say $g$, then $h^n$ has a fundamental domain given by $n$ copies of $\alpha$, and $(gh)^n$ has a fundamental domain given by $n$ copies each of $\alpha$, $\rho$, and $\overline{\rho}$. The estimate given by Lemma \[junctures\] becomes $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ {\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 4n,$$ for a defect of at most $4$. *Case II: $g$ and $h$ are hyperbolic, with positive overlap.* That is, $T_g$ and $T_h$ intersect in a segment, on which $T_g$ and $T_h$ induce the same orientation. Let $e$ be an edge in $T_g \cap T_h$, oriented coherently with $T_g$ and $T_h$. Let $v$ be the terminal vertex of $e$. Since $e \in T_h$, the edges $h^{-1}e$ and $e$ are coherently oriented and $\alpha = [h^{-1}v, v]$ is a fundamental domain for $h$. Similarly, $e$ and $ge$ are coherently oriented and $\beta = [v, gv]$ is a fundamental domain for $g$. Let all the edges of $\alpha$ and $\beta$ be given orientations from $T_h$ and $T_g$ respectively. Since $g$ and $h$ both move $e$ in the same direction (that is, into the same component of $T - \{e\}$), $e$ separates $h^{-1}e$ from $ge$. It follows that the edges of $\alpha$ and of $\beta$ are all coherently oriented in $T$. Hence $\alpha$ and $\beta$ do not overlap, and $\alpha \cdot \beta = [h^{-1}v, gv]$ is a fundamental domain for $gh$. With a total of four junctures (one for $h$, one for $g$, two for $gh$), the estimate given by Lemma \[junctures\] becomes $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 4n,$$ for a defect of at most $4$. *Case III: $g$ and $h$ are hyperbolic, with negative overlap.* That is, $T_g$ and $T_h$ intersect in a segment, on which $T_g$ and $T_h$ induce opposite orientations. Let $\Delta$ be the length (possibly infinite) of $T_g \cap T_h$. There are several sub-cases, according to the relative sizes of ${\left\lvert {g} \right\rvert}$, ${\left\lvert {h} \right\rvert}$, and $\Delta$. *Sub-case III-A: $\Delta {\leqslant}{\left\lvert {g} \right\rvert}, {\left\lvert {h} \right\rvert}$, not all three numbers equal.* Let $\rho$ be the segment $T_h \cap T_g$, oriented coherently with $T_h$. There is a fundamental domain for $h$ of the form $\alpha \cdot \rho$, and similarly, a fundamental domain for $g$ of the form $\overline{\rho} \cdot \beta$; see Figure \[fig:III-A\]. Then $\alpha \cdot \beta$ is a fundamental domain for $gh$. (By assumption, at least one of $\alpha$, $\beta$ is a non-trivial segment, and $gh$ is hyperbolic.) \[tl\] at 102 21 \[tr\] at 137 19 \[r\] at 112 50 \[l\] at 149 97 \[l\] at 183 2 \[l\] at 318 20 ![Case III-A. $T_h$ is green, $T_g$ is purple, $T_{gh}$ is black. The red edges are of the form $e$, $he$, and $ghe$.[]{data-label="fig:III-A"}](III-A "fig:") The quotient circuits have at most six junctures: two for $g$, two for $h$, and two for $gh$. Lemma \[junctures\] leads to an estimate $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 6n,$$ for a defect of at most $6$. *Sub-case III-B: ${\left\lvert {g} \right\rvert} = {\left\lvert {h} \right\rvert} {\leqslant}\Delta$.* In this case, we show that $gh$ is elliptic. Let $\alpha \subset T_h \cap T_g$ be a fundamental domain for $h$. Then $\overline{\alpha}$ is a fundamental domain for $g$, and $gh$ fixes the initial endpoint of $\alpha$. With two junctures in total, we obtain the estimate $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ {\left\lvert {- f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 2n,$$ for a defect of at most $2$. *Sub-case III-C: ${\left\lvert {h} \right\rvert} < {\left\lvert {g} \right\rvert}, \Delta$.* There is a simplicial fundamental domain $\alpha$ for $h$ such that if $e$ is the initial edge of $\alpha$, then $\alpha \cdot he$ is contained in $T_h \cap T_g$. Then, there is a fundamental domain for $g$ of the form $\overline{\alpha} \cdot \beta$; see Figure \[fig:III-CD\]. By considering the location of $ghe$, one finds that $\beta$ is a fundamental domain for $gh$. The three circuits have a total of four junctures, and we obtain $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 4n,$$ for a defect of at most $4$. \[b\] at 58 34 \[tl\] at 26 23 \[bl\] at 185 39 \[t\] at 227 25 \[tl\] at 48 14 \[tl\] at 89 14 \[b\] at 7 30 \[t\] at 165 37 \[b\] at 248 48 \[b\] at 221 46.5 \[tl\] at 128 57 \[r\] at 2 3 \[tl\] at 290 57 \[r\] at 164 3 ![Cases III-C (left) and III-D (right). $T_h$ is green, $T_g$ is purple.[]{data-label="fig:III-CD"}](III-CD "fig:") *Sub-case III-D: ${\left\lvert {g} \right\rvert} < {\left\lvert {h} \right\rvert}, \Delta$.* Fundamental domains $\beta$, $\alpha$, and $\alpha \cdot \overline{\beta}$ for $g$, $gh$, and $h$ respectively can be constructed in a similar fashion as in Case III-C; see Figure \[fig:III-CD\]. Alternatively, this case reduces to Case III-C, replacing $g$ and $h$ by $h^{-1}$ and $g^{-1}$ respectively. *Case IV: $g$ is elliptic, $h$ is hyperbolic, $T_g \cap T_h \not= \emptyset$.* If $T_g \cap T_h$ contains an edge $e$, let $\alpha \subset T_h$ be the fundamental domain starting with $h^{-1}e$. Then $\alpha$ is also a fundamental domain for $gh$. This leads to an estimate $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ {\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 2n,$$ and a defect of at most $2$. If $T_g \cap T_h$ is a single vertex $v$, let $e\in T_h$ be the coherently oriented edge with initial vertex $v$, and let $\alpha$ be the fundamental domain $[h^{-1}v, v]$. If $ge \not\in \alpha$ then $\alpha$ is a fundamental domain for $gh$ also, and we obtain a defect of at most $2$ as above. So now assume that $ge \in \alpha$, i.e. that $ge$ separates $h^{-1}v$ from $v$. Note that $h^{-1}e$ and $ge$ are not coherently oriented, so the characteristic subtree $T_{gh}$ will not contain these edges. We have that $gh(\alpha) \cap \alpha$ contains the edge $ge$. Consider the length of $gh(\alpha) \cap \alpha$. If this length is ${\left\lvert {\alpha} \right\rvert}/2$ or greater, then $gh$ fixes the midpoint of $\alpha$. Then $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ {\left\lvert {- f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ n,$$ giving a defect of at most $1$. Otherwise, there is a subsegment $\beta \subset \alpha$, centered on the midpoint of $\alpha$, of maximal size so that $\beta$ does not overlap $gh\beta$. We can write $\alpha$ as a concatenation $\alpha_1 \cdot \beta \cdot \alpha_2$, where $\alpha_2 = gh\overline{\alpha}_1$. See Figure \[fig:IV\]. \[b\] at 46 13 \[b\] at 117 31 \[b\] at 46 49 \[tl\] at 83 19 \[tr\] at 81 36 \[l\] at 187 26 \[c\] at 147 14 \[r\] at 82 73 ![Case IV. The element $gh$ takes $\alpha_1$ to $\overline{\alpha}_2$. Left to right, the red edges are $h^{-1}e$, $ge$, and $e$.[]{data-label="fig:IV"}](IV "fig:") Now $\beta$ is a fundamental domain for $gh$, and we have a total of four junctures (three in $T_h / \langle h\rangle$ and one in $T_{gh}/\langle gh \rangle$). Thus we have $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ {\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 4n,$$ and a defect of at most $4$. *Case V: $h$ is elliptic, $g$ is hyperbolic, $T_g \cap T_h \not= \emptyset$.* This case is covered by Case IV, replacing $g$ and $h$ by $h^{-1}$ and $g^{-1}$ respectively. *Case VI: the remaining cases.* If $g$ and $h$ are hyperbolic and $T_g$ and $T_h$ intersect in one point, then the configuration closely resembles the first one discussed in Case I, except that the copies of $\rho$ have been shrunk to have length zero. That is, there are fundamental domains $\alpha$ and $\beta$ for $h$ and $g$ respectively, such that $\alpha \cdot \beta$ is a fundamental domain for $gh$. With four junctures, we obtain $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ {\leqslant}\ 4n,$$ for a defect of at most $4$. Lastly, if $g$ and $h$ have a common fixed point, then $${\left\lvert {f_{\gamma}((gh)^n) - f_{\gamma}(g^n) - f_{\gamma}(h^n)} \right\rvert} \ = \ 0$$ for all $n$. \[rem:rtree\] The functions $f_\gamma$ and $h_\gamma$ can be defined in the more general setting of a group acting on an ${{\mathbb R}}$–tree. The proof of Theorem \[defect\] goes through in this setting, with only superficial modifications (essentially, removing any mention of *edges*, and using small segments instead). Well-aligned elements {#well-aligned-elements .unnumbered} --------------------- We now consider elements $g \in G$ for which we can find a segment $\gamma \subset T$ such that $h_\gamma(g) = 1$. \[def:well-aligned\] Given a $G$–tree $T$, a hyperbolic element $g \in G$ is *well-aligned* if there does not exist an element $h\in G$ such that $ghgh^{-1}$ fixes an edge of $T_g$. This property is the $G$–tree analogue of the double coset condition from [@CF Theorem D]. \[th:well-aligned\] Suppose $G$ acts on a simplicial tree $T$. If $g\in G$ is well-aligned then $\operatorname{scl}(g) {\geqslant}1/12$. Let $\gamma = [x, gx] \subset T_g$ be a fundamental domain for $g$ where $x$ is *not* a vertex of $T$. We know that $c_{\gamma}(g^n) = n$ for all $n$. If $c_{\overline{\gamma}}(g^n)> 0$ for some $n$, then there is a copy of $\overline{\gamma}$ in $T_g$. That is, there is an element $h$ such that $h \gamma$ lies in $T_g$ with the opposite orientation. So $h(T_g) = T_{hgh^{-1}}$ has negative overlap with $T_g$ along a segment containing $h \gamma$. The element $ghgh^{-1}$ fixes one of the endpoints of $h\gamma$, since $g$ and $hgh^{-1}$ shift it in opposite directions inside $T_g \cap T_{hgh^{-1}}$. This endpoint is in the interior of an edge $e \subset T_g$, and so $ghgh^{-1}$ fixes $e$. Hence, if $g$ is well-aligned, we must have $h_{\gamma}(g) = 1$. By Theorem \[defect\], $h_{\gamma}$ is a homogeneous quasimorphism with defect at most $6$, and so Proposition \[bavard\] implies that $\operatorname{scl}(g) {\geqslant}1/12$. This bound is in fact *optimal*. Both in HNN extensions and in amalgamated free products, there are examples of elements $g$ with $\operatorname{scl}(g)=1/12$ that are well-aligned with respect to the action on the associated Bass–Serre tree, as we now explain. This answers Question 8.4 from [@CF]. \[th:optimal\] Let $g = tat^{-1}a \in BS(2,3)$ and let $T$ be the Bass–Serre tree associated to the splitting of $BS(2,3)$ as an HNN extension $\langle a \rangle*_{\langle ta^2t^{-1} = a^3\rangle}$. Then $g$ is well-aligned and $\operatorname{scl}(g) = 1/12$. In particular, the bound in Theorem \[th:well-aligned\] is optimal. Denote the vertex of $T$ stabilized by $\langle a \rangle$ by $v_0$ and let $v_1 = tv_0$. The vertices along the axis of $g$ are: $\{ g^nv_0, g^nv_1 \}_{n \in {{\mathbb Z}}}$. If $ghgh^{-1}$ fixes an edge $e \subset T_g$, then we also see that $hgh^{-1}g$ fixes $g^{-1}e \subset T_g$. Replacing $h$ by $hg^k$ for some $k$ (which does not affect $hgh^{-1}$), we can arrange that $h$ fixes a vertex of $T_g$. By further replacing $h$ by a conjugate $g^k h g^{-k}$, we can arrange that the vertex fixed by $h$ is either $v_0$ or $v_1$; the elements $ghgh^{-1}$ and $hgh^{-1}g$ still fix edges of $T_g$. First assume that $h$ fixes $v_0$, and so $h = a^r$ for some $r \in {{\mathbb Z}}$. In this case $$ghgh^{-1} = tat^{-1}a^{1 + r}tat^{-1}a^{1-r}.$$ If $ghgh^{-1}$ is elliptic (which it necessarily is if it fixes an edge), then this expression cannot be cyclically reduced (Remark \[cyc-red\]). Hence we find that $r \equiv \pm 1 \mod 3$. If $r \equiv 1 \mod 3$, then $hgh^{-1}g = a^5$; if $r \equiv -1 \mod 3$ then $ghgh^{-1} = a^5$. In either case, the element does not fix an edge in $T$, giving a contradiction. Similarly, if $h$ fixes $v_1$, then we have $h = ta^rt^{-1}$ for some $r \in {{\mathbb Z}}$, and so $$hgh^{-1}g = ta^{1+r}t^{-1}ata^{1-r}t^{-1}a.$$ Again, this expression cannot be cyclically reduced if $hgh^{-1}g$ is elliptic, and so $r \equiv 1 \mod 2$. Again, we find that $hgh^{-1}g = a^5$, giving a contradiction for the same reason as above. Therefore $g$ is well-aligned as claimed. Finally, $\operatorname{scl}(tat^{-1}a) = 1/12$ by Proposition \[prop:length2\]. The bound in Theorem \[th:well-aligned\] is still optimal if one restricts to amalgamated free products. In the free product ${{\mathbb Z}}/2{{\mathbb Z}}*{{\mathbb Z}}/3{{\mathbb Z}}\cong \operatorname{PSL}(2,{{\mathbb Z}})$, no nontrivial element fixes an edge of the associated Bass–Serre tree, so every hyperbolic element that is not conjugate to its inverse is well-aligned. The group $\operatorname{PSL}(2,{{\mathbb Z}})$ has a finite index free subgroup, and therefore stable commutator length can be computed in this group by using a relationship between stable commutator length in a group and a finite index subgroup from [@Calegari:scl] together with Calegari’s algorithm for computing stable commutator length in free groups [@Calegari:free]. This is described explicitly in [@Louwsma:thesis]. The element $\bigl(\begin{smallmatrix}1&0\\1&1\end{smallmatrix}\bigr)$ is an example of an element that has stable commutator length $1/12$. Acylindrical trees {#acylindrical-trees .unnumbered} ------------------ We conclude this section by adding a moderate restriction, *acylindricity*, to the tree action. We can then say something about hyperbolic elements that are not necessarily well-aligned. Acylindricity has been used previously in the context of counting quasimorphisms on Gromov-hyperbolic spaces, cf. [@CF]. For a tree, the definition is particularly simple to state. A group acts *$K$–acylindrically* on a tree if the stabilizer of any segment of length $K$ is trivial. \[cor:acylindrical\] Suppose $G$ acts $K$–acylindrically on a tree $T$ and let $N$ be the smallest integer greater than or equal to $\frac{K}{2} + 1$. 1. \[g1\] If $g \in G$ is hyperbolic then either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/12N$. 2. \[g2\] If $g \in G$ is hyperbolic and ${\left\lvert {g} \right\rvert} {\geqslant}K$ then either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/24$. In both cases, $\operatorname{scl}(g) = 0$ if and only if $g$ is conjugate to $g^{-1}$. First note that if ${\left\lvert {g} \right\rvert} = 1$ then a fundamental domain for $g$ maps to a single loop in the quotient graph of $T$, which implies that $g$ has infinite order in the abelianization of $G$, and $\operatorname{scl}(g) = \infty$. (In fact, the same conclusion holds whenever ${\left\lvert {g} \right\rvert}$ is odd.) Thus we may assume that ${\left\lvert {g} \right\rvert} {\geqslant}2$. Observe that if for some $h \in G$, we have ${\left\lvert {T_g \cap T_{hgh^{-1}}} \right\rvert} {\geqslant}K + {\left\lvert {g} \right\rvert}$ where $g$ and $hgh^{-1}$ shift in opposite directions, then $ghgh^{-1}$ fixes a segment of length $K$ and hence $g = hg^{-1}h^{-1}$. In particular, $\operatorname{scl}(g) = 0$. For note that ${\left\lvert {g^N} \right\rvert} = N {\left\lvert {g} \right\rvert} {\geqslant}\frac{K}{2}{\left\lvert {g} \right\rvert} + {\left\lvert {g} \right\rvert} {\geqslant}K + {\left\lvert {g} \right\rvert}$. Taking $\gamma$ to be a fundamental domain for $g^N$, if $h_{\gamma}(g^N) < 1$ then there is an $h$ as above and $g = hg^{-1}h^{-1}$, $\operatorname{scl}(g) = 0$. Otherwise, $h_{\gamma}(g^N)=1$ and $\operatorname{scl}(g) = \frac{1}{N}\operatorname{scl}(g^N) {\geqslant}1/12N$ by Theorem \[defect\] and Proposition \[bavard\]. For let $N = 2$ and apply the same reasoning: ${\left\lvert {g^2} \right\rvert} = 2{\left\lvert {g} \right\rvert} {\geqslant}K + {\left\lvert {g} \right\rvert}$, and either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) = \frac{1}{2} \operatorname{scl}(g^2) {\geqslant}1/24$. The gap theorem {#sec:gap} =============== In this section we consider $G$–trees of a particular form, for which we can improve upon the “well-aligned” condition in Theorem \[th:well-aligned\] without any trade-off in the lower bound of $1/12$. Let $G$ be an HNN extension $A \, \ast_C$, with stable letter $t$, such that the edge groups $C$ and $C^t$ are central in $A$. Let $T$ be the Bass–Serre tree associated to this HNN extension. Such a tree has special properties, given below in Lemma \[rotate\] and Proposition \[stabilizer\]. This class of HNN extensions obviously includes the Baumslag–Solitar groups. It is still true (cf. Remark \[t-exp\]) that an element cannot have finite stable commutator length if its $t$–exponent is non-zero. We use the following notation for a $G$–tree $T$: if $X\subset T$ is any subset, then $G_X$ denotes the *stabilizer* of $X$, which is the subgroup of $G$ consisting of those elements that fix $X$ pointwise. \[rotate\] Suppose $S \subset T$ is a subtree and $h\in G$ fixes a vertex $v\in S$. Then $G_{S} = G_{h S}$. Let $e$ be an edge in $S$ with endpoint $v$. Then $G_{S} \subset G_e$. Since $G_e$ is central in $G_v$, $h$ commutes with $G_{S}$, and therefore $G_{h S} = h G_{S} h^{-1} = G_{S}$. Consider the $t$–exponent homomorphism $\phi {\colon \thinspace}G \to {{\mathbb Z}}$ (sending $t$ to $1$ and $A$ to $0$). There is an action of ${{\mathbb Z}}$ on ${{\mathbb R}}$ by integer translations. Letting $G$ act on ${{\mathbb R}}$ via $\phi$, there is also a $G$–equivariant map $F {\colon \thinspace}T \to {{\mathbb R}}$. This map is just the natural map from $T$ to the universal cover of the quotient graph $T/G$. The action of an element $g$ on $T$ projects by $F$ to a translation by $\phi(g)$ on ${{\mathbb R}}$. We think of $F$ as a *height function* on $T$. Then, the elements of $t$–exponent zero act on $T$ by height-preserving automorphisms. \[stabilizer\] If $S \subset T$ is a subtree and $\sigma \subset S$ is a finite subtree such that $F(\sigma) = F(S)$ then $G_{\sigma} = G_{S}$. *Case I: $\sigma$ is a segment mapped by $F$ injectively to ${{\mathbb R}}$.* First suppose that $S$ is a finite subtree. Let $\{S_i\}$ be the subtrees obtained as the closures of the components of $S - \sigma$. Then $G_{S} = G_{\sigma} \cap \bigcap_i G_{S_i}$. Fixing $i$, we will prove by induction on the number of edges of $S_i$ that $G_{\sigma} \subset G_{S_i}$. It then follows that $G_{\sigma} = G_{S}$. The base case is that $S_i$ is a single edge $e$ with one vertex $v$ on $\sigma$. Since $F(S_i) \subset F(\sigma)$, there is an edge $e'$ on $\sigma$ with endpoint $v$ and an element $h \in G_v$ taking $e$ to $e'$. Indeed, edges incident to $v$ are in the same $G_{v}$–orbit if and only if they have the same image under $F$. By Lemma \[rotate\] we have that $G_{S_i} = G_{e'} \supset G_{\sigma}$. For larger $S_i$, let $e \in S_i$ be the edge with endpoint $v \in \sigma$. Again there is an edge $e'$ on $\sigma$ with endpoint $v$ and an element $h \in G_v$ taking $e$ to $e'$. Again, $G_{S_i} = G_{h S_i}$ by Lemma \[rotate\]. But now $h S_i = e' \cup S_i'$ where $S_i'$ has fewer edges than $S_i$. By induction, $G_{\sigma} \subset G_{S_i'}$. Since $G_{\sigma} \subset G_{e'}$, we now have $G_{\sigma} \subset (G_{e'} \cap G_{S_i'}) = G_{S_i}$. Now consider an arbitrary subtree $S$. We need to show that $G_{\sigma}$ fixes $S$ pointwise. But every point $x$ in $S$ is in a finite subtree $S'$ containing $\sigma$, and $G_{\sigma}$ fixes $S'$ pointwise; hence $G_{\sigma}$ fixes $x$. *Case II: $\sigma$ is an arbitrary finite subtree of $S$.* Fixing the image $F(\sigma)$, we proceed by induction on the number of edges of $\sigma$. The base case is when this number is smallest, namely the length of $F(\sigma)$. Then Case I applies. If there are more edges than this, there must be a vertex $v\in \sigma$ and a pair of edges $e_0, e_1\in \sigma$ incident to $v$, with $F(e_0) = F(e_1)$. Decompose $S$ into two subtrees $S = S_0 \cup S_1$ with $S_0 \cap S_1 = \{v\}$ and $e_0 \in S_0$, $e_1 \in S_1$. Let $\sigma_i = S_i \cap \sigma$. There is an element $h \in G_v$ such that $he_0 = e_1$. Let $\sigma' = h\sigma_0 \cup \sigma_1$ and $S' = hS_0 \cup S_1$. Note that $\sigma'$ has fewer edges than $\sigma$. Also, $F(\sigma) = F(\sigma')$ and $F(S) = F(S')$, and $G_{\sigma} = G_{\sigma_0} \cap G_{\sigma_1} = G_{h\sigma_0} \cap G_{\sigma_1} = G_{\sigma'}$ by Lemma \[rotate\]. Similarly, $G_{S} = G_{S_0} \cap G_{S_1} = G_{hS_0} \cap G_{S_1} = G_{S'}$. By the induction hypothesis, $G_{\sigma'} = G_{S'}$, and therefore $G_{\sigma} = G_S$. Now consider a hyperbolic element $g$ with $t$–exponent zero. The axis $T_g$ has the property that $F(T_g)$ is a finite interval. To see this, let $\gamma$ be a fundamental domain, and note that $T_g = \bigcup_n g^n \gamma$. The $t$–exponent condition implies that $F(g^n \gamma) = F(\gamma)$ for all $n$, and hence $F(T_g) = F(\gamma)$. We call a vertex $v$ on $T_g$ *extremal* if $F(v)$ is an endpoint of $F(T_g)$. A segment $\sigma \subset T_g$ is *stable* if $F(\sigma) = F(T_g)$ and $\sigma$ contains no extremal vertex in its interior (equivalently, no proper subsegment $\sigma'$ satisfies $F(\sigma') = F(T_g)$). See Figure \[fig:stable\]. Note that if $\sigma$ and $\tau$ are stable segments, then they do not overlap, unless they are equal. \[c\] at 75 19 \[b\] at 312 30 ![Stable segments along $T_g$, in green and red.[]{data-label="fig:stable"}](stable "fig:") The natural orientation of $T_g$ defines a linear ordering $<_g$ on the stable segments of $T_g$. The “larger” end is the attracting end of $T_g$; that is, $\sigma <_g g\sigma$ always holds. We say that $\sigma {\leqslant}_g \tau$ if $\sigma <_g \tau$ or $\sigma = \tau$. \[cleanremark\] If $\gamma$ is a fundamental domain for $g$ whose endpoints are extremal, then every stable segment either does not overlap with $\gamma$ or is contained in $\gamma$. Moreover, $\gamma$ contains a copy of every stable segment. (Being a fundamental domain, it overlaps with a copy of every non-trivial segment in $T_g$.) If $\gamma$ is a fundamental domain that starts with a stable segment then its endpoints are extremal, as the endpoints of a stable segment are extremal. Proposition \[stabilizer\] immediately implies: \[cor:stable\] If $\sigma \subset T_g$ is stable then $G_{\sigma} = G_{T_g}$. The main technical result of this section is: \[infiniteoverlap\] Let $G = A \, \ast_C$ with stable letter $t$, and $C, C^t$ central in $A$. Let $g\in G$ be a hyperbolic element with $t$–exponent zero. Then either: 1. there is a fundamental domain $\gamma$ for $g$ such that $h_{\gamma}(g) = 1$, or \[one\] 2. there is an element $h$ such that $h(T_g) = \overline{T}_g$. \[two\] Conclusion implies by Proposition \[bavard\] and Theorem \[defect\] that $\operatorname{scl}(g) {\geqslant}1/12$. Let $\alpha \subset T_g$ be a stable segment and let $\gamma$ be the fundamental domain for $g$ that starts with $\alpha$. Note that $\gamma$ has extremal endpoints. If $h_{\gamma}(g) < 1$ then there is an element $h$ such that $h\gamma$ lies in $T_g$ with the opposite orientation and overlaps with $\alpha$. Note that $h$ fixes a point in $\gamma$. In particular $h$ is elliptic, and hence acts as a height-preserving automorphism of $T$. Now $h\gamma$ is a fundamental domain for $g^{-1}$ with extremal endpoints, and so $h\gamma$ contains $\alpha$ (which is stable for $T_{g^{-1}}$ as well as for $T_g$). The segment $\beta = h^{-1}\alpha$ is a stable segment for $T_g$ contained in $\gamma$. Clearly $\alpha {\leqslant}_g \beta$, and as the endpoints of $\alpha$ have different heights, $\alpha \neq \beta$; therefore $\alpha <_g \beta$. Note that $h\alpha = \beta$, since $h$ acts as a reflection on the segment $\gamma \cap h\gamma$. Hence the element $h^2$ fixes the stable segment $\alpha$. Therefore, by Proposition \[stabilizer\], $h^2$ fixes $T_g \cup h(T_g)$. That is, $h$ acts as an involution on this entire subtree of $T$. If there is a stable segment $\rho \subset T_g \cap h(T_g)$ such that either $\rho <_g \alpha$ or $\beta <_g \rho$, then conclusion holds. Since $h$ acts as a reflection on the segment $T_g \cap h(T_g)$, if $\rho \subset T_g \cap h(T_g)$ and $\rho <_g \alpha$, then $h\rho \subset T_g \cap h(T_g)$ and $\beta <_g h\rho$. Thus we only need to verify the claim in the $\beta <_g \rho$ case. Let $\sigma$ be the $<_g$–smallest stable segment in $h\gamma$. Observe that $\sigma \subset T_g \cap h(T_g)$. The translate $g\sigma$ is the $<_g$–smallest stable segment in $gh\gamma$, which has a common endpoint with $\beta$. Hence $g\sigma {\leqslant}_g \tau$ for any $\tau$ satisfying $\beta <_g \tau$. In particular, $g\sigma {\leqslant}_g \rho$. Since $\beta,\rho \subset T_g \cap h(T_g)$ and $\beta <_g g\sigma {\leqslant}_g \rho$, it follows that $g\sigma \subset T_g \cap h(T_g)$. Note that $h(T_g) = T_{hgh^{-1}}$, and $hgh^{-1}$ takes $g\sigma$ to $\sigma$. Thus $ghgh^{-1}$ fixes $g\sigma$. Since this is a stable segment, $ghgh^{-1}$ must fix all of $T_g$ by Corollary \[cor:stable\]. This implies that $hgh^{-1}$ acts on $T_g$ as a translation, of the same amplitude but opposite direction as $g$. Hence $T_{hgh^{-1}} = \overline{T}_g$. Returning to the proof of Theorem \[infiniteoverlap\], assume that the Claim does not apply. Then $\alpha$ and $\beta$ are the $<_g$–smallest and $<_g$–largest stable segments in $T_g \cap h(T_g)$ respectively. It follows that $\beta$ is also the $<_g$–largest stable segment in $\gamma$; otherwise, if $\beta <_g \rho$ and $\rho \subset \gamma$, then $h\rho <_g \alpha$ and $h\rho \subset T_g \cap h(T_g)$, contradicting that $\alpha$ is smallest. Note that $h$ takes stable segments to stable segments, and does not take any stable segment to itself (since the endpoints have different heights). Hence the stable segments of $\gamma$ may be enumerated in order as $\alpha = \alpha_1, \ldots, \alpha_n, \beta_n, \ldots, \beta_1 = \beta$ where $h$ interchanges $\alpha_i$ and $\beta_i$. Now let $\gamma'$ be the fundamental domain for $g$ starting with $\beta_n$. Assuming that conclusion does not hold, we have $h_{\gamma'}(g) <1$, and so there is an elliptic element $k$ such that $k\gamma'$ lies in $T_g$ with the opposite orientation and contains $\beta_n$. The configuration of $T_g$, $k(T_g)$, $\beta_n$, and $k\beta_n$ is exactly analogous to that of $T_g$, $h(T_g)$, $\alpha$, and $\beta$. In particular, the Claim is applicable to this situation. If the Claim does not apply, then we conclude as above that $\beta_n$ and $k\beta_n$ are the $<_g$–smallest and $<_g$–largest stable segments in $T_g \cap k(T_g)$ and that $k\beta_n$ is the $<_g$–largest stable segment in $\gamma'$. The stable segments in $\gamma'$ are, in order: $\beta_n, \ldots, \beta_1, g\alpha_1, \ldots, g\alpha_n$, and the element $k$ interchanges $\beta_i$ and $g\alpha_i$. Thus $$kh\alpha = kh\alpha_1 = k\beta_1 = g\alpha_1 = g\alpha.$$ Since $kh$ and $g$ agree on the stable segment $\alpha$, they agree on all of $T_g \cup h(T_g) \cup k(T_g)$, by Proposition \[stabilizer\]. Similarly, $h$ and $k$ both act as involutions on $T_g \cup h(T_g) \cup k(T_g)$. Now $$h g^{-1} \beta = h (kh)^{-1} \beta = h h k \beta = k\beta = g\alpha,$$ which implies that $g\alpha \subset T_g \cap h(T_g)$. However, $\beta <_g g\alpha$ and $\beta$ is the $<_g$–largest stable segment in $T_g \cap h(T_g)$. This contradiction establishes the theorem. The next proposition concerns conclusion in Theorem \[infiniteoverlap\]. It is a variant of the observation that if an element is conjugate to its inverse, then it has $\operatorname{scl}$ zero. \[sclzero\] Suppose $G$ acts on a tree $T$ and $\operatorname{scl}$ vanishes on the elliptic elements of $G$. If $g$ is hyperbolic and there is an element $h$ such that $h(T_g) = \overline{T}_g$, then $\operatorname{scl}(g) = 0$. Since $h(T_g) = T_{hgh^{-1}}$, the element $ghgh^{-1}$ fixes $T_g$ pointwise. Similarly, $g^n h g^n h^{-1}$ fixes $T_g$ for every $n$. Thus there are elliptic elements $a_n$ such that $g^n hg^nh^{-1} = a_n$. This equation can be realized by a surface of genus zero and three boundary components, labeled by $g^n$, $g^n$, and $a_n^{-1}$ respectively. Lemma \[fill-with-zero\] now implies that $$\operatorname{scl}(g) \ {\leqslant}\ \frac{1}{4n} + \frac{\operatorname{scl}(a_n^{-1})}{2n}.$$ Hence $\operatorname{scl}(g) {\leqslant}1/4n$ for all $n > 0$. \[th:gap\] For every element $g \in BS(m, \ell)$, either $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/12$. Every elliptic element $g$ is conjugate to a power of $a$, and therefore $\operatorname{scl}(g) = 0$ for elliptic elements by Lemma \[scl-a\]. If $g$ is hyperbolic then Theorem \[infiniteoverlap\] and Proposition \[sclzero\] imply that $\operatorname{scl}(g) = 0$ or $\operatorname{scl}(g) {\geqslant}1/12$. If $m$ and $\ell$ are both odd, then conclusion of Theorem \[infiniteoverlap\] can never occur, since the element $h$ would fix a vertex of $T_g$ and exchange two adjacent edges, yielding an element of order two in ${{\mathbb Z}}/m{{\mathbb Z}}$ or ${{\mathbb Z}}/\ell{{\mathbb Z}}$. Therefore, $\operatorname{scl}(g) {\geqslant}1/12$ for *every* hyperbolic element $g$ in $BS(m, \ell)$. If either $m$ or $\ell$ is even, say $m = 2k$, then for $g = ta^k t^{-1} a^r ta^k t^{-1} a^s \in BS(m,\ell)$ where $r+s=\ell$ we have $\operatorname{scl}(g) = 0$. Indeed, taking $h = ta^{-k}t^{-1}$ one checks that $ghgh^{-1} = a^{4\ell} \in \langle a^\ell \rangle = G_{T_g}$. Thus $h(T_g) = \overline{T}_g$ and so by Proposition \[sclzero\] we have $\operatorname{scl}(g) = 0$.

--- abstract: | The binomial and Poisson distributions have interesting relationships with the beta and gamma distributions, respectively, which involve their cumulative distribution functions and the use of conjugate priors in Bayesian statistics. We briefly discuss these relationships and some properties resulting from them which play an important role in the construction of exact nested two-sided confidence intervals and the computation of two-tailed P-values. The purpose of this article is to show that such relationships also exist between the hypergeometric distribution and a special case of the Polya (or beta-binomial) distribution, and to derive some properties of the hypergeometric distribution resulting from these relationships. KEY WORDS: Beta, binomial, gamma, Poisson, and Polya (or beta-binomial) distributions; Conjugate prior distribution; Cumulative distribution function; Posterior distribution. author: - | Peter  H. Peskun, Department of Mathematics and Statistics\ York University, Toronto, Ontario M3J 1P3, Canada\ E-mail: peskun@pascal.math.yorku.ca title: | Some Relationships and Properties\ of the Hypergeometric Distribution --- **1. INTRODUCTION** The binomial and Poisson distributions have interesting relationships with the beta and gamma distributions, respectively, which involve their cumulative distribution functions and the use of conjugate priors in Bayesian statistics. We will briefly discuss these relationships and some properties resulting from them in Sections 2 and 3 for the binomial and Poisson distributions, respectively. The resulting properties play an important role in the construction of exact nested two-sided binomial and Poisson confidence intervals, and the computation of exact two-tailed binomial and Poisson P-values. The purpose of this article is to show that such relationships also exist between the hypergeometric distribution and a special case of the Polya (or beta-binomial) distribution, and to derive some properties of the hypergeometric distribution resulting from these relationships. We shall do this in Section 4. **2. RELATIONSHIPS AND PROPERTIES OF THE BINOMIAL DISTRIBUTION** Suppose that random variable $X$ has a binomial distribution with parameters $n$ and $p$, denoted by $X \sim \text{BIN}(n,p)$, where $n$ is a positive integer and $0 \leq p \leq 1$. Then, for a given $n$ and for $0 < p < 1$, the probability mass function (pmf) of $X$, denoted by $f_{X}(x \mid p)$, is $$\begin{aligned} f_{X}(x \mid p) = P(X=x \mid p) &= \binom{n}{x}p^{x}(1-p)^{n-x}, \quad x = 0,1, \ldots ,n, \\ &=0, \quad \text{otherwise,}\end{aligned}$$ and $f_{X}(0 \mid 0) = f_{X}(n \mid 1) = 1$. Suppose that random variable $Y$ has a beta distribution with parameters $\alpha > 0$ and $\beta > 0$, denoted by $Y \sim \text{BETA}(\alpha,\beta)$. Then the probability density function (pdf) of $Y$, denoted by $f_{Y}(y \mid \alpha,\beta)$, is $$\begin{aligned} f_{Y}(y \mid \alpha,\beta) &= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}y^{\alpha-1}(1-y)^{\beta-1}, \quad 0 \leq y \leq 1,\\ &= 0, \quad \text{otherwise,}\end{aligned}$$ where the gamma function $\Gamma(\kappa) = \int_{0}^{\infty} t^{\kappa-1}e^{-t} \, dt$ for all $\kappa > 0$. Successive integration by parts leads to a relationship between the cumulative distribution functions (cdf’s) of the binomial and beta distributions. If $X \sim \text{BIN}(n,p)$ and $Y \sim \text{BETA}(i+1,n-i)$ for integer $i$, $0 \leq i \leq n-1$, then $$\label{E:1} \sum_{x=0}^{i}\binom{n}{x}p^{x}(1-p)^{n-x} = 1-\frac{n!}{i!(n-i-1)!}\int_{0}^{p}t^{i}(1-t)^{n-i-1} \, dt.$$ That is, $F_{X}(i \mid p) = P(X \leq i \mid p) = 1-P(Y \leq p \mid i+1,n-i) = 1-F_{Y}(p \mid i+1,n-i)$. For fixed integer $i$, $0 \leq i \leq n-1$, it follows from equation (\[E:1\]) that the function $P(X \leq i \mid p)$ is continuous and decreasing in $p$; for fixed integer $j$, $1 \leq j \leq n$, $P(X \geq j \mid p) = 1-P(X \leq j-1 \mid p)$ is continuous and increasing in $p$; and for fixed integers $i$ and $j$, $1 \leq i \leq j \leq n-1$, $P( i \leq X \leq j \mid p)$ is continuous, and increasing for $0 \leq p < p_{n}(i,j)$ and decreasing for $p_{n}(i,j) \leq p \leq 1$ with maximum at $p = p_{n}(i,j) = \{1+[(n-i) \cdots (n-j)/j \cdots i]^{1/(j-i+1)}\}^{-1}$. Also, $p_{n}(0,j) = 0$ for $0 \leq j \leq n-1$ and $p_{n}(i,n) = 1$ for $1 \leq i \leq n$. Suppose that the binomial parameter $p$ is unknown and we wish to estimate it. In Bayesian statistics, information obtained from the data x, a realization of $X \sim \text{BIN}(n,p)$, is combined with prior information about $p$ that is specified in a “prior distribution” with pdf $g(p)$ and summarized in a “posterior distribution” with pdf $h(p \mid x)$ which is derived from the joint distribution $f_{X}(x \mid p)g(p)$, and according to Bayes formula is $$\label{E:2} h(p \mid x) = \frac{f_{X}(x \mid p)g(p)}{\int_{0}^{1}f_{X}(x \mid p)g(p) \, dp}.$$ Because $h(p \mid x)$ is generally not available in closed form, the favoured types of priors until the introduction of Markov chain Monte Carlo methods have been those allowing explicit computations, namely “conjugate priors.” These are prior distributions for which the corresponding posterior distributions are themselves members of the original prior family, the Bayesian updating being accomplished through updating of parameters. For a realization $x$ of $X \sim \text{BIN}(n,p)$, a family of conjugate priors is the family of beta distributions $\text{BETA}(\alpha,\beta)$ where we note from equation (\[E:2\]) that for $x = 0,1, \ldots ,n$, $$\begin{aligned} h(p \mid x) &= \frac{\binom{n}{x}p^{x}(1-p)^{n-x}\frac{\Gamma(\alpha+\beta)} {\Gamma(\alpha)\Gamma(\beta)}p^{\alpha-1}(1-p)^{\beta-1}} {\int_{0}^{1}\binom{n}{x}p^{x}(1-p)^{n-x}\frac{\Gamma(\alpha+\beta)} {\Gamma(\alpha)\Gamma(\beta)}p^{\alpha-1}(1-p)^{\beta-1} \, dp} \\ &= \frac{\Gamma(\alpha+\beta+n)}{\Gamma(\alpha+x)\Gamma(\beta+n-x)} p^{\alpha+x-1}(1-p)^{\beta+n-x-1}, \quad 0 \leq p \leq 1, \\ &= 0, \quad \text{otherwise.}\end{aligned}$$ That is, the posterior distribution is also beta with updated parameters $\alpha+x$ and $\beta+n-x$. **3. RELATIONSHIPS AND PROPERTIES OF THE POISSON DISTRIBUTION** Suppose that random variable $X$ has a Poisson distribution with parameter $\lambda \geq 0$, denoted by $X \sim \text{POI}(\lambda)$. Then, for $\lambda > 0$, the pmf of $X$, denoted by $f_{X}(x \mid \lambda)$, is $$\begin{aligned} f_{X}(x \mid \lambda) = P(X = x \mid \lambda) &= \frac{e^{-\lambda}\lambda^{x}}{x!}, \quad x = 0,1,2, \ldots , \\ &= 0, \quad \text{otherwise,}\end{aligned}$$ and $f_{X}(0 \mid 0) = 1$. Suppose random variable $Y$ has a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$, denoted by $Y \sim \text{GAM}(\alpha,\beta)$. Then the pdf of $Y$, denoted by $f_{Y}(y \mid \alpha,\beta)$, is $$\begin{aligned} f_{Y}(y \mid \alpha,\beta) &= \frac{1}{\beta^{\alpha}\Gamma(\alpha)} y^{\alpha-1}e^{-y/\beta}, \quad y > 0, \\ &= 0, \quad \text{otherwise.}\end{aligned}$$ Successive integration by parts leads to a relationship between the cdf’s of the Poisson and gamma distributions. If $X \sim \text{POI}(\lambda)$ and $Y \sim \text{GAM}(i+1,2)$ for nonnegative integer $i$, then $$\label{E:3} \sum_{x=0}^{i}\frac{e^{-\lambda}\lambda^{x}}{x!} = 1 - \frac{1}{2^{i+1}i!}\int_{0}^{2\lambda}t^{i}e^{-t/2} \, dt.$$ That is, $F_{X}(i \mid \lambda) = P(X \leq i \mid \lambda) = 1 - P(Y \leq 2\lambda \mid i+1,2) = 1 - F_{Y}(2\lambda \mid i+1,2)$. For fixed nonnegative integer $i$, it follows from equation (\[E:3\]) that the function $P(X \leq i \mid \lambda)$ is continuous and decreasing in $\lambda$; for positive integer $j$, $P(X \geq j \mid \lambda) = 1 - P(X \leq j-1 \mid \lambda)$ is continuous and increasing in $\lambda$; and for $1 \leq i \leq j$, $P(i \leq X \leq j \mid \lambda)$ is continuous, and increasing for $0 \leq \lambda < \lambda(i,j)$ and decreasing for $\lambda \geq \lambda(i,j)$ with maximum at $\lambda = \lambda(i,j) = (i \cdots j)^{1/(j-i+1)}$. Also, $\lambda(0,j) = 0$ for $j \geq 0$. Suppose that the Poisson parameter $\lambda$ is unknown and we wish to estimate it using Bayesian methods. For a realization $x$ of $X \sim \text{POI}(\lambda)$, a family of conjugate priors is the family of gamma distributions $\text{GAM}(\alpha, \beta)$ where for $x = 0,1,2, \cdots $, the pdf $h(\lambda \mid x)$ of the posterior distribution is given by $$\begin{aligned} h(\lambda \mid x) &= \frac{\frac{e^{-\lambda}\lambda^{x}}{x!}\frac{1}{\beta^{\alpha}\Gamma(\alpha)} \lambda^{\alpha-1}e^{-\lambda/\beta}}{\int_{0}^{\infty}\frac{e^{-\lambda}\lambda^{x}} {x!}\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\lambda^{\alpha-1}e^{-\lambda/\beta} \, d\lambda} \\ &= \frac{1}{[\beta/(1+\beta)]^{\alpha+x}\Gamma(\alpha+x)} \lambda^{\alpha+x-1}e^{-\lambda/[\beta/(1+\beta)]}, \quad \lambda > 0, \\ &= 0, \quad \text{otherwise.}\end{aligned}$$ That is, the posterior distribution is also gamma with updated parameters $\alpha + x$ and $\beta/(1 + \beta)$. **4. RELATIONSHIPS AND PROPERTIES OF THE HYPERGEOMETRIC DISTRIBUTION** Suppose that integer-valued random variable $X$ has a hypergeometric distribution with parameters $n$, $M$, and $N$, denoted by $X \sim \text{HYP}(n,M,N)$, where $n$, $M$, and $N$ are integers with $1 \leq n \leq N$ and $0 \leq M \leq N$. Then, for given $n$ and $N$, and for $0 < M < N$, the pmf of $X$, denoted by $f_{X}(x \mid M)$, is $$\begin{aligned} \label{E:4} f_{X}(x \mid M) = P(X = x \mid M) &= \frac{\binom{M}{x}\binom{N-M}{n-x}} {\binom{N}{n}}, \quad \text{max}(0,n-N+M) \leq x \leq \text{min}(n,M), \notag \\ &= 0, \quad \text{otherwise,}\end{aligned}$$ and $f_{X}(0 \mid 0) = f_{X}(n \mid N) = 1$. Suppose that random variable $Y$ has a specially defined discrete distribution with parameters $a$, $b$, and $c$, denoted by $Y \sim \text{ABC}(a,b,c)$, where $a$, $b$, and $c$ are nonnegative integers. Then, for $c > 0$, the pmf of $Y$, denoted by $f_{Y}(y \mid a,b,c)$, is $$\begin{aligned} f_{Y}(y \mid a,b,c) = P(Y = y \mid a,b,c) &= \frac{\binom{a+y}{a}\binom{b+c-y}{b}} {\binom{a+b+c+1}{a+b+1}}, \quad y = 0,1, \ldots ,c, \\ &= 0, \quad \text{otherwise,}\end{aligned}$$ and $f_{Y}(0 \mid a,b,0) = 1$. We note that formula (12.16) of Feller (1968, p.65) can be used to prove that $$\sum_{y=0}^{c}\binom{a+y}{a}\binom{b+c-y}{b} = \binom{a+b+c+1}{a+b+1}.$$ We also note that the ABC distribution is just a special case of the Polya (or beta-binomial) distribution (Dyer and Pierce, 1993, p.2130). From equation (\[E:4\]), it easily follows that $P(X \leq n \mid M) = 1$ for $0 \leq M \leq N$. For $0 \leq i < n \leq N$ and $0 \leq M \leq N$, we have from equation (\[E:4\]) that $$\begin{aligned} \binom{N}{n}P(X \leq i \mid M) &= \sum_{x=0}^{i}\binom{M}{x}\binom{N-M}{n-x} \notag \\ &= \sum_{x=0}^{i}\binom{M}{x} \left[ \binom{N-M-1}{n-x-1} + \binom{N-M-1}{n-x} \right] \notag \\ &= \sum_{x=0}^{i}\binom{M}{x}\binom{N-M-1}{n-x-1} + \sum_{x=0}^{i}\binom{M}{x}\binom{N-M-1}{n-x} \notag \\ &= \sum_{x=1}^{i+1}\binom{M}{x-1}\binom{N-M-1}{n-x} + \sum_{x=0}^{i}\binom{M}{x}\binom{N-M-1}{n-x} \notag \\\end{aligned}$$ $$\begin{aligned} \label{E:5} \phantom{\binom{N}{n}P(X \leq i \mid M)} &= \binom{M}{i}\binom{N-M-1}{n-i-1} - \binom{M}{-1}\binom{N-M-1}{n} \notag \\ & \qquad + \sum_{x=0}^{i}\left[ \binom{M}{x-1} + \binom{M}{x} \right]\binom{N-M-1}{n-x} \notag \\ &= \binom{M}{i}\binom{N-M-1}{n-i-1} + \sum_{x=0}^{i}\binom{M+1}{x}\binom{N-M-1}{n-x} \notag \\ &= \binom{M}{i}\binom{N-M-1}{n-i-1} + \binom{N}{n}P(X \leq i \mid M+1), \end{aligned}$$ where by definition $\binom{M}{-1} = 0$, $\binom{M}{i} = 0$ if $M < i$, and $\binom{N-M-1}{n-i-1} = 0$ if $M > N-n+i$. Furthermore, from the recursion relationship in equation (\[E:5\]), it follows that $$\begin{aligned} \label{E:6} P(X \leq i \mid M) &= \sum_{k=M}^{N-n+i}\binom{k}{i}\binom{N-k-1}{n-i-1}\biggr/ \binom{N}{n} \notag \\ &= \sum_{k=M-i}^{N-n}\binom{i+k}{i}\binom{n-i-1+N-n-k}{n-i-1} \biggr/\binom{N}{n} \notag \\ &= 1 - \sum_{k=0}^{M-i-1}\binom{i+k}{i}\binom{n-i-1+N-n-k}{n-i-1} \biggr/\binom{N}{n}.\end{aligned}$$ That is, if $X \sim \text{HYP}(n,M,N)$ and $Y \sim \text{ABC}(i,n-i-1,N-n)$ for integer $i$, $0 \leq i < n \leq N$, then $F_{X}(i \mid M) = P(X \leq i \mid M) = 1 - P(Y \leq M-i-1 \mid i,n-i-1,N-n) = 1 - F_{Y}(M-i-1 \mid i,n-i-1,N-n)$ where, in particular, $$\begin{aligned} \label{E:7} P(X \leq i \mid M) &= 1, \quad \text{if} \quad 0 \leq M \leq i, \notag \\ &= 0, \quad \text{if} \quad N-n+i < M \leq N.\end{aligned}$$ For $0 < i \leq j < n \leq N$ and $0 \leq M \leq N$, we have from equation (\[E:5\]) that $$\begin{aligned} \label{E:8} \binom{N}{n}P(i \leq X \leq j \mid M) &= \binom{N}{n}P(X \leq j \mid M) - \binom{N}{n}P(X \leq i-1 \mid M) \notag \\ &= \binom{M}{j}\binom{N-M-1}{n-j-1} + \binom{N}{n}P(X \leq j \mid M+1) \notag \\ & \qquad - \binom{M}{i-1}\binom{N-M-1}{n-i} - \binom{N}{n}P(X \leq i-1 \mid M+1) \notag \\ &= \binom{M}{j}\binom{N-M-1}{n-j-1} - \binom{M}{i-1}\binom{N-M-1}{n-i} \notag \\ & \qquad + \binom{N}{n}P(i \leq X \leq j \mid M+1).\end{aligned}$$ Similar to the determination of equation (\[E:6\]), it follows from the recursion relationship in equation (\[E:8\]) that $$\begin{aligned} \label{E:9} P(i \leq X \leq j \mid M) &= \sum_{k=M}^{N-n+j}\binom{k}{j}\binom{N-k-1}{n-j-1} \biggr/\binom{N}{n} - \sum_{l=M}^{N-n+i-1}\binom{l}{i-1}\binom{N-l-1}{n-i} \biggr/\binom{N}{n} \notag \\ &= \sum_{k=M-j}^{N-n}\binom{j+k}{j}\binom{n-j-1+N-n-k} {n-j-1}\biggr/\binom{N}{n} \notag \\ & \qquad - \sum_{l=M-i+1}^{N-n}\binom{i-1+l}{i-1} \binom{n-i+N-n-l}{n-i}\biggr/\binom{N}{n} \notag \\ &= \sum_{l=0}^{M-i}\binom{i-1+l}{i-1}\binom{n-i+N-n-l} {n-i}\biggr/\binom{N}{n} \notag \\ & \qquad - \sum_{k=0}^{M-j-1}\binom{j+k}{j} \binom{n-j-1+N-n-k}{n-j-1}\biggr/\binom{N}{n}\end{aligned}$$ where, in particular, $$\label{E:10} P(i \leq X \leq j \mid M) = 0, \quad \text{if either} \quad 0 \leq M < i \quad \text{or} \quad N-n+j < M \leq N.$$ We note in equation (\[E:8\]) that the difference $$\begin{aligned} \label{E:11} \binom{M}{j}\binom{N-M-1}{n-j-1} - \binom{M}{i-1}\binom{N-M-1}{n-i} &= -\binom{N-i}{n-i} < 0, \quad \text{if} \quad M=i-1, \notag \\ &= \binom{N-n+j}{j} > 0, \quad \text{if} \quad M=N-n+j,\end{aligned}$$ and for $i \leq M < N-n+j$, the same difference $$\begin{aligned} \label{E:12} & \binom{M}{j}\binom{N-M-1}{n-j-1} - \binom{M}{i-1}\binom{N-M-1}{n-i} \notag \\ &= \frac{M!}{j!(M-j)!}\frac{(N-M-1)!}{(n-j-1)!(N-M-n+j)!} - \frac{M!}{(i-1)!(M-i+1)!}\frac{(N-M-1)!}{(n-i)!(N-M-n+i-1)!} \notag \\ &= \frac{M!(N-M-1)!}{(i-1)!(M-j)!(n-j-1)!(N-M-n+i-1)!} \notag \\ & \qquad \times \left[ \frac{1}{(j \cdots i)}\frac{1}{(N-M-n+j) \cdots (N-M-n+i)} \right. \notag \\ & \qquad \qquad - \left. \frac{1}{(M-i+1) \cdots (M-j+1)}\frac{1}{(n-i) \cdots (n-j)} \right]\end{aligned}$$ where as $M$ increases, the term $1/(N-M-n+j) \cdots (N-M-n+i)$ increases and the term $1/(M-i+1) \cdots (M-j+1)$ decreases so that as $M$ increases between $i-1$ and $N-n+j$, the difference $\binom{M}{j}\binom{N-M-1}{n-j-1} - \binom{M}{i-1} \binom{N-M-1}{n-i}$ goes from being negative to being positive and staying positive. In summary, $P(X \leq n \mid M)$ equals 1 for $0 \leq M \leq N$, and for fixed integer $i$, $0 \leq i < n \leq N$, we see from equations (\[E:6\]) and (\[E:7\]) that $P(X \leq i \mid M )$ equals 1 for $0 \leq M \leq i$, is decreasing for $i < M \leq N-n+i$, and equals 0 for $N-n+i < M \leq N$; $P(X \geq n+1 \mid M)$ equals 0 for $0 \leq M \leq N$, and for fixed integer $j$, $1 \leq j \leq n \leq N$, $P(X \geq j \mid M) = 1 - P(X \leq j-1 \mid M)$ equals 0 for $0 \leq M \leq j-1$, is increasing for $j-1 < M \leq N-n+j-1$, and equals 1 for $N-n+j-1 < M \leq N$; and we see from equations (\[E:8\]) to (\[E:12\]) that for fixed integers $i$ and $j$, $0 < i \leq j < n \leq N$ where we define $$M_{n,N}(i,j) = \text{min}\{ M \mid i \leq M \leq N-n+j \quad \text{and} \quad \textstyle{\binom{M}{j}\binom{N-M-1}{n-j-1} \geq \binom{M}{i-1}\binom{N-M-1}{n-i}} \},$$ $P(i \leq X \leq j \mid M)$ equals 0 for $0 \leq M < i$, is increasing for $i \leq M < M_{n,N}(i,j)$, is decreasing for $M_{n,N}(i,j) + 1 < M \leq N-n+j$, and equals 0 for $N-n+j < M \leq N$ with maximum at either $M_{n,N}(i,j)$ if $\binom{M}{j}\binom{N-M-1}{n-j-1} > \binom{M}{i-1}\binom{N-M-1}{n-i}$ for $M = M_{n,N}(i,j)$ so that $P(i \leq X \leq j \mid M_{n,N}(i,j)) > P(i \leq X \leq j \mid M_{n,N}(i,j) + 1)$ or maximum at both $M_{n,N}(i,j)$ and $M_{n,N}(i,j) + 1$ if $\binom{M}{j}\binom{N-M-1}{n-j-1} = \binom{M}{i-1}\binom{N-M-1}{n-i}$ for $M = M_{n,N}(i,j)$ so that $P(i \leq X \leq j \mid M_{n,N}(i,j)) = P(i \leq X \leq j \mid M_{n,N}(i,j) + 1)$. Suppose that the hypergeometric parameters $n$ and $N$ are known but $M$ is not and we wish to estimate it using Bayesian methods. For a realization $x$ of $X \sim \text{HYP}(n,M,N)$, a family of conjugate priors for $M - x$ is the family of discrete distributions $\text{ABC}(a,b,N)$ where for $x = 0,1, \ldots ,n$, the pmf $h(M \mid x)$ of the posterior distribution for $M$ is given by $$\begin{aligned} \label{E:13} h(M \mid x) &= \frac{\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}} \frac{\binom{a+M}{a}\binom{b+N-M}{b}}{\binom{a+b+N+1}{a+b+1}}} {\sum_{M=x}^{N-n+x}\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}} \frac{\binom{a+M}{a}\binom{b+N-M}{b}}{\binom{a+b+N+1}{a+b+1}}} \notag \\ &= \frac{\binom{a+M}{a+x}\binom{b+N-M}{b+n-x}} {\binom{a+b+N+1}{a+b+n+1}}, \quad x \leq M \leq N-n+x, \notag \\ &= 0, \quad \text{otherwise,}\end{aligned}$$ from which it easily follows that the pmf $h(M-x \mid x)$ of the posterior distribution for $M-x$ is given by $$\begin{aligned} \label{E:14} h(M-x \mid x) &= \frac{\binom{a+x+M-x}{a+x}\binom{b+n-x+N-n-M+x}{b+n-x}} {\binom{a+x+b+n-x+N-n+1}{a+x+b+n-x+1}}, \quad 0 \leq M-x \leq N-n, \notag \\ &= 0, \quad \text{otherwise.}\end{aligned}$$ That is, the posterior distribution for $M-x$ is also ABC with updated parameters $a+x$, $b+n-x$, and $N-n$. Finally, we note that as a family of conjugate priors for the hypergeometric distribution $\text{HYP}(n,M,N)$, the family of discrete distributions $\text{ABC}(a,b,N)$ has, in addition to unimodal members, strictly increasing members $\text{ABC}(a,0,N)$, strictly decreasing members $\text{ABC}(0,b,N)$, and the discrete uniform distribution $\text{ABC}(0,0,N)$. **REFERENCES** Dyer, D. and Pierce, R. L. (1993), “On the choice of the prior distribution in hypergeometric\ sampling,” *Communications in Statistics - Theory and Methods*, 22(8), 2125-2146.\ Feller, W. (1968), *An Introduction to Probability Theory and Its Applications*, Vol.1, (3rd ed.),\ John Wiley & Sons, Inc.

--- author: - | Jacek Syska[^1]\ \ Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland\ \ \ \ \ \ \ \ \ \ Skrypt dla studentów ekonofizyki[^2]\ \ \ \ title: | Metoda najwiêkszej wiarygodnoœci i informacja Fisher’a w fizyce\ i ekonofizyce\ \ --- Abstract {#abstract .unnumbered} ======== Three steps in the development of the maximum likelihood (ML) method are presented. At first, the application of the ML method and Fisher information notion in the model selection analysis is described (Chapter 1). The fundamentals of differential geometry in the construction of the statistical space are introduced, illustrated also by examples of the estimation of the exponential models.\ At second, the notions of the relative entropy and the information channel capacity are introduced (Chapter 2). The observed and expected structural information principle (IP) and the variational IP of the modified extremal physical information (EPI) method of Frieden and Soffer are presented and discussed (Chapter 3). The derivation of the structural IP based on the analyticity of the logarithm of the likelihood function and on the metricity of the statistical space of the system is given.\ At third, the use of the EPI method is developed (Chapters 4-5). The information channel capacity is used for the field theory models classification. Next, the modified Frieden and Soffer EPI method, which is a nonparametric estimation that enables the statistical selection of the equation of motions of various field theory models (Chapter 4) or the distribution generating equations of statistical physics models (Chapter 5) is discussed. The connection between entanglement of the momentum degrees of freedom and the mass of a particle is analyzed. The connection between the Rao-Cram[é]{}r inequality, the causality property of the processes in the Minkowski space-time and the nonexistence of tachions is shown. The generalization of the Aoki-Yoshikawa sectoral productivity econophysical model is also presented (Chapter 5). Finally, the Frieden EPI method of the analysis of the EPR-Bhom experiment is presented. It differs from the Frieden approach by the use of the information geometry methods. This work is destined mainly for students in physics and econophysics. (At present only Polish version is available). Wstêp {#wstêp .unnumbered} ===== Tematem skryptu jest metoda najwiêkszej wiarygodnoœci (MNW) oraz informacja Fishera (IF) w fizyce i statystyce. Problem dotyczy bardzo aktualnego sposobu konstrukcji modeli fizycznych, który wywodzi siê ze statystycznego opisu zjawisk, którego formalizm pozwala na opis ca³ego spektrum ró¿nych teorii pola, klasycznych i kwantowych. Kluczowe w tym podejœciu pojêcie (oczekiwanej) IF wprowadzi³ Fisher na gruncie w³asnych rozwa¿añ zwi¹zanych z oszacowywaniem parametrów modeli, podlegaj¹cych badaniu statystycznemu w ramach ogólnej metody ekstremalnej wartoœci funkcji wiarygodnoœci $L$. IF opisuje lokalne w³asnoœci funkcji wiarygodnoœci $L \equiv P(y|\Theta)$, która formalnie jest ³¹cznym prawdopodobieñstwem (lub ³¹czn¹ gêstoœci¹ prawdopodobieñstwa) danych $y$, lecz jest rozumiana jako funkcja zbioru parametrów $\Theta$, który z kolei tworzy wspó³rzêdne w przestrzeni statystycznej. Analiza statystyczna modeli fizycznych idzie jak dot¹d dwoma nurtami. Pierwszy z nich, geometryczny, próbuje opisaæ metodê statystyczn¹ wprowadzon¹ w latach 20 poprzedniego wieku przez Fishera [@Fisher], twórcê podstaw techniki statystycznej otrzymywania dobrych estymatorów MNW, w jak siê okaza³o naturalnym dla niej œrodowisku geometrii ró¿niczkowej. Rozwijaj¹c MNW, ju¿ w 1945 roku C.R. Rao [@Rao] zauwa¿y³, ¿e macierz informacji Fishera okreœla metrykê Riemanna i bada³ strukturê modeli statystycznych z punktu wiedzenia geometrii Riemanowskiej. Z kolei B. Efron [@Efron] badaj¹c jednoparametrowe modele i analizuj¹c ich asymptotyczne w³asnoœci dla procedur estymacyjnych, wprowadzi³ i odkry³ u¿ytecznoœæ pojêcia statystycznej krzywizny. A.P. Dawid [@Dawid] wprowadzi³ pojêcie koneksji na przestrzeni wszystkich dodatnio okreœlonych rozk³adów prawdopodobieñstwa, pokazuj¹c, ¿e ze wzglêdu na t¹ koneksjê statystyczna krzywizna jest krzywizn¹ zewnêtrzn¹. Jednak problemem Dawida by³ nieskoñczony wymiar przestrzeni rozk³adów. W roku 1980 S. Amari [@Amari] opublikowa³ systematyczne ujêcie teorii Dawida dla modeli skoñczenie wymiarowych i poda³ spójne okreœlenie $\alpha$-koneksji (wprowadzonej wczeœniej poza kontekstem statystycznej estymacji przez N.N. Chentsova). W 1982 S. Amari wraz z H. Nagaoka [@Amari; @Nagaoka; @book] wykazali dualnoœæ p³askich przestrzeni modeli eksponencjalnych z e-koneksj¹ i modeli mieszanych z m-koneksj¹.\ Procedury estymacyjne statystycznego opisu mechaniki kwantowej (falowej) posz³y dwoma drogami. Pierwsza z nich zwi¹zana jest z naturalnym dla mechaniki kwantowej formalizmem macierzy gêstoœci, druga z konstrukcj¹ zasad informacyjnych (entropijnych). W przypadku formalizmu macierzy gêstoœci, ich zbiór $S = \bigcup_{r=1}^{k} S_{r}$ ($S_{r} \cup S_{i} =\emptyset $, $ i \neq r $) dla przypadku skoñczenie wymiarowych przestrzeni Hilberta ${\cal H}$, tworzy zbiór wypuk³y. Dla stanów czystych, podzbiór $S_{1}$ tego zbioru tworz¹ punkty ekstremalne, a przestrzeñ stanów czystych zwi¹zana z nim mo¿e byæ uto¿samiona z zespolon¹ przestrzeni¹ rzutow¹ $CP^{k-1}$, ($k=dim$ ${\cal H}$). Na przestrzeni tej mo¿na wprowadziæ (z dok³adnoœci¹ do sta³ej) metrykê Riemannowsk¹ nazywan¹ metryk¹ Fubiniego-Study, która jest kwantow¹ wersj¹ metryki Rao-Fishera. Statystyczn¹ estymacj¹ w modelach dla stanów czystych zajmowali siê miêdzy innymi Fujiwara, Nagaoka i Matsumoto [@Fujiwara; @Nagaoka; @Matsumoto]. Natomiast w przypadku podzbioru $S_{k}$ zbioru $S$ dualna struktura z metryk¹ mo¿ne byæ traktowana jako kwantowy analog metryki Rao-Fishera z $\pm \alpha$-koneksj¹. Drugim nurtem, który wy³oni³ siê w ostatnich kilkunastu latach i którym szed³ rozwój zastosowañ MNW oraz pojêcia obserwowanej i oczekiwanej IF w fizyce jest formalizm ekstremalnej fizycznej informacji (EFI) opracowany przez Friedena i jego wspó³pracowników, w szczególnoœci Soffera [@Frieden]. Konstrukcjê modeli fizycznych z wykorzystaniem informacji Fishera zapocz¹tkowa³ Frieden, podaj¹c metodê wyprowadzenia z informacji Fishera cz³onu kinetycznego modeli fizycznych. Nastêpnie zapostulowa³ wprowadzenie dwóch zasad informacyjnych s³u¿¹cych do ustalenia zwi¹zku pomiêdzy pojemnoœci¹ kana³u informacyjnego $I$ oraz informacj¹ strukturaln¹ $Q$, tzn. poprzez zapostulowan¹ now¹ strukturaln¹ zasadê informacyjn¹ skonstruowa³ on cz³ony strukturalne rozwa¿anych przez siebie modeli. W odró¿nieniu od Friedena stosujemy jednak inne [@Dziekuje; @informacja_1], bardziej fizyczne a mniej informacyjne, podejœcie do konstrukcji podstawowych zasad informacyjnych, pos³uguj¹c siê pojêciem ca³kowitej fizycznej informacji $K = I + Q$, a nie wprowadzonym przez Friedena pojêciem zmiany fizycznej informacji. Ró¿nica ta, chocia¿ nie powoduje zasadniczo rachunkowych zmian w sposobie wyprowadzenia równañ ruchu b¹dŸ równañ generuj¹cych rozk³ad dla rozwa¿anych do tej pory problemów, jednak zmieniaj¹c pojêcie informacji fizycznej oraz jej rozk³adu na kinetyczne i strukturalne stopnie swobody, idzie w linii prowadzonych ostatnio badañ nad konstrukcj¹ zasady ekwipartycji dla entropii. To inne ni¿ Friedenowskie podejœcie do pojêcia fizycznej informacji powoduje równie¿ zmiany w pojmowaniu istoty przekazu informacji w procesie pomiaru przy jej przekazie od strukturalnych do kinetycznych stopni swobody. Pomimo ró¿nic sam¹ metodê bêdziemy dalej nazywaæ podejœciem Friedenowskim. Gdyby pomin¹æ chwilowo proces pomiaru, w metodzie Friedena próbkowanie przestrzeni jest wykonywane przez uk³ad, który poprzez w³aœciwe dla niego pole (i zwi¹zane z nim amplitudy) o randze $N$ bêd¹cej wielkoœci¹ próby, próbkuje jego kinetycznymi (Fisherowskimi) stopniami swobody dostêpn¹ mu przestrzeñ konfiguracyjn¹. Nastêpnie, poniewa¿ IF jest infinitezymalnym typem entropii Kulbacka-Leiblera, to zauwa¿aj¹c, ¿e entropia Kulbacka-Leiblera jest wykorzystywana w statystyce do przeprowadzania testów wyboru modeli, pojawia siê przypuszczenie, ¿e IF mo¿e poprzez narzucenie na ni¹ odpowiednich dodatkowych ograniczeñ, zapostulowanych w postaci wspomnianych dwóch zasad informacyjnych, wariacyjnej (skalarnej) oraz strukturalnej (wewnêtrznej), doprowadziæ do wyprowadzenia równañ ruchu b¹dŸ równañ stanu uk³adów fizycznych, najlepszych z punktu widzenia owych informacyjnych zasad. Na tym zasadza siê Friedenowska idea estymacji fizycznych modeli. Zasady informacyjne maj¹ uniwersaln¹ postaæ, jednak ich konkretne realizacje zale¿¹ od fizyki rozwa¿anego zagadnienia. Pierwsza z zasad informacyjnych, strukturalna, opisuje wewnêtrzne charakterystyki uk³adu zwi¹zane np. z jego spinem. Druga, wariacyjna, prowadzi do otrzymania w³aœciwej relacji dyspersyjnej dla uk³adu. Ciekaw¹ spraw¹ jest, ¿e wiele rachunków mo¿na wykonaæ dla przypadku, dla którego ca³kowita fizyczna informacja uk³adu (oraz jej gêstoœæ) dzieli siê na dwie równe (lub z czynnikiem $1/2$) czêœci, tzn. pojemnoœæ kana³u informacyjnego oraz informacjê strukturaln¹, maj¹c swoj¹ ca³kowit¹ wartoœæ równ¹ zero. Frieden poda³ informacyjne wyprowadzenie równania Kleina-Gordona dla ogólnego modelu pola z rang¹ $N$, z szczególnym uwzglêdnieniem przypadku pola skalarnego z $N$=2. Dla pola spinorowego z $N$=8 otrzyma³ równanie Diraca a dla $N$=4 równania Maxwella. Procedura jest na tyle ogólna, ¿e umo¿liwia opis pól Rarity-Schwingera, ogólnej teorii wzglêdnoœci oraz wprowadzenie transformacji cechowania [@Frieden]. W oparciu o wprowadzone zasady informacyjne Frieden poda³ równie¿ informacyjne wyprowadzenie zasady nieoznaczonoœci Heisenberga oparte ze statystycznego punktu widzenia o twierdzenie Rao-Cram[é]{}ra dla informacji Fishera oraz jej relacjê z pojemnoœci¹ informacyjn¹ uk³adu zapisan¹ w reprezentacji pêdowej, czyli po dokonaniu transformacji Fouriera. Transformacja Fouriera pe³ni zreszt¹ w ca³ym formalizmie Friedenowskim rolê wyj¹tkow¹, bêd¹c jednym z typów samospl¹tania wewn¹trz przestrzeni konfiguracyjnej uk³adu, o czym wspomnimy nieco poni¿ej. Frieden poda³ równie¿ wyprowadzenie klasycznej fizyki statystycznej, tzn. jej podstawowych rozk³adów, Boltzmanna dla energii oraz Maxwella-Boltzmanna dla pêdu jak równie¿ pewnych rozk³adów, które zinterpretowa³ jako odpowiadaj¹ce przypadkom nierównowagowym. Kolejn¹ spraw¹ by³o wyprowadzenie górnego ograniczenia na tempo zmiany entropii uk³adu dla przyk³adów klasycznego strumienia cz¹stek, gêstoœci rozk³adu ³adunku, czteropotencja³u elektrodynamicznego oraz strumienia cz¹stek o spinie $1/2$ [@Frieden]. Poda³ równie¿ opis teorii pomiaru z szumem wykorzystuj¹c wariacyjny formalizm EFI pozwalaj¹cy na opis redukcji funkcji falowej w trakcie pomiaru urz¹dzeniem daj¹cym swój w³asny szum. Mianowicie po dokonaniu ekstremalizacji sumy informacji fizycznej $K$ niemierzonego uk³adu oraz funkcjona³u opisuj¹cego w³asnoœci uk³adu pomiarowego (a bêd¹cego splotem funkcji log-wiarygodnoœci dla funkcji przyrz¹du splecionej nieliniowo z rozk³adem uk³adu), otrzyma³ równanie ruchu, które (po przejœciu do nierelatywistycznej granicy Schrödingera) daje równanie typu Feynmana-Mensky’ego z nieliniowym cz³onem opisuj¹cym kolaps funkcji falowej w pomiarze. Ciekawe jest to, ¿e w tym przypadku w pe³ni ujawnia siê traktowanie czasu na równi ze zmiennymi przestrzennymi, czyli jako zmiennej losowej z rozk³adem prawdopodobieñstwa. Przedstawiona w skrypcie postaæ zasad informacyjnych [@Dziekuje; @informacja_2] daje formalnie te same równania ewolucji funkcji falowej uk³adu oplecionej funkcj¹ pomiarow¹ przyrz¹du, jednak otrzymana interpretacja jest zdecydowanie bardziej spójna ni¿ Friedenowska, pozwalaj¹c na jednoznaczne rozró¿nienie uk³adu poza pomiarem od uk³adu w pomiarze.\ Przedstawione w skrypcie, fundamentalna postaæ [*obserwowanej*]{} strukturalnej zasady informacyjnej oraz jej postaæ [*oczekiwana*]{}, [@Dziekuje; @informacja_1; @Dziekuje; @informacja_2], wykorzystywane za Friedenem dla ka¿dego omawianego problemu, zosta³y ostatnio wyprowadzone dla wartoœci tzw. wspó³czynnika efektywnoœci $\kappa=1$ [@Dziekuje; @informacja_2]. Zasada strukturalna sugeruje spl¹tanie przestrzeni danych obserwowanych z nieobserwowan¹ przestrzeni¹ konfiguracyjn¹ uk³adu [@Dziekuje; @informacja_2]. Zatem informacja strukturalna $Q$ [@Dziekuje; @informacja_2] reprezentuje równie¿ informacjê o spl¹taniu widocznym w korelacji danych w przestrzeni pomiarowej, a EFI mo¿e byæ wykorzystywana jako mechanizm w estymacji stanów spl¹tanych. Np. w przypadku problemu EPR-Bohma, splatanie zachodzi pomiêdzy rzutem spinu obserwowanej cz¹stki i nieobserwowan¹ konfiguracj¹ ³¹czn¹ uk³adu, a w przypadku relatywistycznych równañ ruchu otrzymujemy spl¹tanie kinetycznych i strukturalnych (masa) stopni swobody, czego wyrazem jest zwi¹zek stanu obserwowanej cz¹stki w czasoprzestrzeni z jej w³asnym stanem w przestrzeni energetyczno-pêdowej. Ten drugi przypadek jest przyk³adem wspomnianego samospl¹tania opisanego transformat¹ Fouriera. Poniewa¿ $Q$ zwi¹zane jest tu z mas¹ cz¹stki, zatem w podejœciu informacyjnym mo¿na wyci¹gn¹æ równie¿ wniosek, ¿e samospl¹tanie powinno pomóc w odczytaniu struktury wewnêtrznej cz¹stek. W koñcu pojêcie informacji Fishera i jej reinterpretacja przez Friedena jako cz³onu kinetycznego teorii, pozwoli³a na przeprowadzenie informacyjnego dowodu [@Dziekuje; @informacja_1] o niewyprowadzalnoœci mechaniki kwantowej (falowej) oraz ka¿dej teorii pola, dla której ranga pola $N$ jest skoñczona, z mechaniki klasycznej.\ [*Temat skryptu*]{} dotyczy wiêc fundamentalnego zagadnienia zwi¹zanego z okreœleniem statystycznej procedury estymacji modeli fizycznych. Jego realizacja wymaga znajomoœci problemów zwi¹zanych z stosowaniem statystycznej MNW oraz fizycznej EFI dla konstrukcji modeli fizycznych, jak równie¿ podstaw metod geometrii ró¿niczkowej.\ Na koniec uwaga s³ownikowa i podsumowanie treœci metody EFI. Pojêcie “likelihood function” zosta³o wprowadzony przez Fishera jako maj¹ce zwi¹zek z prawdopodobieñstwem. Równie¿ s³ownikowo powinno byæ ono przet³umaczone jako “funkcja mo¿liwoœci”. Zastosowano jednak t³umaczenie “funkcja wiarygodnoœci”. Jako posumowanie istoty przedstawionej metody, powiedzmy, ¿e jest ona wyrazem [*zastosowania informacji Fishera w teorii pola w ujêciu Friedena, którego inspiracja pochodzi z obszaru optyki.*]{} Temat skryptu zwi¹zany jest z dociekaniami, które dane mi by³o prowadziæ wspólnie ze S³awomirem Mani¹, Dorot¹ Mroziakiewicz, Janem S³adkowskim, Robertem Szafronem i Sebastianem Zaj¹cem, którym za te dociekania i rozmowy dziêkujê.\ \ Dziêkujê mojej ¿onie Gra¿ynie za uwa¿ne przeczytanie tekstu skryptu. Metoda najwiêkszej wiarygodnoœci {#MNW} ================================ Z powodu mo¿liwoœci zastosowania [*metody najwiêkszej wiarygodnoœci*]{} (MNW) do rozwi¹zania wielu, bardzo ró¿nych problemów estymacyjnych, sta³a siê ona obecnie zarówno metod¹ podstawow¹ jak równie¿ punktem wyjœcia dla ró¿nych metod analizy statystycznej. Jej wszechstronnoœæ zwi¹zana jest, po pierwsze z mo¿liwoœci¹ przeprowadzenia analizy statystycznej dla ma³ej próbki, opisu zjawisk nieliniowych oraz zastosowania zmiennych losowych posiadaj¹cych zasadniczo dowolny [*rozk³ad prawdopodobieñstwa*]{} [@Nowak], oraz po drugie, szczególnymi w³asnoœciami otrzymywanych przez ni¹ estymatorów, które okazuj¹ siê byæ zgodne, asymptotycznie nieobci¹¿one, efektywne oraz dostateczne [@Nowak]. MNW zasadza siê na intuicyjnie jasnym postulacie przyjêcia za prawdziwe takich wartoœci parametrów rozk³adu prawdopodobieñstwa zmiennej losowej, które maksymalizuj¹ funkcjê wiarygodnoœci realizacji konkretnej próbki. Podstawowe pojêcia MNW ---------------------- Rozwa¿my zmienn¹ losow¹ $Y$ [@Nowak], która przyjmuje wartoœci ${\bf y}$ zgodnie z rozk³adem prawdopodobieñstwa $p\left({\bf y}|\theta \right)$, gdzie $\theta = (\vartheta_{1},\vartheta_{2},...,\vartheta_{k})^{T}\equiv(\vartheta_{s})_{s=1}^{k}$, jest zbiorem $k$ parametrów tego rozk³adu ($T$ oznacza transpozycjê). Zbiór wszystkich mo¿liwych wartoœci ${\bf y}$ zmiennej $Y$ oznaczmy przez ${\cal Y}$.\ Gdy $k>1$ wtedy $\theta$ nazywamy parametrem [*wektorowym*]{}. W szczególnym przypadku $k=1$ mamy $\theta=\vartheta$. Mówimy wtedy, ¿e parametr $\theta$ jest parametrem [*skalarnym*]{}.\ \ [**Pojêcie próby i próbki**]{}: Rozwa¿my [*zbiór danych*]{} $\,{{\bf y}_{1},{\bf y}_{2},...,{\bf y}_{N}}$ otrzymanych w $N$ obserwacjach zmiennej losowej $Y$.\ Ka¿da z danych ${\bf y}_{n}$, $n=1,2,...,N$, jest generowana z rozk³adu $p_{n}({\bf y}_{n}|\theta_{n})$ zmiennej losowej $Y$ w populacji, któr¹ charakteryzuje wartoœæ parametru wektorowego $\theta_{n} = (\vartheta_{1},\vartheta_{2},...,\vartheta_{k})_{n}^{T}\equiv((\vartheta_{s})_{s=1}^{k})_{n}$, $n=1,2,...,N$. St¹d zmienn¹ $Y$ w $n$-tej populacji oznaczymy $Y_{n}$. Zbiór zmiennych losowych $\widetilde{Y} = (Y_{1},Y_{2},...,Y_{N}) \equiv( Y_{n})_{n=1}^{N}$ nazywamy $N$-wymiarow¹ [*prób¹*]{}.\ Konkretn¹ realizacjê $y=({{\bf y}_{1},{\bf y}_{2},...,{\bf y}_{N}})\equiv ({\bf y}_{n})_{n=1}^{N}$ próby $\widetilde{Y}$ nazywamy [*próbk¹*]{}. Zbiór wszystkich mo¿liwych realizacji $y$ próby $\widetilde{Y}$ tworzy przestrzeñ próby (uk³adu) oznaczan¹ jako ${\cal B}$.\ [**Okreœlenie**]{}: Ze wzglêdu na to, ¿e $n$ jest indeksem konkretnego punktu pomiarowego próby, rozk³ad $p_{n}({{\bf y}_{n}|\theta_{n}})$ bêdziemy nazywali rozk³adem [*punktowym*]{} (czego nie nale¿y myliæ z np. rozk³adem dyskretnym). [**Okreœlenie funkcji wiarygodnoœci**]{}: $\;\;\;$ Centralnym pojêciem MNW jest [*funkcja wiarygodnoœci*]{} $L\left(y;\Theta\right)$ (pojawienia siê) próbki $y = ({\bf y}_{n})_{n=1}^{N}$, nazywana te¿ [*wiarygodnoœci¹ próbki*]{}. Jest ona funkcj¹ parametru $\Theta$.\ Przez wzgl¹d na zapis stosowany w fizyce, bêdziemy stosowali oznaczenie $P(y\,|\Theta)\equiv L\left(y;\Theta\right)$, które podkreœla, ¿e formalnie [*funkcja wiarygodnoœci jest ³¹cznym rozk³adem prawdopodobieñstwa*]{}[^3] pojawienia siê realizacji $y \equiv ({\bf y}_{n})_{n=1}^{N}$ próby $\widetilde{Y} \equiv( Y_{n})_{n=1}^{N}$, to znaczy: $$\begin{aligned} \label{funkcja wiarygodnoœci proby - def} P(\Theta) \equiv P\left({y|\Theta}\right) = \prod\limits_{n=1}^{N} {p_{n}\left({{\bf y}_{n}|\theta_{n}}\right)} \; .\end{aligned}$$ Zwrócenie uwagi w (\[funkcja wiarygodnoœci proby - def\]) na wystêpowanie $y$ w argumencie funkcji wiarygodnoœci oznacza, ¿e mo¿e byæ ona rozumiana jako statystyka $P\left({\widetilde{Y}|\Theta}\right)$. Z kolei skrócone oznaczenie $P(\Theta)$ podkreœla, ¿e centraln¹ spraw¹ w MNW jest fakt, ¿e funkcja wiarygodnoœci jest funkcj¹ nieznanych parametrów: $$\begin{aligned} \label{parametr Theta} \Theta = (\theta_{1},\theta_{2},...,\theta_{N})^{T} \equiv (\theta_{n})_{n=1}^{N} \;\;\;\; {\rm przy\; czym} \;\;\; \theta_{n} = (\vartheta_{1n},\vartheta_{2n},...,\vartheta_{kn})^{T} \equiv ((\vartheta_{s})_{s=1}^{k})_{n} \; ,\end{aligned}$$ gdzie $\theta_{n}$ jest wektorowym parametrem populacji okreœlonej przez indeks próby $n$. W toku analizy chcemy oszacowaæ wektorowy parametr $\Theta$.\ Zbiór wartoœci parametrów $\Theta=(\theta_{n})_{n=1}^{N}$ tworzy wspó³rzêdne rozk³adu prawdopodobieñstwa rozumianego jako punkt w $d=k \times N$ - wymiarowej (podprzestrzeni) przestrzeni statystycznej ${\cal S}$ [@Amari; @Nagaoka; @book]. Temat ten rozwiniemy w Rozdziale \[alfa koneksja\].\ \ [**Uwaga o postaci rozk³adów punktowych**]{}: W skrypcie zak³adamy, ¿e “[*punktowe*]{}” rozk³ady $p_{n}\left({{\bf y}_{n}|\theta_{n}}\right)$ dla poszczególnych pomiarów $n$ w $N$ elementowej próbie s¹ [*niezale¿ne*]{}[^4].\ W ogólnoœci w treœci skryptu, rozk³ady punktowe $p_{n} \left({\bf y}_{n}|\theta_{n} \right)$ zmiennych $Y_{n}$ chocia¿ s¹ [*tego samego typu*]{}, jednak nie spe³niaj¹ warunku (\[rozklady pn\]) charakterystycznego dla próby prostej. Taka ogólna sytuacja ma np. miejsce w analizie regresji (Rozdzia³ \[regresja klasyczna\]).\ \ \ [**Pojêcie estymatora parametru**]{}: Za³ó¿my, ¿e dane $y = ({\bf y}_{n})_{n=1}^{N}$ s¹ generowane losowo z punktowych rozk³adów prawdopodobieñstwa $p_{n}({\bf y}_{n}|\theta_{n})$, $n=1,2,...,N$, które chocia¿ nie s¹ znane, to jednak za³o¿ono o nich, ¿e dla ka¿dego $n$ nale¿¹ do okreœlonej, tej samej klasy modeli. Zatem funkcja wiarygodnoœci (\[funkcja wiarygodnoœci proby - def\]) nale¿y do okreœlonej, $d = k \times N$ - wymiarowej, przestrzeni statystycznej ${\cal S}$.\ Celem analizy jest oszacowanie nieznanych parametrów $\Theta$, (\[parametr Theta\]), poprzez funkcjê: $$\begin{aligned} \label{estymator parametrow Theta} \!\!\!\!\!\! \hat{\Theta} \equiv \hat{\Theta}(\widetilde{Y}) =(\hat{\theta}_{1},\hat{\theta}_{2},...,\hat{\theta}_{N})^{T}\equiv (\hat{\theta}_{n})_{n=1}^{N} \; \;\;\; {\rm gdzie} \;\;\; \hat{\theta}_{n} = (\hat{\vartheta}_{1n},\hat{\vartheta}_{2n},...,\hat{\vartheta}_{kn})^{T} \equiv ((\hat{\vartheta}_{s})_{s=1}^{k})_{n} \; , \;\end{aligned}$$ maj¹c¹ $d = k \times N$ sk³adowych.\ Ka¿da z funkcji $\hat{\vartheta}_{kn} \equiv \hat{\vartheta}_{kn}(\widetilde{Y})$ jako funkcja próby jest [*statystyk¹*]{}, któr¹ przez wzgl¹d na to, ¿e s³u¿y do oszacowywania wartoœci parametru $\vartheta_{kn}$ nazywamy estymatorem tego parametru. [*Estymator parametru nie mo¿e zale¿eæ od parametru, który oszacowuje*]{}[^5].\ \ Podsumowuj¹c, odwzorowanie: $$\begin{aligned} \label{estymator jako odwzorowanie} \hat{\Theta}: {\cal B} \rightarrow \mathbf{R}^{d} \; ,\end{aligned}$$ gdzie ${\cal B}$ jest przestrzeni¹ próby, jest estymatorem parametru (wektorowego) $\Theta$.\ \ [**Równania wiarygodnoœci**]{}: Bêd¹c funkcj¹ $\Theta=(\theta_{n})_{n=1}^{N}$, funkcja wiarygodnoœci s³u¿y do konstrukcji estymatorów $\hat{\Theta}=(\hat{\theta}_{1},\hat{\theta}_{2},...,\hat{\theta}_{N})^{T}\equiv(\hat{\theta}_{n})_{n=1}^{N}$ parametrów $\Theta \equiv (\theta_{n})_{n=1}^{N}$. Procedura polega na wyborze takich $(\hat{\theta}_{n})_{n=1}^{N}$ , dla których funkcja wiarygodnoœci przyjmuje maksymaln¹ wartoœæ, sk¹d statystyki te nazywamy estymatorami MNW.\ Zatem warunek konieczny otrzymania estymatorów $\hat{\Theta}$ MNW sprowadza siê do znalezienia rozwi¹zania uk³adu $d=k \times N$ tzw. [*równañ wiarygodnoœci*]{} [@Fisher]: $$\begin{aligned} \label{rown wiaryg} S\left(\Theta\right)_{\left|\Theta = \hat{\Theta} \right.} \equiv\frac{\partial}{\partial\Theta}\ln P(y\,|\Theta)_{\left|\Theta = \hat{\Theta} \right.} = 0 \; ,\end{aligned}$$ gdzie zagadnienie maksymalizacji funkcji wiarygodnoœci $P(y\,|\Theta)$ sprowadzono do (na ogó³) analitycznie równowa¿nego mu problemu maksymalizacji jej logarytmu $\ln P(y\,|\Theta)$.\ \ [**Okreœlenie funkcji wynikowej**]{}: Funkcjê $S\left(\Theta\right)$ bêd¹c¹ gradientem logarytmu funkcji wiarygodnoœci: $$\begin{aligned} \label{funkcja wynikowa} S\left(\Theta\right)\equiv\frac{\partial}{\partial\Theta}\ln P(y\,|\Theta) = \left(\begin{array}{c} \frac{\partial \ln P(y|\Theta)}{\partial \theta_{1}} \\ \vdots \\ \frac{\partial \ln P(y|\Theta)}{\partial \theta_{N}} \\ \end{array}\right) \; \;\;\;\; {\rm gdzie} \;\;\;\; \frac{\partial \ln P(y|\Theta)}{\partial \theta_{n}} = \left(\begin{array}{c} \frac{\partial \ln P(y|\Theta)}{\partial \vartheta_{1n}} \\ \vdots \\ \frac{\partial \ln P(y|\Theta)}{\partial \vartheta_{kn}} \\ \end{array}\right) \; , \end{aligned}$$ nazywamy [*funkcj¹ wynikow¹*]{}.\ \ Po otrzymaniu (wektora) estymatorów $\hat{\Theta}$, [*zmaksymalizowan¹*]{} wartoœæ funkcji wiarygodnoœci definiujemy jako numeryczn¹ wartoœæ funkcji wiarygodnoœci powsta³¹ przez podstawienie do $P(y \,|\Theta)$ wartoœci oszacowanej $\hat{\Theta}$ w miejsce parametru $\Theta$.\ \ [**Przyk³ad**]{}: Rozwa¿my problem estymacji skalarnego parametru, tzn. $\Theta = \theta$ (tzn. $k=1$ oraz $N=1$), dla zmiennej losowej $Y$ opisanej rozk³adem dwumianowym (Bernoulliego): $$\begin{aligned} \label{Bernoulliego rozklad} P \left( y|\theta \right) = \left( \begin{array}{l} m \\ y \end{array} \right) \theta^{y} \left(1 - \theta \right)^{m-y} \; .\end{aligned}$$ Estymacji parametru $\theta$ dokonamy na podstawie [*pojedynczej*]{} obserwacji (d³ugoœæ próby $N=1$) zmiennej $Y$, której iloraz $Y/m$ nazywamy [*czêstoœci¹*]{}. Parametr $m$ charakteryzuje rozk³ad zmiennej Bernoulliego $Y$ (i nie ma zwi¹zku z d³ugoœci¹ $N$ próby).\ Zatem poniewa¿ $y \equiv \left( {\bf y}_{1} \right)$, wiêc $P\left(y|\theta \right)$ jest funkcj¹ wiarygodnoœci dla $N=1$ wymiarowej próby. Jej logarytm wynosi: $$\begin{aligned} \label{ln wiaryg dla Bernoulliego} \ln P\left(y|\theta\right)=\ln\left(\begin{array}{l} m\\ y \end{array}\right) + y \ln \theta + {\left(m-y \right)\ln} \left({1-\theta}\right) \; .\end{aligned}$$\ W rozwa¿anym przypadku otrzymujemy jedno równanie wiarygodnoœci (\[rown wiaryg\]): $$\begin{aligned} \label{r.wiaryg dla Bernoulliego} S(\theta)=\frac{1}{\theta}{y}-\frac{1}{{1-\theta}}{\left({m-y}\right)\left|\begin{array}{l} _{\theta=\hat{\theta}}\end{array}\right.} = 0 \end{aligned}$$ a jego rozwi¹zanie daje estymator MNW parametru $\theta$ rozk³adu dwumianowego, równy: $$\begin{aligned} \hat{\theta}=\frac{y}{m}\label{estymator theta dla Bernoulliego}\end{aligned}$$ Ilustracj¹ powy¿szej procedury znajdowania wartoœci estymatora parametru $\theta$ jest Rysunek 1.1 (gdzie przyjêto $m=5$), przy czym na skutek pomiaru zaobserwowano wartoœæ $Y$ równ¹ $y=1$. \[rysunek dla rozk dwumianowego\] ![Graficzna ilustracja metody najwiêkszej wiarygodnoœci dla $P\left(y|\theta\right)$ okreœlonego wzorem (\[Bernoulliego rozklad\]) dla rozk³adu dwumianowego. Przyjêto wartoœæ parametru $m=5$. Na skutek pomiaru zaobserwowano wartoœæ $Y$ równ¹ $y=1$. Maksimum $P\left(y|\theta\right)$ przypada na wartoœæ $\theta$ równ¹ punktowemu oszacowaniu $\hat{\theta}=y/m=1/5$ tego parametru. Maksymalizowana wartoœæ funkcji wiarygodnoœci wynosi $P\left(y|\hat{\theta}\right)$.[]{data-label="PBernoulli"}](PBernoulli.eps "fig:"){width="65mm"} Wnioskowanie w MNW {#Wnioskowanie w MNW} ------------------ Z powy¿szych rozwa¿añ wynika, ¿e konstrukcja punktowego oszacowania parametru w MNW oparta jest o postulat maksymalizacji funkcji wiarygodnoœci przedstawiony powy¿ej. Jest on wstêpem do statystycznej procedury wnioskowania. Kolejnym krokiem jest konstrukcja przedzia³u wiarygodnoœci. Jest on odpowiednikiem przedzia³u ufnoœci, otrzymywanego w czêstotliwoœciowym podejœciu statystyki klasycznej do procedury estymacyjnej. Do jego konstrukcji niezbêdna jest znajomoœæ rozk³adu prawdopodobieñstwa estymatora parametru, co (dziêki “porz¹dnym” granicznym w³asnoœciom stosowanych estymatorów) jest mo¿liwe niejednokrotnie jedynie asymptotycznie, tzn. dla wielkoœci próby d¹¿¹cej do nieskoñczonoœci. Znajomoœæ rozk³adu estymatora jest te¿ niezbêdna we wnioskowaniu statystycznym odnosz¹cym siê do weryfikacji hipotez.\ W sytuacji, gdy nie dysponujemy wystarczaj¹c¹ iloœci¹ danych, potrzebnych do przeprowadzenia skutecznego czêstotliwoœciowego wnioskowania, Fisher [@Pawitan] zaproponowa³ do okreœlenia niepewnoœci dotycz¹cej parametru $\Theta$ wykorzystanie maksymalizowanej wartoœæ funkcji wiarygodnoœci.\ \ [**Przedzia³ wiarygodnoœci**]{} jest zdefiniowany jako zbiór wartoœci parametru $\Theta$, dla których funkcja wiarygodnoœci osi¹ga (umownie) wystarczaj¹co wysok¹ wartoœæ, tzn.: $$\begin{aligned} \label{przedzial wiarygodnosci} \left\{ {\Theta, \; \frac{{P\left({y|\Theta}\right)}}{{P\left({y|\hat{\Theta}}\right)}}> c} \right\} \; ,\end{aligned}$$ dla pewnego [*parametru obciêcia*]{} $c$, nazywanego [*poziomem wiarygodnoœci*]{}.\ \ [**Iloraz wiarygodnoœci**]{}: $$\begin{aligned} \label{iloraz wiarygodnosci} P(y|\Theta)/P(y|\hat{\Theta})\end{aligned}$$ reprezentuje pewien typ unormowanej wiarygodnoœci i jako taki jest wielkoœci¹ skalarn¹. Jednak z powodu niejasnego znaczenia okreœlonej wartoœci parametru obciêcia $c$ pojêcie to wydaje siê byæ na pierwszy rzut oka za s³abe, aby dostarczyæ tak¹ precyzjê wypowiedzi jak¹ daje analiza czêstotliwoœciowa.\ Istotnie, wartoœæ $c$ nie odnosi siê do ¿adnej wielkoœci obserwowanej, tzn. na przyk³ad $1\%$-we ($c=0,01$) obciêcie nie ma œcis³ego probabilistycznego znaczenia. Inaczej ma siê sprawa dla czêstotliwoœciowych przedzia³ów ufnoœci. W tym przypadku wartoœæ wspó³czynnika $\alpha=0,01$ oznacza, ¿e gdybyœmy rozwa¿yli realizacjê przedzia³u ufnoœci na poziomie ufnoœci $1-\alpha=0,99$, to przy pobraniu nieskoñczonej (w praktyce wystarczaj¹co du¿ej) liczby próbek, $99\%$ wszystkich wyznaczonych przedzia³ów ufnoœci pokry³oby prawdziw¹ (teoretyczn¹) wartoœæ parametru $\Theta$ w populacji generalnej (sk³adaj¹cej siê z $N$ podpopulacji). Pomimo tej s³aboœci MNW zobaczymy, ¿e rozbudowanie analizy stosunku wiarygodnoœci okazuje siê byæ istotne we wnioskowaniu statystycznym analizy doboru modeli i to a¿ po konstrukcjê równañ teorii pola. ### Wiarygodnoœciowy przedzia³ ufnoœci {#Wiarygodnosciowy przedzial ufnosci} [**Przyk³ad rozk³adu normalnego z jednym estymowanym parametrem**]{}: Istnieje przypadek pozwalaj¹cy na prost¹ [*interpretacjê przedzia³u wiarygodnoœciowego jako przedzia³u ufnoœci*]{}. Dotyczy on zmiennej $Y$ posiadaj¹cej rozk³ad Gaussa oraz sytuacji gdy (dla próby prostej) interesuje nas estymacja skalarnego parametru $\theta$ bêd¹cego wartoœci¹ oczekiwan¹ $E(Y)$ zmiennej $Y$. Przypadek ten omówimy poni¿ej. W ogólnoœci, przedzia³ wiarygodnoœci posiadaj¹cy okreœlony poziom ufnoœci jest nazywany przedzia³em ufnoœci.\ Czêstotliwoœciowe wnioskowanie o nieznanym parametrze $\theta$ wymaga okreœlenia rozk³adu jego estymatora, co jest zazwyczaj mo¿liwe jedynie granicznie [@Amari; @Nagaoka; @book]. Podobnie w MNW, o ile to mo¿liwe, korzystamy przy du¿ych próbkach z twierdzeñ granicznych dotycz¹cych rozk³adu ilorazu wiarygodnoœci [@Amari; @Nagaoka; @book; @Pawitan]. W przypadku rozk³adu normalnego i parametru skalarnego okazuje siê, ¿e mo¿liwa jest konstrukcja skoñczenie wymiarowa.\ Niech wiêc zmienna $Y$ ma rozk³ad normalny $N\left({\theta,\sigma^{2}}\right)$: $$\begin{aligned} \label{rozklad norm theta sigma2} p\left({\bf y}|\theta, \sigma^{2}\right) = \frac{1}{\sqrt{2 \pi \, \sigma^2}} \; \exp \left( - \, {\frac{({\bf y} - \theta)^{2}}{2 \, \sigma^2}} \right) \;\, .\end{aligned}$$ Rozwa¿my próbkê $y \equiv ({\bf y}_{1},\ldots, {\bf y}_{N})$, która jest realizacj¹ próby prostej $\widetilde{Y}$ i za³ó¿my, ¿e [*wariancja $\sigma^{2}$ jest znana*]{}. Logarytm funkcji wiarygodnoœci dla $N\left({\theta,\sigma^{2}}\right)$ ma postaæ: $$\begin{aligned} \label{log wiaryg rozklad norm jeden par} \ln P\left(y|\theta\right) = - \frac{N}{2}\ln(2 \pi \sigma^{2}) - \frac{1}{{2\sigma^{2}}} \sum\limits_{n=1}^{N}{\left({{\bf y}_{n} - \theta}\right)^{2}} \;\, ,\end{aligned}$$ gdzie ze wzglêdu na próbê prost¹, w argumencie funkcji wiarygodnoœci wpisano w miejsce $\Theta~\equiv~(\theta)_{n=1}^{N}$ parametr $\theta$, jedyny który podlega estymacji.\ \ Korzystaj¹c z funkcji wiarygodnoœci (\[log wiaryg rozklad norm jeden par\]) otrzymujemy oszacowanie MNW parametru $\theta$ równe[^6] $\hat{\theta} = \bar{{\bf y}} = \frac{1}{{N}}\sum\limits_{n=1}^{N} {\bf y}_{n}$, co pozwala na zapisanie równoœci $\sum_{n=1}^{N}{\left({{\bf y}_{n} - \theta} \right)^{2}}$ $ = \sum_{n=1}^{N}{\left({({\bf y}_{n} - \hat{\theta}) + (\hat{\theta} - \theta)} \right)^{2}}$ $= \sum_{n=1}^{N}({\bf y}_{n} - \hat{\theta})^{2} + \sum_{n=1}^{N} (\hat{\theta} - \theta)^{2}$. W koñcu, nieskomplikowane przekszta³cenia prowadz¹ do nastêpuj¹cej postaci logarytmu ilorazu wiarygodnoœci (\[iloraz wiarygodnosci\]): $$\begin{aligned} \label{iloraz wiaryg dla normalnego} \ln \frac{{P\left( {y|\theta}\right)}}{{P( {y|\hat{\theta}} )}} = - \frac{N}{{2\sigma^{2}}}\left({\hat{\theta} - \theta}\right)^{2} \; .\end{aligned}$$\ [**Statystyka Wilka**]{}: Widaæ, ¿e po prawej stronie (\[iloraz wiaryg dla normalnego\]) otrzymaliœmy wyra¿enie kwadratowe. Poniewa¿ $\bar{Y}$ jest nieobci¹¿onym estymatorem parametru $\theta$, co oznacza, ¿e wartoœæ oczekiwana ${E(\bar{Y})=\theta}$, zatem (dla rozk³adu $Y \sim N\left( \theta, \sigma^{2} \right)$) œrednia arytmetyczna $\bar{Y}$ ma rozk³ad normalny $N\left({\theta,\frac{\sigma^{2}}{N}}\right)$.\ Z normalnoœci rozk³adu $\bar{Y}$ wynika, ¿e tzw. [*statystyka ilorazu wiarygodnoœci Wilka*]{}: $$\begin{aligned} \label{1} W \equiv 2 \ln \frac{{P\left({\widetilde{Y}|\hat{\theta}}\right)}}{{P\left({\widetilde{Y}|\theta}\right)}}\sim\chi_{1}^{2} \; ,\end{aligned}$$ ma rozk³ad $\chi^{2}$, w tym przypadku z jednym stopniem swobody [@Pawitan].\ \ [**Wyskalowanie statystyki Wilka w przypadku normalnym**]{}: Wykorzystuj¹c (\[1\]) mo¿emy wykonaæ wyskalowanie wiarygodnoœci oparte o mo¿liwoœæ powi¹zania przedzia³u wiarygodnoœci z jego czêstotliwoœciowym odpowiednikiem.\ \ Mianowicie z (\[1\]) otrzymujemy, ¿e dla ustalonego (chocia¿ nieznanego) parametru $\theta$ prawdopodobieñstwo, ¿e iloraz wiarygodnoœci znajduje siê w wyznaczonym dla parametru obciêcia $c,$ wiarygodnoœciowym przedziale ufnoœci, wynosi:\ $$\begin{aligned} \label{rownosc prawdop dla zdarzenia z c} P\left({\frac{{P\left({\widetilde{Y}|\theta}\right)}}{{P\left({\widetilde{Y}|\hat{\theta}}\right)}}>c}\right)=P\left({2\ln\frac{{P\left({\widetilde{Y}|\hat{\theta}}\right)}}{{P\left({\widetilde{Y}|\theta}\right)}}<-2\ln c}\right)=P\left({\chi_{1}^{2}<-2\ln c}\right) \; .\end{aligned}$$\ Zatem jeœli dla jakiegoœ $0<\alpha<1$ wybierzemy parametr obciêcia: $$\begin{aligned} \label{2} c = e^{-\frac{1}{2}\chi_{1,\left({1-\alpha}\right)}^{2}} \; ,\end{aligned}$$ gdzie ${\chi_{1,\left({1-\alpha}\right)}^{2}}$ jest kwantylem rzêdu $100(1-\alpha)\%$ rozk³adu $\chi$-kwadrat, to spe³nienie przez $\theta$ zwi¹zku: $$\begin{aligned} \label{1 minus alfa} P\left( {\frac{{P\left({\widetilde{Y}|\theta}\right)}}{ {P\left({\widetilde{Y}|\hat{\theta}}\right)}}>c} \right) = P \left({\chi_{1}^{2}<\chi_{1,\left({1-\alpha}\right)}^{2}}\right) = 1-\alpha \;\end{aligned}$$ oznacza, ¿e przyjêcie wartoœci $c$ zgodnej z (\[2\]) daje zbiór mo¿liwych wartoœci parametru $\theta$: $$\begin{aligned} \label{przedzial wiarygod theta dla rozkl norm} \left\{ {\theta,\frac{{P\left({\widetilde{Y}|\theta}\right)}}{{P\left({\widetilde{Y}|\hat{\theta}}\right)}}>c}\right\} \; , \end{aligned}$$ nazywany 100$(1-\alpha)\%$-owym [*wiarygodnoœciowym przedzia³em ufnoœci*]{}. Jest on odpowiednikiem wyznaczonego na poziomie ufnoœci $(1-\alpha)$ czêstotliowœciowego przedzia³u ufnoœci dla $\theta$. Dla analizowanego przypadku rozk³adu normalnego z estymacj¹ skalarnego parametru $\theta$ oczekiwanego poziomu zjawiska, otrzymujemy po skorzystaniu z wzoru (\[2\]) wartoœæ parametru obciêcia równego $c=0.15$ lub $c=0.04$ dla odpowiednio $95\%$-owego ($1-\alpha=0.95$) b¹dŸ $99\%$-owego ($1-\alpha=0.99$) przedzia³u ufnoœci. Tak wiêc w przypadku, [*gdy przedzia³ wiarygodnoœci da siê wyskalowaæ rozk³adem prawdopodobieñstwa, parametr obciêcia $c$ posiada w³asnoœæ wielkoœci obserwowanej interpretowanej czêstotliwoœciowo poprzez zwi¹zek z poziomem ufnoœci*]{}.\ Zwróæmy uwagê, ¿e chocia¿ konstrukcje czêstotliwoœciowego i wiarygodnoœciowego przedzia³u ufnoœci s¹ ró¿ne, to [*ich losowoœæ wynika*]{} w obu przypadkach [*z rozk³adu prawdopodobieñstwa estymatora*]{} $\hat{\theta}$.\ \ [**Æwiczenie**]{}: W oparciu o powy¿sze rozwa¿ania wyznaczyæ, korzystaj¹c z (\[iloraz wiaryg dla normalnego\]) ogóln¹ postaæ przedzia³u wiarygodnoœci dla skalarnego parametru $\theta$ rozk³adu normalnego. ### Rozk³ady regularne {#Rozklady regularne} Dla zmiennych o innym rozk³adzie ni¿ rozk³ad normalny, statystyka Wilka $W$ ma w ogólnoœci inny rozk³ad ni¿ $\chi^{2}$ [@Pawitan]. Jeœli wiêc zmienne nie maj¹ dok³adnie rozk³adu normalnego lub dysponujemy za ma³¹ próbk¹ by móc odwo³ywaæ siê do (wynikaj¹cych z twierdzeñ granicznych) rozk³adów granicznych dla estymatorów parametrów, wtedy zwi¹zek (\[1\]) (wiêc i (\[2\])) daje jedynie przybli¿one wyskalowanie przedzia³u wiarygodnoœci rozk³adem $\chi^{2}$.\ Jednak¿e w przypadkach wystarczaj¹co [*regularnych rozk³adów*]{}, zdefiniowanych jako takie, w których mo¿emy zastosowaæ przybli¿enie kwadratowe: $$\begin{aligned} \label{log wiaryg dla regularnego} \ln\frac{{P\left({y|\theta}\right)}}{{P\left({y|\hat{\theta}}\right)}} \approx - \frac{1}{2} \texttt{i\!F} \left(\hat{\theta}\right)\left(\hat{\theta}-\theta\right)^{2} \; , \end{aligned}$$ powy¿sze rozumowanie oparte o wyskalowanie wiarygodnoœci rozk³adem $\chi^{2}_{1}$ jest w przybli¿eniu s³uszne. Wielkoœæ $\texttt{i\!F}\left(\hat{\theta}\right)$, która pojawi³a siê powy¿ej jest [*obserwowan¹*]{} informacj¹ Fishera, a powy¿sza formu³a stanowi powa¿ne narzêdzie w analizie doboru modeli. Mo¿na powiedzieæ, ¿e ca³y skrypt koncentruje siê na analizie zastosowania (wartoœci oczekiwanej) tego wyra¿enia i jego uogólnieñ. Do sprawy tej wrócimy dalej.\ \ [**Przyk³ad**]{}: Rozwa¿my przypadek parametru skalarnego $\theta$ w jednym eksperymencie ($N=1$) ze zmienn¹ $Y$ posiadaj¹c¹ rozk³ad Bernoulliego z $m=15$. W wyniku pomiaru zaobserwowaliœmy wartoœæ $Y={\bf y}=3$. Prosta analiza pozwala wyznaczyæ wiarygodnoœciowy przedzia³ ufnoœci dla parametru $\theta$. Poniewa¿ przestrzeñ $V_{\theta}$ parametru $\theta$ wynosi $V_{\theta}=(0,1)$, zatem ³atwo pokazaæ, ¿e dla $c=0,01$, $c=0,1$ oraz $c=0,5$ mia³by on realizacjê odpowiednio $(0,019;0,583)$, $(0,046;0,465)$ oraz $(0,098;0,337)$. Widaæ, ¿e wraz ze wzrostem wartoœci $c$, przedzia³ wiarygodnoœci zacieœnia siê wokó³ wartoœci oszacowania punktowego $\hat{\theta}=y/m=1/5$ parametru $\theta$ i nic dziwnego, bo wzrost wartoœci $c$ oznacza akceptowanie jako mo¿liwych do przyjêcia tylko takich [*modelowych wartoœci parametru*]{} $\theta$, które gwarantuj¹ wystarczaj¹co wysok¹ wiarygodnoœæ próbki.\ Powy¿szy przyk³ad pozwala nabyæ pewnej intuicji co do sensu stosowania ilorazu funkcji wiarygodnoœci. Mianowicie po otrzymaniu w pomiarze okreœlonej wartoœci $y/m$ oszacowuj¹cej parametr $\theta$, jesteœmy sk³onni preferowaæ model z tak¹ wartoœci¹ parametru $\theta$, która daje wiêksz¹ wartoœæ (logarytmu) ilorazu wiarygodnoœci $P(y|\theta)/P(y|\hat{\theta})$. Zgodnie z podejœciem statystyki klasycznej [*nie oznacza to jednak*]{}, ¿e uwa¿amy, ¿e parametr $\theta$ ma jakiœ rozk³ad. Jedynie wobec niewiedzy co do modelowej (populacyjnej) wartoœæ parametru $\theta$ preferujemy ten model, który daje wiêksz¹ wartoœæ ilorazu wiarygodnoœci w próbce. ### Weryfikacja hipotez z wykorzystaniem ilorazu wiarygodnoœci {#weryfikacja hipotez z ilorazem wiaryg} Powy¿ej wykorzystaliœmy funkcjê wiarygodnoœci do [*estymacji wartoœci parametru*]{} $\Theta$. Funkcjê wiarygodnoœci mo¿na równie¿ wykorzystaæ w drugim typie wnioskowania statystycznego, tzn. w [*weryfikacji hipotez statystycznych*]{}.\ \ Rozwa¿my prost¹ hipotezê zerow¹ $H_{0}: \Theta = \Theta_{0}$ wobec z³o¿onej hipotezy alternatywnej $H_{1}: \Theta \neq \Theta_{0}$. W celu przeprowadzenia [*testu statystycznego*]{} wprowadŸmy unormowan¹ funkcjê wiarygodnoœci: $$\begin{aligned} \label{unorm fun wiaryg} \frac{{P\left({y|\Theta_{0}}\right)}}{{P\left({y|\hat{\Theta}}\right)}} \;\, ,\end{aligned}$$ skonstruowan¹ przy za³o¿eniu prawdziwoœci hipotezy zerowej. Hipotezê zerowa $H_{0}$ odrzucamy na rzecz hipotezy alternatywnej, jeœli jej wiarygodnoœæ $P\left({y|\Theta_{0}}\right)$ jest “za ma³a”. Sugerowa³oby to, ¿e z³o¿ona hipoteza alternatywna $H_{1}$ zawiera pewn¹ hipotezê prost¹, która jest lepiej poparta przez dane otrzymane w próbce, ni¿ hipoteza zerowa.\ Jak o tym wspomnieliœmy powy¿ej, np. $5\%$-owe obciêcie $c$ w zagadnieniu estymacyjnym, samo w sobie nie mówi nic o frakcji liczby przedzia³ów wiarygodnoœci pokrywaj¹cych nieznan¹ wartoœæ szacowanego parametru. Potrzebne jest wyskalowanie ilorazu wiarygodnoœci. Równie¿ dla weryfikacji hipotez skalowanie wiarygodnoœci jest istotne. Stwierdziliœmy, ¿e takie skalowanie jest mo¿liwe wtedy gdy mamy do czynienia z jednoparametrowym przypadkiem rozk³adu Gaussa, a przynajmniej z przypadkiem wystarczaj¹co regularnym.\ \ [**Empiryczny poziom istotnoœci**]{}: W przypadku jednoparametrowego, regularnego problemu z ($\Theta \equiv (\theta)_{n=1}^{N})$ jak w Przyk³adzie z Rozdzia³u \[Wiarygodnosciowy przedzial ufnosci\], skalowanie poprzez wykorzystanie statystki Wilka s³u¿y otrzymaniu empirycznego poziomu istotnoœci $p$. Ze zwi¹zku (\[1\]) otrzymujemy wtedy przybli¿ony (a dok³adny dla rozk³adu normalnego) [*empiryczny poziom istotnoœci*]{}: $$\begin{aligned} \label{poziom istotnosci wiaryg} \!\!\!\!\!\!\!\!\!\!\! p &\approx& P\left(\frac{{P\left({\widetilde{Y}|\hat{\theta}}\right)}}{{P\left({\widetilde{Y}|\theta_{0}}\right)}} \geq \frac{{P\left({y|\hat{\theta}_{obs}}\right)}}{{P\left({y|\theta_{0}}\right)}} \right) = P\left({2\ln\frac{{P\left({\widetilde{Y}|\hat{\theta}}\right)}}{{P\left({\widetilde{Y}|\theta_{0}}\right)}} \geq -2\ln c_{obs}}\right) \nonumber \\ &=& P\left({\chi_{1}^{2} \geq -2\ln c_{obs}}\right) \; , \;\;\;\; {\rm gdzie} \;\;\;\;\; c_{obs} \equiv \frac{{P\left({y|\theta_{0}}\right)}}{{P\left({y|\hat{\theta}_{obs}}\right)}} \; ,\end{aligned}$$ przy czym $\hat{\theta}_{obs}$ jest wartoœci¹ estymatora MNW $\hat{\theta}$ wyznaczon¹ w obserwowanej (obs) próbce $y$. Powy¿sze okreœlenie empirycznego poziomu istotnoœci $p$ oznacza, ¿e w przypadku wystarczaj¹co regularnego problemu [@Pawitan], istnieje typowy zwi¹zek pomiêdzy prawdopodobieñstwem (\[1 minus alfa\]), a empirycznym poziomem istotnoœci $p$, podobny do zwi¹zku jaki istnieje pomiêdzy poziomem ufnoœci $1-\alpha$, a poziomem istotnoœci $\alpha$ w analizie czêstotliwoœciowej. I tak, np. w przypadku jednoparametrowego rozk³adu normalnego mo¿emy wykorzystaæ wartoœæ empirycznego poziomu istotnoœci $p$ do stwierdzenia, ¿e gdy $p \leq \alpha$ to hipotezê $H_{0}$ odrzucamy na rzecz hipotezy $H_{1}$, a w przypadku $p > \alpha$ nie mamy podstawy do odrzucenia $H_{0}$.\ \ [**Problem b³êdu pierwszego i drugiego rodzaju**]{}: Jednak¿e podobne skalowanie ilorazu wiarygodnoœci okazuje siê byæ znacznie trudniejsze ju¿ chocia¿by tylko w przypadku dwuparametrowego rozk³adu normalnego, gdy obok $\theta$ estymujemy $\sigma^{2}$ [@Pawitan]. Wtedy okreœlenie co oznacza sformu³owanie ,,zbyt ma³a” wartoœæ $c$ jest doœæ dowolne i zale¿y od rozwa¿anego problemu lub wczeœniejszej wiedzy wynikaj¹cej z innych Ÿróde³ ni¿ prowadzone statystyczne wnioskowanie. Wybór du¿ego parametru obciêcia $c$ spowoduje, ¿e istnieje wiêksze prawdopodobieñstwo pope³nienia [*b³êdu pierwszego rodzaju*]{} polegaj¹cego na odrzuceniu hipotezy zerowej w przypadku, gdy jest ona prawdziwa. Wybór ma³ego $c$ spowoduje zwiêkszenie prawdopodobieñstwa pope³nienia [*b³êdu drugiego rodzaju*]{}, tzn. przyjêcia hipotezy zerowej w sytuacji, gdy jest ona b³êdna. MNW w analizie regresji {#regresja klasyczna} ----------------------- Analiza zawarta w ca³ym Rozdziale \[regresja klasyczna\] oparta jest na przedstawieniu metody MNW w analizie regresji klasycznej podanym w [@Kleinbaum] i [@Mroz].\ [**W metodzie regresji klasycznej**]{}, estymatory parametrów strukturalnych modelu regresji s¹ otrzymane arytmetyczn¹ metod¹ najmniejszych kwadratów (MNK). Zmienne objaœniaj¹ce $X_{n} = x_{n}\,$, $n=1,...,N$, nie maj¹ wtedy charakteru stochastycznego, co oznacza, ¿e eksperyment jest ze wzglêdu na nie kontrolowany.\ \ [**MNK polega na**]{} minimalizacji sumy kwadratów odchyleñ obserwowanych wartoœci zmiennej objaœnianej (tzw. odpowiedzi) od ich wartoœci teoretycznych spe³niaj¹cych równanie regresji. MNK ma znaczenie probabilistyczne tylko w przypadku analizy standardowej, gdy zmienna objaœniana $Y$ ma rozk³ad normalny. Jej estymatory pokrywaj¹ siê wtedy z estymatorami MNW. Poka¿emy, ¿e tak siê sprawy maj¹.\ \ Za³ó¿my, ¿e zmienne $\,Y_{1},Y_{2},...,Y_{N}\,$ odpowiadaj¹ce kolejnym wartoœciom zmiennej objaœniaj¹cej, $x_{1},$ $x_{2},...,x_{N}$, s¹ wzglêdem siebie niezale¿ne i maj¹ rozk³ad normalny ze œredni¹ $\mu_{n}=E\left({Y\left|{x_{n}}\right.}\right) = E\left(Y_{n}\right)$ zale¿n¹ od wariantu zmiennej objaœniaj¹cej $x_{n}$, oraz tak¹ sam¹ wariancjê $\sigma^{2}(Y_{n})=\sigma^{2}(Y)$.\ Funkcja wiarygodnoœci próbki $\left({y_{1},y_{2},...,y_{N}}\right)$ dla normalnego klasycznego modelu regresji z parametrem $\Theta = \mu \equiv (\mu_{n})_{n=1}^{N}$, ma postaæ: $$\begin{aligned} \label{wiaryg dla regr klas} P(\mu) \equiv P\left({y}|\mu \right) &=& \prod\limits_{n=1}^{N} {f\left({{\bf y}_{n}|\mu_{n}} \right)} = \prod\limits_{n=1}^{N}{\frac{1}{{\sqrt{2\pi\sigma^{2}}}} \, \exp\left\{ {-\frac{1}{{2\sigma^{2}}} \left({{\bf y}_{n} - \mu_{n}}\right)^{2}} \right\} } \nonumber \\ &=& \frac{1}{{\left({2\pi\sigma^{2}}\right)^{N/2}}}\exp\left\{ { - \frac{1}{{2\sigma^{2}}} \sum\limits_{n=1}^{N}{\left({{\bf y}_{n}-\mu_{n}}\right)^{2}}}\right\} \; ,\end{aligned}$$ gdzie $f\left({{\bf y}_{n}|\mu_{n}} \right)\,$, $n=1,2...,N$, s¹ punktowymi rozk³adami gestoœci prawdopodobieñstwa Gaussa. Widaæ, ¿e maksymalizacja $P(\mu)$ ze wzglêdu na $(\mu_{n})_{n=1}^{N}$ poci¹ga za sob¹ minimalizacjê sumy kwadratów reszt[^7] ($SKR$): $$\begin{aligned} \label{SKR} SKR = \sum\limits_{n=1}^{N}{\left({{\bf y}_{n}-\mu_{n}}\right)^{2}} \; , \end{aligned}$$ gdzie $\mu_{n}=E\left({Y\left|{x_{n}}\right.}\right)$ jest [*postulowanym modelem regresji*]{}. Zatem w standardowej, klasycznej analizie regresji, estymatory MNW pokrywaj¹ siê z estymatorami MNK. Widaæ, ¿e procedura minimalizacji dla $SKR$ prowadzi do liniowej w $Y_{n}$ postaci estymatorów $\hat{\mu}_{n}$ parametrów $\mu_{n}$.\ \ [**Problem z nieliniowym uk³adem równañ wiarygodnoœci**]{}: Jednak rozwi¹zanie uk³adu równañ wiarygodnoœci (\[rown wiaryg\]) jest zazwyczaj nietrywialne. Jest tak, gdy otrzymany w wyniku ekstremizacji uk³ad algebraicznych równañ wiarygodnoœci dla estymatorów jest nieliniowy, co w konsekwencji oznacza, ¿e mo¿emy nie otrzymaæ ich w zwartej analitycznej postaci. Przyk³adem mo¿e byæ analiza regresji Poissona, w której do rozwi¹zania równañ wiarygodnoœci wykorzystujemy metody iteracyjne. W takich sytuacjach wykorzystujemy na ogó³ jakiœ program komputerowy do analizy statystycznej, np. zawarty w pakiecie SAS. Po podaniu postaci funkcji wiarygodnoœci, program komputerowy dokonuje jej maksymalizacji rozwi¹zuj¹c uk³ad (\[rown wiaryg\]) np. metod¹ Newton-Raphson’a [@Pawitan; @Mroz], wyznaczaj¹c numerycznie wartoœci estymatorów parametrów modelu.\ \ [**Testy statystyczne**]{}: Logarytm ilorazu wiarygodnoœci jest równie¿ wykorzystywany w analizie regresji do przeprowadzania testów statystycznych przy weryfikacji hipotez o nie wystêpowaniu braku dopasowania modelu mniej z³o¿onego, tzw. “ni¿szego”, o mniejszej liczbie parametrów, w stosunku do bardziej z³o¿onego modelu “wy¿szego”, posiadaj¹cego wiêksz¹ liczbê parametrów. Statystyka wykorzystywana do tego typu testów ma postaæ [@Kleinbaum; @Pawitan; @Mroziakiewicz]: $$\begin{aligned} \label{prawie dewiancja} -2\ln\frac{{P\left({\widetilde{Y}|\hat{\Theta}_{1}}\right)}}{{P\left({\widetilde{Y}|\hat{\Theta}_{2}}\right)}}\end{aligned}$$ gdzie ${P\left({\widetilde{Y}|\hat{\Theta}_{1}}\right)}$ jest maksymalizowan¹ wartoœci¹ funkcji wiarygodnoœci dla modelu mniej z³o¿onego, a ${P\left({\widetilde{Y}|\hat{\Theta}_{2}}\right)}$ dla modelu bardziej z³o¿onego. Przy prawdziwoœci hipotezy zerowej $H_{0}$ o braku koniecznoœci rozszerzania modelu ni¿szego do wy¿szego, statystyka (\[prawie dewiancja\]) ma asymptotycznie rozk³ad $\chi^{2}$ z liczb¹ stopni swobody równ¹ ró¿nicy liczby parametrów modelu wy¿szego i ni¿szego.\ \ [**Analogia wspó³czynnika determinacji**]{}: Maksymalizowana wartoœæ funkcji wiarygodnoœci zachowuje siê podobnie jak [*wspó³czynnik determinacji*]{} $R^{2}$ [@Kleinbaum; @Mroz], tzn. roœnie wraz ze wzrostem liczby parametrów w modelu, zatem wielkoœæ pod logarytmem nale¿y do przedzia³u $\left(0,1\right)$ i statystyka (\[prawie dewiancja\]) przyjmuje wartoœci z przedzia³u $\left(0,+\infty\right)$. St¹d (asymptotycznie) zbiór krytyczny dla $H_{0}$ jest prawostronny. Im lepiej wiêc model wy¿szy dopasowuje siê do danych empirycznych w stosunku do modelu ni¿szego, tym wiêksza jest wartoœæ statystyki ilorazu wiarygodnoœci (\[prawie dewiancja\]) i wiêksza szansa, ¿e wpadnie ona w przedzia³ odrzuceñ hipotezy zerowej $H_{0}$, który le¿y w prawym ogonie wspomnianego rozk³adu $\chi^{2}$ [@Kleinbaum; @Mroz]. ### Dewiancja jako miara dobroci dopasowania. Rozk³ad Poissona. {#Dewiancja jako miara dobroci dopasowania} Rozwa¿my zmienn¹ losow¹ $Y$ posiadaj¹c¹ rozk³ad Poissona. Rozk³ad ten jest wykorzystywany do modelowania zjawisk zwi¹zanych z rzadko zachodz¹cymi zdarzeniami, jak na przyk³ad z liczb¹ rozpadaj¹cych siê niestabilnych j¹der w czasie [*t*]{}. Ma on postaæ: $$\label{rozklad Poissona} p \left(Y={\bf y}|\mu \right)=\frac{\mu ^{{\bf y}} e^{-\mu } }{{\bf y}\, !} \; , \;\;\; {\rm oraz} \;\;\; {\bf y} = 0,1,...,\infty \; ,$$ gdzie $\mu $ jest parametrem rozk³adu. Zmienna losowa podlegaj¹ca rozk³adowi Poissona mo¿e przyj¹æ tylko nieujemn¹ wartoœæ ca³kowit¹. Rozk³ad ten mo¿na wyprowadziæ z rozk³adu dwumianowego, b¹dŸ wykorzystuj¹c rozk³ady Erlanga i wyk³adniczy [@Nowak].\ \ [**Zwi¹zek wariancji z wartoœci¹ oczekiwan¹ rozk³ad Poissona**]{}: Rozk³ad Poissona posiada pewn¹ interesuj¹c¹ w³aœciwoœæ statystyczn¹, mianowicie jego wartoœæ oczekiwana, wariancja i trzeci moment centralny s¹ równe parametrowi rozk³adu $\mu$: $$\begin{aligned} \label{E sigma trzeci Poissona} E(Y) = \sigma ^{2} (Y) = \mu _{3} =\mu \; .\end{aligned}$$ Aby pokazaæ dwie pierwsze równoœci w (\[E sigma trzeci Poissona\]) skorzystajmy bezpoœrednio z definicji odpowiednich momentów, otrzymuj¹c: $$\begin{aligned} \label{E Y dla rozkl Poissona} E\left(Y\right)& =& \sum _{{\bf y}=0}^{\infty }{\bf y}\cdot p\left(Y={\bf y}|\mu \right) =\sum _{{\bf y}=0}^{\infty }{\bf y}\cdot \frac{\mu ^{{\bf y}} e^{-\mu } }{{\bf y}!} = e^{-\mu } \sum _{{\bf y}=1}^{\infty }\frac{\mu ^{{\bf y}} }{\left({\bf y}-1\right)!} \nonumber \\ & = & e^{-\mu } \mu \sum _{{\bf y}=1}^{\infty }\frac{\mu ^{{\bf y}-1} }{\left({\bf y}-1\right)!} =e^{-\mu } \mu \sum _{l=0}^{\infty }\frac{\mu ^{l} }{l!} = e^{-\mu } \mu \, e^{\mu } = \mu \; , \end{aligned}$$ oraz, korzystaj¹c z (\[E Y dla rozkl Poissona\]): $$\begin{aligned} \label{sigma2 dla rozkl Poissona} & &\sigma^{2} \left(Y\right) = E\left(Y^{2} \right)-\left[E\left(Y\right)\right]^{2} = E\left(Y^{2} \right) - \mu^{2} =\sum _{{\bf y}=0}^{\infty }{\bf y}^{2} \cdot p\left(Y={\bf y}|\mu \right) - \mu^{2} \nonumber \\ & & = \sum _{{\bf y}=0}^{\infty }{\bf y}^{2} \cdot \frac{\mu ^{{\bf y}} e^{-\mu } }{{\bf y}!} - \mu^{2} = e^{-\mu } \sum _{{\bf y}=1}^{\infty }{\bf y}\frac{\mu ^{{\bf y}} }{\left({\bf y}-1\right)!} - \mu^{2} = \, e^{-\mu } \mu \sum _{l=0}^{\infty }\left(l+1\right)\frac{\mu ^{l} }{l!} - \mu^{2} \nonumber \\ & & = e^{-\mu } \mu \, \left[\sum _{l=0}^{\infty }l\frac{\mu ^{l} }{l!} + \, e^{\mu } \right] - \mu^{2} = e^{-\mu } \mu \, \left[e^{\mu } \mu +e^{\mu } \right] - \mu^{2} = (\mu ^{2} +\mu ) - \mu^{2} = \mu \; . \end{aligned}$$\ [**Uwaga**]{}: [*Zatem otrzymaliœmy wa¿n¹ w³asnoœæ rozk³adu Poissona*]{}, która mówi, ¿e stosunek dyspersji $\sigma$ do wartoœci oczekiwanej $E(Y)$ maleje pierwiastkowo wraz ze wzrostem poziomu zmiennej $Y$ opisanej tym rozk³adem: $$\begin{aligned} \label{sigma do E Poissona} \frac{\sigma}{E(Y)} = \frac{1}{\sqrt{\mu}} \;\; .\end{aligned}$$ Fakt ten oznacza z za³o¿enia [*inne zachowanie siê odchylenia standardowego*]{} w modelu regresji Poissona ni¿ w klasycznym modelu regresji normalnej (w którym zak³adamy jednorodnoœæ wariancji zmiennej objaœnianej w ró¿nych wariantach zmiennej objaœniaj¹cej).\ \ [**Æwiczenie**]{}: Pokazaæ (\[E sigma trzeci Poissona\]) dla trzeciego momentu.\ \ [**Przyczyna nielosowej zmiany wartoœci zmiennej objaœnianej**]{}: Rozwa¿my model regresji dla zmiennej objaœnianej $Y$ posiadaj¹cej rozk³ad Poissona. Zmienne $Y_{n}$, $n=1, 2,...,N$ posiadaj¹ wiêc równie¿ rozk³ad Poissona i zak³adamy, ¿e s¹ [*parami wzajemnie niezale¿ne*]{}. Niech $X$ jest zmienn¹ objaœniaj¹c¹ (tzw. czynnikiem) kontrolowanego eksperymentu, w którym $X$ nie jest zmienn¹ losow¹, ale [*jej zmiana*]{}, jest rozwa¿ana jako mo¿liwa przyczyna warunkuj¹ca [*nielosow¹ zmianê wartoœci zmiennej*]{} $Y$.\ Gdy czynników $X_{1} ,X_{2} ,...X_{k}$ jest wiêcej, wtedy dla ka¿dego punktu $n$ próby podane s¹ wszystkie ich wartoœci: $$\label{wartoœci czynnikow x} x_{1n} ,x_{2n} ,...x_{kn} \; , \;\;\; {\rm gdzie}\;\;\; n=1,2,...,N\; ,$$ gdzie pierwszy indeks w $x_{in}$, $i=1,2,...,k$, numeruje zmienn¹ objaœniaj¹c¹.\ \ [**Brak mo¿liwoœci eksperymentalnej separacji podstawowego kana³u $n$**]{}: Niech $x_{n} = (x_{1n} ,x_{2n} ,...,$ $x_{kn})$ oznacza zbiór wartoœci jednego wariantu zmiennych $\left(X_{1} ,X_{2} ,...,X_{k} \right)$, tzn. dla jednej konkretnej podgrupy $n$. Zwróæmy uwagê, ¿e [*indeks próby*]{} $n$ numeruje podgrupê, co oznacza, ¿e w pomiarze wartoœci $Y_{n}$ nie ma mo¿liwoœci eksperymentalnego siêgniêcia “w g³¹b” indeksu $n$ - tego kana³u, tzn. do rozró¿nienia wp³ywów na wartoœæ ${\bf y}_{n}$ p³yn¹cych z ró¿nych “pod-kana³ów” $i$, gdzie $i=1,2,...,k$.\ \ [**Model podstawowy**]{}: Zak³adaj¹c brak zale¿noœci zmiennej $Y$ od czynników $X_{1} ,X_{2} ,...X_{k}$, rozwa¿a siê tzw. [*model podstawowy*]{}. Dla rozk³adu (\[rozklad Poissona\]) i próby $\widetilde{Y} \equiv (Y_{n})_{n=1}^{N}$, funkcja wiarygodnoœci przy parametrze $\Theta = \mu \equiv (\mu_{n})_{n=1}^{N}$, ma postaæ: $$\label{f wiaryg Poissona dla modelu podstawowego} P\left(\widetilde{Y}|\mu \right) = \prod_{n=1}^{N} \frac{\mu_{n}^{Y_{n} } e^{-\mu_{n} } }{Y_{n} !} = \frac{\left(\prod_{n=1}^{N}\mu_{n}^{Y_{n} } \right)\exp \left(-\sum_{n=1}^{N} \mu_{n} \right)}{\prod_{n=1}^{N} Y_{n} ! } \; ,$$ jest wiêc wyra¿ona jako funkcja wektorowego parametru $\mu \equiv (\mu_{n})_{n=1}^{N}$, gdzie ka¿dy z parametrów $\mu_{n} = E(Y_{n})$ jest parametrem skalarnym. Rozwa¿my uk³ad równañ MNW: $$\label{uklad MNW rozklad Poissona podstawowy} \frac{\partial }{\partial \mu _{n} } \left[\ln P \left(\widetilde{Y}|\mu \right) \right] = 0 \; , \;\;\; n=1,2,...,N \; .$$ Dla funkcji wiarygodnoœci (\[f wiaryg Poissona dla modelu podstawowego\]) otrzymujemy: $$\label{log f wiaryg Poissona dla modelu podstawowego} \ln P \left(\widetilde{Y}|\mu\right)=\sum_{n=1}^{N} Y_{n} \ln \mu_{n} - \sum_{n=1}^{N} \mu_{n} - \sum_{n=1}^{N} \ln Y_{n} ! \;\, .$$ Zatem rozwi¹zanie uk³adu (\[uklad MNW rozklad Poissona podstawowy\]) daje: $$\begin{aligned} \label{estymatory modelu podst dla Poissona} \mu_{n} = \hat{\mu}_{n} = Y_{n} \; , \;\;\; n=1,2,...,N \; ,\end{aligned}$$ jako estymatory modelu postawowego. Zatem funkcja wiarygodnoœci (\[f wiaryg Poissona dla modelu podstawowego\]) modelu podstawowego przyjmuje w punkcie $\mu$ zadanym przez estymatory (\[estymatory modelu podst dla Poissona\]) wartoœæ maksymaln¹: $$\label{wiaryg zmaksym rozklad Poissona model podstawowy} P \left(\widetilde{Y}|\hat{\mu}\right)=\frac{\left(\prod_{n=1}^{N} Y_{n}^{Y_{n}} \right)\exp \left(-\sum _{n=1}^{N} Y_{n} \right)}{\prod _{n=1}^{N} Y_{n} ! } \; ,$$ gdzie zastosowano oznaczenie $\hat{\mu }=\left(\hat{\mu }_{1} ,\hat{\mu }_{2} ,...\hat{\mu }_{N} \right)$. ### Analiza regresji Poissona. {#Analiza regresji Poissona} Niech zmienna zale¿na $Y$ reprezentuje liczbê zliczeñ badanego zjawiska (np. przypadków awarii okreœlonego zakupionego sprzêtu), otrzyman¹ dla ka¿dej z $N$ podgrup (np. klienckich). Ka¿da z tych podgrup wyznaczona jest przez komplet wartoœci zmiennych objaœniaj¹cych $X \equiv \left(X_{1} ,X_{2} ,...,X_{k} \right) = x \equiv \left(x_{1} ,x_{2} ,...,x_{k} \right)$ (np. wiek, poziom wykszta³cenia, cel nabycia sprzêtu). Zmienna $Y_{n} $ okreœla liczbê zliczeñ zjawiska w $n$-tej podgrupie, $n=1,2,...,N$. W konkretnej próbce $(Y_{n})_{n=1}^{N} = ({\bf y}_{n})_{n=1}^{N}$.\ \ [**Okreœlenie modelu regresji Poissona**]{}: Rozwa¿my nastêpuj¹cy model regresji Poissona: $$\label{regresja Poisson} \mu_{n} \equiv E\left(Y_{n} \right) = \ell_{n} \, r\left(x_{n}, \beta \right) \; , \;\;\; n=1,2,...,N \; ,$$ opisuj¹cy zmianê wartoœci oczekiwanej liczby zdarzeñ $Y_{n}$ (dla rozk³adu Poissona) wraz ze zmian¹ [*wariantu*]{} $x_{n} = \left(x_{1n} ,x_{2n} ,...,x_{kn} \right)$.\ \ Funkcja regresji po prawej stronie (\[regresja Poisson\]) ma dwa czynniki. Funkcyjny czynnik funkcji regresji, $r\left(x_{n}, \beta \right)$, opisuje [*tempo zdarzeñ*]{} okreœlanych mianem pora¿ek (np. awarii) w $n$-tej podgrupie (tzn. jest czêstoœci¹ tego zjawiska), sk¹d $r\left(x_{n}, \beta \right)>0$, gdzie $\beta \equiv \left(\beta _{0} ,\beta _{1} ,...,\beta _{k} \right)$ jest zbiorem nieznanych parametrów tego modelu regresji. Natomiast czynnik $\ell_{n}$ jest wspó³czynnikiem okreœlaj¹cym [*dla ka¿dej $\,n$-tej podgrupy*]{} (np. klientów) [*skumulowany czas prowadzenia badañ kontrolnych dla wszystkich jednostek tej podgrupy*]{}.\ Poniewa¿ funkcja regresji[^8] $r\left(x_{n}, \beta \right)$ przedstawia typow¹ liczbê pora¿ek na jednostkê czasu, zatem nazywamy j¹ [*ryzykiem*]{}.\ \ [**Uwaga o postaci funkcji regresji**]{}: Funkcjê $r\left(x_{n}, \beta \right)$ mo¿na zamodelowaæ na ró¿ne sposoby [@Pawitan]. WprowadŸmy oznaczenie: $$\begin{aligned} \label{oznaczenie dla lambda regresji} \lambda_{n}^{*} \equiv \beta _{0} + \sum _{j=1}^{k}\beta _{j} \, x_{jn} \; . \end{aligned}$$ Funkcja regresji $r\left(x_{n}, \beta \right)$ ma ró¿n¹ postaæ w zale¿noœci od typu danych. Mo¿e mieæ ona postaæ charakterystyczn¹ dla regresji liniowej (wielokrotnej), $r\left(x_{n}, \beta \right) = \lambda_{n}^{*}$, któr¹ stosujemy szczególnie wtedy gdy zmienna $Y$ ma [*rozk³ad normalny*]{}. Postaæ $r\left(x_{n} \beta \right) = 1/\lambda_{n}^{*}$ jest stosowana w analizie z danymi pochodz¹cymi z [*rozk³adu eksponentialnego*]{}, natomiast $r\left(x_{n}, \beta \right) = 1/(1+ \exp(-\lambda_{n}^{*}))$ w modelowaniu regresji logistycznej dla opisu zmiennej [*dychotomicznej*]{} [@Kleinbaum; @Pawitan].\ \ [**Postaæ funkcji regresji u¿yteczna w regresji Poissona**]{} jest nastêpuj¹ca: $$\begin{aligned} \label{funkcja reg Poissona} r\left(x_{n}, \beta \right) = \exp(\lambda_{n}^{*}) \; , \;\;\; \lambda_{n}^{*} = \beta _{0} +\sum _{j=1}^{k}\beta _{j} x_{jn} \; .\end{aligned}$$ Ogólniej mówi¹c analiza regresji odnosi siê do modelowania wartoœci oczekiwanej zmiennej zale¿nej (objaœnianej) jako funkcji pewnych czynników. Postaæ funkcji wiarygodnoœci stosowanej do estymacji wspó³czynników regresji $\beta$ odpowiada za³o¿eniom dotycz¹cym rozk³adu zmiennej zale¿nej. Tzn. zastosowanie konkretnej funkcji regresji $r(x_{n}, \beta )$, np. jak w (\[funkcja reg Poissona\]), wymaga okreœlenia postaci funkcji czêstoœci $r(x_{n}, \beta )$, zgodnie z jej postaci¹ dobran¹ do charakteru losowej zmiennej $Y$ przy której generowane s¹ dane w badanym zjawisku. Na ogó³ przy konstrukcji $r(x_{n}, \beta )$ pomocna jest uprzednia wiedza dotycz¹c¹ relacji miêdzy rozwa¿anymi zmiennymi.\ \ [**Funkcja wiarygodnoœci dla analizy regresji Poissona**]{}: Poniewa¿ $Y_{n} $ ma rozk³ad Poissona (\[rozklad Poissona\]) ze œredni¹ $\mu _{n}$, $p\left(Y_{n}|\mu_{n} \right) = \frac{\mu _{n}^{Y_{n} } }{Y_{n} !} \, e^{-\mu _{n} }$, $n=1,\, 2,...,N$, zatem dane $Y_{n} =0, 1,...,\infty $ dla okreœlonego $n=1, 2,...,N$ s¹ generowane z rozk³adów warunkowych: $$\label{prawd Poissona model regresji z li} p\left(Y_{n}| \beta \right)=\frac{\left[\ell_{n} \, r\left(x_{n}, \beta \right)\right]^{\, Y_{n} } }{Y_{n} !} e^{-\ell_{n}\, r\left(x_{n}, \beta \right)} \; ,$$ wokó³ funkcji regresji, (\[regresja Poisson\]), $\mu_{n} = \ell_{n} \, r(x_{n}, \beta )$, dla $n=1, 2,...,N$. Funkcja wiarygodnoœci dla analizy regresji Poissona ma wiêc postaæ: $$\begin{aligned} \label{funkcja wiaryg regresja Poisson} P \left(\widetilde{Y}|\beta \right) &=& \prod_{n=1}^{N} p\left(Y_{n}|\beta \right) = \prod _{n=1}^{N}\frac{\left(\ell_{n} \, r\left(x_{n}, \beta \right)\right)^{Y_{n} } e^{-\ell _{n} \, r\left(x_{n}, \beta \right)} }{Y_{n} !} \nonumber \\ &=& \frac{ \prod _{n=1}^{N}\left(\ell _{n} \, r\left(x_{n}, \beta \right) \right)^{Y_{n} } \, \exp \left[ -\sum _{n=1}^{N} \ell_{n} \, r\left(x_{n}, \beta \right) \right] }{\prod_{n=1}^{N} Y_{n} ! } \; .\end{aligned}$$\ Aby w praktyce pos³u¿yæ siê funkcj¹ regresji $r(x_{n}, \beta )$ bêd¹c¹ okreœlon¹ funkcj¹ zmiennej $\lambda_{n}^{*} = \beta_{0} + \sum_{j=1}^{k} \beta_{j} x_{jn}$, parametry $\beta_{0}, \beta_{1} ,...,\beta_{k}$ musz¹ byæ oszacowane. Estymatory MNW, $\hat{\beta}_{0} ,\hat{\beta}_{1} ,..., \hat{\beta}_{k}$, tych parametrów otrzymuje siê rozwi¹zuj¹c $k+1$ równañ wiarygodnoœci: $$\begin{aligned} \label{ukl row MNW dla beta fun regersji} \frac{\partial}{\partial \beta_{j} } \ln P \left(\widetilde{Y}| \beta \right) = 0 \; , \;\;\; j=0 , 1, 2,...,k \; .\end{aligned}$$ W przypadku regresji Poissona $ P\left(\widetilde{Y}| \beta \right) $ jest okreœlona zgodnie z (\[funkcja wiaryg regresja Poisson\]).\ \ [**Algorytmy IRLS**]{}: Zauwa¿my, ¿e dla rozk³adu Poissona zachodzi zgodnie z (\[E sigma trzeci Poissona\]) oraz (\[regresja Poisson\]), $\sigma^{2} \left(Y_{n} \right) = E\left(Y_{n} \right) = \ell_{n} \, r\left(x_{n}, \beta \right)$, [*co oznacza, ¿e wariancja $\sigma^{2} \left(Y_{n} \right)$ zmiennej objaœnianej nie jest sta³a lecz zmienia siê jako funkcja $\ell_{n} $ oraz $x_{n} $, wchodz¹c w analizê z ró¿nymi wagami wraz ze zmian¹ $n$*]{}. Na fakt ten zwróciliœmy ju¿ uwagê przy okazji zwi¹zku (\[sigma do E Poissona\]). Poniewa¿ uk³ad równañ wiarygodnoœci (\[ukl row MNW dla beta fun regersji\]) jest na ogó³ rozwi¹zywany iteracyjnymi metodami numerycznymi [@Kleinbaum], a wariancja $\sigma^{2} \left(Y_{n} \right)$ jest równie¿ funkcj¹ $\beta$, zatem na ka¿dym kroku procesu iteracyjnego [*wagi*]{} te zmieniaj¹ siê jako funkcja zmieniaj¹cych siê sk³adowych estymatora $\hat{\beta}$. Algorytmy takiej analizy okreœla siê ogólnym mianem [*algorytmów najmniejszych kwadratów[^9] iteracyjnie wa¿onych*]{} (IRLS).\ \ [**Uwaga o programach**]{}: Ró¿ne programy do analiz statystycznych, w tym SAS wykorzystuj¹cy procedurê PROC GENMOD, mog¹ byæ u¿yte do znajdowania estymatorów $\hat{\beta }$ MNW dla funkcji wiarygodnoœci . Równie¿ [*obserwowana macierz kowariancji estymatorów*]{}[^10] oraz miary dobroci dopasowania modelu, takie jak omówiona dalej dewiancja, mog¹ byæ otrzymane przy u¿yciu powy¿ej wspomnianych programów. #### Test statystyczny dla doboru modelu w regresji Poissona {#Testy statystyczne doboru modelu} [**Uwaga o wiêkszej wiarygodnoœci modelu podstawowego**]{}: Maksymalna wartoœæ funkcji wiarygodnoœci $P \left(y|\mu \right)$ wyznaczona w oparciu o bêdzie, dla ka¿dego zbioru danych i dla liczby parametrów $k+1< N $, wiêksza ni¿ otrzymana przez maksymalizacjê funkcji wiarygodnoœci . Jest tak, poniewa¿ w wyra¿eniu na funkcjê wiarygodnoœci modelu podstawowego [*nie narzuca siê ¿adnych ograniczeñ na postaæ*]{} $\mu _{n} $, natomiast wymaga aby $\mu_{n} =\ell _{n} r\left(x_{n}, \beta \right)$.\ \ [**Hipoteza zerowa o nie wystêpowaniu braku dopasowania w modelu ni¿szym**]{}: Zgodnie z powy¿szym zdaniem, analizê doboru modelu regresji mo¿na rozpocz¹æ od postawienia hipotezy zerowej wobec alternatywnej. W hipotezie zerowej wyró¿nimy proponowany model regresji. Wybór modelu badanego oznacza wybór funkcji wiarygodnoœci z nim zwi¹zanej. Stawiamy wiêc hipotezê zerow¹: $$\label{Ho dla regresji Poissona} {\rm H}_{0} :\mu _{n} =\ell _{n} r\left(x_{n}, \beta \right), n=1, 2,...,N,$$ która odpowiada wyborowi modelu z funkcj¹ wiarygodnoœci , wobec hipotezy alternatywnej: $$\label{H1 dla regresji Poissona} {\rm H}_{A}: \;\; \mu _{n} \; {\rm nie \; ma \; ograniczonej\; postaci}, \; n=1, 2,...,N \; ,$$ która odpowiada wyborowi modelu podstawowego zawieraj¹cego tyle parametrów $\mu_{n}$ ile jest punktów pomiarowych, tzn. *N*, z funkcj¹ wiarygodnoœci .\ \ Niech wiêc $P \left(\widetilde{Y}|\hat{\beta }\right)$ jest maksymaln¹ wartoœci¹ funkcji wiarygodnoœci okreœlon¹ jak w . Oznacza to, ¿e w miejsce parametrów $\beta = \left(\beta _{0} ,\beta _{1} ,...,\beta _{k} \right)$ podstawiono ich estymatory $\hat{\beta } = \left( \hat{\beta}_{0}, \hat{\beta}_{1},..., \hat{\beta}_{k} \right)$ wyznaczone przez MNW, jako te które maksymalizuj¹ funkcjê wiarygodnoœci . Podobnie rozumiemy funkcjê wiarygodnoœci $P\left(\widetilde{Y}|\hat{\mu }\right)$ modelu podstawowego.\ \ Poniewa¿ celem ka¿dej analizy jest otrzymanie mo¿liwie najprostszego opisu danych, model $\mu _{n} =\ell _{n} r\left(x_{n}, \beta \right)$ zawieraj¹cy $k+1$ parametrów $\beta $, bêdzie uznany za dobry, jeœli maksymalna wartoœæ funkcji wiarygodnoœci wyznaczona dla niego, bêdzie prawie tak du¿a, jak funkcji wiarygodnoœci dla nie nios¹cego ¿adnej informacji modelu podstawowego z liczb¹ parametrów $\mu_{n}$ równ¹ licznie punktów pomiarowych $N$. Sformu³owanie „prawie tak du¿a” oznacza, ¿e wartoœæ funkcji wiarygodnoœci $P(y|\hat{\beta })$ nie mo¿e byæ istotnie statystycznie mniejsza od $P\left(y|\hat{\mu }\right)$. Zasadniczo powinno to oznaczaæ, ¿e musimy podaæ miary pozwalaj¹ce na okreœlenie statystycznej istotnoœci przy pos³ugiwaniu siê intuicyjnym parametrem obciêcia $c$ (Rozdzia³ \[Wnioskowanie w MNW\]). Okazuje siê, ¿e dla du¿ej próby, miary typu (\[stat ilorazu wiaryg modelu reg\]), podane poni¿ej, uzyskuj¹ cechy pozwalaj¹ce na budownie wiarygodnoœciowych obszarów krytycznych nabywaj¹cych charakteru standardowego (czêstotliwoœciowego).\ \ [**Okreœlenie dewiancji**]{}: WprowadŸmy [*statystykê typu ilorazu wiarygodnoœci*]{}: $$\label{stat ilorazu wiaryg modelu reg} D\left(\hat{\beta }\right) = -2\ln \left[\frac{P(\widetilde{Y}|\hat{\beta })}{P\left(\widetilde{Y}|\hat{\mu }\right)} \right]$$ nazywan¹ *dewiancj¹* (deviance) dla modelu regresji, w tym przypadku dla modelu Poissona z okreœlon¹ postaci¹ $\mu _{n} =\ell _{n} r\left(x_{n} , \beta \right)$. S³u¿y ona do badania dobroci dopasowania modelu z zadan¹ postaci¹ $\mu_{n} =\ell _{n} r\left(x_{n}, \beta \right)$ w stosunku do modelu podstawowego, bez narzuconej postaci na $\mu _{n} $, tzn. do stwierdzenia, czy $P(y|\hat{\beta })$ jest istotnie [*mniejsza*]{} od $P\left(y|\hat{\mu }\right)$, co sugerowa³oby istotny statystycznie brak dopasowania badanego modelu $\mu _{n} =\ell _{n} r\left(x_{n}, \beta \right)$, do danych empirycznych. Jak poka¿emy poni¿ej dewiancja mo¿e byæ rozumiana jako *miara zmiennoœci reszt* (tzn. odchylenia wartoœci obserwowanych w próbie od wartoœci szacowanych przez model) *wokó³ linii regresji*.\ \ Przy prawdziwoœci hipotezy $H_{0} :\mu _{n} =\ell _{n} r\left(x_{n}, \beta \right)$, rozk³ad dewiancji $D\left(\hat{\beta }\right)$ dla regresji Poissona, mo¿na asymptotycznie przybli¿yæ rozk³adem chi-kwadrat (por. dyskusja w [@Pawitan; @Kleinbaum]) z $N-k-1$ stopniami swobody.\ \ Zatem statystyczny test dobroci dopasowania, tzn. niewystêpowania braku dopasowania badanego modelu $H_{0}: \mu_{n} =\ell _{n} r\left(x_{n} , \beta \right)$, w stosunku do modelu podstawowego, przebiega w regresji Poissona nastêpuj¹co: Porównujemy otrzyman¹ w próbie wartoœæ statystyki $D(\hat{ \beta })$ z wartoœci¹ krytyczn¹ le¿¹c¹ w prawym ogonie rozk³adu chi-kwadrat (o $N-k-1$ stopniach swobody). Przyjêcie przez $D\left(\hat{\beta }\right)$ wartoœci równej lub wiêkszej od krytycznej skutkuje odrzuceniem hipotezy zerowej.\ \ [**Wyznaczenie liczby stopni swobody dewiancji**]{}: Podana liczba stopni swobody dewiancji $D\left(\hat{\beta }\right)$ wynika z nastêpuj¹cego rozumowania. Zapiszmy (\[stat ilorazu wiaryg modelu reg\]) w postaci: $$\label{stat ilorazu wiaryg modelu reg 2} D\left(\hat{\beta }\right) + 2\ln P\left(\widetilde{Y}|\hat{\beta }\right) = 2\ln P\left(\widetilde{Y}|\hat{\mu }\right) \; ,$$ co po skorzystaniu z (\[funkcja wiaryg regresja Poisson\]) dla $\beta=\hat{\beta}$ ma postaæ: $$\label{stat ilorazu wiaryg modelu reg 3} D\left(\hat{\beta }\right) - 2\sum _{n=1}^{N} \ell _{n} r\left(x_{n}, \hat{ \beta }\right) = 2\ln P \left(\widetilde{Y}|\hat{\mu }\right) + 2\ln \left(\prod_{n=1}^{N} Y_{n}! \right) - 2\ln \left(\prod _{n=1}^{N} \left(\ell _{n} r \left(x_{n} ,\hat{\beta }\right)\right)^{Y_{n}} \right) \; .$$ Mo¿na zauwa¿yæ, ¿e prawa strona tego równania ma $N$-stopni swobody. Istotnie, ze wzglêdu na (\[estymatory modelu podst dla Poissona\])[^11], $\hat{\mu} \equiv (\hat{\mu}_{n}) = (Y_{n}) \,$, $n=1,2,...,N$, liczba niezale¿nych zmiennych po prawej strony powy¿szego równania, których wartoœci trzeba okreœliæ z eksperymentu, wynosi $N$. Natomiast drugi sk³adnik po lewej stronie ma liczbê stopni swobody równ¹ $k+1$, co jest liczb¹ estymatorów parametrów strukturalnych $\hat{ \beta }$ modelu regresji, których wartoœci trzeba okreœliæ z eksperymentu. Poniewa¿ liczba stopni swobody po prawej i lewej stronie równania musi byæ taka sama, zatem liczba stopni swobody dewiancji $D(\hat{\beta })$ wynosi $N-k-1$.\ \ [**Testy ilorazu wiarygodnoœci**]{}: Dewiancje dla hierarchicznych klas modeli mog¹ s³u¿yæ do budowy testów stosunku wiarygodnoœci. Zwróæmy szczególnie uwagê na funkcjê wiarygodnoœci zawieraj¹c¹ zbiór parametrów $\beta =\left(\beta_{0} ,\beta_{1} ,.....,\beta_{k} \right)$ z *dewiancj¹* $D\left(\hat{\beta }\right)$ dan¹ wyra¿eniem . Przypuœæmy, ¿e chcemy zweryfikowaæ hipotezê o tym, ¿e $k-r$ (gdzie $0<r<k$) ostatnich parametrów bêd¹cych sk³adowymi wektora $\beta $ jest równych [*zeru*]{}.\ \ [*Hipoteza zerowa*]{}, o nieistotnoœci rozszerzenia modelu ni¿szego do wy¿szego, ma wtedy postaæ: $$\label{hip zerowa w testach hierarchicznych} {\rm H}_{0}: \beta _{r+1} =\beta _{r+2} =...=\beta _{k} = 0 \; ,$$ [*Hipoteza alternatywna*]{} $H_{A}$ mówi, ¿e przynajmniej jeden z parametrów strukturalnych $\beta_{r+1}, \beta_{r+2},$ $...,\beta_{k}\,$ jest ró¿ny od [*zera*]{}.\ \ Funkcja wiarygodnoœci przy prawdziwoœci hipotezy zerowej $H_{0}$, , ma postaæ tak¹ jak w , tyle, ¿e zast¹piono w niej parametr $\beta$ parametrem $\beta_{(r)}$: $$\label{parametr modelu nizszego} \beta_{(r)} \equiv \left(\beta _{0} ,\beta _{1} ,...,\beta _{r}; \, 0,0,...,0 \right) \;\;{\rm gdzie \;\, liczba \;\, zer \;\, wynosi} \;\, k-r \; .$$ Oznaczmy funkcje wiarygodnoœci tego modelu jako $P(\widetilde{Y}|\beta_{(r)})$, a $\hat{ \beta}_{(r)} $ niech bêdzie estymatorem MNW wektorowego parametru $\beta_{(r)}$, wyznaczonym przez rozwi¹zanie odpowiadaj¹cego mu uk³adu równañ wiarygodnoœci (oczywiœcie dla niezerowych parametrów $\beta_{0} ,\beta_{1} ,...,\beta_{r}$). Estymator $\hat{ \beta}_{(r)} = (\hat{\beta}_{0}, \hat{\beta}_{1} ,...,$ $\hat{\beta }_{r};$ $0,0,...,0) $ maksymalizuje funkcjê wiarygodnoœci $P\left(\widetilde{Y}|\beta_{(r)} \right)$.\ \ [*Test ilorazu wiarygodnoœci*]{} dla weryfikacji hipotezy $H_{0} $ przeprowadzamy pos³uguj¹c siê *statystyk¹ ilorazu wiarygodnoœci*: $$\label{statystyka ilorazu wiaryg} - 2 \ln \left[\frac{P\left(\widetilde{Y}|\hat{\beta}_{(r)} \right)}{P\left(\widetilde{Y}|\hat{\beta }\right)} \right] \; ,$$ która przy prawdziwoœci hipotezy zerowej ma asymptotycznie rozk³ad chi-kwadrat z $k-r$ stopniami swobody, co widaæ, gdy zapiszemy (\[statystyka ilorazu wiaryg\]) jako ró¿nicê dewiancji: $$\label{roznica dewiancji} -2\ln \left[\frac{P\left(\widetilde{Y}|\hat{\beta}_{(r)} \right)}{P\left(\widetilde{Y}|\hat{\beta }\right)} \right] = - 2\ln \left[\frac{P\left(\widetilde{Y}|\hat{\beta}_{(r)} \right)}{P\left(\widetilde{Y}|\hat{\mu }\right)} \right] + 2\ln \left[\frac{P\left(\widetilde{Y}|\hat{\beta}\right)}{P\left(\widetilde{Y}|\hat{\mu }\right)} \right] = D\left(\hat{\beta}_{(r)} \right) - D\left(\hat{\beta }\right) \; ,$$ oraz skorzystamy z podobnej analizy jak dla (\[stat ilorazu wiaryg modelu reg 3\]).\ \ Zatem, przy prawdziwoœci hipotezy zerowej , któr¹ mo¿na zapisaæ jako ${\rm H}_{0}: \beta_{r+1} =\beta_{r+2} =...=\beta_{k} =0$, ró¿nica $D(\hat{\beta}_{(r)}) - D(\hat{\beta })$ ma dla du¿ej próby w przybli¿eniu rozk³ad chi-kwadrat z $k-r$ stopniami swobody.\ \ [**Wniosek**]{}: Jeœli u¿ywamy regresji Poissona do analizowania danych empirycznych, modele tworz¹ce hierarchiczne klasy mog¹ byæ porównywane miedzy sob¹ poprzez wyznaczenie statystyki ilorazu wiarygodnoœci (\[statystyka ilorazu wiaryg\]), lub co na jedno wychodzi, poprzez wyznaczenie ró¿nicy (\[roznica dewiancji\]) miêdzy parami dewiancji dla tych modeli. Nale¿y przy tym pamiêtaæ o wniosku jaki ju¿ znamy z analizy dewiancji, ¿e [*im model gorzej dopasowuje siê do danych empirycznych tym jego dewiancja jest wiêksza.*]{} #### Podobieñstwo dewiancji do SKR analizy czêstotliwoœciowej {#Podobienstwo dewiancji do SKR} Warunkowe wartoœci oczekiwane $\mu_{n} \equiv E(Y_{n}) = \ell _{n} \, r(x_{n}, \beta)$, $n=1,2,...,N$, (\[regresja Poisson\]), s¹ w analizie regresji przyjmowane jako teoretyczne przewidywania modelu regresji dla wartoœci zmiennej objaœnianej $Y_{n}$, zwanej odpowiedzi¹ (uk³adu). W próbie odpowiadaj¹ im oszacowania, oznaczone jako $\hat{Y}_{n}$, które w $n$-tej komórce s¹ nastêpuj¹ce: $$\begin{aligned} \label{przewidywania modelu regresji Poissona} \hat{Y}_{n} = \ell _{n} \, r(x_{n} ,\hat{\beta }) \; , \;\;\; n=1,2,...,N \; ,\end{aligned}$$ zgodnie z wyestymowan¹ postaci¹ modelu regresji. Wykorzystuj¹c (\[przewidywania modelu regresji Poissona\]) mo¿emy zapisaæ dewiancjê modelu (\[stat ilorazu wiaryg modelu reg\]) nastepuj¹co: $$\begin{aligned} \label{dewiancja poprzez przewidywania rachunek} D\left(\hat{ \beta }\right) &=& - 2 \ln \left[\frac{P\left(\widetilde{Y}|\hat{\beta}\right)}{P\left(\widetilde{Y}|\hat{\mu }\right)} \right] = -2 \ln \left[\frac{\prod _{n=1}^{N} \hat{Y}_{n}^{Y_{n} } \exp \left(-\sum _{n=1}^{N} \hat{Y}_{n} \right) }{\prod_{n=1}^{N}Y_{n}^{Y_{n} } \exp \left(-\sum _{n=1}^{N}Y_{n} \right) } \right] \nonumber \\ &=& -2\left[\sum _{n=1}^{N}Y_{n} \ln \hat{Y}_{n} - \sum_{n=1}^{N}\hat{Y}_{n} - \sum_{n=1}^{N}Y_{n} \ln Y_{n} + \sum_{n=1}^{N} Y_{n} \right]\end{aligned}$$ tzn: $$\begin{aligned} \label{dewiancja poprzez przewidywania} D\left(\hat{\beta }\right) = 2\sum _{n=1}^{N} \left[Y_{n} \ln \left(\frac{Y_{n} }{\hat{Y}_{n} } \right) - \left(Y_{n} -\hat{Y}_{n} \right)\right] \; . \end{aligned}$$ [**Podobieñstwo $D$ do $SKR$**]{}: Powy¿sza postaæ dewiancji oznacza, ¿e $D(\hat{ \beta })$ zachowuje siê w poni¿szym sensie jak suma kwadratów reszt $SKR = \sum_{n=1}^{N} (Y_{n} -\hat{Y}_{n})^{2}$ w standardowej wielorakiej regresji liniowej. Otó¿, gdy dopasowywany model dok³adnie przewiduje obserwowane wartoœci, tzn. $\hat{Y}_{n} =Y_{n} ,\; n=1,2,..,N$ wtedy, jak $SKR$ w analizie standardowej, tak $D(\hat{ \beta })$ w analizie wiarygodnoœciowej jest równe zeru [@Kleinbaum; @Czerwik]. Z drugiej strony wartoœæ $D(\hat{ \beta })$ jest tym wiêksza im wiêksza jest ró¿nica miêdzy wartoœciami obserwowanymi $Y_{n}$ i wartoœciami przewidywanymi $\hat{Y}_{n} $ przez oszacowany model.\ \ [**Asymptotyczna postaæ $D$**]{}: W analizowanym modelu $Y_{n}$, $n=1,2,...,N$ s¹ niezale¿nymi zmiennymi Poissona (np. zmiennymi czêstoœci), natomiast wartoœci $\hat{Y}_{n} $ s¹ ich przewidywaniami. Nietrudno przekonaæ siê, ¿e gdy wartoœci przewidywane maj¹ rozs¹dn¹ wartoœæ[^12], np. $\hat{Y}_{n} >3$ oraz $(Y_{n} -\hat{Y}_{n}) << Y_{n}\,$, $n=1,2,..,N\,$ tak, ¿e $(Y_{n} -\hat{Y}_{n})/Y_{n} << 1$, wtedy wyra¿enie w nawiasie kwadratowym w (\[dewiancja poprzez przewidywania\]) mo¿na przybli¿yæ przez $(Y_{n} -\hat{Y}_{n})^{2}/(2 \,Y_{n})$, a statystykê (\[dewiancja poprzez przewidywania\]) mo¿na przybli¿yæ statystyk¹ o postaci: $$\label{dewinacja jak chi} \chi ^{2} =\sum _{n=1}^{N}\frac{\left(Y_{n} -\hat{Y}_{n} \right)^{2} }{\hat{Y}_{n} } \; ,$$ która (dla du¿ej próby) ma rozk³ad chi-kwadrat z $N-k-1$ stopniami swobody [@Kleinbaum]. Zasada niezmienniczoœci ilorazu funkcji wiarygodnoœci ----------------------------------------------------- Z powy¿szych rozwa¿añ wynika, ¿e funkcja wiarygodnoœci reprezentuje niepewnoœæ dla ustalonego parametru. Nie jest ona jednak gêstoœci¹ rozk³adu prawdopodobieñstwa dla tego parametru. Pojêcie takie by³oby ca³kowicie obce statystyce klasycznej (nie w³¹czaj¹c procesów stochastycznych). Inaczej ma siê sprawa w tzw. statystyce Bayesowskiej. Aby zrozumieæ ró¿nicê pomiêdzy podejœciem klasycznym i Bayesowskim [@Marek_statyst_Bayes] rozwa¿my transformacjê parametru.\ [**Przyk³ad transformacji parametru**]{}: Rozwa¿my eksperyment, w którym dokonujemy jednokrotnego pomiaru zmiennej o rozk³adzie dwumianowym (\[Bernoulliego rozklad\]). Funkcja wiarygodnoœci ma wiêc postaæ $P(\theta)=\left(\!\! \begin{array}{l} m\\ x\end{array} \!\!\! \right) \theta^{x} (1-\theta)^{m-x}$. Niech parametr $m=12$ a w pomiarze otrzymano $x=9$. Testujemy model, dla którego $\theta = \theta_{1} = 3/4$ wobec modelu z $\theta = \theta_{2} = 3/10$. Stosunek wiarygodnoœci wynosi: $$\begin{aligned} \label{stosunek L skalowanie} \frac{P(\theta_{1} = 3/4)}{P(\theta_{2} = 3/10)} = \frac{\left(\begin{array}{l} m\\ x\end{array}\right) \theta_{1}^{9} \, (1-\theta_{1})^{3}}{\left(\begin{array}{l} m\\ x\end{array}\right) \theta_{2}^{9} \, (1-\theta_{2})^{3}} \; = \; 173.774\end{aligned}$$ Dokonajmy hiperbolicznego wzajemnie jednoznacznego przekszta³cenia parametru: $$\begin{aligned} \label{transf parametru} \psi = 1/\theta \; .\end{aligned}$$ Funkcja wiarygodnoœci po transformacji parametru ma postaæ $\tilde{P}(\psi)\!=\!\left( \!\! \begin{array}{l} m\\ x \end{array} \!\!\! \right) (1/\psi)^{x} (1-1/\psi)^{m-x}$. Wartoœci parametru $\psi$ odpowiadaj¹ce wartoœciom $\theta_{1}$ i $\theta_{2}$ wynosz¹ odpowiednio $\psi_{1}=4/3$ oraz $\psi_{2}=10/3$. £atwo sprawdziæ, ¿e transformacja (\[transf parametru\]) nie zmienia stosunku wiarygodnoœci, tzn.: $$\begin{aligned} \label{stosunek L po transformacji skalowanie} \frac{\tilde{P}(\psi_{1} = 4/3)}{\tilde{P}(\psi_{2} = 10/3)} = \frac{P(\theta_{1} = 3/4)}{P(\theta_{1} = 3/10)} = 173.774 \; .\end{aligned}$$\ [**Niezmienniczoœæ stosunku wiarygodnoœci**]{}: Zatem widaæ, ¿e stosunek wiarygodnoœci jest niezmienniczy ze wzglêdu na wzajemnie jednoznacz¹ transformacjê parametru. Gdyby transformacja parametru by³a np. transformacj¹ “logit” $\psi = \ln(\theta/(1 - \theta))$ lub paraboliczn¹ $\psi = \theta^{2}$, to sytuacja tak¿e nie uleg³aby zmianie. Równie¿ w ogólnym przypadku transformacji parametru w³asnoœæ [*niezmienniczoœci stosunku wiarygodnoœci*]{} pozostaje s³uszna. Oznacza to, ¿e informacja zawarta w próbce jest niezmiennicza ze wzglêdu na wybór parametryzacji, tzn. powinniœmy byæ w takiej samej sytuacji niewiedzy niezale¿nie od tego jak zamodelujemy zjawisko, o ile ró¿nica w modelowaniu sprowadza siê jedynie do transformacji parametru. W omawianym przyk³adzie powinniœmy równie dobrze móc stosowaæ parametr $\theta$, jak $1/\theta$, $\theta^{2}$, czy $\ln(\theta/(1 - \theta))$.\ \ [**Uwaga o transformacji parametru w statystyce Bayesowskiej**]{}: Natomiast sytuacja ma siê zupe³nie inaczej w przypadku Bayesowskiego podejœcia do funkcji wiarygodnoœci [@Marek_statyst_Bayes], w którym funkcja wiarygodnoœci uwzglêdnia (Bayesowski) rozk³ad prawdopodobieñstwa $f(\theta|x)$ parametru $\theta$. Oznacza to, ¿e Jakobian transformacji $\theta \rightarrow \psi$ parametru, modyfikuj¹c rozk³ad parametru, zmienia równie¿ funkcjê wiarygodnoœci. Zmiana ta zale¿y od wartoœci parametru, ró¿nie zmieniaj¹c licznik i mianownik w (\[stosunek L skalowanie\]), co niszczy [*intuicyjn¹*]{} w³asnoœæ niezmienniczoœci ilorazu wiarygodnoœci ze wzglêdu na transformacjê parametru [@Pawitan]. Entropia wzglêdna i informacja Fishera {#Entropia wzgledna i IF} ====================================== W pozosta³ych czêœciach skryptu nie bêdziemy zajmowali sie modelami regresyjnymi. Oszacowywany wektorowy parametr $\Theta$, od którego zale¿y funcja wiarygodnoœci $P(\Theta)$ ma postaæ $\Theta = (\theta_{1},\theta_{2},...,\theta_{N})^{T} \equiv (\theta_{n})_{n=1}^{N}$ jak w (\[parametr Theta\]), gdzie $\theta_{n} = (\vartheta_{1n},\vartheta_{2n},...,\vartheta_{kn})^{T} \equiv ((\vartheta_{s})_{s=1}^{k})_{n}$, sk¹d liczba wszystkich parametrów wynosi $d=k \times N$. [*Aby nie komplikowaæ zapisu bêdziemy stosowali oznaczenie*]{} $\Theta \equiv (\theta_{1},\theta_{2},...,\theta_{d})^{T} \equiv (\theta_{i})_{i=1}^{d}$, gdzie indeks $i=1,2,...,d$ zastapi³ parê indeksów $sn$.\ \ Niech $\hat{\Theta}=\left({\hat{\theta}_{1}, \hat{\theta}_{2},..., \hat{\theta}_{d}}\right)$ jest estymatorem MNW wektorowego parametru $\Theta=\left(\theta_{1}, \theta_{2},..., \theta_{d}\right)$, otrzymanym po rozwi¹zaniu uk³adu równañ wiarygodnoœci (\[rown wiaryg\]). Rozwi¹zanie to, jako maksymalizuj¹ce funkcjê wiarygodnoœci, musi spe³niaæ warunek ujemnej okreœlonoœci formy kwadratowej[^13]: $$\begin{aligned} \label{forma kw dla P} \sum_{i, \,j=1}^{d} \frac{\partial^{2} \ln P}{\partial \theta_{i} \partial \theta_{j}} {\left|_{\Theta = \hat{\Theta}} \right.} \Delta\theta_{i}\Delta\theta_{j} \; ,\end{aligned}$$ gdzie przyrosty $\Delta\theta_{i}$, $\Delta\theta_{j}$ nie zeruj¹ siê jednoczeœnie. W przypadku skalarnym (tzn. jednego parametru $\Theta=\theta=\vartheta$) warunek ten oznacza ujemnoœæ drugiej pochodnej logarytmu funkcji wiarygodnoœci w punkcie $\theta=\hat{\theta}$. Wiêksza wartoœæ $-\frac{{\partial^{2}}}{{\partial{\theta}^{2}}} \ln P\left(y|\theta\right)\left|_{\theta=\hat{\theta}} \right.$ oznacza wê¿sze maksimum $\ln P$ w punkcie $\theta=\hat{\theta}$, tzn. wiêksz¹ krzywiznê funkcji wiarygodnoœci, a co za tym idzie mniejsz¹ niepewnoœæ okreœlenia parametru $\theta$. Obserwowana i oczekiwana informacja Fishera {#iF oraz I_definicje} ------------------------------------------- Poniewa¿ $\Theta$ mo¿e byæ w ogólnoœci parametrem wektorowym, wiêc jako uogólnienie przypadku skalarnego zdefiniujmy $d \times d \,$-wymiarow¹ macierz[^14]: $$\begin{aligned} \label{I jako krzywizna dla P} \texttt{i\!F}(\Theta) \equiv -\frac{{\partial^{2}}}{{\partial{\Theta}} \partial{\Theta}^{T}} \ln P\left(y|\Theta\right) \equiv - \left( \frac{\partial^{2}\ln P}{\partial\theta_{i}\partial\theta_{j}} \right)_{d \times d} \; .\end{aligned}$$ [**Okreœlenie obserwowanej informacji Fishera**]{}: Wartoœæ statystyki $\texttt{i\!F}(\Theta)$ zdefiniowanej jak w (\[I jako krzywizna dla P\]) w punkcie $\Theta=\hat{\Theta}$ (oznaczon¹ po prostu jako $\texttt{i\!F}$): $$\begin{aligned} \label{I obserwowana} \texttt{i\!F} = \texttt{i\!F}\left(\hat{\Theta}\right) = \left. \texttt{i\!F}\left(\Theta \right)\right|_{\Theta=\hat{\Theta}} \; ,\end{aligned}$$ nazywamy [*obserwowan¹ informacj¹ Fishera*]{}. W teorii wiarygodnoœci odgrywa ona kluczow¹ rolê. Jako statystyka, czyli funkcja próby $\widetilde{Y}$, jej realizacja w próbce $y \equiv ({\bf y}_{1}, {\bf y}_{2}, ..., {\bf y}_{N})$ jest macierz¹ liczbow¹. Z faktu wyznaczenia $\texttt{i\!F}$ w punkcie estymatora MNW wynika, ¿e jest ona dodatnio okreœlona, natomiast z (\[I jako krzywizna dla P\]) widaæ równie¿, ¿e $\texttt{i\!F}$ jest macierz¹ symetryczn¹.\ \ We wspó³czesnych wyk³adach $\texttt{i\!F}$ jest zazwyczaj zapisane jako: $$\begin{aligned} \label{observed IF Amari} \widetilde{\texttt{i\!F}} = \left(\frac{\partial \ln P(\Theta)}{\partial\theta_{i'}}\frac{\partial \ln P(\Theta)}{\partial\theta_{i}}\right) \; . \end{aligned}$$ Obie definicje, tzn. (\[observed IF Amari\]) oraz (\[I jako krzywizna dla P\]), prowadz¹ na poziomie oczekiwanym do tych samych konkluzji, o ile $\int \! dy$ $P(\Theta)$ $\frac{\partial \ln P(\Theta)}{\partial\theta_{i}} = 0\,$, $\,i=1,2,..,d$ (por. (\[IF 2 poch na kwadrat pierwszej\])). Zasadnicz¹ zalet¹ zdefiniowania $\texttt{i\!F}$ poprzez (\[observed IF Amari\]) jest to, ¿e bardzo naturalne staje siê wtedy wprowadzenie tzw. $\alpha$-koneksji na przestrzeni statystycznej ${\cal S}$ [@Amari; @Nagaoka; @book]. Pojêcie $\alpha$-koneksji omówimy w Rozdziale \[alfa koneksja\].\ \ [**Przyk³ad estymacji wartoœci oczekiwanej w rozk³adzie normalnym**]{}: Jako przyk³ad ilustruj¹cy zwi¹zek wielkoœci obserwowanej informacji Fishera (IF) z niepewnoœci¹ oszacowania parametru, rozwa¿ny realizacjê próby prostej $y$ dla zmiennej $Y$ posiadaj¹cej rozk³ad $N\left({\theta,\sigma^{2}}\right)$.\ \ Za³ó¿my, ¿e [*wariancja $\sigma^{2}$ jest znana*]{}, a [*estymowanym parametrem jest jedynie wartoœæ oczekiwana*]{} $\theta=E\left( Y \right)$. Logarytm funkcji wiarygodnoœci ma postaæ: $$\begin{aligned} \label{log wiaryg dla norm} \ln P\left(y|\theta\right) = -\frac{N}{2} \ln (2 \pi \sigma^{2})-\frac{1}{{2\sigma^{2}}} \sum\limits_{n=1}^{N}{\left({{\bf y}_{n}-\theta}\right)^{2}}\end{aligned}$$ sk¹d funkcja wynikowa (\[funkcja wynikowa\]) jest równa: $$\begin{aligned} \label{f wynikowa_1 wym N_1 par z definicji} S\left(\theta\right) = \frac{\partial}{{\partial\theta}}\ln P\left({y|\theta}\right) = \frac{1}{{\sigma^{2}}}\sum\limits_{n=1}^{N}{\left({{\bf y}_{n}-\theta}\right)} \; .\end{aligned}$$ Rozwi¹zuj¹c jedno równanie wiarygodnoœci, $S\left(\theta\right)_{|\theta=\hat{\theta}} =0$, otrzymujemy postaæ estymatora parametru $\theta$ (por. (\[srednia arytmet z MNW\])): $$\begin{aligned} \label{rozw r wiaryg dla mu w N} \hat{\theta} = \bar{{\bf y}} = \frac{1}{N}\sum_{n=1}^{N}{\bf y}_{n} \; ,\end{aligned}$$ sk¹d: $$\begin{aligned} \label{f wynikowa_1 wym N_1 par} S\left(\theta\right) = \frac{N}{{\sigma^{2}}} \left(\hat{\theta} - \theta \right) \; .\end{aligned}$$ Natomiast z (\[I obserwowana\]) oraz (\[I jako krzywizna dla P\]) otrzymujemy, ¿e obserwowana IF jest równa: $$\begin{aligned} \label{I obserw dla N_parametr mu} \texttt{i\!F}\left({\hat{\theta}}\right) = - \frac{\partial^{2}\, \ln P\left({y|\theta}\right)}{{\partial \theta^{2}}}\left. \right|_{\theta=\hat{\theta}} = - \frac{\partial\, S\left(\theta \right)}{{\partial \theta}}\left. \right|_{\theta=\hat{\theta}} = \frac{N}{{\sigma^{2}}} \; .\end{aligned}$$ Z (\[rozw r wiaryg dla mu w N\]) oraz z klasycznej analizy[^15] [@Nowak] wiemy, ¿e wariancja: $$\begin{aligned} \label{wariancja dla sredniej y} {\sigma^{2}}( \hat{\theta} ) = {\sigma^{2}}({\bar{{\bf y}}})=\sigma^{2}/N \; ,\end{aligned}$$ zatem: $$\begin{aligned} \label{rn} \texttt{i\!F}({\hat{\theta}}) = \frac{1}{{\sigma^{2}}({\hat{\theta}})} \; .\end{aligned}$$ [**Wniosek**]{}: Otrzymaliœmy wiêc wa¿ny zwi¹zek mówi¹cy, ¿e wiêksza obserwowana IF parametru $\theta$ oznacza mniejsz¹ wariancjê jego estymatora $\hat{\theta}$.\ \ Równanie (\[rn\]) mo¿na zapisaæ w postaci: $$\begin{aligned} \label{RC dla 1 N z 1 par} {\sigma^{2}}({\hat{\theta}}) \texttt{i\!F}({\hat{\theta}}) = 1 \; ,\end{aligned}$$ co w przypadku estymacji wartoœci oczekiwanej rozk³adu normalnego, jest sygna³em osi¹gniêcia [*dolnego ograniczenia nierównoœci Rao-Cram[é]{}ra*]{} [@Amari; @Nagaoka; @book]. Temat ten bêdziemy rozwijaæ dalej.\ \ [**Okreœlenie oczekiwanej IF**]{}: Zdefiniujmy oczekiwan¹ informacjê Fishera nastêpuj¹co: $$\begin{aligned} \label{infoczekiwana} I_F \left(\Theta\right) \equiv E_{\Theta} \left(\texttt{i\!F}(\Theta)\right) = \int_{\cal B} dy P(y|\Theta) \, \texttt{i\!F}(\Theta) \; ,\end{aligned}$$ gdzie ${\cal B}$ jest przestrzeni¹ próby (uk³adu). Oznaczenie $\Theta$ w indeksie wartoœci oczekiwanej mówi, ¿e $\Theta$ jest prawdziw¹ wartoœci¹ parametru, przy której generowane s¹ dane $y \equiv ({\bf y}_{1}, {\bf y}_{2}, ..., {\bf y}_{N})$, natomiast element ró¿niczkowy $dy$ oznacza: $$\begin{aligned} \label{element rozniczkowy dy} dy \equiv d^{N}{\bf y} = d{\bf y}_{1} d{\bf y}_{2} ... d{\bf y}_{N} \; .\end{aligned}$$ [**Oczekiwana v.s. obserwowana IF**]{}: Istniej¹ znacz¹ce ró¿nice pomiêdzy oczekiwan¹ a obserwowan¹ informacj¹ Fishera [@Pawitan; @Mania]. Oczekiwana informacja Fishera $I_F$ ma sens jako funkcja dopuszczalnych wartoœci $\Theta$, nale¿¹cych do przestrzeni $V_{\Theta}$ wartoœci $\Theta$. Natomiast jak to wynika z MNW, obserwowana informacja Fishera, $\texttt{i\!F}(\Theta)$, ma zasadniczo sens tylko w pobli¿u $\hat{\Theta}$. Jako zwi¹zana z obserwowan¹ wartoœci¹ wiarygodnoœci, $\texttt{i\!F}$ odnosi siê do pojedynczego zestawu danych i zmienia siê od próbki do próbki. Oznacza to, ¿e nale¿y o niej myœleæ jako o pojedynczej realizacji statystyki $\texttt{i\!F}(\hat{\Theta})$ w próbce, a nie jako o funkcji parametru $\Theta$. Natomiast oczekiwana informacja $I_F$ jest œredni¹ wartoœci¹ dla wszystkich mo¿liwych zestawów danych w ca³ej przestrzeni próby ${\cal B}$, generowanych przy prawdziwej wartoœci parametru. Zatem $I_F(\Theta)$ jest nie tyle u¿ytecznym wskaŸnikiem informacji dla konkretnego zbioru danych, ile funkcj¹ $\Theta$ mówi¹c¹ jak trudno jest estymowaæ $\Theta$, co oznacza, ¿e parametr z wiêksz¹ $I_F$ wymaga mniejszej próbki do osi¹gniêcia wymaganej precyzji jego oszacowania.\ \ Kontynuuj¹c rozpoczêty powy¿ej Przyk³ad rozk³adu normalnego z estymacj¹ skalarnego parametru $\theta$, otrzymujemy po skorzystaniu z (\[I obserw dla N\_parametr mu\]) oraz unormowaniu funkcji wiarygodnoœci: $$\begin{aligned} \label{unormowanie dla f wiaryg P} \int dy\, P(y|\theta) = 1 \; ,\end{aligned}$$ wartoœæ oczekiwan¹ IF dla parametru $\theta$ rozk³adu $N(\theta, \sigma^2$): $$\begin{aligned} \label{I oczekiwana dla N_parametr mu} I_F(\theta) = \int dy \, P(y|\theta) \, \texttt{i\!F}\left({\theta}\right) = \int dy\, P(y|\theta) \frac{N}{{\sigma^{2}}} = \frac{N}{{\sigma^{2}}} \; .\end{aligned}$$ Ze wzglêdu na (\[wariancja dla sredniej y\]) wynik ten oznacza, ¿e zachodzi: $$\begin{aligned} \label{RC dla 1 N z 1 par oczekiwana IF} {\sigma^{2}}({\hat{\theta}}) I_F(\theta) = 1 \; ,\end{aligned}$$ co mówi, ¿e w estymacji wartoœci oczekiwanej rozk³adu normalnego osi¹gamy [*dolne ograniczenie nierównoœci Rao-Cram[é]{}ra*]{} [@Amari; @Nagaoka; @book]. Nierównoœæ ta daje najwa¿niejsze ograniczenie statystyki informacyjnej na jakoœæ estymacji w pojedynczym kanale informacyjnym. Sprawie nierównoœci Rao-Cramera poœwiêcimy czêœæ rozwa¿añ skryptu.\ \ [**Przyk³ad estymacji obu parametrów rozk³adu normalnego**]{}: Rozwa¿my dwuparametrowy ($d=2$) rozk³ad normalny. Niech ${\bf y}_{1},...,{\bf y}_{N}$ jest realizacj¹ próby prostej dla zmiennej $Y$ o rozk³adzie normalnym $N\left({\mu,\sigma}\right)$. Poniewa¿ $\theta_{i}^{n}=\theta_{i}$ dla $n=1,2,...,N$ zatem wektor parametrów przyjmujemy jako $\Theta = ( (\theta_{i}^{\,n})_{i=1}^{2} )_{n=1}^{N} \equiv (\theta_{i})_{i=1}^{2} = \left( \mu, \sigma \right)^{T}$. Funkcja wiarygodnoœci próby ma wtedy postaæ: $$\begin{aligned} \label{fun wiaryg r normalnego 2 par} \ln P \left({y|\Theta} \right) = -N \ln (\sqrt{2 \pi} \; \sigma) - \frac{1}{{2\, \sigma^{2}}} \sum_{n=1}^{N}{\left({{\bf y}_{n}-\mu} \right)^{2}} \; .\end{aligned}$$ Funkcja wynikowa z ni¹ zwi¹zana jest równa: $$\begin{aligned} \label{fun wyn r normalnego 2 par} S \left(\Theta \right) = \left( \begin{array}{l} \frac{\partial}{{\partial \mu}} \ln P \left({y|\Theta} \right)\\ \frac{\partial}{{\partial \sigma}} \ln P \left({y|\Theta} \right) \end{array} \right) = \left( \begin{array}{l} \frac{N}{{\sigma^{2}}} \left({\bar{\bf y}-\mu} \right) \\ -\frac{N}{{\sigma}} + \frac{{\sum_{n=1}^{N} {\left({{\bf y}_{n}-\mu}\right)^{2}}}}{{\sigma^{3}}} \end{array} \right) \; , \end{aligned}$$ gdzie $\bar{\bf y} = \frac{1}{N} \sum_{n=1}^{N} {\bf y}_{n}$ jest œredni¹ arytmetyczn¹ w próbie. Zatem postacie estymatorów MNW, tzn. parametru wartoœci oczekiwanej $\mu$ oraz odchylenia standardowego $\sigma$ zmiennej $Y$, otrzymujemy rozwi¹zuj¹c uk³ad równañ wiarygodnoœci: $$\begin{aligned} \label{fun wyn r normalnego 2 par = 0} S \left(\Theta \right)|_{\Theta={\hat{\Theta}}} = \left( \begin{array}{l} \frac{N}{{\sigma^{2}}} \left({\bar{\bf y}-\mu} \right) \\ -\frac{N}{{\sigma}} + \frac{{\sum_{n=1}^{N} {\left({{\bf y}_{n}-\mu}\right)^{2}}}}{{\sigma^{3}}} \end{array} \right)|_{\Theta={\hat{\Theta}}} = \left( \begin{array}{l} 0 \\ 0 \end{array} \right) \; , \end{aligned}$$ gdzie $\hat{\Theta}=\left( \hat{\mu}, \hat{\sigma} \right)^{T}$ i rozwi¹zanie to ma postaæ: $$\begin{aligned} \label{rozw ukl row MNW dla 2-wym rozkl norm} {\hat{\mu}} = \bar{\bf y} \;\;\;\;\; {\rm oraz} \;\;\;\;\; \hat{\sigma} = \sqrt{\frac{1}{N} \sum_{n=1}^{N} \left({{\bf y}_{n} - \bar{\bf y}}\right)^{2}} \; .\end{aligned}$$ Obserwowana informacja Fishera (\[I jako krzywizna dla P\]) w punkcie ${\hat{\Theta}}$ wynosi wiêc: $$\begin{aligned} \label{obserw iF r normalnego 2 par} \texttt{i\!F}({\hat{\Theta}}) = \left( \begin{array}{cc} {\frac{N}{{\sigma^{2}}}} & {\frac{2 N}{{\sigma^{3}}} \left({\bar{\bf y}-\mu} \right)} \\ {\frac{2 N}{{\sigma^{3}}} \left({\bar{\bf y}-\mu} \right)} & \;\; -\frac{N}{{\sigma^{2}}} + \frac{3}{\sigma^{4}} \sum_{n=1}^{N} \left({{\bf y}_{n}-\mu} \right)^{2} \end{array} \right)|_{\Theta={\hat{\Theta}}} = \left({\begin{array}{cc} {\frac{N}{{\hat{\sigma}^{2}}}} & 0\\ 0 & {\frac{2 N}{{\hat{\sigma}^{2}}}} \end{array}}\right) \; ,\end{aligned}$$\ W koñcu oczekiwana informacja Fishera jest równa: $$\begin{aligned} \label{oczekiw iF r normalnego 2 par} I_{F} \left(\Theta \right) = E_{\Theta} \texttt{i\!F}\left(\Theta \right) = \int_{\cal B} dy \, P(y|\Theta) \, \texttt{i\!F}\left(\Theta \right) = \left({\begin{array}{cc} {\frac{N}{{\sigma^{2}}}} & 0\\ 0 & {\frac{2 N}{{\sigma^{2}}}} \end{array}}\right) \; .\end{aligned}$$\ [**Uwaga o estymatorze $\hat{\sigma^{2}}$ MNW**]{}: Widz¹c powy¿szy rozk³ad jako $N\left({\mu,\sigma^{2}}\right)$ i w konsekwencji przyjmuj¹c wektor parametrów jako $(\mu, \sigma^{2})^{T}$, oraz przeprowadzaj¹c analogiczny rachunek jak powy¿ej, mo¿na pokazaæ, ¿e estymator wartoœci oczekiwanej ma postaæ $\hat{\mu} = \bar{\bf y}$, czyli tak jak w (\[rozw ukl row MNW dla 2-wym rozkl norm\]), natomiast estymator wariancji wynosi $\hat{\sigma^{2}}=\frac{1}{N} \sum_{n=1}^{N} \left({{\bf y}_{n} - \bar{\bf y}}\right)^{2}$. W jego mianowniku wystêpuje czynnik $N$ a nie $N-1$ jak to ma miejsce w przypadku nieobci¹¿onego estymatora wariancji. Estymator $\hat{\sigma^{2}}$ nie jest te¿ efektywny (tzn. nie posiada najmniejszej z mo¿liwych wariancji). Jednak w³asnoœci te posiada on asymptotycznie ($N \rightarrow\infty$), co jest charakterystyczne dla wszystkich estymatorów MNW [@Nowak].\ Sprawdzenie, ¿e przy odpowiednio dobranych tzw. oczekiwanych parametrach rozk³adu normalnego, estymacja obu parametrów jest efektywna dla skoñczonego $N$, pozostawiamy jako æwiczenie na koniec tego rozdzia³u, po tym jak zapoznamy siê z pojêciem dualnych uk³adów wpó³rzêdnych. ### Wartoœæ oczekiwana i wariancja funkcji wynikowej {#E i var funkcji wynikowej} Poka¿my, ¿e wartoœæ oczekiwana funkcji wynikowej $S(\Theta) \equiv S(\widetilde{Y}|\Theta)$, tzn. gradientu logarytmu naturalnego funkcji wiarygodnoœci, jest równa zeru: $$\begin{aligned} \label{znikanie ES} E_{\Theta}S\left(\Theta\right) = 0 \; .\end{aligned}$$ Istotnie, gdy skorzystamy z interpretacji funkcji wiarygodnoœci jako ³¹cznego rozk³adu prawdopodobieñstwa i jej unormowania do jednoœci, wtedy: $$\begin{aligned} \label{dow ES=0} E_{\Theta}S\left(\Theta\right) &=& \int_{\cal B}{dy P\left(y|\Theta\right) S\left(\Theta\right)} = \int_{\cal B} dy P\left({y|\Theta} \right) \left(\frac{\partial}{{\partial\Theta}} \ln P\left({y|\Theta}\right)\right) \nonumber \\ &=& \int_{\cal B} dy P\left({y|\Theta} \right) \frac{{\frac{\partial}{{\partial\Theta}}P\left(\Theta\right)}}{{P\left(\Theta\right)}} = \int_{\cal B}{ dy \frac{\partial}{{\partial\Theta}} P\left({y|\Theta} \right)} = \frac{\partial}{{\partial\Theta}} \int_{\cal B}{d{y}\, P\left({y|\Theta}\right)}=0 \; ,\end{aligned}$$ gdzie zakres ca³kowania obejmuje ca³¹ przestrzeñ próby ${\cal B}$. W przypadku gdy pierwotna zmienna $Y$ jest dyskretna, powy¿szy dowód przebiega podobnie.\ \ [**Za³o¿enie o regularnoœci rozk³adu**]{}: W (\[dow ES=0\]) wyci¹gnêliœmy ró¿niczkowanie po parametrze przed znak ca³ki. Poprawnoœæ takiego przejœcia oznacza spe³nienie ¿¹dania, aby rozk³ad $P\left({y|\Theta}\right)$ by³ wystarczaj¹co g³adki jako funkcja $\Theta$ [@Pawitan]. Oznacza to, ¿e $\Theta$ [*nie mo¿e byæ brzegow¹ wartoœci¹, tzn. istnieje taka funkcja $g\left(y\right)$ (której ca³ka $\int_{\cal B}{g\left({y}\right)dy}$ jest skoñczona), dla której w s¹siedztwie prawdziwej wartoœci parametru $\Theta$ zachodzi $\left|{{\partial P\left({y|\Theta}\right)} /{\partial\Theta}}\right|\le g\left(y\right)$ gdy ${{\partial P}/{\partial\Theta}}$ jest traktowana jako funkcja*]{} $y$ [@Pawitan], sk¹d wynika skoñczonoœæ ca³ki $ \int_{\cal B} dy \, \partial P\left({y|\Theta}\right) /{\partial\Theta} $.\ Zauwa¿my równie¿, ¿e przy odpowiednim warunku regularnoœci: $$\begin{aligned} \label{2 poch po P} \int_{\cal B}{ dy\, \frac{\partial^{2}}{\partial \Theta^{2}} P\left(y|\Theta \right)} = \frac{\partial^{2}}{\partial\Theta^{2}} \int_{\cal B}{ dy\, P\left(y|\Theta \right)} =0 \; .\end{aligned}$$\ [**Twierdzenie o wariancji funkcji wynikowej:**]{} Zak³adaj¹c warunek regularnoœci pozwalaj¹cy na wyci¹gniêcie ró¿niczkowania po parametrze przed znak ca³ki, mo¿na pokazaæ, ¿e wariancja (precyzyjnie, macierz wariancji-kowariancji) funkcji wynikowej $S(\Theta) \equiv S(\widetilde{Y}|\Theta)$ jest równa oczekiwanej IF[^16]: $$\begin{aligned} \label{var S oraz IF} {\sigma^{2}}_{\!\! \Theta} \,S \left( \Theta \right) = I_{F}(\Theta) \; .\end{aligned}$$ Zauwa¿my, ¿e z powy¿szego wynika, ¿e $d\times d$-wymiarowa macierz informacji $I_{F}(\Theta)$ jest macierz¹ kowariancji, co oznacza, ¿e jest ona [*nieujemnie okreœlona*]{}[^17].\ \ [**Dowód**]{} twierdzenia (\[var S oraz IF\]). Korzystaj¹c z (\[znikanie ES\]) otrzymujemy: $$\begin{aligned} \label{dow varS=I 1} {\sigma^{2}}_{\Theta} S\left( \Theta \right) &=& \int_{\cal B} d{y} \, P\left(y|\Theta\right) \left( S\left(\Theta\right) - E_{\Theta} S\left(\Theta \right) \right)^{2} \\ &=& \int{d{y} \, P\left({y|\Theta}\right)\left({S\left(\Theta\right)}\right)^{2}} = \int{d{y}\, P\left({y|\Theta}\right) \left({\frac{\partial}{{\partial\Theta}} \ln P\left({y|\Theta}\right)}\right)}^{2} \nonumber \\ &=&\int{d{y}\, P\left(y|\Theta\right) \left[\left(\frac{\partial}{\partial\Theta}P\left(y|\Theta\right)\right)/P\left(y|\Theta\right)\right]^{2}} = \int{dy\,\left[\frac{\partial}{\partial\Theta}P\left(y|\Theta\right) \right]^{2}/P\left(y|\Theta\right)} \; . \nonumber\end{aligned}$$\ Natomiast korzystaj¹c z (\[2 poch po P\]) otrzymujemy: $$\begin{aligned} \label{dow varS=I 2} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! & & I_{F}(\Theta) = E_{\Theta}\texttt{i\!F}(\Theta) = \int_{\cal B}{d{y} P\left({y|\Theta}\right) \texttt{i\!F}(\Theta)} = -\int{d{y} P\left({y|\Theta}\right) \left(\frac{{\partial^{2}}}{{\partial\Theta^{2}}}\ln P\left({y|\Theta}\right)\right)} \nonumber \\ \!\!\!\!\! &=&-\int dy P\left(y|\Theta\right) {\frac{\partial}{\partial\Theta} }\left[\left(\frac{\partial}{\partial\Theta}P\left(y|\Theta\right)\right)/P\left(y|\Theta\right)\right] \\ \!\!\!\!\! &=&\int dy P\left(y|\Theta\right) \frac{\left[-\left(\frac{\partial^{2}}{\partial\Theta^{2}}P\left(y|\Theta\right)\right)P\left(y|\Theta\right)+ \left(\frac{\partial}{\partial\Theta}P\left(y|\Theta\right)\right)^{2}\right]}{\left(P\left(y|\Theta\right)\right)^{2}} = \int{ d{y} \, \left[\frac{\partial}{\partial\Theta}P\left(y|\Theta\right)\right]^{2}/P\left(y|\Theta\right)} \; , \;\;\; \nonumber $$ co porównuj¹c z (\[dow varS=I 1\]) koñczy dowód twierdzenia (\[var S oraz IF\]).\ \ [**Wniosek**]{}: Dowodz¹c (\[znikanie ES\]) okaza³o siê, ¿e przy za³o¿eniu warunku reguralnoœci, wartoœæ oczekiwana na przestrzeni próby z funkcji wynikowej zeruje siê, tzn. $E_{\Theta}S\left(\Theta\right)=0$ i rezultat ten jest s³uszny dla ogólnego przypadku wektorowego. Z (\[var S oraz IF\]) wynika równie¿, ¿e macierz wariancji-kowariancji funkcji wynikowej jest równa IF, tzn. ${\sigma^{2}}_{\Theta} S \left( \Theta \right)= I_{F}(\Theta)$, co jak sprawdzimy poni¿ej, ma znaczenie dla dodatniej okreœlonoœci metryki Fishera $g_{ij}$ w teorii pola. Skoro równoœci (\[znikanie ES\]) oraz (\[var S oraz IF\]) zachodz¹ dla przypadku wektorowego, zatem s¹ one równie¿ s³uszne w przypadku skalarnym.\ \ [**Æwiczenie**]{}: Przedstawione dowody dla w³asnoœci (\[znikanie ES\]) oraz (\[var S oraz IF\]) by³y ogólne. Sprawdziæ bezpoœrednim rachunkiem, ¿e zachodz¹ one dla (\[fun wyn r normalnego 2 par\]) oraz (\[oczekiw iF r normalnego 2 par\]) w powy¿szym przyk³adzie estymacji obu parametrów rozk³adu normalnego.\ \ [**Wa¿ny zwi¹zek**]{}: Z (\[dow varS=I 1\])-(\[dow varS=I 2\]) widaæ, ¿e przy spe³nieniu wspomnianych w³asnoœci regularnoœci, zachodzi: $$\begin{aligned} \label{IF 2 poch na kwadrat pierwszej} \!\!\!\!\!\!\!\! I_{F}(\Theta) = E_{\Theta}\texttt{i\!F}(\Theta) = - \int_{\cal B} \! {d{y} P\left({y|\Theta}\right) \left(\frac{{\partial^{2}}}{{\partial\Theta^{2}}}\ln P\left({y|\Theta}\right)\right)} = \int_{\cal B} \!{dy P\left(y|\Theta\right) \, \left[\frac{\partial}{\partial \Theta} \ln P\left(y|\Theta\right)\right]^{2} } . \;\;\;\end{aligned}$$\ [**Uwaga o dodatniej okreœlonoœci obserwowanej IF**]{}: Niech $g_{ij}$ s¹ elementami $d\times d$ - wymiarowej macierzy $I_{F}(\Theta)$. Z (\[IF 2 poch na kwadrat pierwszej\]) otrzymujemy dla dowolnego $d$ - wymiarowego wektora $v=(v_{1},v_{2},...,v_{d})^{T}$: $$\begin{aligned} \label{iF polokreslona} v^{T} I_{F}(\Theta) \,v = \! \sum_{i=1}^{d} \sum_{j=1}^{d} v_{i} g_{ij}(\Theta) v_{j} = \!\int_{\cal B} \! dy P(y|\Theta) ( \sum_{i=1}^{d} v_{i} \frac{\partial \ln P(\Theta)}{\partial \theta_{i}} ) ( \sum_{j=1}^{d} \frac{\partial \ln P(\Theta)}{\partial \theta_{j}} \,v_{j} ) \geq 0 \; , \;\;\;\;\;\end{aligned}$$ co oznacza, ¿e oczekiwana informacja Fishera $I_{F}(\Theta)$ jest [*dodatnio pó³okreœlona*]{} [@Pawitan], jak to zaznaczyliœmy poni¿ej (\[var S oraz IF\]). Jednak w teorii pola interesuje nas zaostrzenie warunku (\[iF polokreslona\]) do w³asnoœci dodatniej okreœlonoœci.\ \ Z (\[var S oraz IF\]) (lub (\[IF 2 poch na kwadrat pierwszej\])) widaæ równie¿, ¿e w [*teorii pola, któr¹ da³oby siê sformu³owaæ dla ci¹g³ych, regularnych, unormowanych rozk³adów*]{}, co poci¹ga za sob¹ ci¹g³oœæ rozk³adu funkcji wynikowej na ca³ej przestrzeni próby ${\cal B}$, macierz $I_{F}(\Theta)$ jest okreœlona dodatnio.\ Poniewa¿ w Rozdziale \[alfa koneksja\] zwrócimy uwagê na fakt, ¿e oczekiwana IF okreœla tzw. [*metrykê Fishera-Rao*]{} $(g_{ij}):=I_{F}(\Theta)$ na przestrzeni statystycznej ${\cal S}$, zatem:\ \ [*W teorii pola z ci¹g³ymi, regularnych i unormowanymi rozk³adami, metryka Fishera-Rao $g_{ij}$ jest dodatnio okreœlona.*]{} Wstêp do geometrii ró¿niczkowej na przestrzeni statystycznej i $\alpha$-koneksja {#alfa koneksja} -------------------------------------------------------------------------------- Niech zbiorem punktów $\Omega$ reprezentuj¹cych konfiguracjê uk³adu bêdzie przestrzeñ próby uk³adu ${\cal B}$. Np. w przypadku próby jednowymiarowej ${\cal B} \equiv {\cal Y}$, gdzie ${\cal Y}$ jest zbiorem wszystkich mo¿liwych wartoœci ${\bf y}$ zmiennej losowej $Y$. Niech $P$ jest miar¹ probabilistyczn¹ (prawdopodobieñstwem) na ${\cal B}$. Zbiór wszystkich miar na ${\cal B}$ oznaczmy $\Sigma({\cal B})$ i nazwijmy [*przestrzeni¹ stanów*]{} modelu.\ \ [**Okreœlenie modelu statystycznego**]{}: Rozwa¿my podzbiór ${\cal S} \subset \Sigma({\cal B})$, na którym jest zadany uk³ad wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ [@Amari; @Nagaoka; @book] tak, ¿e ${\cal S}$ jest rozmaitoœci¹[^18]. Niech na ${\cal B}$ okreœlony jest $d$ - wymiarowy model statystyczny (tzn. para $(y, P)$), a precyzyjniej: $$\begin{aligned} \label{model statystyczny S} {\cal S} = \{P_{\Xi} \equiv P(y|\Xi), \Xi \equiv (\xi^{i})_{i=1}^{d} \in {V}_{\Xi} \subset \mathbb{R}^{d} \} \; , \end{aligned}$$ tzn. rodzina rozk³adów prawdopodobieñstwa parametryzowana przez $d$ nielosowych zmiennych o wartoœciach rzeczywistych $(\xi^{i})_{i=1}^{d}$ nale¿¹cych do przestrzeni parametru ${V}_{\Xi}$, bêd¹cej podzbiorem $\mathbb{R}^{d}$. Mówimy, ¿e ${\cal S}$ jest $d$ - wymiarow¹ przestrzeni¹ statystyczn¹.\ \ [**Notacja**]{}: Poniewa¿ w danym modelu statystycznym wartoœæ parametru $\Xi$ okreœla jednoznacznie rozk³ad prawdopodobieñstwa $P$ jako punkt na ${\cal S}$, wiêc ze wzglêdu na wygodê i o ile nie bêdzie to prowadzi³o do nieporozumieñ, sformu³owania (punkt) $P_{\Xi} \in {\cal S}$ oraz (punkt) $\Xi \in {\cal S}$ bêdziemy stosowali zamiennie.\ \ [**Uwaga o niezale¿noœci $P$ od parametryzacji**]{}: Oczywiœcie rozk³ad prawdopodobieñstwa nie zale¿y od wyboru bazy w przestrzeni statystycznej ${\cal S}$, tzn. gdyby np. $\Theta$ by³ innym uk³adem wspó³rzêdnych (inn¹ parametryzacj¹) to $P = P_{\Theta} \equiv P(y|\Theta) = P_{\Xi} \equiv P(y|\Xi)$ dla ka¿dego $P \in {\cal S}$.\ \ [**Okreœlenie macierzy informacyjnej Fishera**]{}: WprowadŸmy oznaczenie: $$\begin{aligned} \label{oznaczenie ln P} \ell_{\Xi} \equiv l(y|\Xi) \equiv \ln P_{\Xi} \; \;\;\;\; {\rm oraz} \;\;\;\; \partial_{i} \equiv \frac{\partial}{\partial \xi^{i}} \;\;\; i=1,2,...,d \; .\end{aligned}$$ Dla ka¿dego punktu $P_{\Xi}$, $d \times d$ - wymiarowa macierz $(g_{ij}(\Xi))$ o elementach: $$\begin{aligned} \label{Fisher inf matrix} \!\!\!\!\!\!\! g_{ij}(\Xi):= E_{\Xi}(\partial_{i} \ell_{\Xi} \partial_{j} \ell_{\Xi}) = \! \int_{{\cal B}} \! dy \, P(y|\Xi) \, \partial_{i} l(y|\Xi) \partial_{j} l(y|\Xi) \; , \;\; i,j = 1,2,...,d \; , \;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ jest nazywana [*macierz¹ informacyjn¹ Fishera*]{} na ${\cal S}$ w punkcie $P_{\Xi}$ [@Amari; @Nagaoka; @book]. Wielkoœæ $E_{\Xi}(.)$ oznacza tutaj wartoœæ oczekiwan¹, a ca³kowanie przebiega po ca³ej przestrzeni próby ${\cal B}$.\ Przy za³o¿eniu spe³nienia warunków regularnoœci (porównaj (\[IF 2 poch na kwadrat pierwszej\])), macierz $(g_{ij}(\Xi))$ mo¿e zostaæ zapisana nastêpuj¹co: $$\begin{aligned} \label{Fisher inf matrix plus reg condition} g_{ij} = - E_{\Xi}(\partial_{i} \partial_{j} \ell_{\Xi}) = - \int_{{\cal B}} dy \, P(y|\Xi) \, \partial_{i} \partial_{j} \ln P(y|\Xi) \; , \;\;\; i,j = 1,2,...,d \; , \;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$\ [**Metryka Fishera**]{}: Macierz informacyjna Fishera okreœla na ${\cal S}$ iloczyn wewnêtrzny nazywany [*metryk¹ Fishera*]{} $\langle , \rangle$ na ${\cal S}$, definuj¹c go w uk³adzie wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ poprzez zwi¹zek: $$\begin{aligned} \label{Fisher metric poprzez macierz informacyjna} \langle \partial_{i} , \partial_{j} \rangle := g_{ij} \; , \; \;\;\;\; {\rm gdzie \;\; wektory \;\; bazowe} \;\;\; \partial_{i} \equiv \frac{\partial}{\partial \xi^{i}} \; , \;\;\;\; i=1,2,...,d \; , \;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Metryka $g_{ij}$ Fishera-Rao jest metryk¹ typu Riemannowskiego[^19]. Warto zauwa¿yæ, ¿e na rozmaitoœci ${\cal S}$ mo¿na zdefiniowaæ nieskoñczon¹ liczbê metryk Riemannowskich. Jednak Chentsov pokaza³, ¿e [*metryka Fishera-Rao jest*]{} wyró¿niona spoœród wszystkich innych tym, ¿e (z dok³adnoœci¹ do sta³ego czynnika) jest ona [*jedyn¹, która jest redukowana*]{} (w znaczeniu zmniejszania siê odleg³oœci dowolnych dwóch stanów) [*w ka¿dym stochastycznym odwzorowaniu*]{} [@Bengtsson_Zyczkowski; @Streater].\ \ [**Okreœlenie koneksji afinicznej**]{}: Oznaczmy przez $T_{P}$ przestrzeñ styczn¹ do ${\cal S}$ w punkcie $P_{\Xi} \in {\cal S}$. Ka¿dy wektor $V \in T_{P}$ mo¿na roz³o¿yæ na wektory bazowe $(\partial_i)_{P}$: $$\begin{aligned} \label{V w T Theta} V = \sum_{i=1}^{d} V^{i} (\partial_{i})_{P} \;\, , \; \;\;\; {\rm gdzie} \;\;\;\; V \in T_{P} \; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ gdzie dolny indeks $_{P}$ oznacza zale¿noœæ uk³adu wspó³rzêdnych $\partial_{i} \equiv \frac{\partial}{\partial \xi^{i}}$, $i=1,2,...,d$, od punktu $P_{\Xi} \in {\cal S}$.\ \ Niech $\gamma_{P P'}$ oznacza œcie¿kê w ${\cal S}$ ³¹cz¹c¹ punkty $P$ oraz $P'$. Przyporz¹dkujmy, ka¿dej œcie¿ce $\gamma_{P P'}$ w ${\cal S}$, odwzorowanie $\Pi_{\gamma_{P P'}}$: $$\begin{aligned} \label{Pi z Tr do Ts} \gamma_{P P'} \rightarrow \Pi_{\gamma_{P P'}}: T_{P} \rightarrow T_{P'} \; , \;\;\;\;\; \forall\, P_{\Xi} \; {\rm i} \; P'_{\Xi} \in {\cal S} \; , \end{aligned}$$ które przekszta³ca wektory z przestrzeni wektorowej $T_{P}$ do przestrzeni wektorowej $T_{P'}$.\ \ Rozwa¿my trzy dowolne punkty $P$, $P'$ oraz $P''$ w ${\cal S}$, oraz œcie¿ki $\gamma_{P P'}$ z punktu $P$ do $P'$ i $\gamma_{P' P''}$ z $P'$ do $P''$. [*Mówimy, ¿e przekszta³cenie $\Pi$ jest koneksj¹ afiniczn¹ jeœli*]{}: $$\begin{aligned} \label{Pi jako koneksja afiniczna} \Pi_{\gamma_{P P''}} = \Pi_{\gamma_{P' P''}} \circ \Pi_{\gamma_{P P'}} \;\;\;\; {\rm oraz} \;\;\;\; \Pi_{\gamma_{0}} \equiv \Pi_{\gamma_{P P}} = id \; , \;\;\;\;\; \forall\, P_{\Xi} \; {\rm i} \;P'_{\Xi} \in {\cal S} \; ,\end{aligned}$$ gdzie $id$ jest przekszta³ceniem identycznoœciowym.\ \ Aby okreœliæ postaæ liniowego odwzorowania $\Pi_{P P'}$ pomiêdzy $T_{P}$ a $T_{P'}$ musimy, dla ka¿dego $i \in 1, 2,...,d$, okreœliæ $\Pi_{P P'}((\partial_i)_{P})$ jako liniow¹ kombinacjê wektorów bazowych $(\partial_1)_{P'}, (\partial_2)_{P'}, ..., (\partial_{d})_{P'}$ w $T_{P'}$.\ \ [**Okreœlenie przesuniêcia równoleg³ego**]{}: Niech $V_{P}$ bêdzie wektorem stycznym w $P$. Wektor $V_{P P'}$ nazywamy [*równoleg³ym przesuniêciem wektora*]{} $V_{P}$ z $P$ do $P'$ wzd³u¿ krzywej (œcie¿ki) $\gamma_{P P'}$, wtedy gdy: $$\begin{aligned} \label{Pi i rownolegle przesuniecie} V_{P} \rightarrow V_{P P'} := \Pi_{\gamma_{P P'}} \, V_{P}\; , \;\;\;\;\; \forall\, P_{\Xi} \; {\rm i} \;P'_{\Xi} \in {\cal S} \; .\end{aligned}$$\ [**Okreœlenie równoleg³oœci wektorów**]{}: [*Dwa wektory $V_{P}$ oraz $V_{P'}$ styczne do ${\cal S}$ w punktach $P$ i $P'$ s¹ równoleg³e*]{}, jeœli równoleg³e przesuniêcie (okreœlone w (\[Pi i rownolegle przesuniecie\])) wzd³u¿ wskazanej krzywej $\gamma$ jednego z nich, powiedzmy $V_{P}$, do punktu $P'$ “zaczepienia” drugiego z nich, da wektor $V_{P P'}$, który jest [*proporcjonalny*]{} do $V_{P'}$ (i na odwrót).\ Koneksja afiniczna pozwala zdefiniowaæ pochodn¹ kowariantn¹ w nastêpuj¹cy sposób. Niech $V$ jest dowolnym wektorem w przestrzeni stycznej $T_{P}$, oraz niech $\gamma(t)$, gdzie $t \in \left\langle 0,1 \right\rangle$, oznacza dowoln¹ œcie¿kê z $P$ do $P'$ w ${\cal S}$, która wychodzi z $P$ w kierunku $W \in T_{P}$.\ \ [**Pochodn¹ kowariantn¹**]{} (koneksji afinicznej $\Pi$) wektora $V$ w punkcie $P$ i w kierunku $W \in T_{P}$ definiujemy nastêpuj¹co: $$\begin{aligned} \label{Pi pochodna kowariantna} \nabla_{W} V := \frac{d}{dt} \left(\Pi_{\gamma_{P \gamma(t)}} V \right)_{|_{t=0}} \; , \;\;\; {\rm gdzie} \;\;\;\; \gamma(t=0) = P \;\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Pojêcie pochodnej kowariantnej pozwala wypowiedzieæ siê co do równoleg³oœci wektorów w punktach $P$ i $P'$.\ \ [**Okreœlenie geodezyjnej**]{}: Geodezyjn¹ nazywamy tak¹ krzyw¹ $\gamma$ w ${\cal S}$, której wszystkie wektory styczne s¹ do siebie równoleg³e. W zwi¹zku z tym mówimy, ¿e geodezyjna jest [*samo-równoleg³¹*]{} krzyw¹ w ${\cal S}$.\ \ \ [**Wspó³czynniki koneksji**]{}: Konkretna analityczna postaæ wspó³czynników koneksji [*musi byæ podana przy ustalonej parametryzacji*]{}[^20] $\Xi \rightarrow P$, tzn. w okreœlonym uk³adzie wspó³rzêdnych $\Xi = (\xi^{i})_{i=1}^{d}$.\ Zak³adaj¹c, ¿e ró¿nica pomiêdzy $\Pi_{P P'}((\partial_j)_{P})$ oraz $(\partial_j)_{P'}$ jest [*infinitezymalna*]{}, i ¿e mo¿e byæ wyra¿ona jako liniowa kombinacja [*ró¿niczek*]{} $d\xi^{1}, d\xi^{2},..., d\xi^{d}$, gdzie: $$\begin{aligned} \label{rozniczka theta dla theta} d\xi^{i} = \xi^{i}(P) - \xi^{i}(P') \; , \;\;\;\; i=1,2,...,d \; , \;\;\;\;\; P_{\Xi} \;{\rm i}\; P'_{\Xi} \in {\cal S} \; ,\end{aligned}$$ mamy: $$\begin{aligned} \label{Pi poprzez partial oraz d theta} \Pi_{P P'}((\partial_{j})_{P}) = (\partial_j)_{P'} - \sum_{i,\,l=1}^{d} d\xi^{i}(\Gamma_{ij}^{l})_{P} (\partial_l)_{P'} \; , \;\;\; j=1,2,...,d \; , \;\;\;\;\; P_{\Xi} \;{\rm i}\; P'_{\Xi} \in {\cal S} \; , \end{aligned}$$ gdzie $(\Gamma_{ij}^{l})_{P}$, $i,j,l=1,2,...,d$, s¹ $d^{3}$ [*wspó³czynnikami koneksji*]{} zale¿¹cymi od punktu $P$. Koneksjê afiniczn¹ $\Pi$, wiêc i wspó³czynniki koneksji, mo¿na (przy ustalonej parametryzacji $\Xi$) okreœliæ na ró¿ne sposoby. Jeden z nich zwi¹zany z koneksj¹ Levi-Civita podany jest w (\[koneksja Levi-Civita\]). Jednak w analizie na przestrzeniach statystycznych szczególnie u¿yteczna okaza³a siê tzw. $\alpha$-koneksja.\ \ [**$\alpha$-koneksja**]{}: [*W ka¿dym punkcie*]{} $P_{\Xi} \in {\cal S}$, [*$\alpha$-koneksja*]{} zadaje $d^{3}$ funkcji $\Gamma^{(\alpha)}_{ij,\,r}\!: \!\Xi \rightarrow \! (\Gamma^{(\alpha)}_{ij,\,r})_{\Xi}$, $i,j,r=1,2,...,d$, przyporz¹dkowuj¹c mu wspó³czynniki koneksji o nastêpuj¹cej postaci [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{affine coefficients} (\Gamma^{(\alpha)}_{ij, \, r})_{\Xi} \equiv (\Gamma^{(\alpha)}_{ij, \, r})_{P_{\Xi}} = E_{\Xi}\left[ \left( \partial_{i} \partial_{j} \ell_{\Xi} + \frac{1-\alpha}{2} \partial_{i} \ell_{\Xi} \partial_{j} \ell_{\Xi} \right) \partial_{r} \ell_{\Xi} \right]\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Koneksja $\alpha$ jest [*symetryczna*]{}, tzn.: $$\begin{aligned} \label{koneksja symetryczna} \Gamma^{(\alpha)}_{ij,\,l} = \Gamma^{(\alpha)}_{ji,\,l} \; , \;\;\;\; i,j,l = 1,2,...,d\; , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ W koñcu, jeœli $g^{ij}(P)$ jest $(i,j)$-sk³adow¹ macierzy odwrotnej do macierzy informacyjnej $(g_{ij}(P))$, to wspó³czynniki $\Gamma^{r\, (\alpha)}_{ij}$ s¹ równe: $$\begin{aligned} \label{wspolczynniki koneksji dla g i g-1} \Gamma^{r\, (\alpha)}_{ij} = \sum_{l=1}^{d} g^{rl} \, \Gamma^{(\alpha)}_{ij,\, l} \; , \;\;\;\; i,j,r = 1,2,...,d\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ [**$\alpha$ - pochodna kowariantna**]{}: Maj¹c metrykê Fishera-Rao $g_{ij}$, (\[Fisher metric poprzez macierz informacyjna\]), zdefiniujmy $\alpha$ - pochodn¹ kowariantn¹ $\nabla^{(\alpha)}$ na ${\cal S}$ poprzez $\alpha$-koneksjê afiniczn¹ nastêpuj¹co: $$\begin{aligned} \label{pochodna kowariantna i wspolczynniki koneksji} \langle \nabla_{\partial_{i}}^{(\alpha)} \partial_{j}, \, \partial_{l}\rangle := \Gamma^{(\alpha)}_{ij,\, l} \; , \;\;\;\; i,j,l = 1,2,...,d\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Powy¿ej, poprzez koneksjê afiniczn¹ zdefiniowaliœmy pochodn¹ kowariantn¹. Ale i odwrotnie, pochodna kowariantna definiuje koneksjê [@Amari; @Nagaoka; @book]. St¹d np. mówimy, ¿e okreœliliœmy koneksjê $\nabla$.\ \ [**Pole wektorowe na ${\cal S}$**]{}: Niech $V: P \rightarrow V_{P}$ jest odwzorowaniem przyporz¹dkowuj¹cym ka¿demu punktowi $P \in {\cal S}$ wektor styczny $V_{P} \in T_{P}({\cal S})$. Odwzorowanie to nazywamy [*polem wektorowym*]{}. Na przyk³ad, jeœli $(\xi^{i})_{i=1}^{d}$ jest uk³adem wspó³rzêdnych, wtedy przyporz¹dkowanie $\frac{\partial}{\partial \xi^{i}}: P \rightarrow \left(\frac{\partial}{\partial \xi^{i}}\right)_{P}$, $i=1,2,...,d$, okreœla $d\,$ pól wektorowych na ${\cal S}$.\ Oznaczmy przez $T({\cal S})$ rodzinê wszystkich pól wektorowych klasy $C^{\infty}$ typu $V_{P} = \sum_{i=1}^{d} V_{P}^{i} \left(\partial_{i}\right)_{P} \in T_{P}({\cal S})$ na ${\cal S}$, gdzie $d$ funkcji $V^{i}: P \rightarrow V^{i}_{P}$ nazywamy wspó³rzêdnymi pola wektorowego $V$ ze wzglêdu na $(\xi^{i})_{i=1}^{d}$.\ \ [**Okreœlenie pochodnej kowariantnej pola wektorowego**]{}: Rozwa¿my dwa pola wektorowe $V, W \in T({\cal S})$. Niech w bazie $(\xi^{i})_{i=1}^{d}$ pola te maj¹ postaæ $V = \sum_{i=1}^{d} V^{i} \partial_{i}$ oraz $W = \sum_{i=1}^{d} W^{i} \partial_{i}$. Pochodn¹ kowariantn¹ pola $V$ ze wzglêdu na $W$ nazywamy pole wektorowe $\nabla_{W} V \in T({\cal S})$, które w bazie $(\xi^{i})_{i=1}^{d}$ ma postaæ: $$\begin{aligned} \label{pochodna kowariantna V wzgledem W} \nabla_{W} V = \sum_{i=1}^{d} W^{i} \, \sum_{k=1}^{d} \, \{ \, \partial_{i} V^{k} + \sum_{j=1}^{d} V^{j} \, \Gamma^{k}_{ij} \, \} \, \partial_{k} \; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Przyjmuj¹c $W = \partial_{i}$ oraz $V=\partial_{j}$ ³atwo sprawdziæ, ¿e zwi¹zek (\[pochodna kowariantna V wzgledem W\]) daje: $$\begin{aligned} \label{pochodna kowariantna partial wzgledem partial} \nabla_{\partial_{i}} \partial_{j} = \sum_{k=1}^{d} \, \Gamma^{k}_{ij} \, \, \partial_{k} \; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ co po skorzystaniu z (\[wspolczynniki koneksji dla g i g-1\]) jest zgodne z okreœleniem koneksji $\nabla$ w (\[pochodna kowariantna i wspolczynniki koneksji\]).\ Mo¿na pokazaæ [@Amari; @Nagaoka; @book], ¿e zachodzi: $$\begin{aligned} \label{alfa conection + -} \nabla^{(\alpha)} = \frac{1 + \alpha}{2} \, \nabla^{(1)} + \frac{1 - \alpha}{2} \, \nabla^{(-1)}\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ [**Uwaga o p³askoœci ${\cal S}$ ze wzglêdu na $\nabla$**]{}: Mówimy, ¿e uk³ad wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ jest [*afinicznym uk³adem wspó³rzêdnych dla koneksji*]{} $\nabla$ gdy zachodz¹ (równowa¿ne) warunki: $$\begin{aligned} \label{uklad afiniczny dla koneksji} \nabla_{\partial_{i}} \partial_{j} = 0 \;\;\;\;\;\; {\rm lub \; r\acute{o}wnowa\dot{z}nie} \;\;\;\;\; \Gamma_{ij}^{l} = 0 \;\;\;\; i,j,l=1,2,...,d \; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Jeœli dla zadanej koneksji $\nabla$ odpowiadaj¹cy jej uk³ad wspó³rzêdnych jest afiniczny, to mówimy, ¿e koneksja $\nabla$ jest p³aska[^21] lub, ¿e [*przestrzeñ statystyczna ${\cal S}$ jest p³aska ze wzglêdu na*]{} $\nabla$.\ P³askoœæ ${\cal S}$ ze wzglêdu na $\nabla$ oznacza, ¿e wszystkie wektory bazowe $\partial_{i} \equiv \frac{\partial}{\partial \xi^{i}}$, $i=1,2,...,d$ s¹ równoleg³e na ca³ej przestrzeni ${\cal S}$.\ \ Modele z koneksj¹ $\nabla^{(\alpha)}$, które s¹ $\alpha = + 1$ b¹dŸ $-1$ p³askie, odgrywaj¹ szczególn¹ rolê w modelowaniu statystycznym [@Pawitan].\ \ [**Dwa przyk³ady rodzin rozk³adów prawdopodobieñstwa**]{}: Istniej¹ dwa przypadki rodzin modeli statystycznych, szczególnie istotnych w badaniu podstawowych geometrycznych w³asnoœci modeli statystycznych. Pierwsza z nich to rodzina rozk³adów eksponentialnych, a druga, rozk³adów mieszanych.\ \ [**Rodzina modeli eksponentialnych**]{} okazuje siê wyj¹tkowo wa¿na, nie tylko dla badania w³asnoœci statystycznych, ale równie¿ w zwi¹zku z jej realizacj¹ w szeregu zagadnieniach fizycznych. Niech wielkoœæ próby $N=1$. Niech zmienna losowa $Y$ bêdzie zmienn¹ typu ci¹g³ego lub dyskretnego. Ogólna postaæ regularnej rodziny rozk³adów eksponentialnych[^22] jest nastêpuj¹ca: $$\begin{aligned} \label{exponential family} p_{\Xi} \equiv p({\bf y}| \Xi) = \exp \left[ C({\bf y}) + \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) - \psi(\Xi) \right] = \exp \left[ \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) - \psi(\Xi) \right] \, h({\bf y}) \; , \end{aligned}$$ gdzie $\xi^{i}$, $\,i=1,2,...,d$, s¹ tzw. [*parametrami kanonicznymi*]{}, natomiast $h({\bf y}) = \exp (C({\bf y}))$ jest nieujemn¹ funkcj¹, która nie zale¿y od wektorowego parametru $\Xi$.\ W (\[exponential family\]) pojawi³a siê $d$-wymiarowa statystyka: $$\begin{aligned} \label{F dla exponential family} F(Y) = (F_{1}(Y),..., F_{d}(Y))^{T} \equiv (F_{i}(Y))_{i=1}^{d} ,\end{aligned}$$ nazywana statystyk¹ [*kanoniczn¹*]{}.\ \ Ca³kuj¹c obustronnie (\[exponential family\]) po przestrzeni próby ${\cal B} = {\cal Y}$, a nastêpnie wykorzystuj¹c w³asnoœæ normalizacji $\int_{\cal Y} d {\bf y}\, p({\bf y}| \Xi) = 1$, otrzymujemy[^23]: $$\begin{aligned} \label{psi dla exponential family} \psi(\Xi) = \ln \int_{\cal Y} d{\bf y} \exp \left[ C({\bf y}) + \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right] \; .\end{aligned}$$\ Modele eksponentialne s¹ $\alpha = 1$ - p³askie, co oznacza, ¿e [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{Gamma 1 dla exponential family} (\Gamma^{(1)}_{ij,\,k})_{\Xi} = 0 \; , \;\;\;\; {\rm dla \; modeli \; eksponentialnych} \; , \;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Istotnie, korzystaj¹c z oznaczenia wprowadzonego w (\[oznaczenie ln P\]), otrzymujemy dla modeli zadanych przez (\[exponential family\]): $$\begin{aligned} \label{partial l w F oraz partial psi dla exponential family} \frac{\partial l({\bf y}|\Xi)}{\partial \xi^{i}} = F_{i}({\bf y}) - \frac{\partial \psi(\Xi)}{\partial \xi^{i}} \; , \end{aligned}$$ sk¹d (przy okazji) otrzymujemy obserwowan¹ informacjê Fishera dla rozk³adów eksponentialnych w parametryzacji kanonicznej: $$\begin{aligned} \label{partial l dla psi dla exponential family} \texttt{i\!F}\left(\Xi \right) = - \frac{\partial^{2} l({\bf y}|\Xi)}{\partial \xi^{j} \partial \xi^{i}} = \frac{\partial^{2} \psi(\Xi)}{\partial \xi^{j} \partial \xi^{i}} \; .\end{aligned}$$ Podstawiaj¹c (\[partial l dla psi dla exponential family\]) do (\[affine coefficients\]), otrzymujemy dla $\alpha=1$ wspó³czynniki $1$-koneksji, równe: $$\begin{aligned} \label{affine coefficients dla eksponentialnych} (\Gamma^{(1)}_{ij, \, r})_{\Xi} = E_{\Xi}\left[ \left(\frac{\partial^{2} l(Y|\Xi)}{\partial \xi^{j} \partial \xi^{i}} \right) \frac{\partial l(Y|\Xi)}{\partial \xi_{r}} \right] = - \frac{\partial^{2} \psi(\Xi)}{\partial \xi^{j} \partial \xi^{i}} E_{\Xi}\left[ \frac{ \partial l(Y|\Xi)}{ \partial \xi_{r} } \right] = 0 \; , \;\;\; \forall\, P_{\Xi} \in {\cal S} \; , \;\;\;\end{aligned}$$ gdzie w ostatniej równoœci skorzystano z (\[znikanie ES\]) dla funkcji wynikowej.\ \ Ponadto dla modeli eksponentialnych z (\[Fisher inf matrix plus reg condition\]) oraz (\[partial l dla psi dla exponential family\]) otrzymujemy nastêpuj¹c¹ postaæ metryki Fishera-Rao: $$\begin{aligned} \label{g dla exponential family} g_{ij} = - E_{\Xi}(\partial_{i} \partial_{j} \ell_{\Xi}) = \frac{\partial^{2} \psi(\Xi)}{\partial \xi^{j} \partial \xi^{i}} \; .\end{aligned}$$ Zwróæmy uwagê na równoœæ prawych stron (\[partial l dla psi dla exponential family\]) oraz (\[g dla exponential family\]) dla rozk³adów eksponentialnych.\ \ Przyk³adami modeli z eksponentialnej rodziny rozk³adów s¹:\ \ i) [*Rozk³ad normalny*]{}, (\[rozklad norm theta sigma2\]), $p\left(Y={\bf y}|\mu, \sigma^{2}\right) = \frac{1}{\sqrt{2 \pi \, \sigma^2}} \; \exp \left( - {\frac{({\bf y} - \mu)^{2}}{2 \, \sigma^2}} \right)$, ${\bf y} \in \mathbf{R}$, dla którego: $$\begin{aligned} \label{rozklad normalny parametry kanoniczne} & & C({\bf y}) = 0 \;, \;\;\; F_{1}({\bf y})={\bf y} \; , \;\; F_{2}({\bf y})={\bf y}^{2} \; , \;\; \xi^{1} = \frac{\mu}{\sigma^{2}} \; , \;\; \xi^{2} = - \frac{1}{2 \sigma^{2}} \; , \nonumber \\ & & \psi(\xi) = \frac{\mu^{2}}{2 \sigma^{2}} + \ln (\sqrt{2 \pi} \, \sigma) = - \frac{(\xi^{1})^{2}}{4 \xi^{2}} + \frac{1}{2}\ln (- \frac{\pi}{\xi^{2}} ) \; . \end{aligned}$$\ ii) [*Rozk³ad Poissona*]{} (\[rozklad Poissona\]), $p \left(Y={\bf y}|\mu \right)=\frac{\mu ^{{\bf y}} \exp({-\mu }) }{{\bf y}\, !}\,$, gdzie ${\bf y} = 0,1,...,\infty$, dla którego: $$\begin{aligned} \label{rozklad Poissona parametry kanoniczne} & & C({\bf y}) = - \ln ({\bf y}!) \;, \;\;\; F({\bf y})={\bf y} \; , \;\; \xi = \ln \mu \; , \;\; \psi(\xi) = \mu = \exp \xi \; . \end{aligned}$$\ iii) ([*Standardowy*]{}) [*rozk³ad eksponentialny*]{}, $p \left(Y={\bf y}|\mu \right)= \mu^{-1} \exp(-{\bf y}/\mu)\,$, gdzie ${\bf y} > 0$, dla którego: $$\begin{aligned} \label{rozklad eksponentialny parametry kanoniczne} & & C({\bf y}) = 0 \;, \;\;\; F({\bf y})= {\bf y} \; , \;\; \xi = - \frac{1}{\mu} \; , \;\; \psi(\xi) = \ln \mu = \ln ( -\frac{1}{\xi}) \; . \end{aligned}$$ Modele eksponentialne wyró¿nia fakt osi¹gania dolnego ograniczenia nierównoœci Rao-Cramera [@Streater] (porównaj zwi¹zek (\[RC dla 1 N z 1 par oczekiwana IF\])).\ \ [**Wymiar statystyki dostatecznej dla**]{} (\[exponential family\]): Dla modeli eksponentialnych zachodzi wa¿na w³asnoœæ zwi¹zana z wymiarem statystyki kanonicznej $F(Y)$, (\[F dla exponential family\]). Rozwa¿my $N$-elementow¹ próbê $\widetilde{Y} \equiv( Y_{n})_{n=1}^{N}$, dla której ka¿dy punktowy rozk³ad ma postaæ (\[exponential family\]). Wtedy funkcja wiarygodnoœci dla próby jest nastêpuj¹ca: $$\begin{aligned} \label{exponential family N fun wiarygodnosci} P(y| \Xi) = \exp \left[ \sum_{i=1}^{d} \xi^{i} \sum_{n=1}^{N} F_{i}({\bf y}_{n}) - N \, \psi(\Xi) \right] \, \exp \left[ \sum_{n=1}^{N} C({\bf y}_{n}) \right] \; .\end{aligned}$$ Dostateczna statystyka dla wektorowego parametru oczekiwanego $(\theta_{i})_{i=1}^{d}$, gdzie $\theta_{i} = E_{\Xi}\left[ F_{i}(Y) \right]$, $i = 1,2,...,d $ (por. (\[wartosi oczekiwane eta model eksponentialny\]) w Rozdziale \[Estymacja w modelach fizycznych na DORC\]) ma zatem postaæ: $$\begin{aligned} \label{F dostateczna dla exponential family dla proby N} \left( \sum_{n=1}^{N} F_{1}({\bf y}_{n}), \sum_{n=1}^{N} F_{2}({\bf y}_{n}), ..., \sum_{n=1}^{N} F_{d}({\bf y}_{n}) \right) \; .\end{aligned}$$ Jej wymiar jest równy $d$ i jak widaæ, dla konkretnych reprezentantów ogólnej rodziny eksponentialnej nie zale¿y on od wymiaru próby $N$. W³asnoœæ ta nie jest spe³niona np. dla takich nieeksponentialnych rozk³adów jak rozk³adu Weibull’a oraz Pareto [@Nowak], dla których wymiar statystyki dostatecznej roœnie wraz z wymiarem próby. Inn¹ wa¿n¹ w³asnoœci¹ rozk³adów nieeksponentialnych jest to, ¿e dziedzina (tzn. noœnik) ich funkcji gêstoœci mo¿e zale¿eæ od parametru.\ \ [**Rodzina mieszanych rozk³adów prawdopodobieñstwa**]{}: $$\begin{aligned} \label{mixture family} p_{\Delta} \equiv p({\bf y}|\Delta) = C({\bf y}) + \sum_{i=1}^{d} \delta^{i} F_{i}({\bf y}) \; , \end{aligned}$$ gdzie $\delta^{i}$ s¹ tzw. [*parametrami mieszanymi*]{}.\ \ [**Æwiczenie**]{}: Pokazaæ, ¿e przestrzeñ statystyczna rodziny rozk³adów mieszanych jest $\alpha = - 1$ - p³aska [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{Gamma 1 dla mieszanej family} (\Gamma^{(-1)}_{ij,\,k})_{\Xi} = 0 \; , \;\;\;\; {\rm dla \; modeli \; mieszanych} \; , \;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$\ [**Geometria przestrzeni ${\cal S}$ a parametryzacja**]{}: Jako w³asnoœci geometryczne przestrzeni ${\cal S}$ przyjmujemy te, które s¹ niezmiennicze ze wzglêdu na zmianê parametryzacji.\ Np. w³asnoœci geometryczne rodziny modeli eksponentialnych nie zale¿¹ od tego czy pos³u¿ymy siê parametrami kanonicznymi $(\xi^{i})_{i=1}^{d}$ czy oczekiwanymi $\theta_{i} = E_{\Xi}\left[ F_{i}(Y) \right]$, $i = 1,2,...,d $ (por. (\[wartosi oczekiwane eta model eksponentialny\])). ### Przestrzeñ statystyczna dualnie p³aska {#Przestrzen dualnie plaska} Z (\[Fisher inf matrix\]) oraz (\[affine coefficients\]) wynika, ¿e: $$\begin{aligned} \label{0 - koneksja} \partial_{r}g_{ij} = \Gamma^{(0)}_{ri, \, j} + \Gamma^{(0)}_{rj, \, i}\; , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; , \end{aligned}$$ co oznacza, ¿e $0$-koneksja jest metryczna[^24] ze wzglêdu na metrykê Fishera-Rao [@Amari; @Nagaoka; @book].\ \ [*Jednak w ogólnoœci dla $\alpha \neq 0$, $\alpha$-koneksja afiniczna nie jest metryczna, natomiast spe³nia warunek dualnoœci*]{} omówiony poni¿ej.\ \ [**Koneksje dualne**]{}: Niech na rozmaitoœci ${\cal S}$ zadana jest pewna metryka Riemannowska $g=\left\langle , \right\rangle$ i dwie koneksje $\nabla$ oraz $\nabla^{*}$. Metryka ta mo¿e byæ np. metryk¹ $g$ Fishera-Rao. Jeœli dla wszystkich pól wektorowych $V,W,Z\in T({\cal S})$, zachodzi: $$\begin{aligned} \label{def dualnych nabla} Z\left\langle V, W \right\rangle = \left\langle \nabla_{Z} V, W \right\rangle + \left\langle V, \nabla^{*}_{Z} W \right\rangle\; , \;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; , \end{aligned}$$ wtedy mówimy, ¿e $\nabla$ oraz $\nabla^{*}$ s¹ ze wzglêdu ma metrykê $\left\langle , \right\rangle$ [*dualne*]{} (sprzê¿one) wzglêdem siebie. W uk³adzie wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ metryka $g$ ma wspó³rzêdne $g_{ij}$, a koneksje $\nabla$ oraz $\nabla^{*}$ maj¹ wspó³czynniki koneksji odpowiednio $\Gamma_{ij,\,r}$ oraz $\Gamma^{*}_{ij, \, r}$.\ Warunek dualnoœci (\[def dualnych nabla\]) mo¿emy teraz zapisaæ w postaci[^25]: $$\begin{aligned} \label{war dualnosci we wspolczynnikach} \partial_{r}g_{ij} = \Gamma_{ri, \, j} + \Gamma^{*}_{rj, \, i}\; , \;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ bêd¹cej uogólnieniem warunku (\[0 - koneksja\]) istniej¹cego dla koneksji metrycznej.\ Ponadto, koneksja $\nabla^{met} \equiv (\nabla+\nabla^{*})/2$ jest koneksj¹ metryczn¹, dla której zachodzi warunek $\partial_{r}g_{ij} = \Gamma^{met}_{ri, \, j} + \Gamma^{met}_{rj, \, i}$ [@Amari; @Nagaoka; @book]. Mo¿na równie¿ sprawdziæ, ¿e (ze wzglêdu na $g$) zachodzi $(\nabla^{*})^{*} = \nabla$.\ \ [**Struktura dualna**]{}: Trójkê $(g, \nabla, \nabla^{*}) \equiv ({\cal S}, g, \nabla, \nabla^{*})$ nazywamy [*struktur¹ dualn¹ na*]{} ${\cal S}$. W ogólnoœci, maj¹c metrykê $g$ oraz koneksjê $\nabla$ okreœlon¹ na ${\cal S}$, koneksja dualna $\nabla^{*}$ jest wyznaczona w sposób jednoznaczny, co jest treœci¹ poni¿szego twierdzenia.\ \ [**Twierdzenie o zwi¹zku pomiêdzy koneksjami dualnymi**]{}: Niech $P$ oraz $P'$ s¹ punktami brzegowymi œcie¿ki $\gamma$, oraz niech przekszta³cenia $\Pi_{\gamma_{P P'}}$ oraz $\Pi^{*}_{\gamma_{P P'}}$ z $T_{P}(\cal S)$ do $T_{P'}(\cal S)$ opisuj¹ [*równoleg³e przesuniêcie wzd³u¿*]{} $\gamma$, odpowiednio ze wzglêdu na koneksje afiniczne $\nabla$ oraz $\nabla^{*}$. Wtedy dla wszystkich $V,W \in T_{P}({\cal S})$ zachodzi [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{koneksja warunek dualnosci ogolnie} g_{P'}(\Pi_{\gamma_{P P'}} V, \, \Pi^{*}_{\gamma_{P P'}} W) = g_{P}(V,\, W)\; , \;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi}\;{\rm i}\;P'_{\Xi} \in {\cal S} \; .\end{aligned}$$ Warunek ten jest uogólnieniem warunku (\[koneksja metryczna\]) istniej¹cego dla koneksji metrycznej o [*niezmienniczoœci iloczynu wewnêtrznego ze wzglêdu na przesuniêcie równoleg³e*]{}. Wyznacza on w sposób jednoznaczny zwi¹zek pomiêdzy $\Pi_{\gamma_{P P'}}$ oraz $\Pi^{*}_{\gamma_{P P'}}$.\ \ [**Niezmienniczoœæ iloczynu wewnêtrznego dla p³askich koneksji dualnych przy przesuniêciu równoleg³ym**]{}: [*Jeœli*]{} $\Pi_{\gamma_{P P'}}$ nie zale¿y na ${\cal S}$ od œcie¿ki $\gamma$, a tylko od punktów koñcowych[^26] $P$ oraz $ P'$, wtedy $\Pi_{\gamma_{P P'}}=\Pi_{P P'}$ na ${\cal S}$. Poniewa¿ przy okreœlonej metryce $g$, koneksja dualna $\nabla^{*}$ jest jednoznacznie wyznaczona dla $\nabla$, zatem równie¿ dla koneksji $\nabla^{*}$ jest na ${\cal S}$ spe³niony warunek $\Pi^{*}_{\gamma_{P P'}} = \Pi^{*}_{P P'}$, sk¹d z (\[koneksja warunek dualnosci ogolnie\]) przy przesuniêciu równoleg³ym otrzymujemy: $$\begin{aligned} \label{koneksja warunek dualnosci ogolnie bez gamma} g_{P'}(\Pi_{P P'} V, \, \Pi^{*}_{P P'} W) = g_{P}(V,\, W)\; , \;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi}\;{\rm i}\;P'_{\Xi} \in {\cal S} \; .\end{aligned}$$ [**Zadanie**]{}: Korzystaj¹c z metryki (\[g dla exponential family\]) modelu eksponentialnego oraz z (\[affine coefficients dla eksponentialnych\]) i (\[affine coefficients\]), sprawdziæ bezpoœrednim rachunkiem w parametryzacji kanonicznej $(\xi^{i})_{i=1}^{d}$ warunek (\[war dualnosci we wspolczynnikach\]), otrzymuj¹c: $$\begin{aligned} \label{koneksja warunek dualnosci dla eksponent} \partial_{r}g_{ij} = \Gamma^{+1}_{ri, \, j} + \Gamma^{-1}_{rj, \, i} = \Gamma^{-1}_{rj, \, i} = \partial_{r} \partial_{i} \partial_{j} \psi (\Xi) \; , \;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ ### Dualne uk³ady wspó³rzêdnych {#Dualne uklady wspolrzednych} [**Okreœlenie dualnie p³askiej przestrzeni**]{}: Mówimy, ¿e $({\cal S}, g, \nabla, \nabla^{*})$ jest [*dualnie p³ask¹ przestrzeni¹*]{}, jeœli obie koneksje dualne, $\nabla$ oraz $ \nabla^{*}$, s¹ p³askie na ${\cal S}$. Oznacza to, ¿e jeœli koneksja $\nabla$ jest p³aska w pewnej bazie $(\xi^{i})_{i=1}^{d}$ to koneksja $\nabla^{*}$ jest p³aska w pewnej bazie $(\xi^{*\,i})_{i=1}^{d}$, któr¹ nazywamy baz¹ dualn¹ do $(\xi^{i})_{i=1}^{d}$.\ \ [**$\alpha$ - koneksja dualnie p³aska**]{}:\ Istotnoœæ pojêcia $\alpha$-koneksji pojawia siê wraz z rozwa¿eniem na przestrzeni statystycznej ${\cal S}$ nie tyle prostej pary $(g, \nabla^{(\alpha)}) \equiv ({\cal S}, g, \nabla^{(\alpha)})$ ale struktury potrójnej $(g, \nabla^{(\alpha)},$ $\nabla^{(-\alpha)}) \equiv ({\cal S}, g, \nabla^{(\alpha)}, \nabla^{(-\alpha)})$. Powodem jest istnienie poprzez metrykê Fishera-Rao [*dualnoœci pomiêdzy koneksjami*]{} $\nabla^{(\alpha)}$ oraz $\nabla^{(-\alpha)}$, która okazuje siê byæ wa¿na przy badaniu modeli statystycznych.\ \ Podsumowuj¹c, [*dla dowolnego modelu statystycznego ${\cal S}$*]{}, zachodzi [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{podwojna plaskosc modelu} ({\cal S} \;\; {\rm jest} \;\; \alpha-p\ell aska) \Leftrightarrow ({\cal S} \;\; {\rm jest} \;\; (-\alpha)-p\ell aska) \; \; .\end{aligned}$$ [**Przyk³ad**]{}: Model statystyczny eksponentialny ${\cal S}$ jest p³aski w parametryzacji kanonicznej $(\xi^{i})_{i=1}^{d}$ dla $\alpha=1$. Istnieje zatem [*parametryzacja dualna*]{}, w której jest on równie¿ $\alpha=-1$ p³aski. Np. w Rozdziale \[Estymacja w modelach fizycznych na DORC\] oka¿e siê, ¿e dla modeli eksponentialnych spe³niaj¹cych warunek maksymalizacji entropii, baz¹ dualn¹ do bazy kanonicznej jest baza parametrów oczekiwanych.\ Podobnie jest dla rodziny rozk³adów mieszanych, tzn. jest ona jednoczeœnie $\pm 1$ p³aska.\ \ [**Okreœlenie dualnych uk³adów wspó³rzêdnych**]{}: Zastanówmy siê nad ogóln¹ struktur¹ przestrzeni $({\cal S}, g, \nabla, \nabla^{*})$, która by³aby dualnie [*p³aska*]{}.\ Z okreœlenia przestrzeni p³askiej ze wzglêdu na okreœlon¹ koneksjê (\[uklad afiniczny dla koneksji\]) oraz z (\[podwojna plaskosc modelu\]) wynika, ¿e jeœli istnieje uk³ad wspó³rzêdnych $\Xi=(\xi^{i})_{i=1}^{d}$ z wektorami bazowymi $\partial_{\xi^i} \equiv \frac{\partial}{\partial \xi^{i}}$, ze wzglêdu na który koneksja $\nabla$ jest p³aska, tzn. $\nabla_{\partial_{\xi^i}} \partial_{\xi^j} = 0$, $i,j=1,2,...,d$, to istnieje równie¿ uk³ad wspó³rzêdnych $\Theta=(\theta^{i})_{i=1}^{d}:=(\xi^{*i})_{i=1}^{d}$ z wektorami bazowymi $\partial_{\theta^i} \equiv \frac{\partial}{\partial \theta^{i}}$, ze wzglêdu na który koneksja $\nabla^{*}$ jest p³aska, tzn. $\nabla_{\partial_{\theta^j}} \partial_{\theta^l} = 0$, $j,l=1,2,...,d$.\ \ [**Wniosek**]{}: Zatem, gdy pole wektorowe $\partial_{\xi^i}$ jest $\nabla$- p³askie, wiêc pole wektorowe $\partial_{\theta^j}$ jest $\nabla^{*}$-p³askie i z (\[koneksja warunek dualnosci ogolnie bez gamma\]) [*wynika sta³oœæ*]{} $\,g_{P_{\Xi}}(\partial_{\xi^i}, \partial_{\theta^j})$ na ${\cal S}$. Fakt ten, bior¹c pod uwagê wszystkie $d$ stopni swobody zawarte w afinicznym uk³adzie wspó³rzêdnych, (\[uklad afiniczny dla koneksji\]), mo¿na zapisaæ jako[^27]: $$\begin{aligned} \label{stalosc il wewn partial i dual partial} \left\langle \partial_{\xi^i}, \partial_{\theta^j} \right\rangle \equiv g_{P_{\Xi}}\!\left(\frac{\partial}{\partial \xi^{i}}, \, \frac{\partial}{\partial \theta^{j}} \right) = \delta_{i j} \; , \; \; \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ gdzie iloczyn wewnêtrzny $\left\langle \cdot, \cdot \right\rangle$ jest wyznaczony w konkretnym uk³adzie wspó³rzêdnych, w tym przypadku w $\Xi = (\xi^{i})_{i=1}^{d}$. Uk³ady wspó³rzêdnych $\Xi = (\xi^{i})_{i=1}^{d}$ oraz $\Theta = (\theta^{j})_{j=1}^{d}$ okreœlone na przestrzeni Riemannowskiej $({\cal S}, g)$ i spe³niaj¹ce warunek (\[stalosc il wewn partial i dual partial\]) nazywamy [*wzajemnie dualnymi*]{}. Warunek (\[stalosc il wewn partial i dual partial\]) oznacza [**sta³oœæ na ${\cal S}$ iloczynu wewnêtrznego dla uk³adów dualnie p³askich**]{}.\ \ W ogólnoœci dla dowolnej przestrzeni Riemannowskiej $({\cal S}, g)$ nie istniej¹ uk³ady wspó³rzêdnych wzajemnie dualne.\ Jeœli jednak przestrzeñ Riemannowska z dualn¹ koneksj¹ $({\cal S}, g, \nabla, \nabla^{*})$ jest dualnie p³aska, to taka para uk³adów wspó³rzêdnych [*istnieje*]{}. Ale i na odwrót. Jeœli na przestrzeni Riemannowskiej $({\cal S}, g)$ istniej¹ dwa uk³ady wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ oraz $(\theta^{j})_{j=1}^{d}$ spe³niaj¹ce warunek (\[stalosc il wewn partial i dual partial\]), wtedy koneksje $\nabla$ oraz $\nabla^{*}$, wzglêdem których uk³ady te s¹ afiniczne, s¹ okreœlone, a $({\cal S}, g, \nabla, \nabla^{*})$ jest dualnie p³aska.\ \ \ \ [**Euklidesowy uk³ad wspó³rzêdnych**]{}: W przypadku Euklidesowego uk³adu wspó³rzêdnych na ${\cal S}$ mamy (z definicji): $$\begin{aligned} \label{stalosc il wewn dla Euklidesowego ukl wsp} \left\langle \partial_{\xi^i}, \partial_{\xi^j} \right\rangle = \delta_{ij}\; , \; \; \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ co oznacza, ¿e jest on [*samo-dualny*]{}.\ \ [**Uwaga o wspó³czesnym zastosowaniu koneksji dualnych**]{}: Interesuj¹cym wydaje siê fakt, ¿e pojêcie koneksji dualnych ma coraz wiêksze zastosowanie w analizie uk³adów liniowych [@Ohara] i szeregów czasowych [@Amari; @Nagaoka; @book]. Przyk³adem mo¿e byæ jej zastosowanie w analizie szeregów czasowych ARMA(p,q) [@Brockwell_Machura], co jest zwi¹zane z faktem, ¿e zbiór wszystkich szeregów czasowych ARMA(p,q) ma skoñczon¹ parametryzacjê i w zwi¹zku z tym tworzy on skoñczenie wymiarow¹ rozmaitoœæ. Aby dokonaæ analizy porównawczej dwóch szeregów czasowych bior¹c pod uwagê problemy ich aproksymacji, estymacji oraz redukcji wymiaru, analizowanie pojedynczego szeregu czasowego jest niewystarczaj¹ce i okazuje siê koniecznym rozwa¿anie w³asnoœci ca³ej przestrzeni tych szeregów wraz z ich struktur¹ geometryczn¹ [@Amari; @Nagaoka; @book]. #### Transformacja Legendre’a pomiêdzy parametryzacjami dualnymi {#Potencjaly ukladow wspolrzednych} Niech $\Xi \equiv (\xi^{i})_{i=1}^{d}$ oraz $\Theta \equiv (\theta^{i})_{i=1}^{d}$ s¹ wzajemnie dualnymi bazami na ${\cal S}$, zgodnie z relacj¹ (\[stalosc il wewn partial i dual partial\]). Zdefiniujmy wspó³rzêdne metryki $g$ ze wzglêdu na uk³ad wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ jako: $$\begin{aligned} \label{ukl wsp ze wzgledu na theta} g^{\xi}_{ij}:= \left\langle \partial_{\xi^i}, \partial_{\xi^j} \right\rangle \; , \; \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ a ze wzglêdu na uk³ad wspó³rzêdnych $(\theta^{j})_{j=1}^{d}$ jako: $$\begin{aligned} \label{ukl wsp ze wzgledu na eta dla theta kowariantne} g^{\theta}_{ij}:= \left\langle \partial_{\theta^i}, \partial_{\theta^j} \right\rangle \; , \; \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Theta} \in {\cal S} \; .\end{aligned}$$ Przejœcie od wspó³rzêdnych kontrawariantnych $(\theta^{j})_{j=1}^{d}$ w bazie $\partial_{\theta^j} \equiv \partial/\partial\theta^j$ do kowariantnych $(\theta_{j})_{j=1}^{d}$ w bazie $\partial^{\theta_j} \equiv \partial/\partial\theta_j$ ma postaæ: $$\begin{aligned} \label{ukl wsp theta kontrawariantne} \theta_j := \sum_{k=1}^{d} g^{\theta}_{jk} \, \theta^k \;\;\; {\rm oraz} \;\;\; \partial^{\theta_j} := \sum_{k=1}^{d} g_{\theta}^{jk} \partial_{\theta^k} \; , \; \;\;\;\; j,k=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Theta} \in {\cal S} \; ,\end{aligned}$$ gdzie $g_{\theta}^{jk}$ jest $(j,k)$-sk³adow¹ macierzy odwrotnej do macierzy informacyjnej $(g^{\theta}_{jk})$. Ze wzglêdu na uk³ad wspó³rzêdnych $(\theta_{j})_{j=1}^{d}$, wspó³rzêdne metryki $g_{\theta}^{ij}$ s¹ równe: $$\begin{aligned} \label{ukl wsp ze wzgledu na eta} g_{\theta}^{ij} = \left\langle \partial^{\theta_i}, \partial^{\theta_j} \right\rangle \; , \; \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Theta} \in {\cal S} \; .\end{aligned}$$ Podobnie okreœlamy macierz informacyjn¹ $(g_{\xi}^{jk})$ jako odwrotn¹ do $(g^{\xi}_{jk})$.\ \ Rozwa¿my transformacje uk³adu wspó³rzêdnych: $$\begin{aligned} \label{transf bazy dualnej eta w theta} \partial^{\theta_{j}} \equiv \frac{\partial }{\partial \theta_{j}} = \sum_{i=1}^{d} \frac{\partial \xi^{i} }{\partial \theta_{j}} \frac{\partial}{\partial \xi^{i}} \equiv \sum_{i=1}^{d} (\partial^{\theta_j} \xi^{i}) \partial_{\xi^i} \; , \;\;\;\; j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P \in {\cal S} \; \end{aligned}$$ oraz $$\begin{aligned} \label{transf bazy dualnej theta w eta} \!\!\!\!\!\!\! \partial_{\xi^i} \equiv \frac{\partial }{\partial \xi^{i}} = \sum_{j=1}^{d} \frac{\partial \theta_{j} }{\partial \xi^{i}} \frac{\partial}{\partial \theta_{j}} \equiv \sum_{j=1}^{d} (\partial_{\xi^i} \theta_{j}) \partial^{\theta_j} = \sum_{j=1}^{d} (\partial_{\xi^i} \theta^{j}) \partial_{\theta^j} \; , \;\;\;\; i=1,2,...,d \; , \;\;\;\; \forall\, P \in {\cal S} \; .\end{aligned}$$ Zatem po skorzystaniu z (\[transf bazy dualnej eta w theta\])-(\[transf bazy dualnej theta w eta\]) oraz warunku (\[stalosc il wewn partial i dual partial\]), mo¿na dualne metryki (\[ukl wsp ze wzgledu na theta\]) oraz (\[ukl wsp ze wzgledu na eta\]) zapisaæ nastêpuj¹co: $$\begin{aligned} \label{g ij we wspol przez eta i theta} g^{\xi}_{ij} = \frac{\partial \theta_{j}}{\partial \xi^{i}} \;\;\;\; {\rm oraz} \;\;\;\; g_{\theta}^{ij} = \frac{\partial \xi^{i}}{\partial \theta_{j}} \; , \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P \in {\cal S} \; ,\end{aligned}$$ co oznacza równie¿, ¿e macierze informacyjne, $I_{F}(\Xi) = (g^{\xi}_{ij})$ w bazie $\Xi \equiv (\xi^{i})_{i=1}^{d}$ oraz $I_{F}(\Theta) = (g_{\theta}^{ij})$ w bazie[^28] dualnej $\Theta \equiv (\theta_{i})_{i=1}^{d}$, s¹ wzglêdem siebie odwrotne: $$\begin{aligned} \label{macierze informacyjne dualne} I_{F}(\Xi) = I_{F}^{-1}(\Theta) \; , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \forall\, P \in {\cal S} \; .\end{aligned}$$\ Rozwa¿my z kolei funkcjê $\psi:{\cal S} \rightarrow \mathbb{R}$ oraz nastêpuj¹ce cz¹stkowe równanie ró¿niczkowe: $$\begin{aligned} \label{row rozn dla psi i eta} \partial_{\xi^i} \psi = \theta_{i} \; , \;\;\;\; i=1,2,...,d \; \;\;\;\; {\rm tzn.} \;\;\;\;\; d\psi = \sum_{i=1}^{d} \theta_{i} \, d \xi^{i} \; , \;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; . \end{aligned}$$ Ze wzglêdu na (\[g ij we wspol przez eta i theta\]) równanie (\[row rozn dla psi i eta\]) daje: $$\begin{aligned} \label{row rozn 2 rzedu dla psi i eta} \partial_{\xi^i} \partial_{\xi^j} \psi =g^{\xi}_{ij} \; , \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; .\end{aligned}$$ Ze wzglêdu na dodatni¹ okreœlonoœæ metryki $g^{\xi}_{ij}$, równanie to oznacza, ¿e druga pochodna $\psi$ tworzy równie¿ [*dodatnio okreœlon¹*]{} macierz. Zatem $\psi$ jest [*œciœle wypuk³¹*]{} funkcj¹ wspó³rzêdnych $\xi^{1},\xi^{2},...,\xi^{d}$, dla ka¿dego $P \in {\cal S}$.\ Podobnie, rozwa¿aj¹c funkcjê $\phi:{\cal S} \rightarrow \mathbb{R}$ oraz cz¹stkowe równanie ró¿niczkowe: $$\begin{aligned} \label{row rozn dla phi i theta} \partial^{\theta_i} \phi = \xi^{i} \; , \;\;\;\; i=1,2,...,d \; \;\;\;\; {\rm tzn.} \;\;\;\;\; d\phi = \sum_{i=1}^{d} \xi^{i} \, d \theta_{i} \; , \;\;\;\;\; \forall\, P_{\Theta} \in {\cal S} \; .\end{aligned}$$ Ze wzglêdu na (\[g ij we wspol przez eta i theta\]) równanie (\[row rozn dla phi i theta\]) daje: $$\begin{aligned} \label{row rozn 2 rzedu dla phi i theta} \partial^{\theta_i} \partial^{\theta_j} \phi =g_{\theta}^{ij} \; , \;\;\;\; i,j=1,2,...,d \; , \;\;\;\;\; \forall\, P_{\Theta} \in {\cal S} \; ,\end{aligned}$$ w zwi¹zku z czym dodatnia okreœlonoœæ dualnej metryki $g_{\theta}^{ij}$ oznacza, ¿e druga pochodna $\phi$ tworzy dodatnio okreœlon¹ macierz. Zatem $\phi$ jest [*œciœle wypuk³¹*]{} funkcj¹ wspó³rzêdnych $\theta^{1},\theta^{2},...,\theta^{d}$, dla ka¿dego $P \in {\cal S}$.\ \ [**Transformacja Legendre’a**]{}: Powiedzmy, ¿e $\psi$ jest pewnym rozwi¹zaniem równania (\[row rozn 2 rzedu dla psi i eta\]). Wtedy po skorzystaniu z (\[g ij we wspol przez eta i theta\]) oraz (\[row rozn dla psi i eta\]), widaæ, ¿e od $\psi \equiv \psi(\Xi)$ do $\phi \equiv \phi(\Theta)$ mo¿na przejœæ przez [*transformacjê Legendre’a*]{}[^29] : $$\begin{aligned} \label{transformacja Legendrea psi w phi} \phi(\Theta) = \sum_{i=1}^{d} \xi^{i} \theta_{i} - \psi(\Xi) \; , \;\;\;\;\;\; \forall\, P \in {\cal S} \; .\end{aligned}$$ Podobnie, powiedzmy, ¿e $\phi$ jest pewnym rozwi¹zaniem równania (\[row rozn 2 rzedu dla phi i theta\]). Wtedy poprzez transformacjê Legendre’a: $$\begin{aligned} \label{transformacja Legendrea phi w psi} \psi(\Xi) = \sum_{i=1}^{d} \xi^{i} \theta_{i} - \phi(\Theta) \; , \;\;\;\;\;\; \forall\, P \in {\cal S} \; ,\end{aligned}$$ mo¿na przejœæ od funkcji $\phi$ do $\psi$.\ \ [**Uwaga**]{}: W ogólnoœci transformacje pomiêdzy uk³adami wspó³rzêdnych $\Xi=(\xi^{i})_{i=1}^{d}$ oraz $\Theta=(\theta_{j})_{j=1}^{d}$, które maj¹ postaæ (\[transformacja Legendrea psi w phi\]) i (\[transformacja Legendrea psi w phi\]) nazywamy transformacjami Legendre’a[^30].\ \ [**Okreœlenie potencja³ów**]{}: Funkcje $\psi$ oraz $\phi$ spe³niaj¹ce odpowiednio warunki (\[row rozn dla psi i eta\]) oraz (\[row rozn dla phi i theta\]), pomiêdzy którymi mo¿na przejœæ transformacj¹ Legendre’a (\[transformacja Legendrea psi w phi\]) lub (\[transformacja Legendrea phi w psi\]), nazywamy [*potencja³ami*]{} uk³adów wspó³rzêdnych (odpowiednio $\Xi$ oraz $\Theta$).\ \ Poni¿ej podamy twierdzenie podsumowuj¹ce powy¿sze rozwa¿ania.\ \ [**Twierdzenie**]{} [*o dualnych uk³adach wspó³rzêdnych*]{}: Niech $\Xi=(\xi^{i})_{i=1}^{d}$ jest $\nabla$-afinicznym uk³adem wspó³rzêdnych na dualnie p³askiej przestrzeni $({\cal S}, g, \nabla, \nabla^{*})$. Wtedy, ze wzglêdu na metrykê $g$, istnieje dualny do $(\xi^{i})_{i=1}^{d}$ uk³ad wspó³rzêdnych $\Theta=(\theta_{i})_{i=1}^{d}$, który jest $\nabla^{*}$-afinicznym uk³adem wspó³rzêdnych. Oba te uk³ady wspó³rzêdnych s¹ zwi¹zane transformacj¹ Legendre’a zadan¹ przy potencja³ach $\psi(\Xi)$ oraz $\phi(\Theta)$ poprzez zwi¹zki (\[transformacja Legendrea psi w phi\]) lub (\[transformacja Legendrea phi w psi\]). Ponadto wspó³rzêdne metryki w tych uk³adach wspó³rzêdnych s¹ zadane jako drugie pochodne potencja³ów, jak w (\[row rozn 2 rzedu dla psi i eta\]) oraz (\[row rozn 2 rzedu dla phi i theta\]).\ \ [**Wspó³czynniki koneksji dla uk³adów dualnych**]{}: Na koniec podajmy wyprowadzone z u¿yciem zwi¹zku (\[war dualnosci we wspolczynnikach\]) oraz (\[row rozn 2 rzedu dla psi i eta\]) postacie wspó³czynników koneksji afinicznej $\Gamma^{\xi\, *}_{ij,\,l}$ [@Amari; @Nagaoka; @book] (por. (\[pochodna kowariantna i wspolczynniki koneksji\])): $$\begin{aligned} \label{pochodna kow i wspolczynniki koneksji dualne *} \Gamma^{\xi\, *}_{ij,\, l}:= \langle \nabla_{\partial_{\xi^i}}^{*} \partial_{\xi^j}, \, \partial_{\xi^l}\rangle = \partial_{\xi^i} \partial_{\xi^j} \partial_{\xi^l} \psi(\Xi) \; , \;\;\;\; i,j,l = 1,2,...,d \; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; \end{aligned}$$ oraz z u¿yciem (\[war dualnosci we wspolczynnikach\]) oraz (\[row rozn 2 rzedu dla phi i theta\]), wspó³czynniki koneksji afinicznej $\Gamma_{\theta}^{ij,\, l}$: $$\begin{aligned} \label{pochodna kow i wspolczynniki koneksji dualne} \Gamma_{\theta}^{ij,\, l}:= \langle \nabla_{\partial^{\theta_i}} \partial^{\theta_j}, \, \partial^{\theta_l}\rangle = \partial^{\theta_i} \partial^{\theta_j} \partial^{\theta_l} \phi(\Theta) \; , \;\;\;\; i,j,l = 1,2,...,d \; , \;\;\;\;\; \forall\, P_{\Theta} \in {\cal S} \; ,\end{aligned}$$ przy czym, poniewa¿ uk³ady wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ oraz $(\theta_{i})_{i=1}^{d}$ s¹ afiniczne, zatem: $$\begin{aligned} \label{pochodna kow i wspolczynniki koneksji dualne zwykle} \Gamma^{\xi}_{ij,\, l} = \Gamma_{\theta}^{* \, ij,\, l} = 0 , \;\;\;\; i,j,l = 1,2,...,d \; , \;\;\;\;\; \forall\, P \in {\cal S} \; .\end{aligned}$$ ### Geometryczne sformu³owanie teorii estymacji dla EFI {#Geometryczne sformulowanie teorii estymacji} Dok³adne sformu³owanie metody ekstremalnej fizycznej informacji (EFI) jest treœci¹ kolejnych rozdzia³ów skryptu. Poni¿ej podajemy jedynie jej wstêpn¹ charakterystykê z punktu widzenia geometrii ró¿niczkowej na ${\cal S}$.\ Teoria estymacji w metodzie EFI mo¿e byæ okreœlona geometrycznie w sposób nastêpuj¹cy. Za³ó¿my, ¿e z pewnych powodów teoretyczny rozk³ad $P$ na ${\cal B}$ le¿y na pewnej podprzestrzeni ([*warstwie*]{}) ${\cal S}_{w} \subseteq {\cal S}$. Warstwa ${\cal S}_{w}$ oraz wymiar ${\cal B}$ nie s¹ z góry okreœlone. Zak³adamy równie¿, ¿e wszystkie rozwa¿ane rozk³ady, ³¹cznie z empirycznym, le¿¹ w przestrzeni statystycznej ${\cal S}$, która (w przeciwieñstwie do estymacji w statystyce klasycznej [@Amari; @Nagaoka; @book; @Streater]) [*nie ma znanej postaci metryki*]{} Fishera-Rao. Na podstawie danych, które da³y rozk³ad empiryczny $P_{Obs}$, szukamy punktu nale¿¹cego do ${\cal S}_{w}$, który spe³nia zasady informacyjne i jest szukanym [*oszacowaniem*]{} $P_{\hat{\,\Xi}}$ rozk³adu teoretycznego $P$.\ \ [**Uwaga o estymacji w statystyce klasycznej**]{}: W statystyce klasycznej wyznaczamy krzyw¹ geodezyjn¹ biegn¹c¹ przez punkt empiryczny $P_{Obs}$ i szukane oszacowanie $P_{\hat{\,\Xi}}$ stanu uk³adu le¿¹cego na ${\cal S}_{w}$. Poniewa¿ za wyj¹tkiem przypadku $\alpha=0$ okazuje siê, ¿e $\alpha$-koneksja nie jest metryczna (tzn. nie jest wyprowadzona jedynie z metryki, w naszym przypadku metryki Fishera-Rao), zatem odleg³oœæ $P_{Obs}$ od $P_{\hat{\,\Xi}}$ nie jest na ogó³ najmniejsza z mo¿liwych [@Amari; @Nagaoka; @book]. Geodezyjna ta przecina ${\cal S}_{w}$ w pewnym punkcie nale¿¹cym do ${\cal S}$, który jest szukanym [*oszacowaniem*]{} $P_{\hat{\,\Xi}}$ stanu uk³adu. Jak wspomnieliœmy, w statystyce klasycznej istnieje jedno u³atwiaj¹ce estymacjê za³o¿enie. Otó¿ znana jest ogólna postaæ modelu statystycznego ${\cal S}$, zatem znana jest i metryka Fishera-Rao na ${\cal S}$.\ \ [**Uwaga o estymacji w EFI**]{}: W przeciwieñstwie do tego estymacja w metodzie EFI nie mo¿e za³o¿yæ z góry znajomoœci postaci metryki $g$ Fishera-Rao. Metoda EFI musi wyestymowaæ $g$ i [*estymacja ta jest dynamiczna*]{}, poprzez konstrukcjê odpowiednich [*zasad informacyjnych*]{}.\ [**Zasada wariacyjna**]{}: Jedna z tych zasad powinna zapewniæ, ¿e po wyestymowaniu metryki Fishera-Rao, znalezione oszacowanie $P_{\hat{\,\Xi}}$ metody EFI bêdzie równie¿ le¿eæ na geodezyjnej ³¹cz¹cej je z $P_{Obs}$. St¹d pojawia siê koniecznoœæ wprowadzenia [*wariacyjnej zasady informacyjnej*]{}.\ [**Zasada strukturalna**]{}: Druga tzw. [*strukturalna zasada informacyjna*]{} zapewni, ¿e szukane oszacowanie bêdzie le¿eæ w klasie rozwi¹zañ [*analitycznych*]{} w parametrze $\Xi\,$, w znaczeniu równowa¿noœci metrycznej otrzymanego modelu z modelem analitycznym.\ Zatem zasada wariacyjna i strukturalna wyznaczaj¹ samospójnie punkt $P_{\hat{\,\Xi}}$ i tym samym wskazuj¹ podprzestrzeñ statystyczn¹ ${\cal S}_{w}$ . Jednak koneksja afiniczna, dla której uk³ad wspó³rzêdnych wzd³u¿ krzywej geodezynej ³¹cz¹cej $P_{Obs}$ z $P_{\hat{\,\Xi}}$ jest p³aski, nie jest $\alpha$-koneksj¹ Amariego.\ Sformu³owaniem i zastosowaniem zasad informacyjnych w estymacji metod¹ EFI zajmiemy siê w kolejnych rozdzia³ach. ### Uwaga o rozwiniêciu rozk³adu w szereg Taylora {#Uwaga o rozwinieciu funkcji w szereg Taylora} W ca³ej treœci skryptu zak³adamy, ¿e rozk³ad prawdopodobieñstwa $P(\Xi)$ (lub jego logarytm $\ln P(\Xi)$), jest wystarczaj¹co g³adki, tzn. posiada [*rozwiniêcie w szereg Taylora*]{} wystarczaj¹co wysokiego rzêdu, w ka¿dym punkcie (pod)przestrzeni statystycznej ${\cal S}$ [@Dziekuje; @informacja_2]. Zatem albo jest spe³niony warunek analitycznoœci rozk³adu we wszystkich sk³adowych estymowanego parametru $\Xi$, albo przynajmniej rozk³ad prawdopodobieñstwa okreœlony w otoczeniu punktu $P \in {\cal S}$ posiada d¿et $J_{P}^{r}({\cal S},\text{R})$ wystarczaj¹co wysokiego, choæ skoñczonego rzêdu $r$, np. wyrazy do drugiego (lub innego okreœlonego, wy¿szego) rzêdu rozwiniêcia w szeregu Taylora, podczas gdy wy¿sze ni¿ $r$ rzêdy rozwiniêcia znikaj¹ [@Murray_differential; @geometry; @and; @statistics]. Do rozwa¿añ na temat rzêdu d¿etów powrócimy w Rozdziale (\[equations of motion\]).\ \ [**Przestrzeñ wektorowa d¿etów**]{}: Istotn¹ spraw¹ jest fakt, ¿e zbiór wszystkich $r$-d¿etów funkcji w punkcie $P \in {\cal S}$ tworzy skoñczenie wymiarow¹ przestrzeñ wektorow¹, natomiast ich suma $J^{r}({\cal S},\text{R}) = \bigcup_{P \in {\cal S}} J_{P}^{r}({\cal S},\text{R})$ jest [*wi¹zk¹ wektorow¹, czyli wi¹zk¹ w³óknist¹[^31], której w³ókno jest przestrzeni¹ wektorow¹ nad przestrzeni¹ bazow¹ ${\cal S}$*]{}.\ Mo¿na pokazaæ, ¿e równie¿ $J_{P}^{\infty}({\cal S},\text{R})$ jest przestrzeni¹ wektorow¹. Zatem d¿ety nale¿¹ce do $J_{P}^{\infty}({\cal S},\text{R})$ mo¿na dodawaæ i mno¿yæ przez liczbê. Tworz¹ one te¿ algebrê co oznacza, ¿e mo¿na je mno¿yæ. Wa¿noœæ przestrzeni $J_{P}^{\infty}({\cal S},\text{R})$ ujawnia siê przy okreœleniu rozwiniêcia funkcji w szereg Taylora.\ [**Klasy równowa¿noœci d¿etów**]{}: O funkcjach mówimy, ¿e s¹ w tej samej klasie równowa¿noœci d¿etów, gdy maj¹ takie samo rozwiniêcie Taylora.\ \ [**Pojêcie odwzorowania Taylora na ${\cal S}$**]{}: Niech $T_{P}^{*}$ jest przestrzeni¹ wektorow¹ dualn¹ do przestrzeni stycznej $T_{P}$ na przestrzeni statystycznej ${\cal S}$, oraz niech $S^{k}(T^{*}_{P})$ jest przestrzeni¹ wektorow¹ wszystkich [*symetrycznych[^32] wieloliniowych odwzorowañ*]{}: $$\begin{aligned} \label{odwzorowanie S k} \underbrace{T_{P} \times \cdots \times T_{P}}_{k-razy} \rightarrow \text{R} \; .\end{aligned}$$ [*Rozwiniêcie w szereg Taylora*]{} $T$ pewnej funkcji (np. $P(\Xi)$ lub $\ln P(\Xi)$) na ${\cal S}$ okreœla wtedy odwzorowanie: $$\begin{aligned} \label{odwzorowanie Taylora} T: J_{P}^{\infty}({\cal S},\text{R}) \rightarrow S(T^{*}_{P}) \;\;\; {\rm gdzie} \;\;\; S(T^{*}_{P}) \equiv \bigoplus_{k \geq 0} S^{k}(T^{*}_{P}) \; ,\end{aligned}$$ nazywane [*odwzorowaniem Taylora*]{}. Szeregi Taylora spe³niaj¹ istotn¹ rolê w analizie statystycznej [@Murray_differential; @geometry; @and; @statistics], o czym przekonamy siê przy wyprowadzeniu podstawowego narzêdzia estymacji metody EFI, a mianowicie strukturalnej zasady informacyjnej (por. Rozdzia³ \[structural principle\]). Twierdzenie Rao-Cramera i DORC {#r-c} ------------------------------ Estymatory MNW maj¹ asymptotycznie optymalne w³asnoœci, tzn. s¹ nieobci¹¿one, zgodne, efektywne i dostateczne [@Nowak]. Poni¿szy rozdzia³ poœwiêcimy efektywnoœci nieobci¹¿onych estymatorów parametru dla dowolnej wielkoœci próby $N$.\ \ [**Estymator efektywny**]{}: Wartoœæ dolnego ograniczenia na wariancjê estymatora, czyli wariancjê estymatora efektywnego, podaje poni¿sze twierdzenie Rao-Cramera. Jego sednem jest stwierdzenie, ¿e osi¹gniêcie przez estymator dolnej granicy wariancji podanej w twierdzeniu oznacza, ¿e w klasie estymatorów nieobci¹¿onych, które spe³niaj¹ warunek regularnoœci (tzn. maj¹ funkcjê rozk³adu prawdopodobieñstwa nie posiadaj¹c¹ punktów nieci¹g³oœci zale¿nych od estymowanego parametru $\Theta$), nie znajdziemy estymatora z mniejsz¹ wariancj¹.\ [*Estymator efektywny ma wiêc najmniejsz¹ z mo¿liwych wariancji, jak¹ mo¿emy uzyskaæ w procesie estymacji parametru.* ]{} ### Skalarne Twierdzenie Rao-Cramera {#r-c-skalarne} **Twierdzenie Rao-Cramera (TRC). Przypadek skalarny**: Niech $F(\widetilde{Y})$ bêdzie nieobci¹¿onym estymatorem funkcji skalarnego parametru $g\left(\theta\right)$, tzn.: $$\begin{aligned} \label{E funkcji par skalarnego} E_{\theta}F\left({\widetilde{Y}}\right)=g\left(\theta\right)\end{aligned}$$ oraz niech $I_{F}(\theta)$ bêdzie informacj¹ Fishera dla parametru $\theta$ wyznaczon¹ na podstawie próby $\widetilde{Y}$. Zak³adaj¹c warunki regularnoœci, otrzymujemy: $$\begin{aligned} \label{tw R-C dla funkcji par skalarnego} {\sigma^{2}}_{\theta}F\left({\widetilde{Y}}\right)\ge \frac{{\left[{g'\left(\theta\right)}\right]^{2}}}{I_{F}\left(\theta\right)} \; ,\end{aligned}$$ co jest tez¹ twierdzenia Rao-Cramera. W szczególnym przypadku, gdy $g(\theta)=\theta$, wtedy z (\[tw R-C dla funkcji par skalarnego\]) otrzymujemy nastêpuj¹c¹ postaæ nierównoœci Rao-Cramera: $$\begin{aligned} \label{tw R-C dla par skalarnego} {\sigma^{2}}_{\theta} F\left({\widetilde{Y}}\right) \ge \frac{{1}}{I_{F}\left(\theta\right)} \; .\end{aligned}$$ Wielkoœæ: $$\begin{aligned} \label{dolne ogr R-C dla funkcji par skalar} \frac{\left[{g'\left(\theta\right)}\right]^{2}}{I_{F}\left(\theta\right)} \;\;\; {\rm lub} \;\;\; \frac{1}{I_{F}\left(\theta\right)} \;\;\; {\rm dla} \;\;\; g(\theta) = \theta \;\end{aligned}$$ nazywana jest dolnym ograniczeniem Rao-Cramera (DORC)[^33]. Przypomnijmy, ¿e poniewa¿ statystyka $F(\widetilde{Y})$ jest estymatorem parametru $\theta$, wiêc wartoœci jakie przyjmuje nie zale¿¹ od tego parametru.\ \ [**Uwaga**]{}: W przypadku, gdy rozk³ad zmiennej $Y$ traktowany jako funkcja estymowanego parametru $\theta$ ma dla pewnych wartoœci tego parametru punkty nieci¹g³oœci, wtedy wariancja estymatora parametru $\theta$ wystêpuj¹ca po lewej stronie (\[tw R-C dla par skalarnego\]) mo¿e okazaæ siê mniejsza ni¿ wartoœæ po stronie prawej. Sytuacji nieci¹g³oœci rozk³adu w parametrze nie bêdziemy jednak rozwa¿ali. Przeciwnie, [*zak³adamy, ¿e rozk³ad $P(\Theta)$, i jej logarytm $\ln P(\Theta)$, jest wystarczaj¹co g³adki*]{}, tzn. posiada [*rozwiniêcie w szereg Taylora*]{} wystarczaj¹co wysokiego rzêdu, w ka¿dym punkcie (pod)przestrzeni statystycznej ${\cal S}$, jak o tym wspomnieliœmy w Rozdziale \[Uwaga o rozwinieciu funkcji w szereg Taylora\]. #### Dowód TRC (wersja dowodu dla przypadku skalarnego) [*Wspó³czynnik korelacji liniowej Pearsona*]{} dla dwóch zmiennych losowych $S(\widetilde{Y})$ i $F(\widetilde{Y})$ zdefiniowany jest nastêpuj¹co: $$\begin{aligned} \label{wsp kor Piersona dla S F} \rho_{\theta}\left({S,F}\right)=\frac{{{\mathop{\rm cov}}_{\theta} \left({S,F}\right)}}{{\sqrt{{\mathop{\sigma_{\theta}^{2}}}\left(S\right)}\sqrt{{\mathop{\sigma_{\theta}^{2}}}\left(F\right)}}} \; .\end{aligned}$$ Z klasycznej analizy statystycznej wiemy, ¿e $\rho_{\theta}\left({S,F}\right)\in\left[{-1,1}\right]$, st¹d z (\[wsp kor Piersona dla S F\]) otrzymujemy: $$\begin{aligned} \label{nier dla F S i covSF} {\sigma_{\theta}^{2}}\left(F\right) \ge \frac{{\left|{{\mathop{\rm cov}}_{\theta}\left({S,F}\right)}\right|^{2}}}{{{\mathop{\sigma_{\theta}^{2}}}\left(S\right)}} \; .\end{aligned}$$ Równoœæ wystêpuje je¿eli wspó³czynnik korelacji liniowej Pearsona jest równy 1, co zachodzi, gdy zmienne $S$ i $F$ s¹ idealnie skorelowane.\ \ Niech teraz zmienna losowa $S$ bêdzie statystyk¹ wynikow¹ $S(\theta) \equiv S(\widetilde{Y}|\theta)$.\ Poka¿my, ¿e: $$\begin{aligned} \label{postac g'} g'(\theta) = {\rm cov}_{\theta} \left(S(\theta),F\right) \; . \end{aligned}$$ Istotnie, poniewa¿ $E_{\theta}\left(S(\theta)\right) = 0$, (\[znikanie ES\]), zatem: $$\begin{aligned} \label{dow covSF = g'theta} \!\!\!\!\!\!\!\!\! & & {\rm cov}_{\theta} \left(S(\theta),F(\widetilde{Y})\right) = E_{\theta}\left(S(\theta) F(\widetilde{Y})\right) = \int_{\cal B} dy \, P(y|\theta) S(\theta) F(y) = \int_{\cal B} dy P(y|\theta) \frac{\frac{\partial}{\partial \theta} P(y|\theta)}{P(y|\theta)} F(y) \nonumber \;\;\; \\ \!\!\!\!\!\!\!\!\! &=& \int_{\cal B} dy \frac{\partial}{\partial \theta} P(y|\theta) F(y) = \frac{\partial}{\partial \theta} \int_{\cal B} dy \, P(y|\theta) F(y) = \frac{\partial}{\partial \theta} E_{\theta}F(\widetilde{Y}) = g'(\theta) \; . \;\;\;\end{aligned}$$ Skoro wiêc zgodnie z (\[var S oraz IF\]) zachodzi, $\sigma_{\theta}^{2}S(\theta)=I_{F}(\theta)$, wiêc wstawiaj¹c (\[dow covSF = g’theta\]) do (\[nier dla F S i covSF\]) otrzymujemy (\[tw R-C dla funkcji par skalarnego\]), co koñczy dowód TRC. #### Przyk³ad skalarny DORC dla rozk³adu normalnego {#DORC dla rozkl norm} Interesuje nas [*przypadek estymacji skalarnego parametru*]{} $\theta=\mu$ w próbie prostej $\widetilde{Y}$, przy czym zak³adamy, ¿e $g(\mu) = \mu$. Rozwa¿my œredni¹ arytmetyczn¹ $\bar{Y}=\frac{1}{N}\sum_{n=1}^{N} Y_{n}$ (z realizacj¹ $\bar{\bf y}=\frac{1}{N}\sum_{n=1}^{N} {\bf y}_{n}$), która, zak³adaj¹c jedynie identyczne rozk³ady zmiennych $Y_{i}$ próby, jest dla dowolnego rozk³adu $p({\bf y})$ zmiennej $Y$, nieobci¹¿onym estymatorem wartoœci oczekiwanej $\mu \equiv E_{\mu}(Y) = \int_{\cal Y} d{\bf y} p({\bf y})\, {\bf y}$, tzn.: $$\begin{aligned} \label{E dla sredniej} E_{\mu}\left(\bar{Y}\right) = \int_{\cal B} dy \, \bar{{\bf y}} \, P(y|\mu) = E_{\mu}(Y) = \mu \; .\end{aligned}$$ Ponadto dla próby prostej, z bezpoœredniego rachunku otrzymujemy: $$\begin{aligned} \label{war dla sredniej y} {\mathop{\sigma_{\mu}^{2}}}\left({\bar{Y}}\right) = \int_{\cal B} dy \,P(y|\mu)\, (\bar{{\bf y}}-E(\bar{Y}))^{2} = \frac{{\sigma^{2}}}{N} \; ,\end{aligned}$$ gdzie $\sigma^2$ jest wariancj¹ ${\sigma_{\mu}^{2}}(Y)$ zmiennej $Y$.\ \ Niech teraz zmienna pierwotna $Y$ ma rozk³ad normalny $N(\mu, \sigma^2)$. Ze zwi¹zków (\[log wiaryg rozklad norm jeden par\]) oraz (\[rown wiaryg skal\])-(\[srednia arytmet z MNW\]) wiemy, ¿e œrednia $\bar{Y}$ jest estymatorem MNW parametru $\mu = E(Y)$, zatem przyjmijmy $F\left(\widetilde{Y}\right) = \hat{\mu} = \bar{Y}$. Z (\[war dla sredniej y\]) widzimy wiêc, ¿e dla zmiennych o rozk³adzie normalnym zachodzi: $$\begin{aligned} \label{war F dla normalnego} \sigma_{\mu}^{2}\left(F\right) = \sigma_{\mu}^{2}\left(\bar{Y}\right) = \frac{\sigma^{2}}{N} \,\; .\end{aligned}$$\ W przypadku [*rozk³adu normalnego*]{} warunek (\[tw R-C dla par skalarnego\]) ³atwo sprawdziæ bezpoœrednim rachunkiem. Istotnie, korzystaj¹c z (\[f wynikowa\_1 wym N\_1 par\]) oraz (\[I oczekiwana dla N\_parametr mu\]), otrzymujemy (por. (\[I oczekiwana dla N\_parametr mu\])): $$\begin{aligned} \label{sprawdzenie RC dla N} {\mathop{\sigma_{\mu}^{2}}}S\left(\mu\right)={\mathop{\sigma_{\mu}^{2}}}\left({\frac{N}{{\sigma^{2}}}\left({\bar{Y} - \mu}\right)}\right)=\left({\frac{N}{{\sigma^{2}}}}\right)^{2}\frac{{\sigma^{2}}}{N}=\frac{N}{{\sigma^{2}}} = I_{F}(\mu) \; .\end{aligned}$$ Z (\[war F dla normalnego\]) oraz (\[sprawdzenie RC dla N\]) otrzymujemy: $$\begin{aligned} \label{DORC dla rozkl norm wzor} {\sigma_{\mu}^{2}}({\hat{\mu}}) = \frac{1}{I_F(\mu)} \; ,\end{aligned}$$ co stanowi DORC (\[dolne ogr R-C dla funkcji par skalar\]) dla nierównoœci Rao-Cramera (\[tw R-C dla par skalarnego\]). Warunek ten otrzymaliœmy ju¿ poprzednio dla rozk³adu normalnego (por. (\[RC dla 1 N z 1 par oczekiwana IF\])). Spe³nienie go oznacza, ¿e œrednia arytmetyczna $\bar{Y}$ jest efektywnym[^34] estymatorem wartoœci oczekiwanej zmiennej losowej opisanej rozk³adem normalnym $N(\mu, \sigma^2)$.\ \ [**Uwaga o rozk³adach eksponentialnych**]{}: Rozk³ad normalny jest szczególnym przypadkiem szerszej klasy rozk³adów, które spe³niaj¹ warunek DORC. Rozk³ady te s¹ tzw. rozk³adami eksponentialnymi (\[exponential family\]) wprowadzonymi w Rozdziale \[alfa koneksja\]. Powy¿ej, w przypadku rozk³adu normalnego, sprawdziliœmy ten fakt bezpoœrednim rachunkiem zak³adaj¹c wpierw typ rozk³adu zmiennej $Y$, a potem sprawdzaj¹c, ¿e estymator $\hat{\mu}$ parametru $\mu = E_{\mu}(Y)$ osi¹ga DORC.\ ### Wieloparametrowe Twierdzenie Rao-Cramera {#r-c-wieloparametrowe} Gdy dokonujemy równoczesnej estymacji $d>1$ parametrów, wtedy funkcja wynikowa $S(\Theta)$ jest $d$-wymiarowym wektorem kolumnowym (\[funkcja wynikowa\]), natomiast obserwowana IF w punkcie ${\hat{\Theta}}$, czyli $\texttt{i\!F}({\hat{\Theta}})$, oraz wartoœæ oczekiwana z $\texttt{i\!F}(\Theta)$, czyli $I_{F}$ (por.(\[infoczekiwana\])), s¹ $d\times d$ wymiarowymi macierzami.\ \ [**Analogia inflacji wariancji**]{}: Poni¿ej poka¿emy, ¿e w³¹czenie do analizy dodatkowych parametrów ma (na ogó³) wp³yw na wartoœæ IF dla interesuj¹cego nas, wyró¿nionego parametru. Sytuacja ta jest analogiczna do problemu inflacji wariancji estymatora parametru w analizie czêstotliwoœciowej [@Kleinbaum]. Poni¿ej przedstawiona zostanie odnosz¹ca siê do tego problemu wieloparametrowa wersja twierdzenia o dolnym ograniczeniu w nierównoœci Rao-Cramera (DORC).\ \ [**Uwaga o wersjach TRC**]{}: Poni¿ej podamy dwie równowa¿ne wersje [@Amari; @Nagaoka; @book] wieloparametrowego Twierdzenia Rao-Cramera (TRC). Pierwsza z nich oka¿e siê byæ bardzo u¿yteczna przy wprowadzeniu w Rozdziale \[Pojecie kanalu informacyjnego\] relacji pomiêdzy tzw. informacj¹ Stam’a a pojemnoœci¹ informacyjn¹ uk³adu. W Rozdziale \[Estymacja w modelach fizycznych na DORC\] przekonamy siê, ¿e wersja druga TRC jest u¿yteczna w sprawdzeniu czy wieloparametrowa estymacja przebiega na DORC. #### Pierwsza wersja wieloparametrowego TRC {#Pierwsza wersja wieloparametrowego TRC} **Wieloparametrowe Twierdzenie RC (wersja pierwsza)**: Niech $F(\widetilde{Y})$ bêdzie funkcj¹ [*skalarn¹*]{} z wartoœci¹ oczekiwan¹: $$\begin{aligned} \label{ET dla DORC wielopar} E_{\Theta}F\left({\widetilde{Y}}\right)=g\left(\Theta \right) \in \mathbf{R} \; \end{aligned}$$ oraz $I_{F}(\Theta)$ niech bêdzie oczekiwan¹ informacj¹ Fishera (\[infoczekiwana\]) dla $\Theta \equiv (\theta_{i})_{i=1}^{d}$ wyznaczon¹ na przestrzeni próby ${\cal B}$. Zachodzi wtedy nierównoœæ: $$\begin{aligned} \label{rc wielop} {\sigma^{2}}_{\!\Theta} \left(F(\widetilde{Y}) \right) \ge {\bf a}^{T} I_{F}^{-1} \left(\Theta \right) {\bf a} \; ,\end{aligned}$$ gdzie $I_{F}^{-1}$ jest macierz¹ odwrotn¹ do macierzy informacyjnej Fishera $I_{F}$, natomiast $$\begin{aligned} \label{alfa} {\bf a} = \frac{\partial g\left(\Theta \right)}{\partial \Theta} \, \; \end{aligned}$$ jest $d$-wymiarowym wektorem.\ \ [**Uwaga**]{}: Warunek (\[ET dla DORC wielopar\]) oznacza, ¿e skalarna funkcja $F({\widetilde{Y}})$ jest nieobci¹¿onym estymatorem skalarnej funkcji $g\left(\Theta \right)$ wektorowego parametru $\Theta$. #### Przyk³ad wektorowego DORC {#Przyklad wektorowego DORC} Jako ilustracjê powy¿szego wieloparametrowego Twierdzenia RC przedstawimy przyk³ad, przyjmuj¹c szczególn¹ postaæ skalarnej funkcji $g(\Theta)$, o której zak³adamy, ¿e jest liniow¹ funkcj¹ sk³adowych $\theta_{i}$ wektora parametrów [@Pawitan]: $$\begin{aligned} \label{uwad} g\left(\Theta \right) = {\bf a}^{T} \Theta = \sum_{i=1}^{d} a_{i} \, \theta_{i} \; ,\end{aligned}$$ gdzie ${\bf a}$ jest pewnym znanym wektorem o sta³ych sk³adowych $a_{i}$, które nie zale¿¹ od sk³adowych wektora $\Theta$. Za³ó¿my chwilowo, ¿e ${\bf a}^{T}=\left({1,0,\ldots,0}\right)$, tzn. jedynie $a_{1} \neq 0$. Wtedy z (\[uwad\]) otrzymujemy $g\left(\Theta \right)=\theta_{1}$, natomiast (\[rc wielop\]) w Twierdzeniu RC, ${\sigma^{2}}_{\! \Theta} \left(F \right) \ge {\bf a}^{T} I_{F}^{-1} \left(\Theta \right) {\bf a}$, przyjmuje dla rozwa¿anego nieobci¹¿onego estymatora $F$ parametru $\theta_{1}$, tzn. $E_{\Theta}(F({\widetilde{Y}})) = \theta_{1}$, postaæ: $$\begin{aligned} \label{RG dla a1} {\sigma_{\Theta}^{2}}\left(F\right) \ge \left[{ I_{F}^{-1}\left( \Theta \right)}\right]_{11} =: I_{F}^{11}\left(\Theta \right) \; ,\end{aligned}$$ gdzie $I_{F}^{11}{(\Theta)}$ oznacza element (1,1) macierzy $I_{F}^{-1}\left(\Theta\right)$. Prawa strona nierównoœci (\[RG dla a1\]) podaje dolne ograniczenie wariancji estymatora $F$, pod warunkiem, ¿e $\theta_{1}$ jest wyró¿nionym parametrem a wartoœci pozosta³ych parametrów [*nie s¹ znane*]{}. Oznaczmy wewnêtrzn¹ strukturê $d \times d\,$-wymiarowych macierzy $ I_{F}(\Theta)$ oraz $I_{F}(\Theta)^{-1}$ nastêpuj¹co: $$\begin{aligned} I_{F} \left(\Theta \right) = \left({\begin{array}{cc} {I_{F 11}} & {I_{F 12}}\\ {I_{F 21}} & {I_{F 22}}\end{array}}\right)\end{aligned}$$ oraz $$\begin{aligned} I_{F}^{-1}\left(\Theta \right)=\left({\begin{array}{cc} {I_{F}^{11}} & {I_{F}^{12}}\\ {I_{F}^{21}} & {I_{F}^{22}}\end{array}}\right)\end{aligned}$$ gdzie $I_{F 11}$ oraz $I_{F}^{11} = \left[{ I_{F}^{-1}\left( \Theta \right)}\right]_{11}$ (zgodnie z oznaczeniem wprowadzonym w (\[RG dla a1\])) s¹ liczbami, $I_{F 22}$, $I_{F}^{22}$ s¹ $(d-1) \times (d-1)$-wymiarowymi macierzami, natomiast $(I_{F 12})_{1 \times (d-1)}$, $(I_{F 21})_{(d-1) \times 1}$, $(I_{F}^{12})_{1 \times (d-1)}$, $(I_{F}^{21})_{(d-1) \times 1}$ odpowiednimi wierszowymi b¹dŸ kolumnowymi wektorami o wymiarze $(d-1)$.\ \ [**Rozwa¿my parametr $\theta_{1}$**]{}. Jego informacja Fishera (patrz poni¿ej [**Uwaga o nazwie**]{}) jest równa $I_{F 11}=I_{F 11}\left(\theta_{1} \right)$. [*Nie oznacza to jednak*]{}, ¿e $\sigma_{\Theta}^{2}\left(F\right)$ oraz $I_{F 11}$ s¹ z sob¹ automatycznie powi¹zane nierównoœci¹ $\sigma_{\Theta}^{2}\left(F\right) \ge 1/I_{F 11}$, która jest treœci¹ Twierdzenia RC (\[tw R-C dla par skalarnego\]). Udowodniliœmy j¹ bowiem tylko dla przypadku parametru skalarnego, tzn. gdy tylko jeden parametr jest estymowany, a reszta parametrów jest znana.\ Okreœlmy relacjê pomiêdzy $(I_{F 11})^{-1}$ oraz $I_{F}^{11}$. Oczywiœcie zachodzi: $$\begin{aligned} I_{F}\left( \Theta \right) \cdot I_{F}^{-1} \left( \Theta \right) = \left({\begin{array}{cc} {I_{F 11}} & {I_{F 12}}\\ {I_{F 21}} & {I_{F 22}}\end{array}}\right)\left({\begin{array}{cc} {I_{F}^{11}} & {I_{F}^{12}}\\ {I_{F}^{21}} & {I_{F}^{22}}\end{array}}\right)=\left({\begin{array}{cc} 1 & 0 \\ 0 & \textbf{1} \end{array}}\right) \; , \end{aligned}$$ gdzie $\textbf{1}$ jest $(d-1) \times (d-1)$-wymiarow¹ macierz¹ jednostkow¹.\ Z powy¿szego mamy: $$\begin{aligned} &(I_{F 11})_{1\times 1}& \!\!\!\! (I_{F}^{11})_{1\times 1} + (I_{F 12})_{1\times (d-1)} (I_{F}^{21})_{(d-1) \times 1} = 1 \nonumber \\ &\Rightarrow& \;\;\;\; \left( {I_{F}^{11}}\right)^{-1} = I_{F 11} + I_{F 12} I_{F}^{21} \left({I_{F}^{11}} \right)^{-1} \; ,\end{aligned}$$ $$\begin{aligned} &(I_{F 21})_{(d-1) \times 1}& \!\!\!\! (I_{F}^{11})_{1\times 1} + (I_{F 22})_{(d-1)\times(d-1)} \; (I_{F}^{21})_{(d-1) \times 1} = (0)_{(d-1)\times 1} \nonumber \\ &\Rightarrow& \;\;\;\; I_{F}^{21} = -\left({I_{F 22}}\right)^{-1} I_{F 21} I_{F}^{11}\end{aligned}$$ sk¹d otrzymujemy: $$\begin{aligned} \label{rownanie macierzowa dla I} \left({I_{F}^{11}}\right)^{-1} = I_{F 11}-I_{F 12}\left({I_{F 22}}\right)^{-1}I_{F 21} \; .\end{aligned}$$ Poniewa¿ $I_{F 22}$ jest macierz¹ informacyjn¹ (dla parametrów $\theta_{2},\theta_{3},...,\theta_{d}$), jest wiêc ona zgodnie z rozwa¿aniami przedstawionymi poni¿ej (\[iF polokreslona\]), symetryczna i nieujemnie okreœlona. Symetryczna i nieujemnie okreœlona jest zatem $(I_{F 22})^{-1}$. Poniewa¿ z symetrii macierzy $I_{F}$ wynika $I_{F 12}=(I_{F 21})^{T}$, zatem ostatecznie forma kwadratowa $I_{F 12} \left({I_{F 22}} \right)^{-1} I_{F 21} \geq 0$, st¹d z (\[rownanie macierzowa dla I\]) otrzymujemy: $$\begin{aligned} \label{porownanie I11 z I11do-1} (I^{F 11})^{-1} \le I_{F 11} \;\; \Rightarrow \;\; I^{F 11} \geq \frac{1}{I_{F 11}} \; ,\end{aligned}$$ co zgodnie z (\[RG dla a1\]) oznacza, ¿e: $$\begin{aligned} \label{porownanie sigma I11 z I11do-1} \sigma_{\Theta}^{2}\left(F\right) \geq I_{F}^{11} \geq \frac{1}{ I_{F 11}} \; .\end{aligned}$$ [**Wniosek**]{}: Zatem widzimy, ¿e $I_{F}^{11}$ daje [*silniejsze ograniczenie*]{} ni¿ $(I_{F 11})^{-1}$. Tzn. w przypadku estymacji wieloparametrowej nale¿y zastosowaæ zwi¹zek $\sigma^{2}\left(F\right) \geq I_{F}^{11}$, (\[RG dla a1\]), gdy¿ to w³aœnie on jest w³aœciwy na podstawie wieloparametrowego Twierdzenia RC. Zastosowanie $\sigma_{\Theta}^{2}\left(F\right) \geq 1/I_{F 11}$, tak jak byœmy mieli do czynienia z przypadkiem skalarnym, mo¿e b³êdnie zani¿yæ wartoœæ dolnego ograniczenia na $\sigma_{\Theta}^{2}\left(F\right)$.\ \ [**Uwaga o nazwie $I_{F 11}$**]{}: W “statycznie” ukierunkowanej analizie statystycznej wielkoœæ $(I_{F}^{11})^{-1}$ jest interpretowana jako informacja Fishera dla $\theta_{1}$ – [**jednak w treœci skryptu odst¹pimy od tej nazwy**]{}. Okazuje siê, ¿e w analizie ukierunkowanej na estymacjê “dynamiczn¹”, tzn. generuj¹c¹ równania ró¿niczkowe dla rozk³adów, bardziej u¿yteczne jest nazwaæ $I_{F}^{11}$ po prostu [*dolnym ograniczeniem RC na wariancjê estymatora parametru*]{} $\theta_{1}$ w sytuacji gdy pozosta³e parametry s¹ nieznane (tzn. trzeba je estymowaæ z próby równoczeœnie z $\theta_{1}$). [*Natomiast $I_{F 11}$ bêdziemy nazywali, zgodnie z tym jak to uczyniliœmy, informacj¹ Fishera parametru $\theta_{1}$*]{} i to niezale¿nie od tego czy inne parametry s¹ równoczeœnie estymowane, czy te¿ nie.\ \ [**Podsumowanie na temat zani¿enia DORC**]{}: Nale¿y pamiêtaæ, ¿e estymuj¹c parametr $\theta_{1}$ nale¿y byæ œwiadomym faktu wystêpowania równoczesnej estymacji innych parametrów, gdy¿ wstawienie wartoœci $I_{F 11}$ do nierównoœci RC mo¿e w przypadku estymacji wieloparametrowej doprowadziæ do zani¿enia wartoœci dolnego ograniczenia wariancji tego parametru.\ \ [**Przypadek “pseudo-skalarny”**]{}: Istnieje jednak pewien wyj¹tek spowodowany dok³adnym zerowaniem siê $I_{F 12}$ dla dowolnego $N$. Wtedy z (\[rownanie macierzowa dla I\]) wynika, ¿e wzrost wariancji estymatora parametru zwi¹zany z dodaniem nowych parametrów o nieznanych wartoœciach by³by równy zeru. Tak te¿ by³o w rozwa¿anym wczeœniej przyk³adzie rozk³adu normalnego $N\left({\mu,\sigma}\right)$ (porównaj (\[oczekiw iF r normalnego 2 par\]) z (\[I oczekiwana dla N\_parametr mu\])).\ Podobnie, taki szczególny przypadek zachodzi, gdy wieloparametrowym rozk³adem prawdopodobieñstwa jest wiarygodnoœæ $N$-wymiarowej próby $P(\theta_1,\theta_2,,...,\theta_{N}) = \prod_{n=1}^{N} p_{\theta_{n}}$, gdzie ka¿dy estymowany parametr $\theta_n$ okreœla tylko jeden punktowy rozk³ad $p_{\theta_{n}}$. Wtedy macierz informacyjna Fishera $I_{F}$ jest diagonalna i zachodzi $I_{F nn}=(I_{F}^{nn})^{-1}$, a w miejsce (\[porownanie sigma I11 z I11do-1\]) otrzymujemy dla ka¿dego parametru $\theta_{n}$: $$\begin{aligned} \label{I11 rowna sie I11do-1 przypadek diagonalny} \sigma_{\Theta}^{2}\left(F_{n}\right) \geq I_{F}^{nn} = \frac{1}{ I_{F nn}} \; , \;\;\; n=1,2,...,N \;, \;\;\;\; {\rm gdy} \;\;\; I_{F} \; \; {\rm diagonalne} \; ,\end{aligned}$$ jako szczególny przypadek nierównoœci RC, gdzie $F_{n}$ jest estymatorem $\hat{\theta}_{n}$ parametru $\theta_{n}$. #### Druga wersja wieloparametrowego TRC {#Druga wersja wieloparametrowego TRC} Niech $\hat{\Theta} \equiv \hat{\Theta}(\widetilde{Y}) = (\hat{\theta}_{i}(\widetilde{Y}))_{i=1}^{d}$ jest nieobci¹¿onym estymatorem parametru $\Theta~=~(\theta_{i})_{i=1}^{d}$, co oznacza, ¿e zachodzi: $$\begin{aligned} \label{wektorowy nieobciazany estymator} \Theta = E_{\Theta}\left[ \hat{\Theta}(\widetilde{Y}) \right] \; , \;\;\;\;\; \forall \, P_{\Theta} \in {\cal S} \; .\end{aligned}$$ [**(Oczekiwan¹) macierz¹ kowariancji**]{}[^35] $V_{\Theta}[ \hat{\Theta} ]$ nieobci¹¿onego estymatora $\hat{\Theta}$ w bazie $\Theta$ nazywamy $d \times d$ - wymiarow¹ macierz o elementach: $$\begin{aligned} \label{oczekiwana macierz kowariancji} V_{\Theta \, ij} [ \hat{\Theta} ] := E_{\Theta}\left[ (\hat{\theta}_{i}(\widetilde{Y}) - \theta_{i})\; (\hat{\theta}_{j}(\widetilde{Y}) - \theta_{j}) \right] \; , \;\;\;\;\; i,j = 1,2,...,d, \;\;\;\;\; \forall \, P_{\Theta} \in {\cal S} \; .\end{aligned}$$ Zachodzi nastêpuj¹ce twierdzenie.\ \ [**Wieloparametrowe Twierdzenie RC (wersja druga)**]{}: Macierz kowariancji $V_{\Theta}[ \hat{\Theta} ]$ nieobci¹¿onego estymatora $\hat{\Theta}$ spe³nia nierównoœæ [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{twierdzenie RC wersja 2} V_{\Theta}[ \hat{\Theta} ] \geq I_{F}^{-1}(\Theta) \;\;\;\;\;\;\;\;\;\; \forall \, P_{\Theta} \in {\cal S} \; ,\end{aligned}$$ co oznacza, ¿e macierz $V_{\Theta}[ \hat{\Theta} ] - I_{F}^{-1}(\Theta)$ jest dodatnio pó³okreœlona.\ \ [**Estymator efektywny**]{}: Nieobci¹¿ony estymator $\hat{\Theta}$ spe³niaj¹cy równoœæ w nierównoœci (\[twierdzenie RC wersja 2\]): $$\begin{aligned} \label{DORC wersja 2 TRC} V_{\Theta}[ \hat{\Theta} ] = I_{F}^{-1}(\Theta) \;\;\;\;\;\;\;\;\;\; \forall \, P_{\Theta} \in {\cal S} \; .\end{aligned}$$ nazywamy estymatorem [*efektywnym*]{} parametru $\Theta$. Entropia informacyjna Shannona i entropia wzglêdna {#shan} -------------------------------------------------- W rozdziale tym omówimy pojêcie, które podaje [*globaln¹ charakterystykê*]{} pojedynczego rozk³adu prawdopodobieñstwa, tzn. entropiê Shannona. Dok³adniejsze omówienie w³asnoœci entropii Shannona mo¿na znaleŸæ w [@Bengtsson_Zyczkowski]. Z treœci poni¿szych przyk³adów wynika jaki jest rozmiar próby $N$.\ \ [**Entropia Shannona**]{}: Niech $P(\omega)$ bêdzie rozk³adem prawdopodobieñstwa[^36] okreœlonym na przestrzeni zdarzeñ $\Omega$, gdzie $\omega$ jest puntem w $\Omega$. Jeœli przestrzeñ zdarzeñ $\Omega$ jest dyskretna, to [*informacyjna entropia Shannona*]{} jest zdefiniowana nastêpuj¹co: $$\begin{aligned} \label{shannon dla omega} S_{H}\left(P\right) = - k \sum\limits_{\omega \in \,\Omega} {P (\omega) \ln P (\omega)} \; ,\end{aligned}$$ gdzie $k$ jest liczb¹ dodatni¹[^37], któr¹ przyjmiemy dalej jako równ¹ 1.\ \ [**Rozk³ad prawdopodobieñstwa jako element sympleksu rozk³adów**]{}: Niech $\Omega$ jest rozpiêta przez skoñczon¹ liczbê $\aleph$ elementów bêd¹cych mo¿liwymi wynikami doœwiadczenia, w rezultacie którego otrzymujemy wartoœci ${\bf y}_{i}$, $i=1,2,...,\aleph$, zmiennej losowej $Y$. Rozk³ad prawdopodobieñstwa $P$ jest wtedy reprezentowany przez wektor $\vec{p} = (p_{i})_{i=1}^{\aleph}\,$, nale¿¹cy do [*sympleksu rozk³adów prawdopodobieñstwa*]{} [@Bengtsson_Zyczkowski], [*tzn. jego $\aleph$ sk³adowych spe³nia warunki*]{}: $$\begin{aligned} \label{rozklad prawdopod} p_{i} \ge 0 \;\;\;\; {\rm oraz} \;\;\;\; \sum \limits_{i=1}^{\aleph} p_{i} = 1 \; .\end{aligned}$$ [*m-Sympleks rozk³adów prawdopodobieñstwa, czyli zbiór*]{} $\Delta^{m}=\{\left(p_{1},p_{2},...,p_{m},p_{m+1} \right)$ $\in \mathbf{R}^{m+1} ;$ $p_{j} \ge 0, \; j=1,...,m+1$ gdzie $m+1 \leq \aleph$, dla $\sum_{j=1}^{m+1}{p_{j}} \le 1 \}$ [*jest zbiorem wypuk³ym*]{}, tzn. ka¿dy punkt $\vec{p}_{m} \in \Delta^{m}$ mo¿na przedstawiæ jako $\vec{p}_{m} = \sum_{i=1}^{m+1} \lambda_{i} p_{i}$, gdzie $\sum_{i=1}^{m+1} \lambda_{i} = 1 \,$.\ \ Entropia Shannona (\[shannon dla omega\]) rozk³adu (\[rozklad prawdopod\]) ma postaæ: $$\begin{aligned} \label{shannon} S_{H}\left(P\right) = - \sum\limits_{i=1}^{\aleph}{p_{i}\ln p_{i}} \; .\end{aligned}$$ Zapis $S_{H}(P)$, gdzie w argumencie pominiêto oznaczenie zmiennej losowej $Y$ podkreœla, ¿e [*jedyn¹ rozwa¿an¹ przez nas cech¹ zmiennej losowej jest jej rozk³ad prawdopodobieñstwa*]{} $P$. ### Interpretacja entropii Shannona Maksymalna mo¿liwa wartoœæ entropii Shannona[^38] wynosi $\ln \aleph$ i jest osi¹gniêta, gdy wszystkie wyniki s¹ równo prawdopodobne ($p_{i}=1/\aleph$), tzn. gdy stan uk³adu jest maksymalnie zmieszany.\ \ [**Uk³ad w stanie czystym**]{}: Gdy jeden z wyników jest pewny, wtedy tylko jedna, odpowiadaj¹ca mu wspó³rzêdna wektora $\vec{p}$ jest równa jeden, a pozosta³e s¹ równe 0. Mówimy wtedy, ¿e uk³ad znajduje siê w stanie czystym, a odpowiadaj¹ca mu wartoœæ entropii Shannona jest minimalna i równa zero.\ \ [**Entropia jako miara niepewnoœci wyniku eksperymentu**]{}: Z powy¿szych przyk³adów mo¿na wnioskowaæ, ¿e entropiê Shannona mo¿na interpretowaæ jako [*miarê niepewnoœci otrzymania wyniku eksperymentu*]{} bêd¹cego realizacj¹ rozk³adu prawdopodobieñstwa $P$ lub inaczej, jako wielkoœæ informacji koniecznej [*do okreœlenia*]{} wyniku, który mo¿e siê pojawiæ w rezultacie przeprowadzenia eksperymentu na uk³adzie.\ Podstawowe w³asnoœci entropii Shanonna mo¿na znaleŸæ w [@Bengtsson_Zyczkowski].\ ### Przypadek ci¹g³ego rozk³adu prawdopodobieñstwa {#Przypadek ci¹glego rozkladu prawdopodobienstwa} Przejœcie z dyskretnego do ci¹g³ego rozk³adu prawdopodobieñstwa polega na zast¹pieniu sumowania w (\[shannon\]) ca³kowaniem po ca³ym zakresie zmiennoœci zmiennej losowej $Y$. W ten sposób otrzymujemy Boltzmanowsk¹ postaæ entropii Shannona: $$\begin{aligned} \label{eboltmana} S_{H}(P) = - \int\limits_{-\infty}^{+\infty}{d{\bf y}\, P\left({\bf y}\right) \ln P\left({\bf y}\right)} \; .\end{aligned}$$ Jednak¿e dla pewnych funkcji rozk³adu $P({\bf y})$, ca³ka (\[eboltmana\]) mo¿e nie byæ okreœlona.\ \ [**Czysty stan klasyczny**]{}: Jako ilustracjê powy¿szego stwierdzenia rozwa¿my sytuacjê [@Bengtsson_Zyczkowski], gdy rozk³ad $P({\bf y})$ przyjmuje w przedziale $[0,t]$ wartoœæ $t^{-1}$ i zero wszêdzie poza tym przedzia³em. Wtedy jego entropia $S_{H}(P)$ jest równa $\ln t$ i dla $t \rightarrow 0$ d¹¿y do [*minus nieskoñczonoœci*]{}. Procedura ta odpowiada przejœciu do punktowego, [*czystego stanu klasycznego*]{} opisanego dystrybucj¹ delta Diraca, dla której $S_{H}(P) = - \infty$.\ Zatem przyjmuj¹c poziom zerowy entropii jako punkt odniesienia, dok³adne okreœlenie stanu opisanego delt¹ Diraca (której fizycznie móg³by odpowiadaæ nieskoñczony skok w gêstoœci rozk³adu substancji cz¹stki) wymaga dostarczenia nieskoñczonej iloœci informacji o uk³adzie. Do sprawy powrócimy w jednym z kolejnych rozdzia³ów.\ \ [**Problem transformacyjny $S_{H}(P)$**]{}: Definicja (\[eboltmana\]) ma pewien formalny minus, zwi¹zany z brakiem porz¹dnych w³asnoœci transformacyjnych entropii Shannona. Omówimy go poni¿ej.\ \ Ze wzglêdu na unormowanie prawdopodobieñstwa do jednoœci, gêstoœæ rozk³adu prawdopodobieñstwa przekszta³ca siê przy transformacji uk³adu wspó³rzêdnych tak jak odwrotnoœæ objêtoœci.\ \ \ [**Przyk³ad zmiennej jednowymiarowej**]{}: Z unormowania $\int_{-\infty}^{+\infty} d{\bf y}\, P\left( {\bf y} \right) = 1$ wynika, ¿e $P({\bf y})$ musi transformowaæ siê tak, jak $1/{\bf y}$.\ Rozwa¿my transformacjê uk³adu wspó³rzêdnych ${\bf y} \rightarrow {\bf y}'$. Ró¿niczka zmiennej $Y$ transformuje siê wtedy zgodnie z $\,d{\bf y}' = J(\frac{{\bf y}}{{\bf y}'}) d{\bf y}$, gdzie $J$ jest jacobianem transformacji, natomiast rozk³ad prawdopodobieñstwa transformuje siê nastêpuj¹co: $\,P'({\bf y}') = \,$ $J^{-1}(\frac{{\bf y}}{{\bf y}'})P({\bf y})$. Zatem tak jak to powinno byæ, unormowanie rozk³adu w transformacji pozostaje niezmiennicze, tzn.: $$\begin{aligned} \label{niezmienniczosc unormowania} \int\limits_{-\infty}^{+\infty} d{\bf y}\, P\left({\bf y}\right) = \int\limits_{-\infty}^{+\infty} d{\bf y}'\, P'\left({\bf y}'\right) = 1 \; . \end{aligned}$$ Rozwa¿my teraz entropiê Shannona uk³adu okreœlon¹ dla rozk³adu ci¹g³ego jak w (\[eboltmana\]), $S_{H}(P)$ $=-\int{d{\bf y}\, P\left({\bf y}\right) \ln P\left({\bf y}\right)}$. Jak to zauwa¿yliœmy powy¿ej, entropia uk³adu jest miar¹ nieuporz¹dkowania w uk³adzie, b¹dŸ informacji potrzebnej do okreœlenia wyniku eksperymentu. Zatem równie¿ i ona powinna byæ niezmiennicza przy rozwa¿anej transformacji. Niestety, chocia¿ miara probabilistyczna pozostaje niezmiennicza, to poniewa¿ $\ln P\left({\bf y}\right) \neq \ln P'\left({\bf y}'\right)$ zatem $S_{H}(P) \neq S_{H}(P')$.\ \ Tak wiêc [*logarytm z gêstoœci rozk³adu prawdopodobieñstwa nie jest niezmienniczy przy transformacji uk³adu wspó³rzêdnych*]{} i w konsekwencji otrzymujemy nastêpuj¹cy wniosek, s³uszny równie¿ w przypadku rozk³adu prawdopodobieñstwa wielowymiarowej zmiennej losowej $Y$.\ \ [**Wniosek**]{}: Entropia Shannona (\[eboltmana\]) nie jest niezmiennicza przy transformacji uk³adu wspó³rzêdnych przestrzeni bazowej ${\cal B}$.\ \ Z drugiej strony, ze wzglêdu na wyj¹tkowe poœród innych entropii w³asnoœci entropii Shannona dla rozk³adu dyskretnego [@Bengtsson_Zyczkowski], zrezygnowanie z jej ci¹g³ej granicy (\[eboltmana\]) mog³oby siê okazaæ decyzj¹ chybion¹. Równie¿ jej zwi¹zek z informacj¹ Fishera omówiony dalej, przekonuje o istotnoœci pojêcia entropii Shannona w jej formie ci¹g³ej.\ [**Entropia wzglêdna jako rozwi¹zanie problemu transformacji**]{}: Proste rozwi¹zanie zaistnia³ego problemu polega na zaobserwowaniu, ¿e poniewa¿ iloraz dwóch gêstoœci $P({\bf y})$ oraz $P_{ref}({\bf y})$ transformuje siê jak skalar, tzn.: $$\begin{aligned} \label{iloraz rozkladow w transformacji} \frac{P'({\bf y}')}{P_{ref}'({\bf y}')} = \frac{J^{-1}(\frac{{\bf y}}{{\bf y}'}) P({\bf y})}{J^{-1}(\frac{{\bf y}}{{\bf y}'}) P_{ref}({\bf y})} = \frac{P({\bf y})}{P_{ref}({\bf y})} \; , \end{aligned}$$ gdzie $P_{ref}({\bf y})$ wystêpuje jako pewien rozk³ad referencyjny, zatem wielkoœæ nazywana [*entropi¹ wzglêdn¹*]{}: $$\begin{aligned} \label{entropia wzgledna roz ciagle} S_{H}(P|P_{ref}) \equiv \int\limits _{-\infty}^{+\infty}{d{\bf y}\, P\left({\bf y}\right) \ln \frac{{P\left({\bf y}\right)}}{{P_{ref}\left({\bf y}\right)}}} \; ,\end{aligned}$$ [*posiada ju¿ w³asnoœæ niezmienniczoœci*]{}: $$\begin{aligned} \label{entropia wzgledna roz ciagle niezm} S_{H}(P'|P'_{ref}) = S_{H}(P|P_{ref}) \; \end{aligned}$$ [*przy transformacji uk³adu wspó³rzêdnych*]{}.\ \ \ [**Entropia wzglêdna jako wartoœæ oczekiwana**]{}: Zwrócmy uwagê, ¿e entropia wzglêdna jest wartoœci¹ oczekiwan¹ logarytmu dwóch rozk³adów, $P({\bf y})$ oraz $P_{ref}({\bf y})$, zmiennej $Y$: $$\begin{aligned} \label{entropia wzgledna jako wartosc oczekiwana ilorazu} S_{H}(P|P_{ref}) = E_{P}\left( \ln \frac{{P\left(Y\right)}}{{P_{ref}\left(Y\right)}} \right)\; ,\end{aligned}$$ wyznaczon¹ przy za³o¿eniu, ¿e zmienna losowa $Y$ ma rozk³ad $P$.\ \ [**Entropia wzglêdna a analiza doboru modelu**]{}: W ten sposób problem logarytmu ilorazu funkcji wiarygodnoœci (czy w szczególnoœci dewiancji) wykorzystywanego w analizie braku dopasowania modelu (por. Rozdzia³y \[Wnioskowanie w MNW\]-\[regresja klasyczna\]), powróci³ w postaci koniecznoœci wprowadzenia entropii wzglêdnej. Istotnie, wiemy, ¿e pojêcie logarytmu ilorazu rozk³adów okaza³o siê ju¿ u¿yteczne w porównywaniu modeli statystycznych i wyborze modelu bardziej “wiarygodnego”. Wybór modelu powinien byæ niezmienniczy ze wzglêdu na transformacjê uk³adu wspó³rzêdnych przestrzeni bazowej ${\cal B}$. Logarytm ilorazu rozk³adów posiada ¿¹dan¹ w³asnoœæ. Jego wartoœci¹ oczekiwan¹, która jest entropi¹ wzglêdn¹, zajmiemy siê w Rozdziale (\[wzg\]). Przez wzgl¹d na zwi¹zek IF z entropi¹ wzglêdn¹ dla rozk³adów ró¿ni¹cych siê infinitezymalnie ma³o, jej pojêcie bêdzie nam towarzyszy³o do koñca skryptu.\ \ [**Nazwy entropii wzglêdnej**]{}: Entropiê wzglêdn¹ (\[entropia wzgledna roz ciagle\]) nazywana siê równie¿ entropi¹ Kullbacka-Leiblera (KL) lub dywergencj¹ informacji. ### Entropia wzglêdna jako miara odleg³oœci {#wzg} Rozwa¿my eksperyment, którego wyniki s¹ generowane z pewnego okreœlonego, chocia¿ nieznanego rozk³adu prawdopodobieñstwa $P_{ref}$ nale¿¹cego do obszaru ${\cal O}$. Obszar ${\cal O}$ jest nie posiadaj¹cym izolowanych punktów zbiorem rozk³adów prawdopodobieñstwa, który jest przestrzeni¹ metryczn¹ zupe³n¹. Oznacza to, ¿e na ${\cal O}$ mo¿na okreœliæ odleg³oœæ oraz ka¿dy ci¹g Cauchy’ego ma granicê nale¿¹c¹ do ${\cal O}$.\ \ [**Twierdzenie Sanova**]{} [@Sanov]: Jeœli mamy $N$ - wymiarow¹ próbkê niezale¿nych pomiarów pochodz¹cych z rozk³adu prawdopodobieñstwa $P_{ref}$ pewnej zmiennej losowej, to prawdopodobieñstwo $Pr$, ¿e empiryczny rozk³ad (czêstoœci) $\hat{{\cal P}}$ wpadnie w obszar ${\cal O}$, spe³nia asymptotycznie zwi¹zek: $$\begin{aligned} \label{Sanov dokladnie} \lim_{N \rightarrow \infty } \frac{1}{N} \ln Pr \left\{ \hat{{\cal P}} \in {\cal O} \right\} = - \beta \;\;\; {\rm gdzie} \;\;\;\; \beta = \inf_{P_{\cal O} \in \, {\cal O}} S_{H} \left( P_{\cal O}|P_{ref} \right) \; , \end{aligned}$$ który w przybli¿eniu mo¿na zapisaæ nastêpuj¹co: $$\begin{aligned} \label{Sanov} Pr(\hat{{\cal P}} \in {\cal O}) \sim e^{- N S_{H} \left( {P_{\cal O}|P_{ref}} \right)} \; ,\end{aligned}$$ gdzie $P_{\cal O}$ jest rozk³adem nale¿¹cym do ${\cal O}$ ró¿nym od $P_{ref}$ z najmniejsz¹ wartoœci¹ entropii wzglêdnej $S_{H} \left( P_{\cal O}|P_{ref} \right)$. Rozk³ad $P_{\cal O}$ uznajemy za rozk³ad wyestymowany na podstawie empirycznego rozk³adu czêstoœci $\hat{{\cal P}}$ otrzymanego w obserwacji.\ \ [**Wniosek**]{}: Zauwa¿my, ¿e Twierdzenie Sanova jest rodzajem prawa wielkich liczb, zgodnie z którym dla wielkoœci próby $N$ d¹¿¹cej do nieskoñczonoœci, [*prawdopodobieñstwo zaobserwowania rozk³adu czêstoœci $\hat{{\cal P}}$ nale¿acego do ${\cal O}$ ró¿nego od prawdziwego rozk³adu*]{} $P_{ref}$ (tzn. tego który generowa³ wyniki eksperymentu), [*d¹¿y do zera*]{}.\ Fakt ten wyra¿a w³aœnie relacja (\[Sanov\]), a poniewa¿ $S_{H} \left(P_{\cal O}|P_{ref} \right)$ jest w jej wyk³adniku, zatem [*entropia wzglêdna okreœla tempo w jakim prawdopodobieñstwo $Pr(\hat{{\cal P}})$ d¹¿y do zera wraz ze wzrostem $N$*]{}.\ [**Entropia wzglêdna dla rozk³adów dyskretnych**]{}: Jeœli $P_{ref} \equiv (p^{i})_{i=1}^{\aleph}$ jest dyskretnym rozk³adem prawdopodobieñstwa o $\aleph$ mo¿liwych wynikach, wtedy rozk³ad $P_{\cal O} \equiv (p^{i}_{\cal O})_{i=1}^{\aleph} \in {\cal O}$ jest te¿ dyskretnym rozk³adem o $\aleph$ wynikach, a entropia wzglêdna $S_{H} \left( P_{\cal O}|P_{ref} \right)$ ma postaæ: $$\begin{aligned} \label{entropia wzgl dyskret} S_{H} \left( {P_{\cal O}|P_{ref}} \right) = \sum\limits_{i=1}^{\aleph}{p_{\cal O}^{i}\ln\frac{{p_{\cal O}^{\, i}}}{{p^{\, i}}}} \; . \end{aligned}$$\ Dla rozk³adów ci¹g³ych entropia wzglêdna zosta³a okreœlona w (\[entropia wzgledna roz ciagle\]). Przekonamy siê, ¿e entropia wzglêdna jest miar¹ okreœlaj¹c¹ jak bardzo dwa rozk³ady ró¿ni¹ siê od siebie.\ \ [**Przyk³ad**]{}: W celu ilustracji twierdzenie Sanova za³ó¿my, ¿e przeprowadzamy doœwiadczenie rzutu niesymetryczn¹ monet¹ z wynikami orze³, reszka, zatem $\aleph=2$. Rozk³ad teoretyczny $P_{ref}$ jest wiêc zero-jedynkowy. [*Natomiast w wyniku pobrania $N$-elementowej próbki dokonujemy jego estymacji na podstawie rozk³adu empirycznego $\hat{{\cal P}}$ czêstoœci pojawienia siê wyników*]{} orze³ lub reszka. Zatem: $$\begin{aligned} \label{zerojeden oraz empir} P_{ref} \equiv (p^{i})_{i=1}^{\aleph=2} = (p, 1-p) \;\;\; {\rm oraz} \;\;\; \hat{{\cal P}} \equiv (\hat{p}^{\,i})_{i=1}^{\aleph=2} = \left( \frac{m}{N}, \,1 - \frac{m}{N} \right) \; . \end{aligned}$$ Twierdzenie Bernoulliego mówi, ¿e prawdopodobieñstwo pojawienia siê wyniku orze³ z czêstoœci¹ $m/N$ w $N$-losowaniach wynosi: $$\begin{aligned} \label{bernoul} Pr(\hat{{\cal P}}) \equiv Pr\left({\frac{m}{N}}\right) = \left(\begin{array}{l} N\\ m\end{array}\right) p^{m}\left({1-p}\right)^{N-m} \; .\end{aligned}$$ Bior¹c logarytm naturalny obu stron (\[bernoul\]), nastêpnie stosuj¹c s³uszne dla du¿ego $n$ przybli¿enie Stirlinga, $\ln n!\approx n\ln n-n$, dla ka¿dej silni w wyra¿eniu $\left(\begin{array}{l} N\\ m\end{array}\right)$, i w koñcu bior¹c eksponentê obu stron, mo¿na otrzymaæ [@Bengtsson_Zyczkowski]: $$\begin{aligned} \label{pew} Pr(\hat{{\cal P}}) \approx e^{-N S_{H}\left( \hat{{\cal P}}|P_{ref} \right)} \; ,\end{aligned}$$ gdzie $$\begin{aligned} \label{bernoul_entropia wzgl} S_{H}\left( \hat{{\cal P}}|P_{ref} \right) &=& \left[{\frac{m}{N}\left({\ln\frac{m}{N} - \ln p}\right) + \left({1-\frac{m}{N}}\right)\left({\ln\left({1-\frac{m}{N}}\right)- \ln\left({1-p}\right)}\right)}\right] \nonumber \\ &=& \sum\limits_{i=1}^{\aleph=2}{\hat{p}^{\,i}\ln\frac{{\hat{p}^{\,i}}}{{p^{\, i}}}} \; .\end{aligned}$$ W ostatnej równoœci skorzystano z (\[zerojeden oraz empir\]) otrzymuj¹c entropiê wzglêdn¹ (\[entropia wzgl dyskret\]) dla przypadku liczby wyników $\aleph=2$.\ \ [**Podsumowanie**]{}: W powy¿szym przyk³adzie entropia wzglêdna $S_{H}\left( \hat{{\cal P}}|P_{ref} \right)$ pojawi³a siê jako pojêcie wtórne, wynikaj¹ce z wyznaczenia prawdopodobieñstwa otrzymania w eksperymencie empirycznego rozk³adu czêstoœci $\hat{{\cal P}} \in {\cal O}$ jako oszacowania rozk³adu $P_{ref}$. Fakt ten oznacza, ¿e [*entropia wzglêdna*]{} nie jest tworem sztucznym, wprowadzonym do teorii jedynie dla wygody jako “jakaœ” miara odleg³oœci pomiêdzy rozk³adami, lecz, ¿e [*jest w³aœciw¹ dla przestrzeni statystycznej rozk³adów miar¹ probabilistyczn¹ tej odleg³oœci*]{}, tzn. “dywergencj¹ informacji” pomiêdzy rozk³adami. Oka¿e siê te¿, ¿e spoœród innych miar odleg³oœci jest ona wyró¿niona poprzez jej zwiazek ze znan¹ ju¿ nam informacj¹ Fishera.\ \ W koñcu podamy twierdzenie o dodatnioœci entropii wzglêdnej, które jeszcze bardziej przybli¿y nas do zrozumienia entropii jako miary odleg³oœci pomiêdzy rozk³adami, rozwijanego w treœci nastêpnego rozdzia³u.\ \ [**Twierdzenie (Nierównoœæ informacyjna)**]{}: Jeœli $P({\bf y})$ oraz $P_{ref}({\bf y})$ s¹ dwoma rozk³adami gêstoœci prawdopodobieñstwa, wtedy entropia wzglêdna spe³nia nastêpuj¹c¹ nierównoœæ: $$\begin{aligned} \label{nierownosc informacyjna} S_{H}\left(P({\bf y})|P_{ref}({\bf y}) \right) \geq 0 \; , \end{aligned}$$ i nierównoœæ ta jest ostra za wyj¹tkiem przypadku, gdy $P({\bf y}) = P_{ref}({\bf y})$ [@Pawitan]. Geometria przestrzeni rozk³adów prawdopodobieñstwa i metryka Rao-Fishera {#geometria i metryka Fishera-Rao} ------------------------------------------------------------------------ Pojêcie metryki Fishera wprowadziliœmy ju¿ w Rozdziale \[alfa koneksja\]. Do ujêcia tam przedstawionego dojdziemy jeszcze raz, wychodz¹c od pojêcia entropii wzglêdnej.\ \ Rozwa¿my dwa rozk³ady prawdopodobieñstwa: $P=(p^{i})_{i=1}^{\aleph}$ oraz $P'=P+dP = (p'^{i}=p^{i}+dp^{i})_{i=1}^{\aleph}$, ró¿ni¹ce siê [*infinitezymalnie ma³o*]{}, przy czym $p_{i}\neq 0$ dla ka¿dego $i$. Rozk³ady $P$ oraz $P'$ spe³niaj¹ warunek unormowania $\sum_{i=1}^{\aleph} p^{i} = 1$ oraz $\sum_{i=1}^{\aleph} p^{i} + dp^{i} = 1$, sk¹d: $$\begin{aligned} \label{suma dp zero} \sum\limits_{i=1}^{\aleph} dp^{i} = 0 \; .\end{aligned}$$ Poniewa¿ $dp^{i}/p^{i}$ jest wielkoœci¹ infinitezymalnie ma³¹, zatem z rozwiniêcia: $$\begin{aligned} \label{rozwiniêcie} \ln ( 1 + \frac{dp^{i}}{p^{i}} ) = \frac{dp^{i}}{p^{i}} - 1/2 (\frac{dp^{i}}{p^{i}})^{2} + ... \end{aligned}$$ otrzymujemy, ¿e entropia KL rozk³adów $P$ oraz $P'$ wynosi: $$\begin{aligned} \label{S dla P oraz P+dP} S_{H} \left( P|P+dP \right) = \sum_{i=1}^{\aleph} p^{i} \ln \frac{p^{i}}{p^{i}+dp^{i}} = - \sum_{i=1}^{\aleph} p^{i} \ln \frac{p^{i}+dp^{i}}{p^{i}} \approx \frac{1}{2} \sum_{i=1}^{\aleph} \frac{dp^{i} dp^{i}}{p^{i}} \; .\end{aligned}$$ Ostatnia postaæ $S_{H} \left( P|P+dP \right)$ sugeruje, ¿e entropia KL okreœla w naturalny sposób na przestrzeni statystycznej ${\cal S}$ infinitezymalny kwadrat odleg³oœci pomiêdzy rozwa¿anymi rozk³adami, co onacza³oby równie¿ [*lokalne*]{} okreœlenie na przestrzeni statystycznej ${\cal S}$ pewnej metryki. Do jej zwi¹zku z metryk¹ Rao-Fisher’a wprowadzon¹ w Rozdziale \[alfa koneksja\] powrócimy póŸniej.\ \ [**Infinitezymalny interwa³ w ${\cal S}$**]{}: Niech $d\vec{p}=\left(dp^{1},....,dp^{\aleph}\right)$ jest infinitezymalnym wektorem w przestrzeni rozk³adów prawdopodobieñstwa, spe³niaj¹cym warunek (\[suma dp zero\]). Wprowadzaj¹c na przestrzeni statystycznej ${\cal S}$ aparat matematyczny geometrii ró¿niczkowej [@Amari; @Nagaoka; @book], zapiszmy kwadrat ró¿niczkowego interwa³u w tej przestrzeni nastêpuj¹co: $$\begin{aligned} \label{ds2 poprzez P} d s^{2} = \sum_{i,\, j =1}^{\aleph} g_{i j} \, dp^{i}\,dp^{j} \; .\end{aligned}$$ W celu uzgodnienia (\[S dla P oraz P+dP\]) z (\[ds2 poprzez P\]) wprowadŸmy na ${\cal S}$ metrykê: $$\begin{aligned} \label{metryka Rao-Fishera z KL} g_{i j} = \frac{\delta_{ij}}{p^{i}} \; .\end{aligned}$$ Zwi¹zek (\[ds2 poprzez P\]) oznacza, ¿e [*liczba mo¿liwych wyników $\aleph$ okreœla wymiar przestrzeni*]{} ${\cal S}$ oraz, ¿e entropia wzglêdna KL definuje dla infinitezymalnie bliskich rozk³adów symetryczn¹ metrykê na przestrzeni statystycznej ${\cal S}$, pozwalaj¹c¹ mierzyæ odleg³oœæ pomiêdzy tymi rozk³adami. Zastêpuj¹c prawdopodobieñstwa $p^{i}$ czêstoœciami, metryka ta staje siê statystyk¹ zwi¹zana z obserwowan¹ informacyjn¹ Fishera wprowadzon¹ poprzednio.\ \ Powy¿ej stan uk³adu okreœlony by³ w reprezentacji rozk³adu prawdopodobieñstwa zmiennej losowej, tzn. okreœlony by³ przez podanie rozk³adu prawdopodobieñstwa. Dwóm infinitezymalnie blisko le¿¹cym stanom opowiada³y rozk³ady $P$ oraz $P'$.\ \ [**Reprezentacja amplitudowa ${\cal S}$**]{}: Gdy interesuj¹ nas czysto geometryczne w³asnoœci metryki Fishera, wtedy wygodne jest u¿ycie innej reprezentacji do opisu stanu uk³adu, okreœlonej nastêpuj¹co. Niech $Q \equiv (q^{i})_{i=1}^{\aleph}$ oraz $Q' \equiv (q'^{i})_{i=1}^{\aleph} = (q^{i} + d q^{i})_{i=1}^{\aleph}$ opisuj¹ te same co poprzednio, dwa infinitezymalnie blisko le¿¹ce stany uk³adu tyle, ¿e zarz¹dajmy, aby infinitezymalny kwadrat interwa³u pomiêdzy nimi (\[ds2 poprzez P\]) by³ równy: $$\begin{aligned} \label{ds2 przy amplitudzie} d s^{2} = 4 \sum_{i=1}^{\aleph} dq^{i}\,dq^{i} \; .\end{aligned}$$ Porównuj¹c formu³ê (\[ds2 przy amplitudzie\]) z (\[ds2 poprzez P\]) zauwa¿amy, ¿e zgadzaj¹ siê one z sob¹ o ile $Q = (q^{i})_{1}^{{\aleph}}$ oraz $P = (p^{i})_{1}^{{\aleph}}$ s¹ powi¹zane zwi¹zkiem: $$\begin{aligned} \label{poch amplituda a poch rozklad} d q^{i} = \frac{d p^{i}}{2 \sqrt{p^{i}}} \;\;\; i = 1, ..., {\aleph} \; ,\end{aligned}$$ co zachodzi wtedy gdy $$\begin{aligned} \label{amplituda a rozklad} q^{i} = \sqrt{p^{i}} \;\;\; i = 1, ..., {\aleph} \; .\end{aligned}$$ Wielkoœci $\; q^{i}$ nazywamy [*amplitudami uk³adu*]{}[^39]. Definuj¹ one na ${\cal S}$ nowe wspó³rzêdne, dla których $q^{i} \ge 0$ dla ka¿dego $i$.\ \ [**${\cal S}$ z geometri¹ jednostkowej sfery**]{}: Otrzymana w bazie $q^{i}$ geometria ${\cal S}$ jest geometri¹ jednostkowej sfery, tzn. ze wzglêdu na unormowanie rozk³adu prawdopodobieñstwa do jednoœci, amplitudy spe³niaj¹ nastêpuj¹cy warunek unormowania na promieniu jednostkowym: $$\begin{aligned} \label{unormowanie amplitud} \;\;\sum_{i=1}^{\aleph} p^{i} = \sum_{i=1}^{\aleph} q^{i}\,q^{i} = 1 \; .\end{aligned}$$ Na sferze tej mo¿emy okreœliæ odleg³oœæ geodezyjn¹ $D_{Bhatt}$, tzw. odleg³oœæ Bhattacharyya’ pomiêdzy dwoma rozk³adami prawdopodobieñstwa $P$ oraz $P'$, jako d³ugoœæ k¹tow¹ liczon¹ wzd³u¿ ko³a wielkiego pomiêdzy dwoma wektorami $Q$ oraz $Q'$ o sk³adowych bêd¹cych amplitudami $q^{i}=\sqrt{p^{i}}$ oraz $q^{'i}=\sqrt{p^{'i}}$.\ \ [**Odleg³oœci Bhattacharyya**]{}: Kwadrat infinitezymalnego interwa³u (\[ds2 przy amplitudzie\]) jest przyk³adem odleg³oœci Bhattacharyya, któr¹ ogólnie okreœlamy nastepuj¹co: Jeœli $P=(p^{i})$ oraz $P'=(p'^{i})$ s¹ rozk³adami prawdopodobieñstwa, wtedy odleg³oœæ Bhattacharyya pomiêdzy $Q$ oraz $Q'$ jest iloczynem wewnêtrznym okreœlonym na przestrzeni statystycznej ${\cal S}$ nastêpuj¹co: $$\begin{aligned} \label{odleglosc Bhattacharyya} \cos D_{Bhatt} = \sum_{i=1}^{\aleph} q^{i}\,q'^{i} = \sum_{i=1}^{\aleph} \sqrt{p^{i}\,p'^{i}} \equiv B(p,p') \; .\end{aligned}$$ [*Statystyczna odleg³oœci¹ Bhattacharyya wygl¹da wiêc jak iloczyn wewnêtrzny mechaniki kwantowej*]{}. W kolejnych rozdzia³ach przekonamy siê, ¿e nie jest to b³êdne skojarzenie.\ \ [**Hessian entropii Shannona**]{}: Zauwa¿my, ¿e metryka $g_{ij}$ jest Hessianem (tzn. macierz¹ drugich pochodnych) entropii Shannona: $$\begin{aligned} \label{hesian z S} g_{ij} = - \partial_{i}\partial_{j}S_{H}\left(p\right) = \frac{\partial}{\partial p^{i}}\frac{\partial}{\partial p^{j}}\sum\limits _{k=1}^{\aleph}{p^{k}\ln p^{k}} = \frac{\delta^{ij}}{p^{j}}\, \geq 0 \, ,\end{aligned}$$ zgodnie z (\[metryka Rao-Fishera z KL\]). Powy¿szy zwi¹zek oznacza, ¿e fakt wklês³oœci entropii Shannona [@Bengtsson_Zyczkowski] daje dodatni¹ okreœlonoœæ metryki $g_{ij}$ na ${\cal S}$.\ \ [**Metryka indukowana z $g_{ij}$**]{}: Uzasadnijmy fakt nazwania czasami metryki $g_{ij}$, (\[metryka Rao-Fishera z KL\]), metryk¹ Rao-Fishera.\ \ Za³ó¿my, ¿e interesuje nas pewna podprzestrzeñ przestrzeni rozk³adów prawdopodobieñstwa ${\cal S}$. WprowadŸmy na niej uk³ad wspó³rzêdnych $\theta^{a}$. Korzystaj¹c z (\[metryka Rao-Fishera z KL\]) widaæ, ¿e [*metryka $g_{ij} = \frac{\delta^{ij}}{p^{j}}$ indukuje w tej podprzestrzeni ${\cal S}$ metrykê*]{}: $$\begin{aligned} \label{metryka Rao-Fishera w ukladzie wsp} \!\!\! g_{ab} &=& \sum\limits _{i,j=1}^{\aleph} {\frac{{\partial p^{i}}}{{\partial\theta^{a}}} \frac{{\partial p^{j}}}{{\partial\theta^{b}}}} \, g_{ij} = \sum\limits_{i=1}^{\aleph}{\frac{{\partial_{a}p^{i} \partial_{b} p^{i}}}{{p^{i}}}} = \sum\limits_{i=1}^{\aleph} p^{i} \, \partial_{a} \ln p^{i} \partial_{b} \ln p^{i} \nonumber \\ &\equiv& E \left( \partial_{a} \ell_{\Theta} \, \partial_{b} \ell_{\Theta} \right) = E_{\Theta} \texttt{i\!F}\left(\Theta \right) = I_{F} \left(\Theta \right) \; .\end{aligned}$$ gdzie skorzystano z zapisu $\ell_{\Theta} \equiv l\left(y|\Theta \right) = \ln P\left(\Theta\right) = (\ln p^{i}\left(\Theta\right))_{i=1}^{\aleph}$ oraz $\partial_{a} \equiv{\partial }/{\partial\theta^{a}}$, (\[oznaczenie ln P\]).\ \ W Rozdziale \[alfa koneksja\] we wzorze (\[Fisher inf matrix\]) zdefiniowaliœmy metrykê Rao-Fishera jako $g_{ab} = E_{\vartheta}(\partial_{a} \ell_{\Theta} \, \partial_{b} \ell_{\Theta})$. Tak wiêc ostateczne otrzymaliœmy zgodnoœæ nazwania metryki (\[metryka Rao-Fishera z KL\]) metryk¹ Rao-Fishera. [*Metryka $g_{ij}$ jest jednak wielkoœci¹ obserwowan¹ a nie oczekiwan¹, tak jak metryka Rao-Fishera $g_{ab}$*]{}.\ \ [**Metryka Roa-Fishera zapisana w amplitudach**]{}: Korzystaj¹c ze zwi¹zku $q^{i} = \sqrt{p^{i}}$, (\[amplituda a rozklad\]), pomiêdzy prawdopodobieñstwami i amplitudami, metrykê Rao-Fishera (\[metryka Rao-Fishera w ukladzie wsp\]) w reprezentacji amplitudowej mo¿na zapisaæ nastêpuj¹co: $$\begin{aligned} \label{metryka Rao-Fishera w ukladzie wsp - amplitudy} g_{ab} = 4 \sum\limits_{i=1}^{\aleph} \frac{\partial q^{i}}{\partial \theta^{a}} \frac{\partial q^{i}}{\partial \theta^{b}} \; .\end{aligned}$$ [**Przejœcie do zmiennej ci¹g³ej**]{}: Gdy liczba parametrów $\theta^{a}$ (wiêc i wektorów bazowych $\vec{\theta}^{a}$) rozpinaj¹cych osie uk³adu wspó³rzêdnych rozwa¿anej podprzestrzeni statystycznej jest skoñczona, wtedy mo¿na dokonaæ nastêpuj¹cego uogólnienia metryki na przypadek ci¹g³ego rozk³adu prawdopodobieñstwa $P(y)$ zmiennej losowej $Y$: $$\begin{aligned} \label{metryka Rao-Fishera w ukl wsp dla r ciaglego} g_{ab} = \int_{\cal Y}{d{\bf y}}\frac{{\partial_{a} P\;\partial_{b} P}}{P} = \int_{\cal Y}{d{\bf y}\; P\; \partial_{a} \ell_{\Theta} \, \partial_{b} \ell_{\Theta}} \equiv E \left( \partial_{a} \ell_{\Theta} \, \partial_{b} \ell_{\Theta} \right) = E_{\Theta} \texttt{i\!F}\left(\Theta \right) = I_{F} \left(\Theta \right) \; ,\end{aligned}$$ gdzie, poniewa¿ przestrzeñ zdarzeñ ci¹g³ego rozk³adu prawdopodobieñstwa jest nieskoñczenie wymiarowa, w miejsce sumowania po $i=1,...,\aleph$ pojawi³o siê ca³kowanie po wartoœciach ${\bf y} \in {\cal Y}$ zmiennej losowej $Y$. Otrzymana postaæ metryki jest jawnie niezmiennicza ze wzglêdu na zmianê uk³adu wspó³rzêdnych ${\bf y} \rightarrow {\bf y}'$ w przestrzeni bazowej ${\cal Y}$.\ Na koniec, dokonuj¹c reparametryzacji i przechodz¹c do amplitud $q({\bf y}|\Theta) = {\sqrt P({\bf y}|\Theta)}\, $, mo¿na metrykê (\[metryka Rao-Fishera w ukl wsp dla r ciaglego\]) dla przypadku rozk³adu ci¹g³ego zapisaæ nastêpuj¹co: $$\begin{aligned} \label{metryka Rao-Fishera w ukl wsp dla r ciaglego - amplitudy} g_{ab} = 4 \int_{\cal Y}{d{\bf y}} \, \partial_{a} q({\bf y}|\Theta) \;\partial_{b} q({\bf y}|\Theta) \; ,\end{aligned}$$ gdzie jawnie zaznaczono zale¿noœæ amplitudy rozk³adu od parametru $\Theta$.\ \ [**Przypadek funkcji wiarygodnoœci**]{}: Powy¿sza analiza dotyczy³a przypadku próby $N=1$ - wymiarowej. Gdyby rozk³adem prawdopodobieñstwa $P$ by³a funkcja wiarygodnoœci próby, wtedy w miejscu zmiennej losowej $Y$ pojawi³aby siê próba $\widetilde{Y} \equiv( Y_{n})_{n=1}^{N}$, a w miejscu przestrzeni ${\cal Y}$, przestrzeñ próby ${\cal B}$. Poza tym, rozwa¿ania w obecnym rozdziale pozosta³yby takie same [@Amari; @Nagaoka; @book].\ \ Omówin¹ sytuacjê odleg³oœci na przestrzeni statystycznej przedstawia graficznie poni¿szy rysunek dla liczby parametrów $d=2$.\ {height="9.5cm"}\ [**Przyk³ad**]{}: Wyznaczyæ kwadrat infinitezymalnego interwa³u dla dwóch stanów posiadaj¹cych rozk³ad normalny $N(\mu, \sigma)$. Rozk³ad normalny ma postaæ: $$\begin{aligned} \label{rozklad 2-wym norm przyklad} P\left({\bf y},\mu,\sigma\right)=\frac{1}{{\sqrt{2\pi}\sigma}}e^{-\frac{{\left({{\bf y} - \mu}\right)^{2}}}{{2\sigma^{2}}}} \; ,\end{aligned}$$ gdzie wektor parametrów $\Theta \equiv (\theta^{a})_{a=1}^{2} = (\mu, \sigma)^{T}$. Macierz informacyjna Fishera dla rozk³adu normalnego $N(\mu, \sigma)$ ma wyznaczon¹ poprzednio postaæ (\[oczekiw iF r normalnego 2 par\]). Zatem metryka Rao-Fishera na 2-wymiarowej przestrzeni normalnych rozk³adów prawdopodobieñstwa z uk³adem wspó³rzêdnych $\mu, \sigma$ ma (dla próby $N=1$), postaæ: $$\begin{aligned} \label{oczekiw iF r normalnego 2 par dla N=1} (g_{ab}) = I_{F} \left(\Theta \right) = E_{\Theta} \texttt{i\!F}\left(\Theta \right) = \left({\begin{array}{cc} {\frac{1}{{ \sigma^{2}}}} & 0\\ 0 & {\frac{2}{{ \sigma^{2}}}} \end{array}}\right) \; ,\end{aligned}$$ tzn. sk³adowe metryki (\[metryka Rao-Fishera w ukl wsp dla r ciaglego\]) s¹ równe: $$\begin{aligned} \label{metryka w przykladzie} g_{\mu\mu} = \frac{1}{{\sigma^{2}}} \; , \;\;\; g_{\mu\sigma} = g_{\sigma \mu} = 0 \; , \;\;\; g_{\sigma\sigma} = \frac{2}{{\sigma^{2}}} \; .\end{aligned}$$ Zatem otrzymany kwadrat infinitezymalnego interwa³u na 2-wymiarowej (pod)przestrzeni statystycznej ${\cal S}$ wynosi [@Bengtsson_Zyczkowski]: $$\begin{aligned} \label{ds2 dla r norm} ds^{2} = g_{\mu\mu} d\mu^{2} + g_{\mu\sigma}d\mu d\sigma + g_{\sigma\sigma}d\sigma^{2} = \frac{1}{{\sigma^{2}}}\left({d\mu^{2} + 2d\sigma^{2}}\right) \; .\end{aligned}$$ Odpowiada on metrce Poincarégo ze sta³¹ ujemn¹ krzywizn¹. W koñcu, z postaci rozk³adu (\[rozklad 2-wym norm przyklad\]) otrzymujemy równie¿ bazê w przestrzeni stycznej do ${\cal S}$: $$\begin{aligned} \label{baza w przestrzeni stycz r norm} \partial_{\mu} \ln P \left({\bf y}|(\mu,\sigma)\right) = \frac{{\bf y} - \mu}{\sigma^{2}} \; ,\;\;\;\; \partial_{\sigma} \ln P\left({\bf y}|(\mu,\sigma)\right) = \frac{\left({\bf y} - \mu \right)^{2}}{\sigma^{3}}-\frac{1}{\sigma} \; .\end{aligned}$$ [**Wniosek z przyk³adu**]{}: [*Na osi $\mu$, na której $\sigma =0$ le¿¹ punkty odpowiadaj¹ce klasycznym stanom czystym i s¹ one zgodnie z (\[ds2 dla r norm\]) nieskoñczenie daleko odleg³e od dowolnego punktu we wnêtrzu górnej pó³p³aszczyzny*]{} $\sigma > 0$. W konsekwencji oznacza to, ¿e wynik pewny, któremu odpowiada rozk³ad klasyczny z $\sigma=0$, jest ³atwy do odró¿nienia od ka¿dego innego. Informacja Fishera ------------------ ### Informacja Fishera jako entropia {#Informacja Fishera jako entropia} W powy¿szych rozwa¿aniach informacja Fishera $I_{F}$ pojawi³a siê poprzez nierównoœæ Rao-Cramera (\[tw R-C dla par skalarnego\]), jako wielkoœæ okreœlaj¹ca graniczn¹ dobroæ procedury estymacyjnej parametru rozk³adu, tzn. o ile estymator efektywny istnieje, to informacja Fishera okreœla minimaln¹ wartoœæ jego wariancji. Poniewa¿ im informacja Fishera mniejsza, tym to graniczne oszacowanie parametru gorsze, zatem jest ona równie¿ miar¹ stopnia nieuporz¹dkowania uk³adu okreœlonego rozk³adem prawdopodobieñstwa. Takie zrozumienie informacji Fishera odnosi siê do typu analizowanego rozk³adu prawdopodobieñstwa i jako miara nieuporz¹dkowania rozk³adu oznacza brak przewidywalnoœci procedury estymacyjnej (spróbuj pomyœleæ o sensie oszacowania wartoœci oczekiwanej rozk³adu jednorodnego).\ Poni¿ej uzasadnimy stwierdzenie, ¿e informacja Fishera okazuje siê byæ proporcjonalna do entropii Kullbacka-Leiblera rozk³adów ró¿ni¹cych siê infinitezymalnie ma³o [*w parametrze*]{} rozk³adu [@Frieden].\ \ Szukany zwi¹zek informacji Fishera z entropi¹ Kullbacka-Leiblera poka¿emy w trzech krokach. Rozwa¿my $N$-wymiarow¹ próbê $Y_{1}, Y_{2}, ..., Y_{N}$, gdzie ka¿da ze zmiennych losowych $Y_{n}$ jest okreœlona na przestrzeni ${\cal Y}$ i posiada rozk³ad prawdopodobieñstwa $p\left({\bf y}_{n}|\theta \right)$. Przyjmijmy, dla uproszczenia rozwa¿añ, ¿e $\Theta = \theta$ jest parametrem skalarnym.\ \ [**(1)**]{} Funkcja wiarygodnoœci $P(y\,|\theta)$ ma postaæ: $$\begin{aligned} \label{fun wiaryg jeden par} P\left({y|\theta}\right) = \prod\limits_{n=1}^{N} {p_{n}\left({{\bf y}_{n}|\theta}\right)} \; ,\end{aligned}$$ gdzie $y\equiv({\bf y}_{n})_{n=1}^{N}$ jest realizacj¹ próby.\ Zauwa¿my, ¿e informacjê Fishera parametru $\theta$ okreœlon¹ w (\[infoczekiwana\]) i (\[I jako krzywizna dla P\]): $$\begin{aligned} \label{inf I jeden parametr - 2 pochodna} I_{F} = -\int_{\cal B}{dy\, P\left({y|\theta}\right)\frac{{\partial^{2}\ln P \left({y|\theta}\right)}}{{\partial\theta^{2}}}} \; ,\end{aligned}$$ gdzie ${\cal B}$ jest przestrzeni¹ próby, a $dy \equiv d^{N}{\bf y} = d{\bf y}_{1} d{\bf y}_{2} ... d{\bf y}_{N}$, mo¿na zapisaæ nastêpuj¹co: $$\begin{aligned} \label{inf I jeden parametr - kwadrat 1 pochodnej} I_{F} = \int_{\cal B}{dy\, P(y\,|\theta)}\left({\frac{{\partial \ln P(y\,|\theta)}}{{\partial\theta}}}\right)^{2} \; .\end{aligned}$$\ Istotnie, postaæ (\[inf I jeden parametr - kwadrat 1 pochodnej\]) otrzymujemy z (\[inf I jeden parametr - 2 pochodna\]) po skorzystaniu z: $$\begin{aligned} \frac{\partial}{\partial\theta}\left({\frac{{\partial\ln P}}{{\partial\theta}}}\right)=\frac{\partial}{\partial\theta}\left({\frac{1}{P}\frac{{\partial P}}{{\partial\theta}}}\right)=-\frac{1}{{P^{2}}}\left({\frac{{\partial P}}{{\partial\theta}}}\right)^{2}+\frac{1}{P}\left({\frac{{\partial^{2}P}}{{\partial\theta^{2}}}}\right) \; \end{aligned}$$ oraz warunku reguralnoœci pozwalaj¹cego na wy³¹czenie ró¿niczkowania po parametrze przed ca³kê (por. Rozdzia³ \[E i var funkcji wynikowej\]): $$\begin{aligned} \int_{\cal B}{dy\left({\frac{{\partial^{2}P}}{{\partial\theta^{2}}}}\right)} = \frac{{\partial^{2}}}{{\partial\theta^{2}}}\int_{\cal B}{dy\, P} = \frac{{\partial^{2}}}{{\partial\theta^{2}}} 1 = 0 \; .\end{aligned}$$\ [**(2) Zadanie**]{}: Pokazaæ, ¿e zachodzi nastêpuj¹cy [*rozk³ad informacji Fishera*]{} parametru $\theta$: $$\begin{aligned} \label{I dla pn jeden parametr} I_{F} = \sum_{n=1}^{N}{I_{F n}} \;\;\;\; {\rm gdzie} \;\;\;\; I_{F n} = {\int_{\cal Y}{d{\bf y}_{n}{p_{n}\left({{\bf y}_{n}|\theta}\right)}\left({\frac{{\partial\ln p_{n}\left({{\bf y}_{n}|\theta}\right)}}{{\partial\theta}}}\right)^{2}}} \; ,\end{aligned}$$ na sk³adowe informacje $I_{F n}$, zale¿ne jedynie od rozk³adów punktowych $p_{n}({\bf y}|\theta)$.\ \ [**Rozwi¹zanie**]{}: Po skorzystaniu z (\[fun wiaryg jeden par\]) zauwa¿amy, ¿e: $$\begin{aligned} \label{poch lnP} \ln P\left({y;\theta}\right) = \sum_{n=1}^{N}{\ln p_{n}} \;, \;\;\; {\rm zatem} \;\;\;\;\; \frac{{\partial\ln P}}{{\partial\theta}}=\sum_{n=1}^{N}{\frac{1}{{p_{n}}}}\frac{{\partial p_{n}}}{{\partial\theta}} \; \end{aligned}$$ i podnosz¹c ostatnie wyra¿enie do kwadratu otrzymujemy: $$\begin{aligned} \label{kwadrat poch lnP} \left({\frac{{\partial\ln P}}{{\partial\theta}}}\right)^{2} = \sum_{{\scriptstyle {{n,m=1}\hfill\atop {\scriptstyle m\ne n\hfill}}}}^{N}{\frac{1}{{p_{m}}}\frac{1}{{p_{n}}}}\frac{{\partial p_{m}}}{{\partial\theta}}\frac{{\partial p_{n}}}{{\partial\theta}} + \sum_{n=1}^{N}{\frac{1}{{p_{n}^{2}}}}\left({\frac{{\partial p_{n}}}{{\partial\theta}}}\right)^{2} \; .\end{aligned}$$ Wstawiaj¹c (\[fun wiaryg jeden par\]) oraz (\[kwadrat poch lnP\]) do (\[inf I jeden parametr - kwadrat 1 pochodnej\]) otrzymujemy: $$\begin{aligned} \label{I przed skorzstaniem z rozkl brzegowych} I_{F} = \int_{\cal B}{dy\prod_{k=1}^{N}{p_{k}}} {\sum_{{\scriptstyle {{n,m=1}\hfill\atop {\scriptstyle m\ne n\hfill}}}}^{N}{\frac{1}{{p_{m}}}\frac{1}{{p_{n}}}}\frac{{\partial p_{m}}}{{\partial\theta}}\frac{{\partial p_{n}}}{{\partial\theta}} + \int_{\cal B}{dy\prod_{k=1}^{N}{p_{k}}} \sum_{n=1}^{N}{\frac{1}{{p_{n}^{2}}}}\left({\frac{{\partial p_{n}}}{{\partial\theta}}}\right)^{2}} \; .\end{aligned}$$ Ze wzglêdu na warunek normalizacji rozk³adów brzegowych: $$\begin{aligned} \label{normalizacja rozkladow brzegowych} \int_{\cal Y} d{\bf y}_{n} \, p_{n}({\bf y}_{n}|\theta) = 1 \; ,\end{aligned}$$ w (\[I przed skorzstaniem z rozkl brzegowych\]) z iloczynu $\prod_{k=1}^{N}{p_{k}}$ pozostaje w pierwszym sk³adniku jedynie $p_{m}p_{n}$ (dla $k\neq m$ oraz $k\neq n$), natomiast w drugim sk³adniku pozostaje jedynie $p_{n}$ (dla $k\neq n$), tzn.: $$\begin{aligned} \label{upr} I_{F} = \sum_{{\scriptstyle {{n,m=1} \hfill \atop {\scriptstyle m\ne n\hfill}}}}^{N}{\int_{\cal Y} {\int_{\cal Y}{d{\bf y}_{m}d{\bf y}_{n}\frac{{\partial p_{m}}}{{\partial\theta}}\frac{{\partial p_{n}}}{{\partial\theta}}}}} + \sum_{n=1}^{N}{\int_{\cal Y}{d{\bf y}_{n}\frac{1}{{p_{n}}}\left({\frac{{\partial p_{n}}}{{\partial\theta}}}\right)^{2}}} \; .\end{aligned}$$ W koñcu, ze wzglêdu na: $$\begin{aligned} \label{poch normalizacji} \int_{\cal Y}{d{\bf y}_{n}\frac{{\partial p_{n}}}{{\partial\theta}}} = \frac{\partial}{{\partial\theta}}\int_{\cal Y}{d{\bf y}_{n}p_{n}}=\frac{\partial}{{\partial\theta}} 1 = 0 \; ,\end{aligned}$$ pierwszy sk³adnik w (\[upr\]) zeruje siê i pozostaje jedynie drugi, zatem: $$\begin{aligned} \label{postac I koncowa w pn} I_{F} = \sum_{n=1}^{N}{\int_{\cal Y}{d{\bf y}_{n}\frac{1}{{p_{n}}}\left({\frac{{\partial p_{n}}}{{\partial\theta}}}\right)^{2}}} = \sum_{n=1}^{N} \int_{\cal Y} d{\bf y}_{n}p_{n}\left({\frac{{\partial\ln p_{n}}}{{\partial\theta}}}\right)^{2} \; .\end{aligned}$$ Ostatecznie otrzymujemy wiêc szukany rozk³ad informacji Fishera (\[I dla pn jeden parametr\]): $$\begin{aligned} I_{F} = \sum_{n=1}^{N}{I_{F n}} \;\;\;\; {\rm gdzie} \;\;\;\; I_{F n} = {\int_{\cal Y}{d{\bf y}_{n}{p_{n}\left({{\bf y}_{n}|\theta}\right)}\left({\frac{{\partial\ln p_{n}\left({{\bf y}_{n}|\theta}\right)}}{{\partial\theta}}}\right)^{2}}} \; . \nonumber\end{aligned}$$\ \ [**(3) Zadanie**]{}: ZnaleŸæ zwi¹zek informacji Fishera z entropi¹ Kullbacka-Leiblera rozk³adów $P \left({y|\theta}\right)$ oraz $P\left({y|\theta + \Delta \theta}\right)$ ró¿ni¹cych siê infinitezymalnie ma³o w parametrze $\theta$.\ \ \ [**Rozwi¹zanie**]{}: Zast¹pmy ca³kê w (\[I dla pn jeden parametr\]) dla $I_{F n}$ sum¹ Riemanna: $$\begin{aligned} \label{dys} I_{F n} &=& {\sum\limits _{k}{\Delta {\bf y}_{nk}\frac{1}{{p_{n}\left({{\bf y}_{nk}|\theta}\right)}}\left[{\frac{{p_{n}\left({{\bf y}_{nk}|\theta+\Delta\theta}\right) - p_{n}\left({{\bf y}_{nk}|\theta}\right)}}{{\Delta\theta}}}\right]^{2}}} \nonumber \\ &=& \left({\Delta\theta}\right)^{-2} \Delta {\bf y}_{n} \sum\limits _{k}{\, p_{n} \left({{\bf y}_{nk}|\theta}\right)\left[{\frac{{p_{n}\left({{\bf y}_{nk}|\theta + \Delta\theta}\right)}}{{p_{n}\left({{\bf y}_{nk}|\theta}\right)}} - 1}\right]^{2}} \; .\end{aligned}$$ Powy¿sza zamiana ca³kowania na sumê jest wprowadzona dla wygody i jest œcis³ym przejœciem w granicy $\Delta {\bf y}_{nk} {\scriptstyle {\;\; \longrightarrow \hfill \atop {\scriptstyle k \rightarrow \infty \hfill}}} 0$. W drugiej równoœci w (\[dys\]) przyjêto równe przyrosty $\Delta {\bf y}_{nk} = \Delta {\bf y}_{n}$ dla ka¿dego $k$, co w tej granicy nie zmienia wyniku. Natomiast przejœcie dla pochodnej po $\theta$ pod ca³k¹ $I_{F n}$ w (\[I dla pn jeden parametr\]) dokonane w (\[dys\]) jest s³uszne w granicy $\Delta\theta \rightarrow 0$.\ \ W granicy $\Delta\theta \to 0$ ka¿de z powy¿szych wyra¿eñ ${{p_{n}\left({{\bf y}_{nk}|\theta + \Delta\theta}\right)}\mathord{\left/{\vphantom{{p_{n}\left({{\bf y}_{nk}|\theta + \Delta\theta}\right)}{p_{n}\left({{\bf y}_{nk}|\theta}\right)}}}\right.\kern -\nulldelimiterspace}{p_{n}\left({{\bf y}_{nk}|\theta}\right)}}$ d¹¿y do 1. Wtedy ka¿da z wielkoœci: $$\begin{aligned} \label{delta} \delta_{\Delta \theta}^{n} \equiv \frac{{p_{n}\left({{\bf y}_{nk}|\, \theta + \Delta \theta}\right)}}{{p_{n}\left({{\bf y}_{nk}| \theta}\right)}} - 1 \end{aligned}$$ staje siê ma³a i rozwijaj¹c funkcjê logarytmu do wyrazu drugiego rzêdu: $$\begin{aligned} {\ln \left({1 + \delta_{\Delta \theta}^{n}}\right) \approx \delta_{\Delta \theta}^{n} - {{(\delta_{\Delta \theta}^{n})^{2}} \mathord{\left/{\vphantom{{(\delta_{\Delta \theta}^{n})^{2}}2}} \right. \kern -\nulldelimiterspace}2}} \; ,\end{aligned}$$ otrzymujemy: $$\begin{aligned} \label{nu2} {(\delta_{\Delta \theta}^{n})^{2} \approx 2 \left[{\delta_{\Delta \theta}^{n} - \ln \left({1 + \delta_{\Delta \theta}^{n}}\right)}\right]} \; .\end{aligned}$$ Korzystaj¹c z (\[delta\]) oraz (\[nu2\]) wyra¿enie (\[dys\]) mo¿na zapisaæ nastêpuj¹co: $$\begin{aligned} \label{I trzy skladniki} I_{F n} &=& -2 \left({\Delta\theta}\right)^{-2} \Delta {\bf y}_{n} \left[ \sum\limits _{k}{p_{n}\left({{\bf y}_{nk}|\, \theta}\right) \; \ln\left({\frac{{p_{n}\left({{\bf y}_{nk}|\, \theta + \Delta \theta} \right)}}{{p\left({{\bf y}_{k}|\theta}\right)}}}\right)} \right. \nonumber \\ &-& \left. \sum\limits_{k} {p_{n}\left({{\bf y}_{nk}|\theta + \Delta \theta}\right)} + \sum\limits_{k}{p_{n} \left({{\bf y}_{nk}|\theta} \right)} \right] \; , \;\;\; {\rm dla} \;\;\; \Delta \theta \rightarrow 0 \; .\end{aligned}$$ Ze wzglêdu na warunek normalizacji zachodzi $\sum\limits_{k} {p_{n}\left({{\bf y}_{nk}|\theta + \Delta \theta}\right)} = \sum\limits_{k}{p_{n} \left({{\bf y}_{nk}|\theta} \right)} = 1$, zatem dwie ostatnie sumy po $k$ w nawiasie kwadratowym znosz¹ siê wzajemnie i (\[I trzy skladniki\]) redukuje siê do postaci: $$\begin{aligned} \label{I porownanie z Sn} I_{Fn} &=& -2\left({\Delta\theta}\right)^{-2} \sum\limits_{k} \Delta {\bf y}_{n} \, {p_{n}\left({{\bf y}_{nk}|\theta}\right) \ln\left({\frac{{p_{n}\left({{\bf y}_{nk}|\, \theta + \Delta \theta}\right)}}{{p_{n}\left({{\bf y}_{nk}|\theta}\right)}}}\right)} \nonumber \\ &=& - 2 \left({\Delta\theta}\right)^{-2} \int_{\cal Y} d{\bf y}_{n}\; p_{n} \left({\bf y}_{n}|\theta\right) \ln\left({\frac{{p_{n}\left({{\bf y}_{n}|\theta + \Delta\theta}\right)}}{{p_{n}\left({{\bf y}_{n}|\theta}\right)}}}\right) \nonumber \\ &=& 2 \left({\Delta\theta}\right)^{-2} S_{H}\left[p_{n} \left({{\bf y}_{n}|\theta}\right)| p_{n}\left({{\bf y}_{n}|\theta + \Delta \theta}\right)\right] \; , \;\;\;\;\; {\rm dla} \;\;\; \Delta \theta \rightarrow 0 \; ,\end{aligned}$$ gdzie w ostatnim przejœciu skorzystaliœmy z (\[entropia wzgledna roz ciagle\]). Postaæ $I_{F n}$ w drugiej lini w (\[I porownanie z Sn\]) mo¿na równie¿, wykorzystuj¹c unormowanie rozk³adów $p_{m}( {{\bf y}_{m}|\theta})$, zapisaæ nastêpuj¹co: $$\begin{aligned} \label{I porownanie z S druga linia} \!\!\!\!\!\!\!\!\!\!\! I_{F n} = - 2 \left({\Delta\theta}\right)^{-2} \!\! \prod_{{\scriptstyle {{m=1} \hfill \atop {\scriptstyle m \ne n\hfill}}}}^{N} \!\! \int_{\cal Y} \!\! {d{\bf y}_{m} {p_{m}( {{\bf y}_{m}|\theta} )}} \!\! \int_{\cal Y} \!\! d{\bf y}_{n}\; p_{n} \left({\bf y}_{n}|\theta\right) \ln \! \left({\frac{{p_{n}\left({{\bf y}_{n}|\theta + \Delta\theta}\right)}}{{p_{n}\left({{\bf y}_{n}|\theta}\right)}}} \right) , \;\, {\rm dla} \;\, \Delta \theta \rightarrow 0 , \;\;\;\;\,\end{aligned}$$ sk¹d po skorzystaniu z $I_{F} = \sum_{n=1}^{N}{I_{F n}} $, (\[I dla pn jeden parametr\]), otrzymujemy: $$\begin{aligned} \label{I porownanie z S} \!\!\!\!\! I_{F} = \sum_{n=1}^{N}{I_{F n}} &=& - 2 \left({\Delta\theta}\right)^{-2} \sum_{n=1}^{N} \prod_{{\scriptstyle {{m=1} \hfill \atop {\scriptstyle m \ne n\hfill}}}}^{N} \int_{\cal Y} {d{\bf y}_{m} {p_{m}\left({{\bf y}_{m}|\theta}\right)}} \int_{\cal Y} d{\bf y}_{n}\; p_{n} \left({\bf y}_{n}|\theta\right) \ln\left({\frac{{p_{n}\left({{\bf y}_{n}|\theta + \Delta\theta}\right)}}{{p_{n}\left({{\bf y}_{n}|\theta}\right)}}}\right) \nonumber \\ &=& - 2 \left({\Delta\theta}\right)^{-2} \int_{\cal B} {dy \prod_{m=1}^{N} {p_{m}\left({{\bf y}_{m}|\theta}\right)}} \ln \left({\frac{\prod_{n=1}^{N}{p_{n}\left({{\bf y}_{n}|\theta + \Delta\theta}\right)}}{{\prod_{n=1}^{N} p_{n}\left({{\bf y}_{n}|\theta}\right)}}}\right) \nonumber \\ &=& 2 \left({\Delta\theta}\right)^{-2} S_{H}\left[P \left({y|\theta}\right)| P\left({y|\theta + \Delta \theta}\right)\right] \; , \;\;\;\;\; {\rm dla} \;\;\; \Delta \theta \rightarrow 0 \; .\end{aligned}$$ Tak wiêc (\[I porownanie z S\]) daje wsparcie dla intuicji wspomnianej na pocz¹tku obecnego Rozdzia³u, która mówi, ¿e skoro entropia jest miar¹ nieuporz¹dkowania uk³adu to informacja Fishera jest równie¿ miar¹ jego nieuporz¹dkowania.\ \ [**Uwaga o lokalnoœci zwi¹zku IF z KL**]{}: O w³asnoœciach globalnych rozk³adu wypowiada siê entropia Shannona, natomiast wniosek wynikaj¹cy z (\[I porownanie z S\]) ma sens tylko dla entropii wzglêdnej. Zatem zwi¹zek ten mo¿e byæ co najwy¿ej sygna³em, ¿e niektóre w³asnoœci uk³adu zwi¹zane z entropi¹ Shannona mog¹ byæ ujête w jêzyku informacji Fishera. Na niektóre sytuacje, w których ma to miejsce zwrócimy uwagê w przysz³oœci.\ \ [**Rozk³ad entropii KL dla rozk³adów punktowych**]{}:\ Ze wzglêdu na $I_{F} = \sum_{n=1}^{N}{I_{F n}} $, (\[I dla pn jeden parametr\]), z porównania (\[I porownanie z S\]) z (\[I porownanie z Sn\]) wynika dodatkowo, ¿e entropia wzglêdna rozk³adów $P \left({y|\theta}\right)$ oraz $P\left({y|\theta + \Delta \theta}\right)$ jest sum¹ entropii wzglêdnych odpowiadaj¹cych im rozk³adów punktowych $p_{n} \left({{\bf y}_{n}|\theta}\right)$ oraz $p_{n}({{\bf y}_{n}| \theta + \Delta \theta})$: $$\begin{aligned} \label{S jako suma Sn delta th inf} \!\!\!\!\!\!\!\! S_{H}\left[P \left({y|\theta}\right)| P\left({y|\theta + \Delta \theta}\right)\right] = \sum_{n=1}^{N} S_{H}\left[p_{n} \left({{\bf y}_{n}|\theta}\right)| \, p_{n}\left({{\bf y}_{n}|\theta + \Delta \theta}\right)\right] \;\;\;\;\; {\rm dla \;\; dowolnego} \;\;\; \Delta \theta \; . \;\;\;\;\;\;\end{aligned}$$ Poniewa¿ warunek $\Delta \theta \rightarrow 0$ nie by³ wykorzystywany w otrzymaniu (\[S jako suma Sn delta th inf\]) z porównania (\[I porownanie z Sn\]) oraz (\[I porownanie z S\]), zatem zwi¹zek ten jest s³uszny dla dowolnego $\Delta \theta$. Natomiast nale¿y pamiêtaæ, ¿e rozk³ady punktowe w definicji funkcji wiarygodnoœci s¹ [*nieskorelowane*]{} dla ró¿nych $n$, dlatego te¿ zwi¹zek (\[S jako suma Sn delta th inf\]) jest s³uszny tylko w tym przypadku.\ Pojêcie kana³u informacyjnego {#Pojecie kanalu informacyjnego} ----------------------------- Niech pierwotna zmienna losowa $Y$ przyjmuje wartoœci wektorowe ${\bf y} \in {\cal Y}$. Wektor ${\bf y}$ mo¿e byæ np. wektorem po³o¿enia. Zatem, aby wprowadzony opis by³ wystarczaj¹co ogólny, wartoœci ${\bf y} \equiv ({\bf y}^{\nu})$ mog¹ posiadaæ np. indeks wektorowy $\nu$. Wartoœci te s¹ realizowane zgodnie z ³¹cznym rozk³adem $p({\bf y}|\Theta)$ w³aœciwym dla badanego uk³adu.\ Rozwa¿my $N$-wymiarow¹ próbê. Oznaczmy przez $y \equiv ({\bf y}_{n})_{n=1}^{N} = ({\bf y}_{1}, ..., {\bf y}_{N})$ dane bêd¹ce realizacjami próby $\widetilde{Y} = (Y_{1}, Y_{2}, ..., Y_{N})$ dla pierwotnej zmiennej $Y$, gdzie ${\bf y}_{n} \equiv ({\bf y}_{n}^{\nu})$ oznacza $n$-t¹ wektorow¹ obserwacjê w próbce ($n=1,2,...,N$). Rozk³ad ³¹czny próby jest okreœlony przez $P\left(y|\Theta\right)$.\ \ [**Dodatkowy indeks parametru**]{}: Podobnie, równie¿ parametry rozk³adu mog¹ mieæ dodatkowy indeks. Niech indeks $\alpha$ okreœla pewn¹ dodatkow¹ wspó³rzêdn¹ wektorow¹ parametru ${\theta}_{i}$, gdzie jak w poprzednich rozdzia³ach $i=1,2,...,d$. Zatem wektor parametrów ma teraz postaæ: $$\begin{aligned} \label{parametr wektorowy} \Theta = \left({{\theta}_{1},{\theta}_{2},...,{\theta}_{d}}\right) \; , \;\;\; {\rm gdzie} \;\;\;\; {\theta}_{i}=\left({\theta_{i}^{\alpha}} \right) \; , \;\;\;\; \alpha = 1,2,... \; .\end{aligned}$$ Wariancja estymatora $\hat{\theta}_{i}^{\alpha}$ parametru $\theta_{i}^{\alpha}$ ma postaæ: $$\begin{aligned} \label{wariancja estymatora theta i alfa} {\sigma^{2}} \,( \hat{\theta}_{i}^{\alpha}) = \int_{\cal B} dy \, P\left(y|\Theta\right) \left(\hat{\theta}_{i}^{\alpha}\left(y\right)-\theta_{i}^{\alpha}\right)^{2} \; ,\end{aligned}$$ gdzie $\hat{\theta}_{i}^{\alpha}\left({y}\right)$ jest estymatorem parametru $\theta_{i}^{\alpha}\,$, a ca³kowanie przebiega po ca³ej przestrzeni próby ${\cal B}$, tzn. po wszystkich mo¿liwych realizacjach $y$.\ Wed³ug Rozdzia³u \[Przyklad wektorowego DORC\], dla ka¿dego wyró¿nionego parametru $\theta_{i}^{\alpha}$ wariancja ${\sigma^{2}} \, (\hat{\theta}_{i}^{\alpha})$ jego estymatora (\[wariancja estymatora theta i alfa\]) jest zwi¹zana z informacj¹ Fishera $I_{F i \alpha}$ parametru $\theta_{i}^{\alpha}$ poprzez nierównoœæ (\[porownanie sigma I11 z I11do-1\]): $$\begin{aligned} \label{krzy3} {\sigma^{2}}\,(\hat{\theta}_{i}^{\alpha}) \ge {I_{F}^{i \alpha}} \ge \frac{1}{I_{F i \alpha}} \; ,\end{aligned}$$ gdzie $I_{F}^{i \alpha} \equiv I_{F}^{i\alpha, \,i\alpha}$ jest dolnym ograniczeniem RC dla parametru ${\theta}_{i}^{\alpha}$ w przypadku wieloparametrowym, natomiast: $$\begin{aligned} \label{pojemnosc jeden kanal inform Fishera} I_{F i \alpha} \equiv I_{F i\alpha, \,i\alpha} \; \end{aligned}$$ jest pojemnoœci¹ informacyjn¹ w [*pojedynczym kanale informacyjnym*]{} $(i, \alpha)$ czyli [*informacj¹ Fishera dla parametru*]{} ${\theta}_{i}^{\alpha}$. Zgodnie z (\[inf I jeden parametr - kwadrat 1 pochodnej\]) jest ona równa[^40]: $$\begin{aligned} \label{krzy4} I_{F i \alpha} \equiv I_{F} \left(\theta_{i \alpha} \right) = \int_{\cal B} {dy \, P\left(y|\Theta\right)} \left({\frac{{\partial \ln P\left(y|\Theta\right)}}{{\partial \theta_{i}^{\alpha}}}}\right)^{2} \; .\end{aligned}$$\ [**Informacja Stama**]{}: Wielkoœæ $\frac{1}{{\sigma^{2}} \, (\hat{\theta}_{i}^{\alpha})}$ odnosi siê do pojedynczego kana³u $\left(i, \alpha\right)$. Sumuj¹c j¹ po indeksach $\alpha$ oraz $i$ otrzymujemy tzw. informacjê Stama $I_{S}$ [@Stam]: $$\begin{aligned} \label{informacja Stama} 0 \leq I_{S} \equiv \sum\limits_{i} {\sum\limits_{\alpha} \frac{1}{{\sigma^{2}}\,(\hat{\theta}_{i}^{\alpha})}} \; ,\end{aligned}$$ która jest skalarn¹ miar¹ jakoœci jednoczesnej estymacji we wszystkich kana³ach informacyjnych. [*Informacja Stama jest z definicji zawsze wielkoœci¹ nieujemn¹.*]{}\ \ [**Pojemnoœæ informacyjna $I$**]{}: W koñcu, sumuj¹c lew¹ i praw¹ stronê (\[krzy3\]) po indeksach $\alpha$ oraz $i$ otrzymujemy nastêpuj¹c¹ nierównoœæ dla $I_{S}$: $$\begin{aligned} \label{informacja Stama vs pojemnosc informacyjna} 0 \leq I_{S} \equiv \sum\limits_{i} \sum\limits_{\alpha} \frac{1}{{\sigma^{2}}\,(\hat{\theta}_{i}^{\alpha})} \le \sum\limits_{i} {\sum\limits_{\alpha}{I_{F i \alpha}}} =: I \equiv C \; ,\end{aligned}$$ gdzie $C$, oznaczana dalej jako $I$, nazywana jest [*pojemnoœci¹ informacyjn¹ uk³adu*]{}. Zgodnie z (\[informacja Stama vs pojemnosc informacyjna\]) i (\[krzy4\]) jest ona równa: $$\begin{aligned} \label{pojemnosc C} I = \sum\limits_{i}{\sum\limits _{\alpha}{I_{F i \alpha}}}=\sum\limits_{i}{\int_{\cal B} {dy \; P\left(y|\Theta\right) \sum\limits _{\alpha}{\left(\frac{\partial\ln P\left(y|\Theta\right)}{\partial\theta_{i}^{\alpha}}\right)^{2}}}} \; .\end{aligned}$$ Jak siê oka¿e, pojemnoœæ informacyja $I$ jest najwa¿niejszym pojêciem statystyki le¿¹cym u podstaw np. cz³onów kinetycznych ró¿nych modeli teorii pola. Jest ona uogólnieniem pojêcia informacji Fishera dla przypadku pojedynczego, skalarnego parametru na przypadek wieloparametrowy. ### Pojemnoœæ informacyjna dla zmiennej losowej po³o¿enia {#Poj inform zmiennej los poloz} ZawêŸmy obszar analizy do szczególnego przypadku, gdy interesuj¹cym nas oczekiwanym parametrem jest wartoœæ oczekiwana zmiennej po³o¿enia uk³adu $Y$: $$\begin{aligned} \label{wartosc oczekiwana EY} \theta \equiv E(Y)= (\theta^{\nu}) \; , \;\;\;\; {\rm gdzie} \;\;\;\;\; \theta^{\nu} = \int_{\cal Y} d {\bf y}\, p({\bf y})\, {\bf y}^{\nu} \; .\end{aligned}$$ Wtedy $N$-wymiarowa próbka $y \equiv ({\bf y}_{n})_{n=1}^{N} = ({\bf y}_{1}, ..., {\bf y}_{N})$ jest realizacj¹ próby $\widetilde{Y}$ dla [*po³o¿eñ uk³adu*]{}, a wartoœæ oczekiwana $\theta_{n}$ po³o¿enia uk³adu w $n$-tym punkcie (tzn. pomiarze) próby wynosi: $$\begin{aligned} \label{wartosc oczekiwana EYn} \theta_{n} \equiv E(Y_{n}) = (\theta_{n}^{\nu}) \; , \;\;\; {\rm gdzie} \;\;\;\; \theta_{n}^{\nu} = \int_{\cal B} dy \, P(y|\Theta) \, {\bf y}_{n}^{\nu} \; .\end{aligned}$$ [**Liczba oczekiwanych parametrów**]{}: Gdy, jak to ma miejsce w rozwa¿anym przypadku, jedynym parametrem rozk³adu, który nas interesuje jest wartoœæ oczekiwana po³o¿enia $\theta_{n} \equiv (\theta_{n}^{\nu})$, gdzie $n =1,2,...,N$ jest indeksem próby, wtedy parametr wektorowy $\Theta=(\theta_{n})_{n=1}^{N}$. Zatem liczba parametrów $\theta_{n}$ pokry³a siê z wymiarem próby $N$, a indeksy parametru $\theta_{i}^{\alpha}$ s¹ nastêpuj¹ce: $i \equiv n$, gdzie $n=1,2,...,N$, oraz $\alpha \equiv \nu$, gdzie $\nu$ jest indeksem wektorowym wspó³rzêdnej ${\bf y}_{n}^{\nu}$. Oznacza to, ¿e wymiar parametru $\Theta$ jest taki sam jak wymiar przestrzeni próby ${\cal B}$.\ \ [**Wspó³rzêdne kowariantne i kontrawariantne**]{}: Rozwa¿ania obecnego Rozdzia³u jak i innych czêœci skryptu s¹ zwi¹zane z analiz¹ przeprowadzan¹ w czasoprzestrzeni Minowskiego. Dlatego koniecznym okazuje siê rozró¿nienie pomiêdzy wspó³rzêdnymi kowariantnymi ${\bf y}_{n \nu}$ i kontrawariantnymi ${\bf y}_{n}^{\, \mu}$. Zwi¹zek pomiêdzy nimi, tak dla wartoœci losowego wektora po³o¿enia jak i dla odpowiednich wartoœci oczekiwanych, jest nastêpuj¹cy: $$\begin{aligned} \label{wsp theta kontra i kowariantne} {\bf y}_{n \nu} = \sum_{\mu=0}^{3} \eta_{\nu \mu} \, {\bf y}_{n}^{\, \mu} \; , \;\;\;\;\;\; \theta_{n \nu} = \sum_{\mu=0}^{3} \eta_{\nu \mu} \, \theta_{n}^{\, \mu} \; , \end{aligned}$$ gdzie $(\eta_{\nu \mu})$ jest tensorem metrycznym przestrzeni ${\cal Y}$. W przypadku wektorowego indeksu Minkowskiego $\nu = 0,1,2,3,...$ przyjmujemy nastêpuj¹c¹ postaæ tensora metrycznego: $$\begin{aligned} \label{metryka M} \eta_{\nu\mu}=\left({\begin{array}{ccccc} {1} & 0 & 0 & 0 & 0\\ 0 & {-1} & 0 & 0 & 0\\ 0 & 0 & {-1} & 0 & 0\\ 0 & 0 & 0 & {-1} & 0\\ 0 & 0 & 0 & 0 & \ddots \end{array}}\right)\end{aligned}$$ lub w skrócie $(\eta_{\nu \mu}) = diag(1,-1,-1,-1,...)$. Symbol “diag” oznacza macierz diagonaln¹ z niezerowymi elementami na przek¹tnej g³ównej oraz zerami poza ni¹. Natomiast dla Euklidesowego indeksu wektorowego $\nu = 1,2,3,...\,$, tensor metryczny ma postaæ: $$\begin{aligned} \label{metryka E} (\eta_{\nu \mu}) = diag(1,1,1,...) \;\, .\end{aligned}$$ [**Za³o¿enie o niezale¿noœci $p_{n}$ od $\theta_{m}$ dla $m \neq n$**]{}: W rozwa¿aniach niniejszego skryptu zmienne $Y_{n}$ próby $\widetilde{Y}$ s¹ niezale¿ne (tzn. zak³adamy, ¿e pomiary dla $m \neq n$ s¹ w próbie niezale¿ne). Oznacza to równie¿, ¿e wartoœæ oczekiwana po³o¿enia $\theta_{m} = \int dy \, P(y|\Theta) \, {\bf y}_{m} $, nie ma wp³ywu na rozk³ad $p_{n}({\bf y}_{n}|\theta_{n})$ dla indeksu próby $m \neq n$. Wtedy dane s¹ generowane zgodnie z punktowymi rozk³adami spe³niaj¹cymi warunek: $$\begin{aligned} \label{rozklady punktowe polozenia} p_{n}({\bf y}_{n}|\Theta) = p_{n}({\bf y}_{n}|{\theta}_{n}) \; , \;\;\;\; {\rm gdzie} \;\;\; n=1,...,N \; ,\end{aligned}$$ a wiarygodnoœæ próby jest iloczynem: $$\begin{aligned} \label{funkcja wiaryg param wektorowy} P\left(y|\Theta\right) = \prod_{n=1}^{N} p_{n}({\bf y}_{n}|\theta_{n}) \; . \end{aligned}$$ [**Pojemnoœæ informacyjna kana³u**]{}: Dla parametru po³o¿enia (czaso)przestrzennego, pojemnoœæ kana³u informacyjnego $I$ dla uk³adu, zdefiniowana ogólnie w (\[pojemnosc C\]), przyjmuje postaæ[^41]: $$\begin{aligned} \label{pojemnosc informacyjna Minkowskiego} I = \sum\limits_{n=1}^{N}{I_{F n}} \; ,\end{aligned}$$ gdzie $$\begin{aligned} \label{pojemnosc C dla polozenia} I_{F n} &\equiv& I_{F n} \left(\theta_{n} \right) = \int_{\cal B} {dy \, P\left(y|\Theta\right)} \, \left( \, \nabla_{\theta_{n}} \ln P\left(y|\Theta\right) \cdot \nabla_{\theta_{n}} \ln P\left(y|\Theta\right) \, \right) \nonumber \\ & = & {\int_{\cal B}{dy \; P\left(y|\Theta\right) \sum\limits_{\nu, \, \mu = (0),1,2,...} \eta^{\nu \mu} {\left(\frac{\partial\ln P\left(y|\Theta\right)}{\partial\theta_{n}^{\nu}} \; \frac{\partial \ln P \left(y|\Theta\right)}{\partial\theta_{n}^{\mu}} \right)}}} \; .\end{aligned}$$ Tensor $(\eta^{\nu \mu}) = diag(1,-1,-1,-1)$ jest tensorem dualnym do $(\eta_{\nu \mu})$, tzn. $\sum_{\mu=0}^{3} \eta_{\nu \mu} \eta^{\gamma \mu}$ $= \delta_{\nu}^{\gamma}$, przy czym $\delta_{\nu}^{\gamma}$ jest delt¹ Kroneckera, a $\nabla_{\theta_{n}} \equiv \frac{\partial}{\partial \theta_{n}} = \! \sum_{\nu} \frac{\partial}{\partial \theta_{n}^{\nu}} \, d {\bf y}_{n}^{\nu} \, $. W zwi¹zku (\[pojemnosc C dla polozenia\]) “$\cdot$” oznacza iloczyn wewnêtrzny zdefiniowany przez tensor metryczny $(\eta^{\nu \mu})$.\ \ [**Uwaga o niefaktoryzowalnoœci czasoprzestrzennych indeksów po³o¿enia**]{}: $\;\;$ W pomiarze wybranej $\nu$-tej wspó³rzêdnej po³o¿enia nie mo¿na wykluczyæ odchyleñ (fluktuacji) wartoœci wspó³rzêdnych do niej ortogonalnych. Oznacza to, ¿e wartoœæ oczekiwana $\nu$-tej wspó³rzêdnej po³o¿enia nie jest w (\[wartosc oczekiwana EY\]) liczona z jakiegoœ rozk³adu typu $p({\bf y}^{\nu})$, lecz musi byæ liczona z ³¹cznego rozk³adu $p({\bf y})$ dla wszystkich wspó³rzêdnych ${\bf y}^{\nu}$. W konsekwencji, w przypadku zmiennych po³o¿enia przestrzennego i ich parametrów naturalnych okreœlonych w (\[wartosc oczekiwana EY\]), ca³kowanie w (\[pojemnosc C dla polozenia\]) nie mo¿e zostaæ sfaktoryzowane ze wzglêdu na wspó³rzêdn¹ wektorow¹ $\nu$.\ Zgodnie z powy¿sz¹ uwag¹, wariancja estymatora $\hat{\theta}_{n} \left({y}\right)$ parametru $\theta_{n}$ powinna przyj¹æ postaæ: $$\begin{aligned} \label{wariancja estymatora theta i alfa} \;\;\;\;\;\;\; {\sigma^{2}} \,( \hat{\theta}_{n}) &=& \int_{\cal B} dy \, P\left(y|\Theta\right) \left(\hat{\theta}_{n} \left(y\right) - \theta_{n} \right) \cdot \left(\hat{\theta}_{n} \left(y\right)-\theta_{n} \right) \\ &=& \int_{\cal B} dy \, P\left(y|\Theta\right) \sum\limits_{\nu, \, \mu = (0),1,2,...} \eta_{\nu \mu} \left(\hat{\theta}_{n}^{\nu} \left(y\right) - \theta_{n}^{\nu} \right) \, \left(\hat{\theta}_{n}^{\mu} \left(y\right) - \theta_{n}^{\mu} \right) \; . \nonumber\end{aligned}$$ Rozwa¿ania przedstawione na koñcu Rozdzia³u \[Przyklad wektorowego DORC\] oznaczaj¹, ¿e ze wzglêdu na (\[rozklady punktowe polozenia\]), dla ka¿dego wyró¿nionego parametru $\theta_{n}\,$ wariancja ${\sigma^{2}} \, (\hat{\theta}_{n})$ jego estymatora (\[wariancja estymatora theta i alfa\]) w jego kanale informacyjnym jest zwi¹zana z informacj¹ Fishera $I_{F n} = I_{F n}(\theta_{n})$ parametru $\theta_{n}$ poprzez nierównoœæ (\[I11 rowna sie I11do-1 przypadek diagonalny\]): $$\begin{aligned} \label{Rao Cram uogolnienie} \frac{1}{{\sigma^{2}}\,\left(\hat{\theta}_{n} \right)} \leq \frac{1}{I_{F}^{n}} \leq I_{F n} \;\;\; {\rm gdzie} \;\;\; n=1,2,...,N \; ,\end{aligned}$$ bêd¹c¹ uogólnieniem nierównoœci informacyjnej Rao-Cramera (\[tw R-C dla par skalarnego\]), gdzie $I^{n}_{F}$ jest DORC dla parameteru ${\theta}_{n}$.\ \ [**Uwaga o zmiennych Fisher’owskich**]{}: Faktu [*zale¿noœci statystycznej*]{} zmiennych po³o¿enia przestrzennego dla ró¿nych indeksów ${\nu}$ nie nale¿y myliæ z posiadan¹ przez nie [*niezale¿noœci¹ analityczn¹*]{}, która oznacza, ¿e zmienne $Y$ s¹ tzw. zmiennymi Fisher’owskimi, dla których: $$\begin{aligned} \label{zmienne Fisherowskie} \frac{\partial {\bf y}^{\nu}}{\partial {\bf y}^{\mu}} = \delta^{\nu}_{\mu} \; .\end{aligned}$$\ [**Zwi¹zek pomiêdzy informacj¹ Stama i pojemnoœci¹ $I$**]{}: Wielkoœæ $\frac{1}{{\sigma^{2}} \, (\hat{\theta}_{n})}$ odnosi siê do estymacji w $n$-tym kanale informacyjnym. Sumuj¹c po indeksie $n$ otrzymujemy [*informacjê Stama*]{} $I_{S}$ [@Stam; @Frieden]: $$\begin{aligned} \label{informacja Stama Minkowskiego} 0 \leq I_{S} \equiv \sum\limits_{n=1}^{N} \frac{1}{{\sigma^{2}}\,(\hat{\theta}_{n})} =: \sum\limits_{n=1}^{N} I_{S n} \;\, .\end{aligned}$$ Poniewa¿ $\theta_{n} = (\theta_{n}^{\nu})$ jest parametrem wektorowym, zatem $I_{S n} = \frac{1}{{\sigma^{2}}\,(\hat{\theta}_{n})}$ jest informacj¹ Stama czasoprzestrzennych kana³ów dla $n$-tego pomiaru w próbie[^42].\ W koñcu, sumuj¹c lew¹ i praw¹ stronê w (\[Rao Cram uogolnienie\]) wzglêdem indeksu $n$ i bior¹c pod uwagê (\[informacja Stama\]), zauwa¿amy, ¿e $I_{S}$ spe³nia nierównoœæ: $$\begin{aligned} \label{informacja Stama vs pojemnosc informacyjna Minkowskiego} 0 \leq I_{S} \equiv \sum\limits_{n=1}^{N} I_{S n} \le \sum\limits_{n=1}^{N} {I_{F n}} = I \; ,\end{aligned}$$ gdzie $I$ jest pojemnoœci¹ kana³u informacyjnego (\[pojemnosc informacyjna Minkowskiego\]). Nierównoœæ (\[informacja Stama vs pojemnosc informacyjna Minkowskiego\]) jest minimalnym uogólnieniem “jednokana³owej” nierównoœci Rao-Cram[é]{}r’a (\[Rao Cram uogolnienie\]), potrzebnym z punktu widzenia przeprowadzanego pomiaru.\ \ Z punktu widzenia modelowania fizycznego, pojemnoœæ kana³u informacyjnego $I$ jest najwa¿niejszym pojêciem statystycznym, le¿¹cym u podstaw cz³onów kinematycznych [@Frieden] ró¿nych modeli teorii pola. Zgodnie z (\[informacja Stama\]) okaza³o siê, ¿e zarówno dla metryki Euklidesowej (\[metryka E\]) jak i metryki Minkowskiego (\[metryka M\]), estymacja jest wykonywana dla dodatniej informacji $I_{S}$. Zatem z (\[informacja Stama vs pojemnosc informacyjna\]) wynika, ¿e $I$ jest równie¿ nieujemna. W Rozdziale \[Informacja Fouriera\] wed³ug (\[I by Fourier to m\]) oka¿e siê, ¿e $I$ jest nieujemnie zdefiniowana dla teorii pola dla cz¹stek, które maj¹ nieujemny kwadrat masy [@dziekuje; @za; @neutron]. Chocia¿ w ka¿dym szczególnym modelu teorii pola z przestrzeni¹ Minkowskiego fakt ten powinien zostaæ sprawdzony, to z punktu widzenia teorii estymacji jest jasne, ¿e: $$\begin{aligned} \label{przyczynowosc} {\sigma^{2}}\,(\hat{\theta}_{n}) \geq 0 \; .\end{aligned}$$ Sytuacja ta ma zawsze miejsce dla [*procesów przyczynowych*]{}.\ \ \ [**Uwaga o indeksie próby i kanale pomiarowym**]{}: Indeks próby $n$ jest najmniejszym indeksem kana³u informacyjnego, w którym dokonywany jest pomiar. Tzn. gdyby indeks próby mo¿na by³o dodatkowo “zaindeksowaæ”, np. indeksem czasoprzestrzennym, wyznaczaj¹c podkana³y, to i tak nie mo¿na by dokonaæ pomiaru tylko w jednym z tak wyznaczonych podkana³ów (nie dokonuj¹c go równoczeœnie w pozosta³ych podkana³ach posiadaj¹cych indeks próby $n$). [*Kana³ niepodzielny z punktu wiedzenia eksperymentu nazwijmy kana³em pomiarowym*]{}.\ \ [**Uwaga o analizie we fragmencie kana³u pomiarowego**]{}: W przypadku ograniczenia analizy do fragmentu kana³u pomiarowego nale¿y siê upewniæ, czy pozosta³a w analizie czêœæ informacji Stama ma wartoœæ dodatni¹. Np. w przypadku zaniedbania czasowo zaindeksowanej czêœci czasoprzestrzennego kana³u pomiarowego, otrzymana nierównoœæ Stama dla sk³adowych przestrzennych ma postaæ: $$\begin{aligned} \label{nierownosc informacja Stama dla podkanalow przestrzennych} \!\!\! 0 \leq I_{S} \! &=& \! \sum_{n=1}^{N} I_{S n} \nonumber \\ & & \leq \sum_{n=1}^{N} {\int{d\vec{y} \, P\left(\vec{y}|\vec{\Theta}\right) \sum \limits_{i=1}^{3}{\frac{\partial\ln P\left(\vec{y}|\vec{\Theta}\right)}{\partial\theta_{n i}} \; \frac{\partial\ln P\left(\vec{y}|\vec{\Theta}\right)}{\partial\theta_{n i}}}}} =: \sum_{n=1}^{N} I_{F n} = I\; ,\end{aligned}$$ gdzie znak wektora oznacza, ¿e analiza zarówno w przestrzeni próby jak i przestrzeni parametrów zosta³a obciêta do czêœci przestrzennej zmiennych losowych i parametrów.\ \ [**Uwaga o symetrii**]{}: Z punktu widzenia pomiaru, b³¹d estymacji sk³adowych wchodz¹cych jednoczeœnie w wyznaczenie [*d³ugoœci*]{} czterowektora w $n$-tym kanale, jest niezale¿ny od uk³adu wspó³rzêdnych w przestrzeni Minkowskiego. Dlatego wielkoœæ $I_{S n} = 1/{\sigma^{2}}\,(\hat{\theta}_{n})$ okreœlona poprzez (\[informacja Stama Minkowskiego\]) oraz (\[wariancja estymatora theta i alfa\]) jest dla tensora metrycznego (\[metryka M\]) niezmiennicza ze wzglêdu na transformacjê Lorentz’a (pchniêcia i obroty). W przypadku tensora metrycznego (\[metryka E\]) jest ona niezmiennicza ze wzglêdu na transformacje Galileusza.\ Jeœli chodzi o pojemnoœæ informacyjn¹ $I$, to w przypadku niezale¿noœci pomiarów w próbie, jest ona równie¿ niezmiennicza ze wzglêdu na transformacjê Lorentz’a w przestrzeni z metryk¹ Minkowskiego (czy transformacjê Galileusza w przestrzeni Euklidesowej), o ile niezmiennicze jest ka¿de $I_{n}$.[^43] Warunki niezmienniczoœci $I_{Sn}$ oraz $I$ schodz¹ siê, gdy w nierównoœci Rao-Cramera (\[Rao Cram uogolnienie\]) osi¹gana jest równoœæ.\ Nierównoœæ Rao-Cramera okazuje siê niezmiennicza ze wzglêdu na podstawowe transformacje [@Frieden; @PPSV]. Istotnie, zgodnie z powy¿szymi rozwa¿aniami, w³aœciwym pomiarem niezale¿nym od przyjêtego uk³adu wspó³rzêdnych jest pomiar kwadratu d³ugoœci $\sum_{\nu=0}^{3} {\bf y}_{\nu} {\bf y}^{\nu}$, a nie pojedynczej wspó³rzêdnej ${\bf y}^{\nu}$. Wiêcej na temat niezmienniczoœci DORC ze wzglêdu na przesuniêcie, odbicie przestrzenne, obroty i transformacjê affiniczn¹ oraz transformacje unitarne mo¿na znaleŸæ w [@PPSV].\ \ [**Kryterium minimalizacji $I$ ze wzglêdu na $N$**]{}: Na koniec zauwa¿my, ¿e sumowanie w (\[pojemnosc C dla polozenia\]) przebiega od $n=1$ do $n=N$. Ka¿dy $n$-ty wyraz w sumie wnosi analityczny wk³ad jako stopieñ swobody dla $I$. O ile dodane stopnie swobody nie wp³ywaj¹ na ju¿ istniej¹ce, to poniewa¿ ka¿dy, ca³y wyraz w sumie po $n$ jest nieujemny, to informacja $I$ ma tendencje do wzrostu wraz ze wzrostem $N$. Kryterium minimalizacji $I$ ze wzglêdu na $N$ pos³u¿y³o Friedenowi i Sofferowi jako dodatkowy warunek przy konstrukcji np. równañ ruchu. Nie znaczy to, ¿e modele z wiêkszym $N$ zosta³y automatycznie wykluczone, tylko ¿e im wiêksze jest $N$ tym wiêcej stopni swobody wchodzi do opisu obserwowanego zjawiska i opisywane zjawisko jest bardziej z³o¿one. Zagadnienie to omówimy w przyk³adach, w dalszej czêœci skryptu. Pomiar uk³adu w podejœciu Friedena-Soffera {#Podstawowe zalozenie Friedena-Soffera} ------------------------------------------ Jak w Rozdziale \[Poj inform zmiennej los poloz\], rozwa¿my zmienn¹ losow¹ $Y$ po³o¿enia uk³adu, przyjmuj¹c¹ wartoœæ ${\bf y}$, która jest punktem zbioru ${\cal Y}$. Mo¿e to byæ punkt czasoprzestrzenny przestrzeni Minkowskiego, co ma miejsce w rozwa¿aniach zwi¹zanych z opisem uk³adu np. w mechanice falowej. Wartoœci ${\bf y} \equiv ({\bf y}^{\nu})_{\nu=0}^{3} \in {\cal Y} \equiv \mathbb{R}^{4}$ s¹ realizowane zgodnie z rozk³adem $p({\bf y})$ w³aœciwym dla uk³adu[^44].\ \ [**Podstawowe za³o¿enie fizyczne podejœcia Friedena-Soffera**]{}: Niech dane $y \equiv ({\bf y}_{n})_{n=1}^{N} = ({\bf y}_{1}, ..., {\bf y}_{N})$ s¹ realizacjami próby dla po³o¿eñ uk³adu, gdzie ${\bf y}_{n} \equiv ({\bf y}_{n}^{\nu})_{\nu=0}^{3}$. [*Zgodnie z za³o¿eniem zaproponowanym przez Frieden’a i Soffer’a*]{} [@Frieden][*, ich zebranie nastêpuje przez sam uk³ad w zgodzie z rozk³adami gêstoœci prawdopodobieñstwa, $p_{n}({\bf y}_{n}|\theta_{n})$, gdzie $n=1,...,N$*]{}.\ \ Treœæ powy¿szego za³o¿enia fizycznego mo¿na wypowiedzieæ nastêpuj¹co: [*Uk³ad próbkuje dostêpn¹ mu czasoprzestrzeñ, “zbieraj¹c dane i dokonuj¹c analizy statystycznej”, zgodnie z zasadami informacyjnymi*]{} (wprowadzonymi w Rozdziale \[Zasady informacyjne\]).\ \ [**Przestrzenie statystyczne próby ${\cal S}$, punktowa ${\cal S}_{4}$ oraz ${\cal S}_{N \times 4} \;$**]{}: Niech rozwa¿ana przestrzeñ jest czasoprzestrzeni¹ Minkowskiego. Wtedy ka¿dy z rozk³adów $p_{n}({\bf y}_{n}|\theta_{n})$ jest punktem modelu statystycznego ${\cal S}_{4} = \{p_{n}({\bf y}_{n}|\theta_{n})\}$ parametryzowanego przez naturalny parametr, tzn. przez wartoœæ oczekiwan¹ $\theta_{n} \equiv (\theta_{n}^{\nu})_{\nu=0}^{3} = E(Y_{n})$, jak w (\[wartosc oczekiwana EYn\]). Zbiór wartoœci $d = 4 \times N$ - wymiarowego parametru $\Theta=(\theta_{n})_{n=1}^{N}$ tworzy wspó³rzêdne dla ³¹cznego rozk³adu $P(y\,|\Theta)$, bêd¹cego punktem na $d=4 \times N$-wymiarowej rozmaitoœci, która jest (pod)przestrzeni¹ statystyczn¹ ${\cal S} \subset \Sigma({\cal B})$ [@Amari; @Nagaoka; @book] (por. (\[model statystyczny S\])): $$\begin{aligned} \label{model statystyczny S z Theta} {\cal S} = \{P_{\Theta} \equiv P(y\,|\Theta), \Theta \equiv (\theta_{n})_{n=1}^{N} \in {V}_{\Theta} \subset \Re^{d} \} \; , \end{aligned}$$ okreœlon¹ na przestrzeni próby ${\cal B}$. Jak wiemy z Rozdzia³u \[alfa koneksja\], rozmaitoœæ ${\cal S}$ jest rodzin¹ rozk³adów prawdopodobieñstwa parametryzowan¹ przez rzeczywist¹, nie losow¹ zmienn¹ wektorow¹ $\Theta \equiv (\theta_{n})_{n=1}^{N} \in {V}_{\Theta}$, która w rozwa¿anym przypadku parametru po³o¿enia, tworzy $N \times 4$-wymiarowy lokalny uk³ad wspó³rzêdnych. Zatem, poniewa¿ próba $\widetilde{Y}$ jest $N \times 4$-wymiarow¹ zmienn¹ losow¹, wiêc wymiary przestrzeni próby ${\cal B}$ i wektorowego parametru $\Theta \equiv (\theta_{n}^{\nu})_{n=1}^{N}$ s¹ takie same[^45].\ W przysz³oœci oka¿e siê, ¿e z powodu zwi¹zku (\[rozklady punktowe polozenia\]) analiza na przestrzeni statystycznej okreœlonej przez (\[model statystyczny S z Theta\]) z parametrami $(\theta_{n}^{\nu})_{n=1}^{N}$ tworz¹cymi $N \times 4$-wymiarowy lokalny uk³ad wspó³rzêdnych, efektywnie redukuje siê do analizy na ${\cal S}_{N \times 4} \equiv \{\bigoplus_{n=1}^{N} p_{n}(y\,|\theta_{n}) \}$. Jednak¿e, poniewa¿ wartoœci parametru $\theta_{n}$ mog¹ siê zmieniaæ od jednego punktu $n$ próby do innego punktu $n'$, zatem nie mo¿e byæ ona w ogólnoœci sprowadzona do analizy na ${\cal S}$ poprzez samo przeskalowanie metryki Riemanna oraz (dualnej) koneksji na ${\cal S}_{4}$ przez czynnik $N$, jak to ma miejsce we wnioskowaniu pojawiaj¹cym siê w statystyce klasycznej [@Amari; @Nagaoka; @book]. W przysz³oœci liczbê $N$ parametrów $\theta_{n}$, bêd¹c¹ wymiarem próby $y \equiv ({\bf y}_{n})_{n=1}^{N}$, bêdziemy nazywali rang¹ pola.\ \ [**Uwaga o podmiocie**]{}: Jednak pamiêtajmy, ¿e estymacji dokonuje tylko cz³owiek, zatem na metodê EFI nale¿y patrzeæ tylko jak na [*pewien model analizy statystycznej*]{}.\ \ Interesuj¹ca nas statystyczna procedura estymacyjna dotyczy wnioskowania o $p_{n}({\bf y}_{n}|\theta_{n})$ na podstawie danych $y$ z wykorzystaniem funkcji wiarygodnoœci $P(y|\Theta)$. Za³ó¿my, ¿e dane zbierane przez uk³ad $y = \left({\bf y}_1,{\bf y}_2,...,{\bf y}_N\right)$ s¹ uzyskane niezale¿nie tak, ¿e ³¹czny rozk³ad prawdopodobieñstwa dla próby faktoryzuje siê na rozk³ady brzegowe: $$\begin{aligned} P(\Theta) \equiv P\left(y|\Theta \right) = \prod\limits_{n=1}^N {p_n \left({\bf y}_n|\Theta \right)} = \prod\limits_{n=1}^N {p_n\left(y_n| \theta_n \right)} \; ,\end{aligned}$$ gdzie w ostatniej równoœci skorzystano z za³o¿enia, ¿e parametr $\theta_{m}$ dla $m \neq n$ nie ma wp³ywu na rozk³ad zmiennej $Y_n$.\ \ [**Wstêpne okreœlenie postaci kinematycznej $I$**]{}: Centralna czêœæ pracy Frieden’a i Soffer’a jest zwi¹zana z przejœciem od pojemnoœci informacyjnej $I$ zadanej równaniem (\[pojemnosc C dla polozenia\]) oraz (\[pojemnosc informacyjna Minkowskiego\]): $$\begin{aligned} \label{pojemnosc C dla polozenia - powtorka wzoru} I = \sum\limits_{n=1}^{N} I_{n} = \sum\limits_{n=1}^{N}{\int_{\cal B}{dy \; P\left(y|\Theta\right)\sum\limits _{\nu=0}^{3}{\left(\frac{\partial\ln P\left(y|\Theta\right)}{\partial\theta_{n \nu}} \; \frac{\partial\ln P\left(y|\Theta\right)}{\partial\theta_{n}^{\nu}} \right)}}} \; ,\end{aligned}$$ gdzie $d y:= d^{4}{\bf y}_{1}...d^{4}{\bf y}_{N}\,$ oraz $d^{4}{\bf y}_{n}=d {\bf y}_{n}^{0} d {\bf y}_{n}^{1} d {\bf y}_{n}^{2} d {\bf y}_{n}^{3} \,$, do tzw. postaci kinematycznej wykorzystywanej w teorii pola oraz fizyce statystycznej.\ \ Rachunek analogiczny jaki doprowadzi³ z (\[inf I jeden parametr - kwadrat 1 pochodnej\]) do (\[I dla pn jeden parametr\]) wygl¹da teraz w skrócie nastêpuj¹co. Przekszta³æmy pochodn¹ $\ln P$ w (\[pojemnosc C dla polozenia - powtorka wzoru\]) do postaci: $$\begin{aligned} \frac{\partial \ln P \left(y|\Theta \right)}{\partial \theta_{n\nu}} = \frac{\partial}{\partial \theta_{n\nu}}\sum\limits_{n=1}^N {\ln p_{n} \left({\bf y}_n|\theta_n \right)} = \sum\limits_{n=1}^N {\frac{1}{p_n \left({\bf y}_n|\theta_n \right)} \frac{\partial{p_n \left({\bf y}_n|\theta_n \right)} }{\partial \theta_{n\nu}}} \; .\end{aligned}$$ Pamiêtaj¹c o unormowaniu ka¿dego z rozk³adów brzegowych, $\int_{\cal Y} d^{4}{\bf y_{n}} $ $ p_n \left({\bf y}_n|\theta_n \right) = 1$, otrzymujemy postaæ pojemnoœci informacyjnej: $$\begin{aligned} \label{postac I bez log p po theta} I = \sum_{n=1}^N {\int_{\cal Y} d^{4}{\bf y}_n \frac{1}{{p_{n} \left( {\bf y}_n|\theta_{n} \right)}} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial p_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n \nu} }}} {\frac{{\partial p_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n}^{ \nu} }}} \right) } } \; ,\end{aligned}$$ bêd¹c¹ uogólnieniem (\[I dla pn jeden parametr\]).\ \ W koñcu przejdŸmy do amplitud $q_{n}\left( {\bf y}_n|\theta_{n} \right)$ okreœlonych jak w (\[amplituda a rozklad\]): $$\begin{aligned} p_{n} \left( {\bf y}_n|\theta_{n} \right) = q_{n}^{2}\left( {\bf y}_n|\theta_{n} \right) \; .\end{aligned}$$ Proste rachunki daj¹: $$\begin{aligned} \label{potrz} I = 4 \sum\limits_{n=1}^N \int_{\cal Y} {d^{4}{\bf y}_{n} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial q_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n \nu} }}} {\frac{{\partial q_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n}^{ \nu} }}} \right) } } \; , \end{aligned}$$ czyli prawie kluczow¹ postaæ pojemnoœci informacyjnej dla rachunku metody EFI Friedena-Soffera. Jedyne co trzeba jeszcze zrobiæ, to przejœæ od przestrzeni statystycznej ${\cal S}$ z baz¹ $(\theta_{n})_{n=1}^{N}$ dla reprezentacji amplitud danych pomiarowych ${\bf y}_{n} \in {\cal Y}$, do przestrzeni amplitud przesuniêæ ${\bf x}_{n} := {\bf y}_{n} - \theta_{n}$ okreœlonych na przestrzeni bazowej ${\cal X}$. Poœwiêcimy temu zagadnienu Rozdzia³ \[The kinematical form of the Fisher information\]. ### Przyk³ad: Estymacja w fizycznych modelach eksponentialnych {#Estymacja w modelach fizycznych na DORC} W Rozdziale \[Dualne uklady wspolrzednych\] wprowadzone zosta³o pojêcie dualnych affinicznych uk³adów wspó³rzêdnych na przestrzeni statystycznej ${\cal S}$. Obecny rozdzia³ poœwiêcony jest zastosowaniu modeli eksponentialnych (\[exponential family\]) i szczególnej roli [*parametrów dualnych*]{} zwi¹zanych z koneksj¹ $\nabla^{(-1)}$. Rodzina modeli eksponentialnych wykorzystywana jest w teorii estymacji szeregu zagadnieñ fizycznych. Jej najbardziej znan¹ realizacj¹ jest estymacja metod¹ maksymalnej entropii, sformu³owana w poni¿szym twierdzeniu dla wymiaru próby $N=1$.\ \ Niech: $$\begin{aligned} \label{entropia w dowodzie o maks entropii} S_{H}(p) = - \int_{\cal Y} d{\bf y} p({\bf y}|\Xi) \ln p({\bf y}|\Xi) \; , \;\;\;\;\; \forall\, p_{\Xi} \in {\cal S} \; \end{aligned}$$ jest entropi¹ Shannona stanu uk³adu zadanego rozk³adem $p({\bf y}|\Xi)$ z parametrami $\Xi$, a $F_{i}(Y)$, $i=1,2,...,d$, uk³adem niezale¿nych zmiennych losowych o okreœlonych wartoœciach oczekiwanych: $$\begin{aligned} \label{wartosi oczekiwane eta model eksponentialny} \theta_{i} = \theta_{i}(\Xi) = E_{\Xi}\left[ F_{i}(Y) \right] = \int_{\cal Y} d{\bf y} p({\bf y}|\Xi) F_{i}({\bf y}) \; , \;\;\;\; i = 1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, p_{\Xi} \in {\cal S} \; .\end{aligned}$$ Z (\[wartosi oczekiwane eta model eksponentialny\]) widaæ, ¿e $F_{i}(Y)$ s¹ [*nieobci¹¿onymi estymatorami parametrów*]{} $\theta_{i}$, tzn.: $$\begin{aligned} \label{Fi jako estymatory wartosi oczekiwane eta model eksponentialny} \hat{\theta}_{i} = F_{i}(Y) \; , \;\;\;\; i=1,2,...,d \; .\end{aligned}$$ [**Twierdzenie o stanie z maksymaln¹ entropi¹ (TME)**]{}. Istnieje jednoznacznie okreœlony unormowany stan uk³adu posiadaj¹cy [*maksymaln¹ entropiê*]{}, zadany nastêpuj¹co: $$\begin{aligned} \label{model eksponentialny dla stanu maks entropii} p_{\Xi} \equiv p({\bf y}|\Xi) = Z^{-1} \exp \left( \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right) \; , \;\;\;\;\; \forall\, p_{\Xi} \in {\cal S} \; \end{aligned}$$ w bazie kanonicznej $\Xi \equiv (\xi^{i})_{i=1}^{d}$ modelu eksponentialnego (\[exponential family\]), gdzie sta³a normalizacyjna $Z$ jest tzw. [*funkcj¹ partycji*]{} (podzia³u) [@Jurek; @Dajka; @kwantowe; @metody; @opisu]: $$\begin{aligned} \label{model eksponentialny funkcja podzialu} Z \equiv Z(\Xi) = \int_{\cal Y} d{\bf y} \exp \left( \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right) \; .\end{aligned}$$\ [**Dowód**]{}: Zgodnie z [*twierdzeniem Lagrange’a*]{} wiemy, ¿e maksimum warunkowe dla funkcji $S_{H}(p)$, przy dodatkowym warunku normalizacyjnym: $$\begin{aligned} \label{normalizacja dla rozkl eksp} \int_{\cal Y} d{\bf y} \,p({\bf y}|\Xi) = 1 \; \end{aligned}$$ oraz $d$ warunkach zwi¹zanych z zadaniem wartoœci oczekiwanych (\[wartosi oczekiwane eta model eksponentialny\]), jest równowa¿ne wyznaczeniu bezwarunkowego ekstremum funkcji: $$\begin{aligned} \label{funkcja S warunkowa z norm i d war} S_{war}(p)\! &=& \! \int_{\cal Y} d{\bf y} s_{war}(p) := S_{H}(p) - \xi^{0} \left (1 - \int_{\cal Y} d{\bf y} \,p({\bf y}|\Xi) \right) - \sum_{i=1}^{d} \xi^{i} \left( \theta_{i} - \int_{\cal Y} d{\bf y} \, p({\bf y}|\Xi) F_{i}({\bf y}) \right) \nonumber \\ &=& \! \int_{\cal Y} d{\bf y} \, p({\bf y}|\Xi) \left( - \ln p({\bf y}|\Xi) + \xi^{0} + \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right) - \xi^{0} - \sum_{i=1}^{d} \xi^{i} \theta_{i} \; ,\end{aligned}$$ ze wzglêdu na wariacjê $p({\bf y}|\Xi)$, gdzie $\xi^{0}$ oraz $(\xi^{i})_{i=1}^{d}$ s¹ czynnikami Lagrange’a. Poniewa¿ $S_{H}(p)$ jest œcisle wklês³a [@Bengtsson_Zyczkowski], zatem otrzymujemy [*maksimum, które jest wyznaczone jednoznacznie*]{}.\ \ Warunek ekstremizacji funkcjona³u[^46] $S_{war}(p)$ ze wzglêdu na $p({\bf y}|\Xi)$, tzn. $\delta_{(p)} S_{war} = 0$, prowadzi do równania Eulera-Lagrange’a: $$\begin{aligned} \label{EL eq dla entropii} \frac{\partial }{\partial {\bf y}} \left({\frac{{\partial s_{war}}}{{\partial \left( \frac{\partial p({\bf y}|\Xi)}{\partial {\bf y}} \right) }}}\right) = \frac{{\partial s_{war}}}{{\partial p({\bf y}|\Xi)}} \;\; ,\end{aligned}$$ gdzie zgodnie z (\[funkcja S warunkowa z norm i d war\]) postaæ funkcji podca³kowej $s_{war}$ wynosi: $$\begin{aligned} \label{postac s_war} s_{war} = p({\bf y}|\Xi) \left( - \ln p({\bf y}|\Xi) + \xi^{0} + \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right) \; .\end{aligned}$$ Podstawiaj¹c $s_{war}$ do (\[EL eq dla entropii\]) otrzymujemy: $$\begin{aligned} \label{wariacja S warunkowanego} - \ln p({\bf y}|\Xi) + \xi^{0} + \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) - 1 = 0 \; .\end{aligned}$$ Równanie (\[wariacja S warunkowanego\]) daje szukan¹ postaæ rozk³adu $p({\bf y}|\Xi)$ maksymalizuj¹cego entropiê $S_{H}(p)$: $$\begin{aligned} \label{rozwiazanie dla stanu maks entropii} p({\bf y}|\Xi) = A \, \exp \left( \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right) \; , \;\;\;\; {\rm gdzie} \;\;\;\; A = 1/\exp(1 - \xi^{0}) = const. \;\; , \end{aligned}$$ który jak to widaæ z (\[exponential family\]) jest typu eksponentialnego z parametrami kanonicznymi $\xi^{i}$, $i=1,2,...,d$ oraz $C({\bf y}) = 0$. Ponadto z warunku normalizacji (\[normalizacja dla rozkl eksp\]) otrzymujemy $A = Z^{-1}$, gdzie $Z$ jest funkcj¹ partycji (\[model eksponentialny funkcja podzialu\]). c.n.d.\ \ [**Parametry dualne**]{}. Po rozpoznaniu, ¿e model maksymalizuj¹cy entropiê stanu uk³adu jest modelem eksponentialnym (\[model eksponentialny dla stanu maks entropii\]) w parametryzacji kanonicznej $\Xi$, mo¿emy (\[model eksponentialny dla stanu maks entropii\]) zapisaæ w postaci (\[exponential family\]): $$\begin{aligned} \label{exponential family dla maks entrop} p_{\Xi} \equiv p({\bf y}| \Xi) = \exp \left[ \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) - \psi(\Xi) \right] \; , \;\;\;\;\; \forall\, p_{\Xi} \in {\cal S} \; ,\end{aligned}$$ gdzie $$\begin{aligned} \label{Z dla exponential family dla maks entrop} Z(\Xi) = \exp \left[\psi(\Xi) \right] \; .\end{aligned}$$ Poniewa¿ zgodnie z (\[psi dla exponential family\]), $\psi(\Xi) = \ln \int_{\cal Y} d{\bf y} \exp \left[ \sum_{i=1}^{d} \xi^{i} F_{i}({\bf y}) \right] $, zatem z (\[wartosi oczekiwane eta model eksponentialny\]) oraz wykorzystuj¹c (\[exponential family dla maks entrop\]), otrzymujemy: $$\begin{aligned} \label{eta dla rozkl eksponentialnego entropia} \theta_{i} = \partial_{\xi^i} \psi(\Xi) \; , \;\;\;\; i=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, p_{\Xi} \in {\cal S} \; ,\end{aligned}$$ gdzie skorzystano z oznaczenia $\partial_{\xi^i} \equiv \partial/\partial \xi^{i}$.\ Korzystaj¹c z $\partial_{\xi^j} \partial_{\xi^i} \ln p({\bf y}|\Xi) = - \partial_{\xi^j} \partial_{\xi^i} \psi(\Xi)$, (\[partial l dla psi dla exponential family\]), oraz $g^{\xi}_{ij} = - E_{\Xi}(\partial_{\xi^i} \partial_{\xi^j} \ln p({\bf y}|\Xi))$, (\[Fisher inf matrix plus reg condition\]), otrzymujemy równie¿: $$\begin{aligned} \label{row rozn 2 rzedu dla psi i eta dla eksponential} g^{\xi}_{ij} = \partial_{\xi^i} \partial_{\xi^j} \psi(\Xi) \; , \;\;\;\; i,j=1,2,...,d \; , \;\;\;\; i,j=1,2,...,d \;\;\;\;\; , \;\;\;\;\; \forall\, p \in {\cal S} \; .\end{aligned}$$\ [**Dualne uk³ady modelu maksymalizuj¹cego entropiê**]{}: Z (\[Gamma 1 dla exponential family\]) wiemy, ¿e uk³ad wspó³rzêdnych $(\xi^{i})_{i=1}^{d}$ jest $\alpha = 1$ - afinicznym uk³adem wspó³rzêdnych modelu eksponentialnego. Zatem z (\[eta dla rozkl eksponentialnego entropia\]) oraz (\[row rozn 2 rzedu dla psi i eta dla eksponential\]) wynika, ¿e $(\theta_{i})_{i=1}^{d}$ [*jest $\alpha = (- 1)$ - afinicznym uk³adem wspó³rzêdnych modelu eksponentialnego dualnym do*]{} $(\xi^{i})_{i=1}^{d}$.\ Parametry $\theta_{i}$, $i=1,2,...,d$, nazywamy z przyczyn podanych powy¿ej [*parametrami dualnymi*]{} do $(\xi^{i})_{i=1}^{d}$ lub [*parametrami oczekiwanymi*]{} modelu statystycznego ${\cal S}$.\ \ [**Estymacja na DORC**]{}: Korzystaj¹c z definicji (\[Fisher inf matrix\]) metryki Rao-Fishera, z postaci (\[exponential family dla maks entrop\]) rozk³adu eksponentialnego oraz z (\[eta dla rozkl eksponentialnego entropia\]), otrzymujemy postaæ macierzy informacyjnej $I_{F}$ w parametryzacji kanonicznej $\Xi$: $$\begin{aligned} \label{Fisher inf matrix eksonent dla max entrop dla Theta} \!\!\!\!\!\! I_{F\, ij}(\Xi) \equiv g^{\xi}_{ij} &=& E_{\Xi}\left[ \partial_{\xi^i} \ln p(Y| \Xi) \; \partial_{\xi^j} \ln p(Y| \Xi) \right] \nonumber \\ &=& E_{\Xi}\left[ (F_{i}(Y) - \theta_{i}) (F_{j}(Y) - \theta_{j}) \right] \; , \;\;\; i,j = 1,2,...,d \; , \;\; \forall\, p_{\Xi} \in {\cal S} \; . \;\;\;\;\;\; \end{aligned}$$ Niech $\hat{\Theta} \equiv ({\hat{\theta}}_{i})_{i=1}^{d}$ s¹ estymatorami parametrów $\Theta \equiv (\theta_{i})_{i=1}^{d}$ bazy dualnej do $\Xi$. Poniewa¿ z (\[Fi jako estymatory wartosi oczekiwane eta model eksponentialny\]) mamy $\hat{\theta}_{i} = F_{i}(Y)$, $i=1,2,...,d$, zatem po prawej stronie (\[Fisher inf matrix eksonent dla max entrop dla Theta\]) stoj¹ elementy macierzy kowariancji $V_{\Xi}(\hat{\Theta})$ estymatorów $\hat{\Theta}$: $$\begin{aligned} \label{macierz kowar estymatorow Eta eksonent dla max entrop} V_{\Xi \, ij}(\hat{\Theta}) = E_{\Xi}\left[ (\hat{\theta}_{i} - \theta_{i}) (\hat{\theta}_{j} - \theta_{j}) \right] \; , \;\;\;\; i,j = 1,2,...,d \; , \;\; \forall\, p_{\Xi} \in {\cal S} \; .\end{aligned}$$ Równoœæ (\[Fisher inf matrix eksonent dla max entrop dla Theta\]) mo¿na wiêc zapisaæ nastêpuj¹co: $$\begin{aligned} \label{rownosc Fisher Theta inf i macierzy kowariancji maks entrop} V_{\Xi}(\hat{\Theta}) = I_{F}(\Xi) \; , \;\;\;\;\;\;\;\;\;\;\; \forall\, p_{\Xi} \in {\cal S} \; .\end{aligned}$$ Poniewa¿ macierze informacyjne w bazach dualnych s¹ wzglêdem siebie odwrotne, tzn. $I_{F}(\Xi) = I_{F}^{-1}(\Theta)$, (\[macierze informacyjne dualne\]), oraz macierz kowariancji okreœlonych zmiennych losowych (w tym przypadku $\hat{\Theta}$) nie zale¿y od bazy w ${\cal S}$: $$\begin{aligned} \label{macierz kowariancji V dla Theta w roznych bazach} V_{\Theta}(\hat{\Theta}) = V_{\Xi}(\hat{\Theta}) \; , \;\;\;\;\;\;\;\;\;\;\; \forall\, p \in {\cal S} \; ,\end{aligned}$$ wiêc z (\[rownosc Fisher Theta inf i macierzy kowariancji maks entrop\]) otrzymujemy nastêpuj¹cy zwi¹zek: $$\begin{aligned} \label{rownosc odwrot Fisher Eta inf i m kowariancji maks entrop} V_{\Theta}(\hat{\Theta}) = I_{F}^{-1}(\Theta) \; , \;\;\;\;\;\;\;\;\;\;\; \forall\, p_{\Theta} \in {\cal S} \; .\end{aligned}$$\ [**Wniosek**]{}: Ze wzglêdu na Twierdzenie Rao-Cramera (\[twierdzenie RC wersja 2\]) powy¿szy warunek oznacza, ¿e estymacja parametrów dualnych jest dla modeli eksponentialnych, spe³niaj¹cych warunek maksymalnej entropii dokonywana na DORC.\ \ \ [**Przyk³ad dualnego uk³adu wspó³rzêdnych. Rozk³ad normalny**]{}: Z Rozdzia³u \[alfa koneksja\], wzór (\[rozklad normalny parametry kanoniczne\]), wiemy, ¿e rozk³ad normalny jest typem modelu eksponentialnego z $C({\bf y}) = 0$. Oznacza to, ¿e mo¿e siê on pojawiæ jako rezultat estymacji spe³niaj¹cej za³o¿enia TME. Korzystaj¹c z postaci rozk³adu normalnego oraz z (\[wartosi oczekiwane eta model eksponentialny\]) i (\[rozklad normalny parametry kanoniczne\]) otrzymujemy dualne parametry tego modelu: $$\begin{aligned} \label{parametry dualne modelu normalnego} \theta_{1} &\equiv& \theta_{1}(\xi^{1},\xi^{2}) = \int_{\cal Y} d{\bf y} p({\bf y}|\Xi) {\bf y} = - \frac{\xi^{1}}{2 \xi^{2}} = \mu \; , \nonumber \\ \theta_{2} &\equiv& \theta_{2}(\xi^{1},\xi^{2}) = \int_{\cal Y} d{\bf y} p({\bf y}|\Xi) {\bf y}^{2} = \frac{(\xi^{1})^{2} - 2 \xi^{2} }{4 (\xi^{2})^{2}} = \mu^{2} + \sigma^{2} \; .\end{aligned}$$ Ponadto estymatory $\hat{\theta}_{1} = F_{1}(Y) = Y$ oraz $\hat{\theta}_{1} = F_{1}(Y) = Y^{2}$ s¹ niezale¿ne. SprawdŸmy, ¿e zachodzi warunek konieczny ich niezale¿noœci, a mianowicie brak korelacji: $$\begin{aligned} \label{niezaleznosc estymatorow F modelu normalnego} & & E_{\Xi}\left[ (\hat{\theta}_{1} - \theta_{1}) \, (\hat{\theta}_{2} - \theta_{2}) \right] = \int_{\cal Y} d{\bf y} \, p({\bf y}|\Xi) \, (F_{1}({\bf y}) - \theta_{1}) \, (F_{2}({\bf y}) - \theta_{2}) \nonumber \\ & & = \frac{1}{\sqrt{2 \pi \, \sigma^2}} \int_{\cal Y} d{\bf y} \, e^{\frac{({\bf y} - \mu)^{2}}{2 \, \sigma^2}} \left( {\bf y} - \mu \right) \, [ {\bf y}^{2} - (\mu^{2} + \sigma^{2}) ] \\ & & = \frac{1}{\sqrt{2 \pi \, \sigma^2}} \left( \int_{\cal Y} d{\bf y} \, e^{\frac{({\bf y} - \mu)^{2}}{2 \, \sigma^2}}\; ({\bf y} - \mu) \, [ {\bf y}^{2} - \mu^{2} ] - \sigma^{2} \int_{\cal Y} d{\bf y} \, e^{\frac{({\bf y} - \mu)^{2}}{2 \, \sigma^2}} \; ({\bf y} - \mu) \right) = 0 \nonumber \; ,\end{aligned}$$ gdzie w drugiej linii skorzystano z postaci (\[rozklad norm theta sigma2\]) rozk³adu normalnego, a w ostatniej z zerowania siê obu ca³ek z osobna.\ \ [**Wniosek**]{}: Z powy¿szych ogólnych rozwa¿añ wnioskujemy wiêc, ¿e estymacja z wieloparametrowym rozk³adem normalnym spe³nia DORC. Poprzednio w Rozdziale \[iF oraz I\_definicje\], w wyniku bezpoœredniego rachunku dla jednoparametrowej estymacji wartoœci oczekiwanej $\mu$, otrzymaliœmy w (\[RC dla 1 N z 1 par oczekiwana IF\]) ten sam wynik.\ Jednak¿e dla rozk³adu normalnego, w którym chcielibyœmy dokonaæ jednoczesnej estymacji parametrów $\mu$ oraz $\sigma^{2}$, pojawi³by siê problem z zastosowaniem TRC wynikaj¹cy z faktu, ¿e dla skoñczonego wymiaru próby $N$ estymator $\hat{\sigma^{2}}$ jest obci¹¿ony. Obecnie wiemy, ¿e parametrami oczekiwanymi, którymi nale¿y siê pos³u¿yæ aby zastosowaæ TR i przekonaæ siê, ¿e model normalny spe³nia DORC s¹ $\mu$ oraz suma $\mu^{2} + \sigma^{2}$ (zamiast $\sigma^{2}$). Chocia¿ powy¿szy rachunek zosta³ przeprowadzony dla $N=1$, jednak wniosek dla dowolnego $N$ nie ulega zmianie.\ \ [**Przyk³ad dualnego uk³adu wspó³rzêdnych. Rozk³ad standardowy eksponentialny**]{}: TME ma swoj¹ reprezentacjê w fizyce statystycznej. Otó¿ stan w równowadze termicznej, który maksymalizuje termodynamiczn¹ entropiê Boltzmanna $S_{B}(p) := k_{B} \,S_{H}(p)$, gdzie $k_{B}$ jest dodatni¹ sta³¹ Boltzmanna, posiada przy warunku $E_{\xi}[E] = \bar{\epsilon}$ na³o¿onym na wartoœæ oczekiwan¹ (zmiennej losowej) energii $E$ cz¹stki gazu, rozk³ad Boltzmanna: $$\begin{aligned} \label{rozklad Boltzmanna z max entropii} p(\epsilon|\xi) = \frac{1}{Z} \, e^{- \frac{2 \, \epsilon}{3 \,k_{B} T}} \; ,\end{aligned}$$ gdzie $\epsilon$ jest realizacj¹ $E$, a $T$ jest (entropijn¹) temperatur¹ (por. Dodatek \[Wyprowadzenie drugiej zasady termodynamiki\]). Rozk³ad (\[rozklad Boltzmanna z max entropii\]) jest standardowym rozk³adem eksponentialnym z $C(\epsilon) = 0$, (\[rozklad eksponentialny parametry kanoniczne\]), dla wymiaru próby $N=1$. Zgodnie z oznaczeniami wprowadzonymi w (\[rozklad eksponentialny parametry kanoniczne\]), jeden parametr kanoniczny $\xi$, jedna funkcja $\hat{\theta}_{\epsilon} = F(E)$ bêd¹ca estymatorem parametru oczekiwanego $\theta_{\epsilon}$ oraz potencja³ $\psi(\xi)$ uk³adu wspó³rzêdnych, maj¹ postaæ: $$\begin{aligned} \label{rozklad eksponentialny parametry kanoniczne 2} F(E)= E \; , \;\; \xi = - \frac{2}{3 k_{B} T} \; , \;\; \theta_{\epsilon} = \bar{\epsilon} \; , \;\; \psi(\xi) = \ln ( -\frac{1}{\xi}) = \ln Z \; . \end{aligned}$$\ Do rozk³adu Boltzmanna powrócimy w Rozdziale \[rozdz.energia\], gdzie wyprowadzimy go odwo³uj¹c siê do wspomnianych w Rozdziale \[Geometryczne sformulowanie teorii estymacji\] zasad informacyjnych. [**Kolejne przyk³ady zastosowania estymacji eksponentialnej z baz¹ dualn¹ $\Theta$**]{}: Estymacja tego typu znajduje swoje zastosowanie wtedy, gdy mikrostan uk³adu jest po krótkim okresie czasu zast¹piony makrostanem, co oznacza estymacjê stanu uk³adu poprzez pewien oszacowuj¹cy stan otrzymany metod¹ maksymalnej entropii na rozmaitoœci modelu eksponentialnego (np. dla wolno zmieniaj¹cych siê) zmiennych makroskopowych [@Streater]. Przyk³adami realizacji tej procedury estymacyjnej s¹:\ \ [**- Metoda analizy nieliniowej dynamiki**]{} Kossakowskiego i Ingardena, którzy zrealizowali powy¿sz¹ procedurê, dokonuj¹c ci¹g³ego rzutowania mikrostanu uk³adu na ³atwiejsze w opisie makrostany uk³adu, le¿¹ce na rozmaitoœci stanów eksponentialnych. Z zaproponowanej analizy statystycznej wynik³a mo¿liwoœæ realizacji nieliniowej dynamiki uk³adu, opisanej jako konsekwencja optymalnej [*estymacji*]{} [**stanu uk³adu**]{}, pojawiaj¹cego siê po up³ywie ka¿dego kolejnego odstêpu czasu (w którym dynamika uk³adu przebiega³a w sposób liniowy), [**stanem**]{} [*le¿¹cym na rozmaitoœci eksponentialnej*]{} [@Streater].\ \ [**- Model Onsagera**]{} realizuj¹cy tego typu estymacjê w badaniu zjawiska przep³ywów energii lub masy, w sytuacji, gdy s¹ one liniowymi funkcjami bodŸców (pe³ni¹cych rolê parametrów) wywo³uj¹cych taki przep³yw. Teoria Onsagera ma zastosowanie do zjawisk maj¹cych charakter [*procesów quasistatystycznych*]{}. Zatem stosuje siê ona do sytuacji, gdy materia³, w którym zachodzi zjawisko jest w lokalnej równowadze, tzn. zwi¹zki zachodz¹ce lokalnie i w tej samej chwili czasu pomiêdzy w³asnoœciami cieplnymi i mechanicznymi materia³u s¹ takie same, jak dla jednorodnego uk³adu znajduj¹cego siê w równowadze termodynamicznej. W ramach jego teorii sformu³owano zasady wariacyjne dla opisu liniowej termodynamiki procesów nieodwracalnych.\ \ [**Podsumowanie**]{}: Z powy¿szej analizy wynika, ¿e metoda maksymalnej entropii dla nieobci¹¿onych estymatorów $\hat{\theta}_{i} = F_{i}(Y)$, $i=1,2,...d$, wykorzystuje $\alpha = (-1)$ – p³ask¹ bazê $\Theta$ dualn¹ do $\alpha = (+1)$ – p³askiej bazy kanonicznej $\Xi$ rozk³adu eksponentialnego z $C({\bf y}) = 0$. Ze wzglêdu na p³askoœæ modelu [*eksponentialnego*]{} w bazie kanonicznej $\Xi$, (\[Gamma 1 dla exponential family\]), macierz informacyjna $I_{F}$ w bazie dualnej $\Theta$ jest odwrotna do $I_{F}$ w bazie $\Xi$, sk¹d w (\[rownosc odwrot Fisher Eta inf i m kowariancji maks entrop\]) przekonaliœmy siê, ¿e estymacja w bazie koneksji affinicznej $\nabla^{(-1)}$ przebiega na DORC. Oznacza to, ¿e równie¿ dla wektorowego parametru oczekiwanego $\Theta$ jego estymacja jest [*efektywna*]{} w klasie entropijnych modeli eksponentialnych.\ Jednak id¹c dalej, dok³adniejsza ni¿ to wynika z TRC, [*modelowa estymacja*]{}, tzn. zwi¹zana z konstrukcj¹ nieobci¹¿onych estymatorów parametrów, nie jest mo¿liwa.\ \ [**O tym co w kolejnej czêœci skryptu**]{}: W kolejnej czêœci skryptu zajmiemy siê estymacj¹ zwi¹zan¹ z zasadami informacyjnymi na³o¿onymi na tzw. [*fizyczn¹ informacjê uk³adu*]{}. Zasady informacyjne {#Zasady informacyjne} =================== Estymacja w statystyce klasycznej a estymacja fizyczna. Postawienie problemu {#physical estim} ---------------------------------------------------------------------------- W dotychczasowej analizie przedstawiona zosta³a MNW w statystyce. Polega ona na estymacji parametrów pewnego zadanego rozk³adu. Na przyk³ad w analizie regresji na podstawie pewnej wczeœniejszej wiedzy na temat zachowania siê zmiennej objaœnianej, zakresu wartoœci jakie mo¿e przyjmowaæ oraz jej charakteru (ci¹g³a czy dyskretna) postulujemy warunkowy rozk³ad i model regresji, a nastêpnie konstruujemy funkcjê wiarygodnoœci, któr¹ maksymalizuj¹c otrzymujemy estymatory parametrów strukturalnych modelu. Opracowanie skutecznego algorytmu znajdowania estymatorów MNW oraz ich odchyleñ standardowych jest centralnym problemem np. w rutynowych aplikacjach s³u¿¹cych do analizy uogólnionych regresyjnych modeli liniowych. W analizie tej najwa¿niejszym wykorzystywanym algorytmem jest ogólny algorytm metody iteracyjnie wa¿onych najmniejszych kwadratów, a jedn¹ z jego g³ównych analitycznych procedur jest procedura Newton-Raphson’a [@Pawitan; @Mroz].\ Niech parametr wektorowy $\Theta \equiv (\theta_{n})_{n=1}^{N}$ jest zbiorem wartoœci oczekiwanych zmiennej losowej po³o¿enia uk³adu w $N$ pomiarach, jak to przyjêliœmy w Rozdziale \[Pojecie kanalu informacyjnego\]. Przypomnijmy wiêc, ¿e MNW jest wtedy skoncentrowana na uk³adzie $N$ równañ wiarygodnoœci (\[rown wiaryg\]): $$\begin{aligned} S(\Theta)\mid_{\Theta=\hat{\Theta}}\;\equiv\frac{\partial}{\partial\Theta} \ln P(\Theta)\mid_{\Theta=\hat{\Theta}}=0\;, \nonumber\end{aligned}$$ których rozwi¹zanie daje $N$ elementowy zbiór $\hat{\Theta}\equiv(\hat{\theta}_{n})_{n=1}^{N}$ estymatorów parametrów. Tzn. uk³ad równañ wiarygodnoœci tworzy $N$ warunków na estymatory parametrów, które maksymalizuj¹ wiarygodnoœæ próbki.\ Estymacja w fizyce musi siê rozpocz¹æ na wczeœniejszym etapie. Wychodz¹c od zasad informacyjnych, którym poœwiêcony bêdzie kolejny rozdzia³, estymujemy odpowiednie dla opisywanego zagadnienia fizycznego równania ruchu, których rozwi¹zanie daje odpowiedni rozk³ad wraz z parametrami. Tak wiêc zastosowanie zasad informacyjnych na³o¿onych na funkcjê wiarygodnoœci zamiast MNW stanowi o podstawowej ró¿nicy pomiêdzy analiz¹ statystyczn¹ wykorzystywan¹ w konstrukcji modeli fizycznych, a statystyk¹ klasyczn¹. Oczywiœcie oznacza to, ¿e informacja Fishera zdefiniowana poprzednio na przestrzeni statystycznej ${\cal S}$ musi zostaæ zwi¹zana z bazow¹ przestrzeni¹ ${\cal Y}$ przestrzeni próby, tak aby mo¿na j¹ wykorzystaæ do konstrukcji równañ ruchu. ### Strukturalna zasada informacyjna. Metoda EFI {#structural principle} Poni¿sze rozwa¿ania prezentuj¹ analizê, le¿¹c¹ u podstaw strukturalnej zasady informacyjnej [@Dziekuje; @informacja_2]. Ta zaœ le¿y u podstaw metody estymacji statystycznej EFI zaproponowanej przez Friedena i Soffera [@Frieden].\ Niech $V_{\Theta}$ jest przestrzeni¹ parametru $\Theta$, tzn. $\Theta \in V_{\Theta}$. Wtedy logarytm funkcji wiarygodnoœci $\ln P: V_{\Theta} \rightarrow \text{R}$ jest funkcj¹ okreœlon¹ na przestrzeni $V_{\Theta}$ o wartoœciach w zbiorze liczb rzeczywistych $\text{R}$. Niech $\tilde{\Theta} \equiv (\tilde{\theta}_{n})_{n=1}^{N} \in V_{\Theta}$ jest inn¹ wartoœci¹ parametru lub wartoœci¹ estymatora $\hat{\Theta}$ parametru $\Theta$. Rozwiñmy w punkcie $\tilde{\Theta}$ funkcjê $\ln P(\tilde{\Theta})$ w szereg Taylora wokó³ prawdziwej wartoœci $\Theta$: $$\begin{aligned} \label{rozwiniecie w szereg Taylora} \ln\frac{P(\tilde{\Theta})}{P(\Theta)} = \sum_{n=1}^{N}\frac{\partial \ln P(\Theta)}{\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n}) + \frac{1}{2} \!\!\sum_{n,n'=1}^{N} \!\frac{\partial^{2} \ln P(\Theta)}{\partial\theta_{n'}\partial\theta_{n}} \, (\tilde{\theta}_{n}- \theta_{n})(\tilde{\theta}_{n'}-\theta_{n'}) + R_{3} \; ,\end{aligned}$$ gdzie u¿yto oznaczenia $\frac{\partial P(\Theta)}{\partial\theta_{n}} \equiv \frac{\partial P(\widetilde{\Theta})}{\partial \tilde{\theta}_{n}}\mid_{\widetilde{\Theta} = \Theta}$, oraz podobnie dla wy¿szych rzêdów rozwiniêcia, a $R_{3}$ jest reszt¹ rozwiniêcia trzeciego rzêdu.\ \ [**Znaczenie zasady obserwowanej**]{}: Wszystkie cz³ony w (\[rozwiniecie w szereg Taylora\]) s¹ statystykami na przestrzeni próby ${\cal B}$, wiêc tak jak i uk³ad równañ wiarygodnoœci, równanie (\[rozwiniecie w szereg Taylora\]) jest okreœlone na poziomie obserwowanym. Jest ono ¿¹daniem analitycznoœci (logarytmu) funkcji wiarygodnoœci na przestrzeni statystycznej ${\cal S}$, co stanowi punkt wyjœcia dla konstrukcji obserwowanej, ró¿niczkowej, strukturalnej zasady informacyjnej (\[micro form of information eq\]), s³usznej niezale¿nie od wyprowadzonej w (\[expected form of information eq\]) postaci ca³kowej. Postaæ obserwowana wraz z zasad¹ wariacyjn¹ (\[var K\]) jest, obok postaci oczekiwanej (\[expected form of information eq\]), podstaw¹ estymacji EFI równañ ruchu teorii pola, lub równañ generuj¹cych rozk³ad.\ \ Zdefiniujmy obserwowan¹ strukturê uk³adu $\texttt{t\!F}$ w nastêpuj¹cy sposób: $$\begin{aligned} \label{structure T} \texttt{t\!F} \equiv \ln \frac{P(\tilde{\Theta})}{P(\Theta)} - R_{3} \; .\end{aligned}$$ Na poziomie obserwowanym (nazywanym czasami mikroskopowym) mo¿emy rozwiniêcie Taylora (\[rozwiniecie w szereg Taylora\]) zapisaæ nastêpuj¹co: $$\begin{aligned} \label{micro structure eq} \Delta_{LHS} \equiv \sum_{n=1}^{N}2\, \frac{\partial \ln P}{\partial\theta_{n}}(\tilde{\theta}_{n} - \theta_{n}) - \sum_{n=1}^{N}2\, \frac{\texttt{t\!F}}{N} \, = \!\! \sum_{n,n'=1}^{N} \texttt{i\!F}_{nn'} \,(\tilde{\theta}_{n} - \theta_{n})(\tilde{\theta}_{n'}-\theta_{n'}) \equiv \Delta_{RHS} \; ,\end{aligned}$$ gdzie $\texttt{i\!F}$ jest znan¹ ju¿ z (\[I jako krzywizna dla P\])-(\[I obserwowana\]) obserwowan¹ macierz¹ informacyjn¹ Fishera: $$\begin{aligned} \label{observed IF} \texttt{i\!F} \equiv \left(-\frac{\partial^{2} \ln P(\Theta)}{\partial \theta_{n'} \partial\theta_{n}}\right) = \left(-\frac{\partial^{2} \ln P(\tilde{\Theta})}{\partial \tilde{\theta}_{n'} \partial \tilde{\theta}_{n}}\right)_{|_{\widetilde{\Theta} = \Theta}} \, ,\end{aligned}$$ która jako macierz odwrotna do macierzy kowariancji, jest symetryczna i dodatnio okreœlona[^47] (Rozdzia³ \[E i var funkcji wynikowej\]). Oznacza to, ¿e istnieje ortogonalna macierz $U$ taka, ¿e $\Delta_{RHS}$ wystêpuj¹ce w (\[micro structure eq\]), a zatem równie¿ $\Delta_{LHS}$, mo¿e byæ zapisane w tzw. postaci normalnej [@kompendium; @matematyki]: $$\begin{aligned} \label{normal form} \Delta_{LHS} = \sum_{n=1}^{N}m_{n}\tilde{\upsilon}_{n}^{2} = \sum_{n,n'=1}^{N} \texttt{i\!F}_{nn'} \,(\tilde{\theta}_{n} - \theta_{n})(\tilde{\theta}_{n'} - \theta_{n'}) \equiv \Delta_{RHS} \, ,\end{aligned}$$ gdzie $\tilde{\upsilon}_{n}$ s¹ pewnymi funkcjami $\tilde{\theta}_{n}$, a $m_{n}$ s¹ elementami dodatnio okreœlonej macierzy $\texttt{m\!F}$ (otrzymanymi dla $\Delta_{LHS}$), która z powodu równoœci (\[normal form\]) musi byæ równa macierzy diagonalnej otrzymanej dla $\Delta_{RHS}$, tzn.: $$\begin{aligned} \label{form of M} \texttt{m\!F} = D^{T} U^{T}\, \texttt{i\!F} \, U \, D \;\; . \end{aligned}$$ Macierz $D$ jest diagonaln¹ macierz¹ skaluj¹c¹ o elementach $d_{n}\equiv\sqrt{\frac{m_{n}}{\lambda_{n}}}$, gdzie $\lambda_{n}$ s¹ wartoœciami w³asnymi macierzy $\texttt{i\!F}$.\ \ Zwi¹zek (\[form of M\]) mo¿na zapisaæ w postaci wa¿nego strukturalnego równania macierzowego bêd¹cego bezpoœredni¹ konsekwencj¹ analitycznoœci logarytmu funkcji wiarygodnoœci na przestrzeni statystycznej ${\cal S}$ oraz postaci normalnej formy kwadratowej (\[normal form\]): $$\begin{aligned} \label{micro form of information eq macierzowe} \texttt{q\!F} + \texttt{i\!F} = 0 \; , \end{aligned}$$ gdzie $$\begin{aligned} \label{micro form of qF} \texttt{q\!F} = -U \, (D^{T})^{-1} \, \texttt{m\!F} \, D^{-1} \, U^{T} \, ,\end{aligned}$$ nazwijmy [*obserwowan¹ macierz¹ struktury*]{}.\ \ [**Dwa proste przypadki $\texttt{q\!F}$**]{}: Istniej¹ dwa szczególne przypadki, które prowadz¹ do prostych realizacji fizycznych.\ Pierwszy z nich zwi¹zany jest z za³o¿eniem, ¿e rozk³ad jest [*reguralny*]{} [@Pawitan]. Wtedy, zak³adaj¹c dodatkowo, ¿e dla wszystkich $n=1,...,N$ zachodzi $\frac{\partial lnP}{\partial\theta_{n}}=0$, z równania (\[micro structure eq\]) widzimy, ¿e: $$\begin{aligned} \label{M for logL zero} \texttt{m\!F} = (2\,\delta_{nn'})\;,\;\;\;\tilde{\upsilon}_{n} = \sqrt{\frac{\texttt{t\!F}}{N}} \; \;\;\; {\rm oraz} \;\;\; d_{n}=\sqrt{2/\lambda_{n}} \; .\end{aligned}$$ Natomiast drugi przypadek zwi¹zany jest z za³o¿eniem, ¿e $\texttt{t\!F}=0$ i wtedy z (\[micro structure eq\]) otrzymuje siê postaæ “równania master” (porównaj dalej (\[L master oczekiwana\])). W przypadku tym: $$\begin{aligned} \label{M for T zero} \texttt{m\!F} = diag\left(2\,\frac{\partial \ln P}{\partial\theta_{n}}\right) \, ,\;\;\tilde{\upsilon}_{n} = \sqrt{\tilde{\theta}_{n} - \theta_{n}} \, , \;\;\texttt{t\!F} = 0 \; \;\;\; {\rm oraz} \;\;\; d_{n}=\sqrt{2\,\frac{\partial lnP}{\partial\theta_{n}}/\lambda_{n}} \; ,\end{aligned}$$ co oznacza, ¿e nie istnieje z³o¿ona struktura uk³adu.\ \ [**Obserwowana strukturalna zasada informacyjna**]{}: Sumuj¹c wszystkie elementy zarówno obserwowanej macierzy informacyjnej Fishera $\texttt{i\!F}$ jak i obserwowanej macierzy struktury $\texttt{q\!F}$, równanie macierzowe (\[micro form of information eq macierzowe\]) prowadzi do [*obserwowanej strukturalnej zasady informacyjnej*]{} Frieden’a: $$\begin{aligned} \label{micro form of information eq} \sum_{n,n'=1}^{N}(\texttt{i\!F})_{nn'} + \sum_{n,n'=1}^{N}(\texttt{q\!F})_{nn'} = 0 \; .\end{aligned}$$\ [**Znaczenie analitycznoœci $P$ oraz postaci $\texttt{i\!F}$**]{} dla EFI: Analitycznoœæ funkcji wiarygodnoœci na przestrzeni statystycznej ${\cal S}$, wyra¿ona istnieniem rozwiniêcia w szereg Taylora (\[rozwiniecie w szereg Taylora\]) oraz symetrycznoœæ i dodatnia okreœlonoœæ obserwowanej macierzy informacyjnej Fishera (\[observed IF\]) s¹ zasadniczymi warunkami, które czyni¹ analizê Friedena-Soffera w ogóle mo¿liw¹. Okazuje siê jednak, ¿e w ogólnoœci, dla otrzymania równañ EFI nale¿y odwo³aæ siê dodatkowo do wprowadzonej poni¿ej ca³kowej zasady strukturalnej.\ \ [**Ca³kowa strukturalna zasada informacyjna**]{}: Ca³kuj¹c obie strony równania (\[micro form of information eq\]) po ca³ej przestrzeni próby ${\cal B}$ (lub na jej podprzestrzeni) z miar¹ $d y\, P(\Theta)$, gdzie jak zwykle stosujemy oznaczenie $dy \equiv d^{N}{\bf y}$, otrzymujemy ca³kow¹ postaæ [*informacyjnej zasady strukturalnej*]{}: $$\begin{aligned} \label{expected form of information eq} Q + I = 0 \; ,\end{aligned}$$ gdzie $I$ jest uogólnieniem pojemnoœci informacyjnej Fishera (\[pojemnosc C\]) (por. (\[IF 2 poch na kwadrat pierwszej\])): $$\begin{aligned} \label{iF and I} I = \int_{\cal B} d y\, P(\Theta) \; \sum_{n,n'=1}^{N}(\texttt{i\!F})_{nn'} \; , \end{aligned}$$ natomiast $Q$ jest informacj¹ strukturaln¹ ($SI$): $$\begin{aligned} \label{qF and Q} Q = \int_{\cal B} d y\, P(\Theta) \; \sum_{n,n'=1}^{N}(\texttt{q\!F})_{nn'} \; .\end{aligned}$$ Pierwotnie, w innej, informatycznej formie i interpretacji, zasada (\[expected form of information eq\]) zosta³a zapostulowana w [@Frieden]. Powy¿sza, fizyczna postaæ zasady strukturalnej (\[expected form of information eq\]) zosta³a zapostulowana w [@Dziekuje; @informacja_1], a nastêpnie wyprowadzona, jak to przedstawiono powy¿ej w [@Dziekuje; @informacja_2].\ \ Obserwowana zasada informacyjna (\[micro form of information eq\]) jest równaniem strukturalnym wspó³czesnych modeli fizycznych wyprowadzanych metod¹ EFI. Natomiast u¿ytecznoœæ oczekiwanej strukturalnej zasady informacyjnej (\[expected form of information eq\]) oka¿e siê byæ jasna przy, po pierwsze okreœleniu zmodyfikowanej obserwowanej zasady strukturalnej, po drugie, przy definicji ca³kowitej fizycznej informacji (\[physical K\]) oraz po trzecie, przy sformu³owaniu informacyjnej zasady wariacyjnej (\[var K\]). Oczekiwana zasada strukturalna jako taka, tzn. w postaci ca³kowej (\[expected form of information eq\]), nie jest rozwi¹zywana jednoczeœnie z zasad¹ wariacyjn¹, co jest czasami jej przypisywane.\ #### Ca³ka rozwiniêcia Taylora {#postac calkowa Taylora} Sca³kujmy (\[rozwiniecie w szereg Taylora\]) na na ca³ej przestrzeni próby ${\cal B}$ (lub na jej podprzestrzeni) z miar¹ $d y\, P(\Theta)$. W wyniku otrzymujemy pewn¹ ca³kow¹ formê strukturalnego równania estymacji modeli: $$\begin{aligned} \label{Freiden like equation} & & \!\!\!\int_{\cal B}\!\! d y P(\Theta) \left(\ln\frac{P(\tilde{\Theta})}{P(\Theta)}- R_{3} - \sum_{n=1}^{N}\frac{\partial \ln P(\Theta)}{\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n}) \right) \nonumber \\ & & \!\! = \frac{1}{2}\!\int_{\cal B}\!\! d y\, P(\Theta) \!\!\sum_{n,n'=1}^{N} \!\frac{\partial^{2} \ln P(\Theta)}{\partial\theta_{n'}\partial\theta_{n}} \, (\tilde{\theta}_{n}- \theta_{n})(\tilde{\theta}_{n'}-\theta_{n'}) \; .\end{aligned}$$ Wyra¿enie po lewej stronie (\[Freiden like equation\]) ma postaæ zmodyfikowanej entropii wzglêdnej.\ Nastêpnie, definuj¹c $\widetilde{{\cal Q}}$ jako: $$\begin{aligned} \label{structure Q} \widetilde{{\cal Q}} = \int_{\cal B}\, d y\, P(\Theta)\,\left(\texttt{t\!F} - \sum_{n=1}^{N}\frac{\partial \ln P}{\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n})\right) \, \end{aligned}$$ otrzymujemy równanie bêd¹ce ca³kow¹ form¹ strukturalnej zasady informacyjnej: $$\begin{aligned} \label{structure eq} - \widetilde{{\cal Q}} = \widetilde{I} \equiv \! \frac{1}{2} \! \int_{\cal B}\!\! d y \, P(\Theta) \!\! \sum_{n,n'=1}^{N} \! \left( - \frac{\partial^{2} \ln P}{\partial\theta_{n'} \partial\theta_{n}} \right) (\tilde{\theta}_{n}-\theta_{n})(\tilde{\theta}_{n'}- \theta_{n'}) \; . \end{aligned}$$\ [**Uwaga**]{}: Równanie (\[structure eq\]) jest wtórne wobec bardziej fundamentalnego równania strukturalnego (\[micro form of information eq\]) s³usznego na poziomie obserwowanym, tzn. pod ca³k¹. Chocia¿ równanie (\[structure eq\]) nie jest bezpoœrednio wykorzystywane w metodzie EFI, to jest ono stosowane do badania w³asnoœci nieobci¹¿onych estymatorów $\tilde{\Theta}$ parametrów $\Theta$ [@Murray_differential; @geometry; @and; @statistics]. Zagadnienie to wykracza poza zakres skryptu. #### $I$ oraz $Q$ dla parami niezale¿nych zmiennych po³o¿eniowych próby {#zmienne Yn niezalezne} Rozwa¿my jeszcze postaæ $SI$ wyra¿on¹ w amplitudach w szczególnym przypadku zmiennych $Y_{n}$ parami niezale¿nych. W takim przypadku amplituda $q_{n}$ nie zale¿y od ${\bf y}_{n}$ dla $n' \neq n$, czyli ma postaæ $q_{n}({\bf y}_{n})$, natomiast $(\texttt{i\!F})$ jest diagonalna, tzn. ma postaæ: $$\begin{aligned} \label{iF diagonalne} (\texttt{i\!F})_{nn'} = \delta_{nn'} \texttt{i\!F}_{nn} \equiv \texttt{i\!F}_{n} \; ,\end{aligned}$$ gdzie $\delta_{nn'}$ jest delt¹ Kroneckera. W takim razie, zgodnie z (\[form of M\]) oraz (\[micro form of qF\]) obserwowana macierz strukturalna jest diagonalna i jej ogólna postaæ jest nastêpuj¹ca: $$\begin{aligned} \label{qF diagonalne} (\texttt{q\!F})_{nn'} = \delta_{nn'} \; \texttt{q\!F}_{nn}\left( q_{n}({\bf y}_{n}), q_{n}^{(r)}({\bf y}_{n}) \right) \equiv \texttt{q\!F}_{n}\left( q_{n}({\bf y}_{n}) \right)\; ,\end{aligned}$$ tzn. nie zale¿y od amplitud $q_{n'}({\bf y}_{n'})$ i jej pochodnych dla $n' \neq n$. Powy¿ej $q_{n}^{(r)}({\bf y}_{n})$ oznaczaj¹ pochodne rzêdu $r=1,2,... \,$. Zobaczymy, ¿e dla teorii pola w $\texttt{q\!F}_{n}$ pojawi¹ siê pochodne co najwy¿ej pierwszego rzêdu. Fakt ten wynika st¹d, ¿e swobodne pola rangi $N$, z którymi bêdziemy mieli do czynienia, bêd¹ spe³nia³y równanie Kleina-Gordona.\ \ [**Uwaga**]{}: Oznaczenie $\texttt{q\!F}_{n}$, jak równie¿ jawne zaznaczenie w argumencie obserwowanej $SI$ tylko amplitudy $q_{n}({\bf y}_{n})$, bêd¹ stosowane w dalszej czêœci skryptu.\ \ Wykorzystuj¹c (\[iF diagonalne\]) oraz (\[qF diagonalne\]), pojemnoœæ informacyjna (\[iF and I\]) przyjmuje w rozwa¿anym przypadku postaæ: $$\begin{aligned} \label{I dla niezaleznych Yn} I = \int_{\cal B} d y\, \textit{i} = \int_{\cal B} d y\, P(\Theta) \; \sum_{n=1}^{N} \texttt{i\!F}_{n} \; , \end{aligned}$$ natomiast informacja strukturalna (\[qF and Q\]) jest nastêpuj¹ca: $$\begin{aligned} \label{Q dla niezaleznych Yn} Q = \int_{\cal B} d y\, \textit{q} = \int_{\cal B} d y\, P(\Theta) \; \sum_{n=1}^{N} \texttt{q\!F}_{n}( q_{n}({\bf y}_{n})) \; .\end{aligned}$$ Powy¿ej $\textit{i}$ jest [*gêstoœci¹ pojemnoœci informacyjnej*]{}: $$\begin{aligned} \label{gestosc i dla niezaleznych Yn} \textit{i} := P(\Theta) \; \sum_{n=1}^{N} \texttt{i\!F}_{n} \; ,\end{aligned}$$ natomiast $\textit{q}$ jest [*gêstoœci¹ informacji strukturalnej*]{}: $$\begin{aligned} \label{gestosc q dla niezaleznych Yn} \textit{q} := P(\Theta) \; \sum_{n=1}^{N} \texttt{q\!F}_{n}( q_{n}({\bf y}_{n})) \; .\end{aligned}$$ [**Obserwowana zasada strukturalna zapisana w gêstoœciach**]{}: Zarówno $\textit{i}$ jak i $\textit{q}$ s¹ okreœlone na poziomie obserwowanym. Zatem korzystaj¹c z (\[iF diagonalne\]) oraz (\[qF diagonalne\]), mo¿emy [*obserwowan¹*]{} informacyjn¹ zasadê strukturaln¹ (\[micro form of information eq\]) zapisaæ w postaci: $$\begin{aligned} \label{obserwowana zas strukt z P} \textit{i} + \textit{q} = 0\; .\end{aligned}$$ Zasada ta, a raczej jej zmodyfikowana wersja, jest obok wariacyjnej zasady informacyjnej, wykorzystywana w celu otrzymania równañ ruchu (b¹dŸ równañ generuj¹cych rozk³ad) metody EFI. Zarówno zmodyfikowana obserwowana zasada strukturalna jak i zasada wariacyjna s¹ okreœlone poni¿ej.\ \ [**Uwaga**]{}: W treœci skryptu gêstoœæ pojemnoœci informacyjnej $\textit{i}$ jest zawsze zwi¹zana z postaci¹ (\[observed IF\]) obserwowanej informacji Fishera $\texttt{i\!F}$.\ \ Na koniec zauwa¿my, ¿e ze wzglêdu na unormowanie rozk³adów brzegowych $\int d^{4}{\bf y}_{n} \, p_{n}({\bf y}_{n}|\theta_{n}) =1$, postaæ $Q$ podan¹ w (\[Q dla niezaleznych Yn\]) mo¿na zapisaæ nastêpuj¹co: $$\begin{aligned} \label{Q dla niezaleznych Yn w d4y} Q = \sum_{n=1}^{N} \int d^{4}{\bf y}_{n} \, p_{n}({\bf y}_{n}|\theta_{n}) \, \texttt{q\!F}_{n}\,( q_{n}({\bf y}_{n})) \; .\end{aligned}$$ Wa¿na kinematyczna postaæ $I$ zostanie wprowadzona w Rozdziale \[The kinematical form of the Fisher information\], natomiast postacie $Q$ bêd¹ pojawia³y siê w toku rozwi¹zywania konkretnych fizycznych problemów. Przep³yw informacji {#information transfer} ------------------- Informacja Fishera $I_{F}$ jest infinitezymalnym typem entropii Kulback-Leibler’a (Rozdzia³ \[Informacja Fishera jako entropia\]) wzór (\[I porownanie z S\]). W statystycznej estymacji KL s³u¿y jako narzêdzie analizy wyboru modelu [@Brockwell_Machura; @Zajac], o czym mo¿emy siê przekonaæ, zauwa¿aj¹c, ¿e jest ona zwi¹zana z wartoœci¹ oczekiwan¹ statystki ilorazu wiarygodnoœci (\[statystyka ilorazu wiaryg\]), wprowadzonej w Rozdziale \[Analiza regresji Poissona\], w³aœnie w celu porównywania wiarygodnoœci modeli. Chocia¿by z tego powodu, pojawia siê przypuszczenie, ¿e pojemnoœæ informacyjna $I$ mog³aby, po na³o¿eniu, jak siê okazuje strukturalnej i wariacyjnej zasady informacyjnej [@Frieden; @Dziekuje; @informacja_1], staæ siê podstaw¹ równañ ruchu (lub równañ generuj¹cych rozk³ad) uk³adu fizycznego. Równania te mia³yby byæ najlepsze z punktu widzenia zasad informacyjnych, co jest sednem metody EFI Friedena-Soffera.\ \ Zgodnie z Rozdzia³em \[Podstawowe zalozenie Friedena-Soffera\], g³ówna statystyczna myœl stoj¹ca za metod¹ EFI jest nastêpuj¹ca: próbkowanie czasoprzestrzeni nastêpuje przez sam uk³ad nawet wtedy, gdy on sam nie jest poddany rzeczywistemu pomiarowi. Sprawê nale¿a³oby rozumieæ tak, ¿e uk³ad dokonuje próbkowania czasoprzestrzeni u¿ywaj¹c charakterystycznego, swojego w³asnego pola (i zwi¹zanej z nim amplitudy) rangi $N$, która jest wymiarem próby, próbkuj¹c swoimi kinematycznymi “Fisherowskimi” stopniami swobody przestrzeñ po³o¿eñ jemu dostêpn¹. Przejœcie od postaci statystycznej pojemnoœci informacyjnej (\[pojemnosc C dla polozenia\]) do jej reprezentacji kinematycznej zostanie omówione poni¿ej w Rozdziale \[The kinematical form of the Fisher information\].\ \ Rozwa¿my nastêpuj¹cy, informacyjny schemat uk³adu. Zanim nast¹pi pomiar, którego dokonuje sam uk³ad, ma on pojemnoœæ informacyjn¹ $I$ zawart¹ w swoich kinematycznych stopniach swobody oraz informacjê strukturaln¹ $Q$ uk³adu zawart¹ w swoich strukturalnych stopniach swobody, jak to przedstawiono symbolicznie na poni¿szym Rysunku.\ \[przeplyw informacji w ukladzie\] {width="47.00000%" height="2cm"} {width="47.00000%" height="2.1cm"} \ “W chwili w³¹czenia” pomiaru, podczas którego transfer informacji ($TI$) przebiega zgodnie z nastêpuj¹cymi zasadami (Rysunek 3.1): $$\begin{aligned} \label{delta Q and I} J \geq 0 \; , \;\;\; {\rm zatem}\;\;\; \delta I=I' - I \geq 0 \; ,\; \; \delta Q = Q' - Q \leq 0 \; ,\end{aligned}$$ gdzie $I'$, $Q'$ s¹ odpowiednio $IF$ oraz $SI$ uk³adu po pomiarze, natomiast $J$ jest dokonanym transferem informacji ($TI$).\ Postulujemy, ¿e w pomiarze $TI$ “w punkcie $\emph{q}$” jest idealny, co oznacza, ¿e: $$\begin{aligned} \label{delta Q and I w idealnym pomiarze} Q = Q' + J = Q + \delta Q + J \, , \;\; {\rm zatem} \;\;\; \delta Q = - J \; . \end{aligned}$$ Oznacza to, ¿e “w punkcie $\emph{q}$” przekazana jest ca³a zmiana $SI$.\ \ Z drugiej strony “w punkcie $\emph{i}$” zasada zwi¹zana z $TI$ jest nastêpuj¹ca: $$\begin{aligned} \label{TI w punkcie i} I' \leq I + J \; \;\; {\rm zatem} \;\;\; 0 \leq \delta I = I' - I \leq J \; .\end{aligned}$$ Dlatego $$\begin{aligned} \label{J > 0 i zwiazek delta dla I oraz Q} {\rm poniewa\dot{z}} \;\;\;\; J \geq 0 \; , \;\;\; {\rm zatem} \;\;\; |\delta I| \leq |\delta Q| \; , \end{aligned}$$ co jest rozs¹dnym resultatem, gdy¿ w pomiarze mo¿e nast¹piæ utrata informacji. Gdyby “w punkcie $\emph{i}$” $TI$ by³ idealny, wtedy ca³y pomiar by³by idealny, tzn.: $$\begin{aligned} \label{caly pomiar idealny} \delta Q = -\delta I \;\; \Leftrightarrow \;\;\; {\rm pomiar \;\; idealny} \; .\end{aligned}$$\ W [@Dziekuje; @informacja_1; @Mroziakiewicz] zosta³o zapostulowane istnienie nieujemnej addytywnej ca³kowitej (totalnej) fizycznej informacji ($TFI$): $$\begin{aligned} \label{physical K} K = I + Q \geq 0 \; .\end{aligned}$$ Wybór intuicyjnego warunku $K \geq 0$ [@Mroziakiewicz] jest zwi¹zany ze [*strukturaln¹ zasad¹ informacyjn¹*]{} zapisan¹ w postaci obserwowanej: $$\begin{aligned} \label{condition from K obserwowana} \sum_{n,n'=1}^{N}(\texttt{i\!F})_{nn'} + \kappa \!\! \sum_{n,n'=1}^{N}(\texttt{q\!F})_{nn'} = 0 \; \end{aligned}$$ lub oczekiwanej: $$\begin{aligned} \label{condition from K} I + \kappa \, Q = 0 \; .\end{aligned}$$ Dla szczególnego przypadku $\kappa = 1$, w Rozdziale \[structural principle\] zosta³a wyprowadzona [@Dziekuje; @informacja_2] postaæ obserwowana zasady strukturalnej (\[micro form of information eq\]): $$\begin{aligned} \label{ideal condition from K obserwowana} \sum_{n,n'=1}^{N}(\texttt{i\!F})_{nn'} + \sum_{n,n'=1}^{N}(\texttt{q\!F})_{nn'} = 0 \;\;\; {\rm dla}\;\;\; \kappa = 1 \; , \end{aligned}$$ oraz jej oczekiwany odpowiednik (\[expected form of information eq\]): $$\begin{aligned} \label{ideal condition from K} I + Q = 0 \;\;\; {\rm dla}\;\;\; \kappa = 1 \; .\end{aligned}$$ Wspó³czynnik $\kappa$ zosta³ nazwany w [@Frieden] wspó³czynnikiem efektywnoœci. W praktyce przyjmuje on dwie mo¿liwe wartoœci [@Frieden]: $$\begin{aligned} \label{wartosc kappa} \kappa = 1 \;\; \vee \;\; \frac{1}{2} \; .\end{aligned}$$ Jego znaczenie zostanie omówione w Rozdziale \[Kryteria informacyjne w teorii pola\]. W przypadku okreœlonym w (\[ideal condition from K\]), otrzymujemy ca³kowit¹ fizyczn¹ informacjê $K$ równ¹: $$\begin{aligned} \label{ideal K} K = I + Q = 0 \;\;\; {\rm dla} \;\;\; \kappa = 1 \; . \end{aligned}$$ W koñcu zauwa¿my, ¿e w zgodzie z zapostulowanym zachowaniem siê uk³adu w pomiarze, otrzymaliœmy z warunków (\[delta Q and I w idealnym pomiarze\]) i (\[TI w punkcie i\]) nierównoœæ $\delta I\leq J = - \delta Q$, z czego wynika, ¿e: $$\begin{aligned} K' = I' + Q' \leq (I + J) + (Q - J) = I + Q = K \; \Rightarrow \;\;\; K' \leq K \; .\end{aligned}$$ Dla pomiaru idealnego (\[caly pomiar idealny\]) otrzymaliœmy $\delta I=-\delta Q$ sk¹d $K'=K$, co oznacza, ¿e informacja fizyczna $TFI$ pozostaje w tym przypadku niezmieniona. Jeœli pomiar idealny by³by wykonany na poziomie próbkowania czasoprzestrzeni przez sam uk³ad, wtedy warunek ten móg³by prowadziæ do wariacyjnej zasady informacyjnej (\[var K\]), tzn.: $$\begin{aligned} \label{zasada wariacyjna} \delta I = - \delta Q \;\; \Rightarrow \;\; \delta(I + Q) = 0 \; .\end{aligned}$$ Chocia¿ rozumowanie powy¿sze wydaje siê byæ rozs¹dne, jednak œciœle mówi¹c s³usznoœæ przyjêcia zasad informacyjnych, strukturalnej oraz wariacyjnej, powinno wynikaæ z dwóch rzeczy. Po pierwsze z ich wyprowadzenia, a po drugie z ich u¿ytecznoœci. Wyprowadzenie zasady strukturalnej (dla $\kappa=1$) zosta³o pokazane w Rozdziale \[structural principle\].\ Natomiast powy¿sze wnioskowanie, które doprowadzi³o do warunku (\[zasada wariacyjna\]) oraz sama implikacja wewn¹trz niego, mo¿e s³u¿yæ jedynie jako przes³anka s³usznoœci zasady wariacyjnej. W Rozdziale \[Geometryczne sformulowanie teorii estymacji\] stwierdziliœmy, ¿e jej s³usznoœæ wynika z ¿¹dania aby rozk³ad empiryczny oraz rozk³ad wyestymowany metod¹ EFI, le¿a³y na wspólnej geodezyjnej w przestrzeni statystycznej ${\cal S}$.\ Co do u¿ytecznoœci zasady wariacyjnej w metodzie EFI, to jest ona oczywista, bowiem prowadzi ona do owocnego w zastosowaniach równania Eulera-Lagrange’a.\ \ [**Analityczny przypadek uk³adu równañ informacyjnych metody EFI**]{}: [*Obserwowana zasada strukturalna*]{} zapisana w gêstoœciach (\[obserwowana zas strukt z P\]), ale uwzglêdniaj¹ca postaæ obserwowanej zasady strukturalnej z $\kappa$ (\[condition from K obserwowana\]), jest nastêpuj¹ca: $$\begin{aligned} \label{obserwowana zas strukt z P i z kappa} \textit{i} + \kappa \, \textit{q} = 0\; .\end{aligned}$$ Drug¹ zasad¹ informacyjn¹ jest [*zasada wariacyjna*]{} (skalarna). Ma ona postaæ [@Dziekuje; @informacja_1]: $$\begin{aligned} \label{var K} \delta K = \delta(I + Q) = 0 \; \, \Rightarrow \; \, K = I + Q \;\;{\rm jest\; ekstremalne} \; .\end{aligned}$$\ [**Warunek geometrycznej struktury na ${\cal S}$**]{}: Kolejnym warunkiem narzuconym na rozwi¹zania metody EFI, a oczywistym od pocz¹tku analizy, jest warunek normalizacji i reguralnoœci rozk³adu prawdopodobieñstwa. Warunek ten oznacza mo¿liwoœæ przejœcia, podanego w (\[IF 2 poch na kwadrat pierwszej\]) i (\[Fisher inf matrix\]), od pierwotnej postaci obserwowanej informacji Fishera (\[observed IF\]) do postaci potrzebnej dla zdefiniowania przestrzeni statystycznej ${\cal S}$ jako przestrzeni metrycznej z metryk¹ Rao-Fishera (\[Fisher inf matrix plus reg condition\]) i $\alpha$-koneksj¹ (\[affine coefficients\]) (por. Rozdzia³ \[alfa koneksja\]). Obie postacie obserwowanej informacji Fishera, pierwsza $\texttt{i\!F}$, (\[observed IF\]), która jest pierwotn¹ form¹ z punktu widzenia analitycznoœci funkcji wiarygodnoœci oraz druga, [*metryczna*]{} $\widetilde{\texttt{i\!F}}$, (\[observed IF Amari\]), która jest istotna dla geometrycznej analizy modelu, s¹ równowa¿ne tylko na poziomie oczekiwanym, tzn. pod ca³k¹ (por. Rozdzia³ \[IF 2 poch na kwadrat pierwszej\]). Powy¿sze rozwa¿ania zostan¹ zilustrowane przyk³adami zawartymi w dalszej czêœci skryptu.\ \ [**Podstawowy uk³ad równañ informacyjnych EFI i zmodyfikowane równanie strukturalne**]{}: Aby wyjaœniæ powy¿szy problem na wstêpnym, symbolicznym poziomie, wprowadŸmy zmodyfikowane równanie strukturalne, uwzglêdniaj¹ce równie¿ wspó³czynnik $\kappa$ wystêpuj¹cy w (\[obserwowana zas strukt z P i z kappa\]). Niech $\widetilde{\texttt{i\!F}}$ jest kwadratow¹ postaci¹ obserwowanej informacji Fishera (\[observed IF Amari\]) tak, ¿e odpowiadaj¹ca jej gêstoœæ pojemnoœci informacyjnej: $$\begin{aligned} \label{gestosc i Amarii} \widetilde{\textit{i}} := P(\Theta) \sum_{n,n'=1}^{N} \widetilde{\texttt{i\!F}}_{n n'} \; ,\end{aligned}$$ daje na poziomie oczekiwanym pojemnoœæ informacyjn¹ $I = \int_{\cal B} \! dy \, \widetilde{\textit{i}}$ $= \int_{\cal B} \! dy \, \textit{i}\,$.\ WprowadŸmy zamiast (\[obserwowana zas strukt z P i z kappa\]) [*zmodyfikowan¹ obserwowan¹ zasadê strukturaln¹*]{} zapisan¹ w nastêpuj¹cy sposób: $$\begin{aligned} \label{zmodyfikowana obserwowana zas strukt z P i z kappa} \widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0 \; , \;\;\; {\rm przy \; czym} \;\;\; I = \int_{\cal B} \! dy \, \widetilde{\textit{i}} = \int_{\cal B} \! dy \, (\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} ) \; , \end{aligned}$$ gdzie $ \widetilde{\mathbf{C}}$ jest pochodn¹ zupe³n¹, która wynika z ca³kowania przez czêœci ca³ki $I = \int_{\cal B} \! dy \, \widetilde{\textit{i}}$. Poniewa¿ $I = \int_{\cal B} \! dy \, \widetilde{\textit{i}}\,$ zatem informacyjna zasada wariacyjna ma postaæ: $$\begin{aligned} \label{var K rozpisana} \delta(I + Q) = \delta \! \int_{\cal B} \! dy \, ( \widetilde{\textit{i}} + \textit{q} ) = 0 \; . \end{aligned}$$ Rozwi¹zanie równañ (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) oraz (\[var K rozpisana\]) jest równowa¿ne rozwi¹zaniu równañ (\[obserwowana zas strukt z P i z kappa\]) oraz (\[var K\]) co najmniej pod ca³k¹, tzn.: $$\begin{aligned} \label{rownowaznosc strukt i zmodyfikowanego strukt} \int_{\cal B} \! dy \, ( \widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} ) = 0 \; \;\; \Leftrightarrow \;\;\; \int_{\cal B} \! dy \, ( \textit{i} + \kappa \, \textit{q} ) = 0 \; . \end{aligned}$$ Powy¿sz¹ symboliczn¹ konstrukcjê (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) zaprezentujemy na przyk³adach w dalszej czêœci skryptu (Rozdzia³ \[Przyklady\]). Jej zrozumienie jest nastêpuj¹ce: Modele “nie do koñca równowa¿ne” pod wzglêdem analitycznym s¹, z dok³adnoœci¹ do wyca³kowania $I$ przez czêœci, równowa¿ne pod wzglêdem metrycznym. To znaczy, istnieje pewien zwi¹zek pomiêdzy ich ró¿niczkowalnoœci¹, a mianowicie wszystkie one s¹ metrycznie (a wiêc na poziomie ca³kowym) równowa¿ne modelowi analitycznemu, tzn. posiadaj¹cemu rozwiniêcie w szereg Taylora.\ \ [**Podsumowanie**]{}. Nale¿y podkreœliæ, ¿e równanie ca³kowe (\[ideal K\]) oraz zasada wariacyjna (\[var K\]) [*nie*]{} tworz¹ pary równañ metody EFI rozwi¹zywanych samospójnie.\ Natomiast obie zasady, obserwowana zmodyfikowana zasada strukturalna (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) oraz zasada wariacyjna (\[var K rozpisana\]) s¹ podstaw¹ metody estymacyjnej EFI. Tworz¹ one uk³ad dwóch równañ ró¿niczkowych dla wprowadzonych w Rozdziale \[geometria i metryka Fishera-Rao\] amplitud uk³adu (\[amplituda a rozklad\]). Uk³ad ten mo¿e byæ zgodny, daj¹c samospójne rozwi¹zanie dla amplitud [@Frieden] i prowadz¹c przy $\kappa=1$ lub $1/2$ do dobrze znanych modeli teorii pola (Rozdzia³ \[Kryteria informacyjne w teorii pola\]) lub modeli fizyki statystycznej (Rozdzia³ \[Przyklady\]). Ponadto, strukturalna (wewnêtrzna) zasada informacyjna (\[condition from K\]) [@Dziekuje; @informacja_1] jest operacyjnie równowa¿na zapostulowanej przez Frieden’a [@Frieden], wiêc jako wyprowadzona powinna mieæ przynajmniej tak¹ sam¹ moc przewidywania jak i ona. Wiele z podstawowych modeli zosta³o ju¿ wyliczonych [@Frieden], jednak ich ponowne przeliczenie [@Mroziakiewicz] przy powy¿ej podanej interpretacji informacji fizycznej $K$ mo¿e daæ lepsze zrozumienie samej metody EFI i jej zwi¹zku z istniej¹cym ju¿ modelowaniem zjawisk w fizyce oraz jej ograniczeñ.\ \ [**Zasada ekwipartycji entropii wzglêdnej**]{}: W koñcu, w strukturalnej zasadzie informacyjnej ciekawe jest równie¿ to, ¿e stanowi ona warunek zerowego podzia³u dla $TFI$, który jest dawno poszukiwanym warunkiem zasady ekwipartycji entropii (w tym przypadku infinitezymalnej entropii wzglêdnej).\ \ [**Uwaga o podejœciu Friedena**]{}: Wspomnieliœmy o tym, ¿e pomys³ metody EFI pochodzi od Friedena. Jednak mówi¹c w skrócie, Frieden i Soffer [@Frieden] podeszli inaczej do informacji strukturalnej. W [@Frieden] wprowadzono tzw. informacjê zwi¹zan¹ J, która ma interpretacjê informacji zawartej w uk³adzie przed pomiarem. Chocia¿, aksjomaty Frieden’a s¹ równowa¿ne powy¿szym warunkom (\[condition from K\]) oraz (\[var K\]), o ile ${\rm J} = -Q$, to jednak¿e ró¿nica pomiêdzy podejœciami jest widoczna. A mianowicie, o ile w podejœciu Friedena-Soffera uk³ad doœwiadcza transferu informacji ${\rm J} \rightarrow I$, maj¹c w ka¿dej chwili czasu tylko jeden z tych typów informacji, o tyle w naszym podejœciu system jest charakteryzowany jednoczeœnie przez $I$ oraz $Q$ w ka¿dej chwili czasu.\ \ [**Uwaga o podobieñstwie EFI i teorii Jaynes’a**]{}: Metoda EFI zaproponowana przez Friedena i Soffera [@Frieden] jest konsekwencj¹ postulatu podobnego do zasady Jaynes’a[^48]. Mianowicie podobieñstwo obu teorii le¿y w tym, ¿e poprzez zasadê wariacyjn¹ wi¹¿¹ one strukturalne (Boltzmann’owskie) stopnie swobody z kinematycznymi (Shannona) stopniami swobody[^49].\ Wed³ug podejœcia Jaynes’a, maksymalizacja entropii Shannona wzglêdem prawdopodobieñstw mikro-stanu uk³adu, posiadaj¹cego znane w³asnoœci, np. ustalon¹ energiê, umo¿liwia identyfikacjê termodynamicznej entropii Boltzmanna jako zmaksymalizowanej entropii Shannona, a nastêpnie na konstrukcjê funkcji stanu, np. energii swobodnej. Kinetyczna postaæ informacji Fishera {#The kinematical form of the Fisher information} ------------------------------------ Centralna czêœæ pracy Frieden’a i Soffer’a zwi¹zana jest z transformacj¹ postaci pojemnoœci informacyjnej $I$ zadanej równaniem (\[pojemnosc C dla polozenia\]) oraz (\[pojemnosc informacyjna Minkowskiego\]) do tzw. postaci kinematycznej wykorzystywanej w teorii pola oraz fizyce statystycznej. W obecnym rozdziale zaprezentujemy podstawowe za³o¿enia, które doprowadzi³y do konstrukcji kinematycznego cz³onu (ca³ki) dzia³ania dla czterowymiarowych modeli teorii pola. Przejœcie to ma nastêpuj¹c¹ postaæ [@Frieden].\ \ Zgodnie z podstawowym za³o¿eniem Friedena-Soffera, $N$-wymiarowa próbka ${\bf y}_{n} \equiv ({\bf y}_{n}^{\nu})$ jest pobierana przez uk³ad posiadaj¹cy rozk³ad $p_{n}({\bf y}_{n})$, gdzie obok indeksu próby $n=1,2,...,N$ wprowadzono indeks (czaso)przestrzenny $\nu = (0),1,2,3$. Zgodnie z Rozdzia³em \[geometria i metryka Fishera-Rao\], wzór (\[amplituda a rozklad\]), metryka Fishera na (pod)rozmaitoœci ${\cal S}$ prowadzi w naturalny sposób do pojêcia rzeczywistej amplitudy $q({\bf y}_{n}|\theta_{n}) \equiv \sqrt{p({\bf y}_{n}|\theta_{n}})$ pola uk³adu. Od razu skorzystano te¿ z zapisu, który sugeruje niezale¿noœæ rozk³adu dla $Y_{n}$ od $\theta_{m}$, gdy $m \neq n$.\ \ Jak przedstawiliœmy w Rozdziale \[Podstawowe zalozenie Friedena-Soffera\] pojemnoœæ informacyjna (\[pojemnosc C dla polozenia - powtorka wzoru\]) mo¿e zostaæ zapisana jako (\[potrz\]): $$\begin{aligned} \label{Fisher_information with q dla przejscia} I \equiv I(\Theta) = 4 \sum\limits_{n=1}^N \int_{\cal Y} {d^{4}{\bf y}_{n} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial q_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n \nu} }}} {\frac{{\partial q_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n}^{ \nu} }}} \right) } } \; .\end{aligned}$$\ \ [**Addytywny rozk³ad po³o¿eñ i regu³a ³añcuchowa**]{}: Niech ${\bf x}_{n} \equiv({\bf x}^{\nu}_{n})$ s¹ przesuniêciami (np. addytywnymi fluktuacjami) danych ${\bf y}_{n} \equiv ({\bf y}_{n}^{\nu})$ od ich wartoœci oczekiwanych $\theta^{\nu}_{n}$, tzn.: $$\begin{aligned} \label{parameters separation} {\bf y}_{n}^{\nu} = \theta^{\nu}_{n} + {\bf x}^{\nu}_{n} \; . $$ Przesuniêcia ${\bf x}^{\nu}_{n}$ s¹ zmiennymi Fisher’owskimi, spe³niaj¹c warunek $\frac{\partial {\bf x}^{\nu}}{\partial {\bf x}^{\mu}} = \delta^{\nu}_{\mu}$, (\[zmienne Fisherowskie\]).\ \ Odwo³uj¹c siê do “regu³y ³añcuchowej” dla pochodnej: $$\begin{aligned} \label{chain rule} \frac{\partial}{\partial {\bf \theta_{n}^{\nu}}} = \frac{\partial ({\bf y_{n}^{\nu}} - \theta_{n}^{\nu})}{\partial \theta_{n}^{\nu}} \, \frac{\partial}{\partial ({\bf y_{n}^{\nu}} - \theta_{n}^{\nu})} = - \; \frac{\partial}{\partial ({\bf y_{n}^{\nu}} - \theta_{n}^{\nu})} = - \; \frac{\partial}{\partial {\bf x_{n}^{\nu}}} \; \end{aligned}$$ oraz uwzglêdniaj¹c $d^{4}{\bf x}_{n}=d^{4}{\bf y}_{n}$, co wynika z tego, ¿e parametry $\theta_{n}$ s¹ sta³ymi, mo¿emy przejœæ od postaci statystycznej (\[Fisher\_information with q dla przejscia\]) do [*postaci kinematycznej $IF$*]{} : $$\begin{aligned} \label{Fisher_information-kinetic form} I = 4 \sum_{n=1}^{N} \int_{{\cal X}_{n}} \!\! d^{4}{\bf x}_{n} \sum_{\nu} \frac{\partial q_{n}({\bf x}_{n})}{\partial {\bf x}_{n \nu}} \frac{\partial q_{n}({\bf x}_{n})}{\partial {\bf x}^{\nu}_{n}} \; ,\end{aligned}$$ gdzie $d^{4}{\bf x}_{n}=d {\bf x}_{n}^{0} d {\bf x}_{n}^{1} d {\bf x}_{n}^{2} d {\bf x}_{n}^{3} $. W (\[Fisher\_information-kinetic form\]) wprowadzono oznaczenie: $$\begin{aligned} \label{zapis dla qn w xn} q_{n}({\bf x}_{n}) \equiv q_{n}({\bf x}_{n}+\theta_{n}|\theta_{n}) = q_{n}({\bf y}_{n}|\theta_{n}) \; ,\end{aligned}$$ [*pozostawiaj¹c ca³¹ informacjê o $\theta_{n}$ w indeksie $n$ amplitudy $q_{n}({\bf x}_{n})$*]{}.\ \ [**Kinematyczna postaæ IF dla $q_{n}$**]{}: Zak³adaj¹c, ¿e zakres zmiennoœci wszystkich ${\bf x}_{n}^{\nu}$ jest dla ka¿dego $n$ taki sam, mo¿emy pomin¹æ indeks $n$ przy tej zmiennej (ale nie przy amplitudzie $q_{n}$), otrzymuj¹c postaæ: $$\begin{aligned} \label{Fisher_information-kinetic form bez n} I = 4 \sum_{n=1}^{N} \int_{\cal X} \!\! d^{4}{\bf x} \sum_{\nu} \frac{\partial q_{n}({\bf x})}{\partial {\bf x}_{\nu}} \frac{\partial q_{n}({\bf x})}{\partial {\bf x}^{\nu}} \; ,\end{aligned}$$ któr¹ wykorzystamy przy wyprowadzeniu równañ generuj¹cych fizyki statystycznej [@Frieden], ale która zosta³a równie¿ wykorzystana do wyprowadzenia elektrodynamiki Maxwella metod¹ EFI [@Frieden].\ \ [**Uwaga**]{}: [*Wymiar próby $N$ jest rang¹ pola uk³adu zdefiniowanego jako zbiór amplitud $\left(q_{n}({\bf x}_{n})\right)_{n=1}^{N}$*]{}.\ \ W Rozdziale \[structural principle\] pokazaliœmy, ¿e strukturalna zasada informacyjna $I + Q =0$ jest artefaktem istnienia rozwiniêcia $\ln P(\tilde{\Theta})$ w szereg Taylora[^50] wokó³ prawdziwej wartoœci parametru $\Theta$. Obecnie znamy ju¿ ogóln¹ postaæ kinematyczn¹ $I$ czêœci pomiarowej zasady strukturalnej. W metodzie EFI, jej czêœæ strukturalna $Q$ ma postaæ zale¿n¹ od np. fizycznych wiêzów na³o¿onych na uk³ad. Zagadnieniem tym zajmiemy siê w kolejnych Rozdzia³ach \[Kryteria informacyjne w teorii pola\] oraz \[Przyklady\].\ \ [**Amplitudy zespolone**]{}: Kolejnym za³o¿eniem jest konstrukcja [*sk³adowych funkcji falowej*]{} sk³adanych z amplitud[^51] w nastêpuj¹cy sposób [@Frieden]: $$\begin{aligned} \label{amplitudapsi dla roznych xn} \psi_{n}({\bf x}_{2n-1},{\bf x}_{2n}) \equiv \frac{1}{{\sqrt{N}}}\left( q_{2n-1}({\bf x}_{2n-1}) + i \, q_{2n}({\bf x}_{2n}) \right)\; , \quad\quad n=1,...,{N/2} \; .\end{aligned}$$ Powy¿sza postaæ jest uogólnieniem konstrukcji Friedena, który tworz¹c funkcjê falow¹ uk³adu z³o¿y³ $n$-t¹ [*sk³adow¹ funkcji falowej*]{} z amplitud w nastêpuj¹cy sposób [@Frieden]: $$\begin{aligned} \label{amplitudapsi} \psi_{n}({\bf x}) \equiv \frac{1}{{\sqrt{N}}}\left( q_{2n-1}({\bf x}) + i \, q_{2n}({\bf x}) \right)\; , \quad\quad n=1,...,{N/2} \; .\end{aligned}$$ Dok³adniej mówi¹c, aby pos³u¿enie siê funkcj¹ falow¹ (\[amplitudapsi\]) mia³o sens, musi przynajmniej pod ca³k¹ zachodziæ równowa¿noœæ zmiennych: $$\begin{aligned} \label{x n rownowaznosc} {\bf x}_{n} \equiv {\bf x} \;\;\; {\rm dla \;\; wszystkich} \;\;\; n =1,2,...,N \; .\end{aligned}$$ Za³o¿enie to ca³kiem wystarcza przy liczeniu wartoœci oczekiwanych oraz prawdopodobieñstw.\ Przy za³o¿eniu postaci (\[amplitudapsi\]) dla $n$-tej sk³adowej[^52], postaæ [*funkcji falowej*]{}  Friedena jest nastêpuj¹ca: $$\begin{aligned} \label{psi zespolona} \psi({\bf x}) \equiv (\psi_{n}\left({\bf x})\right)_{n=1}^{N/2} \; .\end{aligned}$$ Zbiór $N/2$ sk³adowych funkcji falowych $\psi_{n}$ nazwijmy [*funkcj¹ falow¹ uk³adu rangi $N$*]{}.\ Zauwa¿my, ¿e zachodz¹ nastêpuj¹ce równoœci: $$\begin{aligned} \label{analizaq dla q} \!\sum\limits _{n=1}^{N}{q_{n}^{2}}= \left({q_{1}^{2}+q_{3}^{2}+...+q_{N-1}^{2}}\right)+\left({q_{2}^{2}+q_{4}^{2}+...+q_{N}^{2}}\right) = \sum\limits_{n=1}^{{N/2}}{\left({q_{2n-1}}\right)^{2}+\left({q_{2n}}\right)^{2}} \; \end{aligned}$$ i analogicznie: $$\begin{aligned} \label{analizaq} \sum\limits_{n=1}^{N} \frac{\partial q_{n}}{\partial {\bf x}_{n \nu}} \frac{\partial q_{n}}{\partial {\bf x}_{n}^{\nu}} = \sum\limits_{n=1}^{N/2} \left(\frac{\partial q_{2n-1}}{\partial {\bf x}_{n \nu}} \frac{\partial q_{2n-1}}{\partial {\bf x}_{n}^{\nu}} + \frac{\partial q_{2n}}{\partial {\bf x}_{n \nu}} \frac{\partial q_{2n}}{\partial {\bf x}_{n}^{\nu}} \right) \; .\end{aligned}$$ [*Zak³adaj¹c dla wszystkich poni¿szych rozwa¿añ s³usznoœæ (\[x n rownowaznosc\]), przynajmniej pod ca³k¹, oraz postaæ funkcji falowych (\[amplitudapsi\])*]{}, dokonajmy nastêpuj¹cego ci¹gu przekszta³ceñ dla (\[Fisher\_information-kinetic form bez n\]): $$\begin{aligned} \label{krok zamiany I dla q na psi} I &=& 4 \sum\limits_{n=1}^{N} \int_{\cal X} d^{4}{\bf x} \sum\limits_{\nu} \frac{\partial q_{n}}{\partial {\bf x}_{\nu}} \frac{\partial q_{n}}{\partial {\bf x}^{\nu}} = 4 \sum\limits_{n=1}^{N/2} \int_{\cal X} d^{4}{\bf x} \sum\limits_{\nu} \left[ \frac{\partial q_{2n-1}}{\partial {\bf x}_{\nu}} \frac{\partial q_{2n-1}}{\partial {\bf x}^{\nu}} + \frac{\partial q_{2n}}{\partial {\bf x}_{\nu}} \frac{\partial q_{2n}}{\partial {\bf x}^{\nu}} \right] \nonumber \\ &=& 4 N \sum\limits_{n=1}^{{N/2}} \int_{\cal X} d^{4}{\bf x} \sum\limits_{\nu} \frac{1}{{\sqrt{N}}} \frac{\partial\left({q_{2n-1} - i q_{2n}} \right)}{\partial {\bf x}_{\nu}} \frac{1}{\sqrt{N}} \frac{\partial\left({q_{2n-1} + iq_{2n}}\right)}{\partial {\bf x}^{\nu}} \; ,\end{aligned}$$ gdzie w ostatnim przejœciu skorzystano z przekszta³cenia typu: $$\begin{aligned} \label{q2 na q q} q_{2n-1}^{k} \,q_{2n-1}^{k} + q_{2n}^{k}\, q_{2n}^{k} = (q_{2n-1}^{k} - i \,q_{2n}^{k})( q_{2n-1}^{k} + i \,q_{2n}^{k}) \; , \end{aligned}$$ z indeksem $k$ oznaczaj¹cym pochodn¹ rzêdu $k=0,1,...\,$.\ \ [**Kinematyczna postaæ IF dla $\psi$**]{}: Odwo³uj¹c si¹ do definicji (\[amplitudapsi\]) funkcji falowej, otrzymujemy: $$\begin{aligned} \label{inf F z psi} I = 4 N \sum\limits_{n=1}^{{N/2}} \int_{\cal X} d^{4}{\bf x} \sum\limits_{\nu}{\frac{\partial \psi_{n}^{*}({\bf x})}{\partial {\bf x}_{\nu}}}{\frac{\partial \psi_{n}({\bf x})}{\partial {\bf x}^{\nu}}} \; .\end{aligned}$$ Pojemnoœc informacyjna (\[inf F z psi\]) ma typow¹ postaæ np. dla relatywistycznej mechaniki falowej, odpowiadaj¹c¹ czêœci kinetycznej ca³ki dzia³ania. Dlatego w³aœnie oczekiwan¹ informacjê Fishera nazwa³ Frieden [*informacj¹ kinetyczn¹*]{}. W [@Frieden] u¿yto jej do wyprowadzenia równañ Kleina-Gordona oraz Diraca metod¹ EFI [@Frieden].\ \ [**Rozk³adu prawdopodobieñstwa przesuniêcia w uk³adzie**]{}: Korzystaj¹c z twierdzenia o prawdopodobieñstwie ca³kowitym, [*gêstoœæ*]{} rozk³adu prawdopodobieñstwa przesuniêcia (lub fluktuacji) w uk³adzie mo¿e byæ zapisana nastêpuj¹co [@Frieden]: $$\begin{aligned} \label{p jako suma po qn2 przez N} p\left({\bf x}\right) &=& \sum_{n=1}^{N} p\left({\bf x}|{\theta}_{n} \right) r\left({\theta}_{n}\right) = \sum_{n=1}^{N} {p_{n}\left( {\bf x}_{n}|{\theta}_{n}\right) r\left({\theta}_{n}\right)} = \frac{1}{N} \sum_{n=1}^{N} q_{n}^{2} \left({\bf x}_{n}|{\theta}_{n}\right) \nonumber \\ &=& \frac{1}{N} \sum_{n=1}^{N} q_{n}^{2} \left({\bf x}\right) \; ,\end{aligned}$$ gdzie skorzystano z za³o¿enia, ¿e $n$-ta wartoœæ oczekiwana ${\theta}_{n}$ nie ma dla $m \neq n$ wp³ywu na rozk³ad przesuniêcia ${\bf x}_{m}$ oraz jak zwykle z postaci amplitudy $q_{n}^{2} = p_{n}$. Prawdopodobieñstwo $p_{n}$ jest prawdopodobieñstwem pojawienia siê wartoœci ${\bf x}_{n}$ zmiennej losowej przesuniêcia (lub fluktuacji) z rozk³adu generowanego z parametrem ${\theta}_{n}$, tzn. ma ono interpretacjê prawdopodobieñstwa warunkowego $p_{n}\left({\bf x}_{n}|{\theta}_{n}\right)$. Funkcjê $r\left({\theta}_{n}\right) = \frac{1}{N}$ mo¿na nazwaæ funkcj¹ “niewiedzy”, gdy¿ jej postaæ jest odzwierciedleniem ca³kowitego braku wiedzy odnoœnie tego, która z $N$ mo¿liwych wartoœci ${\theta}_{n}$ pojawi siê w konkretnym $n$-tym z $N$ eksperymentów próby.\ [**Postaæ rozk³adu dla $\psi$**]{}: W koñcu, korzystaj¹c z (\[amplitudapsi\]), (\[analizaq\]), (\[q2 na q q\]) oraz (\[p jako suma po qn2 przez N\]) widaæ, ¿e: $$\begin{aligned} \label{prawdpsi} p\left({\bf x}\right) = \sum_{n=1}^{N/2} {\psi_{n}^{*}\left({\bf x}\right) \psi_{n}}\left({\bf x}\right) \; \end{aligned}$$ jest gêstoœci¹ rozk³adu prawdopodobieñstwa przesuniêcia (lub fluktuacji) ${\bf x}$ w uk³adzie opisanym funkcj¹ falow¹ (\[psi zespolona\]).\ \ [**Uwaga o ró¿nicy z podejœciem Friedena**]{}: W ca³ym powy¿szym wyprowadzeniu nie u¿yliœmy podstawowego za³o¿enia Friedena-Soffera o [*niezmienniczoœci rozk³adu ze wzglêdu na przesuniêcie*]{}, tzn.: $$\begin{aligned} \label{shift inv property} p_{n} ({\bf x}_{n}) = p_{x_{n}} ({\bf x}_{n}|\theta_{n}) = p_{n} ({\bf y}_{n}|\theta_{n}) \; , \;\;\; {\rm gdzie} \;\;\; {\bf x}_{n}^{\nu} \equiv {\bf y}_{n}^{\nu} - \theta_{n}^{\nu} \; ,\end{aligned}$$ gdzie ${\bf y}_{n} \equiv ({\bf y}_{n}^{\nu})$, ${\bf x}_{n} \equiv ({\bf x}_{n}^{\nu})$ oraz $\theta_{n} \equiv (\theta_{n}^{\nu})$. Za³o¿enie to nie jest potrzebne przy wyprowadzeniu postaci (\[Fisher\_information-kinetic form\]) pojemnoœci informacyjnej.\ [**Uwaga**]{}: Co wiêcej, [*informacja o $\theta_{n}$ musi pozostaæ w rozk³adzie $p_{n}$ oraz jego amplitudzie $q_{n}$. Wczeœniej umówiliœmy siê, ¿e indeks $n$ zawiera t¹ informacjê*]{}. Po umiejscowieniu informacji o $\theta_{n}$ w indeksie $n$ mo¿na, w razie potrzeby wynikaj¹cej np. z fizyki zjawiska, za¿¹daæ dodatkowo niezmienniczoœci ze wzglêdu na przesuniêcie. ### Postaæ kinematyczna pojemnoœci zapisana w prawdopodobieñstwie {#Postac kinematyczna pojemnosci zapisana w prawdopodobienstwie} Poni¿ej podamy postaæ kinematyczn¹ pojemnoœci zapisan¹ w (punktowych) prawdopodobieñstwach próby. Postaæ ta jest bardziej pierwotna ni¿ (\[Fisher\_information with q dla przejscia\]), chocia¿ w treœci skryptu wykorzystywana jedynie w Dodatku.\ \ Puntem wyjœcia jest pojemnoœæ (\[postac I bez log p po theta\]): $$\begin{aligned} I \equiv I(\Theta) = \sum_{n=1}^N {\int_{\cal Y} d^{4}{\bf y}_n \frac{1}{{p_{n} \left( {\bf y}_n|\theta_{n} \right)}} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial p_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n \nu} }}} {\frac{{\partial p_{n} \left( {\bf y}_n|\theta_{n} \right)}}{{\partial \theta_{n}^{ \nu} }}} \right) } } \; . \nonumber\end{aligned}$$ Korzystaj¹c z przejœcia do addytywnych przesuniêæ ${\bf x}_{n} \equiv({\bf x}^{\nu}_{n})$ ,(\[parameters separation\]), oraz z “regu³y ³añcuchowej” (\[chain rule\]) dla pochodnej, otrzymujemy (podobnie do (\[Fisher\_information-kinetic form\])) nastêpuj¹c¹ [*kinematyczn¹ postaæ pojemnoœci informacyjnej*]{}, wyra¿on¹ w prawdopodobieñstwach: $$\begin{aligned} \label{postac I dla p po x} I = \sum_{n=1}^N {\int_{{\cal X}_{n}} d^{4}{\bf x}_{n} \frac{1}{{p_{n} \left( {\bf x}_{n} \right)}} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial p_{n} \left( {\bf x}_{n} \right)}}{{\partial {\bf x}_{n \nu} }}} {\frac{{\partial p_{n} \left( {\bf x}_{n} \right)}}{{\partial {\bf x}_{n}^{ \nu} }}} \right) } } \; , \end{aligned}$$ gdzie, podobnie jak poprzednio dla amplitud, pozostawiliœmy ca³¹ informacjê o [*$\theta_{n}$ w indeksie $n$ rozk³adu $p_{n}({\bf x}_{n})$*]{}.\ \ [**Postaæ kinematyczna $I$ zapisana w prawdopodobieñstwie**]{}: W koñcu, zak³adaj¹c, ¿e zakres zmiennoœci wszystkich ${\bf x}_{n}^{\nu}$ jest dla ka¿dego $n$ taki sam, pomijamy indeks $n$ przy tej zmiennej (ale nie przy rozk³adzie $p_{n}$), otrzymuj¹c: $$\begin{aligned} \label{postac I dla p po x bez n} I = \sum_{n=1}^N {\int_{\cal X} d^{4}{\bf x} \frac{1}{{p_{n} \left( {\bf x} \right)}} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial p_{n} \left( {\bf x} \right)}}{{\partial {\bf x}_{\nu} }}} {\frac{{\partial p_{n} \left( {\bf x} \right)}}{{\partial {\bf x}^{ \nu} }}} \right) } } \; . \end{aligned}$$ Postaæ t¹ wykorzystamy w Dodatku jako pierwotn¹ przy wyprowadzeniu elektrodynamiki Maxwella, granicy s³abego pola w teorii grawitacji oraz twierdzenia $I$ fizyki statystycznej. Równania master {#master eq} --------------- PodejdŸmy nieco inaczej ni¿ w Rozdziale \[structural principle\] do problemu estymacji. Rozwiñmy $P(\tilde{\Theta})$ w szereg Taylora wokó³ prawdziwej wartoœci parametru $\Theta$ i wyca³kujmy po ca³ej przestrzeni próby ${\cal B}$, otrzymuj¹c: $$\begin{aligned} \label{rozw w szereg T dla P} & & \!\!\!\!\!\!\!\!\int_{\cal B}\!\! d y \!\left(P(\tilde{\Theta})-P(\Theta)\right) = \! \int_{\cal B} \! d y \!\left(\sum_{n=1}^{N}\!\frac{\partial P(\Theta)}{\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n})\right.\nonumber \\ & + & \left.\frac{1}{2}\sum_{n,n'=1}^{N} \frac{\partial^{2}P(\Theta)} {\partial\theta_{n'}\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n})(\tilde{\theta}_{n'}-\theta_{n'})+\cdots\right) \; , \end{aligned}$$ gdzie u¿yto oznaczenia $\frac{\partial P(\Theta)}{\partial\theta_{n}} \equiv \frac{\partial P(\widetilde{\Theta})}{\partial \tilde{\theta}_{n}}\mid_{\widetilde{\Theta} = \Theta}$ oraz podobnie dla wy¿szych rzêdów rozwiniêcia. Poniewa¿ ca³kowanie zostaje wykonane po ca³ej przestrzeni próby ${\cal B}$, zatem bior¹c pod uwagê warunek normalizacji $\int_{\cal B} d y \, P(\Theta) = \int_{\cal B} d y \, P(\tilde{\Theta})=1$ widzimy, ¿e lewa strona równania (\[rozw w szereg T dla P\]) jest równa zero. Pomijaj¹c cz³ony wy¿szego rzêdu[^53], otrzymujemy: $$\begin{aligned} \label{L expand macroscop bez P P} \int_{\cal B} \! d y \!\left(\sum_{n=1}^{N}\!\frac{\partial P(\Theta)}{\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n}) + \frac{1}{2}\sum_{n,n'=1}^{N} \frac{\partial^{2}P(\Theta)} {\partial\theta_{n'}\partial\theta_{n}}(\tilde{\theta}_{n}-\theta_{n})(\tilde{\theta}_{n'}-\theta_{n'}) \right) = 0 \; .\end{aligned}$$ Dla estymatorów $\tilde{\Theta}$ lokalnie nieobci¹¿onych [@Amari; @Nagaoka; @book] równanie (\[L expand macroscop bez P P\]) przyjmuje dla konkretnych $n$ oraz $n'$ nastêpuj¹c¹ postaæ [**równania master**]{}: $$\begin{aligned} \label{L master oczekiwana} \int_{\cal B} \! d y \; \frac{\partial^{2}P(\Theta)} {\partial\theta_{n'}\partial\theta_{n}}\,(\tilde{\theta}_{n}-\theta_{n})(\tilde{\theta}_{n'}-\theta_{n'}) = 0 \; , \;\;\;\;\;\; n, n' =1,2,...,N \; .\end{aligned}$$ Gdy parametr $\theta_{n}^{\nu}$ ma index Minkowskiego $\nu$, wtedy mo¿na pokazaæ, ¿e wykorzystuj¹c $P=\prod_{n=1}^{N}p_{n}({\bf y}_{n})$ w (\[L master oczekiwana\]) otrzymujemy, po przejœciu do zmiennych Fisherowskich (porównaj (\[zmienne Fisherowskie\])), równanie maj¹ce w granicy $\tilde{\theta}_{n} \rightarrow \theta_{n}$ nastêpuj¹c¹ postaæ [**obserwowan¹ równania master**]{}: $$\begin{aligned} \label{conservation flow eq} \frac{\partial p_{n}({\bf y}_{n})}{\partial t_{n}} + \sum_{i=1}^{3}\frac{\partial\, p_{n}({\bf y}_{n})}{\partial {\bf y}_{n}^{i}}\, v_{n}^{i}=0\;,\;\;\; \;\;\; {\rm gdzie} \;\;\;\; v_{n}^{i} = \lim_{\tilde{\theta}_{n} \rightarrow \theta_{n}} \hat{v}_{n}^{i}\equiv\frac{\tilde{\theta}_{n}^{i} - \theta_{n}^{i}}{\tilde{\theta}_{n}^{0} - \theta_{n}^{0}} \; , \;\;\; n=1,2,...,N \; , \;\;\;\end{aligned}$$ bêd¹cego typem [**równania ci¹g³oœci strumienia**]{}, gdzie $t_{n}\equiv {\bf y}_{n}^{0}$. W (\[conservation flow eq\]) $\, \theta_{n}^{i}$ oraz $\theta_{n}^{0}$ s¹ odpowiednio wartoœciami oczekiwanymi po³o¿enia oraz czasu uk³adu. Podsumowanie rozwa¿añ {#Podsumowanie rozwazan} --------------------- Podstawowym przes³aniem wyniesionym z metody estymacyjnej Friedena-Soffera jest to, ¿e $TFI$ jest poprzednikiem Lagrangianu uk³adu [@Frieden]. Temat ten rozwiniemy w kolejnym rozdziale. Pewnym minusem teorii Friedena-Soffera mog³a wydawaæ siê koniecznoœæ zapostulowania nowych zasad informacyjnych. Co prawda z puntu widzenia fenomenologii skutecznoœæ tych zasad w wyprowadzeniu du¿ej liczby modeli u¿ytecznych do opisu zjawisk wydaje siê byæ ca³kiem satysfakcjonuj¹ca, jednak wyprowadzenie tych zasad przesunê³oby teoriê do obszaru bardziej podstawowego. Pozwoli³oby to zarówno na podanie jej przysz³ych ograniczeñ fenomenologicznych jak i jej mo¿liwych teoretycznych uogólnieñ.\ W tym kontekstcie, t¹ w³aœnie rolê spe³nia wyprowadzenie strukturalnej zasady informacyjnej jako konsekwencji analitycznoœci logarytmu funkcji wiarygodnoœci w otoczeniu prawdziwej wartoœci parametru $\Theta$ oraz wskazanie geometrycznego znaczenia informacyjnej zasady wariacyjnej, która le¿u u podstaw zasady ekstremizacji dzia³ania fizycznego. W obecnym rozdziale zwrócono uwagê, ¿e u podstaw informacyjnego zrozumienia zasady wariacyjnej mo¿e le¿eæ idea idealnego pomiaru [@Dziekuje; @informacja_2], przy której wariacja pojemnoœci informacyjnej $I$ jest równa (z wyj¹tkiem znaku) wariacji informacji strukturalnej $Q$.\ W powy¿szym rozdziale wyprowadzono te¿ równanie master (\[L master oczekiwana\]) dla funkcji wiarygodnoœci, które prowadzi do równania ci¹g³oœci strumienia dla punktowego rozk³adu w próbie (\[conservation flow eq\]). Ciekawe jest to, ¿e równanie master pojawia siê z rozwiniêcia funkcji wiarygodnoœci w szereg Taylora wokó³ prawdziwej wartoœci parametru $\Theta$. Si³¹ rzeczy (por. (\[rozw w szereg T dla P\])) nie pojawia siê wiêc w nich (logarytmiczna) czêœæ nieliniowa struktury uk³adu. Ta ga³aŸ uogólnienia MNW w klasycznej statystycznej estymacji le¿y bli¿ej teorii procesów stochastycznych [@Sobczyk_Luczka] ni¿ EFI.\ \ Wyprowadzaj¹c w Rozdziale \[structural principle\] strukturaln¹ zasadê informacyjn¹ [@Dziekuje; @informacja_2] wykazano, ¿e metoda Friedena-Soffera jest pewn¹ modyfikacj¹ MNW, pozwalaj¹c¹, jak siê oka¿e, na nieparametryczn¹ estymacjê równañ ruchu teorii pola lub równañ generuj¹cych rozk³ad fizyki statystycznej [@Frieden]. Wiele z tych równañ otrzymano ju¿ w [@Frieden] zgodnie z informatycznym zrozumieniem Friedena-Soffera wspomnianym na koñcu Rozdzia³u \[information transfer\]. W [@Mroziakiewicz] wyprowadzenia te zosta³y sprawdzone dla przyjêtej w obecnym skrypcie fizycznej postaci zasad informacyjnych [@Dziekuje; @informacja_1].\ Jednak¿e dopiero wyprowadzenie strukturalnej zasady informacyjnej pozwala na faktoryzacjê z obserwowanej $SI$ czêœci, bêd¹cej miar¹ probabilistyczn¹ i w zwi¹zku z tym na prawid³owe umieszczenie rozk³adów spe³niaj¹cych równania ró¿niczkowe metody EFI w odpowiednich podprzestrzeniach przestrzeni statystycznej. Dlatego omówieniu b¹dŸ przeliczeniu niektórych rozwi¹zañ EFI z uwzglêdnieniem tego faktu poœwiêcimy dwa nastêpne rozdzia³y.\ Nale¿y jednak podkreœliæ, ¿e Frieden, Soffer i ich wspó³pracownicy Plastino i Plastino, podali metodê rozwi¹zania uk³adu (ró¿niczkowych) zasad informacyjnych dla problemu EFI, która jest bardzo skuteczna, gdy¿ poza warunkami brzegowymi i ewentualnymi równaniami ci¹g³oœci nie jest ograniczona przez ¿adn¹ konkretn¹ postaæ rozk³adu. Metodê t¹ wykorzystamy w dalszym ci¹gu analizy. Kryteria informacyjne w teorii pola {#Kryteria informacyjne w teorii pola} =================================== [*G³ówne estymacyjne przes³anie metody*]{} EFI. Jak stwierdziliœmy poprzednio, poniewa¿ podstawowa myœl stoj¹ca za metod¹ EFI jest nastêpuj¹ca: Skoro $IF$ jest infinitezymalnym typem entropii wzglêdnej Kulback-Leibler’a, która s³u¿y do statystycznego wyboru pomiêdzy zaproponowanymi [*rêcznie*]{} modelami, zatem po dodatkowym [*rêcznym*]{}, aczkolwiek uzasadnionym, na³o¿eniu ró¿niczkowych zasad strukturalnych na uk³ad, staje siê ona metod¹ estymuj¹c¹ równania ruchu i ich wyboru drog¹ wymogu spe³nienia zasad[^54] informacyjnych. B¹dŸ, jeœli ktoœ woli, metoda EFI jest metod¹ estymuj¹c¹ rok³ady, które s¹ rozwi¹zaniami tych równañ. Jest wiêc to metoda estymacji nieparametrycznej. Wspomniane równania to np. równania ruchu teorii pola b¹dŸ równania generuj¹ce rozk³ady fizyki statystycznej. Informacja Fishera i klasyfikacja modeli {#Informacja Fishera i klasyfikacja modeli} ----------------------------------------- Obecny rozdzia³ poœwiêcony jest g³ównie przedstawieniu wstêpnej klasyfikacji modeli fizycznych ze wzglêdu na skoñczonoœæ (b¹dŸ nieskoñczonoœæ) pojemnoœci informacyjnej $I$. Ponadto, poni¿sze rozwa¿ania dla modeli ze skoñczonym $I$ dotycz¹ wy³¹cznie modeli metody EFI. Kolejna, bardziej szczegó³owa klasyfikacja pozwala sklasyfikowaæ modele ze wzglêdu na wielkoœæ próby $N$.\ Jak poka¿emy mechanika klasyczna posiada nieskoñczon¹ pojemnoœæ informacyjn¹ $I$. Œciœle mówi¹c, mechanika klasyczna jest teori¹ z symplektyczn¹ struktur¹ rozmaitoœci i nie posiada struktury statystycznej. Czasami jednak s³yszy siê stwierdzenie, ¿e jest ona stochastyczn¹ granic¹ mechaniki kwantowej. Ale i na odwrót, wed³ug von Neumann’a [@Neumann] teoria kwantowa jest niespójna z istnieniem zespo³ów nie posiadaj¹cych rozmycia (rozproszenia). W zwi¹zku z tym, doœæ powszechnie uwa¿a siê, ¿e wystêpowanie odstêpstw od klasycznego zachowania siê uk³adów mo¿na uchwyciæ jedynie na poziomie statystycznym [@Peres].\ Poni¿ej udowodnimy twierdzenie klasycznej statystyki mówi¹ce o niemo¿liwoœci wyprowadzenia mechaniki falowej[^55] metody EFI z mechaniki klasycznej. W tym celu wykorzystamy statystyczne pojêcie pojemnoœci informacyjnej, które jest narzêdziem dla dwóch sprzê¿onych z sob¹ zagadnieñ, a mianowicie powy¿ej wspomnianego statystyczego dowodu o niewyprowadzalnoœci mechaniki kwantowej z klasycznej i zwi¹zanego z nim problemu konsystencji samospójnego formalizmu. Ostatni fakt wykorzystywany jest w takich ga³êziach badañ fizycznych jak nadprzewodnictwo [@superconductivity], fizyka atomowa i cz¹stek elementarmych [@bib; @B-K-1] oraz astrofizyka [@Bednarek]. ### Podzia³ modeli ze wzglêdu na $N$ oraz kategorie $I$ {#rozdzielnosci mech fal i klas} Jak dot¹d nie odnieœliœmy siê do wartoœci $N$ wymiaru próby. Pierwsza klasyfikacja zwi¹zana z $N$ jest ogólna. Tzn. poka¿emy, ¿e modele nale¿¹ do dwóch ró¿nych, ogólnych kategorii z ró¿n¹ wartoœci¹ $N$. Pierwsza z nich posiada skoñczon¹ wartoœæ $N$ i jest zwi¹zana ze skoñczon¹ wartoœci¹ $I$. Obejmuje ona modele mechaniki falowej i klasycznych teorii pola, gdy¿ jedno skoñczenie wymiarowe, polowe rozwi¹zanie równañ ruchu okreœla ewolucjê uk³adu wraz z pe³nym okreœleniem jego struktury w przestrzeni i czasie. Natomiast mechanika klasyczna nale¿y do drugiej kategorii z nieskoñczonym $N$, gdy¿ rozwi¹zanie równania ruchu nie okreœla struktury cz¹stki, która musi byæ niezale¿nie od tego równania okreœlona poprzez zdefiniowanie, w ka¿dym punkcie toru cz¹stki, jej punktowej struktury (np. poprzez dystrybucjê $\delta$-Diraca). Mechanika klasyczna okazuje siê posiadaæ nieskoñczon¹ wartoœæ $I$. #### Dowód podzia³u na dwie kategorie $I$ {#kategorie I - dowod} Zobrazujmy powy¿sze s³owa nastêpuj¹c¹ analiz¹. Dla uproszczenia rozwa¿my uk³ad jednowymiarowy w po³o¿eniu[^56]. Za³ó¿my wpierw, ¿e uk³ad jest opisany przez nieosobliw¹ dystrybucjê. Wtedy dla $N\rightarrow\infty$ pojemnoœæ informacyjna $I$, (\[I dla pn jeden parametr\]), rozbiega siê do nieskoñczonoœci. Taka sama sytuacja zachodzi jednak dla ka¿dej [*osobliwej*]{} dystrybucji jak np. dystrybucja $\delta$-Diraca. SprawdŸmy, ¿e tak jest istotnie. Rozwa¿my punktow¹ cz¹stkê swobodn¹, dla uproszczenia w spoczynku, w po³o¿eniu $\theta$, oraz $\delta$-Diracowski ci¹g funkcji, np. ci¹g funkcji Gaussa: $$\begin{aligned} \label{ciag Gaussa} \left\{\delta_{k}(y_{n}|\theta) = \frac{k}{\sqrt{\pi}}\, \exp(-k^{2}(y_{n}-\theta)^{2})\right\} \; , \;\;\; {\rm gdzie} \;\;\; k = 1,2,3,... \;\; .\end{aligned}$$ Wtedy, poniewa¿ dla okreœlonego indeksu $k$ ci¹gu (\[ciag Gaussa\]), pojemnoœæ informacyjna (\[I dla pn jeden parametr\]): $$\begin{aligned} \label{pojemnosc I dla ciag Gaussa} I_{k} = \sum_{n=1}^{N} {\int{dy_{n} {\delta_{k}(y_{n}|\theta)}\left({\frac{{\partial \ln \delta_{k}(y_{n}|\theta)}}{{\partial\theta}}}\right)^{2}}} \; , \;\;\; {\rm gdzie} \;\;\; k = 1,2,3,... \;\; \end{aligned}$$ jest równa: $$\begin{aligned} \label{wartosc pojemnosci I_k dla ciag Gaussa} I_{k} = \frac{N}{\sigma_{k}^{2}} \; , \;\;\; {\rm dla} \;\;\; k = 1,2,3,... \;\; ,\end{aligned}$$ gdzie $\sigma_{k}^{2}=\frac{1}{2k^{2}}$ opisuje wariancjê po³o¿enia cz¹stki dla $k$-tego elementu ci¹gu (\[ciag Gaussa\]), wiêc widzimy, ¿e $I_{k}$ rozbiega siê do nieskoñczonoœci dla $N \rightarrow \infty\,$ i nawet jeszcze mocniej, gdy dodatkowo $k \rightarrow \infty$.\ Posumowuj¹c, dla $N \rightarrow \infty$ pojemnoœæ informacyjna $I$ nie istnieje, obojêtnie z jak¹ dystrybucj¹ mielibyœmy do czynienia.\ Istniej¹ wiêc [*dwie, powy¿ej wymienione, rozdzielne kategorie modeli odnosz¹ce siê do wymiaru $N$ próby*]{}. Tzn. dla jednych, takich jak mechanika falowa i teorie pola, $N$ oraz $I$ s¹ skoñczone, podczas gdy mechanika klasyczna tworzy osobn¹ klasê z nieskoñczonym $N$ oraz $I$. To koñczy dowód [@Dziekuje; @informacja_1] o niewyprowadzalnoœci modeli falowych i teorio-polowych z mechaniki klasycznej.\ \ Powy¿szy dowód nie obejmuje mo¿liwoœci wyprowadzenia mechaniki falowej (czy te¿ teorii kwantowych) z klasycznej teorii pola b¹dŸ samospójnej teorii pola [@bib; @B-K-1; @Dziekuje_Jacek_nova_2; @Dziekuje_Jacek_nova_1].\ [**Uwaga**]{}: Oznacza to, ¿e mechanika klasyczna nie ma skoñczonego statystycznego pochodzenia[^57], chyba, ¿e tak jak w (\[ciag Gaussa\]) wprowadzi siê nieskoñczon¹ liczbê statystycznych parametrów, co jednak poci¹ga za sob¹ nieskoñczonoœæ pojemnoœci informacyjnej $I$. ### Podzia³ modeli ze skoñczonym $I$ na podklasy z ró¿nym $N$ {#podzial modeli ze wzgledu na N} Jak ju¿ wspomninaliœmy, Frieden i Soffer [@Frieden] wyprowadzili modele falowe pos³uguj¹c siê pojêciem pojemnoœci informacyjnej $I$ oraz zasadami informacyjnymi estymacyjnej metody EFI. Rozwiniemy ten temat w dalszej czêœci obecnego rozdzia³u. Na razie zauwa¿my, ¿e stosuj¹c zasadniczo[^58] jednoczeœnie obie zasady informacyjne, strukturaln¹ $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0$, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), oraz wariacyjn¹ $\delta(I + Q) = 0$, (\[var K rozpisana\]), oraz uwzglêdniaj¹c odpowiednie fizyczne wiêzy (wyra¿one narzuceniem na uk³ad np. równania ci¹g³oœci, symetrii oraz warunków brzegowych), otrzymujemy zró¿nicowanie ze wzglêdu na $N$ modeli posiadaj¹cych skoñczone wartoœci $N$ oraz $I$. I tak równanie Kleina-Gordona oraz równanie Schr[ö]{}dingera jako jego nierelatywistyczna granica (por. Dodatek \[Rownanie Schrodingera\]) posiadaj¹ rangê pola $N=2$, równanie Diraca posiada $N=8$, równania Maxwell’a posiadaj¹ $N=4$, a teoria grawitacji, zasadniczo bardziej w ujêciu Logunova [@Denisov-Logunov] ni¿ ogólnej teorii wzglêdnoœci, posiada $N=10$ (Dodatek \[general relativity case\]). ### Konkluzje i konsekwencje podzia³u modeli na kategorie $I$ {#konkluzje o dwoch kategoriach I} Powy¿ej otrzymaliœmy rezultat mówi¹cy, ¿e wszystkie modele opisane strukturaln¹ zasad¹ informacyjn¹ nale¿¹ do kategorii skoñczonej wartoœci pojemnoœci informacyjnej $I$ oraz, ¿e mechanika klasyczna nale¿y do kategorii nieskoñczonego $I$. Zatem w ramach zagadnieñ rozwa¿anych w skrypcie, granica nie le¿y pomiêdzy tym co micro a makro, ale przebiega pomiêdzy teoriami, które maj¹ pochodzenie statystyczne oraz tymi, które maj¹ pochodzenie klasyczno-mechaniczne. Albo lepiej, pomiêdzy tym co ma pochodzenie falowe lub szerzej, teorio-polowe, oraz tym co ma pochodzenie œciœle punktowe.\ Poniewa¿ w konstrukcji modeli klasycznej teorii pola oraz mechaniki falowej, u¿yty jest ten sam statystyczny formalizm informacji Fishera, dlatego jest ona równie¿ w³aœciwym narzêdziem w konstrukcji samospójnych teorii pola [@bib; @B-K-1], ³¹cz¹c modele mechaniki falowej i klasycznej teorii pola w jeden, logicznie spójny aparat matematyczny.\ Jak wiemy, aby otrzymaæ jak¹kolwiek teoriê pola, metoda EFI u¿ywa dwóch nowych zasad, wariacyjnej (\[var K rozpisana\]), która minimalizuje ca³kowit¹ fizyczn¹ informacjê uk³adu oraz obserwowanej (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) i oczekiwanej (\[rownowaznosc strukt i zmodyfikowanego strukt\]) zasady strukturalnej, która t¹ informacjê zeruje. Frieden i Soffer [@Frieden] zwrócili uwagê, ¿e pojêcie informacji poprzedza pojêcie fizycznego dzia³ania, a wprowadzony formalizm mo¿na s³usznie nazwaæ podejœciem Friedena do równañ ruchu. Sporo te¿ na tej drodze konstrukcji modeli fizycznych ju¿ zrobiono. Jednak¿e liczne zagadnienia, ze wzglêdu na odmienne ni¿ w [@Frieden] zrozumienie zasady strukturalnej (patrz [@Dziekuje; @informacja_1; @Mroziakiewicz; @Dziekuje; @informacja_2] oraz obecny skrypt), wymagaj¹ ponownego zinterpretowania i zrozumienia. Ci¹gle na ogólne opracowanie czeka wprowadzenie do formalizmu informacyjnych poprzedników Ÿróde³ oraz lepsze zrozumienie fizyki le¿¹cej u podstaw znaczenia wymiaru próby $N$. Poni¿sze rozwa¿ania s³u¿¹ usystematyzowaniu istniej¹cego ju¿ statystycznego aparatu pojêciowego informacji kinetycznej i strukturalnej metody EFI oraz lepszemu opisowi zwi¹zku informacji fizycznej z ca³k¹ dzia³ania. Równania ró¿niczkowe metody EFI {#equations of motion} -------------------------------- Kolejna czêœæ obecnego rozdzia³u poœwiêcona jest omówieniu rozwi¹zañ zasad informacyjnych metod¹ EFI dla modeli mechaniki falowej i teorii pola [@Dziekuje; @za; @models; @building]. Punktem wyjœcia jest pojemnoœæ informacyjna $I$ w jej kinematycznych postaciach (\[Fisher\_information-kinetic form bez n\]): $$\begin{aligned} I = 4 \sum_{n=1}^{N} \int_{\cal X} \!\! d^{4}{\bf x} \sum_{\nu=0}^{3} \frac{\partial q_{n}({\bf x})}{\partial {\bf x}_{\nu}} \frac{\partial q_{n}({\bf x})}{\partial {\bf x}^{\nu}} \; \nonumber\end{aligned}$$ b¹dŸ (\[inf F z psi\]): $$\begin{aligned} I = 4 N \sum\limits_{n=1}^{{N/2}} \int_{\cal X} d^{4}{\bf x} \sum\limits_{\nu=0}^{3} {\frac{\partial\psi_{n}^{*}({\bf x})}{\partial {\bf x}_{\nu}}}{\frac{\partial \psi_{n}({\bf x})}{\partial {\bf x}^{\nu}}} \; , \nonumber\end{aligned}$$ wyprowadzonych w Rozdziale \[The kinematical form of the Fisher information\], gdzie ${\bf x}^{\nu}_{n}$ s¹ zgodnie z (\[parameters separation\]) przesuniêciami wartoœci pomiarowych po³o¿enia zebranymi przez uk³ad od ich wartoœci oczekiwanych. Wyprowadzenie (\[Fisher\_information-kinetic form bez n\]) oraz (\[inf F z psi\]) zosta³o zaprezentowane w Rozdziale \[The kinematical form of the Fisher information\] i nie zak³ada ono (w przeciwieñstwie do orginalnego wyprowadzenia Friedena-Soffera) koniecznoœci istnienia niezmienniczoœci przesuniêcia rozk³adów prawdopodobieñstwa. Pewne informacje na temat niezmienniczoœci Lorentzowskiej pojemnoœci informacyjnej $I$ zosta³y podane w Rozdziale \[Poj inform zmiennej los poloz\].\ Uogólnienia powy¿szych kinematycznych postaci na przypadek wystêpowania w uk³adzie pól cechowania omówimy w dalszej czêœci rozdzia³u. ### Ogólna postaæ funkcji gêstoœci TFI oraz obserwowane zasady informacyjne {#ogolna postac TFI i zasad obserwowanych} Wyprowadzenie strukturalnej zasady informacyjnej zosta³o przedstawione w Rozdziale \[structural principle\]. Odwo³uje siê ono do pe³nych danych pomiarowych $({\bf y}_{n})_{n=1}^{N}$, ale jego postaæ dla przesuniêæ $({\bf x}_{n})_{n=1}^{N}$ jest dok³adnie taka sama [@Dziekuje; @informacja_2]. Tak wiêc, poni¿ej stosowane zasady informacyjne, strukturalna oraz wariacyjna, bêd¹ odwo³ywa³y siê do miary probabilistycznej $ d{\bf x} \, p_{n}(\bf x)$ okreœlonej na przestrzeni przesuniêæ ${\bf x} \in {\cal X}$ jako przestrzeni bazowej, gdzie ${\cal X}$ jest czasoprzestrzeni¹ Minkowskiego $R^{4}$.\ \ Przyst¹pmy do przedstawienia konstrukcji mechaniki falowej i teorii pola zgodnie z metod¹ EFI. Wed³ug równania (\[physical K\]) TPI zosta³a okreœlona jako $K=Q+I$. Poniewa¿ przesuniêcie ${\bf x}_{n}$ nie zale¿y od parametru $\theta_{m}$ dla $m\neq n$ oraz zakres ca³kowania dla wszystkich ${\bf x}_{n}$ jest taki sam, dlatego $I$ redukuje siê do diagonalnych postaci (\[Fisher\_information-kinetic form bez n\]) b¹dŸ (\[inf F z psi\]), a $Q$ do postaci: $$\begin{aligned} \label{Q diag z q2} Q = \sum_{n=1}^{N}\int_{\cal X} d^{4}{\bf x}\, q_{n}^{2}({\bf x})\,\texttt{q\!F}_{n}(q_{n}({\bf x})) \; ,\end{aligned}$$ zgodnie z oznaczeniem w (\[Q dla niezaleznych Yn w d4y\]), b¹dŸ w przypadku pola $\psi({\bf x})$, (\[psi zespolona\]), do ogólnej (jak zwykle rzeczywistej) postaci: $$\begin{aligned} \label{Q diag z psi2} Q \equiv Q_{\psi} = \int_{\cal X} d^{4}{\bf x} \sum_{n,n'=1}^{N/2} \, \psi_{n}^{*}({\bf x}) \psi_{n'}({\bf x}) \, \texttt{q\!F}_{nn'}^{\psi}(\psi({\bf x}), \psi^{*}({\bf x}), \psi^{(l)}({\bf x}), \psi^{*(l)}({\bf x})) \; ,\end{aligned}$$ przy czym ca³a funkcja podca³kowa jest wielomianem pól $\psi({\bf x})$ oraz $\psi^{*}({\bf x})$, stopnia nie mniejszego ni¿ 2, oraz ich pochodnych rzêdu $l = 1,2,...\;$ (por. (\[qF diagonalne\])), natomiast $\texttt{q\!F}^{\psi}_{nn'}$ jest pewn¹ obserwowan¹ (w ogólnoœci zespolon¹) informacj¹ strukturaln¹ uk³adu. Konkretn¹, jak sie okazuje prost¹ postaæ $Q$ dla przypadku pól skalarnych Kleina-Gordona oraz pola Diraca omówimy poni¿ej.\ \ [**Gêstoœæ TFI**]{}: Korzystaj¹c z (\[Fisher\_information-kinetic form bez n\]), (\[Q diag z q2\]) oraz (\[physical K\]), mo¿emy zapisaæ TFI w postaci: $$\begin{aligned} \label{TPI diag} \mathbb{S} \equiv K = \int_{\cal X} d^{4}{\bf x}\, k\;,\end{aligned}$$ gdzie dla pola opisanego amplitudami $q_{n}$: $$\begin{aligned} \label{k form} k = 4\sum_{n=1}^{N}\left[\;\sum_{\nu=0}^{3}\frac{\partial q_{n}({\bf x})}{\partial {\bf x}_{\nu}} \frac{\partial q_{n}({\bf x})}{\partial {\bf x}^{\nu}} \, + \, \frac{1}{4}\, q_{n}^{2}({\bf x})\,\texttt{q\!F}_{n}(q_{n}({\bf x}))\right] \; .\end{aligned}$$ natomiast dla pola opisanego amplitudami $\psi_{n}$: $$\begin{aligned} \label{k form dla psi} k &=& 4 N \sum_{n, n'=1}^{N/2}\left[\;\sum_{\nu=0}^{3} \delta_{nn'}{\frac{\partial\psi_{n}^{*}({\bf x})}{\partial {\bf x}_{\nu}}}{\frac{\partial \psi_{n'}({\bf x})}{\partial {\bf x}^{\nu}}} \, \right. \nonumber \\ &+& \left. \, \frac{1}{4}\, \, \psi_{n}^{*}({\bf x}) \psi_{n'}({\bf x}) \, \texttt{q\!F}_{nn'}^{\psi}(\psi({\bf x}), \psi^{*}({\bf x}), \psi^{(l)}({\bf x}), \psi^{*(l)}({\bf x})) \right] \; .\end{aligned}$$ Równowa¿noœæ w (\[TPI diag\]) sugeruje, ¿e $K$ [*pe³ni funkcjê statystycznego poprzednika*]{} ([*ca³ki*]{}) [*dzia³ania*]{} $\mathbb{S}$, natomiast $k$, bêd¹ce [*funkcj¹ gêstoœci*]{} TFI, jest statystycznym poprzednikiem [*gêstoœci Lagrangianu*]{} ${\cal L}$. Sprawie tej poœwiêcimy jeden z poni¿szych rozdzia³ów.\ \ [**Uwaga o sformu³owaniu Lagrange’a i rz¹d d¿etów funkcji wiarygodnoœci**]{}: W dalszej czêœci skryptu za³o¿ymy, ¿e obserwowana informacja strukturalna nie zawiera pochodnych pól rzêdu wy¿szego ni¿ $l=1$. Za³o¿enie to ma charakter fizyczny. Oznacza ono, ¿e jeœli wspó³rzêdne uogólnione (u nas amplitudy) oraz prêdkoœci uogólnione (u nas pochodne amplitud) uk³adu s¹ zadane w pewnej chwili czasu, to ewolucja uk³adu jest ca³kowicie okreœlona, o ile równania ruchu s¹ 2-giego rzêdu. Odpowiada to sformu³owaniu Lagrange’a wykorzystywanemu w badaniu dynamicznych i termodynamicznych w³asnoœci uk³adów.\ Fakt ten z punktu widzenia statystycznego oznacza, ¿e interesuj¹ nas tylko takie (pod)przestrzenie statystyczne, dla których wszystkie mo¿liwe logarytmy funkcji wiarygodnoœci posiadaj¹ $r$-jety $J_{p}^{\,r}({\cal S},\text{R})$ w otoczeniu $U_{p}$ punktu $p \equiv P(\Theta)$ $\in$ ${\cal S}$ rzêdu $r \leq 2$ (por. Rozdzia³ \[r-c\]). Jest to istotne z punktu widzenia obserwowanej IF (\[observed IF\]) zdefiniowanej pierwotnie poprzez drugie pochodne logarytmu funkcji wiarygodnoœci po parametrach, co z kolei uwo¿liwia konstrukcjê strukturalnej zasady informacyjnej (Rozdzia³ \[structural principle\]), która jest równaniem metody EFI.\ Jednoczeœnie oczekiwana IF (\[iF and I\]) wchodzi w nierównoœæ Rao-Cramera, której pewn¹ postaci¹ jest, po dokonaniu w informacji Fishera transformacji Fouriera do przestrzeni pêdowej, zasada nieoznaczonoœci Heisenberga (Dodatek \[Zasada nieoznaczonosci Heisenberga\]). Zatem fakt wystêpowania w funkcji Lagrange’a kwadratu pierwszych pochodnych by³by (z tego punktu widzenia) artefaktem koniecznoœci wykonania przez uk³ad estymacji jego czasoprzestrzennych po³o¿eñ, która to estymacja posiada dolne ogranicze Rao-Cramera na dok³adnoœæ jednoczesnej estymacji po³o¿enia oraz prêdkoœci.\ \ \ [**Warunek sta³oœci metryki Rao-Fishera**]{}: Odpowiedzmy jeszcze na pytanie, co z punktu widzenia [*statystycznego*]{} oznacza niewystêpowanie w rozwiniêciu $\ln P(\tilde{\Theta})$ w szereg Taylora (\[Freiden like equation\]) wyrazów rzêdu wy¿szego ni¿ drugi (tzn. brak jetów o rzêdzie $r>2$). Sytuacja ta ma miejsce, gdy obserwowana informacja Fishera $\texttt{i\!F}$, (\[observed IF\]), nie zale¿y od parametru $\Theta \in V_{\Theta}$, gdzie $V_{\Theta}$ jest przestrzeni¹ parametru $\Theta$. Wtedy bowiem jej pochodne po parametrze $\Theta=(\theta_{n})$ s¹ w równe zero dla ka¿dego punktu $p'=P(\Theta')$ w otoczeniu $U_{p}$. Zatem: $$\begin{aligned} \label{observed IF wynika nie jet} \!\!\!\!\!\!\!\! \frac{\partial }{\partial \Theta'} \texttt{i\!F}\left|_{p'\in \,U_{p}} \right. = \frac{\partial }{\partial \Theta'} \left(-\frac{\partial^{2} \ln P(\Theta')}{\partial \Theta'^{\,2}}\right)\left|_{p'\in \,U_{p}} \right. = 0 \; \Rightarrow \; j_{p'}^{\,r}({\cal S},\text{R}) - j_{p'}^{\,2}({\cal S},\text{R})=0 \; {\rm dla} \; r>2 \,, \end{aligned}$$ gdzie $j_{p}^{\,r}({\cal S},\text{R})$ jest dowolnym elementem przestrzeni jetów $J_{p}^{\,r}({\cal S},\text{R})$. Lewa strona powy¿szej implikacji oznacza, ¿e: $$\begin{aligned} \label{observed IF stala} (\texttt{i\!F})_{nn'}\left|_{p'\in \,U_{p}} \right. = const. \;\;\; {\rm na } \;\;\; U_{p} \; ,\end{aligned}$$ tzn. obserwowana informacja Fishera $\texttt{i\!F}$ nie zale¿y w $U_{p}$ od parametru $\Theta$, z czego wynika [*niezale¿noœæ oczekiwanej*]{} IF, [*czyli metryki Rao-Fishera*]{}, [*od parametru*]{} $\Theta$ w $U_{p}$.\ Takie zachowanie siê metryki Rao-Fishera ma nastêpuj¹c¹ ciekaw¹ konsekwencjê. Otó¿ w Rozdziale \[Informacja strukturalna EPR\] oka¿e siê, ¿e fakt [*sta³oœci metryki Rao-Fishera*]{} (\[metryka Rao-Fishera dla EPR\]) jest odpowiedzialny za otrzymanie w ramach metody EFI znanych formu³ mechaniki kwantowej (\[wynikEPR\]), opisuj¹cych spl¹tanie w problemie EPR-Bohm’a. #### Postaæ obserwowana zasad informacyjnych {#obserw zasady inform} Na³ó¿my na uk³ad informacyjn¹ obserwowan¹ zasadê strukturaln¹ $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0$, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), oraz informacyjn¹ zasadê wariacyjn¹ $\delta(I + Q)=0$, (\[var K rozpisana\]).\ \ [**Przypadek z amplitud¹ $q$**]{}: Ze zmodyfikowanej obserwowanej zasady strukturalnej (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), bior¹c pod uwagê wczeœniejsze przejœcia pomiêdzy (\[IF 2 poch na kwadrat pierwszej\]), (\[pojemnosc C dla polozenia - powtorka wzoru\]), (\[potrz\]) oraz (\[Fisher\_information-kinetic form bez n\]), wynika warunek zerowy: $$\begin{aligned} \label{eq zero} \sum_{\nu=0}^{3}\frac{\partial q_{n}({\bf x})}{\partial {\bf x}_{\nu}} \frac{\partial q_{n}({\bf x})}{\partial {\bf x}^{\nu}} + \,\frac{\kappa}{4}\, q_{n}^{2}({\bf x})\,\texttt{q\!F}_{n}(q_{n}({\bf x}))=0 \; , \;\;\; n =1,2,...,N \; .\end{aligned}$$ Natomiast z zasady wariacyjnej (\[var K rozpisana\]) wynika uk³ad równañ Eulera-Lagrange’a: $$\begin{aligned} \label{EL eq} \sum_{\nu=0}^{3} \frac{\partial }{\partial {\bf x^{\nu}}} \left({\frac{{\partial k}}{{\partial ( \frac{\partial q_{n}({\bf x})}{\partial {\bf x}^{\nu}} ) }}}\right) = \frac{{\partial k}}{{\partial q_{n}({\bf x})}} \; , \;\;\; n =1,2,...,N \; ,\end{aligned}$$ gdzie $k$ zosta³o podane w (\[k form\]).\ \ [**Przypadek z amplitud¹ $\psi$**]{}: Odpowiednia dla pola $\psi$ obserwowana, zmodyfikowana postaæ zasady strukturalnej (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), która bierze pod uwagê (\[inf F z psi\]), jest nastêpuj¹ca: $$\begin{aligned} \label{strukt row dla psi} &&\sum_{\nu=0}^{3} {\frac{\partial\psi_{n}^{*}({\bf x})}{\partial {\bf x}_{\nu}}}{\frac{\partial \psi_{n}({\bf x})}{\partial {\bf x}^{\nu}}} \\ &+& \frac{\kappa}{4}\, \sum_{n'=1}^{N/2} \, \psi_{n}^{*}({\bf x}) \psi_{n'}({\bf x}) \, \texttt{q\!F}_{nn'}^{\psi}(\psi({\bf x}),\psi({\bf x}), \frac{\partial \psi({\bf x})}{\partial {\bf x}}, \frac{\partial \psi^{*}({\bf x})}{\partial {\bf x}} ) = 0 \; , \;\;\;\;\;\; n=1,2,...,N/2 \; . \nonumber\end{aligned}$$ Natomiast uk³ad równañ Eulera-Lagrang’a wynikaj¹cy z zasady wariacyjnej (\[var K rozpisana\]) jest nastêpuj¹cy: $$\begin{aligned} \label{EL eq dla psi} \sum_{\nu=0}^{3} \frac{\partial }{\partial {\bf x^{\nu}}} \left({\frac{{\partial k}}{{\partial ( \frac{ \partial \psi_{n}^{*}({\bf x})}{\partial {\bf x}^{\nu}} ) }}}\right) = \frac{{\partial k}}{{\partial \psi_{n}^{*}({\bf x})}} \; , \;\;\; n =1,2,...,N/2 \; ,\end{aligned}$$ gdzie $k$ zosta³o podane w (\[k form dla psi\]).\ Zauwa¿my, ¿e powy¿sza postaæ $k$, (\[k form dla psi\]), jest na tyle ogólna, ¿e aby zobaczyæ dzia³anie metody EFI wynikaj¹ce z równañ (\[strukt row dla psi\]) i (\[EL eq dla psi\]), nale¿y podaæ konkretn¹ postaæ $k$, dla ka¿dego zagadnienia z polem typu $\psi$ z osobna.\ Natomiast, jak siê przekonamy, równania (\[eq zero\]) oraz (\[EL eq\]) z amplitudami $q_{n}$ s¹ ju¿ zapisane w postaci bliskiej ich bezpoœredniego u¿ycia i otrzymania jawnej postaci $\texttt{q\!F}_{n}$ oraz rozwi¹zañ metody EFI, czyli odpowiednich fizycznych równañ ruchu (b¹dŸ równañ generuj¹cych, por. Rozdzia³ \[Przyklady\]) dla amplitud $q_{n}({\bf x})$.\ \ Let us summarize this Section. Postacie kinetyczne (\[Fisher\_information-kinetic form bez n\]) oraz (\[inf F z psi\]), oparte o informacjê Fishera (\[observed IF Amari\]), s¹ wykorzystywane do konstrukcji równañ ruchu (lub równañ generuj¹cych rozk³ad) modeli fizycznych. Wystêpuj¹ one w (\[eq zero\])-(\[EL eq dla psi\]). Natomiast pierwotna postaæ pojemnoœci $I$ ma swój pocz¹tek w (\[I dla niezaleznych Yn\]) oraz w (\[observed IF\]) i (\[iF diagonalne\]). Zgodnie z (\[rownowaznosc strukt i zmodyfikowanego strukt\]), postacie te s¹ równowa¿ne na poziomie oczekiwanym.\ Przy samospójnym rozwi¹zywaniu strukturalnej i wariacyjnej zasady informacyjnej metody EFI, wykorzystywana jest postaæ obserwowana zasady strukturalnej (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), wynikaj¹ca z ¿¹dania analitycznoœci logarytmu funkcji wiarygodnoœci oraz metrycznoœci przestrzeni statystycznej ${\cal S}$ (por. Rozdzia³ \[structural principle\]). Natomiast oczekiwana strukturalna zasada informacyjna (\[expected form of information eq\]) jest narzêdziem pomocniczym w definicji ca³kowitej fizycznej informacji $K$, (\[physical K\]), oraz informacyjnej zasady wariacyjnej (\[var K rozpisana\]).\ \ Poni¿ej przekonamy siê, ¿e wszystkie modelowe ró¿nice le¿¹ po stronie $\texttt{q\!F}_{n}$, której postaæ zale¿y od konkretnego fizycznego scenariusza, w³¹czaj¹c w to symetrie oraz warunki brzegowe.\ Po raz pierwszy równania (\[eq zero\]) oraz (\[EL eq\]) otrzymali Frieden i Soffer [@Frieden]. Jednak powy¿sza ich forma uwzglêdnia inn¹ interpretacjê $Q$ (jako obecnej stale podczas ewolucji uk³adu) oraz faktoryzacjê probabilistycznego czynnika $q_{n}^{2}({\bf x})$ zawartego obligatoryjnie w mierze ca³kowej[^59]. TFI oraz specyficzne formy $Q$ w teorii pola {#structural inf} -------------------------------------------- Poni¿ej zebrano i rozwiniêto wyniki EFI otrzymane poprzednio w [@Frieden]. Jednak¿e zapisano je, szczególnie dla $Q$, w otwartej formie z punktu widzenia analizy porównawczej modeli [@Dziekuje; @za; @models; @building]. Metoda EFI prowadzi do takiego sformu³owania metody teorii pola, która jest zgodna z dzisiejszym opisem mechaniki falowej dla szerokiej klasy struktur.\ Jednak¿e, czasami metoda EFI wraz z ca³ym towarzysz¹cym jej statystycznym aparatem pojêciowym mo¿e doprowadziæ do korekty istniej¹cej analizy. Z sytuacj¹ tak¹ mo¿emy mieæ do czynienia np. w przypadku sformu³owania zasady nieoznaczonoœci Heisenberga dla pola [*œwietlnego*]{}. Otó¿ w œwietle nowych eksperymentów, w wyniku których otrzymano [*za w¹ski impuls œwietlny w czêstotliwoœci*]{} [@Roychoudhuri], standardowa Fourierowska podstawa zasady nieoznaczonoœci jest ostatnio kwestionowana. Wyt³umaczenie istoty nierównoœci Heisenberga w oparciu o nierównoœæ Rao-Cramera wraz z rozró¿nieniem pomiêdzy estymacj¹ parametru w przypadku skalarnym [@Frieden] (por. Dodatek \[Zasada nieoznaczonosci Heisenberga\]) i wektorowym, dla którego zachodzi ci¹g nierównoœci (\[porownanie sigma I11 z I11do-1\]), mo¿e okazaæ siê kluczem do zrozumienia pytañ, narastaj¹cych na skutek nowych eksperymentów ze œwiat³em.\ \ W metodzie EFI, informacja strukturalna $Q$ musi zostaæ wyprowadzona z u¿yciem zasady strukturalnej, wariacyjnej i czasami pewnych dodatkowych warunków symetrii, bior¹cych pod uwagê specyficzny fizyczny scenariusz teorii. Szeroki opis metod stosowanych przy rozwi¹zywaniu równañ (\[EL eq\]) oraz (\[eq zero\]) mo¿na znaleŸæ w [@Frieden].\ Jednak poni¿sze rozwa¿ania powinny okazaæ siê pomocne w zrozumieniu metody, szczególnie dla uk³adu zasad informacyjnych (\[strukt row dla psi\]) oraz (\[EL eq dla psi\]) dla pola $\psi$. Poni¿ej zostanie pokazane jak mo¿liwe rozwi¹zania zasad informacyjnych, strukturalnej i wariacyjnej, przewiduje pojawienia siê trzech typów pól: $N$-skalarów [@Dziekuje; @za; @models; @building], fermionów oraz bozonów [@Frieden].\ ### Informacja Fouriera {#Informacja Fouriera} Rozwa¿my cz¹stkê jako uk³ad opisany polem rangi $N$ poprzez zbiór zespolonych funkcji falowych $\psi_{n}({\bf x})$, $n=1,2,...,N/2$, okreœlonych w czasoprzestrzeni ${\cal X}$ po³o¿eñ ${\bf x} \equiv ({\bf x}^{\mu})_{\mu=0}^{\, 3} = (c t,\, {\bf x}^{1},{\bf x}^{2},{\bf x}^{3})$, zgodnie z konstrukcj¹ (\[amplitudapsi\]) oraz (\[x n rownowaznosc\]) z Rozdzia³u \[The kinematical form of the Fisher information\]. Ich transformaty Fouriera $\phi_{n}({\bf p})$ w sprzê¿onej do przestrzeni przesuniêæ ${\cal X}$, energetyczno-pêdowej przestrzeni ${\cal P}$ czteropêdów ${\bf p} \equiv (\wp^{\mu})_{\mu=0}^{\,3} = (\frac{E}{c},\, \wp^{1},\wp^{2},\wp^{3})$, maj¹ postaæ: $$\begin{aligned} \label{Fourier transf} \phi_{n}({\bf p}) = \frac{1}{(2\pi\hbar)^{2}} \int_{\cal X} d^{4} {\bf x} \; \psi_{n}({\bf x}) \, e^{i\,(\,\sum_{\nu=0}^{3}{\bf x}^{\nu} \wp_{\nu})/\hbar} \; ,\end{aligned}$$ gdzie $\sum_{\nu=0}^{3} {\bf x}^{\nu} \wp_{\nu} = E t - \sum_{l=1}^{3} {\bf x}^{l} \wp^{l}$, a $\hbar$ jest sta³¹ Plancka.\ \ [**Uwaga**]{}: Transformacja Fouriera jest [*unitarn¹ transformacj¹ zachowuj¹c¹ miarê*]{} na przestrzeni $L^{2}$ funkcji ca³kowalnych z kwadratem, tzn.: $$\begin{aligned} \label{miara zachowana} \int_{\cal X} d^{4}{\bf x}\,\psi_{n}^{*}({\bf x})\,\psi_{m}({\bf x}) = \int_{\cal P} d^{4}{\bf p}\,\phi_{n}^{*}({\bf p})\,\phi_{m}({\bf p}) \; ,\end{aligned}$$ zatem wykorzystuj¹c warunek normalizacji prawdopodobieñstwa (\[prawdpsi\]) otrzymujemy[^60]: $$\begin{aligned} \label{norm condition} \sum_{n=1}^{N/2} \int_{\cal X} d^{4}{\bf x} \,|\psi_{n}({\bf x})|^{2} = \sum_{n=1}^{N/2}\int_{\cal P} d^{4} {\bf p}\,|\phi_{n}({\bf p})|^{2} = 1 \; ,\end{aligned}$$ gdzie $|\psi_{n}({\bf x})|^{2} \equiv \psi_{n}^{*}({\bf x})\,\psi_{n}({\bf x})$ oraz $|\phi_{n}({\bf p})|^{2} \equiv \phi_{n}^{*}({\bf p})\,\phi_{n}({\bf p})$. Korzystaj¹c z (\[Fourier transf\]) mo¿emy zapisaæ $I$ podane wzorem (\[inf F z psi\]) w nastêpuj¹cy sposób: $$\begin{aligned} \label{I by Fourier tr} I\left[\psi({\bf x})\right] = I\left[\phi({\bf p})\right] = \frac{4N}{\hbar^{2}} \int_{\cal P} d^{4} {\bf p} \sum_{n=1}^{N/2} |\phi_{n}({\bf p})|^{2}\,(\frac{E^{2}}{c^{2}} - \vec{\wp}^{\,2}\,) \, ,\end{aligned}$$ gdzie $\vec{\wp}^{\,2} = \sum_{k=1}^{3}\wp_{k} \wp^{k}\,$.\ \ [**Okreœlenie kwadratu masy cz¹stki**]{}: Poniewa¿ $I$ jest z definicji sum¹ po wartoœciach oczekiwanych (por. (\[krzy4\]) i (\[pojemnosc C\])), dlatego kwadrat [*masy*]{} cz¹stki zdefiniowany jako [@Frieden]: $$\begin{aligned} \label{m E p} m^{2} := \frac{1}{c^{2}} \int_{\cal P} d^{4} {\bf p} \sum_{n=1}^{N/2} |\phi_{n}({\bf p})|^{2} \,(\frac{E^{2}}{c^{2}}-\vec{\wp}^{\,2}\,) \;\end{aligned}$$ jest sta³¹ niezale¿nie od statystycznych fluktuacji energii $E$ oraz pêdu $\vec{\wp}$, tzn. przynajmniej wtedy, gdy ca³kowanie jest wykonane (czyli jako œrednia). Tak wiêc, dla cz¹stki swobodnej mo¿emy (\[I by Fourier tr\]) zapisaæ nastêpuj¹co: $$\begin{aligned} \label{I by Fourier to m} I\left[\psi({\bf x})\right] = I\left[\phi({\bf p})\right] = 4N(\frac{m\, c}{\hbar})^{2} = const. \;\; .\end{aligned}$$ [**Informacja Fouriera z $\psi$**]{}: Powy¿szy warunek oznacza, ¿e: $$\begin{aligned} \label{free field eq all 1} K_{F} \equiv I\left[\psi({\bf x}^{\mu})\right] - I\left[\phi(p^{\mu})\right]=0 \; , \end{aligned}$$ co korzystaj¹c ze sta³oœci $4N(\frac{m\, c}{\hbar})^{2}$ oraz (\[norm condition\]) mo¿na zapisaæ jako warunek spe³niony przez pole swobodne rangi $N$: $$\begin{aligned} \label{free field eq all 2} K_{F} = 4 \, N \int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2} \left[\;\sum_{\nu=0}^{3}\left( \frac{\partial\psi_{n}}{\partial {\bf x}_{\nu}} \right)^{\!\!*}\; \frac{\partial\psi_{n}}{\partial {\bf x}^{\nu}}-(\frac{m\, c}{\hbar})^{2}\psi_{n}^{*}\,\psi_{n}\right] = 0 \; .\end{aligned}$$ Wielkoœæ $K_{F}$ definiuje tzw. [*informacjê Fouriera*]{} ($F$), a $k_{F}$ jej gêstoœæ: $$\begin{aligned} \label{gestosc kF inf Fouriera} k_{F} = 4 \, N \sum_{n=1}^{N/2} \left[\;\sum_{\nu=0}^{3}\left( \frac{\partial\psi_{n}}{\partial {\bf x}_{\nu}} \right)^{\!\!*}\; \frac{\partial\psi_{n}}{\partial {\bf x}^{\nu}}-(\frac{m\, c}{\hbar})^{2}\psi_{n}^{*}\,\psi_{n}\right] \; .\end{aligned}$$\ Pomijaj¹c fakt, ¿e z powy¿szych rachunków $m^2$ wy³ania siê jako œrednia, równanie (\[free field eq all 1\]), a zatem (\[free field eq all 2\]), jest odbiciem twierdzenia Parseval’s (\[tw Parsevala\]) i jako takie jest ono zdaniem tautologicznym. Fakt ten oznacza, ¿e transformacja Fouriera odzwierciedla jedynie zmianê bazy w przestrzeni statystycznej ${\cal S}$. Dlatego te¿, sam z siebie, warunek (\[free field eq all 2\]) nie nak³ada ¿adnego dodatkowego wiêzu na uk³ad, [*chyba, ¿e $4N(\frac{m\, c}{\hbar})^{2}$ jest zadana jako informacja strukturala $Q$ uk³adu*]{}. Tylko wtedy (\[free field eq all 2\]) staje siê informacyjn¹ zasad¹ strukturaln¹ dla uk³adu definuj¹cego szczególny typ pola omawianego w Rozdziale \[Klein-Gordon scalars\]. #### Informacja Fouriera dla amplitudy rzeczywistej {#Informacja Fouriera dla amplitudy rzeczywistej} Rozwa¿my z kolei cz¹stkê, jako uk³ad opisany polem rangi $N$ okreœlonym poprzez zbiór rzeczywistych amplitud $q_{n}({\bf x})$, $n=1,2,...,N$, na czasoprzestrzeni przesuniêæ ${\bf x} \equiv ({\bf x}^{\mu})_{\mu=0}^{\, 3} \in {\cal X}$ i posiadaj¹c¹ pojemnoœæ kana³u informacyjnego jak w (\[Fisher\_information-kinetic form bez n\]). Zespolone transformaty Fouriera $\tilde{q}_{n}({\bf p})$ rzeczywistych funkcji $q_{n}({\bf x})$, gdzie ${\bf p} \equiv (\wp^{\mu})_{\mu=0}^{\,3} \in {\cal P}$ jest czteropêdem, maj¹ postaæ: $$\begin{aligned} \label{Fourier transf dla q} \tilde{q}_{n}({\bf p}) = \frac{1}{(2\pi\hbar)^{2}} \int_{\cal X} d^{4} {\bf x} \; q_{n}({\bf x}) \, e^{i\,(\,\sum_{\nu=0}^{3} {\bf x}^{\nu} \wp_{\nu})/\hbar} \; .\end{aligned}$$ Podobnie jak w (\[miara zachowana\]), transformacja Fouriera spe³nia zwi¹zek: $$\begin{aligned} \label{miara zachowana dla q} \int_{\cal X} d^{4}{\bf x}\,q_{n}^{*}({\bf x})\,q_{m}({\bf x}) = \int_{\cal P} d^{4}{\bf p}\,\tilde{q}_{n}^{\,*}({\bf p})\,\tilde{q}_{m}({\bf p}) \; .\end{aligned}$$ Wykorzystuj¹c warunek unormowania prawdopodobieñstwa: $$\begin{aligned} \label{norm condition dla q} \frac{1}{N}\sum_{n=1}^{N} \int d^{4}{\bf x} \, q_{n}^{2}({\bf x}) = 1 \; , \end{aligned}$$ otrzymujemy, jako konsekwencjê twierdzenia Parseval’a: $$\begin{aligned} \label{norm condition dla q i tw Parseval} \frac{1}{N}\sum_{n=1}^{N} \int_{\cal X} d^{4}{\bf x} \, q_{n}^{2}({\bf x}) = \frac{1}{N} \sum_{n=1}^{N} \int_{\cal P} d^{4} {\bf p}\,| \tilde{q}_{n}({\bf p})|^{2} = 1 \; ,\end{aligned}$$ gdzie $|q_{n}({\bf x})|^{2} \equiv q_{n}^{2}({\bf x})$ i $| \tilde{q}_{n}({\bf p})|^{2} \equiv \tilde{q}_{n}^{*}({\bf p})\,\tilde{q}_{n}({\bf p})$.\ \ [**Informacja Fouriera dla $q$**]{}: Podobne rachunki jak wykonane poprzednio dla zespolonego pola rangi $N$, prowadz¹ w przypadku pola okreœlonego poprzez zbiór $N$ rzeczywistych amplitud $q_{n}({\bf x})$ i dla pojemnoœci informacyjnej kana³u $I$, (\[Fisher\_information-kinetic form bez n\]), do nastêpuj¹cej jego postaci: $$\begin{aligned} \label{I by Fourier tr dla q} I\left[q({\bf x})\right] = I\left[\tilde{q}({\bf p})\right] = \frac{4}{\hbar^{2}} \int_{\cal P} d^{4} {\bf p} \sum_{n=1}^{N} |\tilde{q}_{n}({\bf p})|^{2}\,(\frac{E^{2}}{c^{2}} - \vec{\wp}^{\,2}\,) \, ,\end{aligned}$$ gdzie $\vec{\wp}^{\,2} = \sum_{k=1}^{3}\wp_{k} \wp^{k}\,$.\ \ [**Okreœlenie kwadratu masy cz¹stki**]{}: Zatem podobnie jak dla pola zespolonego, równie¿ dla pola rzeczywistego, kwadrat masy cz¹stki zdefiniowany nastêpuj¹co: $$\begin{aligned} \label{m E p dla q} m^{2} := \frac{1}{N\, c^{2}} \int_{\cal P} d^{4} {\bf p} \sum_{n=1}^{N} |\tilde{q}_{n}({\bf p})|^{2} \,(\frac{E^{2}}{c^{2}}-\vec{\wp}^{\,2}\,) \;\end{aligned}$$ jest sta³¹, która nie zale¿y od statystycznych fluktuacji energii $E$ oraz pêdu $\vec{\wp}$. Tak wiêc i dla cz¹stki opisanej polem rzeczywistym rangi $N$, mo¿emy (porównaj (\[I by Fourier to m\])) zapisaæ (\[I by Fourier tr dla q\]) w postaci: $$\begin{aligned} \label{I by Fourier to m dla q} I\left[q({\bf x})\right] = I\left[\tilde{q}({\bf p})\right] = 4 N \, (\frac{m\, c}{\hbar})^{2} = const. \;\; ,\end{aligned}$$ co oznacza, ¿e warunek (\[informacja Stama vs pojemnosc informacyjna\]), $I \geq 0$, poci¹ga za sob¹ w zgodzie z (\[I by Fourier to m dla q\]) warunek $m^{2} \geq 0$, mówi¹cy o nieobecnoœæ tachionów w teorii [@dziekuje; @za; @neutron], a wynikaj¹cy z jej przyczynowoœci, zgodnie z (\[przyczynowosc\]) oraz (\[informacja Stama vs pojemnosc informacyjna Minkowskiego\]).\ Zauwa¿my, ¿e w zgodzie z (\[m E p dla q\]), zerowanie siê masy cz¹stki by³oby niemo¿liwe dla czasoprzestrzeni z metryk¹ Euklidesow¹ (\[metryka E\]).\ \ [**Informacja Fouriera dla pola typu $q$**]{}: Zwi¹zek (\[I by Fourier to m dla q\]) pozwala na zapisanie informacji Fouriera, w przypadku rzeczywistego swobodnego pola rangi $N$, w nastêpuj¹cej postaci: $$\begin{aligned} \label{free field eq all 2 real amplitudes} K_{F} = 4\int_{\cal X} d^{4}{\bf x}\sum_{n=1}^{N} \left[\;\sum_{\nu=0}^{3}\frac{\partial q_{n}}{\partial {\bf x}_{\nu}} \frac{\partial q_{n}}{\partial {\bf x}^{\nu}} - \,(\frac{m\, c}{\hbar})^{2}q_{n}^{\,2}\right]=0 \; .\end{aligned}$$ Wnioski p³yn¹ce z zastosowania powy¿szej postaci informacji Fouriera w przypadku pola bezmasowego, mo¿na znaleŸæ w Dodatku \[Maxwell field\]. ### Skalary Kleina-Gordona {#Klein-Gordon scalars} Rozwa¿my pole skalarne, którego TFI oznaczmy jako $K_{S}$. Równanie ruchu Kleina-Gordona dla swobodnego pola skalarnego rangi $N$ wynika z wariacyjnej zasady informacyjnej (\[var K rozpisana\]): $$\begin{aligned} \label{wariacyjna zas dla pola skalarnego} \delta_{(\psi^{*})}K_{S} = 0 \; , $$ gdzie na $k=k_{S}$, okreœlone ogólnie zwi¹zkiem (\[k form dla psi\]), na³o¿ony jest dodatkowy warunek, wynikaj¹cy z nastêpuj¹cej postaci informacji strukturalnej $Q$: $$\begin{aligned} \label{Q for N scalar} Q \left[\phi({\bf p})\right] = Q_{S} = \int_{\cal X} d^{4}{\bf x} \, \textit{q}_{S} \equiv - \, 4\, N(\frac{m\, c}{\hbar})^{2} \; .\end{aligned}$$ Dla swobodnego skalarnego pola Kleina-Gordona, TFI jest równa jego informacji Fouriera (\[free field eq all 2\]), tzn. $K=K_{S}=K_{F}$. Powy¿ej $\textit{q}_{S}$ jest gêstoœci¹ informacji strukturalnej dla pola skalarnego: $$\begin{aligned} \label{gestosc inf strukt dla pola skalarnego} \textit{q}_{S} = - 4 \, N (\frac{m\, c}{\hbar})^{2} \sum_{n=1}^{N/2} \psi_{n}^{*}\,\psi_{n} \; .\end{aligned}$$ Zatem z wariacyjnej zasady informacyjnej $\delta_{(\psi^{*})}K_{S} = 0$ wynika $N/2$ równañ Eulera-Lagrange’a (\[EL eq dla psi\]), które dla gêstoœci TFI równej $k = k_{S} = k_{F}$, (\[gestosc kF inf Fouriera\]), prowadz¹ do $N/2$ równañ Kleina-Gordona[^61]. Ich wyprowadzenie mo¿na znaleŸæ w [@Frieden; @Mroziakiewicz].\ Uzasadnienie faktu, ¿e pole $(\psi_{n})$ z gêstoœci¹ informacji strukturalnej $\textit{q}_{S}$ zadan¹ przez (\[gestosc inf strukt dla pola skalarnego\]) jest polem skalarnym, wymaga rozwa¿añ zwi¹zanych z badaniem reprezentacji transformacji izometrii pojemnoœci kana³u informacyjnego $I$, co odk³adamy do Rozdzia³u \[Dirac field\].\ \ [**Niezmienniczoœæ Fouriera zasady strukturalnej**]{}: Poniewa¿ dla (\[Q for N scalar\]) warunek (\[free field eq all 2\]) stanowi oczekiwan¹ strukturaln¹ zasadê informacyjn¹ $I + Q = 0$, (\[expected form of information eq\]), zatem dla swobodnego pola skalarnego transformacja Fouriera jest transformacj¹ unitarn¹, ze wzglêdu na któr¹ warunek (\[free field eq all 2\]) pozostaje niezmienniczy.\ \ [**Masa uk³adu a Fourierowskie spl¹tanie**]{}: Powy¿szy fakt oznacza, ¿e transformacja Fouriera tworzy rodzaj samospl¹tania pomiêdzy reprezentacj¹ po³o¿eniow¹ a pêdow¹ realizowanych wartoœci zmiennych uk³adu wystêpuj¹cych w $I$ [@Frieden], a wyprowadzenie oczekiwanej zasady strukturalnej $I+Q = 0$, (por. (\[expected form of information eq\])), jako konsekwencji analitycznoœci logarytmu funkcji wiarygodnoœci [@Dziekuje; @informacja_2], wyjaœnia je jako spl¹tanie pêdowych stopni swobody uk³adu spowodowane jego mas¹ (\[m E p\]).\ \ [**Uwaga**]{}: Podkreœlmy, ¿e w przypadku swobodnego pola skalarnego, oczekiwana zasada structuralna, $I + Q = 0$, jest jedynie odbiciem warunku (\[Q for N scalar\]). Jest on warunkiem brzegowym i nie jest on rozwi¹zywany samo-spójnie wraz z wariacyjn¹ zasad¹ informacyjn¹ (\[wariacyjna zas dla pola skalarnego\]).\ \ [**Typy pól skalarnych**]{}: Podajmy wynikaj¹ce z powy¿szej analizy metody EFI dwa typy pól skalarnych:\ \ [*Zwyk³e (na³adowane) pole skalarne*]{}: Pole skalarne maj¹ce rangê $N=2$ ma tylko jedn¹ sk³adow¹ zespolon¹, tzn. $\psi\equiv(\psi_{n})=\psi_{1}$ [@Frieden]. W przypadku tym informacja strukturalna (\[Q for N scalar\]) jest równa $Q_{S} = - 8(\frac{m\, c}{\hbar})^{2}$. Na³adowane pole skalarne Higgsa $H^{+}$ [@ASSzZZ] mog³oby teoretycznie byæ jego przyk³adem.\ \ [*$N$-skalary*]{}: W przypadku $n>1$ sk³adowe $\psi_{n}$ podlegaj¹ ewolucji opisanej przez $n=N/2$ nie sprzê¿onych równañ Kleina-Gordona z dwoma dodatkowymi wiêzami. Pierwszy z nich oznacza, ¿e wszystkie pola $\psi_{n}$ maj¹ tak¹ sam¹ masê $m$, a drugim jest warunek normalizacji (\[norm condition\]). Informacja strukturalna $Q$ takiego uk³adu jest okreœlona przez ogóln¹ postaæ (\[Q for N scalar\]) dla pola skalarnego rangi $N$. Owe skalarne pola Kleina-Gordona rangi $N$ nazwijmy [*$N$-skalarami*]{} [@Dziekuje; @za; @models; @building]. S¹ one teoretycznie realizowane w ramach tzw. $\sigma$-modeli teorii pola [@sigma-field; @theory; @Dziekuje; @za; @models; @building].\ \ W kolejnym rozdziale omówimy postaæ TFI oraz $Q$ dla równania Diraca. Podstawowe fakty metody EFI dla pól cechowania w elektrodynamice Maxwella [@Frieden] omówione s¹ w Dodatku \[Maxwell field\]. Równie¿ w Dodatku \[general relativity case\] zamieszczona jest postaæ $Q$ w teorii grawitacji [@Frieden]. W opracowaniu jest postaæ $Q$ dla pól nieabelowych [@Dziekuje; @za; @models; @building]. ### TFI równania Kleina-Gordona dla pól rangi $N$ {#TPI of the Klein-Gordon equation} Rozdzia³ ten poœwiêcony jest konstrukcji równania Diraca metod¹ EFI, z uwzglêdnieniem pól cechowania[^62]. Szczególn¹ uwagê zwrócono na problem kwadratury TFI pola Kleina-Gordona [@Frieden; @Dziekuje; @za; @models; @building]. #### Wstêpna foliacja ${\cal S}$ oraz pochodna kowariantna. Ogólny zarys problemu {#foliation of S} [**Wybór przestrzeni bazowej cz³onu kinetycznego**]{}: Wyjœciowa struktura modelu EFI opiera³a siê o analizê wartoœci oczekiwanej informacji fizycznej $K = I + Q$ na przestrzeni bazowej próby ${\cal B} = {\cal Y}_{1} \times {\cal Y}_{1} \times ... \times {\cal Y}_{N}$. Niezale¿ne zmienne losowe $Y_{n}$, $n=1,2,...,N\,$, przyjmowa³y wartoœci ${\bf y}_{n} \in {\cal Y}_{n}$. Po przejœciu od pe³nych danych $y \equiv ({\bf y}_{n})_{n=1}^{N} \in {\cal B}$ do niezale¿nych zmiennych losowych przesuniêæ $X_{n} = Y_{n} - \theta_{n}$, $n=1,2,...,N\,$, oraz uto¿samieniu zbiorów wartoœci tych zmiennych losowych, tzn. przyjêciu, ¿e ${\cal X}_{n=1} \equiv {\cal X}$, $n=1,2,...,N\,$, zosta³y skonstruowane kinematyczne postacie pojemnoœci informacyjnej (\[Fisher\_information-kinetic form bez n\]) oraz (\[inf F z psi\]). [*Zatem przestrzeni¹ bazow¹ cz³onów kinetycznych jest zbiór przesuniêæ*]{} ${\cal X}$.\ \ [**Estymacja na w³óknach**]{}: Zatem model EFI, z którego wy³oni siê model teorii pola jest budowany na [*przestrzeni bazowej*]{} ${\cal X}$, bêd¹cej w rozwa¿anych przez nas przypadkach czasoprzestrzeni¹ Minkowskiego ${\cal X} \equiv R^{4}$. Pojawi³a siê ona jako konsekwencja transformacji modelu statystycznego ${\cal S}$ z okreœlonego w przestrzeni parametru $V_{\Theta} \equiv R^{4}$ do przestrzeni przesuniêæ ${\cal X}$. Jednak¿e jednoczeœnie, tak przed jak i po jego przedefiniowaniu, model statystyczny pozostaje zdefiniowany ponad przestrzeni¹ bazow¹ ${\cal B}$, która jest oryginaln¹ przestrzeni¹ próby. Nastêpnie, w celu uczynienia kinematycznej postaci pojemnoœci informacyjnej $I$ niezmiennicz¹ ze wzglêdu na lokalne transformacje cechowania, musimy, poprzez zdefiniowanie pochodnej kowariantnej na przestrzeni bazowej ${\cal X}$, zapisaæ $I$ w postaci wspó³zmienniczej. Z kolei, zdefiniowanie tej pochodnej kowariantnej oznacza koniecznoœæ podania uk³adu wspó³rzêdnych na przestrzeni statystycznej ${\cal S}$, co wynika z tego, ¿e pola cechowania s¹ (jak siê oka¿e) amplitudami typu $q_{n}$ (por. Dodatek \[Maxwell field\]).\ Zatem wprowadzenie uk³adu wspó³rzêdnych na ${\cal S}$ nie jest zadaniem trywialnym [@Amari; @Nagaoka; @book]. Sytuacja ta wynika z koniecznoœci wykonania analizy EFI dla pól rangi $N$, które z góry przekszta³caj¹ siê zgodnie z transformacjami, których parametry mog¹ zale¿eæ lokalnie od po³o¿enia w przestrzeni bazowej ${\cal X}$. Zatem na ${\cal X}$ okreœlana jest [*strukturalna grupa symetrii*]{} $G$ wspomnianej powy¿ej transformacji, która w teorii pola jest grup¹ Liego pola cechowania. Konstruuje siê wiêc g³ówn¹ wi¹zkê w³óknist¹ $E({\cal X},G)\,$, a fizyczn¹ estymacjê przeprowadza siê na w³óknach[^63].\ Zauwa¿my, ¿e skoro kwadrat pola cechowania jest elementem ${\cal S}$, wiêc dla okreœlonej algebry grupy cechowania, wybór cechowania dokonuje czêœciowej foliacji[^64] przestrzeni ${\cal S}$ na pierwsze warstwy. Dopiero w tym momencie, zasady informacyjne umo¿liwiaj¹ dokonanie wyboru kolejnych foliacji przestrzeni ${\cal S}$ na warstwy zwi¹zane z wszystkimi szczególnymi reprezentacjami grupy $G$ [@Dziekuje; @za; @models; @building], których wymiar jest œciœle zwi¹zany z rang¹ pól $N$ [@Frieden].\ \ [**Sens powy¿szego rozwa¿ania**]{} ujmijmy tak: Ca³a procedura EFI musi nie tylko od pocz¹tku wybraæ typ amplitudy (tzn. $q_{n}$ lub $\psi_{n}$), okreœliæ zasady informacyjne i warunki brzegowe, ale musi byæ wykonana od pocz¹tku we w³aœciwym uk³adzie wspó³rzêdnych, który uwzglêdnia istnienie strukturalnej grupy symetrii $G$ oraz zwi¹zanych z ni¹ pól cechowania. Pole cechowania umo¿liwia bowiem wybór podprzestrzeni $T_{{\cal P}}H$ przestrzeni stycznej $T_{{\cal P}}E$ wi¹zki g³ównej $E\equiv E({\cal X},G)$ w ka¿dym jej punkcie ${\cal P}$, tak, ¿e $T_{{\cal P}}E=T_{{\cal P}}G\otimes T_{{\cal P}}H$, przy czym baza na $T_{{\cal P}}H$ mo¿e byæ zdefiniowana jako liniowa kombinacja czterowymiarowej bazy: $$\begin{aligned} \label{baza kowariantna} (\partial_{\mu}) \equiv \left(\frac{\partial}{\partial {\bf x}^{\mu}} \right) \equiv \left( \frac{\partial}{\partial (c\,t)} \,, \, \frac{\partial}{\partial {\bf x}^{1}}, \frac{\partial}{\partial {\bf x}^{2}}, \frac{\partial}{\partial {\bf x}^{3}} \right) \equiv \left( \frac{\partial}{\partial (c\,t)} \,, \, \vec{\nabla} \right) \end{aligned}$$ oraz generatorów $D_{G}$ infinitezymalnych transformacji grupy $G$.\ \ [**Przyk³ad**]{}: W przypadku pól cechowania $A_{\mu}$ grupy $G=U(1)$ pochodna kowariantna ma postaæ: $$\begin{aligned} \label{poch kowariantna na H} D_{\mu}\equiv(D_{0}, D_{l})=\partial_{\mu} - i \, \frac{e}{c\,\hbar} \, A_{\mu} \, ,\end{aligned}$$ gdzie $e$ jest ³adunkiem elektronu. Zatem, wprowadzaj¹c pochodn¹ kowariantn¹ do zasad informacyjnych i stosuj¹c metodê EFI, otrzymujemy równania ruchu, które rozwi¹zuj¹c daj¹ bazê na znalezionych (pod)rozmaitoœciach przestrzeni statystycznej ${\cal S}$.\ \ Jest wiele sposobów, na które wspó³zmiennicza postaæ pojemnoœci informacyjnej $I$ mo¿e byæ odczytana. Poni¿ej skoncentrujemy siê na dwóch z nich, jednej dla pól skalarnych rangi $N$ (tzn. $N$-skalarów) oraz drugiej, dla pól fermionowych rangi $N$. #### Równanie ruchu Diraca dla pola swobodnego rangi $N$. {#Dirac field} Zgodnie z powy¿szymi rozwa¿aniami, wspó³zmiennicza forma (\[inf F z psi\]) pojemnoœci informacyjnej z pochodn¹ kowariantn¹ $D_{\mu}$ ma postaæ: $$\begin{aligned} \label{I every psi N field} I = 4 \, N\int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2} \sum_{\mu=0}^{3} (D_{\mu}\psi_{n})^{*}D^{\mu}\psi_{n} \; .\end{aligned}$$ Zatem jedyna TFI (\[physical K\]) dostêpna w metodzie EFI dla równania Kleina-Gordona i ka¿dego pola typu $\psi$ rangi $N$ (skalarnego czy fermionowego) ma postaæ: $$\begin{aligned} \label{TPI every field} K = K_{KG} = \int_{\cal X} d^{4}{\bf x} \; k_{KG} \equiv 4 \, N\int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2} \sum_{\mu=0}^{3} (D_{\mu}\psi_{n})^{*}D^{\mu}\psi_{n} + Q \; ,\end{aligned}$$ gdzie $k_{KG}$ jest gêstoœci¹ TFI dla równania Kleina-Gordona (por. (\[k form dla psi\])).\ \ Podobnie, wykorzystuj¹c (\[Fisher\_information-kinetic form bez n\]) w miejsce (\[inf F z psi\]) moglibyœmy zapisaæ $K_{KG}$ dla pola bosonowego, do czego powrócimy jednak póŸniej.\ S¹ dwie drogi, którymi analiza oparta o TFI zadan¹ przez (\[TPI every field\]) mo¿e pod¹¿aæ [@Frieden; @Dziekuje; @za; @models; @building]. Pierwsza zwi¹zana jest z $N$-skalarami a druga z polami Diraca. #### TFI Kleina-Gordona {#tfi-kleina-gordona .unnumbered} Wtedy, gdy rozwa¿amy pole $N$-skalara, to jak wiemy $Q$ jest równe $Q_{S}$ zadanemu przez (\[Q for N scalar\]). Fakt ten oznacza, ¿e TFI dla równania ruchu $N$-skalara, sprowadza siê do postaci Kleina-Gordona, tzn.: $$\begin{aligned} \label{TPI for N-scalar} K = K_{S} = K_{KG} \;\;\;\;{\rm dla}\;\;\;\; Q = Q_{S} \; , \end{aligned}$$ a wariacyjna zasada informacyjna: $$\begin{aligned} \label{var IP for N-scalar} \delta_{(\psi^{*})}K_{S} \equiv \delta_{(\psi^{*})}(I + Q_{S}) = 0 \; ,\end{aligned}$$ zadana przez (\[var K\]), a w konsekwencji przez (\[EL eq dla psi\]), prowadzi do równania Kleina-Gordona dla pola skalarnego [@Frieden; @Mroziakiewicz] (patrz (\[row KL dla swobodnego\])). #### TFI Diraca {#tfi-diraca .unnumbered} Jak wiemy, dla pola Diraca, równanie Kleina-Gordona jest otrzymane drog¹ kwadratury równania Diraca [@Fecko-fizyka; @matematyczna]. Zatem, mog³oby siê wydawaæ, ¿e metoda informacyjna nie wybiera sama z siebie w³aœciwej postaci TFI dla pola fermionowego. Sytuacja ma siê jednak zgo³a inaczej. Przedstawimy j¹ poni¿ej [@Dziekuje; @za; @models; @building].\ \ [**Transformacje izometrii $I$**]{}: Zapiszmy (\[I every psi N field\]) pól $\psi$, zarówno dla $N$-skalara jak i pola fermionowego rangi $N$, w nastêpuj¹cej postaci: $$\begin{aligned} \label{I with g} I = 4 N \sum_{n=1}^{N/2} \int_{\cal X} d^{4}{\bf x} \,\sum_{\mu,\,\nu=0}^{3} (D_{\mu}\,\psi_{n}^{*}({\bf x}))\, \eta^{\mu\nu}\,(D_{\nu}\,\psi_{n}({\bf x})) \; ,\end{aligned}$$ która to postaæ pozwala zauwa¿yæ, ¿e jest ona niezmiennicza ze wzglêdu na transformacje izometrii dzia³aj¹ce w $N/2$ - wymiarowej, zespolonej przestrzeni $\mathcal{C}^{N/2}$ pól $\psi\equiv(\psi_{n}({\bf x}))$ rangi $N$.\ \ [**Dwa typy izometrii $I$**]{}:\ [**Przypadek skalarów**]{}: W przypadku $N$-skalarów $\psi$ tworz¹cych podprzestrzeñ w $\mathcal{C}^{N/2}$ izometrie te s¹ transformacjami identycznoœciowymi, pozostawiaj¹c pola $\psi$ niezmienionymi.\ [**Przypadek pól Diraca**]{}: Natomiast dla pól fermionowych rangi $N$ okreœlonych na przestrzeni bazowej ${\cal X}$ Minkowskiego, transformacje izometrii tworz¹ grupê Clifforda $Pin(1,3)$, bêd¹c¹ podzbiorem algebry Clifforda $C(1,3)$). Elementy grupy Clifforda $Pin(1,3)$ dzia³aj¹ w podprzetrzeni $\mathcal{C}^{N/2}$ spinorów $\psi$.\ \ [**Macierze Diraca**]{}: Okazuje siê, ¿e macierze Diraca $\gamma^{\mu}$, $\mu=0,1,2,3\,$, (por. (\[kowariantna postac r Diraca\])-(\[m Pauliego\])), tworz¹ spinorow¹ reprezentacjê ortogonalnej bazy w $C(1,3)$ i spe³niaj¹ to¿samoœci: $$\begin{aligned} \label{antykomutator dla m Diraca} \eta^{\mu\nu} = 1/2 \, \{\gamma^{\mu},\gamma^{\nu}\} \; , \;\;\; \mu, \nu = 0,1,2,3 \; ,\end{aligned}$$ które s¹ podstawowym zwi¹zkiem dla iloczynu Clifforda. W przypadku spinorów rangi $N=8$, baza w $Pin(1,3)$ jest $2^{N/2}=16$ wymiarowa [@Fecko-fizyka; @matematyczna].\ \ [**Faktoryzacja $K_{KG}$**]{}: Dla trywialnego przypadku $N$-skalarów, postaæ (\[TPI every field\]) jest form¹ podstawow¹. Jednak w przypadku spinorów rangi $N$, po skorzystaniu z (\[antykomutator dla m Diraca\]) w (\[I with g\]), mo¿na dokonaæ rozk³adu fizycznej informacji $K_{KG}$, (\[TPI every field\]), na sk³adowe. Postaæ jawnego rozk³adu $K_{KG}$ na sk³adowe i ich faktoryzacjê, gdzie ka¿dy z otrzymanych czynników jest elementem grupy Clifforda $Pin(1,3)$, poda³ Frieden [@Frieden].\ \ G³ówny rezultat rozk³adu fizycznej informacji $K_{KG}$, z uwzglêdnieniem pól cechowania w pochodnej kowariantnej $D_{\mu}$, który czyni zadoœæ “ograniczeniu” Kleina-Gordona (co oznacza, ¿e wymno¿enie czynników i dodanie ich da³oby na powrót (\[TPI every field\])), ma postaæ: $$\begin{aligned} \label{TPI Dirac field} K = K_{D}\equiv4\, N\int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2}\sum_{\mu=0}^{3}(D_{\mu}\psi_{n})^{*}D^{\mu}\psi_{n} + Q \, , \end{aligned}$$ gdzie informacja strukturalna $Q$ jest równa: $$\begin{aligned} \label{Q in Dirac} \!\!\!\!\!\!\!\!\! Q = Q_{D} = \int_{\cal X} d^{4}{\bf x} \, \textit{q}_{D} \equiv -\,4\, N\int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2}\left[v_{1n}\, v_{2n} + (\frac{m\, c}{\hbar})^{2}\psi_{n}^{*}\,\psi_{n}\right] + (pozosta{\ell}e\; cz{\ell}ony) \; , \;\;\;\;\end{aligned}$$ gdzie $\textit{q}_{D}$ jest zgodnie z (\[gestosc q dla niezaleznych Yn\]) gêstoœci¹ informacji strukturalnej, a $\frac{N}{2}=4$ wymiarowe wektory kolumnowe $v_{i} = (v_{i1},v_{i2},v_{i3},v_{i4})^{T}$, $i=1,2$, maj¹ sk³adowe: $$\begin{aligned} v_{1n} = \sum_{n'=1}^{4} \left(i\, {\mathbf 1} \, D_{0}-\beta\frac{m\, c}{\hbar} + \sum_{l=1}^{3} i\, \alpha^{l}\, D_{l}\right)_{n n'} \!\! \psi_{n'}\;\label{free field eq 1} \; , \;\;\;\;\;\;\;\; n=1,2,3,4 \; \end{aligned}$$ oraz $$\begin{aligned} \label{free field eq 2} v_{2n} = \sum_{n'=1}^{4} \left(-i\, {\mathbf 1}\, D_{0}+\beta^{*}\frac{m\, c}{\hbar}+\sum_{l=1}^{3} i\, \alpha^{l *}\, D_{l}\right)_{n n'} \!\! \psi_{n'}^{*} \; , \;\;\;\;\; n=1,2,3,4 \; ,\end{aligned}$$ gdzie macierze $\alpha^{l}$, $l=1,2,3$, oraz $\beta$ s¹ macierzami Diraca (\[m Diraca alfa beta\])[^65], a ${\mathbf 1}$ jest $4\times4$ - wymiarow¹ macierz¹ jednostkow¹.\ \ [**Zasady informacyjne dla równania Diraca**]{}: W koñcu, informacyjna zasada strukturalna (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) dla $\kappa=1$ i gêstoœci informacji strukturalnej $\textit{q}_{D}$ okreœlonej zgodnie z (\[Q in Dirac\]) oraz zasada wariacyjna (\[var K rozpisana\]) maj¹ postaæ: $$\begin{aligned} \label{IPs for Dirac} \widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \textit{q}_{D} = 0 \; \;\; {\rm dla} \;\;\; \kappa=1 \;\; \;\;\;\;\; {\rm oraz} \;\;\;\;\;\;\; \delta_{(\psi^{*})} K_{D} = \delta_{(\psi^{*})} (I + Q_{D}) = 0 \; . \;\;\;\end{aligned}$$ Powy¿sze zasady daj¹ na poziomie obserwowanym warunki (\[strukt row dla psi\]) oraz (\[EL eq dla psi\]) metody EFI, czyli uk³ad dwóch równañ ró¿niczkowych, które s¹ rozwi¹zywane samospójnie. W wyniku otrzymujemy równanie Diraca: $$\begin{aligned} \label{Dirac eq} v_{1} = \left(i\, D_{0}-\beta\frac{m\, c}{\hbar} + \sum_{l=1}^{3}\alpha^{l}\, i\, D_{l}\right)\,\psi\; = 0 \; , \;\;\; {\rm gdzie} \;\;\; \psi = \left({\begin{array}{c} \psi_{1} \\ \psi_{2} \\ \psi_{3} \\ \psi_{4} \end{array}}\right) \; .\end{aligned}$$ [**Generacja masy**]{}: Zwróæmy uwagê, ¿e w trakcie powy¿szego rozk³adu $K_{KG}$ z cz³onu kinetycznego $I$, (\[I with g\]), generowane s¹ wszystkie sk³adniki informacji strukturalnej $Q$, (\[Q in Dirac\]). W trakcie tej procedury cz³on masowy generowany jest poprzez sprzê¿enie Fourierowskie (\[Fourier transf\]) pomiêdzy reprezentacj¹ po³o¿eniow¹ i pêdow¹ oraz uœrednienie dokonane w reprezentacji energetyczno-pêdowej, tak jak to mia³o miejsce dla (\[m E p\]). Zatem równie¿ masa pola Diracowskiego jest przejawem istnienia Fourierowskiego samospl¹tania pomiêdzy reprezentacj¹ po³o¿eniow¹ a pêdow¹, a ca³a procedura jest odbiciem:\ [**i)**]{} za³o¿enia analitycznoœci logarytmu funkcji wiarygodnoœci,\ [**ii)**]{} podzia³u na czêœæ Fisherowsk¹ $I$ oraz (pocz¹tkowo nieznan¹) czêœæ strukturaln¹ $Q$ z wysumowaniem po kana³ach informacyjnych i uœrednieniem po przestrzeni próby ${\cal B}$ opisanym powy¿ej (\[expected form of information eq\]),\ [**iii)**]{} przejœcia z pojemnoœci¹ informacyjn¹ $I$ do Friedenowskiej postaci kinematycznej z ca³kowaniem po zakresie przesuniêæ ${\cal X}$ amplitud rozk³adu uk³adu, opisanym w Rozdziale \[The kinematical form of the Fisher information\],\ [**iv)**]{} tautologicznego wygenerowania, zgodnie z zasad¹ Macha, informacji strukturalnej $Q$ z informacji Fishera $I$, z uwzglêdnieniem niezmienniczoœci $I$ ze wzglêdu na transformacjê amplitud bêd¹c¹ jej izometri¹ oraz transformacjê Fouriera pomiêdzy reprezentacj¹ po³o¿eniow¹ i pêdow¹.\ \ Przedstawiony schemat generacji masy nie obejmuje wyznaczenia jej wartoœci. Mówi on raczej o tym czego jej pojawienie siê jest wyrazem.\ \ [**Warunek zerowania siê $pozosta{\ell}ych \; cz\ell on\acute{o}w$**]{}: W równaniach (\[free field eq 1\]) oraz (\[free field eq 2\]) macierz jednostkowa ${\mathbf 1}$, która stoi przy $D_{0}$ oraz $\frac{N}{2}\times \frac{N}{2} = 4\times4$ wymiarowe macierze Diraca $\beta$, $\alpha_{l}$, s¹ jednymi z elementów grupy Clifforda $Pin(1,3)$. Jak wspomnieliœmy, Frieden [@Frieden] przeprowadzi³ opisan¹ decompozycjê $K_{KG}$, (\[TPI every field\]), do postaci $K_{D}$ podanej w (\[TPI Dirac field\]). Równoczeœnie pokaza³, ¿e wyra¿enie oznaczone w (\[Q in Dirac\]) jako “$pozosta{\ell}e \; cz\ell ony$” zeruje siê przy za³o¿eniu, ¿e [*macierze $\beta$ i $\alpha_{l}$ spe³niaj¹ relacje algebry Clifforda*]{}.\ \ [**Podsumowanie**]{}: Zauwa¿yliœmy, ¿e pojemnoœæ informacyjna $I$ w TFI jest zadana przez (\[I with g\]) zarówno dla pól $N$-skalarów jak i pól fermionowych rangi $N$. Kluczow¹ spraw¹ jest, ¿e dla ka¿dego równania ruchu, okreœlona jest tylko jemu charakterystyczna postaæ $\textit{q}$. Zauwa¿yliœmy, ¿e dla $N$-skalarów, jedynym cz³onem, który tworzy ca³kowit¹ fizyczn¹ informacyjnê jest $K_{KG}$ w postaci (\[TPI every field\]) bez ¿adnego ukrytego rozk³adu i faktoryzacji, pochodz¹cych z nie-Fourierowskiego spl¹tania. W przypadku równania Diraca, informacja fizyczna Kleina-Gordona $K_{KG}$, (\[TPI every field\]), równie¿ wchodzi w ca³kowit¹ informacjê fizyczn¹ Diraca $K_{D}$, ale tylko jako jej szczególna czêœæ (jak to mo¿na równie¿ zauwa¿yæ z porównania (\[TPI Dirac field\]), (\[Q in Dirac\]) oraz (\[TPI every field\]) i (\[Q for N scalar\])), spl¹tana Fourierowsko w $K_{F}$ (\[free field eq all 2\]). Fakt ten jest blisko zwi¹zany z efektem EPR-Bohm’a [@Khrennikov] opisanym w Rozdziale \[Pojemnosc informacyjna zagadnienia EPR\].\ \ \ [**Postaæ $Q$ dla równania Diraca**]{}: Tak wiêc, w zgodzie z (\[Q in Dirac\]), szczegó³owa postaæ $Q$ dla równania ruchu Diraca okaza³a siê byæ inna ni¿ dla $N$-skalarów i mo¿na j¹ zapisaæ nastêpuj¹co: $$\begin{aligned} \label{Q for psi field} Q = Q_{D} \equiv \mathbb{S}_{q} - \,4\, N\int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2}\left[ (\frac{m\, c}{\hbar})^{2}\psi_{n}^{*}\,\psi_{n}\right] \, , \end{aligned}$$ gdzie $\mathbb{S}_{q}$ jest charakterystyczn¹ czêœci¹ Diracowsk¹ dzia³ania dla kwadratury równania Diraca, wyznaczon¹ dla pola Diraca o randze $N=8$ przy wziêciu pod uwagê wszystkich symetrii uk³adu.\ \ [**Ca³ka dzia³ania kwadratury równania Diraca**]{}: Poniewa¿ jednak ca³ka dzia³ania (por. Rozdzia³ \[dzialanie v.s. zasady informacyjne\]) dla kwadratury specyficznej czêœci Diracowskiej zachowuje siê nastêpuj¹co: $$\begin{aligned} \label{Sq for psi field} \!\!\!\!\! \mathbb{S}_{q} = - \,4\, N\int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2}\left(v_{1n}\, v_{2n}\right) + (pozosta{\ell}e\; cz{\ell}ony) = 0 \, ,\end{aligned}$$ tzn. spe³nia warunek zerowy, zatem $K\equiv K_{D}$ redukuje siê i (w kwadraturze) okreœla postaæ $K_{KG}$ dla (\[TPI every field\])[^66]: $$\begin{aligned} \label{TPI every field jawna postac} K = K_{KG} \equiv 4 \, N \int_{\cal X} d^{4}{\bf x} \sum_{n=1}^{N/2} \left[ \sum_{\mu=0}^{3} (D_{\mu}\psi_{n})^{*}D^{\mu}\psi_{n} - (\frac{m\, c}{\hbar})^{2}\psi_{n}^{*}\,\psi_{n}\right] \; .\end{aligned}$$ [**Uwaga o fundamentalnej ró¿nicy pola skalarnego i pola Diraca**]{}: Wa¿n¹ spraw¹ jest zauwa¿enie, ¿e chocia¿ postaæ $I$ dla $N$-skalarów oraz pól fermionowych rangi $N$ wygl¹da “powierzchniowo” tak samo, to jednak uk³ady te ró¿ni¹ siê istotnie. W samej bowiem rzeczy, podczas gdy [*$N$-skalar jest rozwi¹zaniem rówania ruchu, które wynika jedynie z wariacyjnej zasady informacyjnej*]{} (\[var IP for N-scalar\]), to [*pole fermionowe jest samospójnym rozwi¹zaniem zasad informacyjnych, zarówno wariacyjnej jak i strukturalnej*]{}, podanych w (\[IPs for Dirac\]). ### Koñcowe uwagi o wk³adzie $Q$ w zasadê strukturaln¹ {#few comments on Q} Wynik Rozdzia³u \[structural principle\] [@Dziekuje; @informacja_2] zwi¹zany z wyprowadzeniem strukturalnej zasady informacyjnej jest ogólny, o ile tylko, w celu zagwarantowania s³usznoœci rozwiniêcia w szereg Taylora, funkcja log-wiarygodnoœci $\ln P(\Theta)$ jest analityczn¹ funkcj¹ wektorowego parametru po³o¿enia $\Theta$ w zbiorze jego wartoœci $V_{\Theta}$. Z kolei, w Rozdziale \[information transfer\] zauwa¿yli¿my, ¿e model rozwi¹zany przez metodê EFI jest modelem metrycznym z metryk¹ Rao-Fisher’a oraz, ¿e na poziomie ca³kowym, model metryczny jest równowa¿ny modelowi analitycznemu.\ Dla uk³adów, które nie posiadaj¹ dodatkowych wiêzów ró¿niczkowych, wszystkie po³o¿eniowe stopnie swobody s¹ zwi¹zane jedynie poprzez analizê modelu metrycznego, wynikaj¹c¹ z równañ (\[zmodyfikowana obserwowana zas strukt z P i z kappa\])-(\[var K rozpisana\]) oraz wspomnian¹ zasadê Macha, generuj¹c¹ cz³on strukturalny z Fisherowskiego cz³onu kinematycznego, co prowadzi do czynnika efektywnoœci $\kappa = 1$. Równania Kleina-Gordona oraz Diraca omawiane w Rozdzia³ach \[Klein-Gordon scalars\] oraz \[TPI of the Klein-Gordon equation\] s¹ modelami tego typu.\ Jeœli na uk³ad na³o¿ony jest dodatkowy warunek, który nie wynika ani ze strukturalnej zasady informacyjnej (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \textit{q} = 0$ z $\kappa =1$, ani z zasady wariacyjej (\[var K rozpisana\]), wtedy wzrasta zwi¹zek strukturalny pomiêdzy po³o¿eniowymi stopniami swobody co powoduje, ¿e $\kappa$ wi¹¿¹ca w równaniu (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) gêstoœæ informacji strukturalnej $\textit{q}$ z gêstoœci¹ pojemnoœci informacyjnej $\widetilde{\textit{i'}}\,$, musi maleæ poni¿ej wartoœci 1. W³asnoœæ ta wynika z wklês³oœci pojemnoœci informacyjnej $I$ przy mieszaniu uk³adów [@Frieden], które pojawiaja siê np. na skutek wprowadzenia dodatkowego ró¿niczkowego wiêzu. Na przyk³ad, omawiany w Dodatku \[Maxwell field\] warunek Lorentza dla pola cechowania Maxwella, który jest równaniem typu równania ci¹g³oœci strumienia, pojawia siê nie jako konsekwencja równañ ruchu badanego pola, ale jako ograniczenie szukane na drodze niezale¿nej statystycznej estymacji. Najprawdopodobniej ten dodatkowy warunek[^67] pojawia siê jako rezultat rozwiniêcia w szereg Taylora “go³ej” funkcji wiarygodnoœci $P(\Theta)$ [@Dziekuje; @informacja_2], podobnie jak wyprowadzone w Rozdziale \[master eq\] równanie master (co sygnalizowa³oby spójnoœæ ca³ej statystycznej metody estymacyjnej[^68] ). Zatem warunek Lorentza na³o¿ony na uk³ad redukuje jego symetriê, co poci¹ga za sob¹ pojawienie siê zasady strukturalnej $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0$, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), z czynnikiem efektywnoœci $\kappa$ mniejszym ni¿ 1. W przypadku równañ Maxwella omówionych w Dodatku \[Maxwell field\], zwrócimy uwagê [@Dziekuje; @za; @models; @building], ¿e wartoœæ $\kappa =1/2$ pojawia siê automatycznie w metodzie EFI na skutek samospójnego rozwi¹zania równania strukturalnego, wariacyjnego i na³o¿onego warunku Lorentza. ### Zasada najmniejszego dzia³ania v.s. zasady informacyjne {#dzialanie v.s. zasady informacyjne} Powy¿sze rozwa¿ania Rozdzia³u \[structural inf\] s¹ czêœciowo poœwiêcone omówieniu dodatkowego wyniku zastosowania metody EFI, tzn. ustaleniu ró¿nych postaci informacji strukturalnej $Q$. Zauwa¿ono, ¿e przy umiarkowanie rozbudowanym aparacie metody EFI, nastêpuje nie tylko wyprowadzenie, tzn. estymacja, jej wynikowego równania ruchu (lub równania generuj¹cego rozk³ad), ale jakby przy okazji, pojawienie siê postaci obserwowanej informacji strukturalnej $\texttt{q\!F}_{n}(q_{n}({\bf x}))$.\ \ [**Konstrukcja ca³ki dzia³ania**]{}: Z powy¿szego faktu wynika okreœlenie zwi¹zku pomiêdzy ca³k¹ dzia³ania wraz z zasad¹ najmniejszego dzia³ania, a ca³kowit¹ informacj¹ fizyczn¹ wraz z zasadami informacyjnymi. Okazuje siê bowiem, ¿e po wstawieniu $\texttt{q\!F}_{n}(q_{n}({\bf x}))$ z powrotem do $K$ otrzymujemy ca³kê dzia³ania $\mathbb{S}$ modelu [@Dziekuje; @za; @models; @building]: $$\begin{aligned} \label{S and K connection} \mathbb{S}(q_{n}({\bf x}),\partial q_{n}({\bf x})) = K(q_{n}({\bf x}),\partial q_{n}({\bf x}),\texttt{q\!F}_{n}(q_{n}({\bf x}))) \, .\end{aligned}$$\ Sens równoœci (\[S and K connection\]) okreœla poni¿sza konstrukcja $\mathbb{S}(q_{n}({\bf x}),\partial q_{n}({\bf x}))$. Wpierw rozwi¹zujemy zasady informacyjne, strukturaln¹ i wariacyjn¹, znajduj¹c obserwowan¹ informacjê strukturaln¹ $\texttt{q\!F}_{n}(q_{n}({\bf x}))$. Nastêpnie mo¿emy otrzymaæ równanie ruchu na dwa sposoby:\ \ [**(a) Pierwszy sposób analizy**]{}: W metodzie EFI wstawiamy otrzymane $\texttt{q\!F}_{n}(q_{n}({\bf x}))$ od razu do równañ [*Eulera-Lagrange’a wynikaj¹cych z wariacyjnej zasady informacyjnej*]{} z gêstoœci¹ informacji fizycznej $k(q_{n}({\bf x}),\partial q_{n}({\bf x}),$ $\texttt{q\!F}_{n}(q_{n}({\bf x})))$.\ \ [**(b) Drugi sposób analizy**]{}: W drodze do ca³ki dzia³ania teorii pola wstawiamy otrzymane $\texttt{q\!F}_{n}(q_{n}({\bf x}))$ do $K(q_{n}({\bf x}),\partial q_{n}({\bf x}),$ $\texttt{q\!F}_{n}(q_{n}({\bf x})))$ i stosuj¹c zasadê najmniejszego dzia³ania dla tak skonstruowanego $\mathbb{S}(q_{n}({\bf x}),\partial q_{n}({\bf x}))$, [*otrzymujemy równania Eulera-Lagrange’a teorii pola*]{}.\ \ [**Rezultat**]{}: Otrzymane w konsekwencji analizy typu (a) lub (b) równanie ruchu, b¹dŸ równanie generuj¹ce rozk³ad[^69], jest w obu podejœciach takie samo[^70].\ \ [**Struktura statystyczna teorii**]{}: Jednak¿e, z powodu definicji $\mathbb{S}$ danej przez [*lew¹ stronê*]{} równania (\[S and K connection\]), pojawia siê pytanie, czy metoda EFI, która wykorzystuje $K$ dane przez [*praw¹ stronê*]{} równania (\[S and K connection\]), daje jakieœ dodatkowe informacje w porównaniu do zasady najmniejszego dzia³ania dla $\mathbb{S}$. Potwierdzaj¹ca odpowiedŸ jest nastêpuj¹ca. Wed³ug Rozdzia³u \[structural inf\] metoda EFI dokonuje (zazwyczaj) rozk³adu $K$ na podstawowe bloki podaj¹c otwarcie wszystkie fizyczne i statystyczne pojêcia, i narzêdzia analizy. Równoœæ (\[S and K connection\]), która jest jedynie definicj¹ $\mathbb{S}$, wyra¿a bowiem jednoczeœnie fakt, ¿e $K$ ma bardziej z³o¿on¹ strukturê ni¿ $\mathbb{S}$.\ Powy¿sze stwierdzenie oznacza, ¿e $\mathbb{S}$ oraz zasada najmniejszego dzia³ania nios¹ zarówno mniejsz¹ [*fizyczn¹*]{} informacjê ni¿ oryginalne zasady informacyjne (uzupe³nione fizycznymi wiêzami modelu), jak i ni¿sz¹ informacjê o pojêciach [*statystycznych*]{} .\ [**Zmniejszenie informacji statystycznej**]{}: Po otrzymaniu rozwi¹zania równañ informacyjnych i “œci¹gniêciu” wyniku do $\mathbb{S}$, znikaj¹ z $\mathbb{S}$ nie tylko pojêcia informacji Fishera oraz automatycznie pojemnoœci informacyjnej, lecz i pojêcie ca³kowitej wewnêtrznej dok³adnoœci modelu, tzn. informacji Stama.\ \ [**Zmniejszenie informacji fizycznej**]{}: W koñcu, jeœli rzeczywistoœæ (w znaczeniu matematycznych podstaw modelowych) le¿¹ca poza modelami teorii pola by³aby statystyczna, wtedy odrzucaj¹c ich oryginalny zwi¹zek z analiz¹ na przestrzeni statystycznej ${\cal S}$ oraz estymacyjn¹ metod¹ EFI, która wybiera fizycznie w³aœciw¹ (pod)przestrzeñ ${\cal S}$, tracimy równie¿ oryginalne narzêdzia tej analizy s³u¿¹ce do konstrukcji dzia³ania $\mathbb{S}$. Zamiast tego zadowalamy siê zgadywaniem jego postaci jedynie na podstawie fizycznych warunków wstêpnych, zastanawiaj¹c siê np. jak szczególnym jest pojêcie amplitudy oraz jak wyj¹tkowa jest zasada nieoznaczonoœci Heisenberga.\ \ [**Przyk³ad**]{}: Obok uwagi na wstêpie Rozdzia³u \[structural inf\], ostatnie eksperymenty ze œwiat³em wydaj¹ siê mówiæ, ¿e Fourierowskie czêstoœci nie s¹ widoczne w optycznej lokalizacji najmniejszych jego impulsów. W takiej sytuacji zasada Heisenberga $\delta\nu\delta t\geq 1/2$ mog³aby nie reprezentowaæ sob¹ ¿adnej granicznej fizycznej rzeczywistoœci, gdzie $\delta\nu$ oraz $\delta t$ s¹ poprzez transformacjê Fouriera zwi¹zanymi z sob¹ szerokoœciami po³ówkowymi impulsu w czêstoœci oraz w czasie. Gdyby eksperymenty te potwierdzi³y siê, wtedy zasada Heisenberga straci³aby swoje oparcie w transformacji Fouriera dla zmiennych komplementarnych [@Roychoudhuri]. Natomiast jest mo¿liwe, ¿e znalaz³aby ona wtedy swoje oparcie w nierównoœci Rao-Cramera, która podaje dolne ograniczenie na wariancjê estymatora parametru w przypadku pomiaru jednokana³owego [@Frieden] (por. Dodatek \[Zasada nieoznaczonosci Heisenberga\]). Przyk³ady z fizyki statystycznej i ekonofizyki oraz efekt EPR-Bohm’a {#Przyklady} ==================================================================== Ogólne statystyczne podstawy estymacji MNW zosta³y przedstawione w Rozdzia³ach \[MNW\]-\[Entropia wzgledna i IF\]. Natomiast w Rozdzia³ach \[Zasady informacyjne\] oraz \[Kryteria informacyjne w teorii pola\] skryptu przedstawiono metodê EFI jako szczególny typ estymacyjnej procedury statystycznej, opracowanej w ramach teorii pomiaru. Metoda EFI pokazuje w jaki sposób wychodz¹c z pojêcia funkcji wiarygodnoœci oraz pojemnoœci informacyjnej $I$ (por. Rozdzia³ \[The kinematical form of the Fisher information\]) otrzymaæ wiêzy strukturalne wynikaj¹ce z analitycznej informacyjnej zasady strukturalnej, która wraz z wariacyjn¹ zasad¹ informacyjn¹ oraz równaniem ci¹g³oœci (lub ogólniej, równaniem typu master por. Rozdzia³ \[Geometryczne sformulowanie teorii estymacji\])), prowadzi do równañ ró¿niczkowych teorii [@Frieden]. W trakcie procedury otrzymujemy informacjê strukturaln¹ $Q$ opisywanego uk³adu, która wraz z pojemnoœci¹ informacyjn¹ $I$ tworzy informacjê fizyczn¹ uk³adu $K$, bêd¹c¹ statystycznym poprzednikiem ca³ki dzia³ania (por. Rozdzia³ \[dzialanie v.s. zasady informacyjne\]).\ Wyprowadzenia równañ ruchu lub równañ generuj¹cych rozk³ad zasadzaj¹ siê na potraktowaniu wszystkich warunków na³o¿onych na uk³ad jako zwi¹zków na odchylenia (fluktuacje) wartoœci pomiarowych od wartoœci oczekiwanych. Przy tym, analizowane dane pojawiaj¹ siê jako efekt pomiaru dokonanego przez uk³ad (por. Rozdzia³ \[Podstawowe zalozenie Friedena-Soffera\]).\ Obecny Rozdzia³ dzieli siê na dwie czêœci, pierwsz¹ maj¹c¹ zastosowanie termodynamiczne oraz drug¹, opisuj¹c¹ zjawisko EPR-Bohm’a. Za wyj¹tkiem krótkiej analizy rozwi¹zañ metody EFI dla równañ transportu Boltzmann’a, obie czêœci ³¹czy wymiar próby $N=1$. Wyprowadzenie klasycznej fizyki statystycznej z informacji Fishera {#fizykastatystyczna} ------------------------------------------------------------------ Celem obecnego rozdzia³u jest wyprowadzenie metod¹ EFI podstawowych rozk³adów klasycznej fizyki statystycznej. Otrzymamy wiêc równania generuj¹ce, z których wyprowadzone zostan¹: rozk³ad Boltzmanna dla energii, a nastêpnie rozk³ad Maxwella-Boltzmanna dla pêdu. Jako przyk³ad zastosowania analizy w ekonofizyce, podamy przyk³ad produkcyjnoœci bran¿ w modelu Aoki-Yoshikawy. Przedstawione rachunki id¹ œladem analizy Friedena, Soffera, Plastino i Plastino [@Frieden], jednak zostan¹ one wykonane w oparciu o wprowadzon¹ w Rozdziale \[structural principle\] strukturaln¹ zasadê informacyjn¹ [@Dziekuje; @informacja_2]. Ró¿nicê interpretacyjn¹ pomiêdzy obu podejœciami podano w Rozdziale \[information transfer\].\ Dodatkowo wyprowadzony zostanie warunek informacyjny na górne ograniczenie tempa wzrostu entropii Shannona [@Frieden]. ### Fizyczne sformu³owanie zagadnienia {#Fizyczne sformulowanie zagadnienia dla predkosci} Rozwa¿my gaz sk³adaj¹cy siê z $M$ identycznych cz¹steczek o masie $m$ zamkniêty w zbiorniku. Temperatura gazu ma sta³¹ wartoœæ $T$. Ruch cz¹steczek jest losowy i oddzia³uj¹ one ze sob¹ poprzez si³y potencjalne, zderzaj¹c siê ze sob¹ i œciankami naczynia, przy czym zak³adamy, ¿e s¹ to zderzenia sprê¿yste. Œrednia prêdkoœæ ka¿dej cz¹steczki jest równa zero.\ Oznaczmy przez $\theta_{\wp} = \left(\theta_{\epsilon}, \vec{\theta}_{\wp}\right)$ czterowektor wartoœci oczekiwanej energii oraz pêdu cz¹steczki, gdzie indeks $\wp$ oznacza pêd. We wspó³rzêdnych kartezjañskich $\vec{\theta}_{\wp}=\left(\theta_{\wp_1},\theta_{\wp_2}, \theta_{\wp_3}\right)$. Podobnie jak w (\[parameters separation\]) wprowadzamy zmienn¹ losow¹ $Y = \left( Y_{\epsilon} \equiv \frac{E}{c}, \;\vec{Y}_{\wp}\right) $, przyjmuj¹c¹ waroœci ${\bf y}=\left({\bf y}_{\epsilon}\equiv \epsilon/c\,,\;\vec{\bf y}_{\wp}\right)$, której sk³adowe spe³niaj¹ zwi¹zki: $$\begin{aligned} \label{E} {\bf y}_{\epsilon} \equiv \frac{\epsilon}{c} = \theta_{\epsilon} + {\bf x}_{\epsilon} \; , \;\;\;\; \frac{\epsilon_{0}}{c} \le {\bf y}_{\epsilon} \le \infty \end{aligned}$$ $$\begin{aligned} \label{y_p} \vec{\bf y}_{\wp}=\vec{\theta}_{\wp} + \vec{\bf x}_{\wp} \; , \;\;\;\; \vec{\bf y}_{\wp} = \left({\bf y}_{\wp_1},{\bf y}_{\wp_2},{\bf y}_{\wp_3}\right)\; , \end{aligned}$$ gdzie $$\begin{aligned} \label{xp to p} \vec{\bf x}_{\wp} = \vec{\wp} \; \end{aligned}$$ oznacza fluktuacjê pêdu, natomiast $c$ oznacza prêdkoœæ œwiat³a. Zmienne i parametry energetyczne $ {\bf y}_{\epsilon}$, $\,{\bf x}_{\epsilon}$ oraz $\theta_{\epsilon}$ zosta³y wyra¿one w jednostce wspó³rzêdnych pêdowych $energia/c$. Parametry $\theta_{\epsilon}$ oraz $\vec{\theta}_{\wp}$ s¹ odpowiednimi wartoœciami oczekiwanymi energii (z dok³adnoœci¹ do 1/c) oraz pêdu, a ${\bf x}_{\epsilon}$ oraz $\vec{\bf x}_{\wp}$ fluktuacjami wzglêdem wartoœci oczekiwanych.\ \ Znajdziemy rozk³ad prawdopodobieñstwa dla fluktuacji energii $X_{\epsilon}$ przyjmuj¹cej wartoœci ${\bf x}_{\epsilon}$ oraz fluktualcji pêdu $\vec{X}_{\wp}$ przyjmuj¹cej wartoœci $\vec{\wp}$ dla jednej, dowolnej cz¹steczki gazu w dowolnej chwili czasu $t$. Poniewa¿ wartoœæ $t$ nie musi byæ du¿a, zatem rozwa¿amy gaz, który nie koniecznie jest w stanie równowagi. Bêdziemy wiêc szukaæ postaci nierównowagowego rozk³adu prawdopodobieñstwa, odpowiadaj¹cego tak postawionemu problemowi.\ \ [**Uwaga o czterowektorze energii-pêdu**]{}: Jednak po pierwsze, wspó³rzêdne czterowektora fluktuacji energii-pêdu $\left(X_{\epsilon}, \vec{X}_{\wp} \right)$ nie s¹ statystycznie niezale¿ne, tzn. nie s¹ niezale¿nymi stopniami swobody uk³adu. Po drugie, jak siê oka¿e, spe³niaj¹ one zasadê dyspersyjn¹ typu (\[m E p\]), wiêc nie tworz¹ uk³adu zmiennych Fishera (porównaj (\[zmienne Fisherowskie\])). Zatem ogólny problem wymaga³by estymacji odpowiednich równañ generuj¹cych dla skomplikowanego czasoprzestrzennego zagadnienia, tzn. nale¿a³oby wyznaczyæ ³¹czny rozk³ad prawdopodobieñstwa wspó³rzêdnych $\left(X_{\epsilon}, \vec{X}_{\wp} \right)$, co wykracza poza zakres skryptu. Niemniej np. w przypadku relatywistycznych zjawisk astrofizycznych, taka estymacja mo¿e okazaæ siê niezbêdna.\ W skrypcie ograniczymy siê jedynie do wyznaczenia brzegowych rozk³adów prawdopodobieñstwa dla $X_{\epsilon}$ (i w konsekwencji dla $E$) oraz dla $\vec{X}_{\wp}$, co w nierelatywistycznej granicy jest uzasadnione.\ ### Informacja kinetyczna i strukturalna oraz sformu³owanie zasad informacyjnych {#zasady inf dla energii i predkosci} Tak wiêc, statystycznie jedna cz¹steczka podlega ³¹cznemu rozk³adowi prawdopodobieñstwa $p\left({\bf x}_{\epsilon},\vec{\bf x}_{\wp}\right)$, przy czym wspó³rzêdne czterowektora pêdu nie s¹ niezale¿ne. Uproszczona analiza skoncentruje siê na rozk³adach brzegowych, których analiza ze wzglêdu na wspomniany brak niezale¿noœci nie odtwarza analizy ³¹cznej, chocia¿ jest s³uszna w przybli¿eniu nierelatywistycznym.\ Wyznaczymy wiêc [*brzegowe amplitudy prawdopodobieñstwa*]{} $q\left({\bf x}_{\epsilon}\right)$ oraz $q\left(\vec{\bf x}_{\wp}\right)$: $$\begin{aligned} q_{n}\left({\bf x}_{\epsilon}\right),\quad n=1,...,N_{\epsilon} \;\;\;\; {\rm oraz} \;\;\;\; q_{n}\left(\vec{\bf x}_{\wp}\right), \;\;\;\; n=1,...,N_{\wp} \; ,\end{aligned}$$ a nastêpnie powrócimy do brzegowych rozk³adów prawdopodobieñstwa $p\left({\bf x}_{\epsilon}\right)$ oraz $p\left(\vec{\bf x}_{\wp}\right)$. W koñcu po wyznaczeniu $p\left({\bf x}_{\epsilon}\right)$ skorzystamy z (\[E\]), aby otrzymaæ wymagany rozk³ad $p\left(\epsilon\right)$. Podobnie, korzystaj¹c z (\[y\_p\]) otrzymamy po wyznaczeniu $p\left(\vec{\bf x}_{\wp}\right)$ rozk³ad prêdkoœci $p\left(\vec{\bf y}_{\wp}\right)$, przy czym te dwa ostatnie s¹ równe w naszych rozwa¿aniach, ze wzglêdu na œredni¹ wartoœæ prêdkoœci cz¹steczki $\vec{\theta}_{\wp} = 0$.\ \ [**Pojemnoœæ informacyjna dla parametrów czterowektor energii-pêdu**]{}: Chocia¿ $\left({\bf x}_{\epsilon}, \vec{\bf x}_{\wp}\right)$ nie jest czterowektorem we wspomnianym ujêciu Fishera, to jest on czterowektorem w sensie Lorentza. Korzystamy wiêc z metryki czasoprzestrzeni Minkowskiego w postaci $(\eta_{\nu \mu}) = diag(1,-1,-1,-1)$ zgodnie z (\[metryka M\]).\ W ogólnym przypadku rozk³adu ³¹cznego oraz próbkowania czterowektora energii i pêdu, pojemnoœæ informacyjna ma dla parametru wektorowego $\Theta = \left((\theta_{\nu n})_{\nu=0}^{3}\right)_{n=1}^{N}$ postaæ (\[pojemnosc C dla polozenia - powtorka wzoru\])[^71]: $$\begin{aligned} \label{I dla lacznego E p} I = \sum\limits_{n=1}^{N} \int_{\cal B} dy \; P\left(y|\Theta\right) \left[ {\left(\frac{\partial\ln P\left(y|\Theta\right)}{\partial\theta_{\epsilon \, n}} \right)^{2}} - \sum\limits_{k=1}^{3} {\left(\frac{\partial\ln P\left(y|\Theta\right)}{\partial\theta_{\wp_{k} n}} \right)^{2}} \right] \; ,\end{aligned}$$ gdzie $y=({\bf y})_{n=1}^{N}$ jest $N$-wymiarow¹ prób¹, a ${\cal B}$ przestrzeni¹ próby. Zatem pojemnoœci informacyjne $I\left(\Theta_\epsilon\right)$ oraz $I\left(\Theta_{\vec{\wp}}\right)$ dla rozk³adów brzegowych fluktuacji energii oraz pêdu maj¹ postaæ[^72]: $$\begin{aligned} \label{min} I\left(\Theta_{\epsilon}\right) = 4 \int_{{\cal X}_{\epsilon}}{d{\bf x}_{{\epsilon}} \sum\limits_{n=1}^{N_{{\epsilon}}}{\left({\frac{{dq_{n}\left({{\bf x}_{{\epsilon}}}\right)}} {{d{\bf x}_{{\epsilon}}}}}\right)^{2}}}\end{aligned}$$ oraz $$\begin{aligned} \label{minp} I\left( \Theta_{\vec{\wp}} \right) = - 4 \int_{{\cal X}_{\wp}} d \vec{\bf x}_{\wp} \sum\limits_{n=1}^{N_{\wp}} \sum\limits_{k=1}^{3} \left( \frac{ dq_{n} \left( \vec{\bf x}_{\wp} \right) }{ dx_{\wp_{k}}} \right)^{2} \; ,\end{aligned}$$ gdzie minus w (\[I dla lacznego E p\]) i w konsekwencji w (\[minp\]), wynika zgodnie z Rozdzia³em \[Poj inform zmiennej los poloz\] z uwzglêdnienia metryki Minkowskiego (\[metryka M\]), natomiast ${\cal X}_{\epsilon}$ oraz ${\cal X}_{\wp}$ s¹ zbiorami wartoœci odpowiednio zmiennych $X_{\epsilon}$ oraz $X_{\wp}$. Z poni¿szych rachunków przekonamy siê, ¿e nieuwzglêdnienie metryki Minkowskiego (gdy $\eta_{00} =1$ to $\eta_{kk} =-1$ dla $k=1,2,3$), doprowadzi³oby w konsekwencji w termodynamicznych rozwa¿aniach do b³êdnego rozk³adu prêdkoœci[^73].\ \ [**Zasady informacyjne**]{}: Zasady informacyjne, strukturalna (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) oraz wariacyjna (\[var K rozpisana\]) maj¹ poni¿sz¹ postaæ. Dla energii: $$\begin{aligned} \label{epiE} \widetilde{\textit{i'}}(\Theta_{\epsilon}) + \widetilde{\mathbf{C}}_{\epsilon} + \kappa \, \textit{q}(\Theta_{\epsilon}) = 0 \; , \;\;\; \delta_{(q_{n})}\left(I(\Theta_{\epsilon}) + Q(\Theta_{\epsilon})\right) = 0 \; , \end{aligned}$$ oraz dla pêdu: $$\begin{aligned} \label{epip} \widetilde{\textit{i'}}(\Theta_{\vec{\wp}}) + \widetilde{\mathbf{C}}_{\vec{\wp}} + \kappa \, \textit{q}(\Theta_{\vec{\wp}}) = 0 \; , \;\;\; \delta_{(q_{n})}\left(I(\Theta_{\vec{\wp}}) + Q(\Theta_{\vec{\wp}})\right) = 0 \; . \end{aligned}$$ Gêstoœci pojemnoœci informacyjnych, $\widetilde{\textit{i'}}(\Theta_{\epsilon})$ oraz $\widetilde{\textit{i'}}(\Theta_{\vec{\wp}})$, s¹ okreœlone zgodnie z (\[gestosc i Amarii\]) i (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), a gêstoœci informacji strukturalnych, $\textit{q}(\Theta_{\epsilon})$ oraz $\textit{q}(\Theta_{\vec{\wp}})$, s¹ okreœlone zgodnie z (\[gestosc q dla niezaleznych Yn\]), natomiast $Q(\Theta_{\epsilon})$ oraz $Q(\Theta_{\vec{\wp}})$ s¹ odpowiednimi informacjami strukturalnymi.\ W pierwszej kolejnoœci rozwa¿ymy problemem (\[epiE\]) dla $p({\epsilon})$.\ \ [**Przypomnienie roli zasad informacyjnych**]{}: W rachunkach metody EFI prowadz¹cych do równania generuj¹cego rozk³ad, obok zasady wariacyjnej (\[var K rozpisana\]) wykorzystywana jest postaæ (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) zmodyfikowanej obserwowanej zasady strukturalnej. Warto pamiêtaæ, ¿e zasada (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) wynika z ¿¹dania istnienia rozwiniêcia Taylora logarytmu funkcji wiarygodnoœci wokó³ prawdziwej wartoœci parametru (por. Rozdzia³ \[structural principle\]) oraz z metrycznoœci przestrzeni statystycznej ${\cal S}$. Natomiast oczekiwana strukturalna zasada informacyjna (\[expected form of information eq\]) jest narzêdziem pomocniczym w definicji ca³kowitej fizycznej informacji $K$, (\[physical K\]), oraz informacyjnej zasady wariacyjnej (\[var K rozpisana\]). ### Rozk³ad Boltzmanna dla energii {#rozdz.energia} Poni¿ej podamy rozwi¹zanie zasad informacyjnych (\[epiE\]), strukturalnej oraz wariacyjnej, otrzymuj¹c w pierwszym kroku analizy równanie generuj¹ce amplitudy dla rozk³adu Boltzmanna.\ \ Za³ó¿my wstêpnie, ¿e wartoœæ fluktuacji energii ${\bf x}_{{\epsilon}}$ zmienia siê w pewnym zakresie $\left\langle {\bf x}_{{\epsilon}}^{min}, {\bf x}_{{\epsilon}}^{max}\right\rangle$: $$\begin{aligned} \label{zakres} {\bf x}_{{\epsilon}}^{min} \le {\bf x}_{{\epsilon}} \le {\bf x}_{{\epsilon}}^{max} \;\; .\end{aligned}$$ W ten sposób wpierw uchwycimy ogóln¹ zale¿noœæ amplitudy rozk³adu od ${\bf x}_{{\epsilon}}^{max}$, a nastêpnie dokonamy przejœcia granicznego, przechodz¹c z górn¹ granic¹ fluktuacji energii ${\bf x}_{{\epsilon}}^{max}$ do nieskoñczonoœci.\ \ [**Pojemnoœæ informacyjna**]{} ma postaæ (\[Fisher\_information-kinetic form bez n\]): $$\begin{aligned} \label{informacjaE} I\left(\Theta_{{\epsilon}}\right) = 4 \int\limits_{{\bf x}_{{\epsilon}}^{min}}^{{\bf x}_{{\epsilon}}^{max}} {d{\bf x}_{{\epsilon}} \sum\limits _{n=1}^{N}{q_{n}^{'2}\left({\bf x}_{{\epsilon}} \right)}} \; , \;\;\;\; {\rm gdzie} \;\;\;\; \; q_{n}^{'}\left({\bf x}_{{\epsilon}}\right) \equiv \frac{dq_{n}\left({\bf x}_{{\epsilon}}\right)}{d{\bf x}_{{\epsilon}}} \; ,\end{aligned}$$ gdzie rozk³ady prawdopodobieñstwa $p_{n}$ dla fluktuacji energii s¹ powi¹zane z amplitudami $q_{n}$ zale¿noœci¹ (\[amplituda a rozklad\]): $$\begin{aligned} p_{n}({\bf x}_{{\epsilon}}) = q_{n}^{2}({\bf x}_{{\epsilon}}) \; .\end{aligned}$$ [**Informacja strukturalna**]{} zgodnie z (\[Q dla niezaleznych Yn w d4y\]) jest nastêpuj¹ca: $$\begin{aligned} \label{Q diag dla E} Q(\Theta_{{\epsilon}}) = \int\limits_{{\bf x}_{{\epsilon}}^{min}}^{{\bf x}_{{\epsilon}}^{max}} d{\bf x}_{{\epsilon}} \, \textit{q}(\Theta_{{\epsilon}}) = \int\limits_{{\bf x}_{{\epsilon}}^{min}}^{{\bf x}_{{\epsilon}}^{max}} d{\bf x}_{{\epsilon}} \, \sum\limits _{n=1}^{N} q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})) \; .\end{aligned}$$\ [**Informacyjna zasada wariacyjna**]{} w (\[epiE\]) dla pojemnoœci informacyjnej $I$, (\[informacjaE\]), oraz informacji strukturalnej $Q$, (\[Q diag dla E\]), ma postaæ: $$\begin{aligned} \label{gggg} \delta_{(q_{n})} K = \delta_{(q_{n})} (I(\Theta_{{\epsilon}}) + Q(\Theta_{{\epsilon}})) = \delta_{(q_{n})} \left(\;\int\limits_{{\bf x}_{{\epsilon}}^{min}}^{{\bf x}_{{\epsilon}}^{max}} {d{\bf x}_{{\epsilon}} \, k} \right) = 0 \; , \end{aligned}$$ gdzie $k$ jest równe: $$\begin{aligned} \label{k dla Boltzmanna} k = 4 \sum\limits_{n=1}^{N} {\left(q_{n}^{'2} + \frac{1}{4}q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})) \right)} \; ,\end{aligned}$$ zgodnie z ogóln¹ postaci¹ gêstoœci obserwowanej informacji fizycznej (\[k form\]) dla amplitud $q_{n}$.\ \ [**Rozwi¹zaniem problemu wariacyjnego**]{} (\[gggg\]) wzgêdem $q_{n}({\bf x}_{{\epsilon}})$ jest [*równanie Eulera-Lagrange’a*]{} (\[EL eq\]): $$\begin{aligned} \label{euler} \frac{d}{{d{\bf x}_{{\epsilon}}}}\left({\frac{{\partial k}}{{\partial q_{n}^{'}({\bf x}_{{\epsilon}})}}}\right) = \frac{{\partial k}}{{\partial q_{n}({\bf x}_{{\epsilon}})}} \; \;\;\;\; {\rm dla} \;\; n = 1, 2,..., N \; ,\end{aligned}$$ gdzie ’prim’ oznacza pochodn¹ po ${\bf x}_{{\epsilon}}$, tzn. $q_{n}^{'}({\bf x}_{{\epsilon}}) \equiv d q_{n}({\bf x}_{{\epsilon}})/d{\bf x}_{{\epsilon}}$.\ \ Zatem dla rozwa¿anego problemu, równanie (\[euler\]) dla $k$ jak w (\[k dla Boltzmanna\]), przyjmuje postaæ: $$\begin{aligned} \label{rownanie wariacyjne E} 2 \, q_{n}^{''}({\bf x}_{{\epsilon}}) = \frac{{\partial (\frac{1}{4}q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})))}}{{\partial q_{n}}} = \frac{{d (\frac{1}{4} q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})))}}{{dq_{n}}} \; . \label{rozweulera}\end{aligned}$$ [**Zmodyfikowana obserwowana zasada strukturalna**]{}: Po wyca³kowaniu (\[informacjaE\]) przez czêœci, pojemnoœæ $I$ wynosi: $$\begin{aligned} \label{postac I po calk czesci} I(\Theta_{{\epsilon}}) = - 4 \int\limits_{{\bf x}_{{\epsilon}}^{min}}^{{\bf x}_{{\epsilon}}^{max}} {d{\bf x}_{{\epsilon}} \sum\limits _{n=1}^{N}{q_{n}({\bf x}_{{\epsilon}}) \, q_{n}^{''}({\bf x}_{{\epsilon}})}} + C \; , \end{aligned}$$ gdzie $$\begin{aligned} \label{postac IC z pradem} C \equiv 4 \int\limits_{{\bf x}_{{\epsilon}}^{min}}^{{\bf x}_{{\epsilon}}^{max}} d{\bf x}_{{\epsilon}} \, \left( q_{n}\left({\bf x}_{{\epsilon}}\right) q^{'}_{n}\left({\bf x}_{{\epsilon}}\right) \right)^{'} = 4 \sum \limits_{n=1}^{N}{C_{n}} \; \end{aligned}$$ oraz $$\begin{aligned} \label{Cn} C_{n} = {q_{n}\left({\bf x}_{{\epsilon}}^{max}\right)q_{n}^{'}\left({\bf x}_{{\epsilon}}^{max}\right)-q_{n}\left({\bf x}_{{\epsilon}}^{min}\right)q_{n}^{'}\left({\bf x}_{{\epsilon}}^{min}\right)} \; .\end{aligned}$$ Zatem widzimy, ¿e [*zmodyfikowana obserwowana zasada strukturalna*]{} w (\[epiE\]) jest ze wzglêdu na (\[postac I po calk czesci\])-(\[postac IC z pradem\]) oraz (\[Q diag dla E\]), nastêpuj¹ca: $$\begin{aligned} \label{obserwowana zas strukt dla Boltzmann suma n} \sum\limits_{n=1}^{N}{\left( - {q_{n}({\bf x}_{{\epsilon}}) q_{n}^{''}({\bf x}_{{\epsilon}}) + \tilde{C}_{n} + \kappa \frac{1}{4}q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}}))}\right)} = 0 \; , \;\;\;\end{aligned}$$ gdzie $$\begin{aligned} \label{Cn tilde} \tilde{C}_{n} = C_{n}/({\bf x}_{{\epsilon}}^{max} - {\bf x}_{{\epsilon}}^{min}) \; . \end{aligned}$$ Zatem na poziomie obserwowanym, dla ka¿dego $n=1,2,...,N$, otrzymujemy zasadê strukturaln¹ w postaci[^74]: $$\begin{aligned} \label{rownanie strukt E} - q_{n}({\bf x}_{\epsilon}) q_{n}^{''}({\bf x}_{\epsilon}) + \tilde{C}_{n} + \kappa \frac{1}{4}q_{n}^{2}({\bf x}_{\epsilon}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{\epsilon})) = 0 \; .\end{aligned}$$ Korzystaj¹c z (\[rownanie wariacyjne E\]) w (\[rownanie strukt E\]) otrzymujemy: $$\begin{aligned} \label{row rozn z q oraz qF} \frac{1}{2} q_{n} \frac{{d (q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})))}}{{dq_{n}}} = \kappa q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})) + 4 \,\tilde{C}_{n} \; .\end{aligned}$$ Poni¿ej, dla uproszczenia zapisu pominiemy przy rozwi¹zywaniu równania (\[row rozn z q oraz qF\]) oznaczenie argumentu ${\bf x}_{{\epsilon}}$ w amplitudzie $q_{n}$. Zapiszmy (\[row rozn z q oraz qF\]) w postaci: $$\begin{aligned} \label{row rozn z q oraz qF do wycalkowania} 2\frac{{dq_{n}}}{{q_{n}}}=\frac{{d\left[\frac{1}{4}q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}) \right]}}{{\kappa \, \left[\frac{1}{4} \, q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n})\right] + \tilde{C}_{n}}} \; .\end{aligned}$$ Rozwi¹zuj¹c powy¿sze równanie ró¿niczkowe otrzymujemy kolejno: $$2\ln q_{n} + \alpha_{n}^{'} = \frac{1}{\kappa} \ln \left({ \frac{\kappa }{4} \, q_{n}^{2} \, \texttt{q\!F}_{n}(q_{n}) + \, \tilde{C}_{n}}\right)$$ i po przekszta³ceniu: $$2\kappa\ln\alpha_{n}^{''} q_{n} = \ln \left({ \frac{\kappa}{4}q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}) + \, \tilde{C}_{n}}\right) \; ,$$ gdzie $\alpha_{n}^{'}$ jest w ogólnoœci zespolon¹ sta³¹ ca³kowania oraz sta³a $ \alpha_{n}^{''} = \exp(\alpha_{n}^{'}/2)$, sk¹d: $$\begin{aligned} \label{wyn-j} q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}) = \frac{4}{\kappa}\left({\alpha_{n}^{2}q_{n}^{2\kappa} - \, \tilde{C}_{n}}\right) \; ,\end{aligned}$$ gdzie (w ogólnoœci zespolona) sta³a $\alpha_{n}^{2}=\alpha_{n}^{'' \, 2\kappa}$. Zatem w rezultacie otrzymaliœmy obserwowan¹ informacjê strukturaln¹, $\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}}))$, jako znan¹ funkcjê amplitudy $q_{n}({\bf x}_{{\epsilon}})$.\ \ [**Równanie generuj¹ce z $\kappa$**]{}: W koñcu, korzystaj¹c z (\[rownanie wariacyjne E\]) oraz z (\[wyn-j\]), otrzymujemy równanie ró¿niczkowe dla $q_{n}({\bf x}_{{\epsilon}})$: $$\begin{aligned} \label{row generujace ampl dla E} q_{n}^{''}({\bf x}_{{\epsilon}}) = \alpha_{n}^{2} q_{n}^{2\kappa-1}({\bf x}_{{\epsilon}}) \, .\end{aligned}$$ [**Podsumowanie**]{}: Wariacyjna oraz strukturalna obserwowana zasada informacyjna zosta³y zapisane w postaci uk³adu równañ (\[rownanie wariacyjne E\]) oraz (\[rownanie strukt E\]), który rozwi¹zuj¹c, da³ w rezultacie szukan¹ postaæ (\[row generujace ampl dla E\]) [*równania generuj¹cego*]{} amplitudê $q_{n}({\bf x}_{{\epsilon}})$. Znalezienie postaci tego równania by³o poœrednim celem metody EFI.\ \ [**Warunek analitycznoœci i metrycznoœci**]{}: Z postaci (\[row generujace ampl dla E\]) widaæ, ¿e skoro amplituda $q_{n}({\bf x}_{{\epsilon}})$ jest funkcj¹ rzeczywist¹, zatem sta³a $\alpha_{n}^{2}$ musi byæ rzeczywista. Rozwi¹zanie równania (\[row generujace ampl dla E\]) znajdziemy wtedy, gdy wspó³czynnik efektywnoœci wynosi: $$\begin{aligned} \label{kappa1} \kappa = 1 \;\; ,\end{aligned}$$ czyli dla przypadku, gdy zasada strukturalna dla uk³adu jest konsekwencj¹ analitycznoœci logarytmu funkcji wiarygodnoœci (\[rozwiniecie w szereg Taylora\]) oraz metrycznoœci przestrzeni statystycznej ${\cal S}$.\ W przypadku $\kappa = 1 $ obserwowana informacja strukturalna (\[wyn-j\]) przyjmuje prost¹ postaæ: $$\begin{aligned} \label{postac mikro Q dla E} q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})) = 4 ( {\alpha_{n}^{2}q_{n}^{2}({\bf x}_{{\epsilon}}) - \tilde{C}_{n}} ) \; ,\end{aligned}$$ a ca³kowa postaæ $SI$ jest nastêpuj¹ca: $$\begin{aligned} \label{postac calkowa Q dla E} Q(\Theta_{{\epsilon}}) = 4 \int{d{\bf x}_{{\epsilon}} \sum\limits_{n=1}^{N}{\alpha_{n}^{2}q_{n}^{2}\left({\bf x}_{{\epsilon}}\right)} - 4 \sum\limits_{n=1}^{N}{C_{n}}} \; ,\end{aligned}$$ co po skorzystaniu z unormowania kwadratu amplitudy: $$\begin{aligned} \label{unormowanie q2} \int{d{\bf x}_{{\epsilon}} \, q_{n}^{2}\left({\bf x}_{{\epsilon}}\right)} = \int{d{\bf x}_{{\epsilon}} \, p_{n}\left({\bf x}_{{\epsilon}}\right)} = 1 \; ,\end{aligned}$$ daje: $$\begin{aligned} \label{postac Q dla E po war normalizacji qn} Q(\Theta_{{\epsilon}}) = 4 \sum\limits_{n=1}^{N} ( \alpha_{n}^{2} - C_{n} ) = 4 \sum\limits_{n=1}^{N} \alpha_{n}^{2} - C \; .\end{aligned}$$ W ostatnim przejœciu w (\[postac Q dla E po war normalizacji qn\]) skorzystano z postaci sta³ej $C \equiv 4 \sum \limits_{n=1}^{N}{C_{n}}$ wprowadzonej w (\[postac I po calk czesci\]).\ \ [**Równanie generuj¹ce**]{}: Dla rozwa¿anego przypadku $\kappa =1$, równanie generuj¹ce (\[row generujace ampl dla E\]) ma postaæ: $$\begin{aligned} \label{falowe kappa 1} q_{n}^{''}({\bf x}_{{\epsilon}}) = \alpha_{n}^{2}q_{n}({\bf x}_{{\epsilon}}) \; .\end{aligned}$$\ [**Sprawdzenie rachunków**]{}: Wstawiaj¹c (\[falowe kappa 1\]) do (\[postac I po calk czesci\]), otrzymujemy nastêpuj¹c¹ postaæ pojemnoœci informacyjnej: $$\begin{aligned} \label{I wraz z Ic dla E} I(\Theta_{{\epsilon}}) = - 4 \sum\limits_{n=1}^{N} {\alpha_{n}^{2}} + C \; ,\end{aligned}$$ co wraz z (\[postac Q dla E po war normalizacji qn\]) oznacza sprawdzenie poprawnoœci rachunku, poprzez spe³nienie przez otrzymane rozwi¹zanie oczekiwanego strukturalnego warunku $I+Q =0$, zgodnie z (\[rownowaznosc strukt i zmodyfikowanego strukt\]).\ \ [**Rozwi¹zanie równania generuj¹cego**]{}: Kolejnym etapem analizy jest rozwi¹zanie równania generuj¹cego (\[falowe kappa 1\]).\ Najogólniejsza postaæ amplitudy $q_{n}({\bf x}_{{\epsilon}})$ bêd¹cej rozwi¹zaniem równania (\[falowe kappa 1\]) jest przy warunku ${\bf x}_{{\epsilon}}^{min} \le {\bf x}_{{\epsilon}} \le {\bf x}_{{\epsilon}}^{max} $, jak w (\[zakres\]), nastêpuj¹ca: $$\begin{aligned} \label{rozw2q} q_{n}({\bf x}_{{\epsilon}}) = B_{n} \exp \left(\alpha_{n} {\bf x}_{{\epsilon}} \right) + D_{n}\exp\left(-\alpha_{n} {\bf x}_{{\epsilon}} \right) \, , \;\;\;\; {\bf x}_{{\epsilon}}^{min} \le x\le {\bf x}_{{\epsilon}}^{max} \, , \;\;\; B_{n},D_{n} = const. \end{aligned}$$ Poniewa¿ sta³a $\alpha_{n}^{2}$ jest rzeczywista, zatem wprowadzaj¹c now¹ rzeczywist¹ sta³¹ $\beta_{n}$, mo¿na $\alpha_{n}$ przedstawiæ jako $\alpha_{n}=\beta_{n}$ i wtedy rozwi¹zanie (\[rozw2q\]) ma charakter czysto eksponencjalny: $$\begin{aligned} \label{rozw ekspo dla E} q_{n}({\bf x}_{{\epsilon}})=B_{n}\exp\left(\beta_{n}{\bf x}_{{\epsilon}}\right)+D_{n}\exp\left(-\beta_{n}{\bf x}_{{\epsilon}}\right) \; ,\end{aligned}$$ b¹dŸ jako $\alpha_{n}=i\beta_{n}$, i wtedy rozwi¹zanie (\[rozw2q\]) ma charakter czysto trygonometryczny: $$\begin{aligned} \label{rozw trygonometryczne dla E} q_{n}({\bf x}_{{\epsilon}})=B_{n}\exp\left(i\beta_{n}{\bf x}_{{\epsilon}}\right)+D_{n}\exp\left(-i\beta_{n}{\bf x}_{{\epsilon}}\right) \; .\end{aligned}$$ [**Warunek normalizacji dla amplitud**]{}: Tak okreœlone funkcje musz¹ byæ dopuszczalne jako amplitudy prawdopodobieñstwa, zatem musz¹ spe³niaæ [*warunek normalizacji*]{} dla gêstoœci prawdopodobieñstwa (\[unormowanie q2\]).\ Za³ó¿my, ¿e wartoœæ fluktuacji energii ${\bf x}_{{\epsilon}}$ nie jest ograniczona od góry, co zrealizujemy jako d¹¿enie ${\bf x}_{{\epsilon}}^{max}$ do nieskoñczonoœci. Jednak kwadrat funkcji trygonometrycznej nie mo¿e byæ unormowany do jednoœci dla ${\bf x}_{{\epsilon}}^{max} \rightarrow\infty$, zatem funkcja trygonometryczna (\[rozw trygonometryczne dla E\]) nie jest dopuszczalnym rozwi¹zaniem.\ [**Pozostaje wiêc rozwi¹zanie eksponencjalne**]{} (\[rozw ekspo dla E\]). Poniewa¿ jednak warunek unormowania (\[unormowanie q2\]) ma byæ zadany na przedziale otwartym ${\bf x}_{{\epsilon}}^{min} \le {\bf x}_{\epsilon} < \infty$, zatem czêœæ rozwi¹zania z dodatni¹ eksponent¹ musi byæ odrzucona ze wzglêdu na jej rozbie¿noœæ do nieskoñczonoœci. Sk¹d otrzymujemy ¿¹danie, ¿e dla $\beta_{n} \ge 0$ sta³a $B_{n}=0$.\ \ Podsumowuj¹c, szukana postaæ amplitudy jest wiêc nastêpuj¹ca: $$\begin{aligned} \label{qn rozwiazanie dla E} q_{n}\left({\bf x}_{{\epsilon}}\right) = {D_{n}\exp\left({-\beta_{n} \, {\bf x}_{{\epsilon}}}\right)}\; , \quad \beta_{n}\in\mathbf{R}_{+} \; , \quad {\bf x}_{{\epsilon}}^{min} \le {\bf x}_{{\epsilon}} < \infty \; .\end{aligned}$$ Z powy¿szego i z warunku normalizacji (\[unormowanie q2\]) $\int_{{\bf x}_{{\epsilon}}^{min}}^{\infty}{d{\bf x}_{{\epsilon}} \, q_{n}^{2}\left({\bf x}_{{\epsilon}}\right)} = 1$, wyznaczamy sta³¹ $D_{n}$, otrzymuj¹c: $$\begin{aligned} D_{n} = \sqrt{2\beta_{n}}\exp\left({\beta_{n} \, {\bf x}_{{\epsilon}}^{min}}\right)\; .\end{aligned}$$\ [**Ostateczna postaæ amplitudy**]{}: Rozwi¹zanie to zosta³o otrzymane dla przypadku $\beta_{n} = \alpha_{n} \in \mathbf{R}_{+}$, zatem ostateczn¹ postaci¹ (\[rozw2q\]) jest: $$\begin{aligned} \label{rozw cosinus dla ampl dla E} q_{n}\left({\bf x}_{{\epsilon}}\right)=\sqrt{2\alpha_{n}}\exp\left[{\alpha_{n}\left({{\bf x}_{{\epsilon}}^{min}-{\bf x}_{{\epsilon}}}\right)}\right] \; ,\quad \alpha_{n} \in \mathbf{R}_{+} \; .\end{aligned}$$ Zauwa¿my, ¿e $\alpha_{n}$ jest w jednostkach $\left[c/energia\right]$.\ \ [**Koñcowa postaæ pojemnoœci informacyjnej**]{}: W koñcu mo¿emy wyznaczyæ pojemnoœæ informacyjn¹. Wstawiaj¹c (\[rozw cosinus dla ampl dla E\]) do (\[informacjaE\]), otrzymujemy: $$\begin{aligned} \label{I dla E z war > 0} I(\Theta_{{\epsilon}}) = 4 \sum\limits_{n=1}^{N} {\alpha_{n}^{2}} > 0 \; , \quad \alpha_{n} \in \mathbf{R}_{+} \; ,\end{aligned}$$ co po porównaniu z (\[I wraz z Ic dla E\]) daje wartoœæ sta³ej $C$ równ¹: $$\begin{aligned} \label{wartosc stalej I_C} C = 8 \sum\limits_{n=1}^{N} {\alpha_{n}^{2}} \, .\end{aligned}$$ [**Uwaga o stabilnoœci rozwi¹zania**]{}: Warunek dodatnioœci pojemnoœci informacyjnej otrzymany w (\[I dla E z war &gt; 0\]), dla pojemnoœci informacyjnej zwi¹zanej z dodatni¹ czêœci¹ sygnatury metryki Minkowskiego, jest istotnym wynikiem z punktu widzenia teorii pomiaru. W rozwa¿anym przyk³adzie analizy estymacyjnej wartoœci oczekiwanej energii cz¹stki gazu, zosta³ on otrzymany na gruncie samospójnego rozwi¹zania równañ ró¿niczkowych [@Arnold] informacyjnej obserwowanej zasady strukturalnej oraz zasady wariacyjnej. Niespe³nienie tego warunku oznacza niestabilnoœæ badanego uk³adu.\ \ Z kolei, wstawiaj¹c otrzyman¹ wartoœæ sta³ej $C$ do (\[postac Q dla E po war normalizacji qn\]) otrzymujemy: $$\begin{aligned} \label{Q dla E z war < 0} Q(\Theta_{{\epsilon}}) = - 4 \sum\limits_{n=1}^{N} {\alpha_{n}^{2}} <0 \; .\end{aligned}$$ Natomiast sam problem wariacyjny (\[gggg\]), który mo¿na wyraziæ po skorzystaniu z postaci $Q(\Theta_{{\epsilon}})$ w (\[postac Q dla E po war normalizacji qn\]) w postaci: $$\begin{aligned} \label{sam problem wariacyjny} \delta_{(q_{n})}\left(I(\Theta_{{\epsilon}}) + 4 \sum\limits_{n=1}^{N} \alpha_{n}^{2} - C \right) = 0 \; ,\end{aligned}$$ jest ze swojej natury nieczu³y na wartoœæ sta³ej $C$.\ \ [**Uwaga o randze amplitudy $N$**]{}: Istotnym zagadnieniem metody EFI jest wielkoœæ próby $N$ pobranej przez uk³ad, tzn. ranga $N$ amplitudy. Do sprawy liczby amplitud $q_{n}$ wchodz¹cych w opis uk³adu podejdziemy w najprostszy z mo¿liwych sposobów, sugeruj¹c stosowanie dwóch prostych i niewykluczaj¹cych siê kryteriów:\ [**(1)**]{} [**Kryterium minimalizacji $I$ ze wzglêdu na rangê amplitudy $N$**]{} [@Frieden]. Tzn. przy zachowaniu warunku $I>0$, (\[informacja Stama vs pojemnosc informacyjna Minkowskiego\]), ranga $N$ mo¿e na tyle spaœæ, ¿eby istnia³o jeszcze samospójne rozwi¹zanie równañ cz¹stkowych strukturalnej i wariacyjnej zasady informacyjnej.\ Kryterium to nie oznacza nie realizowania rozwi¹zañ z wiêksz¹ ni¿ minimalna liczb¹ $N$.\ [**(2) Kryterium obserwacyjne wyboru rangi amplitudy**]{} wi¹¿e siê z wyborem takiej wartoœci $N$, dla której otrzymane rozwi¹zanie ma realizacjê obserwowan¹ w eksperymencie. We wspó³czesnych teoriach fizyki statystycznej oraz teoriach pola, realizowane s¹ rozwi¹zania z niskimi wartoœciami rangi. Fakt ten zauwa¿yliœmy ju¿ w Rozdziale \[Kryteria informacyjne w teorii pola\] dla modeli teorii pola.\ \ Rozwa¿any w bie¿¹cym rozdziale przyk³ad rozk³adu energii pozwala na ilustracjê kryterium (1). W kolejnym z rozdzia³ów znajdziemy rozwi¹zanie EFI dla rozk³adu prêdkoœci cz¹steczki gazu, stosuj¹c równie¿ kryterium (1). Obok stacjonarnego rozwi¹zania Maxwella-Boltzmanna z $N=1$, wska¿emy na fizyczn¹ interpretacjê rozwi¹zañ z $N>1$.\ \ [**Zastosowanie kryterium (1) dla rozk³adu energii**]{}: Zauwa¿my, ¿e poniewa¿ wszystkie $\alpha_{n}$ w (\[I dla E z war &gt; 0\]) s¹ rzeczywiste, zatem, przy [*za³o¿eniu*]{} braku wp³ywu nowych stopni swobody na poprzednie, pojemnoœæ informacyjna $I$ wzrasta wraz ze wzrostem $N$. Zgodnie z kryterium (1), przyjmijmy dla rozwa¿anego przypadku rozk³adu fluktuacji energii, ¿e: $$\begin{aligned} \label{minimalne N 1 dla rozkladu E} N=1 \; .\end{aligned}$$ Jedyny wspó³czynnik $\alpha_{1}$ oznaczmy teraz $\alpha$, natomiast parametr $\Theta_{{\epsilon}} = (\theta_{{\epsilon}1}) \equiv \theta_{{\epsilon}}$.\ Zatem z (\[rozw cosinus dla ampl dla E\]) mamy amplitudê: $$\begin{aligned} \label{qn dla N=1} q\left({\bf x}_{{\epsilon}}\right) = \sqrt{2\alpha}\exp\left[{\alpha\left({{\bf x}_{{\epsilon}}^{min} - {\bf x}_{{\epsilon}}}\right)}\right] \; ,\end{aligned}$$ której odpowiada rozk³ad gêstoœci prawdopodobieñstwa fluktuacji energii ${\bf x}_{{\epsilon}}$: $$\begin{aligned} \label{pn dla N=1} p\left({\bf x}_{{\epsilon}}\right) = q^{2}\left({\bf x}_{{\epsilon}}\right) = 2\alpha\exp\left[{2\alpha\left({{\bf x}_{{\epsilon}}^{min} - {\bf x}_{{\epsilon}}}\right)}\right] \; , \quad \alpha \in \mathbf{R}_{+} \; .\end{aligned}$$ [**Rozk³ad gêstoœci prawdopodobieñstwa energii ${\epsilon}$ cz¹steczki**]{} jest koñcowym punktem analizy obecnego rozdzia³u. Poniewa¿ zgodnie z (\[E\]) mamy: $$\begin{aligned} \label{zakres E} {\bf y}_{{\epsilon}} \equiv \frac{{\epsilon}}{c} = \theta_{{\epsilon}} + {\bf x}_{{\epsilon}} \; , \;\;\;\; \frac{{\epsilon}_{0}}{c} \le {\bf y}_{{\epsilon}} < \infty \; , \;\;\;\; {\rm dla} \;\;\;\; {\bf x}_{{\epsilon}}^{min} \leq {\bf x}_{{\epsilon}} < \infty \; , \end{aligned}$$ gdzie ${\epsilon}_{0}/c = \theta_{{\epsilon}} + {\bf x}_{{\epsilon}}^{min}$. Zatem $d{\epsilon}/d{\bf x}_{{\epsilon}}=c$, sk¹d rozk³ad dla zmiennej ${\epsilon}$ ma postaæ: $$\begin{aligned} \label{rozklad p od E} p\left({\epsilon}\right) = p\left({\bf x}_{{\epsilon}}\right) \frac{1}{|d{\epsilon}/d{\bf x}_{{\epsilon}}|} = 2 \frac{\alpha}{c} \exp \left[{-2\alpha\left({{\epsilon}-{\epsilon}_{0}}\right)/c}\right] \; , \quad {\epsilon}_{0} \le {\epsilon} <\infty \; .\end{aligned}$$\ Pozosta³o jeszcze okreœlenie sta³ej $\alpha$. Poniewa¿ wartoœæ oczekiwana energii wynosi: $$\begin{aligned} \label{srednia E} \langle {E} \rangle \equiv c\, \theta_{\epsilon} = \int\limits_{{\epsilon}_{0}}^{+\infty}{d{\epsilon} \, p\left({\epsilon}\right) {\epsilon}} \; ,\end{aligned}$$ wiêc wstawiaj¹c (\[rozklad p od E\]) do (\[srednia E\]) otrzymuje siê: $$\begin{aligned} \label{stala alfa} 2 \alpha = c \left(\langle E \rangle - {\epsilon}_{0}\right)^{-1} \; .\end{aligned}$$ Zwróæmy uwagê, ¿e z (\[srednia E\]) wynika po pierwsze, ¿e $E$ jest estymatotem wartoœci oczekiwanej $\langle {E} \rangle$ energii cz¹stki: $$\begin{aligned} \label{estymator sredniej E} \widehat{\langle E \rangle} = E \; ,\end{aligned}$$ a po drugie, ¿e jest on nieobci¹¿ony.\ \ [**Szukany rozk³ad gêstoœci prawdopodobieñstwa energii cz¹stki**]{} ma zatem postaæ: $$\begin{aligned} \label{rozklad koncowy E} p\left( {\epsilon} \right) = \left\{ \begin{array}{l} (\left\langle E \right\rangle -{\epsilon}_{0})^{-1} \; \exp\left[- \left({{\epsilon}-{\epsilon}_{0}}\right)/(\left\langle E \right\rangle - {\epsilon}_{0})\right] \; \;\;\;\; {\rm dla} \quad \;\;\; {\epsilon} \ge {\epsilon}_{0} \\ \quad \quad 0 \;\; \quad \quad \quad \quad\quad \quad\quad \quad \;\; \quad \quad \quad \quad \quad \quad \quad \;\, {\rm dla} \quad \;\;\; {\epsilon} < {\epsilon}_{0}\end{array} \right. \; ,\end{aligned}$$ gdzie w drugiej linii po prawej stronie zaznaczono fakt nie wystêpowania cz¹stek z energi¹ mniejsz¹ ni¿ ${\epsilon}_{0}$. Rozk³ad (\[rozklad koncowy E\]) jest koñcowym rezultatem metody EFI. Jego postaæ daje zasadniczo [rozk³ad Boltzmanna]{} dla energii cz¹steczki w gazie.\ \ [**Rozk³ad Boltzmanna dla energii cz¹steczki**]{}: Aby domkn¹æ temat od strony fizycznej zauwa¿my, ¿e energia ca³kowita cz¹steczki wynosi ${\epsilon} = {\epsilon}_{kin} + V$, gdzie ${\epsilon}_{kin}$ jest energi¹ kinetyczn¹ cz¹steczki a $V$ jej energi¹ potencjaln¹. Do potencja³u $V$ mo¿emy dodaæ pewn¹ sta³¹ np. ${\epsilon}_{0}$, nie zmieniaj¹c przy tym fizycznego opisu zjawiska, zatem po przesuniêciu ${\epsilon}$ o ${\epsilon}_{0}$ otrzymujemy: $$\begin{aligned} \label{E bez E0} {\epsilon}_{0} = 0 \; \;\;\;\; {\rm oraz} \;\;\;\; {\epsilon} \ge 0 \; .\end{aligned}$$ Z kolei, dla cz¹stki gazu poruszaj¹cej siê bez obrotu, [*twierdzenie o ekwipartycji energii*]{} mówi, ¿e: $$\begin{aligned} \label{ekwipart} \left\langle E \right\rangle = \frac{3kT}{2} \; ,\end{aligned}$$ gdzie $T$ jest tempetatur¹ bezwzglêdn¹ gazu. Wstawiaj¹c (\[E bez E0\]) wraz z (\[ekwipart\]) do (\[rozklad koncowy E\]) otrzymujemy: $$\begin{aligned} \label{rozklad Boltzmanna z kT} p\left({\epsilon}\right) = (3kT/2)^{-1} e^{-2{\epsilon}/3kT} \; ,\quad\quad {\epsilon} \ge 0 \; ,\end{aligned}$$ czyli w³aœciw¹ postaæ rozk³adu Boltzmnna dla energii cz¹steczki w gazie o temperaturze $T$.\ \ [**Informacja Fishera dla $\theta_{\epsilon}$ i DORC dla estymatora $\left\langle E \right\rangle$**]{}: W przypadku $N=1$ oraz skalarnego parametru $\theta_{\epsilon}$, pojemnoœæ informacyjna (\[I dla E z war &gt; 0\]) jest równa informacji Fishera $I_{F}$ dla tego parametru. Gdy dolne ograniczenie na energiê cz¹stki wynosi $\epsilon_{0} = 0$, otrzymujemy po skorzystaniu z (\[stala alfa\]) oraz (\[Q dla E z war &lt; 0\]) nastêpuj¹cy zwi¹zek: $$\begin{aligned} \label{IF I oraz Q dla E oraz N=1} I_{F} (\theta_{\epsilon}) = I(\theta_{\epsilon}) = - Q(\theta_{\epsilon})= 4 \alpha^{2} = \frac{c^{2}}{\left\langle E \right\rangle^{2}} > 0 \; .\end{aligned}$$ Z Rozdzia³u \[Estymacja w modelach fizycznych na DORC\] wiemy, ¿e estymacja powy¿szego parametru oczekiwanego $\theta_{\epsilon}$, dualnego do $2 \alpha$, dla standardowego rozk³adu eksponentialnego, a takim jest rozk³ad Boltzmanna, spe³nia DORC w Twierdzeniu Rao-Cramera. Zatem wariancja estymatora tego parametru wynosi: $$\begin{aligned} \label{wariancja estymatora theta dla E} \sigma^{2}(\hat{\theta}_{\epsilon}) = \frac{1}{I_{F} (\theta_{{\epsilon}})} = \frac{\left\langle E \right\rangle^{2}}{c^{2}} \; ,\end{aligned}$$ sk¹d otrzymujemy wariancjê estymatora (\[estymator sredniej E\]) œredniej energii cz¹stki, równ¹: $$\begin{aligned} \label{wariancja estymatora sredniej E} \sigma^{2}\left(\widehat{\left\langle E \right\rangle}\right) = \left\langle E \right\rangle^{2} \; .\end{aligned}$$ Na tym koñczymy analizê metody EFI dla rozk³adu Boltzmanna.\ \ [**Uwaga o równowadze statystycznej w metodzie**]{} EFI: Bior¹c pod uwagê ograniczenia narzucone na normalizacjê (\[unormowanie q2\]) i skoñczonoœæ wartoœci oczekiwanej (\[srednia E\]) rozk³adu, metoda EFI wyznacza rozk³ad przy narzuceniu zasad informacyjnych. Poprzez obserwowan¹ zasadê strukturaln¹ dokonuje ona, dla g³adkiej funkcji wiarygodnoœci, separacji cz³onu estymacyjnego dla parametru $\theta_{\epsilon}$ zwi¹zanego z gêstoœci¹ pojemnoœci $\textit{i}$, której ca³ka $I$ jest œladem po metryce Rao-Fishera przestrzeni statystycznej ${\cal S}$, od cz³onu strukturalnego $\textit{q}$. Jest to wyrazem zasady Macha. Natomiast poprzez zasadê wariacyjn¹, metoda EFI dokonuje stabilizacji rozwi¹zania, wybieraj¹c najmniejsz¹ odleg³oœæ (liczon¹ wzd³u¿ geodezyjnej w przestrzeni statystycznej ${\cal S}$) wyestymowanego stanu uk³adu od stanu zaobserwowanego. Geometria ${\cal S}$ zale¿y od pojemnoœci kana³u informacyjnego $I$ wyliczonej z uwzglêdnieniem jej zale¿noœci od informacji strukturalnej $Q$.\ \ [**Uwaga**]{}: W zwi¹zku z powy¿szym oraz poprzednio podanym kryterium (1) doboru rangi amplitudy $N$ poprzez minimalizacjê $I$, pojawia siê nastêpuj¹ce, ogólne spojrzenie na estymacyjny charakter metody EFI.\ [**Estymacja metod¹ EFI**]{} oznacza wybór równañ generuj¹cych rozk³ad (lub równañ ruchu) na skutek dzia³ania dwóch czynników. Pierwszy z nich wymaga, po pierwsze wzrostu informacji o uk³adzie zawartej w gêstoœci pojemnoœci informacyjnej $\widetilde{\textit{i'}}\,$ okreœlonej w (\[gestosc i Amarii\]) oraz (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), ale tylko na tyle na ile wymaga tego, zawarta w zasadzie strukturalnej $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0$, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), struktura uk³adu opisana gêstoœci¹ informacji strukturalnej $\textit{q}$, (\[gestosc q dla niezaleznych Yn\]), przy, po drugie jednoczesnej minimalizacji ca³kowitej fizycznej informacji $K = I + Q$, (\[physical K\]). Oszacowanie poszukiwanego równania nastêpuje wiêc na skutek ¿¹dania samospójnoœci rozwi¹zania dwóch ró¿niczkowych zasad informacyjnych, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) oraz (\[var K rozpisana\]).\ Natomiast oczekiwana zasada strukturalna $I + \kappa Q =0$, (\[condition from K\]), jest w myœl zwi¹zku (\[rownowaznosc strukt i zmodyfikowanego strukt\]) drugim, metrycznym czynnikiem, domykaj¹cym powy¿sz¹ ró¿niczkow¹ analizê statystyczn¹ doboru modelu. Dzieli ona modele statystyczne na dwie grupy: modele metryczne, równowa¿ne zgodnie z (\[rownowaznosc strukt i zmodyfikowanego strukt\]) modelowi analitycznemu z metryk¹ Rao-Fishera oraz modele, które nie spe³niaj¹ zasady (\[rownowaznosc strukt i zmodyfikowanego strukt\]), czyli nierównowa¿ne modelowi z metryk¹ Rao-Fishera.\ \ [**Zwi¹zek EFI oraz ME dla standardowego rozk³adu eksponentialnego**]{}: Poprzednio otrzymaliœmy rozk³ad Boltzmanna (\[rozklad Boltzmanna z max entropii\]) jako szczególny przypadek rozwi¹zania zasady maksymalnej entropii (ME), realizowany w sytuacji warunku normalizacji oraz istnienia jedego parametru obserwowanego, którym jest œrednia wartoœæ energii cz¹stki. Szczególna postaæ standardowego rozk³adu eksponentialnego oraz te same warunki brzegowe s¹ przyczyn¹ otrzymania tego samego rozwi¹zania w metodzie EFI oraz ME. W ogólnoœci metoda EFI wykracza poza ca³¹ klasê modeli eksponentialnych metody ME. ### Model Aoki-Yoshikawy dla produkcyjnoœci bran¿ {#Model Aoki-Yoshikawy - ekonofizyka} Model Aoki i Yoshikawy (AYM) zosta³ opracowany w celu opisu produkcyjnoœci bran¿ kraju [@Aoki-Yoshikawy; @Garibaldi-Scalas]. Rozwa¿my $g$ ekonomicznych sektorów. Sektor $i$-ty jest scharakteryzowany przez [*czynnik wielkoœci produkcji*]{} $n_{i}$, tzn. liczebnoœæ, oraz [*zmienn¹ poziomu produkcyjnoœci*]{} $A$, tzn. wydajnoœæ jednostkow¹, przyjmuj¹c¹ wartoœci $a_{i}$.\ Niech liczebnoœæ $n_{i}$ bêdzie liczb¹ [*aktywnych*]{} pracowników w $i$-tym sektorze, co oznacza, ¿e praca (robocizna) jest jedynym czynnikiem produkcyjnym. Zmienn¹ losow¹ w AYM jest poziom produkcyjnoœci $A$, której rozk³ad jest okreœlony poprzez parê $(a_{i}, n_{i})$, $i=1,2...,g$.\ \ [**Unormowanie jako pierwszy warunek modelu**]{}: Za³ó¿my, ¿e [*ca³kowity zasób czynnika wielkoœci produkcji*]{}, tzn. liczba dostêpnych pracowników, jest w ekonomii zadany jako wielkoœæ egzogeniczna, czyli nie kontrolowana od wewn¹trz lecz zadana z zewnatrz. Niech jego wielkoœæ jest równa $n$, co traktujemy jako [*pierwsze ograniczenie*]{} w modelu, tak, ¿e zachodzi: $$\begin{aligned} \label{calkowita produkcji} \sum_{i=1}^{g} n_{i} = n \; .\end{aligned}$$\ [**Sektory**]{}. Uporz¹dkujmy wielkoœæ produkcyjnoœci $a_{i}$ w sektorach od najmniejszej do najwiêkszej: $$\begin{aligned} \label{ciag produkcyjnosci} a_{1} < a_{2} < ... < a_{g} \; .\end{aligned}$$ Poniewa¿ $a_{i}$ jest poziomem produkcyjnoœci $i$-tego sektora, zatem uzysk (wartoœæ produkcji) w $i$-tym sektorze wynosi: $$\begin{aligned} \label{produkcji w i tym sektorze} z_{i} = a_{i} \, n_{i} \; .\end{aligned}$$ Zatem ca³kowity uzysk $z$ w ekonomii kraju wynosi: $$\begin{aligned} \label{calkowity uzysk z} z = \sum_{i=1}^{g} z_{i} = \sum_{i=1}^{g} a_{i} \, n_{i} \; .\end{aligned}$$ [**Drugi warunek modelu**]{}: Wiekoœæ $z$ jest interpretowana jako [*produkt krajowy brutto*]{} (PKB).\ Za³o¿eniem AYM dla wartoœci PKB jest ustalenie wartoœci $z$ poprzez egzogenicznie zadany agregatowy popyt $D$ (demand), tzn.: $$\begin{aligned} \label{popyt} z = D \; .\end{aligned}$$ Zatem [*drugie ograniczenie*]{} w modelu ma postaæ: $$\begin{aligned} \label{calkowity uzysk} \sum_{i=1}^{g} a_{i} \, n_{i} = D \; .\end{aligned}$$ [**Celem metody EFI dla modelu AYM**]{} jest, po pierwsze wyznaczenie równania generuj¹cego, a po drugie, teoretycznego rozk³adu liczebnoœci dla zmiennej poziomu produkcyjnoœci $A$, tzn. okreœlenie wektora obsadzeñ: $$\begin{aligned} \label{rozklad n} {\bf n} = (n_{1} , n_{2} , ... , n_{g}) \; .\end{aligned}$$ [**Porównanie analizy dla AYM oraz rozk³adu Boltzmanna**]{}: Rozwa¿ane zagadnienie jest odpowiednikiem poprzedniego zagadnienia zwi¹zanego z okreœleniem rozk³adu prawdopodobieñstwa energii cz¹stki gazu. Mo¿na je okreœliæ jako zagadnienie rozmieszczenia $n$ cz¹stek gazu na $g$ poziomach energetycznych ${\epsilon}_{i}$ w warunkach równowagi statystycznej, w taki sposób, ¿e zachowane s¹, liczba cz¹stek: $$\begin{aligned} \label{calkowita liczba czastek} \sum_{i=1}^{g} n_{i} = n \; \end{aligned}$$ oraz ca³kowita energia gazu ${\cal E}$: $$\begin{aligned} \label{calkowita energia E} \sum_{i=1}^{g} {\epsilon}_{i} \, n_{i} = {\cal E} \; \;\;\; {\rm lub} \;\;\;\; \left\langle E \right\rangle = \sum_{i=1}^{g} \frac{n_{i}}{n} \,{\epsilon}_{i} = \frac{{\cal E}}{n} \; .\end{aligned}$$ Zatem poziom produkcyjnoœci $a_{i}$ jest analogiem poziomu energetycznego ${\epsilon}_{i}$, natomiast ograniczenie, które da³ w AYM popyt $D$ jest analogiem ograniczenia pochodz¹cego od wartoœci ca³kowitej energii ${\cal E}$ gazu.\ \ Dokonajmy nastêpuj¹cego przyporz¹dkowania pomiêdzy wielkoœciami opisuj¹cymi rozk³ad energii cz¹stki gazu oraz rozk³ad poziomu produkcyjnoœci. Po lewej stronie przyporz¹dkowania “$\leftrightarrow$” jest wielkoœæ dla energii cz¹stki, po prawej dla produkcyjnoœci pracownika. Strza³ki $\rightarrow$ w nawiasach oznaczaj¹ [*przejœcie od ci¹g³ego do dyskretnego*]{} rozk³adu zmiennej (lub na odwrót).\ \ Zatem, zmiennej energii cz¹stki $E$ odpowiada poziom produkcyjnoœci pracownika $A$: $$\begin{aligned} \label{przyporzadkowanie A E} E = ({\epsilon} \rightarrow {\epsilon}_{i}) \leftrightarrow A = (a_{i} \rightarrow a) \; .\end{aligned}$$ Rozk³adowi prawdopodobieñstwa zmiennej energii cz¹stki $p(\epsilon)$ odpowiada rozk³ad poziomów produkcyjnoœci pracownika $p(a)$: $$\begin{aligned} \label{przyporzadkowanie Pa Pe} \left( p (\epsilon) \rightarrow p_{\epsilon_{i}}=\frac{n_{i}}{n} \right) \leftrightarrow \left( p_{i}=\frac{n_{i}}{n} \rightarrow p (a)\right) \; .\end{aligned}$$ Normalizacje rozk³adów s¹ sobie przyporz¹dkowane nastêpuj¹co: $$\begin{aligned} \label{przyporzadkowanie Na Ne} \left(\int_{{\cal Y}_{{\epsilon}}} d\epsilon \, p(\epsilon) = 1 \rightarrow \sum_{i=1}^{g} p_{\epsilon_{i}} = 1 \right) \leftrightarrow \left( \sum_{i=1}^{g} p_{i} = 1 \rightarrow \int_{{\cal Y}_{a}} da \, p(a) =1 \right) \; .\end{aligned}$$ Wartoœci oczekiwanej energii cz¹stki odpowiada wartoœæ oczekiwana wartoœæ produkcyjnoœci pracownika: $$\begin{aligned} \label{przyporzadkowanie SRa SRe} \theta_{\epsilon} \equiv \left\langle E \right\rangle = \left( \int_{{\cal Y}_{{\epsilon}}} d{\epsilon} \, p({\epsilon}) \,\epsilon \rightarrow \sum_{i=1}^{g} \frac{n_{i}}{n} \,{\epsilon}_{i} \right) \leftrightarrow \left( \sum_{i=1}^{g} p_{i} \, a_{i} \rightarrow \int_{{\cal Y}_{a}} da \; p(a) \,a \right) = \left\langle A \right\rangle \equiv \theta_{A} \, ,\end{aligned}$$ przy czym w AYM wartoœæ oczekiwana produkcyjnoœci jest zadana jako: $$\begin{aligned} \label{wartosc oczekiwana produkcyjnosci} \left\langle A \right\rangle = D/n \; .\end{aligned}$$\ [**Okreœlenie zmiennej addytywnych fluktuacji**]{}: Aby analiza w AYM mog³a przebiegaæ dok³adnie tak samo jak dla rozk³adu Boltzmanna, musimy dokonaæ jeszcze jednego przejœcia po stronie produkcyjnoœci, a mianowicie przejœæ od poziomu produkcyjnoœci $A$ do jej fluktuacji $X_{a}$ od wartoœci oczekiwanej $\left\langle A \right\rangle$, tzn. dokonaæ addytywnego rozk³adu: $Y_{a} \equiv A = \left\langle A \right\rangle + X_{a}$.\ \ Odpowiedni analog pomiêdzy fluktuacjami energii i produkcyjnoœci ma wiêc nastêpuj¹c¹ postaæ: Dla energii cz¹stki zachodzi (\[E\]): $$\begin{aligned} \label{przyporzadkowanie xe} {\bf y}_{\epsilon} = \frac{\epsilon}{c} = \theta_{\epsilon} + {\bf x}_{\epsilon} \; , \;\; \frac{\epsilon_{0}}{c} \le {\bf y}_{\epsilon} \le \infty \; , \;\;\; {\bf x}_{{\epsilon}}^{min} \le {\bf x}_{{\epsilon}} < \infty \; ,\end{aligned}$$ gdzie skorzystano z za³o¿enia o nieograniczonoœci od góry fluktuacji energii (por. (\[qn rozwiazanie dla E\]) wraz z dyskusj¹ zawart¹ powy¿ej), natomiast dla produkcyjnoœci pracownika zachodzi: $$\begin{aligned} \label{przyporzadkowanie xa} {\bf y}_{a} \equiv a = \theta_{A} + {\bf x}_{a} \; , \;\;\;\; a_{0} \le {\bf y}_{a} \le \infty \; , \;\;\; {\bf x}_{a}^{min} = a_{0} -\theta_{A} \le {\bf x}_{a} < \infty\; .\end{aligned}$$\ [**Równanie generuj¹ce amplitudê produkcyjnoœci**]{}: Mo¿emy teraz przenieœæ powy¿sz¹ analizê EFI dla rozk³adu Boltzmanna na grunt modelu AYM. Zatem wychodz¹c z zasady wariacyjnej (\[gggg\]) oraz strukturalnej (\[obserwowana zas strukt dla Boltzmann suma n\]) i odpowiednich analogów pojemnoœci informacyjnej $(\ref{informacjaE})$ oraz informacji strukturalnej (\[Q diag dla E\]), otrzymujemy zgodnie z analiz¹ Rozdzia³u \[rozdz.energia\] równanie generuj¹ce rozk³ad produkcyjnoœci (\[falowe kappa 1\]), które dla wielkoœci próby $N=1$ ma postaæ: $$\begin{aligned} \label{produkcyjnosc row generujace} \frac{d^{2}q({\bf x}_{a})}{d\,{\bf x}_{a}^2} = \alpha^{2} q({\bf x}_{a}) \; ,\end{aligned}$$ gdzie $q({\bf x}_{a})$ jest amplitud¹ rozk³adu fluktuacji produkcyjoœci, a $\alpha$ rzeczywist¹ sta³¹.\ \ Rachunki analogiczne do przeprowadzonych pomiêdzy (\[falowe kappa 1\]) a (\[rozklad koncowy E\]), z warunkiem normalizacji (\[przyporzadkowanie Na Ne\]), które poprzednio doprowadzi³y do amplitudy (\[qn dla N=1\]) rangi $N=1$, daj¹ nastepuj¹ce rozwi¹zanie równania (\[produkcyjnosc row generujace\]) na amplitudê fluktuacji produkcyjnoœci $X_{a}$ w zakresie (\[przyporzadkowanie xa\]): $$\begin{aligned} \label{qn dla N=1 dla produkcyjnosci} q\left({\bf x}_{a}\right) = \frac{1}{\sqrt{ D/n }}\exp \left[ - \,{\frac{{(D/n) - a_{0} + {\bf x}_{a} }}{2 \, D/n } }\right] \; \;\;\;{\rm dla} \;\;\;\; {\bf x}_{a}^{min} = a_{0} -\theta_{A} \le {\bf x}_{a} < \infty\; \end{aligned}$$ oraz w analogii do (\[rozklad koncowy E\]), rozk³ad gêstoœci produkcyjnoœci $A$: $$\begin{aligned} \label{rozklad koncowy A} p\left( a \right) = \left\{ \begin{array}{l} \frac{1}{(D/n) - a_{0}} \; \exp\left( - \, \frac{a - a_{0}}{(D/n) - a_{0}} \right) \;\;\; \;\;\;\;\;\; {\rm dla} \quad \;\;\; a \ge a_{0} \\ \quad \quad 0 \;\, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\; {\rm dla} \quad \;\;\; a < a_{0} \end{array} \right. \; ,\end{aligned}$$ gdzie w drugiej linii po prawej stronie zaznaczono fakt nie wystêpowania produkcyjnoœci mniejszej ni¿ $a_{0}$. Rozk³ad (\[rozklad koncowy A\]) jest koñcowym rezultatem metody EFI dla modelu AYM produkcyjnoœci.\ \ [**Porównanie wyników metody AYM oraz EFI**]{}: Aby postaæ rozk³adu (\[rozklad koncowy A\]) odda³a w pe³ni wynik Aoki i Yoshikawy[^75], [**nale¿y powróciæ do dyskretyzacji wartoœci**]{} $a \rightarrow a_{i}$ zmiennej $A$.\ Rozwi¹zali oni powy¿szy problem stosuj¹c metodê mno¿ników Lagrange’a z warunkami ograniczaj¹cymi (\[calkowita produkcji\]) oraz (\[calkowity uzysk\]). Nastêpnie za³o¿yli [@Aoki-Yoshikawy], ¿e wartoœci produkcyjnoœci s¹ dyskretne, tworz¹c ci¹g arytmetyczny: $$\begin{aligned} \label{ciag a i} a_{i} = i \; a_{0} \; \;\;\; {\rm gdzie} \;\;\; i=1,2,...,g \;,\end{aligned}$$ gdzie $a_{0}$ jest najmniejsz¹ produkcyjnoœci¹.\ \ [**Wzglêdny popyt agregatowy $r$**]{}: W koñcu, przy za³o¿eniach, po pierwsze, ¿e liczba dostêpnych sektorów produkcyjnoœci jest bardzo du¿a, tzn. $g >> 1$ oraz po drugie, ¿e: $$\begin{aligned} \label{defin r} r \equiv \frac{D/n}{a_{0}} \; ,\end{aligned}$$ tzn. [*agregatowy popyt przypadaj¹cy na jednego pracownika $D/n$ odniesiony do najmniejszej produkcyjnoœæ*]{} $a_{0}$, jest bardzo du¿y, Aoki i Yoshikawy otrzymali wynik [@Aoki-Yoshikawy; @Garibaldi-Scalas]: $$\begin{aligned} \label{wynik AY na p} P(i|{\bf n}^{*}) = \frac{n_{i}^{*}}{n} \approx \frac{1}{r-1} \left( \frac{r-1}{r} \right)^{i}\approx (\frac{1}{r} + \frac{1}{r^2}) \; e^{ - \frac{i}{r} } \; , \;\;\; i=1,2,... \; , \;\;\; r >> 1 \; ,\end{aligned}$$ gdzie $n_{i}^{*}$, $i=1,2,...,g$, s¹ wspó³rzêdnymi $n_{i}$ wektora obsadzeñ (\[rozklad n\]), przy których prawdopodobieñstwo pojawienia siê tego wektora obsadzeñ jest maksymalne.\ \ [**Wynik analizy metod¹ AYM**]{}: Rezultat (\[wynik AY na p\]) podaje [*prawdopodobieñstwo, ¿e losowo wybrany pracownik jest w $i$-tym sektorze produkcyjnoœci, o ile gospodarka znajduje siê w stanie okreœlonym wektorem obsadzeñ*]{} ${\bf n}^{*} = (n_{1}^{*},n_{2}^{*},...,n_{g}^{*})$.\ \ [**Przejœcie do rozk³adu dyskretnego dla wyniku EFI**]{}: Aby porównaæ wynik (\[wynik AY na p\]) otrzymany w AYM z wynikiem (\[rozklad koncowy A\]) otrzymanym w metodzie EFI, przejdŸmy w (\[rozklad koncowy A\]) do rozk³adu dyskretnego. W tym celu musimy wyca³kowaæ wynik EFI w przedziale $(a_{i},a_{i+1})$, przy czym od razu za³o¿ymy, ¿e zachodzi $a_{i} = i \; a_{0}$, (\[ciag a i\]). W rezultacie otrzymujemy[^76]: $$\begin{aligned} \label{rozklad koncowy A dysktretny} P\left( i \right) = \int_{i a_{0}}^{(i+1) a_{0}} \! da \; p\left( a \right) = \left( 1 - e^{-1/(r - 1 )} \right) e^{- (i - 1)/(r-1) } \;\;\;\;\; {\rm dla} \quad \;\;\; i=1,2,... \; .\end{aligned}$$\ [**Porównanie modeli dla $a_{0}=0$**]{}. Niech $\delta a$ jest sta³¹ szerokoœci¹ sektorów produkcyjnoœci. Wtedy, w przypadku gdy $a_{0}=0$, wzór (\[rozklad koncowy E\]) metody EFI prowadzi w miejsce (\[rozklad koncowy A dysktretny\]) do rozk³adu: $$\begin{aligned} \label{rozklad koncowy A dysktretny a0 = 0} P\left( i \right) = \int_{(i-1) \delta a}^{i \delta a} \! da \; p\left( a \right) = \left( -1 + e^{1/\,\tilde{r}} \, \right) e^{- i/\,\tilde{r} } \; , \;\;\;\; i=1,2,... \; \;\;\; {\rm dla} \quad \; a_{0}=0 \; ,\end{aligned}$$ gdzie zamiast (\[defin r\]) wprowadziliœmy: $$\begin{aligned} \label{defin r tilda} \tilde{r} \equiv \frac{D/n}{\delta a} \; ,\end{aligned}$$ jako [*agregatowy popyt przypadaj¹cy na jednego pracownika $D/n$ odniesiony do szerokoœci sektora produkcyjnoœæ*]{} $\delta a$. W granicy $\tilde{r} >> 1$ z (\[rozklad koncowy A dysktretny a0 = 0\]) otrzymujemy: $$\begin{aligned} \label{rozklad koncowy A dysktretny a0 = 0 oraz r duze} P\left( i \right) \approx \left( \frac{1}{\tilde{r}} + \frac{1}{2 \;\tilde{r}^2} \right) \, e^{- i/\,\tilde{r} } \; , \;\;\;\; i=1,2,... \; , \;\;\; {\rm dla} \quad \; a_{0}=0 \; , \;\;\; \tilde{r} >> 1 \; .\end{aligned}$$\ \ \ [**Wniosek z porównania wyników metody AYM oraz EFI**]{}: W granicy du¿ych $\tilde{r}$ oba wyniki siê schodz¹. Jednak¿e granicê najmniejszej produkcyjnoœci $a_{0}$ metoda EFI ujmuje inaczej ni¿ AYM. To znaczy, formu³a EFI (\[rozklad koncowy A dysktretny a0 = 0 oraz r duze\]) daje dla $a_{0}=0$ inn¹ kwadratow¹ poprawkê w $\tilde{r}$ ni¿ rezultat (\[wynik AY na p\]) modelu AYM dla $r=\tilde{r}$ oraz $a_{0} \rightarrow 0$.\ \ [**Uwaga**]{}: Ponadto wynik (\[rozklad koncowy A dysktretny a0 = 0\]) jest dok³adny, natomiast w AYM dyskretyzacja poziomu produktywnoœci jest tylko wybiegiem technicznym, gdy¿ zmienna ta jest z natury ci¹g³a, do czego i tak w koñcu odwo³uje siê metoda AYM przy wyznaczaniu mno¿ników Lagrange’a, przechodz¹c z powodów rachunkowych w (\[calkowita produkcji\]) oraz (\[calkowity uzysk\]) z $g$ do nieskoñczonoœci.\ \ [: [*Analiza Aoki i Yoshikawy dla produkcyjnoœci.*]{} Podajmy sposób wyprowadzenia rozk³adu wektora obsadzeñ w AYM metod¹ czynników Lagrange’a [@Garibaldi-Scalas]. WprowadŸmy wektor ${\bf H}^{(n)}$ [*indywidualnych*]{} przypisañ, posiadaj¹cy tyle sk³adowych ilu jest pracowników w ca³ej gospodarce kraju: $$\begin{aligned} \label{wektor idywidualnych przypisan} {\bf H}^{(n)} \equiv ( H_{1}, H_{2},..., H_{n} ) = {\bf h} \equiv ( h_{1}, h_{2},..., h_{n} ) \; .\end{aligned}$$ Ka¿da ze wspó³rzêdnych $H_{i}$, $i=1,2,...,n$, mo¿e przyjmowaæ wartoœci $h_{i} = s$, gdzie $s \in \left\{ 1,2,...,g \right\}$, co oznacza, ¿e $i$-ty pracownik jest aktywny w $s$-tym sektorze gospodarki. Zatem wektor ${\bf h}$ podaje jedn¹ konfiguracjê indywidualnych przypisañ. Jeœli ustalimy wektor obsadzeñ ${\bf n}$, (\[rozklad n\]), to liczba $W$ ró¿nych ${\bf h}$ (indywidualnych konfiguracji przypisañ), realizuj¹cych ten sam ustalony wektor obsadzeñ ${\bf n}$, wynosi [@podrecznik; @z; @kombinatoryki]: $$\begin{aligned} \label{liczba konfiguracji W dla wekt obsadzen} W({\bf H}|{\bf n}) = \frac{n!}{\prod_{i=1}^{g} n_{i}!} \; .\end{aligned}$$ Boltzman zauwa¿y³, ¿e gdy uk³ad znajduje siê w równowadze statystycznej to, prawdopodobieñstwo $\pi({\bf n})$, ¿e znajduje siê on w stanie o okreœlonym wektorze obsadzeñ ${\bf n}$ (tzn. ¿e pojawi³ siê taki w³aœnie wektor obsadzeñ), jest proporcjonalne do liczby jego mo¿liwych realizacji, tzn. do $W({\bf H}|{\bf n})$. Zatem: $$\begin{aligned} \label{prawdopodobienstwo pojawieniem sie wektora n} \pi({\bf n}) = W({\bf H}|{\bf n}) \;P({\bf H}|{\bf n}) = \frac{n!}{\prod_{i=1}^{g} n_{i}!} \;\prod_{i=1}^{g} p^{n_{i}} = \frac{n!}{\prod_{i=1}^{g} n_{i}!} \;p^{n} = {\cal K} \, \frac{n!}{\prod_{i=1}^{g} n_{i}!} \; ,\end{aligned}$$ gdzie ${\cal K}$ jest w³aœciw¹ normalizacyjn¹ sta³¹, a $p$ jest [*prawdopodobieñstwem*]{} zajêcia przez $i$-tego pracownika okreœlonego $s$-tego sektora obsadzeñ, [*które zosta³o przyjête jako takie samo dla wszystkich indywidualnych konfiguracji tych obsadzeñ*]{}. W celu rozwi¹zania postawionego problemu maksymalizacji prawdopodobieñstwa $\pi({\bf n})$ z warunkami (\[calkowita produkcji\]) oraz (\[calkowity uzysk\]), rozwi¹zujemy poni¿szy uk³ad $g$ równañ: $$\begin{aligned} \label{MNW z warunkami} \frac{\partial }{\partial n_{i}} \left[ \ln \pi({\bf n}) + \nu \left( \sum_{i=1}^{g}n_{i} -n \right) - \beta \left( \sum_{i=1}^{g} a_{i}n_{i} - D \right) \right] = 0\; ,\end{aligned}$$ otrzymuj¹c jako rozwi¹zanie wektor obsadzeñ ${\bf n}$: $$\begin{aligned} \label{n*} n_{i} = n_{i}^{*} = e^{\nu} \, e^{-\beta a_{i}} \; , \;\;\; i=1,2,...,g \; .\end{aligned}$$ Sta³e $\nu$ oraz $\beta$ otrzymujemy wykorzystuj¹c (\[n\*\]) w (\[calkowita produkcji\]) oraz (\[calkowity uzysk\]).\ Uwaga: Aby zapisaæ $\prod_{i=1}^{g} n_{i}!$ w formie nadaj¹cej siê do minimalizacji, korzystamy z przybli¿enia Stirling’a: $$\begin{aligned} \label{wzor Stirlinga} \ln \left[ \prod_{i=1}^{g} n_{i}! \right] \approx \sum_{i=1}^{g} n_{i} (\ln n_{i} -1) \; ,\end{aligned}$$ s³usznego dla du¿ego uk³adu z $n>>1$ oraz $n_{i}>>1$, $i=1,2,...,g$.\ \ [**Uk³ad w równowadze statystycznej**]{}: Przedstawiona w tym przypisie metoda znajdowania [*rozk³adu w równowadze statystycznej*]{} zak³ada spe³nienie hipotezy wyra¿onej równaniem (\[prawdopodobienstwo pojawieniem sie wektora n\]) i mówi¹cej, ¿e [*wszystkie stany opisane wektorem indywidualnych przypisañ ${\bf H}^{(n)}\,$, a spe³niaj¹ce warunki (\[calkowita produkcji\]) oraz (\[calkowity uzysk\]) s¹ równie prawdopodobne*]{}. Rozk³ad opisany wektorem ${\bf n}^{*}$ jest w tym ujêciu sednem definicji rozk³adu, bêd¹cego w równowadze statystycznej. ]{} ### Rozk³ad Maxwella-Boltzmanna dla prêdkoœci Poni¿ej wyznaczymy rozk³adu prêdkoœci cz¹steczki w gazie. Wychodz¹c z informacji Fishera (\[minp\]) dla pêdu oraz pos³uguj¹c siê wariacyjn¹ i strukturaln¹ zasad¹ informacyjn¹ (\[epip\]) otrzymamy równanie generuj¹ce i znajdziemy jego rozwi¹zania, tzn. postaæ amplitud dla rozk³adu prêdkoœci.\ \ [**Wartoœæ wspó³czynnika efektywnoœci**]{}: Ze wzglêdu na spójnoœæ rozwa¿añ dla czterowektora pêdu, przyjêcie wspó³czynnika $\kappa=1$ w rozwa¿aniach dla energii skutkuje przyjêciem $\kappa=1$ w analizie dla rozk³adu pêdu.\ \ [**Pojemnoœæ informacyjna parametru**]{} $\Theta_{\vec{\wp}}$ zosta³a podana w (\[minp\]): $$\begin{aligned} I\left(\Theta_{\vec{\wp}}\right) = - 4 \int_{{\cal X}_{\wp}} {\! d\vec{\bf x}_{\wp} \sum\limits _{n=1}^{N}{ {\sum\limits_{k=1}^{3}{\left(\! {\frac{{\partial q_{n}(\vec{\bf x}_{\wp})}}{{\partial x_{\wp_{k}}}}}\right)^{2} \! }}}} \, , \;\;\; \nonumber\end{aligned}$$ gdzie znaczenie znaku “minus” w definicji pojemnoœci informacyjnej dla czêœci pêdowej zosta³o omówione w Rozdziale \[zasady inf dla energii i predkosci\].\ \ [**Informacja strukturalna i zale¿noœæ $\texttt{q\!F}_{n}$ od amplitudy oraz prêdkoœci**]{}: $Q$ dla parametru $\Theta_{\vec{\wp}}$ ma postaæ: $$\begin{aligned} \label{inform strukt dla B-M} Q\left(\Theta_{\vec{\wp}}\right) = \int_{{\cal X}_{\wp}} {\! d\vec{\bf x}_{\wp} \sum\limits _{n=1}^{N} q_{n}^{2}(\vec{\bf x}_{\wp}) \,\texttt{q\!F}_{n}(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) } \, . \;\;\;\end{aligned}$$ Wprowadzenie do obserwowanej informacji strukturalnej $\texttt{q\!F}_{n}(q_{n}(\vec{\bf x}_{\wp}),\vec{\bf x}_{\wp})$ jej jawnej zale¿noœci nie tylko od amplitud $q_{n}$, ale równie¿ od pêdu $\vec{\bf x}_{\wp}$, pozwala na rozwa¿enie szerszego zakresu zagadnieñ ni¿ to mia³o miejsce dla przypadku rozk³adu energii, mianowicie pozwala na rozwa¿enie rozwi¹zañ [*nierównowagowych*]{}.\ \ [**Obserwowana, strukturalna zasada informacyjna**]{}: Podobnie jak to uczyniliœmy w (\[postac I po calk czesci\]) dla energii, tak i teraz dla pojemnoœci informacyjnej $I\left(\Theta_{\vec{\wp}}\right)$, (\[minp\]), dokonamy ca³kowania przez czêœci. Zak³adaj¹c dodatkowo, ¿e amplitudy dla prêdkoœci $q_{n}(\vec{\bf x}_{\wp})$ znikaj¹ w $\pm$ nieskoñczonoœci, [**oczekiwana strukturalna zasada informacyjna**]{} $\,\widetilde{\textit{i'}}(\Theta_{\vec{\wp}}) + \widetilde{\mathbf{C}}_{\vec{\wp}} + \kappa \, \textit{q}(\Theta_{\vec{\wp}}) = 0$ w (\[epip\]) przyjmuje postaæ: $$\begin{aligned} \label{strukt zas dla pedu calkow czesci} \!\!\!\!\! \widetilde{\textit{i'}}\left(\Theta_{\vec{\wp}}\right) + \textit{q}\left(\Theta_{\vec{\wp}}\right) = 4 \sum\limits _{n=1}^{N} \left[ q_{n}(\vec{\bf x}_{\wp})\sum_{k=1}^{3}{\frac{{\partial^{2} q_{n}(\vec{\bf x}_{\wp})}}{{\partial x_{\wp_{k}}^{2}}}} \! + \frac{1}{4}\; q_{n}^{2}(\vec{\bf x}_{\wp}) \,\texttt{q\!F}_{n}(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) \right] = 0 \, , \;\;\,\end{aligned}$$ gdzie fakt, ¿e $\widetilde{\mathbf{C}}_{\vec{\wp}} = 0$, nie wnosz¹c tym samym wk³adu w powy¿sz¹ zasadê strukturaln¹, wynika ze znikania amplitud w nieskoñczonoœci.\ \ [**Wariacyjna zasada informacyjna**]{} w (\[epip\]) przyjmuje postaæ: $$\begin{aligned} \label{wariacyjna zas dla pedu} \!\!\!\!\! \delta_{(q_{n})} \! \left(I\left(\Theta_{\vec{\wp}}\right) + Q\left(\Theta_{\vec{\wp}}\right)\right) = 4 \!\! \int_{{\cal X}_{\wp}}{\!\! d\vec{\bf x}_{\wp}\sum\limits _{n=1}^{N}{\left[ - {\sum\limits_{k=1}^{3}{ \!\left( \!{\frac{{\partial q_{n}(\vec{\bf x}_{\wp})}}{{\partial x_{\wp_{k}}}}}\right)^{2} + \! \frac{1}{4} q_{n}^{2}(\vec{\bf x}_{\wp}) \,\texttt{q\!F}_{n}(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) }}\right]}} = 0 \, . \;\;\, \end{aligned}$$\ [**Równania Eulera-Lagrange’a**]{}: Rozwi¹zaniem $N$-funkcyjnego problemu wariacyjnego (\[wariacyjna zas dla pedu\]) jest uk³ad równañ Eulera-Lagrange’a: $$\begin{aligned} \label{row E-L dla pedu} \sum\limits _{m=1}^{3}{\frac{\partial}{{\partial x_{\wp_{m}}}}\left({\frac{{\partial \, k \,(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) }}{{\partial \left(\frac{{\partial\, q_{n}(\vec{\bf x}_{\wp})}}{{\partial x_{\wp_{m}}}} \right) }}}\right)}=\frac{{\partial k(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp})}}{{\partial q_{n}(\vec{\bf x}_{\wp})}},\quad\quad n=1, 2,..., N \; ,\end{aligned}$$ gdzie $$\begin{aligned} \label{gestosc k dla pedu} k \,(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) = 4 {\left[ - {\sum\limits_{k=1}^{3}{ \!\left( \!{\frac{{\partial q_{n}(\vec{\bf x}_{\wp})}}{{\partial x_{\wp_{k}}}}}\right)^{2} + \! \frac{1}{4} q_{n}^{2}(\vec{\bf x}_{\wp}) \,\texttt{q\!F}_{n}(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) }}\right]} \; \end{aligned}$$ jest gêstoœci¹ informacji fizycznej (\[k form\]) zdefiniowan¹ w Rozdziale \[equations of motion\].\ \ [**Nieujemnoœæ $k$**]{}: Zauwa¿my, ¿e z (\[gestosc k dla pedu\]) i z ¿¹dania nieujemnoœci informacji fizycznej $k$ na poziomie obserwowanym, wynika: $$\begin{aligned} \label{qF dodatnia dla rozkl pedu} \texttt{q\!F}_{n}(q_{n}(\vec{\bf x}_{\wp}), \vec{\bf x}_{\wp}) \geq 0 \; , \end{aligned}$$ tzn. [*nieujemnoœæ obserwowanej informacji strukturalnej w analizie estymacyjnej wartoœci oczekiwanej pêdu cz¹steczki gazu.*]{}\ \ [**Uwaga**]{}: Poni¿ej, ze wzglêdu na uproszczenie zapisu, pominiemy zaznaczenie fluktuacji pêdu $\vec{\bf x}_{\wp}$ w argumencie amplitudy $q_{n}(\vec{\bf x}_{\wp})$.\ \ [**Uk³ad równañ ró¿niczkowych**]{}: Dla ka¿dego $n=1,2,...,N$, zasada strukturalna (\[strukt zas dla pedu calkow czesci\]) jest na poziomie obserwowanym nastêpuj¹ca: $$\begin{aligned} \label{uppro} q_{n}\sum_{m=1}^{3}{\frac{{\partial^{2}q_{n}}}{{\partial x_{\wp_{m}}^{2}}}} + \frac{1}{4} q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp}) = 0 \; .\end{aligned}$$ Natomiast ka¿de z $N$ równañ Eulera-Lagrange’a (\[row E-L dla pedu\]) ma postaæ nastêpuj¹cego równania ró¿niczkowego: $$\begin{aligned} \label{roznp} \sum\limits_{m=1}^{3}{\frac{{\partial^{2}q_{n}}}{{\partial x_{\wp_{m}}^{2}}}} = - \frac{1}{2}\frac{{\partial ( \frac{1}{4} q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp}) ) }}{{\partial q_{n}}} \; .\end{aligned}$$ Równanie strukturalne (\[uppro\]) wraz z równaniem Eulera-Lagrange’a (\[roznp\]) pos³u¿y do wyprowadzenia równania generuj¹cego rozk³ad Maxwella-Boltzamnna.\ \ [**Wyprowadzenie równania generuj¹cego**]{}: Równania (\[uppro\]) oraz (\[roznp\]) pozwalaj¹ wyeliminowaæ wystêpuj¹c¹ w nich sumê, daj¹c równanie: $$\begin{aligned} \label{row rozniczkowe na qF dla p} \frac{1}{2}\frac{{\partial (\frac{1}{4} q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp}))}}{{\partial q_{n}}} = \frac{{\frac{1}{4} q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp}) }}{{q_{n}}} \; ,\end{aligned}$$ które po obustronnym sca³kowaniu prowadzi do rozwi¹zania: $$\begin{aligned} \label{rozw dla qF} \frac{1}{4} \, q_{n}^{2} \,\texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp}) = q_{n}^{2} \, f_{n}\left({\vec{\bf x}_{\wp}}\right) \; ,\end{aligned}$$ lub $$\begin{aligned} \label{rozw dla samego qF} \texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp}) = 4 \, f_{n}\left({\vec{\bf x}_{\wp}}\right) \geq 0\; .\end{aligned}$$ Funkcja $f_{n}(\vec{\bf x}_{\wp})$ nie zale¿y od amplitudy $q_{n}$ i pojawi³a siê w wyniku ca³kowania równania (\[row rozniczkowe na qF dla p\]) jako pewna sta³a ca³kowania w znaczeniu jej niezale¿noœci od amplitudy $q_{n}$. Zaznaczona nieujemnoœæ funkcji $f_{n}\left({\vec{\bf x}_{\wp}}\right)$ w jej dziedzinie wynika z nieujemnoœci $\texttt{q\!F}_{n}(q_{n}, \vec{\bf x}_{\wp})$ otrzymanej w (\[qF dodatnia dla rozkl pedu\]).\ \ [**Równanie generuj¹ce**]{}: Wykorzystuj¹c (\[rozw dla qF\]) w (\[uppro\]) eliminujemy obserwowan¹ informacjê strukturaln¹, otrzymuj¹c [*równanie generuj¹ce*]{} dla amplitudy rozk³adu (fluktuacji) prêdkoœci $\vec{\bf x}_{\wp}$ cz¹steczki w gazie: $$\begin{aligned} \label{rpw generujace dla q pedu} \nabla^{2} q_{n}(\vec{\bf x}_{\wp}) = - q_{n}(\vec{\bf x}_{\wp}) \, f_{n}\left({\vec{\bf x}_{\wp}}\right) \; ,\end{aligned}$$ gdzie $\nabla^{2} \equiv \sum_{m=1}^{3} {\partial^{2}}/{\partial x_{\wp_{m}}^{2}}$ jest operatorem Laplace’a ze wzglêdu na wspó³rzêdne pêdowe $x_{\wp_{m}}$.\ \ Równanie (\[rpw generujace dla q pedu\]), podobnie jak (\[falowe kappa 1\]), pojawi³o siê jako rozwi¹zanie strukturalnej i wariacyjnej zasady informacyjnej.\ Rozwi¹zanie $q_{n}$ równania (\[rpw generujace dla q pedu\]) jest [*samospójnym*]{} rozwi¹zaniem sprzê¿onego uk³adu równañ ró¿niczkowych (\[uppro\]) i (\[roznp\]), utworzonych przez parê zasad informacyjnych.\ \ [**Rozwi¹zanie równania generuj¹cego**]{}: Poni¿ej rozwi¹¿emy równanie generuj¹ce (\[rpw generujace dla q pedu\]), czyni¹c kilka fizycznych za³o¿eñ odnoœnie postaci funkcji $f_{n}(\vec{\bf x}_{\wp})$. Jej postaæ wp³ywa na postaæ otrzymanych amplitud $q_{n}(\vec{\bf x}_{\wp})$, a zatem równie¿ na rozk³ad $p(\vec{\bf x}_{\wp})$.\ \ [**Fizyczne za³o¿enia o postaci $f_{n}(\vec{\bf x}_{\wp})$**]{}: Za³ó¿my, ¿e ka¿da gêstoœæ rozk³adu prawdopodobieñstwa $p_{n}(\vec{x})$ jest tak¹ sam¹ funkcj¹ parzyst¹ ka¿dej wspó³rzêdnej $x_{\wp_{i}}$ (fluktuacji) pêdu $\vec{\bf x}_{\wp}$. W konsekwencji [*uk³ad jest izotropowy*]{}, tzn. rozk³ad prawdopodobieñstwa dla pêdu nie zale¿y od kierunku w bazowej przestrzeni po³o¿eñ.\ Nastêpnie zak³adamy [*nierelatywistyczne przybli¿enie*]{}, co oznacza, ¿e prêdkoœci cz¹steczek s¹ du¿o mniejsze od prêdkoœci œwiat³a $c$, czyli fluktuacja pêdu cz¹steczki $x_{\wp_{i}}$ jest równie¿ ma³a w porównaniu z $mc$.\ Z powy¿szych za³o¿eñ wynika ogólna postaæ funkcji $f_{n}(\vec{\bf x}_{\wp})$. Otó¿ jej rozwiniêcie w szereg potêgowy ma tylko sk³adowe parzyste modu³u fluktuacji pêdu $|\vec{\bf x}_{\wp}|$, a poniewa¿ wartoœæ $|\vec{\bf x}_{\wp}|$ jest ma³a, zatem szereg ten obetniemy na drugim wyrazie: $$\begin{aligned} \label{postac funkcji ABp} f_{n}(\vec{\bf x}_{\wp}) = A_{n} + B \, |\vec{\bf x}_{\wp}|^{2} \; ,\quad \quad A_{n}, B = const. \end{aligned}$$ [**Równanie generuj¹ce z $f_{n}$**]{}: Podstawiaj¹c (\[postac funkcji ABp\]) do (\[rpw generujace dla q pedu\]), otrzymujemy nastêpuj¹c¹ postaæ równania generuj¹cego: $$\begin{aligned} \label{rownanie gener z f kwadrat} \nabla^{2}q_{n}\left({\vec{\bf x}_{\wp}}\right)+\left(A_{n} + B |\vec{\bf x}_{\wp}|^{2}\right)q_{n}\left({\vec{\bf x}_{\wp}}\right) = 0 \; \;\; {\rm dla} \;\;\; n=1,2,...,N \; .\end{aligned}$$ PrzejdŸmy do nowego indeksu: $$\begin{aligned} \label{index n'} n^{'}:= n-1 \; = \; 0, 1,..., N-1 \; .\end{aligned}$$ Wtedy w miejsce (\[rownanie gener z f kwadrat\]) otrzymujemy równanie: $$\begin{aligned} \label{rownanie gener z f kwadrat z n'} \nabla^{2} q_{n'} \left( \vec{\bf x}_{\wp} \right) + \left(A_{n'} + B |\vec{\bf x}_{\wp}|^{2} \right) q_{n'} \left( \vec{\bf x}_{\wp} \right) = 0 \; \;\; {\rm dla} \;\;\; n' = 1 , 2,..., N-1 \; , \end{aligned}$$ które po dokonaniu separacji zmiennych kartezjañskich i faktoryzacji amplitudy: $$\begin{aligned} \label{separacja} q_{n'}\left({\vec{\bf x}_{\wp}}\right) = q_{n'_{1}}\left(x_{{\wp}_{1}}\right) q_{n'_{2}}\left(x_{{\wp}_{2}}\right) q_{n'_{3}}\left(x_{{\wp}_{3}}\right) \; , $$ przechodzi w równowa¿ny mu uk³ad trzech równañ ró¿niczkowych: $$\begin{aligned} \label{row gen po faktoryzacji dla p} q_{n'_{i}}^{''}\left({x_{\wp_i}}\right)+\left({A_{n'_{i}} + B \, x_{\wp_i}^{2}}\right)q_{n'_{i}}\left({x_{\wp_i}}\right) = 0 \; , \quad i=1,2,3, \;\;\; \sum\limits _{i=1}^{3}{A_{n'_{i}}} \equiv A_{n'} \; .\end{aligned}$$ Gdy sta³e równania (\[row gen po faktoryzacji dla p\]) maj¹ postaæ: $$\begin{aligned} \label{stale row gener dla ni'} A_{n_{i}^{'}}=\frac{n_{i}^{'} + 1/2}{a_{0}^{2}} \, , \;\;\; B = -\frac{1}{{4a_{0}^{4}}} \, , \;\;\; a_{0}=const. \, , \;\;\; n_{i}^{'}=0,1,... \; ,\end{aligned}$$ wtedy ma ono rozwi¹zanie.\ \ [**Postaæ rozwi¹zania**]{}: Poniewa¿ ka¿de z równañ (\[row gen po faktoryzacji dla p\]) jest [**równaniem Helmholtz’a**]{}, zatem jego rozwi¹zaniami s¹ paraboliczno-cylindryczne funkcje [@Ab; @kamke]: $$\begin{aligned} \label{rozwiazanie dla q z ni' dla pedu} q_{n_{i}^{'}}\left({x_{\wp_{i}}}\right) = e^{-x_{\wp_{i}}^{2}/\left(4 a_{0}^{2} \right)} \; 2^{- n_{i}^{'}/2} \; H_{n_{i}^{'}} \left( \frac{x_{\wp_{i}}}{a_{0} \sqrt{2}} \right) \, , \;\;\; {\rm gdzie} \;\;\; n_{i}^{'}=0,1,... \; , \;\;\; i=1,2,3 \; ,\end{aligned}$$ gdzie $H_{n_{i}^{'}}$ s¹ wielomianami Hermite’a: $$\begin{aligned} \label{wiel Hermite} H_{j}\left( t \right) = j! \sum\limits_{m=0}^{\left[j/2\right]} \left({-1}\right)^{m} \, \frac{\left( 2 \,t \right)^{j-2m}}{m!\left( j-2m \right)! } \; , \;\;\; j = 0,1,... \; ,\end{aligned}$$ gdzie $\left[j/2\right]$ oznacza liczbê ca³kowit¹ nie przekraczaj¹c¹ $j/2$.\ \ [**Amplitudy fluktuacji pêdu**]{}: Wstawiaj¹c (\[rozwiazanie dla q z ni’ dla pedu\]) do (\[separacja\]) otrzymujemy szukan¹ postaæ amplitudy fluktuacji pêdu: $$\begin{aligned} \label{rozwiazanie dla q z n' dla pedu} \!\!\!\!\!\!\!\!\!\! q_{n^{'}}\left({\vec{\bf x}_{\wp}}\right) & = & e^{ - |\vec{\bf x}_{\wp}|^{2}/\left(4 a_{0}^{2} \right)} \; 2^{- n^{'}/2} \; \cdot \nonumber \\ & & \nonumber \\ & \cdot & \!\!\!\!\!\!\!\!\!\! \sum\limits _{{\scriptstyle {\quad ijk\hfill\atop {\scriptstyle i+j+k=n^{'}\hfill}}}}{\!\!\!\!\!\! a_{n_{ijk}^{'}}} \; H_{i}\left(\frac{x_{\wp_{1}}}{a_{0}\sqrt{2}}\right)H_{j}\left(\frac{x_{\wp_{2}}}{a_{0}\sqrt{2}}\right)H_{k}\left(\frac{x_{\wp_{3}}}{a_{0}\sqrt{2}}\right) \, , \;\;\; {\rm gdzie} \;\;\; a_{n_{ijk}^{'}} = const. \; , \;\;\end{aligned}$$ przy czym: $$\begin{aligned} \label{zwiazek n z ni'} \sum\limits _{i=1}^{3}{n_{i}^{'} = n^{'}} = 0, 1,..., N-1 \; .\end{aligned}$$\ [**Rozk³ad fluktuacji pêdu**]{}: Funkcja rozk³adu prawdopodobieñstwa ma wiêc zgodnie z (\[p jako suma po qn2 przez N\]) postaæ: $$\begin{aligned} \label{rozwiazanie dla prawdop z n' dla pedu} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! p\left({\vec{\bf x}_{\wp}}\right) &=& \frac{1}{N}\sum_{n'=0}^{N-1}{q_{n'}^{2}}\left({\vec{\bf x}_{\wp}}\right) = p_{0} \, e^{- \left| \vec{\bf x}_{\wp} \right|^{2}/ \left({2a_{0}^{2}}\right)} \;\; \cdot \nonumber \\ &\cdot& \left\{ 1+\sum_{n^{'}=1}^{N-1}{2^{-n^{'}}\left[\sum\limits _{{\scriptstyle {\quad ijk\hfill\atop {\scriptstyle i+j+k=n^{'}\hfill}}}}{b_{n_{ijk}^{'}} H_{i}\left(\frac{x_{\wp_{1}}}{a_{0} \sqrt{2}}\right) H_{j}\left(\frac{x_{\wp_{2}}}{a_{0} \sqrt{2}}\right) H_{k}\left(\frac{x_{\wp_{3}}}{a_{0} \sqrt{2}}\right)}\right]^{2}}\right\} \; ,\end{aligned}$$ gdzie $p_{0}=a_{0_{000}}^{2}/N$, a liczba 1 w nawiasie klamrowym pochodzi z $n^{'}=0$ w sumie w pierwszej równoœci, natomiast nowe sta³e $b_{n_{ijk}^{'}}$ s¹ proporcjonalne do sta³ych $a_{n_{ijk}^{'}}$.\ Po raz pierwszy takie wyprowadzenie postaci funkcji rozk³adu gêstoœci prawdopodobieñstwa dla fluktuacji pêdu zosta³o podane w [@Frieden].\ \ [**Rozwi¹zania równowagowe i nierównowagowe**]{}: Wartoœæ $N=1$, dla której (\[rozwiazanie dla prawdop z n’ dla pedu\]) jest rozk³adem Gaussa, daje równowagowy [*rozk³ad Maxwella-Boltzmanna dla (fluktuacji) pêdu cz¹steczki gazu*]{}. Pozosta³e rozwi¹zania dla $N\ge2$ daj¹ rozwi¹zania nierównowagowe [@Frieden]. Jednak i one s¹ stacjonarne ze wzglêdu na fakt bycia samospójnymi rozwi¹zaniami uk³adu sprzê¿onych równañ ró¿niczkowych zasady strukturalnej (\[uppro\]) i wariacyjnej (\[roznp\]).\ Wczeœniej, rozk³ady (\[rozwiazanie dla prawdop z n’ dla pedu\]) z $N\ge2$ zosta³y odkryte przez Rumer’a i Ryskin’a [@Rumer; @and; @Ryskin-1980] jako rozwi¹zania równania transportu Boltzmann’a.\ \ [**Interferencja rozwi¹zañ**]{}: Zwróæmy uwagê, ¿e rozwi¹zanie (\[rozwiazanie dla prawdop z n’ dla pedu\]) implikuje interferencjê pomiêdzy wyra¿eniami iloczynowymi wystêpuj¹cymi w amplitudach (\[rozwiazanie dla q z n’ dla pedu\]). Interferencja pojawi³a siê wiêc jako cecha charakterystyczna dla rozwi¹zywanego równania ró¿niczkowego oraz wprowadzenia amplitud do opisu uk³adu, a nie jako cecha charakterystyczna wy³¹cznie mechaniki kwantowej.\ \ [**Rozk³ad Maxwella-Boltzmanna dla prêdkoœci z $N=1$**]{}: Za³o¿yliœmy na wstêpie, ¿e zbiornik zawieraj¹cy cz¹steczki gazu jest w spoczynku, w pewnym inercjalnym uk³adzie wspó³rzêdnych. Zatem œrednia prêdkoœæ cz¹steczki wynosi zero, tzn. $\vec{\theta}_{\wp} = 0$ i z (\[y\_p\]) otrzymujemy $\vec{\bf y}_{\wp} = \vec{\bf x}_{\wp} = \vec{\wp}$.\ \ Tak wiec, dla $N=1$ rozk³ad gêstoœci prawdopodobieñstwa pêdu (\[rozwiazanie dla prawdop z n’ dla pedu\]) przyjmuje postaæ: $$\begin{aligned} \label{row} p \left( \vec{\wp} \right) = p_{0} \; e^{- |\vec{\wp}\,|^{2}/(2a_{0}^{2})} = p_{0}\; e^{- \wp_{1}^{2}/(2a_{0}^{2})} e^{- \wp_{2}^{2}/(2a_{0}^{2})} e^{- \wp_{3}^{2}/(2a_{0}^{2})} \; ,\end{aligned}$$ gdzie $\vec{\wp} = (\wp_{1}, \wp_{2}, \wp_{3})$ oraz $-\infty<\wp_{i}<\infty$, $i=1,2,3$. Wyznaczmy wartoœæ sta³ej $a_{0}$ dla przypadku zerowej energii potencjalnej. Z warunku normalizacji $\int d \vec{\wp} \, p\left(\vec{\wp} \right)=1\,$ wyznaczamy, ¿e sta³a $p_{0}$ w (\[row\]) wynosi: $$\begin{aligned} p_{0}=\frac{1}{{\left({2\pi}\right)^{{3\mathord{\left/{\vphantom{32}}\right.\kern -\nulldelimiterspace}2}}a_{0}^{3}}}\label{p0} \; .\end{aligned}$$ Poniewa¿ wtedy $\left\langle E\right\rangle = \left\langle \vec{\wp}^{\,2}\right\rangle/(2m)\,$, a z twierdzenia o ekwipartycji energii wiemy, ¿e $\left\langle E\right\rangle = 3kT/2$, zatem $$\begin{aligned} \label{wartoczekp} \left\langle {\vec{\wp}^{\,2}}\right\rangle = 3 \, m \, k T .\end{aligned}$$ Maj¹c wartoœæ $p_{0}$, wyznaczmy wartoœæ oczekiwan¹: $$\begin{aligned} \left\langle {\vec{\wp}^{2}}\right\rangle =\int{d\vec{\wp} \, p\left({\vec{\wp}}\right)\vec{\wp}^{2}} \, \end{aligned}$$ i przyrównajmy j¹ do (\[wartoczekp\]). W rezultacie otrzymujemy: $$\begin{aligned} \label{a0} {a_{0}^{2}=mc^{2}kT} \; . \end{aligned}$$ [**Rozk³adu wartoœci pêdu**]{}: Kolejnym krokiem jest wyznaczenie rozk³adu prawdopodobieñstwa wartoœci pêdu $\wp = |\vec{\wp}\,|$. Przechodz¹c od wspó³rzêdnych kartezjañskich pêdu $\left(\wp_{1},\wp_{2},\wp_{3}\right)$ do wspó³rzêdnych sferycznych $\left(\wp,\theta,\phi\right)$, otrzymujemy: $$\begin{aligned} \label{pj} p\left({\wp,\theta,\phi}\right)=\left|J \,\right|p\left({\wp_{1},\wp_{2},\wp_{3}}\right) \; , \;\;\; J = \wp^{2}\sin\theta \; ,\end{aligned}$$ gdzie $J$ jest jakobianem przejœcia. Wstawiaj¹c teraz (\[p0\]) wraz z (\[a0\]) do (\[row\]) otrzymujemy rozk³ad $p\left({\wp_{1},\wp_{2},\wp_{3}}\right)$ ze znanymi ju¿ sta³ymi $a_{0}$ oraz $p_{0}$. Wynik ten wstawiaj¹c do (\[pj\]) i wyca³kowuj¹c po zmiennych $\theta$ oraz $\phi$, otrzymujemy szukany rozk³ad gêstoœci prawdopodobieñstwa dla wartoœci pêdu: $$\begin{aligned} \label{maxbol} p\left(\wp\right)=\sqrt{{2\mathord{\left/{\vphantom{2\pi}}\right.\kern -\nulldelimiterspace}\pi}}\left({mkT}\right)^{{{-3}\mathord{\left/{\vphantom{{-3}2}}\right.\kern -\nulldelimiterspace}2}}\wp^{2}e^{{{-\wp^{2}}\mathord{\left/{\vphantom{{-\wp^{2}}{2mkT}}}\right.\kern -\nulldelimiterspace}{2mkT}}} \, .\end{aligned}$$ [**Prawo Maxwella-Boltzmanna**]{}: Rozk³ad (\[maxbol\]) wyra¿a tzw. prawo Maxwella-Boltzmanna, a podstawiaj¹c $\wp =m v$ w (\[maxbol\]), w miejsce wartoœci pêdu, otrzymujemy rozk³ad Maxwella-Boltzmanna dla wartoœci prêdkoœci $v$ cz¹steczki gazu. ### Informacja Fishera jako ograniczenie dla wzrostu entropii Rozwa¿my uk³ad zawieraj¹cy *jedn¹* lub *wiêcej* cz¹stek poruszaj¹cych siê w sposób losowy, w pewnym zamkniêtym obszarze. Niech rozwa¿any obszar bêdzie izolowany, tzn. ¿adna cz¹stka ani z obszaru nie ucieka ani do niego nie przechodzi. Brzeg ($\vec{B}$) obszaru jest zadany wektorem wodz¹cym $\vec{b} \in \vec{B}$. W pewnej chwili uk³ad dokonuje pomiaru jego czterowektora po³o¿enia $t, \vec{y}$, zgodnie z rozk³adem gêstoœci prawdopodobieñstwa $p(\vec{y}, t)$[^77].\ \ Niech $(y^{1},y^{2},y^{3})$ s¹ wspó³rzêdnymi kartezjañskimi wektora $\vec{y}$ [**po³o¿enia przestrzennego**]{}, a $r=|\vec{y}|$ jego d³ugoœci¹. Poniewa¿ za³o¿yliœmy, ¿e cz¹stka znajduje siê gdzieœ wewn¹trz obszaru wiêc: $$\begin{aligned} \label{pt brzegowe = 1} p\left(t \right) = \int d \vec{y} \, p\,(\vec{y}, t) = 1 \; .\end{aligned}$$ Oznaczmy przez $S_{H}\left(t\right)$ entropiê Shannona uk³adu (\[eboltmana\]) w chwili czasu $t$: $$\begin{aligned} S_{H}\left(t\right) = - \int d\vec{y}\; p(\vec{y}, t) \ln p(\vec{y}, t) \; . \label{st}\end{aligned}$$ [**Drug¹ zasada termodynamiki**]{}: Mo¿na pokazaæ (por. Dodatek \[Wyprowadzenie drugiej zasady termodynamiki\]), ¿e entropia Shannona spe³nia drug¹ zasadê termodynamiki: $$\begin{aligned} \frac{{dS_{H}\left(t\right)}}{{dt}} \ge 0 \; , \label{2zasterm}\end{aligned}$$ która okreœla [*doln¹ granicê tempa zmiany entropii*]{}. Poni¿sze rozwa¿ania poœwiêcone s¹ znalezieniu ograniczenia na jego górn¹ granicê.\ \ [**Równanie ci¹g³oœci strumienia**]{}: Poniewa¿ ¿adna cz¹stka nie opuszcza ani nie wp³ywa do obszaru, zatem spe³nione jest równanie ci¹g³oœci strumienia prawdopodobieñstwa: $$\begin{aligned} \label{ciaglosc} \frac{\partial p(\vec{y}, t)}{\partial t} + \vec{\nabla} \cdot \vec{P}\left({\vec{y},t}\right)=0 \; , \end{aligned}$$ gdzie $\vec{P}\left({\vec{y},t}\right)$ jest [*pr¹dem prawdopodobieñstwa*]{}, którego konkretna postaæ zale¿y od rozwa¿anego uk³adu.\ \ [**Warunki brzegowe**]{}: Z kolei okreœlmy warunki brzegowe spe³niane przez gêstoœæ prawdopodobieñstwa i jego pr¹d. Bêd¹ nam one przydatne dla rozwi¹zania postawionego sobie zadania.\ Poniewa¿ ¿adne cz¹stki nie przechodz¹ przez granicê obszaru, zatem $\vec{P}$ spe³nia warunek brzegowy Dirichleta: $$\begin{aligned} \label{warbrzegP} \vec{P} \left( {\vec{y},t} \right) \left|\begin{array}{l} \!\!\!_{{\vec{y} \in {\vec{B}}}}\end{array} \right. = 0 \; .\end{aligned}$$ Za³ó¿my dodatkowo, ¿e jeœli brzeg obszaru znajduje siê w nieskoñczonoœci, to: $$\begin{aligned} \label{Pr0} \mathop{\lim}\limits _{\vec{y}\to\infty}\vec{P}\left({\vec{y},t}\right) \to 0 \; , \;\;\;\; {\rm szybciej \;\; ni\dot{z}} \;\;\;\; 1/r^{2} \; .\end{aligned}$$ Poniewa¿ obszar jest odizolowany, zatem prawdopodobieñstwo, ¿e cz¹stka znajduje siê na jego granicy znika, co oznacza, ¿e $p(\vec{y}, t)$ równie¿ spe³nia warunek brzegowy Dirichleta: $$\begin{aligned} \label{warbrzegp} p(\vec{y}, t)\left|\begin{array}{l} \!\!\!_{{\vec{y} \in \vec{B}}}\end{array}\right. = 0 \; .\end{aligned}$$ Ze wzglêdu na normalizacjê: $$\begin{aligned} \int{d\vec{y}\; p(\vec{y}, t)} = 1 \; ,\end{aligned}$$ zak³adamy, ¿e w przypadku brzegu nieskoñczenie oddalonego, $p(\vec{y}, t)$ spe³nia warunek: $$\begin{aligned} \label{pr0} \mathop{\lim}\limits _{\vec{y}\to\infty}p\left({\vec{y}\left|\; t\right.}\right)\to 0 \; , \;\;\;\; {\rm szybciej \;\; ni\dot{z}} \;\;\;\; 1/r^{3} \;\; .\end{aligned}$$ W koñcu dla domkniêcia potrzebnych warunków brzegowych przyjmujemy: $$\begin{aligned} \label{zeroPp} \vec{P}\ln p\left|\begin{array}{l} \!\!\!_{{\vec{y} \in \vec{B}}}\end{array}\right. = 0 \; .\end{aligned}$$\ [**Wyprowadzenie ograniczenia na tempo wzrostu entropii**]{}: Przyst¹pmy teraz do sedna rachunków. Ró¿niczkowanie (\[st\]) po $\frac{\partial }{\partial t}$ daje: $$\begin{aligned} \frac{\partial S_{H}}{\partial t} = - \frac{\partial}{{\partial t}}\int{d\vec{y}\; p\ln p} = - \int{d\vec{y}\; \frac{\partial p}{\partial t}\ln p} - \int{ d\vec{y}\; p \frac{1}{p} \frac{\partial p}{\partial t}} \; . \label{dr}\end{aligned}$$ Druga ca³ka po prawej stronie daje po skorzystaniu z warunku unormowania: $$\begin{aligned} \frac{\partial}{{\partial t}}\int{d\vec{y}\; p} = 0 \; .\end{aligned}$$ Oznaczmy przez $\left(P^{1},P^{2},P^{3}\right)$ kartezjañskie sk³adowe pr¹du $\vec{P}$. Podstawiaj¹c (\[ciaglosc\]) do pierwszej ca³ki po prawej stronie w (\[dr\]) otrzymujemy: $$\begin{aligned} \label{pierwsza} \frac{\partial S_{H}}{\partial t} = \int{d\vec{y}\; \vec{\nabla} \cdot\vec{P}\ln p}=\int{\int{\int{dy^{3} \,dy^{2}\,dy^{1} \left[{\frac{\partial}{{\partial y^{1}}}P^{1} + \frac{\partial}{{\partial y^{2}}}P^{2} + \frac{\partial}{{\partial y^{3}}}P^{3}}\right]\ln p}}} \; ,\end{aligned}$$ gdzie $\nabla_{i} = \frac{\partial}{\partial y^{i}} \,$.\ \ [**Ca³kuj¹c przez czêœci**]{} wewnêtrzne ca³ki $\int{dy^{i} \frac{\partial}{\partial y^{i}} P^{i} }$ w (\[pierwsza\]) dla trzech sk³adników $i=1,2,3$, otrzymujemy dla ka¿dego z nich: $$\begin{aligned} \int{dy^{i} (\frac{\partial}{{\partial y^{i}}} P^{i}) \ln p} = P^{i} \ln p \left|\begin{array}{l} \!\!\!_{{\vec{y} \in \vec{B}}} \end{array}\right. - \int{dy^{i}\frac{{P^{i}}}{p} \frac{\partial p}{\partial y^{i}} } = - \int{dy^{i} P^{i} \frac{\partial p}{\partial y^{i}} \frac{1}{p} } \; , \;\;\; i = 1,2,3 \; ,\end{aligned}$$ gdzie skorzystano z (\[zeroPp\]). Zatem: $$\begin{aligned} \label{dS po dt iloczyn} \frac{\partial S_{H}}{\partial t} = - \int{dy^{3}dy^{2}dy^{1} \left( P^{1} \frac{\partial p}{\partial y^{1}} + P^{2} \frac{\partial p}{\partial y^{2}} + P^{3} \frac{\partial p}{\partial y^{3}}\right) \frac{1}{p} } = - \int{d\vec{y} \left( \vec{P} \cdot \vec{\nabla} p \right) \frac{1}{p} } \; .\end{aligned}$$ Z powy¿szego, po prostych przekszta³ceniach mamy: $$\begin{aligned} \label{kwadrat dS po dt} \left(\frac{\partial S_{H}}{\partial t}\right)^{2} = \left[ \int d\vec{y} \left(\frac{\vec{P}}{\sqrt{p}}\right) \cdot \left(\frac{\sqrt{p}\, \vec{\nabla} p}{p} \right) \right]^{2} = \left[ \, \sum_{i=1}^{3} \int d\vec{y} \left(\frac{P^{i}}{\sqrt{p}}\right) \; \left(\frac{\sqrt{p}\; \nabla_{i} p}{p} \right) \right]^{2} \; .\end{aligned}$$\ [**Z nierównoœci Schwartza**]{} otrzymujemy: $$\begin{aligned} \label{nierownosc Schwartza} \left[ \, \sum_{i=1}^{3} \int d\vec{y} \left(\frac{P^{i}}{\sqrt{p}}\right) \; \left(\frac{\sqrt{p}\; \partial_{i} p}{p} \right) \right]^{2} \leq \left[ \sum_{i=1}^{3} \int d\vec{y} \left(\frac{P^{i}}{\sqrt{p}}\right)^{2} \right] \left[ \sum_{i=1}^{3} \int d\vec{y} \left(\frac{\sqrt{p}\; \nabla_{i} p}{p} \right)^{2} \right] \; .\end{aligned}$$ [**Ogólna postaæ ograniczenia na tempo wzrostu entropii**]{}: Ostatecznie z (\[kwadrat dS po dt\]) i (\[nierownosc Schwartza\]) oraz po zastêpieniu sumowañ po $i$ znakiem iloczynów skalarnych, dostajemy: $$\begin{aligned} \label{st2} \left(\frac{\partial S_{H}}{\partial t}\right)^{2} \le \int{d\vec{y}\;\frac{{\vec{P}\cdot\vec{P}}}{p}}\int{d\vec{y}\;\frac{{\vec{\nabla} p\cdot\vec{\nabla} p}}{p}} \; .\end{aligned}$$\ [**Przejœcie do postaci z pojemnoœci¹ informacyjn¹**]{}: Poniewa¿ gradiant $\vec{\nabla} \equiv (\frac{\partial }{\partial y^{1}}, \frac{\partial }{\partial y^{2}}, \frac{\partial }{\partial y^{3}})$ zawiera ró¿niczkowanie po wartoœciach pomiarowych $y^{i}$, a nie fluktuacjach $x^{i}$, jak to jest w kinematycznej postaci pojemnoœci informacyjnej Friedena-Soffera, zatem musimy uczyniæ dodatkowe za³o¿enie.\ [**Za³o¿enie niezmienniczoœci rozk³adu ze wzglêdu na przesuniêcie**]{}: PrzejdŸmy do addytywnych przesuniêæ, $\vec{x} = \vec{y} - \vec{\theta}$. Poniewa¿ $\vec{x} = (x^{i})_{i=1}^{3}$ oraz $\vec{y} = (y^{i})_{i=1}^{3}$ ró¿ni¹ siê o sta³¹ wartoœæ oczekiwan¹ po³o¿enia $\vec{\theta}=(\theta^{i})_{i=1}^{3}$, zatem [*zak³adaj¹c dodatkowo , ¿e rozk³ad $p$ jest niezmienniczy ze wzglêdu na przesuniêcie*]{} $\vec{\theta}$, otrzymujemy: $$\begin{aligned} \label{niezm na przes} \frac{\partial p }{\partial y^{i}} = \frac{\partial p}{\partial x^{i}} \; , \;\;\; i=1,2,3 \; .\end{aligned}$$ Przy powy¿szym za³o¿eniu, zachodzi: $$\begin{aligned} \label{poj info od t} \int{d\vec{y}\;\frac{{\vec{\nabla} p(\vec{y}, t)\cdot\vec{\nabla} p(\vec{y}, t)}}{p(\vec{y}, t)}} = \int{d\vec{x}\;\frac{{\vec{\nabla} p(\vec{x}, t)\cdot\vec{\nabla} p(\vec{x}, t)}}{p(\vec{x}, t)}} \; \end{aligned}$$ i okazuje siê, ¿e druga ca³ka po prawej stronie nierównoœci w (\[st2\]) jest pojemnoœci¹ informacyjn¹ trzech kana³ów przestrzennych dla $N=1$ w chwili czasu $t$: $$\begin{aligned} \label{pojI} I \equiv I \left(t\right)=\int{d\vec{x}\;\frac{\nabla p\left(\vec{x},t \right) \cdot \nabla p\left(\vec{x},t \right)}{p\left(\vec{x},t \right)}} \; .\end{aligned}$$\ [**Ograniczenie na tempo wzrostu $S_{H}$ uk³adu z niezmienniczoœci¹ przesuniêcia**]{}: Korzystaj¹c z powy¿szej postaci pojemnoœci informacyjnej, mo¿emy zapisaæ (\[st2\]) nastêpuj¹co: $$\begin{aligned} \label{ograniczenieS} \left(\frac{\partial S_{H}}{\partial t}\right)^{2} \le I(t) \int{d\vec{x}\;\frac{{\vec{P}\cdot\vec{P}}}{p}} \; , \;\;\; {\rm lub} \;\;\; \left(\frac{\partial S_{H}}{\partial t}\right) \le \sqrt{I(t)} \;\; \sqrt{\int{d\vec{x}\;\frac{{\vec{P}\cdot\vec{P}}}{p}}} \; ,\end{aligned}$$\ Pokazaliœmy zatem, ¿e [*przy za³o¿eniu niezmienniczoœci rozk³adu za wzglêdu na przesuniêcie*]{}, tempo wzrostu entropii jest obustronnie ograniczone. Ograniczenie dolne (\[2zasterm\]) wyra¿a zasadê niemalenia entropii Shannona w czasie. Jej termodynamicznym odpowiednikiem jest twierdzenie H Boltzmanna.\ \ [**Wniosek**]{}: Nierównoœæ (\[ograniczenieS\]) oznacza, ¿e [*ograniczenie górne tempa wzrostu entropii jest proporcjonalne do pierwiastka z pojemnoœci informacyjnej*]{} (\[pojI\]) [*dla pomiaru po³o¿enia $\vec{y}$*]{}. Jest to jednen z nowych wyników teorii pomiaru otrzymany przez Friedena, Soffera, Plastino i Plastino [@Frieden]. Jego termodynamiczne konsekwencje czekaj¹ na weryfikacjê.\ \ W [@Frieden] podano przyk³ady zastosowania tego twierdzenia dla strumienia cz¹stek klasycznych, strumienia w elektrodynamice klasycznej i strumienia cz¹stek ze spinem 1/2. Poni¿ej zostanie podany wynik analizy dla tego ostatniego przypadku. #### Wynik dla strumienia cz¹stek ze spinem 1/2 W Rozdziale \[Dirac field\] pokazaliœmy, ¿e metoda EFI daje dla relatywistycznej cz¹stki o spinie po³ówkowym równanie ruchu Diraca (\[Dirac eq\]). Pod nieobecnoœæ pola elektromagnetycznego wynikaj¹ce z niego równanie ci¹g³oœci ma postaæ: $$\begin{aligned} \label{rownciaglpsi} \frac{\partial}{{\partial t}}p\left({\vec{y}, t}\right) + \vec{\nabla}\cdot\vec{P}\left({\vec{y}, t}\right) = 0 \; ,\end{aligned}$$ gdzie gêstoœæ prawdopodobieñstwa $p$ oraz gêstoœæ pr¹du prawdopodobieñstwa s¹ równe odpowiednio: $$\begin{aligned} \label{psi} p\left({\vec{y}, t}\right)=\psi^{\dagger}\psi,\quad\psi\equiv\psi\left({\vec{y}, t}\right) \; ,\end{aligned}$$ oraz $$\begin{aligned} \label{psi+} \vec{P}\left({\vec{y}, t}\right) = c \, \psi^{\dagger} \, \vec{\alpha} \,\psi \; ,\end{aligned}$$ gdzie $\vec{\alpha} \equiv \left(\alpha^{1},{\alpha^{2}},{\alpha^{3}}\right)$ s¹ macierzami Diraca (\[m Diraca alfa beta\]), natomiast $\psi^{\dagger} = (\psi_{1},\psi_{2},\psi_{3},\psi_{4})^{*}$ jest polem sprzê¿onym hermitowsko do bispinora Diraca $\psi$ (Rozdzia³ \[Dirac field\]).\ \ [**Ograniczenie na tempo wzrostu entropii**]{}: W [@Frieden] pokazano, ¿e tempo wzrostu entropii (\[ograniczenieS\]) ma w tym przypadku postaæ: $$\begin{aligned} \label{nierownosc S I c} {\frac{{\partial S_{H}}}{{\partial t}}}\le c\sqrt{I(t)} \; .\end{aligned}$$\ [**Wniosek**]{}: Nierównoœæ ta oznacza, ¿e dla uk³adu, który posiada niezmienniczoœæ przesuniêcia, wzrost entropii Shannona rozk³adu w jednostce czasu (czyli tempo spadku informacji) jest ograniczony przez skoñczon¹ prêdkoœæ œwiat³a jak równie¿ przez pojemnoœæ informacyjn¹, jak¹ posiada uk³ad (np. swobodny elektron) o swoich wspó³rzêdnych czasoprzestrzennych.\ Inny sposób interpretacji (\[nierownosc S I c\]) polega na zauwa¿eniu, ¿e dostarcza ona definicji prêdkoœci œwiat³a $c$ jako górnego ograniczenia stosunku tempa zmiany entropii do pierwiastka informacji Fishera, która jest przecie¿ na wskroœ statystycznym pojêciem informacyjnym. Zastosowanie wprowadzonego formalizmu do analizy paradoksu EPR {#paradoks EPR} -------------------------------------------------------------- Przedstawiona w tym rozdziale analiza paradoksu EPR pochodzi od Friedena [@Frieden]. Zamieszczamy j¹ ze wzglêdu na to, ¿e efekt ten jest doœæ powszechnie uto¿samiany z w³asnoœci¹ teorii kwantowych, ale równie¿ z powodu uporz¹dkowania warunków brzegowych zagadnienia zawartego w pracy orginalnej [@Frieden], a przedstawionego w pracy [@Mroziakiewicz] oraz wnioskowania uzgodnionego z przedstawion¹ w skrypcie fizyczn¹ interpretacj¹ informacji fizycznej $K$ [@Dziekuje; @informacja_1; @Dziekuje; @informacja_2].\ \ [**Opis eksperymentu EPR-Bohm’a**]{}: Rozpocznijmy od omówienia eksperymentu EPR-Bohm’a. Rozwa¿my Ÿród³o moleku³ o spinie zero, które rozpadaj¹ siê na parê identycznych cz¹stek o spinie $1/2$ lec¹cych w przeciwnych kierunkach. Taka, pocz¹tkowa dla rozwa¿añ eksperymentu EPR-Bohm’a, konfiguracja dwucz¹stkowej moleku³y mo¿e byæ efektywnie przygotowana jako stan koñcowy w rozpraszaniu $e^{-}e^{-} \to e^{-}e^{-}$, gdzie spiny pocz¹tkowych elektronów procesu s¹ ustawione przeciwnie (równolegle i antyrównolegle wzglêdem osi $z$), a ich pocz¹tkowe pêdy wzd³u¿ osi $y$ wynosz¹ $\vec{p}\,$ oraz $-\vec{p}$ [@Manoukian]. Istnieje niezerowa amplituda, ¿e dwie rozproszone cz¹stki (tu wyprodukowane elektrony), poruszaj¹ siê z pêdami wzd³u¿ osi $x$, jak na Rysunku \[fig:5\]. ![Eksperyment EPR-Bohma. Zaznaczono elektrony rejestrowane w urz¹dzeniach Sterna-Gerlacha; opis w tekœcie Rozdzia³u. W lewym górnym rogu rysunku zaznaczono k¹t $\vartheta$ pomiêdzy kierunkami $\vec{a}$ oraz $\vec{b}$ urz¹dzeñ Sterna-Gerlacha. []{data-label="fig:5"}](Epr2.eps){width="70mm" height="35mm"} \ W eksperymencie EPR-Bohm’a dokonywany jest pomiar spinu rozproszonych cz¹stek, spinu cz¹stki 1 wzd³u¿ wektora jednostkowego $\vec{a}$, tworz¹cego z osi¹ $z$ k¹t $\chi_{1}$ oraz spinu cz¹stki 2 wzd³u¿ wektora jednostkowego $\vec{b}$, tworz¹cego z osi¹ $z$ k¹t $\chi_{2}$. Niech analizator ,,$a$”, bêd¹cy urz¹dzeniem typu Sterna-Gerlacha, mierzy rzut $S_{a}$ spinu $\vec{S}_{1}$ cz¹stki 1 na kierunek $\vec{a}$ i podobnie analizator ,,$b$” mierzy rzut $S_{b}$ spinu $\vec{S}_{2}$ cz¹stki 2 na kierunek $\vec{b}$. K¹t pomiêdzy p³aszczyznami wektorów $\vec{a}$ oraz $\vec{b}$, zawieraj¹cymi oœ $x$, wynosi $\vartheta = \chi_{1}-\chi_{2}$, $0\le \vartheta < 2\pi$.\ Poni¿ej wyprowadzimy warunki brzegowe dla metody EFI, uwa¿aj¹c aby nie odwo³ywaæ siê do widzenia rzeczywistoœci przez pryzmat mechaniki kwantowej [^78]. ### Warunki brzegowe {#Warunki brzegowe} W celu rozwi¹zania równañ ró¿niczkowych EFI konieczne jest ustalenie warunków brzegowych na prawdopodobieñstwa. Wynikaj¹ one z przes³anek fenomenologicnych, zasad zachowania i symetrii przestrzennej badanego uk³adu.\ \ [**Za³o¿enie**]{} [*o istnieniu dok³adnie dwóch mo¿liwych rzutów spinu cz¹stki ze spinem $\hbar/2$ na dowolny wybrany kierunek w przestrzeni*]{}: Niech “$+$” oznacza obserwowan¹ wartoœæ rzutu spinu $S_{a}=+\hbar/2$ natomiast “$-$” oznacza $S_{a} = - \hbar/2$.\ \ [**Uwaga o fenomenologii rejestracji spinu cz¹stki**]{}: Jest to jedyne miejsce, w którym uciekamy siê do opisu fenomenologicznego, odwo³uj¹c siê do nieklasycznej fizyki zjawiska, mianowicie zak³adamy, ¿e rzutowanie spinu cz¹stki na okreœlony kierunek przestrzenny jest skwantowane zgodnie z wymiarem reprezentacji grupy obrotów (fakt istnienia takich reprezentacji wynika z rozwa¿añ entropijnych [@Frieden]).\ Jednak nie oznacza to, ¿e nie istnieje model teoriopolowy, który by takie kwantowanie rzutowania opisywa³. Musia³by on jedynie zak³adaæ, ¿e po pierwsze spinowe stopnie swobody (z ci¹g³ym rozk³adem jego kierunku przed pomiarem) nie maj¹ (prostego) charakteru czasoprzestrzennego [@dziekuje; @za; @neutron], a po drugie, ¿e bezw³adnoœæ cz¹stki zwi¹zana ze spinowymi stopniami swobody jest bardzo ma³a w porównaniu z si³¹ sprzê¿enia tych spinowych stopni swobody z otoczeniem (np. aparatur¹ traktowan¹ jako rezerwuar), którego moment pêdu ignorujemy. Wtedy przejœcie cz¹stki ze spinem przez jak¹kolwiek aparaturê pomiarow¹ typu Sterna-Gerlacha porz¹dkowa³oby jej spin w sposób dyskretny. Oczywiœcie zmiana momentu pêdu aparatury ju¿ nas “nie interesuje”.\ \ [**£¹czn¹ przestrzeñ zdarzeñ**]{} $\Omega_{ab}$ stanów spinowych pary cz¹stek 1 i 2 przyjmujemy jako nastêpuj¹c¹: $$\begin{aligned} \label{4 zdarzenia EPR} S_{a}S_{b} \equiv S_{ab} \in \Omega_{ab} = \left\{ S_{++},S_{--},S_{+-},S_{-+} \right\} \equiv \left\{(++),(--),(+-),(-+)\right\} \; .\end{aligned}$$ Za³ó¿my, ¿e mo¿emy zdefiniowaæ cztery ³¹czne warunkowe prawdopodobieñstwa[^79] $P\left(S_{ab}|\vartheta \right)$: $$\begin{aligned} \label{spinprawdop} P\left(++|\vartheta\right) \; , \;\; P\left(--|\vartheta\right) \; , \;\; P\left(+-|\vartheta\right) \; , \;\; P\left(-+|\vartheta\right) \; . $$ [**Warunek normalizacji prawdopodobieñstwa**]{} $P\left(S_{a}S_{b}|\vartheta\right)$ w eksperymencie EPR-Bohm’a mo¿na, ze wzglêdu na wykluczanie siê ró¿nych zdarzeñ $S_{ab}$, (\[4 zdarzenia EPR\]), zapisaæ nastêpuj¹co: $$\begin{aligned} \label{normalizacja P daje wsp w qab} P\left(\bigcup\limits _{ab}{S_{a}S_{b}|\vartheta}\right)} = \sum\limits_{ab}{P\left(S_{a}S_{b}|\vartheta\right) = 1 \; \;\;\;\;\; {\rm dla \;\; ka\dot{z}dego} \;\; \vartheta \in \langle 0, 2\pi)\; .\end{aligned}$$ W analizie estymacyjnej postulujemy, ¿e prawdopodobieñstwo $P\left(S_{ab}\left|\vartheta \right.\right)\left|\begin{array}{l} _{\!\!\! \vartheta = \hat{\vartheta}} \end{array}\right.$ jest funkcj¹ estymatora $\hat{\vartheta}$ parametru $\vartheta$.\ \ [**Metoda estymacji**]{}: Korzystaj¹c z proporcjonalnoœci prawdopodobieñstw(\[spinprawdop\]) do obserwowanej liczebnoœci zdarzeñ w detektorze, oszacowuje siê wartoœæ k¹ta $\vartheta$. Jednak, aby to uczyniæ, trzeba mieæ model analitycznych formu³ na prawdopodobieñstwa (\[spinprawdop\]). Poni¿ej wyprowadzimy je metod¹ EFI.\ \ [**Sformu³owanie warunków brzegowych**]{}.\ \ (1) Poniewa¿ zdarzenia (\[4 zdarzenia EPR\]) wykluczaj¹ siê wzajemnie i rozpinaj¹ ca³¹ przestrzeñ zdarzeñ, wiêc pierwszym warunkiem brzegowym jest [**warunek normalizacji**]{}: $$\begin{aligned} \label{normalizacja prawd dla EPR} \sum_{ab}{P\left(S_{ab}\right)} = P\left(S_{++}\right)+P\left(S_{+-}\right)+P\left(S_{-+}\right)+P\left(S_{--}\right) = 1 \; ,\end{aligned}$$ spe³niony niezale¿nie od wartoœci k¹ta $\vartheta$.\ \ Dla ka¿dego zdarzenia $ab$ prawdopodobieñstwa $P\left(S_{ab}\right)$ oraz $P\left(S_{ab}|\vartheta\right)$ s¹ ze sob¹ zwi¹zane nastêpuj¹co: $$\begin{aligned} \label{srednia z PSab theta} P\left(S_{ab}\right) = \int\limits_{0}^{2\pi} {P\left(S_{ab}|\vartheta\right) r\left(\vartheta\right) d\vartheta} \; ,\end{aligned}$$ gdzie uœrednienie nast¹pi³o z tzw. [*funkcj¹ niewiedzy*]{} $r(\vartheta)$. Ze wzglêdu na warunki unormowania (\[normalizacja P daje wsp w qab\]) oraz (\[normalizacja prawd dla EPR\]) otrzymujemy jej mo¿liw¹ postaæ: $$\begin{aligned} \label{prawdkat} r(\vartheta)=\frac{1}{2\pi} \; ,\quad\quad0\le \vartheta < 2 \pi \; ,\end{aligned}$$ która oznacza, ¿e w zakresie aparaturowej zmiennoœci ustawienia wartoœci k¹ta $\vartheta \in \left\langle 0, 2 \pi \right)$, z powodu naszej niewiedzy, jest mo¿liwa w równym stopniu ka¿da jego wartoœæ.\ \ [**Uwaga o wartoœci k¹ta $\vartheta$**]{}: Wartoœæ $\vartheta$ jest zwi¹zana z ustawieniem aparatury pomiarowej Sterna-Gerlacha ,,$a$” i ,,$b$”, i trudno j¹ (na serio) traktowaæ jako wielkoœæ posiadaj¹c¹ rozproszenie. Zasadnicza $\vartheta$ jest parametrem charakterystycznym dla przeprowadzonego eksperymentu. Niemniej przez niektórych $r(\vartheta)$ jest widziane jako ,,prawdopodobieñstwo” znane a priori, co oznacza, ¿e wyprowadzon¹ w ten sposób mechanikê kwantow¹ nale¿a³oby traktowaæ jako statystyczn¹ teoriê Bayesowsk¹ [@stany; @koherentne].\ \ (2) Kolejne warunki wynikaj¹ z [**symetrii uk³adu i zasady zachowania ca³kowitego spinu**]{} przy czym w eksperymencie [*wzglêdny orbitalny moment pêdu wynosi zero*]{}.\ \ Rozwa¿my prosty przypadek $\vartheta=0$, gdy obie p³aszczyzny, w których ustawione s¹ urz¹dzenia Sterna-Gerlacha s¹ tak samo zorientowane. Z warunku zachowania ca³kowitego spinu wynika: $$\begin{aligned} \label{vartheta zero} P\left(++|0\right)=P\left(--|0\right)=0 \; ,\quad \vartheta=0\end{aligned}$$ co oznacza, ¿e w tym ustawieniu aparatury nigdy nie zobaczymy obu spinów jednoczeœnie skierowanych w górê czy w dó³. Analogicznie, jeœli k¹t $\vartheta=\pi$, to warunek: $$\begin{aligned} \label{vartheta pi} P\left(+-|\pi\right)=P\left(-+|\pi\right)=0,\quad \vartheta = \pi \; \end{aligned}$$ oznacza, ¿e w tym przypadku nigdy nie zobaczymy spinów ustawionych jeden w górê drugi w dó³. W konsekwencji z zasady zachowania ca³kowitego spinu otrzymujemy, ¿e jeœli $\vartheta=0$ lub $\vartheta=\pi$, to zawsze obserwacja jednego spinu daje nam ca³kowit¹ wiedzê o drugim. W tym przypadku stany spinów wyraŸnie nie s¹ niezale¿ne, s¹ stanami skorelowanymi. Wniosek ten jest intuicyjnie zawarty w sposobie ich przygotowania.\ \ Nastêpnie, poniewa¿ $P\left(S_{b}|\vartheta\right)$ jest prawdopodobieñstwem brzegowym wyst¹pienia okreœlonej wartoœci rzutu spinu cz¹stki 2, zatem nie zale¿y ono od $S_{a}$, czyli od orientacji rzutu spinu cz¹stki 1, wiêc nie zale¿y równie¿ od k¹ta $\vartheta$ pomiêdzy wektorami $\vec{a}$ oraz $\vec{b}$. Mamy wiêc: $$\begin{aligned} \label{Sb} P\left(S_{b}|\vartheta\right) = C = const. \end{aligned}$$ St¹d $$\begin{aligned} \label{warunek poczatkowy dla Sb} \!\!\!\!\!\!\! P\left(S_{b}\right) = \int\limits_{0}^{2\pi} {P\left(S_{b}|\vartheta\right) r\left(\vartheta\right) d\vartheta} = C \int\limits_{0}^{2\pi} {r\left(\vartheta\right) d\vartheta} = C \quad \;\; {\rm dla} \;\;\; S_{b} = +,- \;\, , \end{aligned}$$ gdzie $r(\vartheta)$ jest okreœlone w (\[prawdkat\]).\ \ Z warunku unormowania prawdopodobieñstwa zdarzenia pewnego, mamy $$\begin{aligned} \label{cpol} P\left(S_{b}=+\right)+P\left(S_{b} = - \right) = C + C = 2C = 1 \;\;\; {\rm czyli} \;\;\; C = \frac{1}{2} \; .\end{aligned}$$ Zatem z ( \[Sb\]) otrzymujemy, ¿e[^80] : $$\begin{aligned} \label{pol} P\left(S_{b}|\vartheta\right) = \frac{1}{2} \; ,\end{aligned}$$ natomiast z (\[warunek poczatkowy dla Sb\]): $$\begin{aligned} \label{Sb wycalkowane} P\left(S_{b}\right) = \frac{1}{2} \; .\end{aligned}$$\ Inn¹ wa¿n¹ w³asnoœci¹ symetrii przestrzennej eksperymentu jest brak preferencji wystêpowania spinu skierowanego w górê czy w dó³, tzn: $$\begin{aligned} \label{symetriaEPR} P\left(S_{+-}|\vartheta\right)=P\left(S_{- +}|\vartheta\right)\quad {\rm oraz} \quad P\left(S_{++}|\vartheta\right)=P\left(S_{--}|\vartheta\right) \; ,\end{aligned}$$ co oznacza, ¿e jeœli obserwowalibyœmy eksperyment dla uk³adu odwróconego wzglêdem osi $x$ o k¹t $\pi$, to statystyczny wynik by³by dok³adnie taki sam. Z (\[symetriaEPR\]) otrzymujemy: $$\begin{aligned} \label{++ = - -} P\left(S_{++}\right)=\int{P\left(S_{++}|\vartheta\right)r\left(\vartheta\right)d\vartheta}=\int{P\left(S_{--}|\vartheta\right)r\left(\vartheta\right)d\vartheta}=P\left(S_{--}\right)\end{aligned}$$ oraz w sposób analogiczny: $$\begin{aligned} P\left(S_{+-}\right)=P\left(S_{-+}\right)\label{22} \; .\end{aligned}$$ Dodatkowo otrzymujemy: $$\begin{aligned} \label{doda} P\left(S_{+-}\right) &=& \int{P\left(S_{+-}|\vartheta\right) r\left(\vartheta\right)d\vartheta} = \int{P\left(S_{+-}|\vartheta+\pi\right) r\left(\vartheta+\pi\right)d\vartheta} \nonumber \\ & = & \int{P\left(S_{++}|\vartheta\right) r\left(\vartheta+\pi\right) d\vartheta} = P\left(S_{++}\right) \; , \end{aligned}$$ gdzie w drugiej równoœci skorzystano ze zwyk³ej zamiany zmiennych $\vartheta \to \vartheta + \pi$, w trzeciej równoœci z tego, ¿e $P\left(S_{+-}|\vartheta+\pi\right) = P\left(S_{++}|\vartheta\right)$, a w ostatniej z $r(\vartheta)=1/{(2\pi)} = r(\vartheta+\pi)$.\ \ Jako konsekwencja (\[++ = - -\])-(\[doda\]) otrzymujemy: $$\begin{aligned} P\left(S_{-+}\right) = P\left(S_{--}\right) \; . \label{doda2}\end{aligned}$$ Ze wzglêdu na (\[++ = - -\])-(\[doda2\]) oraz (\[normalizacja prawd dla EPR\]) otrzymujemy: $$\begin{aligned} P\left(S_{ab}\right) = \frac{1}{4} \; . \label{laczneEPR}\end{aligned}$$ W koñcu wzór Bayes’a na [*prawdopodobieñstwo warunkowe*]{} daje: $$\begin{aligned} \label{warunkSab} P\left(S_{a}|S_{b}\right)=\frac{P\left(S_{ab}\right)}{P\left(S_{b}\right)} = \frac{1/4}{1/2} = \frac{1}{2}\quad {\rm dla\; ka\dot{z}dego }\;\; S_{a}, S_{b} \; .\end{aligned}$$\ [**Koñcowe uwagi o warunkach brzegowych**]{}: Zauwa¿my, ¿e jak dot¹d w wyprowadzeniu zale¿noœci (\[vartheta zero\]) do (\[warunkSab\]) nie skorzystano z EFI. S¹ to bowiem warunki brzegowe dla równañ metody EFI. Zale¿noœci te wynika³y z pocz¹tkowej obserwacji istnienia dla cz¹stki o spinie po³ówkowym dok³adnie dwóch mo¿liwych rzutów spinu na dowolny kierunek (por. Uwaga na pocz¹tku rozdzia³u), zasady zachowania ca³kowitego momentu pêdu oraz z symetrii uk³adu. Przy wyborze rozwi¹zania równania generuj¹cego rozk³ad, skorzystamy jeszcze z warunku geometrycznej symetrii uk³adu przy obrocie o k¹t $\vartheta = 2 \pi$.\ ### Pojemnoœæ informacyjna dla zagadnienia EPR-Bohm’a {#Pojemnosc informacyjna zagadnienia EPR} Poni¿ej wyprowadzimy wyra¿enia dla prawdopodobieñstw (\[spinprawdop\]) jako wynik estymacji rozk³adów metod¹ EFI.\ \ [**Wielkoœæ mierzona przez obserwatora**]{}: Obserwator zewnêtrzny mierzy jedynie wartoœæ rzutu spinu jednej cz¹stki *powiedzmy, ¿e cz¹stki* 1, to znaczy $S_{a}$, natomiast nie mierzy wartoœci rzutu spinu drugiej cz¹stki $S_{b}$, jest ona traktowana jako wielkoœæ nieznana, ale ustalona. Równie¿ wartoœæ $S_{b}$ nie jest estymowana z obserwacji $S_{a}$.\ \ [**Okreœlenie przestrzeni po³o¿eñ dla EFI**]{}: Podobnie jak poprzednio uk³ad sam próbkuje swoimi Fisherowskimi kinetycznymi stopniami swobody dostêpn¹ mu przestrzeñ po³o¿eñ. Tym razem jest to jednowymiarowa przestrzeñ k¹ta $\vartheta$.\ Na podstawie danych pomiarowych wartoœci rzutu spinu $S_{a}$, estymujemy metod¹ EFI k¹t $\vartheta$ pomiêdzy dwoma p³aszczyznami zaznaczonymi na rysunku \[fig:5\]. K¹t ten jest wiêc traktowany jako nieznany parametr i jest estymowany z obserwacji $S_{a}$.\ \ [**Okreœlenie funkcji wiarygodnoœci i przestrzeni próby**]{}: Poniewa¿ dokonujemy pomiaru $S_{a}$, a $S_{b}$ oraz $\vartheta$ s¹ nieznanymi, ale ustalonymi wielkoœciami, zatem funkcjê wiarygodnoœci dla powy¿ej postawionego problemu zapiszemy nastêpuj¹co: $$\begin{aligned} \label{finkcja wiaryg dla vartheta} P\left(S_{a}|S_{b}, \vartheta\right) \; \rightarrow \;\;\;\; {\rm funkcja \;\; wiarygodno\acute{s}ci \;\; pr\acute{o}by \;\; dla} \;\; \vartheta \, .\end{aligned}$$ Jej postaci szukamy, wykorzystuj¹c metodê EFI. Jednak postaæ informacji Fishera, która by³aby miar¹ precyzji estymacji k¹ta $\vartheta$ opartej o pomiar $S_{a}$, wynikaæ bêdzie z jej pierwotnej postaci (\[postac I koncowa w pn\]) dla wymiaru próby $N=1$.\ \ Skoro wiêc $\vartheta$ jest estymowanym parametrem, zatem pe³ni on teraz tak¹ rolê jak parametrem $\theta$ w wzorze (\[postac I koncowa w pn\]), w którym, obok podstawienia $\theta\to \vartheta$, nale¿y dokonaæ nastêpuj¹cych podstawieñ: $$\begin{aligned} \label{N oraz wartosci y w funkcji wiaryg dla EPR} n = N = 1 \; \;\;\; {\rm oraz} \;\;\; \int{dy^{1}} \to \sum\limits_{a} \; ,\end{aligned}$$ gdzie przestrzeni¹ (jednowymiarowej) próby jest zbiór $\left\{-1,+1\right\}$ wartoœci zmiennej losowej $S_{a}$.\ \ [**Amplitudy prawdopodobieñstwa**]{} $q_{ab}$ s¹ zdefiniowane jak zwykle nastêpuj¹co: $$\begin{aligned} \label{inffishEPR2} P\left(S_{a}\left|{S_{b},\vartheta}\right.\right) = q_{ab}^{2}\left(\vartheta\right) \; .\end{aligned}$$\ [**Oczekiwana IF parametru $\vartheta$. Pojemnoœæ informacyjna kana³u $(\vartheta, S_{b})$**]{} jest równa: $$\begin{aligned} \!\!\!\!\!\!\!\! I_{b} &= & \sum\limits_{a=-}^{+}{\frac{1}{P\left(S_{a}\left|{S_{b}, \vartheta} \right.\right)}\left(\frac{\partial P \left(S_{a}\left|{S_{b}, \vartheta}\right.\right)}{\partial \vartheta}\right)^{2}} = \sum\limits_{a=-}^{+}{\frac{1}{q_{ab}^{2}} \left(\frac{\partial q_{ab}^{2}}{\partial \vartheta}\right)^{2}} = \sum\limits_{a=-}^{+}{\frac{1}{q_{ab}^{2}}\left(2q_{ab}\frac{\partial q_{ab}}{\partial \vartheta}\right)^{2}} \;\;\;\; \nonumber \\ \!\!\!\!\!\!\!\! &=& 4\sum\limits_{a=-}^{+}{\left(\frac{\partial q_{ab}}{\partial \vartheta}\right)^{2}} \; \;\;\;\;\end{aligned}$$ i podsumowuj¹c: $$\begin{aligned} \label{inffishEPR} I_{b} \equiv I\left({S_{b},\vartheta}\right) = 4 \sum\limits_{a=-}^{+}{q_{ab}^{'2}\left(\vartheta\right)} \; ,\quad S_{b} = \left({+,-}\right) \; , \;\;\; {\rm gdzie} \;\;\; q_{ab}^{'} \equiv \frac{dq_{ab}}{d\vartheta} \, ,\end{aligned}$$ gdzie sumowanie po $a$ odpowiada ca³kowaniu w informacji Fishera po przestrzeni bazowej.\ Zwrócmy uwagê, ¿e $\vartheta$ w powy¿szej formule na $I_{b} \equiv I\left({S_{b},\vartheta}\right)$ jest wyznaczone dla konkretnej wartoœci $\vartheta$ i konkretnej wartoœci $S_{b}$ rzutu spinu cz¹stki 2. Zatem jest to pojemnoœæ informacyjna jednego kana³u $(\vartheta, S_{b})$ czyli [*informacja Fishera parametru*]{} $\vartheta$ w tym kanale: $$\begin{aligned} \label{IF dla vartheta w EPR} I_{F}(\vartheta) = I_{b} \, . \end{aligned}$$ Powróæmy do warunków (\[N oraz wartosci y w funkcji wiaryg dla EPR\]). Pierwszy z nich oznacza, ¿e ranga amplitudy pola jest równa $N=1$, drugi oznacza, ¿e [*przestrzeñ bazowa to teraz dwupunktowy zbiór wartoœci rzutu spinu cz¹stki*]{} 1 [*na kierunek ’$a$’.*]{} W poni¿szych uwagach odniesiemy siê do tych elementów analizy.\ \ [**Uwaga o sumowaniu po $a$ w przestrzeni próby**]{}: Jako, ¿e $\vartheta$ jest parametrem, a pomiary w próbie s¹ zwi¹zane z obserwacjami $S_{a}$, zatem zapis w (\[inffishEPR\]) wymaga pewnego wyjaœnienia. Wyra¿enie (\[inffishEPR\]) reprezentuje dwa równania dla dwóch mo¿liwych wartoœci $S_{b}$. Poniewa¿ zmienn¹ w pomiarze jest rzut spinu cz¹stki 1, zatem sumowanie przebiega po dwóch mo¿liwych wartoœciach spinu $S_{a}=+,-.$ Tak wiêc, zapis $I\left({S_{b},\vartheta}\right)$ wskazuje na [*informacjê zawart¹ w przestrzeni próby zmiennej*]{} $S_{a}$ dla cz¹stki $1$ na temat nieznanego k¹ta $\vartheta$ w obecnoœci pewnej nieznanej, lecz ustalonej wartoœci rzutu spinu $S_{b}$ cz¹stki $2$.\ \ [**Ca³kowita pojemnoœæ informacyjna $I_{1_{a}}$ dla parametru $\vartheta$**]{}: Poniewa¿ k¹t $\vartheta$ jest parametrem, którego wartoœæ mo¿e siê zmieniaæ w sposób ci¹g³y w przedziale $\langle0,2\pi)$, st¹d zgodnie z (\[inffishEPR\]) mamy nieskoñczon¹ liczbê kana³ów informacji Fishera zwi¹zanych z wartoœciami $\vartheta$. Dla ka¿dego z tej nieskoñczonej liczby kana³ów s¹ jeszcze dwa kana³y informacyjne zwi¹zane z mo¿liwymi wartoœciami dla $S_{b}$. Aby poradziæ sobie z tak¹ sytuacj¹ metoda EFI wykorzystuje pojedyncz¹, skalarn¹ informacjê (oznaczmy j¹ $I_{1_a}$), nazywan¹ pojemnoœci¹ informacyjn¹, wprowadzon¹ w Rozdziale \[Pojecie kanalu informacyjnego\]. Wielkoœæ t¹ konstruujemy dokonuj¹c sumowania informacji po wszystkich mo¿liwych kana³ach. Zatem, suma przebiegaæ bêdzie po wszystkich wartoœciach k¹ta $\vartheta_{k}$ przy nieznanym rzucie spinu $S_{b}$ cz¹stki 2.\ Tak wiêc, po pierwsze, pojemnoœæ informacyjna $I_{n k}$ dla jednego kana³u, wprowadzona po raz pierwszy w (\[krzy4\]), przyjmuje postaæ: $$\begin{aligned} \label{pojemnosc jednego kanalu w EPR} I_{b k} = I \left(S_{b},\vartheta_{k}\right) \; .\end{aligned}$$ Po drugie, w wyniku sumowania po wszystkich mo¿liwych wartoœciach k¹ta $\vartheta_{k}$ oraz rzutach spinu $S_{b}$ cz¹stki 2, otrzymujemy [*ca³kowit¹ pojemnoœæ informacyjn¹ dla parametru*]{} $\vartheta$: $$\begin{aligned} \label{pojemnoscEPR} & & I_{1_{a}} \equiv \sum\limits_{b=-}^{+} \sum\limits_{k}{I\left({S_{b},\vartheta_{k}}\right)} \rightarrow \nonumber \\ & & \rightarrow I_{1_{a}} = \sum\limits_{b=-}^{+}{\int\limits_{0}^{2\pi} d\vartheta\;I\left({S_{b},\vartheta}\right)} = 4 \sum\limits_{b=-}^{+}\sum\limits_{a=-}^{+} {\int\limits _{0}^{2\pi}{d\vartheta}\; q_{ab}^{'2}\left(\vartheta\right)} \equiv 4\sum\limits _{ab}{\int\limits _{0}^{2\pi}{d\vartheta}\; q_{ab}^{'2}\left(\vartheta\right)} \; ,\end{aligned}$$ gdzie ca³kowanie pojawia siê z powodu zast¹pienia sumy dla dyskretnego indeksu $k$ ca³kowaniem po ci¹g³ym zbiorze wartoœci parametru $\vartheta$. Oznacza to, ¿e po zrozumieniu czym jest pojedynczy $k$ - ty kana³ zwi¹zany z $\vartheta$, wykonujemy w celu otrzymania ca³kowitej pojemnoœci informacyjnej ca³kowanie, które przebiega zgodnie z (\[prawdkat\]) od $0$ do $2\pi$. W ostatniej linii wykorzystano (\[inffishEPR\]).\ Sumowanie w (\[pojemnoscEPR\]) po $ab$ przebiega po wszystkich mo¿liwych kombinacjach ³¹cznej przestrzeni stanów $S_{ab}$ okreœlonej w (\[4 zdarzenia EPR\]). Indeks ’$1_a$’ w $I_{1_a}$ oznacza, ¿e jest to pojemnoœæ informacyjna jednej cz¹stki, przy czym wyró¿niliœmy cz¹stkê 1, dla której pomiar dokonywany jest w analizatorze $a$. [**Pojemnoœæ informacyjna $I_{1_a}$ wchodzi do estymacyjnej procedury**]{} EFI.\ \ [**Uwaga o randze $N=1$**]{}. Zachodzi pytanie: Jeœli oka¿e siê, ¿e tradycyjne formu³y mechaniki kwantowej dla eksperymentu EPR-Bohm’a pojawi¹ siê dla powy¿ej okreœlonej estymacji z $N=1$ (a tak siê istotnie stanie), to czy¿by pomiar odzia³ywania cz¹stki z aparatur¹ Sterna-Gerlach’a mia³ mieæ termodynamiczny charakter oddzia³ywania ma³ego uk³adu z termostatem, w zrozumieniu podanym na samym pocz¹tku Rozdzia³u \[Warunki brzegowe\]? W koñcu, jak wiemy z poprzednich rozdzia³ów, rozwi¹zania z $N=1$ odnosz¹ siê do zjawisk termodynamicznych. ### Informacja strukturalna. Amplituda prawdopodobieñstwa {#Informacja strukturalna EPR} Bior¹c pod uwagê ogóln¹ postaæ informacji strukturalnej (\[Q dla niezaleznych Yn w d4y\]) oraz uwagi Rozdzia³u \[Pojemnosc informacyjna zagadnienia EPR\] odnoœnie konstrukcji postaci $I_{1_a}$, (\[pojemnoscEPR\]), zauwa¿amy, ¿e [*informacja strukturalna*]{} $Q_{1_a}$ dla obserwowanej cz¹stki ma postaæ: $$\begin{aligned} \label{strukturalnaEPR} Q_{1_a} \equiv \sum\limits_{ab}{\int\limits_{0}^{2\pi}{d\vartheta}\; q_{ab}^{2}(\vartheta) \, \texttt{q\!F}_{ab}(q_{ab})} \; .\end{aligned}$$ Poni¿sza analiza zwi¹zana z wyprowadzeniem amplitud $q_{ab}$ jest podobna do analizy dla rozk³adu Boltzmanna w Rozdziale \[fizykastatystyczna\].\ \ [**Wariacyjna zasada informacyjna**]{}: Dla pojemnoœci informacyjnej $I$, (\[pojemnoscEPR\]), oraz informacji strukturalnej $Q$, (\[strukturalnaEPR\]) wariacyjna zasada informacyjna ma postaæ: $$\begin{aligned} \label{zskalarnaEPR} \delta_{(q_{ab})} K \equiv \delta_{(q_{ab})}\left( I_{1_a} + Q_{1_a}\right) = \delta_{(q_{ab})} \left(\;\int\limits _{0}^{2\pi} {d\vartheta \, k} \right) = 0 \; ,\end{aligned}$$ gdzie zgodnie z ogóln¹ postaci¹ (\[k form\]), gêstoœæ ca³kowitej informacji fizycznej $k$ dla amplitud $q_{ab}$ jest równa: $$\begin{aligned} \label{k EPR} k = 4 \sum\limits_{ab}{\left(q_{ab}^{'2} + \frac{1}{4} q_{ab}^{2}(\vartheta) \texttt{q\!F}_{ab}(q_{ab})\right)} \; .\end{aligned}$$ Rozwi¹zaniem problemu [*wariacyjnego*]{} (\[zskalarnaEPR\]) wzglêdem $q_{ab}$ jest [*równanie Eulera-Lagrange’a*]{} (\[EL eq\]) (por. (\[euler\])): $$\begin{aligned} \label{row E-L dla EPR} \frac{d}{d\vartheta}\left(\frac{\partial k}{\partial q_{ab}^{'}(\vartheta)}\right)=\frac{\partial k}{\partial q_{ab}} \; .\end{aligned}$$ Z równania (\[row E-L dla EPR\]) dla $k$ jak w (\[k EPR\]) otrzymujemy, dla ka¿dego dwucz¹stkowego ³¹cznego stanu spinowego $S_{ab}$: $$\begin{aligned} \label{rozweularaEPR} q_{ab}^{''}=\frac{1}{2}\frac{{d (\frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab}))}}{{dq_{ab}}} \; .\end{aligned}$$ Poniewa¿ $q_{ab}^{2}(\vartheta) \texttt{q\!F}_{ab}(q_{ab})$ jest jawnie jedynie funkcj¹ $q_{ab}$, wiêc ró¿niczka zupe³na zast¹pi³a pochodn¹ cz¹stkow¹ po $q_{ab}$ wystêpuj¹c¹ w (\[row E-L dla EPR\]).\ \ [**Zmodyfikowana obserwowana zasada strukturalna**]{}: Po wyca³kowaniu (\[pojemnoscEPR\]) przez czêœci, pojemnoœæ $I$ wynosi (por. (\[postac I po calk czesci\]), (\[Cn\]), (\[Cn tilde\])): $$\begin{aligned} \label{IEPR} I_{1_a} = 4\sum\limits _{ab}{\int\limits_{0}^{2\pi} {d\vartheta\left({\tilde{C}_{ab}-q_{ab}q_{ab}^{''}}\right)}} \; ,\end{aligned}$$ gdzie $$\begin{aligned} \label{stalaCEPR} \tilde{C}_{ab} = \frac{1}{{2\pi}}\left({q_{ab}\left({2\pi}\right)q_{ab}^{'} \left({2\pi}\right)-q_{ab}\left(0\right)q_{ab}^{'}\left(0\right)}\right) \; .\end{aligned}$$ Z powodu braku dodatkowych wiêzów $\kappa =1$, zatem [*zmodyfikowana obserwowana zasada strukturalna*]{} $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}}_{EPR} + \textit{q} = 0$, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), jest ze wzglêdu na (\[IEPR\]) oraz (\[strukturalnaEPR\]), nastêpuj¹ca: $$\begin{aligned} \label{u} 4\sum\limits_{ab} \left(- q_{ab}q_{ab}^{''} + \tilde{C}_{ab} + \frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab}) \right) = 0 \; .\end{aligned}$$ Poni¿ej przekonamy siê, ¿e $\widetilde{\mathbf{C}}_{EPR} = 4 \sum \tilde{C}_{ab} = 0$.\ \ Dla ka¿dego dwucz¹stkowego ³¹cznego stanu spinowego $S_{ab}$, obserwowana zasada strukturalna (\[u\]) ma wiêc postaæ (por. \[rownanie strukt E\]): $$\begin{aligned} \label{mikroEPR} - q_{ab}q_{ab}^{''} + \tilde{C}_{ab} + \frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab}) = 0 \; .\end{aligned}$$ Wraz z równaniem Eulera-Lagrange’a (\[rozweularaEPR\]), równanie (\[mikroEPR\]) pos³u¿y do wyprowadzenia równania generuj¹cego rozk³ad.\ \ [**Wyprowadzenia równania generuj¹cego**]{}: Wykorzystuj¹c w (\[mikroEPR\]) zwi¹zek (\[rozweularaEPR\]), otrzymujemy (por. (\[row rozn z q oraz qF\])) dla ka¿dego dwucz¹stkowego ³¹cznego stanu spinowego $S_{ab}$: $$\begin{aligned} \frac{1}{2}q_{ab}\frac{{d(\frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab}))}}{{dq_{ab}}} = \frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab}) + \tilde{C}_{ab} \; .\end{aligned}$$ Zapiszmy powy¿sze równanie w wygodniejszej formie: $$\begin{aligned} \frac{{2dq_{ab}}}{{q_{ab}}}=\frac{{d \left(\frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab})\right)}}{{\frac{1}{4} q_{ab}^{2} \texttt{q\!F}_{ab}(q_{ab}) + \tilde{C}_{ab}}} \; ,\end{aligned}$$ z której po obustronnym wyca³kowaniu otrzymujemy: $$\begin{aligned} \label{jEPR} \frac{1}{4} q_{ab}^{2}(\vartheta) \texttt{q\!F}_{ab}(q_{ab}) = \frac{{q_{ab}^{2}(\vartheta)}}{{A_{ab}^{2}}} - \tilde{C}_{ab} \; ,\end{aligned}$$ gdzie $A_{ab}^{2}$ jest w ogólnoœci zespolon¹ sta³¹.\ \ [**Równanie generuj¹ce**]{}: Podstawiaj¹c (\[jEPR\]) do (\[rozweularaEPR\]) otrzymujemy szukane ró¿niczkowe równanie generuj¹ce dla amplitud $q_{ab}$: $$\begin{aligned} \label{row generujace dla amplitud w EPR} q_{ab}^{''}(\vartheta) = \frac{{q_{ab}(\vartheta)}}{{A_{ab}^{2}}} \; ,\end{aligned}$$ bêd¹ce konsekwencj¹ obu zasad informacyjnych, strukturalnej i wariacyjnej.\ \ [**Rozwi¹zanie równania generuj¹cego**]{}: Poniewa¿ amplituda $q_{ab}$ jest rzeczywista, wiêc $A_{ab}^{2}$ te¿ musi byæ rzeczywiste. Podobnie jak w rozdziale \[rozdz.energia\], poniewa¿ sta³a $A_{ab}^{2}$ jest rzeczywista, wiêc mo¿na j¹ przedstawiæ za pomoc¹ innej rzeczywistej sta³ej $a_{ab}$ jako $A_{ab}=a_{ab}$ lub $A_{ab} = i \, a_{ab}$. Zatem istniej¹ dwie klasy rozwi¹zañ równania (\[row generujace dla amplitud w EPR\]).\ \ Dla $A_{ab} = a_{ab}$, rozwi¹zanie (\[row generujace dla amplitud w EPR\]) ma charakter czysto [*eksponencjalny*]{}: $$\begin{aligned} q_{ab}(\vartheta) = B_{ab}^{''}\exp\left(-\frac{\vartheta}{a_{ab}}\right) + C_{ab}^{''}\exp\left(\frac{\vartheta}{a_{ab}}\right) \; ,\quad A_{ab}=a_{ab} \; ,\end{aligned}$$ gdzie sta³e $B_{ab}^{''}$ oraz $C_{ab}^{''}$ s¹ rzeczywiste.\ \ Natomiast dla $A_{ab} = i\, a_{ab}$, rozwi¹zanie (\[row generujace dla amplitud w EPR\]) ma charakter [*trygonometryczny*]{}: $$\begin{aligned} \label{qabEPR} q_{ab}(\vartheta) = B_{ab}^{'}\sin\left(\frac{\vartheta}{a_{ab}}\right) + C_{ab}^{'}\cos\left(\frac{\vartheta}{a_{ab}}\right)\; ,\quad A_{ab} = i \, a_{ab} \; ,\end{aligned}$$ gdzie $a_{ab}$, $B_{ab}^{'}$, $C_{ab}^{'}$ s¹ rzeczywistymi sta³ymi.\ \ [**Warunek niezmienniczoœci przy obrocie o $2 \pi$**]{}: W rozwa¿anym obecnie przypadku[^81] wartoœci $\vartheta$ s¹ k¹tem z ograniczonego zbioru $\langle0, 2\pi)$. Zatem z geometrycznej symetrii uk³adu przy obrocie o k¹t $2\pi$ wynika, ¿e rozk³ad $P\left(S_{a}|S_{b},\vartheta\right)$ jest równie¿ funkcj¹ okresow¹ zmiennej $\vartheta$. Ze wzglêdu na (\[inffishEPR2\]) warunek ten oznacza, ¿e funkcja $q_{ab}(\vartheta)$ powinna byæ okresowa, *zatem wybieramy rozwi¹zanie o charakterze* [**trygonometrycznym**]{}, przy czym Funkcje sin oraz cos w (\[qabEPR\]) s¹ funkcjami bazowymi tworz¹cymi amplitudê prawdopodobieñstwa $q_{ab}(\vartheta)$.\ \ [**Funkcje bazowe**]{} na przestrzeni amplitud powinny byæ ortogonalne. Z postaci $q_{a}$ w (\[qabEPR\]) wynika, ¿e funkcjami bazowymi s¹ funkcje ’sin’ oraz ’cos’. Poniewa¿ $q_{a}$ jest okreœlone na przestrzeni parametru $\vartheta \in \left\langle 0, 2 \pi\right)$, zatem [*warunek ortogonalnoœci funkcji bazowych*]{} na tej przestrzeni: $$\begin{aligned} \int\limits _{0}^{2\pi}{d\vartheta\sin\left({\vartheta\mathord{\left/{\vphantom{\vartheta{a_{ab}}}}\right.\kern -\nulldelimiterspace}{a_{ab}}}\right)}\cos\left({\vartheta\mathord{\left/{\vphantom{\vartheta{a_{ab}}}}\right.\kern -\nulldelimiterspace}{a_{ab}}}\right)=0\label{ortogonalEPR}\end{aligned}$$ daje po wyca³kowaniu postaæ sta³ych $a_{ab}$: $$\begin{aligned} \label{warAEPR} a_{ab}=\frac{2}{{n_{ab}}} \; ,\quad\quad {\rm gdzie} \;\;\; n_{ab} = 1,2,... \;\; .\end{aligned}$$\ [**Warunek minimalizacji pojemnoœci $I$**]{}: Warunek (\[warAEPR\]) jest warunkiem na dopuszczalne wartoœci $a_{ab} = -i A_{ab}$ (por. (\[qabEPR\])), ale jak widaæ nie wskazuje on na ¿adn¹ z nich jednoznacznie. Mo¿e to nast¹piæ, gdy ustalimy wartoœæ $n_{ab}$, co uczynimy ograniczaj¹c rozwa¿ania do zapostulowanego w Rozdziale \[Poj inform zmiennej los poloz\] warunku minimalizacji informacji kinetycznej $I \rightarrow min$.\ \ Zacznijmy od wyznaczenia sta³ej $\tilde{C}_{ab}$, (\[stalaCEPR\]). Po wstawieniu do (\[stalaCEPR\]) amplitudy (\[qabEPR\]) z (\[warAEPR\]), otrzymujemy: $$\begin{aligned} \label{postac stalej Cab} \!\!\!\!\!\!\!\!\!\!\!\! \tilde{C}_{ab} \!\! &=& \!\! \frac{n_{ab}}{4 \pi} \, \sin (\pi \, n_{ab}) \\ &\cdot& \!\! \left((B_{ab}^{'}-C_{ab}^{'}) (B_{ab}^{'}+C_{ab}^{'}) \cos (\pi \, n_{ab}) - 2 B_{ab}^{'} C_{ab}^{'} \sin (\pi \, n_{ab}) \right) = 0 \, , \;\;\; {\rm gdzie} \;\;\; n_{ab} = 1,2,... \;\; , \;\; \nonumber\end{aligned}$$ sk¹d po skorzystaniu z $I_{1_a} = 4\sum\limits{\int {d\vartheta({\tilde{C}_{ab}-q_{ab}q_{ab}^{''}})}}$, (\[IEPR\]), oraz równania generuj¹cego (\[row generujace dla amplitud w EPR\]) pozwalaj¹cego wyeliminowaæ $q_{ab}^{''}$, otrzymujemy u¿yteczn¹ postaæ pojemnoœci informacyjnej: $$\begin{aligned} \label{postac Iab w Aab i qab} I_{1_{a}} = - 4\sum\limits _{ab}{\frac{1}{{A_{ab}^{2}}}\int\limits _{0}^{2\pi}{d\vartheta\; q_{ab}^{2}(\vartheta)}} \; .\end{aligned}$$ Nale¿y jeszcze wyznaczyæ ca³kê w (\[postac Iab w Aab i qab\]): $$\begin{aligned} \label{piEPR} \int\limits _{0}^{2\pi}{d\vartheta\; q_{ab}^{2}\left(\vartheta\right)} \!\! &\equiv& \!\! \int\limits_{0}^{2\pi} {d\vartheta\; P \left({S_{a}\left|{S_{b}, \vartheta} \right.} \right)} = \int\limits_{0}^{2\pi}{d\vartheta\frac{P\left(S_{a},S_{b},\vartheta\right)}{p\left(\vartheta,S_{b}\right)}} = \int\limits_{0}^{2\pi} {d\vartheta \frac{{p\left({\vartheta,S_{a}\left|{S_{b}}\right.}\right)P\left({S_{b}}\right)}}{{p\left({\vartheta,S_{b}}\right)}}} \nonumber \\ &=& \int{d\vartheta\frac{{p\left({\vartheta,S_{a}\left|{S_{b}}\right.}\right)\frac{1}{2}}}{{p\left({S_{b}\left|\vartheta\right.}\right)r\left(\vartheta\right)}}}=\int\limits _{0}^{2\pi}{d\vartheta\frac{{p\left({\vartheta,S_{a}\left|{S_{b}}\right.}\right)\frac{1}{2}}}{{\frac{1}{2}\frac{1}{{2\pi}}}}} \nonumber \\ &=& 2\pi\int\limits _{0}^{2\pi}{d\vartheta\; p\left({\vartheta,S_{a}\left|{S_{b}}\right.}\right)} = 2\pi P \left({S_{a}\left|{S_{b}}\right.}\right)=2\pi\frac{1}{2} = \pi \; ,\end{aligned}$$ gdzie w pierwszej linii i) wpierw skorzystano z definicji amplitudy, $q_{ab}^{2}\left(\vartheta\right) = P\left(S_{a}\left|{S_{b},\vartheta}\right.\right)$, (\[inffishEPR2\]), ii) nastêpnie z definicji prawdopodobieñstwa warunkowego, iii) znowu z definicji prawdopodobieñstwa warunkowego: $$\begin{aligned} \label{p od vartheta Sa warunek Sb} p\left({\vartheta,S_{a}\left|{S_{b}}\right.}\right) = \frac{P\left(S_{a},S_{b},\vartheta\right)}{ P\left({S_{b}}\right)} \; ,\end{aligned}$$ nastêpnie w drugiej linii z $p\left(\vartheta, S_{b} \right) = p\left(S_{b}\left|\vartheta \right. \right) r(\vartheta)\,$ (por. Uwaga poni¿ej (\[prawdkat\])) oraz z wyra¿eñ (\[prawdkat\]), (\[pol\]) oraz w trzeciej linii z: $$\begin{aligned} \label{calka z p od vartheta Sa warunek Sb} P(S_{a}|S_{b}) = \int_{0}^{2 \pi} d \vartheta \, p\left({\vartheta,S_{a}\left|{S_{b}}\right.}\right) \; ,\end{aligned}$$ a na koniec z (\[warunkSab\]).\ Wykorzystuj¹c (\[postac Iab w Aab i qab\]), (\[piEPR\]) oraz $A_{ab} = i \, a_{ab}$ i (\[warAEPR\]), mo¿emy wyraziæ informacjê $I_{1_a}$ poprzez sta³e $n_{ab}$, otrzymuj¹c: $$\begin{aligned} \label{I1piEPR} \!\!\! I_{1_a} = -Q_{1_a} = - 4\sum\limits _{ab}{\frac{1}{{A_{ab}^{2}}}\int\limits _{0}^{2\pi}{d\vartheta\; q_{ab}^{2}}} = 4\pi\sum\limits_{ab}{\frac{1}{{a_{ab}^{2}}}} = \pi\sum\limits_{ab}{n_{ab}^{2}} \; , \;\; {\rm gdzie} \;\; n_{ab} = 1,2,... \;\, , \;\;\end{aligned}$$ przy czym drug¹ z powy¿szych równoœci, a zatem i pierwsz¹, otrzymano korzystaj¹c z postaci informacji strukturalnej (\[strukturalnaEPR\]), (\[jEPR\]) oraz z (\[postac stalej Cab\]). Zwi¹zek $Q_{1_a} = - I_{1_a}$ jest wyrazem ogólnego warunku (\[rownowaznosc strukt i zmodyfikowanego strukt\]) spe³nienia przez metodê EFI oczekiwanej zasady strukturalnej (\[ideal condition from K\]).\ \ Z powy¿szego zwi¹zku wynika, ¿e warunek minimalizacji $I_{1_a}$ bêdzie spe³niony dla: $$\begin{aligned} \label{n_ab minimal I} n_{ab}=1 \; , \;\;\;\; I_{1_a} \rightarrow min \;\; , \end{aligned}$$ dla dowolnych $S_{a}$ oraz $S_{b}$. Warunek $n_{ab}=1$ zgodnie z (\[warAEPR\]) odpowiada nastêpuj¹cej wartoœci $a_{ab}$: $$\begin{aligned} \label{ss} a_{ab} = 2 \quad\quad {\rm dla\;\; dowolnych} \;\;\; S_{a}\;\; {\rm oraz} \;\; S_{b} \; .\end{aligned}$$ Sumowanie w (\[I1piEPR\]) przebiega po wszystkich $S_{a}$ i $S_{b}$, zatem dla $n_{ab} = 1$, otrzymujemy wartoœæ minimaln¹ $I_{1_a}$: $$\begin{aligned} \label{I EPR minimalna} I_{1_a \,(min)} = \pi \sum\limits_{a=-}^{+} \sum\limits_{b=-}^{+} {n_{ab}^{2}} = 4 \pi \; , \;\; {\rm gdzie} \;\; n_{ab} = 1 \; . \end{aligned}$$ Tak wiêc otrzymaliœmy minimaln¹ wartoœæ pojemnoœci informacyjnej dla parametru $\vartheta$. [**Wyznaczenie sta³ych w amplitudzie**]{}: W wyra¿eniu (\[qabEPR\]), które dla $a_{ab} = 2$ przyjmuje postaæ: $$\begin{aligned} \label{qab dla a=2} q_{ab}(\vartheta) = B_{ab}^{'}\sin\left(\frac{\vartheta}{2}\right) + C_{ab}^{'}\cos\left(\frac{\vartheta}{2}\right) \; , \end{aligned}$$ wystêpuj¹ jeszcze sta³e $B_{ab}^{'}$ oraz $C_{ab}^{'}$, które równie¿ musimy wyznaczyæ. Wyznaczymy je korzystaj¹c z wczeœniej ustalonych w (\[vartheta zero\]), wartoœci ³¹cznego prawdopodobieñstwa warunkowego $P\left(S_{ab}|\vartheta\right)$ dla $\vartheta=0$.\ WyraŸmy wpierw $P\left(S_{ab}|\vartheta\right)$ poprzez amplitudê $q_{ab}$: $$\begin{aligned} \label{PqEPR} P\left(S_{ab}|\vartheta\right) \equiv P\left(S_{a}S_{b}|\vartheta\right) = \frac{P\left(S_{a}S_{b},\vartheta\right)}{r(\vartheta)} = P\left(S_{a}|S_{b},\vartheta\right)\frac{P\left(S_{b} \, , \vartheta\right)}{r(\vartheta)}\nonumber \\ =P\left(S_{a}|S_{b},\vartheta\right)P\left(S_{b}|\vartheta\right)=q_{ab}^{2}(\vartheta)P\left(S_{b}|\vartheta\right)=\frac{1}{2}\, q_{ab}^{2}(\vartheta) \; ,\end{aligned}$$ gdzie skorzystano z (\[pol\]). Z równania (\[PqEPR\]) wynika, ¿e: $$\begin{aligned} \label{q Sab zero} q_{ab}(\vartheta) = 0 \; \;\; {\rm je\acute{s}li} \;\;\; P\left(S_{ab}|\vartheta\right) = 0 \; .\end{aligned}$$ Korzystaj¹c z (\[q Sab zero\]) oraz $P\left(++|0\right)=P\left(--|0\right)=0$, (\[vartheta zero\]), i wstawiaj¹c (\[qab dla a=2\]) dla $\vartheta=0$ do (\[PqEPR\]), otrzymujemy: $$\begin{aligned} \label{ce} C_{++}^{'}=C_{--}^{'}=0 \; .\end{aligned}$$ Ponadto, korzystaj¹c z symetrii geometrycznej eksperymentu, $P\left(S_{+-}|\vartheta\right)=P\left(S_{- +}|\vartheta\right)$ oraz $P\left(S_{++}|\vartheta\right)$ $=P\left(S_{--}|\vartheta\right)$, zapisanej w (\[symetriaEPR\]), otrzymujemy: $$\begin{aligned} \label{wspol B prim dla q w EPR} B_{++}^{'} = B_{--}^{'} \;\;\; {\rm oraz} \;\;\; B_{+-}^{'} = B_{-+}^{'} \; ,\end{aligned}$$ oraz $$\begin{aligned} \label{C+- rowne C+- EPR} C_{+-}^{'} = C_{-+}^{'} \; .\end{aligned}$$ [**Analiza z wykorzystaniem metryki Rao-Fishera**]{}: Poni¿ej przekonamy siê, ¿e wyznaczenie pozosta³ych sta³ych $B_{ab}^{'}$ oraz $C_{ab}^{'}$ wymaga dodatkowego za³o¿enia, odnosz¹cego siê do postaci metryki Rao-Fishera na przestrzeni statystycznej ${\cal S}$, szukanego statystycznego modelu.\ Szukanym rozk³adem prawdopodobieñstwa w eksperymencie EPR-Bohm’a jest dyskretny rozk³ad (\[spinprawdop\]) okreœlony na przestrzeni zdarzeñ $S_{a}S_{b} \equiv S_{ab} \in \Omega_{ab}$, (\[4 zdarzenia EPR\]), i unormowany zgodnie z (\[normalizacja P daje wsp w qab\]) do jednoœci. Zatem zbiór mo¿liwych wyników to: $$\begin{aligned} \label{wyniki lacznego pomiaru Sa i Sb} i \equiv ab = (++),(--),(+-),(-+) \;\; .\end{aligned}$$ Aplitudy prawdopodobieñstwa zwi¹zane z rozk³adem (\[spinprawdop\]) maj¹ zgodnie z (\[PqEPR\]) postaæ: $$\begin{aligned} \label{rozklad na przestrzeni statystycznej EPR} \tilde{q}_{i} \equiv \frac{1}{\sqrt{2}}\, q_{ab}(\vartheta) = \sqrt{P\left(S_{ab}|\vartheta\right)} \; .\end{aligned}$$ Zgodnie z (\[qab dla a=2\]) amplitudy $q_{ab}(\vartheta)$ maj¹ nastêpuj¹ce pochodne: $$\begin{aligned} \label{pochodna qab dla EPR} \frac{\partial q_{ab}}{\partial \vartheta} = \frac{1}{2}\left(B_{ab}^{'}\cos\left(\frac{\vartheta}{2}\right) - C_{ab}^{'}\sin\left(\frac{\vartheta}{2}\right) \right)\; .\end{aligned}$$ Dla $\aleph=4$ wyników (\[wyniki lacznego pomiaru Sa i Sb\]), z ogólnego zwi¹zku (\[metryka Rao-Fishera w ukladzie wsp - amplitudy\]) okreœlaj¹cego metrykê Rao-Fishera $g_{ab} = 4 \sum_{i=1}^{\aleph} \frac{\partial q^{i}}{\partial \theta^{a}} \frac{\partial q^{i}}{\partial \theta^{b}} \, $, otrzymujemy po skorzystaniu z (\[rozklad na przestrzeni statystycznej EPR\]) oraz (\[pochodna qab dla EPR\]) nastêpuj¹c¹ postaæ metryki $g_{\vartheta \, \vartheta}$ indukowanej z rozk³adu (\[spinprawdop\]): $$\begin{aligned} \label{metryka Rao-Fishera dla EPR} g_{\vartheta \, \vartheta}(\vartheta) &=& 4 \sum_{i=1}^{\aleph=4} \frac{\partial \tilde{q}_{i}}{\partial \vartheta} \frac{\partial \tilde{q}_{i}}{\partial \vartheta} = 4 \sum\limits_{ab} \frac{\partial (\frac{1}{\sqrt{2}} q_{ab})}{\partial \vartheta} \frac{\partial (\frac{1}{\sqrt{2}} q_{ab})}{\partial \vartheta} \nonumber \\ &=& \frac{1}{2} \sum_{ab} \left( (B_{ab}^{'})^{2} + \left((C_{ab}^{'})^{2} - (B_{ab}^{'})^{2}\right) \sin^{2} (\frac{\vartheta}{2}) - B_{ab}^{'} \, C_{ab}^{'}\sin(\vartheta) \right) \; \end{aligned}$$ na jednowymiarowej przestrzeni statystycznej ${\cal S}$ parametryzowanej w bazie $\vartheta$.\ \ Po skorzystaniu z (\[wspol B prim dla q w EPR\]), postaæ metryki $g_{\vartheta \, \vartheta}(\vartheta)$, (\[metryka Rao-Fishera dla EPR\]), prowadzi dla $\vartheta=0$ do warunku: $$\begin{aligned} \label{g dla vartheta = 0} g_{\vartheta \, \vartheta}(\vartheta=0) = \frac{1}{2} \sum_{ab} (B_{ab}^{'})^{2} = (B_{++}^{'})^{2} + (B_{+-}^{'})^{2} \; \;\;\; {\rm dla } \;\; \vartheta=0 \; ,\end{aligned}$$ natomiast dla $\vartheta = \pi$ do warunku: $$\begin{aligned} \label{g dla vartheta = pi} g_{\vartheta \, \vartheta}(\vartheta=\pi) = \frac{1}{2} \sum_{ab} (C_{ab}^{'})^{2} = (C_{+-}^{'})^{2} \;\;\;\; {\rm dla } \;\; \vartheta = \pi \; ,\end{aligned}$$ gdzie skorzystano równie¿ z (\[ce\]).\ \ [**Centralne za³o¿enie statystyczne dla eksperymentu EPR-Bohm’a**]{}: Za³ó¿my, ¿e [*w eksperymencie* ]{} EPR-Bohm’a [*metryka Rao-Fishera $g_{\vartheta \, \vartheta}$ na ${\cal S}$ jest niezale¿na od wartoœci parametru*]{} $\vartheta$, tzn.: $$\begin{aligned} \label{Centralne zal stat dla EPR} g_{\vartheta \, \vartheta}(\vartheta) = g_{\vartheta \, \vartheta} = const. \; . \end{aligned}$$ W szczególnym przypadku warunek (\[Centralne zal stat dla EPR\]) oznacza, ¿e $g_{\vartheta \, \vartheta}(\vartheta=0) = g_{\vartheta \, \vartheta}(\vartheta=\pi)$, co uwzglêdniaj¹c w zale¿noœciach (\[g dla vartheta = 0\]) oraz (\[g dla vartheta = pi\]) daje: $$\begin{aligned} \label{rownanie dla B i C} (B_{++}^{'})^{2} + \, (B_{+-}^{'})^{2} = (C_{+-}^{'})^{2} \geq 0 \; .\end{aligned}$$ Warunek ten oznacza, ¿e $C_{+-}^{'} \neq 0 \, $, gdy¿ w przeciwnym wypadku, tzn. dla $C_{+-}^{'}=0$, z warunku (\[rownanie dla B i C\]) oraz z (\[ce\]) otrzymalibyœmy zerowanie siê wszystkich wspó³czynników $B_{ab}^{'}$ oraz $C_{ab}^{'}$, co odpowiada³oby trywialnemu przypadkowi braku rozwi¹zania dla zagadnienia EPR. Zatem otrzymujemy: $$\begin{aligned} \label{C+- niezerowe EPR} C \equiv C_{+-}^{'} = C_{-+}^{'} \neq 0 \; ,\end{aligned}$$ gdzie równoœci wspó³czynników wynika z (\[C+- rowne C+- EPR\]).\ \ W koñcu niezale¿noœæ $g_{\vartheta \, \vartheta}$ od wartoœci $\vartheta$, (\[Centralne zal stat dla EPR\]), daje po skorzystaniu z postaci $g_{\vartheta \, \vartheta}$, (\[metryka Rao-Fishera dla EPR\]), oraz z warunków (\[ce\]), (\[wspol B prim dla q w EPR\]) i (\[C+- rowne C+- EPR\]), warunek: $$\begin{aligned} \label{warunek z g dla dowolnego vartheta 1} \sum_{ab} \left( (C_{ab}^{'})^{2} - (B_{ab}^{'})^{2} \right) = 2 \,( \,(C_{+-}^{'})^{2} - (B_{++}^{'})^{2} - \,(B_{+-}^{'})^{2} ) = 0 \; ,\end{aligned}$$ czyli warunek pokrywaj¹cy siê z (\[rownanie dla B i C\]) oraz: $$\begin{aligned} \label{warunek z g dla dowolnego vartheta 2} \sum_{ab} B_{ab}^{'} \,C_{ab}^{'} = 2 (B_{+-}^{'} \,C_{+-}^{'}) = 0 \; .\end{aligned}$$ Warunek (\[warunek z g dla dowolnego vartheta 2\]) wraz z (\[C+- niezerowe EPR\]) oznacza: $$\begin{aligned} \label{zerowanie B+-} B_{+-}^{'} = B_{-+}^{'} = 0 \; ,\end{aligned}$$ gdzie skorzystano równie¿ z (\[wspol B prim dla q w EPR\]). Po uwzglêdnieniu (\[zerowanie B+-\]) w (\[warunek z g dla dowolnego vartheta 1\]), otrzymujemy: $$\begin{aligned} \label{rownosc B2 i C2} (B_{++}^{'})^{2} = (C_{+-}^{'})^{2} \; .\end{aligned}$$ W koñcu zauwa¿my, ¿e ze wzglêdu na (\[zerowanie B+-\]) oraz (\[C+- niezerowe EPR\]), warunek istnienia nietrywialnego rozwi¹zania oznacza, ¿e: $$\begin{aligned} \label{nie zerowanie B++} B \equiv B_{++}^{'} = B_{--}^{'} \neq 0,\end{aligned}$$ gdzie ponownie w równoœci skorzystano z (\[wspol B prim dla q w EPR\]).\ Podstawiaj¹c otrzymane wyniki dla wspó³czynników $B_{ab}^{'}$ oraz $C_{ab}^{'}$ do (\[qab dla a=2\]), otrzymujemy: $$\begin{aligned} \label{wynikqEPR} q_{++}(\vartheta) = q_{--}(\vartheta)=B\sin\left(\vartheta/2\right) \; , \;\; q_{-+}(\vartheta) = q_{+-}(\vartheta)=C\cos\left(\vartheta/2\right) \; .\end{aligned}$$ Jak widaæ, równoœæ wspó³czynników w zwi¹zkach (\[C+- niezerowe EPR\]) oraz (\[nie zerowanie B++\]) jest odbiciem równoœci odpowiednich amplitud, wynikaj¹cej z symetrii odbicia przestrzennego (\[symetriaEPR\]) oraz wzoru (\[PqEPR\]).\ \ Musimy jeszcze wyznaczyæ sta³e $B$ oraz $C$. Z powodu warunku normalizacji prawdopodobieñstwa $P\left(S_{a}S_{b}|\vartheta\right)$, (\[normalizacja P daje wsp w qab\]), otrzymujemy ze wzglêdu na (\[PqEPR\]) równanie: $$\begin{aligned} \label{qkwa} \frac{1}{2}\left(q_{++}^{2}(\vartheta)+q_{--}^{2}(\vartheta)+q_{-+}^{2}(\vartheta)+q_{+-}^{2}(\vartheta)\right) = 1 \; .\end{aligned}$$ Korzystaj¹c z (\[wynikqEPR\]) i (\[qkwa\]) mamy: $$\begin{aligned} B^{2}\sin^{2}\left(\vartheta/2\right)+C^{2}\cos^{2}\left(\vartheta/2\right)=\left(B^{2}-C^{2}\right)\sin^{2}\left(\vartheta/2\right)+C^{2} = 1 \; .\end{aligned}$$ [**Koñcowa postaæ amplitud**]{}: Porównuj¹c wspó³czynniki stoj¹ce przy odpowiednich funkcjach zmiennej $\vartheta$ po lewej i prawej stronie drugiej równoœci powy¿szego wyra¿enia, otrzymujemy: $$\begin{aligned} \label{postac B oraz C} B^{2} = C^{2} = 1 \; ,\end{aligned}$$ co po wstawieniu do (\[wynikqEPR\]) daje ostatecznie rozwi¹zanie równania generuj¹cego (\[row generujace dla amplitud w EPR\]): $$\begin{aligned} \label{qEPR} q_{++}(\vartheta) = q_{--}(\vartheta) = \pm \sin\left(\vartheta/2\right) \; , \quad \;\; q_{-+}(\vartheta) = q_{+-}(\vartheta) = \pm \cos\left(\vartheta/2\right) \; .\end{aligned}$$ [**Wynik na prawdopodobieñstwo w eksperymencie EPR-Bohm’a**]{}: Podstawienie amplitudy (\[qEPR\]) do $P\left(S_{ab}|\vartheta\right) = \frac{1}{2} \, q_{ab}^{2}(\vartheta) \,$, (\[PqEPR\]), daje wynik na ³¹czne prawdopodobieñstwo otrzymania okreœlonej kombinacji rzutów spinów przy zadanej wartoœci k¹ta $\vartheta$: $$\begin{aligned} \label{wynikEPR} P\left(++|\vartheta\right) = P\left(--|\vartheta\right)=\frac{1}{2}\sin^{2}\left(\vartheta/2\right) , \;\; P\left(+-|\vartheta\right) = P\left(-+|\vartheta\right)=\frac{1}{2}\cos^{2}\left(\vartheta/2\right) ,\end{aligned}$$ [*który jest przewidywaniem mechaniki kwantowej*]{} [@Manoukian].\ \ Na koniec, podstawiaj¹c wartoœci otrzymanych wspó³czynników $B_{ab}^{'}$ oraz $C_{ab}^{'}$ do (\[metryka Rao-Fishera dla EPR\]), otrzymujemy wartoœæ sta³ej $g_{\vartheta \vartheta}$ dla jedynej sk³adowej metryki Rao-Fishera w (\[Centralne zal stat dla EPR\]): $$\begin{aligned} \label{metryka w EPR} g_{\vartheta \, \vartheta}(\vartheta) = g_{\vartheta \, \vartheta} = 1 \; ,\end{aligned}$$ która zgodnie (\[stalosc il wewn dla Euklidesowego ukl wsp\]) jest metryk¹ samo-dualnego (Euklidesowego) uk³adu wspó³rzêdnych $\vartheta$ na jednowymiarowej przestrzeni statystycznej ${\cal S}$.\ \ [**Ró¿nica pomiêdzy przedstawionym wyprowadzeniem a analiz¹ w [@Frieden]**]{}: Wynik (\[wynikEPR\]) w ramach medody EFI, zosta³ oryginalnie wyprowadzony w [@Frieden], jednak¿e powy¿szy sposób wyprowadzenia ró¿ni siê w dwóch miejscach. Po pierwsze, warunki brzegowe zosta³y ujêtne w sposób bardziej przejrzysty [@Mroziakiewicz], a po drugie w [@Frieden] odwo³ano siê do w³asnoœci ortogonalnoœci wprowadzonych tam kwantowych amplitud, o czym wspomnimy na koñcu rozdzia³u, a czego w powy¿szym wyprowadzeniu unikniêto, wprowadzaj¹c w to miejsce [*warunek niezale¿noœci metryki Rao-Fishera od wartoœci parametru $\vartheta$*]{}.\ \ [**Wnioski**]{}: W (\[laczneEPR\]) wyznaczono ³¹czne prawdopodobieñstwo $P\left({S_{ab}}\right)=\frac{1}{4}$. Z drugiej strony w (\[Sb wycalkowane\]) otrzymano $P(S_{b})=1/2$ (i analogicznie $P(S_{a})=1/2$), z czego wynika, ¿e: $$\begin{aligned} \label{niezalezne} P\left({S_{ab}}\right)=P\left({S_{a}}\right)P\left({S_{b}}\right) \; ,\end{aligned}$$ który to warunek oznacza niezale¿noœæ spinowych zmiennych $S_{a}$ oraz $S_{b}$.\ \ Natomiast z (\[wynikEPR\]) widaæ, ¿e efekt korelacji spinu zmienia siê bardzo mocno wraz z wartoœci¹ k¹ta $\vartheta$ i w samej rzeczy porównanie (\[wynikEPR\]) z (\[pol dla Sa\]) i (\[pol\]) daje warunek: $$\begin{aligned} \label{nierownosc P z brzegowymi dla zaleznosci od kata w EPR} P\left(S_{ab}|\vartheta\right)\neq P\left(S_{a}|\vartheta\right)P\left(S_{b}|\vartheta\right) \; .\end{aligned}$$ Relacje (\[niezalezne\]) oraz (\[nierownosc P z brzegowymi dla zaleznosci od kata w EPR\]) nie s¹ jednak w sprzecznoœci. Istotnie, poniewa¿ prawdopodobieñstwa $P\left({S_{ab}}\right)$, $P\left({S_{a}}\right)$ oraz $P\left({S_{b}}\right)$ s¹ wyznaczone na skutek uœrednieñ po wszystkich wartoœciach k¹ta $\vartheta$, zatem mo¿na by³o oczekiwaæ, ¿e po dokonaniu tych uœrednieñ korelacja zmniejszy siê. Zauwa¿my, ¿e w (\[niezalezne\]) nie ma zale¿noœci od $\vartheta$, co wynika z tego, ¿e prawdopodobieñstwo $P\left({S_{ab}}\right)$ z definicji okreœla równoczesne pojawienie siê kombinacji rzutów spinów $S_{ab}$ niezale¿nie od informacji o zmiennej k¹towej $\vartheta$. Podobnie, $P\left({S_{a}}\right)$ oraz $P\left({S_{b}}\right)$ okreœlaj¹ prawdopodobieñstwa odpowiadaj¹cych im zdarzeñ w sytuacji pozbycia siê informacji o k¹cie $\vartheta$.\ Równanie (\[niezalezne\]) mówi wiêc, ¿e w sytuacji uœrednienia po k¹cie $\vartheta$, czyli wtedy gdy zmienna ta jest pod kontrol¹, zmienne $S_{a}$ oraz $S_{b}$ rzutów spinów s¹ niezale¿ne, wiêc s¹ równie¿ ca³kowicie nieskorelowane.\ Natomiast wynik (\[nierownosc P z brzegowymi dla zaleznosci od kata w EPR\]) dla eksperymentu EPR-Bohm’a zachodzi wtedy, gdy dokonujemy estymacji k¹ta $\vartheta$ metod¹ EFI, czyli w sytuacji oczywistego braku uœrednienia po $\vartheta$.\ \ [**Porównanie stwierdzeñ o uœrednieniu**]{}: Powy¿szy wynik ma interesuj¹c¹ fizyczn¹ interpretacjê. Mianowicie twierdzenie Ehrenfesta mówi, ¿e œrednie wartoœci kwantowych operatorów s¹ równe ich klasycznym odpowiednikom. W jego œwietle wyra¿enie (\[niezalezne\]) mówi, ¿e w przypadku uœrednienia po k¹cie $\vartheta$ stany spl¹tane EPR-Bohm’a, na powrót ulegaj¹ klasycznej separacji. W analizie statystycznej mówimy, ¿e po wyeliminowaniu wp³ywu zmiennej trzeciej, któr¹ jest k¹t $\vartheta$, okaza³o siê, ¿e zmienne rzutów spinów $S_{a}$ i $S_{b}$ s¹ nieskorelowane.\ \ [**Uwaga o wyprowadzeniu Friedena [@Frieden]**]{}: Warunek niezale¿noœci wyprowadzenia formu³ (\[wynikqEPR\]) od mechaniki kwantowej mo¿na by nieco os³abiæ, tzn. na tyle, aby statystycznoœæ teorii by³a dalej widoczna. Frieden uczyni³ to, jak nastêpuje [@Frieden]:\ Poka¿my, ¿e $q_{ab}(\vartheta)$ jest proporcjonalna do “[*kwantowej*]{}” [*amplitudy*]{} $\psi_{ab}(\vartheta)$, parametryzowanej parametrem $\vartheta$ dla stanów (\[4 zdarzenia EPR\]). Niech $P\left(S_{ab},\vartheta\right)$ jest prawdopodobieñstwem pojawienia siê konfiguracji $(a,b)$ rzutu spinów, [*podczas gdy*]{} parametr wynosi $\vartheta$. Nie jest to prawdopodobieñstwo ³¹czne zajœcia zdarzenia: “pojawi³a siê konfiguracji $(a,b)$ rzutu spinów oraz k¹t $\vartheta$”, gdy¿ ten ostatni nie jest zmienn¹ losow¹.\ \ Funkcjê $\psi_{ab}(\vartheta)$ okreœlimy tak, ¿e kwadrat jej modu³u spe³nia zwi¹zek: $$\begin{aligned} \label{k} |\psi_{ab}(\vartheta)|^{2} \equiv |\psi_{ab}\left({\vartheta\left|{S_{ab}} \right.}\right)|^{2} \equiv p\left({\vartheta\left|{S_{ab}}\right.}\right) = \frac{P\left(S_{ab},\vartheta\right)}{P\left(S_{ab}\right)} \; ,\end{aligned}$$ tzn. [**jest prawdopodobieñstwem, ¿e skoro pojawi³aby siê ³¹czna konfiguracja spinów $S_{ab}$, to wartoœæ k¹ta wynosi**]{} $\vartheta$.\ Problem polega na tym, ¿e $q_{ab}(\vartheta)$ opisuje losowe zachowanie zmiennej spinowej $S_{a}$, a nie $\vartheta$, co oznacza, ¿e $q_{ab}(\vartheta)$ jest amplitud¹ typu $\psi_{\vartheta}({ab})$, a nie $\psi_{ab}({\vartheta})$.\ Aby pokazaæ, ¿e $q_{ab}(\vartheta)$ jest proporcjonalna do $\psi_{ab}(\vartheta)$, skorzystajmy z (\[k\]), a nastêpnie z twierdzenia Bayesa oraz definicji prawdopodobieñstwa warunkowego: $$\begin{aligned} \label{d} \!\!\!\!\!\!\!\! |\psi_{ab}(\vartheta)|^{2} & = & \frac{P\left(S_{ab},\vartheta\right)}{P\left(S_{ab}\right)} = \frac{P(S_{ab}\left|{\vartheta}\right.) r(\vartheta)}{P(S_{ab}\left|{S_{b}}\right.)P\left(S_{b}\right)} = \frac{P(S_{ab}\left|{\vartheta}\right.) r(\vartheta)}{P(S_{a}\left|{S_{b}}\right.)P\left(S_{b}\right)} = \frac{{P(S_{a}\left|{S_{b},\vartheta}\right.)P\left({S_{b}}\right)r(\vartheta)}}{{P(S_{a}\left|{S_{b}}\right.)P\left({S_{b}}\right)}} \;\;\; \nonumber \\ \!\!\!\!\!\!\!\! &=& \frac{{q_{ab}^{2}\left(\vartheta\right) r(\vartheta)}}{{P(S_{a}\left|{S_{b}}\right.) }} \; . \;\;\; \end{aligned}$$ Korzystaj¹c z (\[prawdkat\]), (\[warunkSab\]) oraz (\[inffishEPR2\]), otrzymujemy wiêc z (\[d\]), ¿e $$\begin{aligned} |\psi_{ab}(\vartheta)|^{2} = \frac{1}{\pi}\, q_{ab}^{2}(\vartheta) \; , \label{proporcEPR}\end{aligned}$$ sk¹d $$\begin{aligned} \psi_{ab}(\vartheta) = \frac{e^{i\alpha}}{\pi}\, q_{ab}(\vartheta) \; , \quad \alpha \in \mathbf{R} \; .\end{aligned}$$ Zatem, $\psi_{ab}(\vartheta) \propto q_{ab}(\vartheta)$, co oznacza, ¿e[^82], skoro pojawi³a siê ³¹czna konfiguracja spinów $S_{ab}$, to amplituda prawdopodobieñstwa $\psi_{ab}(\vartheta)$, ¿e wartoœæ k¹ta wynosi $\vartheta$, jest proporcjonalna do amplitudy prawdopodobieñstwa $q_{ab}(\vartheta)$ zaobserwowania rzutu spinu $S_{a}$ cz¹stki 1 pod warunkiem, ¿e rzut spinu cz¹stki 2 wynosi³by $S_{b}$, a k¹t $\vartheta$. Zdanie to jest wyrazem spl¹tania, które pojawi³o siê w kwanto-mechaniczno opisie eksperymentu EPR-Bohm’a.\ \ W koncu, ze wzglêdu na (\[proporcEPR\]), amplitudy $q_{ab}(\vartheta)$ s¹ proporcjonalne do amplitud $\psi_{ab}(\vartheta)$. Zatem Frieden zarz¹da³ [**ortogonalnoœci amplitud kwantowych**]{} $\psi_{++}$ i $\psi_{+-}$, sk¹d automatycznie wyniknê³a ortogonalnoœæ amplitud $q_{++}$ oraz $q_{+-}$: $$\begin{aligned} \int\limits _{0}^{2\pi}{d\vartheta\; q_{++}q_{+-}}=\int\limits _{0}^{2\pi}{d\vartheta\; B_{++}^{'}\sin\left({\vartheta\mathord{\left/{\vphantom{\vartheta2}}\right.\kern -\nulldelimiterspace}2}\right)\left[{B_{+-}^{'}\sin\left({\vartheta\mathord{\left/{\vphantom{\vartheta2}}\right.\kern -\nulldelimiterspace}2}\right)+C_{+-}^{'}\cos\left({\vartheta\mathord{\left/{\vphantom{\vartheta2}}\right.\kern -\nulldelimiterspace}2}\right)}\right]} = 0 \; , \label{ortogEPR}\end{aligned}$$ co pozwoli³o na wyprowadzenie zerowania siê $B_{+-}^{'}$ oraz $B_{-+}^{'}$, (\[zerowanie B+-\]), a dalej (postêpuj¹c ju¿ jak powy¿ej) otrzymanie formu³ (\[wynikEPR\]). ### Niepewnoœæ wyznaczenia k¹ta {#Niepewnosc wyznaczenia kata} Istnieje jeszcze jedna sprawa dotycz¹ca analizy estymacyjnej, opisanej powy¿szymi rachunkami, która wymaga podkreœlenia.\ \ [**Rozk³ad eksperymentalny**]{}: Otó¿ w rzeczywistym pomiarze badacz otrzymuje wartoœci rzutu spinu $S_{a}$ z okreœlonymi czêstoœciami, które s¹ oszacowaniami rozk³adu prawdopodobieñstwa (\[wynikEPR\]), co z kolei pozwala na punktowe oszacowanie k¹ta $\vartheta$.\ Analiza statystyczna metody EFI, która doprowadzi³a do (\[wynikEPR\]) jest, zgodnie z postulatem EFI, spraw¹ uk³adu dokonuj¹cego próbkowania przestrzeni pomiarowej po³o¿eñ i estymacji oczekiwanych parametrów[^83]. [*Jednak jako metoda statystyczna i ona podlega pod ograniczenie Rao-Cramera dok³adnoœci oszacowania estymowanego parametru, którym w tym przypadku jest $\vartheta$*]{}. Sytuacja ta ma nastêpuj¹ce konsekwencje.\ \ [**Wewnêtrzny b³¹d estymacji metody EFI parametru $\vartheta$**]{}: Zauwa¿my, ¿e nierównoœæ Rao-Cramera ma postaæ, $\sigma^{2}_{\theta} \, F \ge 1/I_{F}\left(\vartheta\right) \,$, (\[tw R-C dla par skalarnego\]), gdzie tym razem estymator $F \equiv \hat{\vartheta}$. Z nierównoœci tej otrzymujemy warunek na wariancjê estymatora $\hat{\vartheta}$ k¹ta $\vartheta$: $$\begin{aligned} \label{R-C dla IF dla EPR} \mathop{\sigma^{2}}\left(\hat{\vartheta}\right)\ge\frac{1}{I_{F}(\vartheta)} \; ,\end{aligned}$$ gdzie $I_{F}(\vartheta) = I_{b}$ jest informacj¹ Fishera (\[IF dla vartheta w EPR\]), a nie pojemnoœci¹ informacyjn¹ $I_{1_a}$, (\[pojemnoscEPR\]), czy w konsekwencji jej wartoœci¹ minimaln¹ (\[I EPR minimalna\]). Bez analizowania postaci estymatora $\hat{\vartheta}$, nierównoœæ Rao-Cramera (\[R-C dla IF dla EPR\]) okreœla DORC dla jego wariancji, o ile tylko estymator ten jest nieobci¹¿ony.\ \ Wiemy te¿, ¿e poniewa¿ pojemnoœæ informacyjna jest sum¹ po kana³ach z odpowiadaj¹cych im informacji Fishera, wiêc: $$\begin{aligned} \label{relacja pojemnosci i IF dla EPR} I_{1_a} \ge I_{F}(\vartheta) \; . \label{okok}\end{aligned}$$ Z (\[relacja pojemnosci i IF dla EPR\]) oraz z (\[R-C dla IF dla EPR\]) otrzymujemy: $$\begin{aligned} \label{I oraz IF oraz var dla EPR} \frac{1}{I_{1_a}}\le\frac{1}{I_{F}(\vartheta)} \le \mathop{\sigma^{2}}\left(\hat{\vartheta}\right) \; .\end{aligned}$$ Poniewa¿ pojemnoœæ kana³u $I_{1_a}$ wed³ug (\[I EPR minimalna\]) wynosi: $$\begin{aligned} I_{1_a} = 4 \pi \; ,\end{aligned}$$ wiêc podstawiaj¹c t¹ wartoœæ do nierównoœci (\[I oraz IF oraz var dla EPR\]) otrzymujemy: $$\begin{aligned} \label{Rao-Cramer w EPR} \mathop{\sigma^{2}}\left(\hat{\vartheta}\right)\ge\frac{1}{4\pi} \approx 0.08 \; {\rm rad}^{\,2} \; .\end{aligned}$$ [**Wniosek**]{}: Nierównoœæ (\[Rao-Cramer w EPR\]) stwierdza, ¿e obserwacja jednej wartoœci rzutu spinu *przy kompletnej nieznajomoœci k¹ta* $\vartheta$ (st¹d (\[prawdkat\])) daje o nim ma³¹, lecz skoñczon¹ informacjê. B³¹d jego estymacji $\sqrt{0,08}\; rad=0,28\; rad$ jest doœæ du¿y, co jest zwi¹zane z p³ask¹ funkcj¹ “niewiedzy” $r(\vartheta)$ okreœlon¹ w (\[prawdkat\]). #### Wp³yw zaszumienia pomiaru {#zaszumienie pomiaru} Analiza EFI w obliczu pomiaru i p³yn¹cego z tego faktu zaszumienia danych wewnêtrznych EFI nie jest przedmiotem tego skryptu. Zainteresowany czytelnik znajdzie omówienie tego tematu w [@Frieden; @Mroziakiewicz]. Poni¿ej zamieszczam wniosek z analizy Mroziakiewicz [@Mroziakiewicz] dotycz¹cy tego problemu dla powy¿szej analizy eksperymentu EPR-Bohm’a.\ \ W pomiarze stanu uk³adu przez zewnêtrznego obserwatora, otrzymujemy dane zaszumione przez uk³ad pomiarowy. Tzn. dane pomiarowe rzutu spinu (oznaczmy je $\bar{S}_{a}$) s¹ generowane przez prawdziwe wartoœci wielkoœci $S_{a},S_{b},\vartheta$ w obecnoœci szumu aparatury pomiarowej. Szum ten powstaje co prawda w urz¹dzeniach Sterna-Gerlacha $a$ oraz $b$, ale ze wzglêdu na za³o¿enie podane na samym pocz¹tku Rozdzia³u \[Warunki brzegowe\], z innych przyczyn ni¿ (przyjête jako równe zeru) fluktuacje rzutu spinu.\ \ [**IF uwzglêdniaj¹ca zaszumienie**]{}: Uzyskana przy tych danych informacja Fishera $I_{zasz}$ o parametrze $\vartheta$, uwzglêdniaj¹ca [*zaszumienie*]{} pomiarowe, jest generowana przez informacjê Fishera $I_{F}(\vartheta) = I_{b}$ (\[IF dla vartheta w EPR\]). W tym sensie [*informacja Fishera $I_{F}(\vartheta)$, (\[IF dla vartheta w EPR\]), procedury estymacyjnej EFI jest czêœci¹ informacji, która przejawia siê w pomiarze*]{}. Zatem obok nierównoœci (\[okok\]) zachodzi równie¿: $$\begin{aligned} I_{F}(\vartheta) \ge I_{zasz}(\vartheta) \; .\end{aligned}$$ Ze wzglêdu na to, ¿e $I_{zasz}(\vartheta)$ wchodzi w (\[I oraz IF oraz var dla EPR\]) w miejsce $I_{F}(\vartheta)$, warunek ten oznacza pogorszenie jakoœci estymacji w porównaniu z (\[Rao-Cramer w EPR\]). ### Informacja $Q$ jako miara spl¹tania Tak jak we wszystkich problemach estymacyjnych rozwi¹zanych metod¹ EFI, tak i w eksperymencie EPR-Bohm’a wykorzystano przy estymacji k¹ta $\vartheta$, obserwowan¹ zmodyfikowan¹ informacyjn¹ zasadê strukturaln¹ $\widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0$, (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]), która ze wzglêdu na $\kappa=1$ w eksperymencie EPR-Bohm’a[^84], daje na poziomie oczekiwanym zwi¹zek $Q_{1_a} = - I_{1_a}$, (\[I1piEPR\]), zgodnie z ogólnym warunkiem (\[rownowaznosc strukt i zmodyfikowanego strukt\]) metody EFI.\ \ [**Dostêpnoœæ pomiarowa $I_{1_a}$**]{}: Poniewa¿ nie mierzony by³ stan cz¹stki $2$, tzn. rzut spinu $S_{b}$ w czasie naszej analizy móg³ przyj¹æ wartoœæ $+$ lub $-$, zatem pojemnoœæ informacyjna $I_{1_a}$ jest informacj¹ o k¹cie $\vartheta$, zawart¹ w obserwacji rzutu spinu cz¹stki $1$ dokonanej przez uk³ad i zgodnej z dok³adnoœci¹ do szumu z obserwacj¹ dokonan¹ przez zewnêtrznego obserwatora[^85].\ \ [**Nieobserwowalna wewnêtrza struktura uk³adu odbita w $Q_{1_a}$**]{}: Natomiast informacja strukturalna $Q_{1_a}$ (\[strukturalnaEPR\]) dla cz¹stki $1$, jako pozosta³a czêœæ informacji fizycznej $K$, jest informacj¹ zawart¹ w nieobserwowanej wewnêtrznej strukturze informacyjnej ca³ego uk³adu. Sugeruje to istnienie ³¹cznej, [*nieseparowalnej*]{} informacji strukturalnej dla cz¹stek $1$ i $2$. Zatem informacja o ³¹cznej strukturze uk³adu, to nie to samo co suma informacji o jego sk³adowych.\ \ [**Spl¹tanie stanów cz¹stek**]{}: Ze wzglêdu na $Q_{1_a} = - I_{1_a}$, przestrzeñ danych odbita w $I_{1_a}$, tzn. obserwacje czêstoœci rzutu spinu cz¹stki $1$, jest na poziomie informacji, zwi¹zana z wewnêtrz¹ przestrzeni¹ konfiguracyjn¹ uk³adu odbit¹ w $Q_{1_a}$. Co prawda, ze wzglêdu na brak bezpoœredniego wgl¹du w relacje wewn¹trz uk³adu, mo¿emy jedynie ze skoñczon¹ dok³adnoœci¹ (Rozdzia³ \[Niepewnosc wyznaczenia kata\]) wnioskowaæ o k¹cie $\vartheta$, jednak sam fakt mo¿liwoœci takiego wnioskowania okreœla sytuacjê nazywan¹ [*spl¹taniem stanów*]{} obu cz¹stek.\ \ [**Wniosek ogólny**]{}: Wynik ten prowadzi do stwierdzenia, ¿e strukturalna (wewnêtrzna) zasada informacyjna $I = - \kappa Q$, stosowana dla ka¿dego rozwa¿anego problemu EFI, opisuje[^86] spl¹tanie przestrzeni danych obserwowanych[^87] z nieobserwowan¹ konfiguracj¹ uk³adu cz¹stek[^88]. Postuluje to wykorzystywanie EFI jako formalizmu do predykcji wyst¹pienia stanów spl¹tanych i to nie tylko w przypadku problemu EPR-Bohm’a. Tak wiêc, informacja strukturala $Q$ jest informacj¹ o ,,spl¹taniu” widocznym w korelacji danych przestrzeni pomiarowej z przestrzeni¹ konfiguracyjn¹ uk³adu.\ \ Na koniec wspomnijmy, ¿e inne traktowanie informacji fizycznej $K$ w [@Frieden] ni¿ w obecnym skrypcie, sprawia, ¿e wnioski tam otrzymane s¹ inne[^89]. Zakoñczenie =========== Celem skryptu by³a prezentacja uogólnienia MNW oraz zastosowania IF w jej czêœci wnioskowania statystycznego dotycz¹cego estymacji niepartametrycznej metod¹ EFI, zaproponowan¹ przez Friedena i Soffera do opisu zjawisk fizycznych i ekonofizycznych. Podstawy takiej analizy statystycznego opisu zjawisk zosta³y wprowadzonych w latach 20 ubieg³ego wieku przez Fishera.\ Pocz¹tkowo Fisher wprowadzi³ MNW poszerzon¹ o pojêcie IF w celu rozwi¹zania problemu estymacji punktowej i przedzia³owej parametru rozk³adu zmiennej losowej w sytuacji ma³ej próby. Jego statystystyczna metoda doboru modeli, konstruowana niezale¿nie od ówczeœnie rozwijanych teorii fizycznych, okaza³a siê jednak siêgaæ o wiele dalej, w obszar estymacji równañ fizycznych teorii pola, przy czym okaza³o siê, ¿e wielkoœæ (ma³ej) próby jest w tej estymacji cech¹ charakterystyczn¹ modeli. Celowi temu poœwiêcona jest g³ówna czêœæ niniejszego skryptu.\ \ Jesteœmy w punkcie, w którym mo¿na ju¿ daæ pewne wstêpne podsumowanie metody EFI jako procedury statystycznej budowania modeli. Jest wiêc EFI metod¹ statystycznej estymacji równañ ruchu teorii pola lub równañ generuj¹cych rozk³ady fizyki statystycznej. Gdy szukane równanie ruchu, wyestymowane metod¹ EFI poprzez rozwi¹zanie jej równañ (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) oraz (\[var K rozpisana\]), zosta³o otrzymane, wtedy mo¿e byæ dalej prowadzone poszukiwanie rozk³adu prawdopodobieñstwa na analitycznych warstwach przestrzeni statystycznej ${\cal S}$, tzn. na (pod)rozmaitoœci w ${\cal S}$ z metryk¹ Rao-Fishera.\ Mo¿e siê zdarzyæ, ¿e ³¹czny rozk³ad prawdopodobieñstwa jest analizowany przez EFI na ca³ej po³o¿eniowo-pêdowej przestrzeni fazowej uk³adu[^90] i to dla wiêcej ni¿ jednego pola. Ma to np. miejsce, gdy obok pola Diraca, obecne jest równie¿ pole cechowania (Rozdzia³ \[foliation of S\]). Wtedy na warstwie rozk³adu fermionowego, w jego pochodnych kowariantnych, pole cechowania[^91] musi byæ samospójnie wziête pod uwagê.\ Sedno metody EFI jest zawarte w ogólnej postaci informacji fizycznej $K$ zadanej przez równania (\[TPI diag\]) oraz (\[k form\]) (lub (\[k form dla psi\])). Jest ona funkcj¹ zarówno obserwowanej informacji strukturalnej $\texttt{q\!F}_{n}(q_{n}({\bf x}))$ jak i amplitud $q_{n}({\bf x})$ (lub $\psi_{n}({\bf x})$) rozk³adów, wraz z ich pochodnymi. Przy tak ogólnym zrozumieniu $K$, ró¿norodnoœæ równañ EFI jest konsekwencj¹ ró¿nych warunków wstêpnych dyktowanych przez fizykê. Mog¹ siê one wyra¿aæ np. poprzez równania ci¹g³oœci, które same s¹ wynikiem estymacji statystycznej [@Dziekuje; @informacja_2] (Rozdzia³ \[master eq\]), pewne symetrie charakterystyczne dla zjawiska (Rozdzia³ \[Poj inform zmiennej los poloz\]) oraz warunki normalizacyjne. Wstêpne za³o¿enia fizyczne mog¹, poza ewentualnoœci¹ na³o¿enia dodatkowych równañ ci¹g³oœci, wskazywaæ równie¿ na zastosowanie jedynie wariacyjnej zasady informacyjnej, jak to ma miejsce w przypadku równania Kleina-Gordona dla pola skalarnego (Rozdzia³ \[Klein-Gordon scalars\]), b¹dŸ na obie zasady i ich samospójne rozwi¹zanie, jak to mia³o miejsce w pozosta³ych przypadkach.\ \ Podsumowuj¹c, przeszliœmy w skrypcie przez trzy etapy statystycznej rozbudowy zastosowania MNW oraz IF. Na pierwszym, wstêpnym etapie, przedstawiono zastosowanie MNW oraz IF w analizie doboru modeli (Rozdzia³ \[MNW\]) oraz omówiono podstawy podejœcia geometrii ró¿niczkowej do konstrukcji przestrzeni statystycznej, obrazuj¹c poznany aparat statystyczny przyk³adami estymacji modeli eksponentialnych. Na drugim etapie, wprowadzono pojêcie entropii wzglêdnej i pojemnoœci informacyjnej oraz wprowadzono strukturaln¹ i wariacyjn¹ zasadê informacyjn¹ (Rozdzia³ \[Entropia wzgledna i IF\]-\[Zasady informacyjne\]) metody EFI, a w trzecim przedstawiono wykorzystanie tych pojêæ do wyprowadzenia podstawowych równañ modeli fizycznych (Rozdzia³y \[Kryteria informacyjne w teorii pola\]-\[Przyklady\], Dodatek \[Maxwell field\]-\[general relativity case\]), w³¹czaj¹c w to przyk³ad ekonofizycznego opisu zjawiska (Rozdzia³ \[Model Aoki-Yoshikawy - ekonofizyka\]). Dodatki ======= Dodatek: Zasada nieoznaczonoœci Heisenberga {#Zasada nieoznaczonosci Heisenberga} ------------------------------------------- Poni¿sze wyprowadzenie zasady nieoznaczonoœci Heisenberga zosta³o przedstawione w [@Frieden; @Mania]. Zasada Heisenberga stwierdza, ¿e w okreœlonej chwili czasu *t* nie mo¿na z dowoln¹ dok³adnoœci¹ wyznaczyæ jednoczeœnie po³o¿enia ${\bf y}$ i pêdu ${\bf y}_{p}$ cz¹stki, tzn. ¿e po³o¿enie i pêd cz¹stki s¹ [*rozmyte*]{}, co mo¿emy zapisaæ nastêpuj¹co: $$\begin{aligned} \label{U1.1} \sigma_{\theta}^{2} \cdot \sigma _{p}^{2} \ge \left(\frac{\hbar}{2} \right)^{2} \; ,\end{aligned}$$ gdzie $\sigma_{\theta}^{2} $ i $\sigma_{p}^{2} $ s¹ wariancjami po³o¿enia i pêdu cz¹stki wzglêdem ich wartoœci oczekiwanych $\theta$ oraz $\theta_{p}$. #### Wyprowadzenie nierównoœci (\[U1.1\]) dla pola rangi $N=2$ {#wyprowadzenie-nierównoœci-u1.1-dla-pola-rangi-n2 .unnumbered} Za³ó¿my, ¿e dokonujemy estymacji tylko w jednym przestrzennym kanale informacyjnym przy za³o¿eniu, ¿e pozosta³e czasoprzestrzenne parametry rozk³adu s¹ znane. Powy¿sza relacja mo¿e byæ wyprowadzona przy odwo³aniu siê do w³asnoœci informacji Fishera. Wartoœci po³o¿enia ${\bf y}$ zmiennej $Y$ s¹ [*rozmyte*]{} (b¹dŸ fluktuuj¹) wokó³ jej wartoœci oczekiwanej $\theta$, sk¹d wartoœci ${\bf x} \in {\cal X}$ zmiennej odchyleñ $X$ spe³niaj¹ (jak zwykle w skrypcie), zwi¹zek: $$\begin{aligned} \label{U1.0} {\bf x} = {\bf y} - \theta \; . $$ Za³ó¿my, ¿e jest spe³niona nierównoœæ Rao-Cramera (\[tw R-C dla par skalarnego\]): $$\begin{aligned} \label{U1.2} \sigma_{\theta}^{2} \cdot I_{F}(\theta) \ge 1 \; , \;\;\; {\rm gdzie} \;\;\;\; \sigma_{\theta}^{2} \equiv E\left((\hat{\theta }(Y)-\theta )^{2}\right) \; ,\end{aligned}$$ gdzie $I_{F}(\theta) $ jest informacj¹ Fishera parametru $\theta$ dla [*pojedynczego pomiaru*]{} (z powodu odwo³ania siê do skalarnej wersji twierdzenia Rao-Cramera), która po skorzystaniu z postaci kinematycznej (\[Fisher\_information-kinetic form bez n\]) w Rozdziale \[The kinematical form of the Fisher information\], ma postaæ: $$\begin{aligned} \label{U1.3} I_{F} = 4 \int_{\cal X} d{\bf x}\left(\frac{\partial q({\bf x})}{\partial {\bf x}} \right)^{2} \, .\end{aligned}$$ Rozwa¿my zespolon¹ amplitudê $\psi$, (\[amplitudapsi\]), dla pola rangi $N=2$: $$\begin{aligned} \label{pole psi rangi 2} \psi ({\bf x}) = \psi_{1}({\bf x}) = \frac{1}{{\sqrt{2}}} \left( q_{1}({\bf x}) + i \, q_{2}({\bf x}) \right) \; , \end{aligned}$$ która ma jako czêœæ rzeczywist¹ i urojon¹ dwie rzeczywiste amplitudy $q_{i}$, $i=1,2$. Za³o¿ymy, ¿e $N=2$ - wymiarowa próba dla zmiennej odchyleñ $X$ jest prosta, sk¹d amplitudy $q_{i}({\bf x})$, $i=1,2$, s¹ takie same i równe $q({\bf x})$: $$\begin{aligned} \label{prosta proba} q_{1}({\bf x}) = q_{2}({\bf x}) = q({\bf x}) \; .\end{aligned}$$ Zespolon¹ funkcjê $\psi \left({\bf x}\right)$ mo¿na zapisaæ poprzez jej transformatê Fouriera, przechodz¹c z reprezentacji po³o¿eniowej ${\bf x}$ do pêdowej ${\bf p}$: $$\begin{aligned} \label{U1.4} \psi \left({\bf x}\right) = \frac{1}{\sqrt{2\pi \hbar} } \int d{\bf p} \; \phi({\bf p}) \, \exp \left(i \,{\bf p}\, {\bf x} \right) \, ,\end{aligned}$$ gdzie ${\bf p}$ jest rozmyciem (fluktuacj¹) pêdu ${\bf y}_{p}$ wokó³ jego wartoœci oczekiwanej $\theta_{p}\,$, tzn.: $$\begin{aligned} \label{rozmycie pedu} \;\;\;\; {\bf p} = {\bf y}_{p} - \theta_{p} \; ,\end{aligned}$$ a pêd cz¹stki jest mierzony w tej samej chwili co jego po³o¿enie.\ Dla amplitudy $N=2$ pojemnoœæ informacyjna $I$, (\[inf F z psi\]), ma postaæ: $$\begin{aligned} \label{U1.5} I = 8 \int d{\bf x} \frac{d\psi^{\,*}({\bf x})}{d{\bf x}} \frac{d\psi ({\bf x})}{d{\bf x}} = 8 \int d{\bf x}\left|\frac{d\psi \left({\bf x}\right)}{d{\bf x}} \right|^{2} \; .\end{aligned}$$ Przedstawiaj¹c $\psi$ w postaci: $$\begin{aligned} \label{U1.6} \psi({\bf x}) =\left|\psi({\bf x}) \right|\exp \left(i S({\bf x})\right) \; \end{aligned}$$ i wykonuj¹c ró¿niczkowanie: $$\begin{aligned} \label{U1.7} \frac{d\psi \left({\bf x}\right)}{d{\bf x}} =\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} \;e^{iS\left({\bf x}\right)} +i\left|\psi \left({\bf x}\right)\right|\; e^{i S\left({\bf x}\right)} \; \frac{d S\left({\bf x}\right)}{d{\bf x}} \; ,\end{aligned}$$ mo¿emy (\[U1.5\]) przekszta³ciæ nastêpuj¹co: $$\begin{aligned} \label{U1.8} \begin{array}{l} {I = 8\int d{\bf x}\left|\frac{d\psi \left({\bf x}\right)}{d{\bf x}} \right|^{2} =8\int d{\bf x}\frac{d\psi ^{*} \left({\bf x}\right)}{d{\bf x}} \frac{d\psi \left({\bf x}\right)}{d{\bf x}} =} \\ {\quad = 8 \int d{\bf x}\left(\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} e^{-iS\left({\bf x}\right)} -i\left|\psi \left({\bf x}\right)\right|e^{-iS\left({\bf x}\right)} \frac{dS\left({\bf x}\right)}{d{\bf x}} \right)\left(\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} e^{iS\left({\bf x}\right)} +i\left|\psi \left({\bf x}\right)\right|e^{iS\left({\bf x}\right)} \frac{dS\left({\bf x}\right)}{d{\bf x}} \right) =} \\ {\quad =8\int d{\bf x}\left[\left(\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} \right)^{2} +i\left|\psi \left({\bf x}\right)\right|e^{iS\left({\bf x}\right)} \frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} \frac{dS\left({\bf x}\right)}{d{\bf x}} -i\left|\psi \left({\bf x}\right)\right|e^{iS\left({\bf x}\right)} \frac{dS\left({\bf x}\right)}{d{\bf x}} \frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} +\right. } \\ {\quad \left. +\left|\psi \left({\bf x}\right)\right|^{2} \left(\frac{dS\left({\bf x}\right)}{d{\bf x}} \right)^{2} \right]=8\int d{\bf x}\left[\left(\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} \right)^{2} +\left|\psi \left({\bf x}\right)\right|^{2} \left(\frac{dS\left({\bf x}\right)}{d{\bf x}} \right)^{2} \right] } \; . \end{array}\end{aligned}$$ Zajmijmy siê teraz pierwszym sk³adnikiem pod ostatnia ca³k¹ w (\[U1.8\]). Poniewa¿ norma amplitudy $\psi$ jest równa: $$\begin{aligned} \label{norma psi} |\psi| = \sqrt{\frac{1}{2} \,( q_{1}^{2} + q_{2}^{2} )} \; ,\end{aligned}$$ zatem: $$\begin{aligned} \label{U1.9} \left(\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} \right)^{2} = \frac{1}{2} \left(\frac{d}{d{\bf x}} \sqrt{q_{1}^{2} + q_{2}^{2} } \,\right)^{2} =\left|q_{1} = q_{2} = q\right|=\left(\frac{dq}{d{\bf x}} \right)^{2} \; .\end{aligned}$$ Podstawiaj¹c powy¿szy wynik do ca³ki (\[U1.8\]) i korzystaj¹c z (\[U1.3\]), otrzymamy nastêpuj¹c¹ równoœæ: $$\begin{aligned} \label{U1.10} I &=& 8\int d{\bf x} \left|\frac{d\psi \left({\bf x}\right)}{d{\bf x}} \right|^{2} = 8\int d{\bf x}\left[\left(\frac{d\left|\psi \left({\bf x}\right)\right|}{d{\bf x}} \right)^{2} +\left|\psi \left({\bf x}\right)\right|^{2} \left(\frac{dS\left({\bf x}\right)}{d{\bf x}} \right)^{2} \right] \nonumber \\ & =& 8\int d{\bf x}\left(\frac{dq}{d{\bf x}} \right)^{2} +8\int d{\bf x}\left|\psi \left({\bf x}\right)\right|^{2} \left(\frac{dS\left({\bf x}\right)}{d{\bf x}} \right)^{2} = 2I_{F} + 8 \, E\left[ \left(\frac{dS\left({\bf x}\right)}{d{\bf x}} \right)^{2} \right] \; .\end{aligned}$$ Z (\[U1.10\]) oraz (\[U1.5\]) wynika wiêc nastêpuj¹ca nierównoœæ: $$\begin{aligned} \label{U1.11} 2 I_{F} \le I \;\;\; {\rm lub} \;\;\; I_{F} \le 4\int d{\bf x}\left|\frac{d\psi \left({\bf x}\right)}{d{\bf x}} \right|^{2} \; \; .\end{aligned}$$ Wykorzystuj¹c transformatê Fouriera (\[U1.4\]), otrzymujemy z (\[U1.11\]) po przejœciu do reprezentacji pêdowej: $$\begin{aligned} \label{U1.12} I_{F} \le \frac{4}{\hbar^{2} } \int d{\bf p}\, \left|\phi \left({\bf p}\right)\right|^{2} \, {\bf p}^{2} \; .\end{aligned}$$ gdzie $\left|\phi \left({\bf p}\right)\right|^{2} $ jest brzegow¹ gêstoœci¹ prawdopodobieñstwa $P({\bf p})$ odchyleñ pêdu.\ Zatem ca³ka po prawej stronie (\[U1.12\]) jest wartoœci¹ oczekiwan¹ $E({\bf p}^{2})$ dla ${\bf p}^{2}$: $$\begin{aligned} \label{U1.13} I_{F} \le \frac{4}{\hbar^{2} } \, E\left( {\bf p}^{2} \right) = \left(\frac{2}{\hbar} \right)^{2} \,E\left[ ({\bf y}_{p} - \theta_{p})^{2} \right] \equiv \left(\frac{2}{\hbar} \right)^{2} \sigma_{p}^{2} \; ,\end{aligned}$$ gdzie $\sigma_{p}^{2} $ jest wariancj¹ pêdu cz¹stki, a pierwsza równoœæ wynika z tego, ¿e ${\bf p}$ jest odchyleniem (fluktuacj¹) pêdu ${\bf y}_{p}$ od jego wartoœci oczekiwanej $\theta_{p}\,$, (\[rozmycie pedu\]). Podstawiaj¹c powy¿szy wynik do nierównoœci Rao-Cramera, $\sigma_{\theta}^{2} \cdot I_{F}(\theta) \ge 1$, (\[U1.2\]), otrzymujemy (\[U1.1\]): $$\begin{aligned} \label{U1.14} \sigma_{\theta}^{2} \cdot \sigma _{p}^{2} \ge \left({\raise0.7ex\hbox{$ \hbar $}\!\mathord{\left/{\vphantom{\hbar 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ 2 $}} \right)^{2} \; . \end{aligned}$$ c.n.d. Dodatek: Równanie Schrödingera {#Rownanie Schrodingera} ------------------------------ Równania Schrödingera wyprowadzimy jako nierelatywistyczn¹ granicê równania Kleina-Gordona. Wyprowadzenie go jako nierelatywistycznej granicy równania Diraca by³oby merytorycznie bardziej uzasadnione [@Sakurai; @2], jednak celem poni¿szego wyprowadzenia jest zwrócenie uwagi na relatywistyczne pochodzenie spinu elektronu.\ \ Rozpoczniemy od separacji zmiennych czasowych i przestrzennych wystêpuj¹cych w równaniu (\[row KL dla swobodnego\]) dla pola rangi $N=2$, zapisuj¹c po lewej stronie wszystkie wyrazy zawieraj¹ce pochodn¹ czasow¹, a po prawej pochodn¹ po wspó³rzêdnych przestrzennych: $$\begin{aligned} \label{klseparacja} & &\left({\hbar^{2}\frac{{\partial^{2}}}{{\partial t^{2}}} + 2\,i\,\hbar \,e\,\phi\frac{\partial}{{\partial t}} + i \,\hbar \,e\,\frac{{\partial\phi}}{{\partial t}} - e^{2}\phi^{2}}\right) \psi \nonumber \\ & & = \left({ c^{2}\hbar^{2} \vec{\nabla}^{2} - 2\,i\,e\,c\, \hbar\left({\vec{A} \cdot \vec{\nabla}}\right) - i\,e\,c \,\hbar \left({\vec{\nabla} \cdot \vec{A}}\right) - e^{2}{\vec{A}}^{2} - m^{2}c^{4}} \right) \psi \, ,\end{aligned}$$ gdzie skorzystano z wyra¿enia $$\begin{aligned} {\vec{\nabla}\cdot\left({\vec{A}\psi}\right)=\psi\vec{\nabla}\cdot\vec{A}+\vec{A}\cdot\vec{\nabla}\psi} \; .\end{aligned}$$ Nastêpnie skorzystajmy z nierelatywistycznej reprezentacji funkcji falowej [@Sakurai; @2]: $$\begin{aligned} \label{nonrelat} \psi\left({{\bf x},t}\right) = \tilde{\psi}\left({{\bf x},t}\right)e^{-imc^{2}t/\hbar} \; ,\end{aligned}$$ gdzie wydzieliliœmy z $\psi$ wyraz zawieraj¹cy energiê spoczynkow¹ $mc^{2}$, otrzymuj¹c now¹ funkcjê falow¹ $\tilde{\psi}$. Kolejnym krokiem jest zastosowanie (\[nonrelat\]) w (\[klseparacja\]). Ró¿niczkuj¹c (\[nonrelat\]) po czasie otrzymujemy: $$\begin{aligned} \label{1roz} \frac{\partial}{{\partial t}} \psi = \left({\frac{\partial}{{\partial t}}\tilde{\psi}-\frac{{imc^{2}}}{\hbar}\tilde{\psi}}\right) e^{{{-imc^{2}t}/\hbar}} \;\; ,\end{aligned}$$ a po kolejnym ró¿niczkowaniu: $$\begin{aligned} \label{2roz} \frac{{\partial^{2}}}{{\partial t^{2}}}\psi = \left({\frac{{\partial^{2}}}{{\partial t^{2}}}\tilde{\psi} - 2\frac{{imc^{2}}}{\hbar}\frac{\partial}{{\partial t}}\tilde{\psi} - \frac{{m^{2}c^{4}}}{{\hbar^{2}}}\tilde{\psi}}\right) e^{{{-imc^{2}t}/\hbar}} \;\; .\end{aligned}$$ Stosuj¹c nierelatywistyczne ($n.r$) przybli¿enie: $$\begin{aligned} \label{enr} \frac{E_{n.r}}{mc^{2}} << 1 \; ,\end{aligned}$$ gdzie $E_{n.r}$ jest okreœlone poprzez poni¿sze zagadnienie w³asne: $$\begin{aligned} - \hbar^{2} \frac{\partial^{2}}{\partial t^{2}}\, \tilde{\psi} = E_{n.r}^{2} \, \tilde{\psi} \; ,\end{aligned}$$ mo¿emy pomin¹æ pierwszy wyraz po prawej stronie (\[2roz\]).\ Odwo³ajmy siê do przybli¿enia sta³ego, s³abego potencja³u, dla którego zachodzi: $$\begin{aligned} \label{ephi} e \phi << mc^{2} \; ,\end{aligned}$$ oraz $$\begin{aligned} \label{stacphi} \frac{\partial\phi}{\partial t} = 0 \; .\end{aligned}$$ Nastêpnie, wstawiaj¹c (\[1roz\]) oraz (\[2roz\]) do (\[klseparacja\]), i wykorzystuj¹c (\[enr\]), (\[ephi\]) oraz (\[stacphi\]), otrzymujemy po przemno¿eniu przez $-1/{2mc^{2}}$ i [*oznaczeniu*]{} $\tilde{\psi}$ jako $\psi$, równanie: $$\begin{aligned} \label{juz} {i\hbar\frac{\partial}{\partial t}\psi-e\phi\psi = -\frac{\hbar^{2}}{2m}\vec{\nabla}^{2}\psi+\frac{ie\hbar}{mc}\vec{A}\cdot\vec{\nabla}\psi+\frac{ie\hbar}{2mc}\left(\vec{\nabla}\cdot\vec{A}\right)\psi + \frac{e^{2}{\vec{A}}^{2}}{2mc^{2}}\,\psi} \; .\end{aligned}$$ Jest to równanie Schödingera dla nierelatywistycznej funkcji falowej $\psi$ z potencja³em elektromagnetycznym $\vec{A}$ oraz $\phi$.\ W przypadku zerowania siê czêœci wektorowej potencja³u $\vec{A}$ i dla niezerowego potencja³u skalarnego $\phi$, równanie to przyjmuje znan¹ postaæ: $$\begin{aligned} \label{schrodinger} i\hbar\frac{\partial}{{\partial t}}\psi = -\frac{{\hbar^{2}}}{{2m}}\vec{\nabla}^{2}\psi + e \,\phi \, \psi \; .\end{aligned}$$\ \ [**Pytanie**]{}: [*Powiedz dlaczego powy¿sze wyprowadzenie równania Schödingera z równania Kleina-Gordona, œwiadczy o relatywistycznym charakterze spinu elektronu?*]{} Dodatek: Rezultaty EFI dla elektrodynamiki Maxwella oraz teorii grawitacji -------------------------------------------------------------------------- ### Dodatek: Pole cechowania Maxwella {#Maxwell field} Poni¿ej zaprezentujemy rezultat metody EFI otrzymany w zapisie Friedena-Soffera dla wyprowadzenia równañ Maxwella ($M$). Punktem wyjœcia jest pojemnoœæ informacyjna (\[postac I dla p po x bez n\]): $$\begin{aligned} I = \sum_{n=1}^N {\int_{\cal X} d^{4}{\bf x} \frac{1}{{p_{n} \left( {\bf x} \right)}} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial p_{n} \left( {\bf x} \right)}}{{\partial {\bf x}_{\nu} }}} {\frac{{\partial p_{n} \left( {\bf x} \right)}}{{\partial {\bf x}^{ \nu} }}} \right) } } \; .\nonumber\end{aligned}$$ Rozwa¿my równania ruchu Maxwella dla pola rangi $N=4$ z amplitud¹ rzeczywist¹ $q_{n}$, $n=1,2,3,4$. Zak³adamy, ¿e pola cechowania s¹ proporcjonalne do tych rzeczywistych amplitud [@Frieden]: $$\begin{aligned} \label{amplitudy dla Maxwella} q_{\nu}({\bf x})=a\, A_{\nu}({\bf x}) \; , \;\;\; {\rm gdzie} \;\;\; \nu \equiv n-1 = 0,1,2,3 \; , \end{aligned}$$ gdzie $a$ jest pewn¹ sta³¹.\ Wykorzystuj¹c metrykê Minkowskiego $(\eta^{\nu\mu})$, definujemy amplitudy $q^{\nu}({\bf x})$ [*dualne*]{} do $q_{\nu}({\bf x})$: $$\begin{aligned} \label{amplitudy dla Maxwella dual} q^{\nu}({\bf x}) \equiv \sum_{\mu=0}^{3} \eta^{\nu \mu} q_{\mu}({\bf x}) = a\, \sum_{\mu=0}^{3} \eta^{\nu \mu} A_{\mu}({\bf x}) \equiv a\,A^{\nu}({\bf x}) \; , \;\;\; {\rm gdzie} \;\;\; \nu \equiv n-1 = 0,1,2,3 \; ,\end{aligned}$$ gdzie wprowadzono dualne pola cechowania $A^{\mu}({\bf x})$: $$\begin{aligned} \label{pola cech dualne dla Maxwella} A^{\nu}({\bf x}) \equiv \sum_{\mu=0}^{3} \eta^{\nu \mu} A_{\mu}({\bf x}) \; , \;\;\; {\rm gdzie} \;\;\; \nu = 0,1,2,3 \; .\end{aligned}$$ Amplitudy $q_{\nu}({\bf x})$ s¹ zwi¹zane z punktowymi rozk³adami prawdopodobieñstwa $p_{n} \left({\bf x} \right)$ nastêpuj¹co: $$\begin{aligned} \label{rozklad p n dla Maxwella} p_{n} \left({\bf x} \right) \equiv p_{q_{\nu}} \left( {\bf x} \right) = q_{\nu}({\bf x}) q_{\nu}({\bf x}) = a^{2} A_{\nu}({\bf x}) A_{\nu}({\bf x}) \; , \;\;\; {\rm gdzie} \;\;\; \nu \equiv n-1 = 0,1,2,3 \; .\end{aligned}$$ Widzimy wiêc, ¿e w metodzie EFI dla elektrodynamiki Maxwella, indeks próby $n$ staje siê indeksem czasoprzestrzennym. Zatem postaæ pojemnoœci informacyjnej musi uwzglêdniaæ fakt dodatkowej estymacji w kana³ach czasoprzestrzennych, przyjmuj¹c zgodnie z ogólnymi zaleceniami Rozdzia³u \[Poj inform zmiennej los poloz\], postaæ wspó³zmiennicz¹: $$\begin{aligned} \label{postac I dla p po x suma po Mink Maxwell} I &=& \sum_{\mu=0}^{3} {\int_{\cal X} d^{4}{\bf x}\, \eta^{\mu\mu}\frac{1}{{p_{q_{\mu}} \left( {\bf x} \right)}} \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial p_{q_{\mu}} \left( {\bf x} \right)}}{{\partial {\bf x}_{\nu} }}} {\frac{{\partial p_{q_{\mu}} \left( {\bf x} \right)}}{{\partial {\bf x}^{ \nu} }}} \right) } } \nonumber \\ &=& 4 \sum_{\mu=0}^{3} {\int_{\cal X} d^{4}{\bf x}\, \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial q_{\mu} \left( {\bf x} \right)}}{{\partial {\bf x}_{\nu} }}} {\frac{{\partial q^{\mu} \left( {\bf x} \right)}}{{\partial {\bf x}^{ \nu} }}} \right) } } \; .\end{aligned}$$ Dalej postêpujemy jak w [@Frieden]. Stosujemy obie zasady informacyjne, strukturaln¹ (\[zmodyfikowana obserwowana zas strukt z P i z kappa\]) z $\kappa=1/2$ oraz wariacyjn¹ (\[var K rozpisana\]): $$\begin{aligned} \label{IPs for Maxwell} \widetilde{\textit{i'}} + \widetilde{\mathbf{C}} + \kappa \, \textit{q} = 0 \;, \;\;\; {\rm gdzie} \;\;\; \kappa=1/2 \; \;\;\;\; {\rm oraz } \;\;\;\; \delta_{(q)}(I + Q) = 0 \; ,\end{aligned}$$ które rozwi¹zujemy samospójnie, wraz z na³o¿onym dodatkowo warunkiem Lorentza: $$\begin{aligned} \label{warunek Lorentza} \partial_{\mu} A^{\mu}=0 \; .\end{aligned}$$ Zgodnie z (\[postac I dla p po x suma po Mink Maxwell\]) oraz (\[amplitudy dla Maxwella dual\]) pojemnoœæ informacyjna $I$ jest nastêpuj¹ca: $$\begin{aligned} \label{IF for Maxwell} I = 4\, a^{2} \sum_{\mu=0}^{3} \int_{\cal X} \! d^{4}{\bf x}\, \sum\limits_{\nu=0}^{3} {\left( {\frac{{\partial A_{\mu} \left( {\bf x} \right)}}{{\partial {\bf x}_{\nu} }}} {\frac{{\partial A^{\mu} \left( {\bf x} \right)}}{{\partial {\bf x}^{ \nu} }}} \right) } \, .\end{aligned}$$ W³aœciwa postaæ informacji strukturalnej $Q\equiv Q_{M}$ dla równañ ruchu Maxwella, zosta³a wyprowadzona w [@Frieden].\ \ [*Warunki fizyczne*]{}: W celu otrzymania zarówno $Q_{M}$, jak i wartoœci $\kappa$, musz¹ byæ przyjête pewne fizyczne za³o¿enia [@Frieden]. [*Po pierwsze*]{} jest to warunek Lorentza, [*po drugie*]{}, pewna wstêpna postaæ $Q_{M}$. [*Po trzecie*]{}, wymagane jest równie¿ za³o¿enie o braku dodatkowych Ÿróde³ pola elektromagnetycznego w przestrzeni wolnej od czterowektora pr¹du $j^{\,\mu}=(c\rho,\vec{j}\,)$.\ \ Drugi z tych warunków, zapisany zgodnie z notacj¹ zawart¹ w (\[k form\]), wyra¿a siê ¿¹daniem nastêpuj¹cej faktoryzacji obserwowanej informacji strukturalnej: $$\begin{aligned} \label{qF M} \,a^{2}\, A_{\nu}\,A^{\nu}\,\texttt{q\!F}_{\nu}(A_{\nu}(x)) = \frac{a}{\kappa}\, A^{\nu}\, F_{\nu}(j_{\nu}) \; ,\end{aligned}$$ która jest nastêpnie wykorzystana w (\[IPs for Maxwell\]). Je¿eli funkcja $F_{\nu}(j_{\nu})$ spe³nia za³o¿enie o zale¿noœci jedynie od czterowektora pr¹du $j_{\,\nu}$ [@Frieden], wtedy z zasad informacyjnych metody EFI, (\[IPs for Maxwell\]), wynika zarówno wartoœæ wspó³czynnika efektywnoœci $\kappa=1/2$, jak i równanie ci¹g³oœci strumienia $\partial^{\nu}j_{\nu}=0$ [@Dziekuje; @za; @models; @building]. W efekcie informacja strukturalna dla równania Maxwella jest równa [@Frieden]: $$\begin{aligned} \label{Q in electrodyn} Q = Q_{M} \equiv\,- \frac{64\pi}{c}\, a\,\sum_{\mu=0}^{3}\int_{\cal X} d^{4}{\bf x}\, j_{\mu}\, A^{\mu} \; . \end{aligned}$$ W koñcu, dla pojemnoœci informacyjnej $I$ jak w (\[IF for Maxwell\]) i informacji strukturalnej $Q$ jak w (\[Q in electrodyn\]) oraz dla $a=2$, metoda EFI daje wektorowe równanie falowe w cechowaniu Lorentza: $$\begin{aligned} \label{wave eq for A} \Box A^{\mu} = (4\pi/c)j^{\mu} \; ,\end{aligned}$$ co w konsekwencji prowadzi do znanej postaci równañ Maxwella dla pola magnetycznego i elektrycznego [@Jackson; @Frieden].\ \ Podkreœlmy, ¿e proporcjonalnoœæ $q_{\nu}({\bf x}) = a A_{\nu}({\bf x})$ oraz warunek normalizacji: $$\begin{aligned} \label{A normalization} (1/4)\sum_{\nu=0}^{3}\int_{\cal X} d^{4} {\bf x} \, q_{\nu}^{2}({\bf x})=\sum_{\nu=0}^{3}\int_{\cal X} d^{4} {\bf x} \, A_{\nu}^{2}({\bf x}) = 1 \; ,\end{aligned}$$ stawiaj¹ pytanie o znaczenie lokalizacji fotonu oraz istnienie jego funkcji falowej, które s¹ ostatnio mocno dyskutowane w literaturze dotycz¹cej optyki. Dyskusja zawarta w [@Roychoudhuri] wspiera g³ównie pogl¹d, który stawia równania Maxwella na tych samych podstawach co równanie Diraca w sformu³owaniu pierwszej kwantyzacji. Fakt ten by³ wczeœniej zauwa¿ony w pracy Sakurai [@Sakurai; @2]. Przyjmuj¹c interpretacjê funkcji falowej fotonu jako maj¹c¹ te same podstawy co wystêpuj¹ca w (\[free field eq all 2\]) funkcja falowa cz¹stek materialnych z mas¹ $m=0\,$, z trzeciego z powy¿szych [*fizycznych warunków*]{} mo¿na by zrezygnowaæ [@Frieden].\ \ Zauwa¿my, ¿e normalizacja[^92] (\[A normalization\]) czteropotencja³u $A_{\nu}$ uzgadnia wartoœæ sta³ej proporcjonalnoœci[^93] $a=2$ z wartoœci¹ $N=4$ dla pola œwietlnego.\ \ Normalizacja (\[A normalization\]), na³o¿ona jako warunek na rozwi¹zanie równania (\[wave eq for A\]), jest spójna z narzuceniem warunku pocz¹tkowego Cauchy’ego we wspó³rzêdnej czasowej (por. dyskusja w [@Frieden]). Gdy jednoczeœnie we wspó³rzêdnych przestrzennych narzucony jest warunek Dirichlet’a (lub Neumann’a), wtedy te mieszane czasowo-przestrzenne warunki brzegowe nie s¹ wspó³zmiennicze, co skutkuje tym, ¿e rozwi¹zanie równania (\[wave eq for A\]) nie jest wspó³zmiennicze.\ Jednak¿e tylko z mieszanymi warunkami brzegowymi rozwi¹zanie to jest jednoznaczne [@Morse-Feshbach; @Frieden], co dla metody EFI jest warunkiem koniecznym, gdy¿ jest ona metod¹ estymacji statystycznej. Fakt ten stoi w opozycji do przypadku, gdy warunek Dirichlet’a (lub Neumann’a) jest na³o¿ony wspó³zmienniczo zarówno we wspó³rzêdnych przestrzennych jak i wspó³rzêdnej czasowej, gdy¿ co prawda otrzymane rozwi¹zanie jest wtedy wspó³zmiennicze, jednak nie jest ono jednoznaczne.\ \ W Rozdziale \[Klein-Gordon scalars\] powiedzieliœmy, ¿e transformacja Fouriera tworzy rodzaj samospl¹tania pomiêdzy dwoma reprezentacjami, po³o¿eniow¹ i pêdow¹, dla realizowanych wartoœci zmiennych uk³adu wystêpuj¹cych w $I$ [@Frieden]. Strukturalna zasada informacyjna wyjaœnia to spl¹tanie, zachodz¹ce pomiêdzy pêdowymi stopniami swobody, jako spowodowane mas¹ uk³adu zgodn¹ z (\[m E p\]) lub (\[m E p dla q\]).\ Rozwa¿my bezmasow¹ cz¹stkê, np. foton. Jeœli w zgodzie z powy¿szymi rozwa¿aniami dla pola Maxwella, relacja (\[free field eq all 2 real amplitudes\]) okreœlaj¹ca informacjê Fouriera mia³aby byæ spe³niona dla cz¹stki o masie $m=0\,$ oraz dla amplitud interpretowanych zgodzie z (\[amplitudy dla Maxwella\]) i (\[A normalization\]) jako charakteryzuj¹cych foton, wtedy równie¿ ze strony eksperymentu nale¿a³oby siê spodziewaæ zarówno weryfikacji kwestii zwi¹zanej z natur¹ funkcji falowej fotonu [@Roychoudhuri] jak i sygnatur¹ metryki czasoprzestrzeni. W samej rzeczy, w metryce czasoprzestrzeni Minkowskiego (\[metryka M\]), zgodnie z (\[m E p dla q\]) jedyn¹ mo¿liwoœci¹, aby cz¹stka by³a bezmasowa, jest zachodzenie warunku $E^{2}/c^{2}-\vec{\wp}^{\,2} = 0$ dla wszystkich jej monochromatycznych, Fourierowskich mod, o ile tylko mody te mia³yby posiadaæ fizyczn¹ interpretacjê dla cz¹stki bezmasowej. Warunek ten oznacza³by jednak, ¿e mody Fourierowskie nie by³yby ze sob¹ spl¹tane [@Dziekuje; @za; @channel] (w przeciwieñstwie do tego co zachodzi dla cz¹stki masowej) i w zasadzie powinna istnieæ mo¿liwoœæ dokonania detekcji ka¿dego indywidualnego moda rozk³adu Fouriera. Zatem, jeœli czêstoœæ indywidualnego moda Fouriera impulsu œwietlnego nie zosta³aby zarejestrowana, to mog³oby to oznaczaæ, ¿e nie jest on obiektem fizycznym. Fakt ten móg³by doprowadziæ do problemów dla kwantowej interpretacji fotonu, okreœlonego jako fizyczna realizacja konkretnego Fourierowskiego moda. Problem ten zosta³ ostatnio zauwa¿ony w zwi¹zku z przeprowadzonymi eksperymentami optycznymi [@Roychoudhuri], z których wynika, ¿e Fourierowski rozk³ad czêstoœci impulsu œwietlnego nie reprezentuje rzeczywistych optycznych czêstoœci, co sugeruje, ¿e byæ mo¿e rzeczywisty foton jest “ziarnem elektromagnetycznej substancji” nie posiadaj¹cym Fourierowskiego przedstawienia. [@Roychoudhuri; @Dziekuje_Jacek_nova_2]. ### Dodatek: Metoda EFI dla teorii grawitacji {#general relativity case} Poni¿ej przedstawimy jedynie g³ówne wyniki zwi¹zane z konstrukcj¹ EFI dla teorii grawitacji. Wychodz¹c z ogólnej postaci pojemnoœci (\[postac I dla p po x bez n\]) i postêpuj¹c analogicznie jak powy¿ej dla pola Maxwella przy definicji amplitud dualnych, otrzymujemy amplitudow¹ postaæ dla pojemnoœci informacyjnej metody EFI. Nastêpnie postêpujemy ju¿ jak w [@Frieden], gdzie zosta³o podane wyprowadzenie s³abej (tzn. falowej) granicy równañ ruchu Einsteina, dla przypadku pól z rang¹ $N=10$, a amplitudy $q_{n}({\bf x}) \equiv q_{\nu\mu}({\bf x})$ w liczbie dziesiêæ, s¹ rzeczywiste i symetryczne w indeksach $\nu,\mu=0,1,2,3$. Zatem, pojemnoœci informacyjna jest nastêpuj¹ca: $$\begin{aligned} \label{I in gen rel} I = 4 \int_{\cal X} d^{4}{\bf x} \sum_{\nu,\mu=0}^{3} \sum_{\gamma=0}^{3} \frac{\partial q_{\nu\mu}({\bf x})}{\partial {\bf x}_{\gamma}}\frac{\partial q^{\nu\mu}({\bf x})}{\partial {\bf x}^{\gamma}} \; ,\end{aligned}$$ gdzie amplitudy dualne maj¹ postaæ $q^{\delta \tau}({\bf x}) =\sum_{\nu,\mu=0}^{3} \eta^{\delta \nu} \eta^{\tau \mu} q_{\nu\mu}({\bf x})$. Rozwi¹zuj¹c samospójne równania ró¿niczkowe obu zasad informacyjnych, strukturalnej i wariacyjnej, wraz z na³o¿onym warunkiem Lorentza, $\sum_{\nu=0}^{3}\partial_{\nu} q^{\nu\mu}({\bf x})=0$, który redukuje wspó³czynnik efektywnoœci do wartoœci $\kappa=1/2$, otrzymujemy nastêpuj¹c¹ postaæ informacji strukturalnej: $$\begin{aligned} \label{Q in gen rel} Q = - \,8 \int_{\cal X} d^{4}{\bf x}\,\frac{1}{L^{4}}\sum_{\nu\,\mu=0}^{3} \bar{h}_{\nu\mu}\left[\frac{16\,\pi\, G}{c^{4}} \, T^{\nu\mu} - 2\,\Lambda\,\eta^{\nu\mu}\right] \; ,\end{aligned}$$ gdzie $$\begin{aligned} \label{h and q in gen rel} \bar{h}_{\nu\mu}({\bf x})\equiv L^{2}q_{\nu\mu}({\bf x}) \; ,\end{aligned}$$ natomiast $\eta_{\nu\mu}$ jest metryk¹ Minkowskiego, $G$ jest sta³¹ grawitacyjn¹, $T_{\nu\mu}$ jest tensorem energii-pêdu, $\Lambda$ jest tzw. sta³¹ kosmologiczn¹, a sta³a $L$ jest charakterystyczn¹ skal¹, na której amplitudy $\bar{h}_{\nu\mu}$ s¹ uœrednione.\ Rozwi¹zanie metody EFI pojawia siê w postaci równania falowego dla amplitud $\bar{h}_{\nu\mu}$: $$\begin{aligned} \label{h equation in gravit} \Box \bar{h}_{\nu\mu}=\frac{16\,\pi\, G}{c^{4}}T_{\nu\mu}-2\,\Lambda\,\eta_{\nu\mu} \; ,\end{aligned}$$ gdzie $\Box \equiv \sum_{\nu=0}^{3} \partial^{\nu}\partial_{\nu} = \sum_{\mu, \,\nu=0}^{3} \eta^{\mu \nu} \partial_{\mu}\partial_{\nu}$ jest operatorem d’Alemberta. Równania (\[h equation in gravit\]) maj¹ postaæ w³aœciw¹ dla równañ ruchu w granicy s³abego pola, w ogólnej teorii wzglêdnoœci. Rezultatem EFI jest równie¿ równanie ci¹g³oœci strumienia, $\sum_{\nu=0}^{3}\partial^{\nu} T_{\nu\mu}({\bf x})=0$, dla tensora $T_{\nu\mu}$.\ Pozostaje pytanie o rozwi¹zanie dla dowolnie silnego pola. Jedna z odpowiedzi jest nastêpuj¹ca. Poniewa¿ $\Box\bar{h}_{\nu\mu}$ jest jednoznaczn¹ liniow¹ aproksymacj¹ tensora rangi drugiej $2 R_{\nu\mu} - g_{\mu\nu}R$, gdzie $R_{\nu\mu}$ jest tensorem Ricci’ego [@Misner-Thorne-Wheeler], $g_{\nu\mu}$ jest tensorem metrycznym, a $R = \sum_{\nu=0}^{3} R^{\nu}_{\,\nu}$, zatem mo¿na by uznaæ, ¿e równanie Einsteina wy³ania siê jako jedyne mo¿liwe, w tym sensie, ¿e jego linearyzacj¹ jest równanie falowe s³abego pola (\[h equation in gravit\]) otrzymane w EFI [@Frieden].\ Jednak¿e postaæ rozwi¹zania (\[h equation in gravit\]) metody EFI mog³aby równie¿ sugerowaæ inn¹ selekcjê przysz³ego modelu grawitacji. Otó¿ w ca³ym formalizmie EFI amplitudy s¹ podstaw¹ do definicji pola, a nie metryki czasoprzestrzeni. Zatem bardziej naturalnym wydaje siê zinterpretowanie $\bar{h}_{\nu\mu}$ jako pola grawitacyjnego[^94]. Tak wiêc równanie (\[h equation in gravit\]) metody EFI dla grawitacji le¿y bli¿ej innej teorii grawitacji, nazywanej “szczególn¹ teori¹ wzglêdnoœci grawitacji”, która prowadzi do efektywnej teorii grawitacji typu Logunov’a [@Denisov-Logunov], co poprzez widoczny zwi¹zek z teoriami cechowania czyni j¹ “bogatsz¹” ni¿ sam¹ ogóln¹ teoriê wzglêdnoœci. Problem statystycznego porównania obu teorii grawitacji jest kwesti¹ przysz³ych prac. Dodatek: Informacyjny odpowiednik drugiej zasady termodynamiki:\ Twierdzenie $I$ {#Wyprowadzenie drugiej zasady termodynamiki} ---------------------------------------------------------------- Niech próba bêdzie $N=1$-wymiarowa. Podobnie jak przechodzi siê z (\[I dla pn jeden parametr\]) do (\[dys\]) w celu otrzymania dyskretnej postaci informacji Fishera[^95], tak mo¿na te¿ pokazaæ, ¿e mo¿na przejœæ do jej nastêpuj¹cej dyskretnej postaci: $$\begin{aligned} \label{zwiazek I oraz S} I_{F} = \Delta x\sum_{k}{\frac{1}{p\left(x_{k}\right)}\left[\frac{p\left(x_{k}+ \Delta x \right) - p\left(x_{k}\right)}{\Delta x_{k}}\right]^{2}} \; .\end{aligned}$$ Postêpuj¹c w sposób analogiczny jak w przypadku wyprowadzenia (\[I porownanie z Sn\]) z (\[dys\]), otrzymujemy: $$\begin{aligned} \label{iewf} I_{F} = -\frac{2}{{\left({\Delta x}\right)^{2}}} \; S_{H}\left({p\left(x\right)| p\left({x+\Delta x}\right)}\right) \ .\end{aligned}$$ Niech zmiana rozk³adu prawdopodobieñstwa z $p \equiv p(x)$ w chwili $t$ do $p_{\Delta x} \equiv p\left({x+\Delta x}\right)$ nastêpuje na skutek infinitezymalnej zmiany czasu o $dt$. Jeœli wiêc dla rozk³adów $p$ oraz $p_{\Delta x}$ entropia wzglêdna spe³nia warunek: $$\begin{aligned} \label{wzrost S wzglednego z czasem} \frac{ d S_{H} \left({p\,| p_{\Delta x}}\right)(t) }{dt} \ge 0 \; ,\end{aligned}$$ wtedy ze wzglêdu na (\[iewf\]) informacja Fishera spe³nia [**warunek**]{}: $$\begin{aligned} \label{twierdzenie I} \frac{{dI_{F} \left(t\right)}}{{dt}} \le 0\end{aligned}$$ [**nazywny $I$-twierdzeniem**]{}. Oznacza on, ¿e informacja Fishera $I_{F}$ dla parametru $\theta$ maleje monotonicznie z czasem.\ \ [**Uwaga**]{}: Dowód (\[wzrost S wzglednego z czasem\]) oraz faktu, ¿e relacja ta istotnie prowadzi do drugiej zasady termodynamiki w rozumienia twierdzenia $H$, czyli dowód, który trzeba przeprowadziæ dla entropii Shannona, mo¿na znaleŸæ w pracach na temat dynamiki uk³adów otwartych [@uklady; @otwarte] .\ \ Wprowadzenie twierdzenia $I$ jako maj¹cego zwi¹zek z twierdzeniem $H$ jest jak widaæ nieprzypadkowe. Ale równie¿ nieprzypadkowe jest podobieñstwo dzia³aj¹ce w drug¹ stronê. A mianowicie, pojemnoœæ informacja jest zwi¹zana z rang¹ pola $N$, co by³o widoczne w ca³ej treœci skryptu. W istniej¹cej literaturze mo¿na znaleŸæ wyprowadzenia pokazuj¹ce, ¿e istnieje entropijny odpowiednik zasady nieoznaczonoœci Heisenberga, oraz zwi¹zek informacji Shannona z wymiarem reprezentacji grupy obrotów dla pól fermionowych rangi $N$. ### Dodatek: Temperatura Fishera {#Temperatura Fishera} Jako kolejn¹ ilustracjê nieprzypadkowego podobieñstwa rachunków informacyjnych i entropijnych rozwa¿my definicjê temperatury Fishera dla parametru $\theta$. Otó¿, informacja Fishera parametru $\theta$ pozwala na zdefiniowanie [*temperatury Fishera*]{} $T_{\theta}$ zwi¹zanej z estymacj¹ tego parametru: $$\begin{aligned} \label{temperatura I} \frac{1}{T_{\theta}} \equiv - k_{\theta} \frac{\partial I_{F}}{\partial \theta} \; .\end{aligned}$$ Tak okreœlona temperatura $T_{\theta}$ jest miar¹ czu³oœci informacji Fishera na zmianê parametru $\theta$, podobnie jak temperatura Boltzmanna $T$ (dla której zachodzi $\frac{1}{T} \equiv \frac{\partial H_{B}}{\partial E} $) jest miar¹ czu³oœci entropii Boltzmanna $H_{B}$ na zmianê energii $E$ uk³adu. Temperatura Fishera znalaz³a swoje zastosowanie miêdzy innymi w badaniu rynków finansowych [@temperature; @1; @temperature; @ostatnia]. R.A. Fisher, Phil. Trans. R. Soc. Lond.[**222**]{}, 309 (1922). R.A. Fisher, *Statistical methods and scientific inference*, 2 nd edn. (London, Oliver and Boyd, 1959). C.R. Rao, Bulletin of the Calcutta Mathem. Soc. [**37**]{}, pp.81-91 (1945). *Information and accuracy attainable in the estimation of statistical parameters*. B. Efron, The Annals of Statistics [**3**]{}, pp.1189-1242 (1975). *Defining the curvature of statistical problem (with application to second order efficiency) (with discussion*. A.P. Dawid, The Annals of Statistics [**3**]{}, pp.1231-1234 (1975). *Discussion to Efron’s paper*. A.P. Dawid, The Annals of Statistics [**5**]{}, pp.1249 (1977). *Further comments on a paper by Bradley Efron*. S. Amari, POST RAAG Report [**106**]{} (1980). *Teory of information spaces-a geometrical foundation of statistics*. S. Amari, The Annals of Statistics [**10**]{}, pp.357-385 (1982). *Differential geometry of curved exponential families-curvature and information loss.* S. Amari, H. Nagaoka, *Methods of information geometry, transletions of Mathematical monographs*, Vol.191, Oxford Univ. Press, (2000).\ A. Fujiwara and H. Nagaoka, *Quantum Fisher metric and estimation for pure state models*, Phys. Lett. A **201**, 119-124, (1995). A. Fujiwara and H. Nagaoka, *An estimation theoretical characterization of coherent states*, J. Math. Phys. **40**, 4227-4239, (1999). K. Matsumoto, *A geometrical approach to quantum estimation theory*, PhD thesis, Univ. of Tokyo, (1998). A. Fujiwara and H. Nagaoka, Phys. Lett. A [**201**]{}, pp.119-124 (1995). *Quantum Fisher metric and estimation for pure state models*. A. Fujiwara and H. Nagaoka, Journal of Mathem. Phys. [**40**]{}, pp. 4227-4239 (1999).\ *An estimation theoretical characterization of coherent states*. K. Matsumoto, Technical Report METR 96-09, Dept. of Math. Eng. and Inform. Phys, Univ. of Tokyo, (1996). *A new approach to the Cramer-Raotype bound of the pure state model*. K. Matsumoto, PhD thesis, Univ. of Tokyo, (1998). *A geometrical approach to quantum estimation theory*. B.R. Frieden, Found.Phys.**16**, No.9, *A probability law for the fundamental constants*, 883-903 (1986). B.R. Frieden, Phys.Rev.A**41**, *Fisher information, disorder, and the equilibrium distributions of physics*, 4265-4276 (1990). B.R. Frieden, B.H. Soffer, Phys.Rev.E**52**, *Lagrangians of physics and the game of Fisher-information transfer*, 2274-2286 (1995). B.R. Frieden, Phys.Rev.A**66**, *Relations between parameters of a decoherent system and Fisher information*, 022107, (2002). B.R. Frieden, A. Plastino, A.R. Plastino and B.H. Soffer, *Schr[ö]{}dinger link between nonequilibrium thermodynamics and Fisher information*, Phys.Rev.E**66**, 046128 (2002). B.R. Frieden, A. Plastino, A.R. Plastino and B.H. Soffer, *Non-equilibrium thermodynamics and Fisher information: An illustrative example*, *Phys.Lett.A* **304**, (2002), pp.73-78.\ B.R. Frieden, Science from Fisher information: A unification, (Cambridge University Press, 2004). J. Syska, *The Fisher information and quantum models-classical field theory, classical statistics similarity*, Phys. Stat. Sol.(b), **244**, No.7, 2531-2537 (2007)/DOI 10.1002/pssb.200674646. E.W. Piotrowski, J. S[ł]{}adkowski, J. Syska, S. Zaj[a]{}c, *The method of the likelihood and the Fisher information in the construction of physical models*, Phys. Stat. Sol.(b), [**246**]{}, No.5, 1033-1037, (2009)/DOI 10.1002/pssb.200881566, arXiv:physics/0811.3554. R. Nowak, *Statystyka dla fizyków*, PWN, Warszawa 2002. J. Jakubowski, R. Sztencel, *Wstêp do teorii prawdopodobieñstwa*, wydanie 2, Script, Warszawa, (2001). Y. Pawitan, *In all likelihood: Statistical modelling and inference using likelihood*, (Oxford Univ. Press, 2001). D.G. Kleinbaum, L.L. Kupper, K.E. Muller, A. Nizam, *Applied Regression Analysis and Other Multivariable Method*, Duxbury Press, (1998). D. Mroziakiewicz, *Analiza regresji Poissona z estymatorami metody najwiêkszej wiarygodnoœci z wykorzystaniem programu statystycznego SAS*, Praca licencjacka, Inst. Fizyki, Univ. Œl¹ski (Rybnik), Polska, (2006). D. Mroziakiewicz, *Metoda najwiêkszej wiarygodnoœci i informacja Fishera w fizyce i statystyce*, Praca mgr., (*The maximum likelihood method and Fisher information in physics and statistics*), unpublished, Uniwersytet Œl¹ski, Inst. Fizyki, (2008); M. Czerwik, *Wykorzystanie programu SAS jako narzêdzia do analizy wspó³zale¿noœci zmiennych metod¹ regresji*, Praca licencjacka, Uniwersytet Œl¹ski, Inst. Fizyki, Jastrzêbie Zdrój, (2004). M. Biesiada, *Statystyka w ujêciu Bayesowskim*, Skrypt dla studentów ekonofizyki, Uniwersytet Œl¹ski, Instytut Fizyki, (2011).\ J-P. Gazeau, *Coherent states in quantum physics*, Wiley-VCH, (2009). S. Mania, *Informacja Fishera. Wyprowadzenie równania Kleina-Gordona*, Praca licencjacka, Uniwersytet Œl¹ski, Inst. Fizyki, Jastrzêbie Zdrój, (2007). I. Bengtsson, K. [¯]{}yczkowski, *Geometry of quantum states*, Cambridge Univ. Press, (2006). R.F. Streater, *Statistical dynamics. A stochastic approach to nonequilibrium thermodynamics*, 2nd ed., Imperial College Press, (2009). A. Ohara and T. Kitamori, *Geometric structures of stable state feedback systems*, IEEE Transactions on Automatic Control, [**38(10)**]{}, pp.1579-1583, (1993). A. Ohara and S. Amari, *Differential geometric structures of stable state feedback systems with dual connections*, Kybernetica, [**30(4)**]{}, pp.369-386, (1994). A. Ohara, N. Suda, and S. Amari, *Dualistic differential geometry of positive definite matrices and its applications*. Linear Algebra and its Applications, [**247**]{}, pp.31-53, (1996). P.J. Brockwell, R.A. Davis , *Time Series: Theory and Methods*, 2nd edition, Springer, (2009).\ £. Machura, *Analiza szeregów czasowych*, Skrypt dla studentów ekonofizyki, Uniwersytet Œl¹ski, Instytut Fizyki, (2011). M.K. Murray, J.W. Rice, *Differential geometry and statistics*, Chapman and Hall/CRC, (1993). M. Nakahara, *Geometry, topology and physics*, 2nd edition, IOP Publishing Ltd, (2003). A.P. Dembo and O. Zeitouni, *Large deviations techniques and applications*, 2nd ed. Springer, New York, (1998). A.J. Stam, *Some mathematical properties of quantities of information*, Ph.D. dissertation. Technical University of Delf, (1959). A. J. Stam, *Some inequalities satisfied by the quantities of information of Fisher and Shannon*, Inform. and Control [**2**]{}, 101–112 (1959). F. Pennini, A. Plastino, B.H. Soffer, C. Vignat, *Physical symmetries and Fisher’s information measure*, Phys.Lett. A**373**, pp.817-820, (2009). J.J. Sakurai, *Advanced quantum mechanics*, p.169, Addison-Wesley Publishing Company, (1967). J. Syska, *Kaluza-Klein Type Model for the Structure of the Neutral Particle-like Solutions*, International Journal of Theoretical Physics, Volume 49, Issue 9, p.2131-2157, DOI:10.1007/s10773-010-0400-8, Open Access, (2010). J. Dajka, *Kwantowe metody opisu dynamiki instrumentów finansowych*, Skrypt dla studentów ekonofizyki, Uniwersytet Œl¹ski, Instytut Fizyki, (2010). I.N. Bronsztejn, K.A. Siemiendiajew, G. Musiol, H. M[ü]{}hlig, *Nowoczesne kompendium matematyki*, 305-306 Wydawnictwo PWN, 305-306, (2004). S. Zaj[a]{}c, *Modelowanie szeregów czasowych za pomoc¹ procesów ARMA i ARIMA, (aplikacje w systemie SAS)*, Praca licencjacka, Inst. Fizyki, Univ. Œl¹ski (Rybnik), Polska, (2005). K. Sobczyk, *Stochastyczne równania ró¿niczkowe. (Teoria i zastosowania)*, Wydawnictwo NT, (1996).\ J. [Ł]{}uczka, *Procesy i zjawiska losowe*, Skrypt dla studentów ekonofizyki, Uniwersytet Œl¹ski, Instytut Fizyki, (2010). J. von Neumann, *Mathematische Grundlagen der Quantenmechanic*, (Springer, Berlin), 171 (1932); transl. by E.T. Beyer, Mathematical foundations of quantum mechanics, Princeton Univ. Press, Princeton, 324 (1955). A. Peres, *Quantum Theory: Concepts and Methods*, Kluwer Academic Publishers, (1995). M.R. Esfandiary, H.J. Fink, Phys.Lett.A**54**, 383 (1975). V.A. Schweigert, F.M. Peeters et.al., Phys.Rev.Lett.**79**, 4653 (1997); Phys.Rev.Lett.**83**, 2409 (1999); Phys.Rev.B**59**, 6039 (1999); Phys.Rev.B**62**, 9663 (2000). J.J. Palacios, Phys.Rev.B**57**, 10873 (1998); Phys.Rev.Lett.**83**, 2409 (1999); Phys.Rev.Lett.**84**, 1796 (2000). G.F. Zharkov, V.G. Zharkov and A.Yu. Zvetov, Phys.Rev.B**61**, 12293 (2000). G.F. Zharkov, Phys.Rev.B**63**, 214502 (2001); Phys.Rev.B**63**, 224513 (2001). A.O. Barut, J.F. Van Huele, Phys.Rev.A**32**, 3187 (1985). A.O. Barut and N. Ünal, J.Math.Physics**27**, 3055 (1986). A.O. Barut, J.P. Dowling, Phys.Rev.A**36**, 649 (1987). A.O. Barut and N. Ünal, Physica **142 A**, 467, 488 (1987). A.O. Barut, Phys.Scr.**T21**, 18 (1988). A.O. Barut, Y.I. Salamin, Phys.Rev.A**37**, 2284 (1988).\ $\;$ J. Syska, Self-consistent classical fields in gauge field theories, PhD thesis, (University of Silesia, 1995/99), (unpublished); $\;$ Trends in Boson Research, e.d. A.V. Ling, (Nova Science Publishers), 163-181 (2006); Frontiers in field theory, ed. O. Kovras, (Nova Science Publishers), 125-154 (2005). I. Bednarek, R. Ma[ñ]{}ka, J.Phys.G **31**, 1, (2005). J. Syska, *Boson ground state fields in the classical counterpart of the electroweak theory with non-zero charge densities*, in *Frontiers in field theory*, ed. O. Kovras, (Nova Science Publishers), (Nowy Jork 2005), chapter 6, 125-154 and in *Focus on Boson Research*, (Nova Publishers), 233-258, (2006). J. Syska, *Remarks on self-consistent models of a particle*, in *Trends in Boson Research*, ed. A.V. Ling, (Nova Science Publishers), 163-181, (2006). V.I. Denisov and A.A. Logunov, *The Theory of Space-Time and Gravitation*, in *Gravitation and Elementary Particle Physics*, Physics Series, ed. A.A. Logunov, (MIR Publishers, Moscow), 14-130, (1983). A.A. Logunov, *The Theory of Gravity*, arXiv:gr-qc/0210005v2, (2002). J. Syska, R. Szafron, *Informational criteria for the field theory models building*, in preparation. Edited by Ch. Roychoudhuri, A.F. Kracklauer, K. Creath, *The nature of light. What is a photon?*, (CRC Press, Taylor&Francis Group), (2008). F.del Aguila, J. Syska, R. Szafron, S. Zaj¹c, M. Zra³ek, *The dynamics of the neutrino flavour transition with the general effective interaction*, in preparation. R. Ticciati, *Quantum field theory for mathematicians*, (Cambridge University Press), (1999). M. Fecko, *Differential geometry and Lie groups for physicists*, (Cambridge University Press), (2006). A. Khrennikov, *Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?*, Entropy [**10**]{}, pp.19-32, (2008). W.I. Arnold, *Teoria równañ ró¿niczkowych*, PWN, Warszawa, (1983). M. Aoki, H. Yoshikawa, *Reconstructing macroeconomics. A perspective from statistical physics and combinatorial stochastic processes*, (Cambridge University Press), (2007). U. Garibaldi, E. Scalas, *Finitary Probabilistic Methods in Econophysics*, Cambridge Univ. Press, 1st ed., (2010). J. Flachsmeyer, *Kombinatoryka*, PWN, (1977). M. Abramowitz, I. A. Stegun. *Handbook of Mathematical Functions* Washington, DC.: National Bureau of Standards, 1965. E. Kamke, *Sprawocznik po obyknowiennym differiencialnym urawnieniam*, red. C.W. Fomin, Izdatielstwo inostrannoj litieratury, Moskwa, 1951. I.B. Rumer and M.Sh. Ryvkin, *Thermodynamics, Statistical Physics and Kinetics*, Moscow: Mir Publishers, p.526, (1980). E.B. Manoukian, *Quantum Theory A Wide Spectrum*, Springer, (2006). J.S. Bell, *Speakable and unspeakable in quantum mechanics*, Cambridge Univ. Press: Cambridge, UK, (1987). L. Accardi, *The probabilistic roots of the quantum mechanical paradoxes*, Proc. Conf. The wave–particle dualism. A tribute to Louis de Broglie on his 90th Birthday, Reidel Publ. Company: Dordrecht, Netherlands, pp. 47–55, (1970). A. Fine, *Hidden Variables, Joint Probability, and the Bell Inequalities*, Phys. Rev. Lett. [**48**]{}, pp.291–295, (1982). I. Pitowsky, *Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes*, Phys. Rev. Lett. [**48**]{}, pp.1299-1302, 1982. I. Pitowsky, *Deterministic Model of Spin and Statistics*, Phys. Rev. D [**27**]{}, pp.2316-2326, (1983). P. Rastall, *The Bell inequalities*, Found. Phys. [**13**]{}, 555-575, (1983). K. Hess, W. Philipp, *A possible loophole in the theorem of Bell*, Proc. Natl. Acad. Sci. [**98**]{}, 14224, (2001). K. Hess, W. Philipp, *Exclusion of time in the theorem of Bell*. Europhys. Lett. [**57**]{}, 775, (2002). A. Khrennikov, *Statistical measure of ensemble nonreproducibility and correction to Bell’s inequality*, Il Nuovo Cimento B, [**115**]{}, pp.179–184, (1999). A. Khrennikov, *Non-Kolmogorov probability models and modified Bell’s inequality*, J. Math. Phys. [**41**]{}, pp.1768–1777, (2000). A. Khrennikov, *A perturbation of CHSH inequality induced by fluctuations of ensemble distributions*, J. Math. Phys. [**41**]{}, pp.5934–5944, (2000). J. D. Jackson, *Elektrodynamika klasyczna*, PWN, Warszawa, (1982). U. Leonhardt, *Essential quantum optics. From quantum measurements to black holes*, (Cambridge Univ.Press.), (2010). P.M. Morse and H. Feshbach, *Methods of Theoretical Physics*, (New York: McGraw-Hill), (1953). J. S[ł]{}adkowski, J. Syska, *The information channel capacity in the field theory estimation*, sent to Phys.Lett.A, (2012). C.W. Misner, K.S. Thorne, J.A. Wheeler, *Gravitation*, New York: Freeman, (1973). A. Wehrl, Rev. Mod. Phys. [**50**]{}, 221, (1978); R. Alicki and K. Lendi, *Quantum Dynamical Semigroups and Applications*, Lect. Notes Phys. [**717**]{}, (Springer, Berlin Heidelberg), (2007), DOI 10.1007/b11976790. E.W.  Piotrowski, J. S³adkowski, *The thermodynamics of portfolios*, Acta Physica Polonica B, Vol.[**32**]{}, No.2, (2001); H. Kleinert, X.J. Chen, *Boltzmann distribution and market temperature*, arXiv:physics/0609209v2, (2007). René Brand, *An econophysical investigation: Using the Boltzmann-distribution to determine market temperature as applied to the JSE All Share Index*, Stellenbosch University, (16 March 2009). [^1]: jacek.syska@us.edu.pl [^2]: Wspó³finansowanie projektu “Uniwersytet partnerem gospodarki opartej na wiedzy” w ramach Europejskiego Funduszu Spo³ecznego. Kapita³ Ludzki Narodowa Strategia Spójnoœci. [^3]: [**Miara produktowa**]{}: Niech bêdzie danych $N$ przestrzeni probabilistycznych $\left\{\Omega_{1}, {\cal F}_{1}, P_{1}\right\}$, ... , $\left\{\Omega_{N}, {\cal F}_{N}, P_{N}\right\}$, gdzie $\Omega_{n}$, ${\cal F}_{n}$, $P_{n}$ s¹, dla ka¿dego $n=1,2,...,N$, odpowiednio $n$-t¹ przestrzeni¹ zdarzeñ, $\sigma$ - cia³em na $\Omega_{n}$ oraz miar¹ probabilistyczn¹. WprowadŸmy na produkcie $\Omega = \Omega_{1}\times ... \times \Omega_{N}$ tzw. $\sigma$ - cia³o produktowe ${\cal F}\equiv {\cal F}_{1} \otimes ... \otimes{\cal F}_{N}$ bêd¹ce najmniejszym $\sigma$ - cia³em zawieraj¹cym zbiory postaci $A_{1}\times ... \times A_{N}$ gdzie $A_{1} \in {\cal F}_{1}$, ... , $A_{N} \in {\cal F}_{N}$.\ Na produkcie $\Omega$ mo¿na zdefiniowaæ miarê produktow¹ $P$ tak¹, ¿e: $$\begin{aligned} \label{Miara produktowa} P(A_{1} \times A_{2} \times... \times A_{N}) = P_{1}(A_{1}) \, P_{2}(A_{2}) \, ... \, P_{N}(A_{N}) \; .\end{aligned}$$ [**Zmienne niezale¿ne**]{}: Zmienne losowe $Y_{1}, ... , Y_{N}$ o wartoœciach w $\mathbb{R}$ okreœlone odpowiednio na $\left\{\Omega_{1}, {\cal F}_{1}, P_{1}\right\}$, ... , $\left\{\Omega_{N}, {\cal F}_{N}, P_{N}\right\}$ nazywamy [*niezale¿nymi*]{}, gdy dla ka¿dego ci¹gu zbiorów borelowskich $B_{1}, ... , B_{N}$ zachodzi równoœæ: $$\begin{aligned} \label{Miara produktowa zmienne losowe} P(Y_{1} \in B_{1}, \, Y_{2} \in B_{2}, ... \, Y_{N} \in B_{N}) = P_{1}(Y_{1} \in B_{1}) \, P_{2}(Y_{2} \in B_{2}) \, ... \, P_{N}(Y_{N} \in B_{N}) \; .\end{aligned}$$ Poniewa¿ po prawej stronie (\[Miara produktowa zmienne losowe\]) stoj¹ zdarzenia losowe $Y_{n}^{-1}(B_{n})$, gdzie $B_{n}$ nale¿y do $\sigma$ - cia³a $\mathbb{B(R)}$, zatem stwierdzenie (\[Miara produktowa zmienne losowe\]) oznacza, ¿e [*zmienne losowe s¹ niezale¿ne wtedy i tylko wtedy, gdy $\sigma$ - cia³a ${\cal F}_{n}$ generowane przez zmienne losowe $Y_{n}$, $n=1,2,...,N$, s¹ niezale¿ne*]{} [@Jakubowski-Sztencel]. [^4]: W przypadku analizy jednej zmiennej losowej $Y$, rozk³ady te obok niezale¿noœci spe³niaj¹ dodatkowo warunek: $$\begin{aligned} \label{rozklady pn} p_{n} \left({{\bf y}_{n}|\theta_{n}}\right) = p \left({{\bf y}|\theta} \right) \; ,\end{aligned}$$ co oznacza, ¿e próba jest [*prosta*]{}. [^5]: Natomiast rozk³ad estymatora oszacowywanego parametru, zale¿y od tego parametru. [^6]: [**Postaæ estymatora parametru skalarnego $\theta$ rozk³adu $N\left({\theta,\sigma^{2}}\right)$**]{}: Korzystaj¹c z równania wiarygodnoœci (\[rown wiaryg\]) dla przypadku skalarnego parametru $\theta$, otrzymujemy: $$\begin{aligned} \label{rown wiaryg skal} S\left(\theta\right)_{|_{\theta = \hat{\theta}}} \equiv\frac{\partial}{\partial\theta} \ln P(y\,|\theta)_{|_{\theta = \hat{\theta}}} = 0 \; , \end{aligned}$$ sk¹d dla log-funkcji wiarygodnoœci (\[log wiaryg rozklad norm jeden par\]), otrzymujemy: $$\begin{aligned} \label{srednia arytmet z MNW} \hat{\theta} = \bar{{\bf y}} = \frac{1}{{N}}\sum\limits_{n=1}^{N} {\bf y}_{n} \; .\end{aligned}$$ Zatem estymatorem parametru $\theta$ jest œrednia arytmetyczna: $$\begin{aligned} \label{srednia arytmet estymator z MNW} \hat{\theta} = \overline{Y} = \frac{1}{{N}}\sum\limits_{n=1}^{N} Y_{n} \; .\end{aligned}$$ Estymator i jego realizowan¹ wartoœæ bêdziemy oznaczali tak samo, tzn. $\hat{\theta}$ dla przypadku skalarnego i $\hat{\Theta}$ dla wektorowego. [^7]: SSE w literaturze angielskiej. [^8]: Czynnik $r\left(x_{n}, \beta \right)$ nazywany dalej funkcj¹ regresji, chocia¿ w³aœciwie nazwa ta odnosi siê do ca³ej $E\left(Y_{n} \right)$. [^9]: Nale¿y jednak pamiêtaæ, ¿e zwrotu “najmniejszych kwadratów” nie nale¿y tu braæ dos³ownie, gdy¿ metoda najmniejszych kwadratów ma sens jedynie wtedy, gdy rozk³ad zmiennej $Y$ jest normalny (por. Rozdzia³ \[regresja klasyczna\]). [^10]: Obserwowana macierz (wariancji-) kowariancji $\hat{V}(\hat{\beta })$ estymatorów $\hat{\beta }$ MNW jest zdefiniowana jako odwrotnoœæ macierzy obserwowanej informacji Fishera (\[I obserwowana\]) [@Pawitan]: $$\begin{aligned} \label{macierz kowariancji estymatorów} \hat{V}(\hat{\beta }) := \texttt{i\!F}^{-1}(\hat{\beta}) \; .\end{aligned}$$ [^11]: W przyjêtym przedstawieniu danych jak dla diagramu punktowego, $N$ jest ogólnie liczb¹ punktów pomiarowych (równ¹ liczbie wariantów czy komórek). Tylko dla modelu podstawowego jest $N$ równie¿ liczb¹ parametrów. [^12]: Zauwa¿my, ¿e statystyka (\[dewinacja jak chi\]) mo¿e mieæ myl¹co du¿¹ wartoœæ gdy wielkoœci $\hat{Y}_{n} $ s¹ bardzo ma³e. [^13]: W pe³nym zapisie indeksów, ze wzglêdu na to, ¿e jeden punktowy parametr $\theta_{n} = (\vartheta_{1n},\vartheta_{2n},...,\vartheta_{kn})^{T} \equiv ((\vartheta_{s})_{s=1}^{k})_{n}$ mo¿e byæ parametrem wektorowym, co ma miejsce gdy $k>1$, zapis (\[forma kw dla P\]) oznacza: $$\begin{aligned} \label{forma kw dla P wszystkie parametry} \sum_{n, \,n'=1}^{N} \; \sum_{s, s'=1}^{k} \frac{\partial^{2} \ln P}{\partial \vartheta_{sn} \partial \vartheta_{s'n'}} {\left|_{\Theta = \hat{\Theta}} \right.} \, \Delta\vartheta_{sn} \, \Delta\vartheta_{s'n'} \; .\end{aligned}$$ Wkrótce i tak ograniczymy siê do sytuacji gdy $k=1$, tzn. $\theta_{n} = \vartheta_{1n}=\vartheta_{n}$. [^14]: Bêdziemy powszechnie stosowali skrócony zapis typu: $\,\Theta^{2} \equiv \Theta \Theta^{T} = (\theta_{i} \theta_{j})_{d \times d}$ oraz $\frac{\partial^{2}}{\partial \Theta^{2}} \equiv \frac{\partial^{2}}{\partial \Theta \,\partial \Theta^{T}} = (\frac{\partial^{2}}{\partial \theta_{i} \partial \theta_{j}})_{d \times d}$. [^15]: Poniewa¿ dla próby prostej wszystkie $Y_{n}$, $n=1,2,...,N$, maja taki sam rozk³ad jak $Y$ oraz dla $n \neq n'$ zachodzi ${\rm cov}(Y_{n},Y_{n'}) = 0$, zatem: $\, \sigma^{2}({\bar{Y}}) = \sigma^{2}(\frac{1}{N}\sum_{n=1}^{N} Y_{n}) $ $= \frac{1}{N^{2}}\sigma^{2}(\sum_{n=1}^{N} Y_{n}) = \frac{1}{N^{2}}\sum_{n=1}^{N} \sigma^{2}(Y_{n}) = \frac{1}{N^{2}} N \,\sigma^{2}(Y) = \frac{\sigma^{2}}{N}\, $. [^16]: Poniewa¿ funkcja wynikowa $S(\Theta) \equiv S(\widetilde{Y}|\Theta) = \frac{\partial P(\widetilde{Y}|\Theta)}{\partial \Theta}$ jest $d$-wymiarowym kolumnowym wektorem losowym (\[funkcja wynikowa\]) z wartoœci¹ oczekiwan¹ równ¹ zero, zatem ${\rm Cov}_{\Theta}\left[ \,S \left( \widetilde{Y}|\Theta \right) \right]$ jest $d\times d$-wymiarow¹ macierz¹ kowariancji (wspó³rzêdnych) wektora $S(\widetilde{Y}|\Theta)$. W skrypcie bêdziemy stosowali oznaczenie ${\sigma^{2}}_{\!\! \Theta} \,S \left( \Theta \right) \equiv {\rm Cov}_{ \Theta}\left[ \,S \left( \widetilde{Y}|\Theta \right) \right]$. [^17]: Dla dowolnego wektora ${\bf a} = \left(a_{1},...,a_{d} \right)^{T} \in \mathbf{R}^{d}$ oraz macierzy kowariancji $C = E \left(\left(Z-E(Z) \right) \left(Z-E(Z) \right)^{T}\ \right)$, gdzie $Z$ jest $d$-wymiarowym wektorem losowym, zachodzi ${\bf a}^{T} E\left[\left(Z-E(Z) \right)\left(Z - E(Z) \right)^{T} \right] {\bf a} = {\bf a}^{T} E \left(W W^{T} \right) {\bf a}$ $ = E \left[({\bf a}^{T} W )({\bf a}^{T} W )^{T}\right] ={\mathop{\sigma^{2}}}\left({\bf a}^{T} W \right)\ge0$, gdzie $W=Z-E(Z)\;$ i $E(W)=0$, tzn. macierz $C$ jest nieujemnie okreœlona. [^18]: Niech ${\cal S}$ jest zbiorem punktów, na którym okreœlony jest uk³ad wspó³rzêdnych $\phi_{\Xi}: {\cal S} \rightarrow \mathbb{R}^{d}$. [**W skrypcie interesuj¹ nas tylko globalne uk³ady wspó³rzêdnych**]{}. Wtedy $\phi_{\Xi}$ odwzorowuje ka¿dy punkt $P \in {\cal S}$ w zbiór $d$ liczb rzeczywistych $\phi_{\Xi}(P) \equiv (\xi^{1}(P), \xi^{2}(P),...,\xi^{d}(P))^{T} = ( \xi^{1}, \xi^{2},...,\xi^{d})^{T} \equiv \Xi$.\ \ Niech istnieje zbiór uk³adów wspó³rzêdnych (czyli atlas) ${\cal A}$ spe³niaj¹cy nastêpuj¹ce warunki:\ 1) Ka¿dy element $\phi_{\Xi} = \left[(\xi^{i})_{i=1}^{d}\right] \in {\cal A}$, jest wzajemnie jednoznacznym odwzorowaniem $\phi_{\Xi}: {\cal S} \rightarrow \mathbb{R}^{d}$ z ${\cal S}$ w pewien otwarty podzbiór w $\mathbb{R}^{d}$.\ 2) Dla ka¿dego $\phi_{\Xi} \in {\cal A}$ oraz wzajemnie jednoznacznego odwzorowania $\phi_{\Xi'}$ z ${\cal S}$ w $\mathbb{R}^{d}$, zachodzi\ równowa¿noœæ: $(\phi_{\Xi'} = (\xi^{'\,i})_{i=1}^{d} \in {\cal A})$ $\Leftrightarrow$ $(\phi_{\Xi'} \circ \phi_{\Xi}^{-1}$ jest dyfeomorfizmem rzêdu $C^{\infty})$. $$\begin{aligned} \label{uklad wspolrzednych na S} Zbi\acute{o}r \; {\cal S} \; z \; tak \; okre\acute{s}lonym \; atlasem \; {\cal A} \; to \; (C^{\infty} \; r\acute{o}\dot{z}niczkowalna) \; rozmaito\acute{s}\acute{c} \; .\end{aligned}$$ [^19]: [**Okreœlenie przestrzeni Riemannowskiej**]{}: Niech ${\cal S}$ jest rozmaitoœci¹ i za³ó¿my, ¿e dla ka¿dego punktu $P_{\Xi} \in {\cal S}$ w jej przestrzeni stycznej $T_{P} \equiv T_{P}({\cal S})$ jest okreœlony iloczyn wewnêtrzny $\left\langle , \right\rangle_{P}$ taki, ¿e $\left\langle V, W \right\rangle_{P} \in \mathbb{R}$ oraz posiadaj¹cy dla dowolnych wektorów $V,W \in T_{P}$ nastêpuj¹ce w³asnoœci: $$\begin{aligned} \label{metryka Riemanna} & & (\forall \,a,b \in \mathbb{R}) \;\;\;\; \left\langle a V + b W, Z \right\rangle_{P} = a \left\langle V, Z \right\rangle_{P} + b \left\langle W, Z \right\rangle_{P} \;\;\;\;\;\;\; \quad {\rm liniowo\acute{s}\acute{c}} \; \\ & &\left\langle V, W \right\rangle_{P} = \left\langle W, V \right\rangle_{P} \;\;\;\;\;\;\;\;\;\; \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad {\rm symetryczno\acute{s}\acute{c}} \; \\ & & {\rm Je\acute{s}li} \;\;\;\; V \neq 0 \, , \;\;\; {\rm wtedy} \;\;\; \left\langle V, V \right\rangle_{P} > 0 \;\;\;\;\;\;\;\; \;\;\quad \quad\quad {\rm dodatnia \;okre\acute{s}lono\acute{s}\acute{c}} \; .\end{aligned}$$ Pierwsze dwie w³asnoœci oznaczaj¹, ¿e $\left\langle , \right\rangle_{P}$ jest form¹ dwuliniow¹. Odwzorowanie $g:{\cal S} \ni P \rightarrow \left\langle , \right\rangle_{P} $ jest na ${\cal S}$ polem wektorowym (kowariantnym) rangi 2. Nazywamy je [*metryk¹ Riemanna na ${\cal S}$, a przestrzeñ ${\cal S}$, z tak okreœlon¹ metryk¹ $g$, nazywamy przestrzeni¹ Riemannowsk¹*]{} $({\cal S}, \, g)$.\ Gdy dla ka¿dego $P \in {\cal S}$ wspó³rzêdne $V^{i}$, $i=1,2,...,d$ dowolnego wektora $V \in T_{P}$ s¹ $C^{\infty}$ (tzn. s¹ analityczne) wzglêdem pewnego uk³adu wspó³rzêdnych $\Xi$, wtedy metryka Riemanna jest $C^{\infty}$. W skrypcie rozwa¿amy tylko przypadek $C^{\infty}$.\ Maj¹c metrykê $g$ mo¿emy zdefiniowaæ d³ugoœæ wektora stycznego $V$ przez ni¹ indukowan¹: $$\begin{aligned} \label{dlugosc z metryki Riemanna} \left\| V \right\| = \sqrt{\left\langle V, V \right\rangle_{P}} = \sqrt{\sum_{i,j=1}^{d} g_{ij}(\Xi) V^{i} V^{j}} \; .\end{aligned}$$ [^20]: Niech $\Delta=(\delta^{i})_{i=1}^{d}$ jest innym ni¿ $\Xi=(\xi^{i})_{i=1}^{d}$ uk³adem wspó³rzêdnych (inn¹ parametryzacj¹) i niech $\tilde{\partial}_{j} \equiv \frac{\partial}{\partial \delta^{j}} = \sum_{i=1}^{d} \frac{\partial \xi^{i}}{\partial \delta^{j}}\, \partial_{i} $. Mo¿na pokazaæ, ¿e wspó³czynniki koneksji w uk³adach wspó³rzêdnych $\Delta$ oraz $\Xi$ s¹ zwi¹zane ze sob¹ nastêpuj¹co [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{nowy wspolczynnik koneksji} \tilde{\Gamma}^{t}_{rs} = \sum_{i, j, \,l=1}^{d} \left(\Gamma^{l}_{ij} \frac{\partial \xi^{i} }{\partial \delta^{r}} \frac{\partial \xi^{j} }{\partial \delta^{s}} + \frac{\partial^{2} \xi^{\,l}}{\partial \delta^{r} \partial \delta^{s}} \right) \frac{\partial \delta^{t}}{\partial \xi^{l}} \; , \;\;\;\;\;\; t,r,s=1,2,...,d \; .\end{aligned}$$ Drugi sk³adnik w (\[nowy wspolczynnik koneksji\]) zale¿y tylko od postaci transformacji wspó³rzêdnych i jest niezale¿ny od koneksji. Zatem, za wyj¹tkiem transformacji liniowej, dla której drugi sk³adnik znika, wspó³czynniki koneksji nie transformuj¹ siê przy przejœciu do nowego uk³adu wspó³rzêdnych jak wielkoœæ tensorowa. Gdyby wiêc w uk³adzie wspó³rzêdnych $\Xi=(\xi^{i})_{i=1}^{d}$ wszystkie $\Gamma^{l}_{ij} = 0$, to (za wyj¹tkiem transformacji liniowej) w nowym uk³adzie wspó³rzêdnych $\Delta=(\delta^{i})_{i=1}^{d}$ nie wszystkie wspó³czynniki $\tilde{\Gamma}^{t}_{rs}$ by³yby równie¿ równe zero.\ Jednak¿e postaæ koneksji afinicznej $\Pi$ jest w nowej parametryzacji taka sama jak (\[Pi poprzez partial oraz d theta\]), tzn. jest ona wspó³zmiennicza, co oznacza, ¿e: $$\begin{aligned} \label{Pi poprzez partial oraz d xi} \Pi_{P P'}((\tilde{\partial}_{j})_{P}) = (\tilde{\partial}_j)_{P'} - \sum_{i,\,l=1}^{d} d\delta^{i}(\tilde{\Gamma}_{ij}^{l})_{P} (\tilde{\partial}_l)_{P'} \; , \;\;\;{\rm gdzie} \;\;\;\; j=1,2,...,d \;\;\; {\rm oraz} \;\;\; d\delta^{i} = \delta^{i}(P) - \delta^{i}(P') \; .\end{aligned}$$ [^21]: W ogólnoœci, znikanie koneksji jest warunkiem wystarczaj¹cym lecz niekoniecznym [*afinicznej p³askoœci, która w bardziej fundamentalnym okreœleniu ma miejsce, gdy tensor krzywizny (Riemanna) $R$ zeruje siê na ca³ej rozmaitoœci $\cal S$*]{}, tzn. zeruj¹ siê wszystkie jego sk³adowe. Sk³adowe tensora krzywizny $R$ w bazie $(\xi^{i})_{i=1}^{d}$ maj¹ nastêpuj¹c¹ postaæ: $$\begin{aligned} \label{tensor krzywizny R} R^{l}_{ijk} \equiv R^{l}_{ijk}(\Gamma) = \partial_{i} \Gamma^{l}_{jk} - \partial_{j} \Gamma^{l}_{ik} + \sum_{s=1}^{d} \Gamma^{l}_{is} \Gamma^{s}_{jk} - \sum_{s=1}^{d} \Gamma^{l}_{js} \Gamma^{s}_{ik} \; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} .\end{aligned}$$ Gdy na rozmaitoœci $\cal S$ istnieje globalny uk³adu wspó³rzêdnych, wtedy dla koneksji, która jest symetryczna (tak jak np. $\alpha$-koneksja), znikanie tensora krzywizny poci¹ga istnienie uk³adu wspó³rzêdnych, w którym znika koneksja i oba okreœlenia afinicznej p³askoœci pokrywaj¹ siê.\ [^22]: Zlogarytmowane modele eksponentialne $\{ \ln p_{\Xi}, \; \Xi \equiv (\xi^{i})_{i=1}^{d} \in {V}_{\Xi} \subset \mathbb{R}^{d}\}$ s¹ przestrzeniami afinicznymi. Dlatego z punktu widzenia ich geometrycznej charakterystyki odgrywaj¹ tak¹ rolê jak proste i powierzchnie w $3$-wymiarowej geometrii Euklidesowej. [^23]: Funkcja $\psi(\Xi)$ okazuje siê byæ tzw. potencja³em transformacji Legendre’a pomiêdzy affinicznymi uk³adami wspó³rzêdnych modeli eksponentialnych (por. Rozdzia³y \[Potencjaly ukladow wspolrzednych\] oraz \[Estymacja w modelach fizycznych na DORC\]). [^24]: [**Koneksja metryczna**]{}: Poprzez metrykê Riemannowsk¹ $g$, której przyk³adem jest metryka Fishera-Rao, (por. Rozdzia³ \[alfa koneksja\]), mo¿na okreœliæ koneksjê $\gamma_{P P'} \rightarrow \Pi_{\gamma_{P P'}}$, która jest metryczna na ${\cal S}$. Z [*koneksj¹ metryczn¹*]{} mamy do czynienia gdy: $$\begin{aligned} \label{koneksja metryczna} g_{P'}(\Pi_{\gamma_{P P'}} V, \; \Pi_{\gamma_{P P'}} W) = g_{P}(V,\, W)\; , \;\;\;\;\; \forall\, P_{\Xi} \;{\rm i} \;P'_{\Xi} \in {\cal S}\; , \end{aligned}$$ tzn. gdy [*iloczyn wewnêtrzny jest niezmienniczy ze wzglêdu na przesuniêcie równoleg³e*]{} dla wszystkich wektorów stycznych $V$ oraz $W$ i wszystkich œcie¿ek $\gamma$ z $P$ do $P'$.\ Koneksjê metryczn¹ nazywamy [*koneksj¹ Levi-Civita*]{} (lub Riemannowsk¹), jeœli jest ona zarówno metryczna jak i symetryczna. Dla zadanego $g$ koneksja taka jest okreœlona jednoznacznie jako: $$\begin{aligned} \label{koneksja Levi-Civita} \Gamma_{ij,\,k} = \frac{1}{2} \left( \partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij} \right)\; , \;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; ,\end{aligned}$$ gdzie jej [*symetria*]{} oznacza spe³nienie warunku (\[koneksja symetryczna\]). Koneksja (\[koneksja Levi-Civita\]) spe³nia warunek (\[0 - koneksja\]).\ \ Geodezyjne koneksji Levi-Civita s¹ krzywymi o najmniejszej (mierzonej przez metrykê $g$) d³ugoœci, tzn. jej d³ugoœæ pomiêdzy punktem $P$ a $P'$ okreœlona nastêpuj¹co: $$\begin{aligned} \label{dlugosc wzdluz gamma} \left\| \gamma \right\| = \int_{P}^{P'} \left\| \frac{d \gamma}{dt} \right\| dt = \int_{P}^{P'} \sqrt{ \sum_{i,j=1}^{d} g_{ij} \frac{d \gamma^{i}}{dt} \frac{d \gamma^{j}}{dt}} \;dt \; ,\end{aligned}$$ jest najmniejsza, gdzie $\gamma^{i}(t)$ jest $i$-t¹ wspó³rzêdn¹ punktu na krzywej $\gamma(t)$, tzn. $\gamma^{i}(t) := \xi^{i}(\gamma(t))$. [^25]: Wystarczy zapisaæ (\[def dualnych nabla\]) dla wektorów bazowych $(\partial_{i})_{i=1}^{d}$, otrzymuj¹c: $$\begin{aligned} \label{def dualnych w bazach nabla} \partial_{r}\left\langle \partial_{i}, \partial_{j} \right\rangle = \left\langle \nabla_{\partial_{r}} \partial_{i}, \partial_{j} \right\rangle + \left\langle \partial_{i}, \nabla^{*}_{\partial_{r}} \partial_{j} \right\rangle\; , \;\;\;\;\;\;\;\;\;\; \forall\, P_{\Xi} \in {\cal S} \; , \end{aligned}$$ sk¹d wykorzystuj¹c (\[pochodna kowariantna i wspolczynniki koneksji\]) dla koneksji $\nabla$ oraz $\nabla^{*}$, otrzymujemy (\[war dualnosci we wspolczynnikach\]). [^26]: Mówimy wtedy, ¿e koneksja na ${\cal S}$ jest ca³kowalna. Gdy przestrzeñ ${\cal S}$ jest jedno-spójna, to warunek ten oznacza znikanie tensora krzywizny Riemanna na ${\cal S}$. [^27]: Niech $\Xi = (\xi^{i})_{i=1}^{d}$ bêdzie uk³adem wspó³rzêdnych afinicznych. Wtedy, zgodnie z (\[nowy wspolczynnik koneksji\]) wspó³czynniki koneksji w uk³adzie wspó³rzêdnych $\Theta=(\theta^{i})_{i=1}^{d}$ s¹ równe: $$\begin{aligned} \label{gdy Gamma zero w afin uk wsp} \tilde{\Gamma}^{t}_{rs} = \sum_{i, j, \,l=1}^{d} \left( \frac{\partial^{2} \xi^{\,l}}{\partial \theta^{r} \partial \theta^{s}} \right) \frac{\partial \theta^{t}}{\partial \xi^{l}} \; , \;\;\;\;\;\; t,r,s=1,2,...,d \; . \nonumber\end{aligned}$$ Za¿¹dajmy aby nowy uk³ad wspó³rzêdnych $\Theta$ by³ równie¿ afiniczny. Koniecznym i wystarczaj¹cym warunkiem spe³nienia tego ¿¹dania jest zerowanie siê wszystkich drugich pochodnych $ \frac{\partial^{2} \xi^{\,l}}{\partial \theta^{r} \partial \theta^{s}} = 0\,$, gdzie $l,r,s=1,2,...,d$. Warunek ten oznacza, ¿e pomiêdzy wspó³rzêdnymi $(\xi^{i})_{i=1}^{d}$ oraz $(\theta^{i})_{i=1}^{d}$ istnieje [*afiniczna transformacja*]{}: $$\begin{aligned} \label{transformacja affiniczna} \xi^{\,l} \longrightarrow \theta^{\,l} = \sum_{k=1}^{d} A^{l}_{\; k} \, \xi^{k} + V^{l} \; , \;\;\;\;\;\; l = 1,2,...,d \; \; , \;\;\;\;\;\;\;\;\;\; \forall\, P \in {\cal S} , \nonumber\end{aligned}$$ gdzie $A = (A^{l}_{\; k}) = (\partial \theta^{l}/\partial \xi^{k}) \in GL(d)$ jest niezale¿n¹ od wspó³rzêdnych $d \times d$ - wymiarow¹ nieosobliw¹ macierz¹ ogólnej grupy liniowych transformacji, natomiast jest $V = \sum_{l=1}^{d} V^{l} \partial_{\xi^l}$ jest sta³ym $d$-wymiarowym wektorem.\ Metryka $g$ jest tensorem kowariantnym rangi 2, jest to wiêc forma dwu-liniowa, zatem iloczyn wewnêtrzny wektorów uk³adów dualnych $\partial/\partial \xi^{s}$ oraz $\partial/\partial \theta^{r}$, mo¿na zapisaæ nastêpuj¹co: $$\begin{aligned} \label{dzialanie tensora metrycznego} \left\langle \partial/\partial \xi^{s} , \; \partial/\partial \theta^{r} \right\rangle &=& \left\langle \partial/\partial \xi^{s} , \, \sum_{l=1}^{d} \frac{\partial \xi^{l}}{\partial \theta^{r}} \frac{\partial}{\partial \xi^{l}} \right\rangle = \sum_{l=1}^{d} \frac{\partial \xi^{l}}{\partial \theta^{r}} \left\langle \partial/\partial \xi^{s} , \; \partial/\partial \xi^{l} \right\rangle \nonumber \\ &=& \sum_{l=1}^{d} (A^{-1})^{l}_{\;\,r} \, g^{\xi}_{sl} = \delta_{sr} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; s, r = 1,2,...,d \; , \;\;\;\;\;\;\; \forall\, P \in {\cal S} \, , \;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{aligned}$$ gdzie $g^{\xi}_{sl} = \left\langle \partial/\partial \xi^{s} , \; \partial/\partial \xi^{l} \right\rangle$, a ostatnia równoœæ w (\[dzialanie tensora metrycznego\]) wynika z ¿¹dania sta³oœci iloczynu wewnêtrznego $\left\langle \partial/\partial \xi^{s} , \; \partial/\partial \theta^{r} \right\rangle $ na ${\cal S}$ dla uk³adów dualnie p³askich zgodnie z (\[uklad afiniczny dla koneksji\]) oraz z nieosobliwoœci transformacji afinicznej, tak ¿e uk³ad wspó³rzêdnych $\Theta$ ma tyle samo stopni swobody co uk³ad $\Xi$. Zatem otrzymaliœmy warunek (\[stalosc il wewn partial i dual partial\]) dla uk³adów dualnie p³askich.\ Przechodz¹c zgodnie z (\[ukl wsp theta kontrawariantne\]) od wspó³rzêdnych kontrawariantnych $\theta^{r}$ do kowariantnych, $\theta_{r}$, $r=1,2,..,d$, widaæ, ¿e powy¿szy zwi¹zek (\[dzialanie tensora metrycznego\]) dla wektorów bazowych uk³adów dualnych mo¿na zapisaæ w postaci $\, \left\langle \partial/\partial \xi^{s} , \; \partial/\partial \theta_{r} \right\rangle = \delta_{s}^{r}$ podanej w [@Amari; @Nagaoka; @book].\ [**Uwaga**]{}: Przestrzeñ wektorów dualnych $(\partial/\partial \theta^{r})_{r=1}^{d}$ uk³adu, który jest afiniczny wzglêdem koneksji $\nabla^{*}$ jest izomorficzna z przestrzeni¹ 1-form $({\rm d} \theta^{r})_{r=1}^{d}$ sprzê¿on¹ liniowo do uk³adu przestrzeni wektorów $(\partial/\partial \theta^{r})_{r=1}^{d}$. Dla 1-form sprzê¿onych do wektorów bazowych z definicji zachodzi ${\rm d} \theta^{r} (\partial/\partial \theta^{s}) = \delta^{r}_{s}$, $\,s,r =1,2,...,d$. \[1-form ${\rm d} \theta^{r}$, $r =1,2,...,d$, nie nale¿y myliæ z przyrostami typu (\[rozniczka theta dla theta\]) czy jak w (\[Pi poprzez partial oraz d xi\])\]. [^28]: W zale¿noœci od kontekstu i aby nie komplikowaæ zapisu, przez $\Theta$ bêdziemy rozumieli parametr wektorowy we wspó³rzêdnych kowariantnych $(\theta_{i})_{i=1}^{d}$ b¹dŸ kontrawariantnych $(\theta^{i})_{i=1}^{d}$. Analogicznie post¹pimy dla $\Xi$. [^29]: SprawdŸmy zgodnoœæ (\[transformacja Legendrea psi w phi\]) z warunkami (\[row rozn dla psi i eta\]) oraz (\[row rozn dla phi i theta\]). Z (\[transformacja Legendrea psi w phi\]) otrzymujemy: $$\begin{aligned} \label{d phi w theta eta i d psi} d \phi = \sum_{i=1}^{d} \xi^{i} d\theta_{i} + \sum_{i=1}^{d} d\xi^{i} \theta_{i} - d\psi \; , \;\;\;\;\; \forall\, P \in {\cal S} \; .\end{aligned}$$ Korzystaj¹c z (\[row rozn dla psi i eta\]) otrzymujemy $d \phi = \sum_{i=1}^{d} d\xi^{i} \theta_{i}$ jak w (\[row rozn dla phi i theta\]). [^30]: Jeœli $\psi$ oraz $\phi$ s¹ wypuk³ymi funkcjami na wypuk³ych przestrzeniach parametrów $V_{\Xi}$ oraz $V_{\Theta}$, gdzie parametry wektorowe maj¹ postaæ $\Xi \equiv (\xi^{i})_{i=1}^{d}$ oraz $\Theta \equiv (\theta_{i})_{i=1}^{d}$, to transformacje Legendre’a mo¿na sformu³owaæ w sposób bardziej ogólny [@Amari; @Nagaoka; @book]: $$\begin{aligned} \label{transformacja Legendrea psi w phi ogolna} \phi(\Theta) = max_{(\Xi \in V_{\Xi})} \left\{ \sum_{i=1}^{d} \xi^{i} \theta_{i} - \psi(\Xi) \right\} \; , \;\;\;\;\; \forall\, P \in {\cal S} \; \end{aligned}$$ dla $\psi(\Xi)$ bêd¹cego rozwi¹zaniem równania (\[row rozn 2 rzedu dla psi i eta\]). Podobnie, powiedzmy, ¿e $\phi(\Theta)$ jest pewnym rozwi¹zaiem równania (\[row rozn 2 rzedu dla phi i theta\]). Wtedy poprzez transformacjê Legendre’a: $$\begin{aligned} \label{transformacja Legendrea phi w psi ogolna} \psi(\Xi) = max_{(\Theta \in V_{\Theta})} \left\{ \sum_{i=1}^{d} \xi^{i} \theta_{i} - \phi(\Theta) \right\} \; , \;\;\;\;\; \forall\, P \in {\cal S} \; ,\end{aligned}$$ otrzymujemy potencja³ $\psi(\Xi)$. [^31]: [**Okreœlenie wi¹zki w³óknistej**]{}: (Ró¿niczkowalna) wi¹zka w³óknista $(E, \pi, {\cal M}, F, G)$ nad ${\cal M}$ sk³ada siê z nastêpuj¹cych siedmiu elementów [@Nakahara]:\ 1. Ró¿niczkowalnej rozmaitoœci $E$ nazywanej [*przestrzeni¹ totaln¹*]{}.\ 2. Ró¿niczkowalnej rozmaitoœci ${\cal M}$ nazywanej [*przestrzeni¹ bazow¹*]{}.\ 3. Ró¿niczkowalnej rozmaitoœci $F$ nazywanej [*(typowym) w³óknem*]{}.\ 4. Odwzorowania suriektywnego $\pi: E \rightarrow {\cal M}$ nazywanego [*rzutowaniem*]{}, którego odwrotny obraz $\pi^{-1}(p) = F_{p} \cong F$ nazywamy w³óknem w $p$ gdzie $p \in {\cal M}$.\ 5. Grupy Liego $G$ nazywanej [*grup¹ strukturaln¹*]{}, która dzia³a lewostronnie na $F$.\ 6. Zbioru otwartych pokryæ $\{ U_{i} \}$ rozmaitoœci ${\cal M}$ z dyfeomorfizmem $\phi_{i}: U_{i} \times F \rightarrow \pi^{-1}(U_{i})$ takim, ¿e $\pi \circ \phi_{i}(p, \,f) = p\,$, gdzie $f \in F$. Poniewa¿ $\phi_{i}^{-1}$ odwzorowuje $\pi^{-1}(U_{i})$ [*na*]{} [*iloczyn prosty*]{} $\,U_{i} \times F$ dlatego odwzorowanie $\phi_{i}$ jest nazywane [*lokaln¹ trywializacj¹*]{} .\ 7. WprowadŸmy oznaczenie $\phi_{i,p}(f) \equiv \phi_{i}(p, \, f)$. Odwzorowanie $\phi_{i,p}: F \rightarrow F_{p}$ jest dyfeomorfizmem. Rz¹damy aby na $U_{i} \bigcap U_{j}\neq \emptyset$ odwzorowanie $t_{ij} \equiv \phi_{i,p}^{-1} \circ \phi_{j,p}: F \rightarrow F$ by³o elementem grupy $G$. Wtedy $\phi_{i}$ oraz $\phi_{j}$ s¹ zwi¹zane poprzez g³adkie odwzorowanie $t_{ij}: U_{i} \bigcap U_{j} \rightarrow G$ w nastêpuj¹cy sposób: $\phi_{j}(p, \, f) = \phi_{i}(p, \, t_{ij}(p) f)$.\ Odwzorowania $t_{ij}$ s¹ nazwane [*funkcjami przejœcia*]{}.\ [**Oznaczenie**]{}: Czasami na oznaczenie wi¹zki w³óknistej $(E, \pi, {\cal M}, F, G)$ u¿ywa siê skróconego zapisu $E \stackrel{\pi}{\rightarrow} {\cal M}$ lub nawet tylko $E$.\ \ [**Przyk³ad**]{}: W powy¿szych rozwa¿aniach przestrzeni¹ bazow¹ ${\cal M}$ jest przestrzeñ statystyczna ${\cal S}$. Gdy $T_{P}$ jest przestrzeni¹ styczn¹ do ${\cal S}$ w $P$ a $T_{P}^{*}$ jest przestrzeni¹ wektorow¹ dualn¹ do $T_{P}$, wtedy typowe w³ókno $F_{P}$ mo¿e byæ np. przestrzeni¹ tensorow¹: $$\begin{aligned} \label{tensor q r} \left[T_{P}\right]^{q}_{r} \equiv \underbrace{T_{P} \otimes \cdots \otimes T_{P}}_{q-razy} \otimes \underbrace{T^{*}_{P} \otimes \cdots \otimes T^{*}_{P}}_{r-razy} \; , \end{aligned}$$ gdzie $q$ jest indeksem stopnia kontrawariantnego iloczynu tensorowego $T_{P}$ a $r$ indeksem stopnia kowariantnego iloczynu tensorowego $T_{P}^{*}$. Informacja Fishera jest szczególnym przyk³adem tensora $\left[T_{P}\right]^{0}_{2}$ na ${\cal S}$. Gdy typowe w³ókno $F$ jest przestrzeni¹ tensorow¹ a przestrzeñ wektorowa $T_{P}$ jest $d$ - wymiarowa, to grupa strukturalna $G$ jest w ogólnoœci ogóln¹ grup¹ liniowych transformacji $GL(d)$. [^32]: Odwzorowanie nazywamy symetrycznym jeœli jest symetryczne ze wzglêdu na permutacjê zmiennych. [^33]: W jêz. angielskim [*Cram$\acute{e}$r-Rao lower bound*]{} (CRLB). [^34]: Tzn. poœród estymatorów nieobci¹¿onych parametru $\mu = E_{\mu}(Y)$ i regularnych, posiada najmniejsz¹ z mo¿liwych wariancji. [^35]: Czyli tzw. macierz¹ oczekiwanego b³êdu kwadratowego. [^36]: [**Przestrzeñ stanów modelu**]{}: Mówimy, ¿e na przestrzeni zdarzeñ $\Omega$ zosta³a okreœlona funkcja $\omega \rightarrow P(\omega)$ spe³niaj¹ca warunki, $P(\omega) \ge 0$ oraz $ \sum \limits_{\omega}P(\omega) = 1$, nazywana wtedy miar¹ probabilistyczn¹. [*Zbiór wszystkich miar probabilistycznych okreœlonych na $\Omega$ tworzy przestrzeñ stanów modelu*]{}. [^37]: W przypadku statystycznej entropii fizycznej $\aleph$ mo¿e byæ np. [*liczb¹ konfiguracji okreœlonej liczy moleku³ przy zadanej energii ca³kowitej uk³adu*]{}. Wtedy $k$ jest uto¿samiane ze sta³¹ Boltzmann’a $k_{B}$. Dla uk³adu okreœlonego w przestrzeni ci¹g³ej $\mathbb{R}^{3}$ liczba konfiguracji jest nieskoñczona. Gdyby ograniczyæ siê do skoñczonej podprzestrzeni i podzieliæ j¹ na komórki o skoñczonej wielkoœci, i podobnie uczyniæ w przestrzeni pêdowej, to liczba mo¿liwych konfiguracji uk³adu by³aby skoñczona a jego entropia mog³aby byæ policzona. Jednak¿e poprawny rachunek entropii wymaga wtedy uto¿samienia konfiguracji powiedzmy $n$ cz¹stek ró¿ni¹cych siê jedynie ich permutacj¹, w ramach jednej klasy równowa¿noœci. Na fakt, ¿e w³aœciwa przestrzeñ próby ma w tym przypadku nie $\aleph$ lecz $\aleph/n!$ punktów zwróci³ uwagê Gibbs, a otrzyman¹ przestrzeñ próby nazywa siê przestrzeni¹ próby Gibbsa. Np. dla 1 $cm^{3}$ cieczy w zwyk³ych warunkach $n \approx 10^{23}$. Problem ten nie bêdzie rozwa¿any dalej w niniejszym skrypcie. [^38]: Ze wzglêdu na warunek $\sum\limits_{i=1}^{\aleph} p_{i} = 1$ maksymalizujemy funkcjê $S_{H\,war}\left(P\right) = S_{H}\left(P\right) - \lambda \left( 1 - \sum\limits_{i=1}^{\aleph} p_{i} \right) \, $, licz¹c pochodn¹ po $p_{j}$, $j=1,2,...,\aleph\,$, gdzie $\lambda$ jest czynnikiem Lagrange’a. [^39]: Fisher korzysta³ z amplitud prawdopodobieñstwa niezale¿nie od ich pojawienia siê w mechanice kwantowej. [^40]: Porównaj przejœcie od (\[inf I jeden parametr - 2 pochodna\]) do (\[inf I jeden parametr - kwadrat 1 pochodnej\]). [^41]: Lokalne w³asnoœci funkcji wiarygodnoœci $P(y|\Theta)$ opisuje obserwowana macierz informacji Fishera: $$\begin{aligned} \label{observed IF empirical statistics} \texttt{i\!F} = \left(- \frac{\partial^{2} ln P(\Theta)}{\partial \theta_{n'}^{\nu} \partial \theta_{n}^{\mu}}\right) \, ,\end{aligned}$$ która pozostaje symetryczna i dodatnio okreœlona. Jak wiemy (por. Rozdzia³ \[alfa koneksja\]) jej wartoœæ oczekiwana na ${\cal B}$ zadaje geometryczn¹ strukturê nazywan¹ metryk¹ Rao-Fishera na przestrzeni statystycznej $S$ [@Amari; @Nagaoka; @book]. [^42]: Uwzglêdnienie metryki Minkowskiego w definicji informacji Stama mo¿na zrozumiæ równie¿ jako konsekwencjê ogólnego wskazania przy liczeniu œredniej kwadratowej wielkoœci mierzalnej w dowolnej metryce Euklidesowej. W sytuacji gdy obok indeksów przestrzennych $x_{i}$, $i=1,2,3$, wystêpuje indeks czasowy $t$, nale¿y w rachunkach w czterowymiarowej czasoprzestrzeni Euklidesowej uwzglêdniæ we wspó³rzêdnych przestrzennych jednostkê urojon¹ $i$, [*³¹cznie z uwzglêdnieniem tego faktu w prawie propagacji b³êdów*]{}. W zwi¹zku z tym w oryginalnej analizie EFI Friedena-Soffera, zmienna losowa czterowektora po³o¿enia oraz jej wartoœæ oczekiwana maj¹ odpowiednio postaæ $(Y_{0} = \,c\,T, i\, \vec{Y})$ oraz $(\theta_{0} = \,c \,E_{\Theta}(T), \,\vec{\theta} = i \, E(\vec{Y}))$, co nie zmienia rezultatów analizy zawartej w skrypcie, odnosz¹cej siê do równañ mechaniki falowej oraz termodynamiki. Nie zmienia to równie¿ rezultatów analizy relatywisycznej mechaniki kwantowej [@Sakurai; @2]. Jednak¿e opis wykorzystuj¹cy metrykê Minkowskiego wydaje siê autorowi skryptu korzystniejszy z punktu widzenia zrozumienia konstrukcji niepodzielnego ekperymantalnie kana³u informacyjnego (por. Uwaga o indeksie próby). [^43]: Pojemnoœæ informacyjna: $$\begin{aligned} \label{one channel Fisher_information - I z iF} I_{n} = \int_{\cal B} d y \, P(y|\Theta) \, \sum_{\nu=0}^{3} \texttt{i\!F}_{n n \nu}^{\;\;\;\;\, \nu} \; \end{aligned}$$ jest niezmiennicza ze wzglêdu na g³adkie odwracalne odwzorowania $Y \rightarrow X$, gdzie $X$ jest now¹ zmienn¹ [@Streater]. Jest ona równie¿ niezmiennicza ze wzglêdu na odbicia przestrzenne i czasowe. [^44]: Przy wyprowadzaniu równañ generuj¹cych rozk³ad w fizyce statystycznej ${\bf y}$ mo¿e byæ np. wartoœci¹ energii $\epsilon$ uk³adu [@Frieden] i wtedy ${\bf y} \equiv \epsilon \in {\cal Y} \equiv \mathbb{R}$. [^45]: Jednak¿e przypomnijmy, ¿e w ogólnym przypadku estymacji, wymiar wektora parametrów $\Theta \equiv (\theta_{i})_{i=1}^{d}$ oraz wektora próby $y \equiv ({\bf y}_{n})_{n=1}^{N}$ mo¿e byæ inny. [^46]: Funkcjona³, w tym przypadku $S_{war}(p)$, jest liczb¹, której wartoœæ zale¿y od funkcji $p({\bf y}|\Xi)$. [^47]: Co oznacza, ¿e zak³adamy, ¿e funkcja $\ln P$ jest wypuk³a w otoczeniu prawdziwej wartoœci parametru $\Theta$. [^48]: E. Jaynes, Information Theory and Statistical Mechanics, *Phys.Rev.* **106**, 620–630 (1957). E. Jaynes, Information Theory and Statistical Mechanics. II, *Phys.Rev.* **108**, 171–190 (1957). [^49]: Metod tych nie nale¿y jednak uto¿samiaæ. Nale¿y pamiêtaæ, ¿e macierz informacyjna Fishera, wykorzystywana w EFI, oraz entropia Shannona, wykorzystywana w podejœciu Jaynes’a, s¹ ró¿nymi pojêciami. Macierz informacyjna Fishera jest Hessianem (\[hesian z S\]) entropii Shannona. [^50]: W [@Dziekuje; @informacja_2] by³a u¿yta miara $d^{N}{\bf x} \, P(\Theta)$ zamiast $d^{N}{\bf y} \, P(\Theta)$. Nie zmienia to jednak dowodu strukturalnej zasady informacyjnej, lecz poszerza jego zastosowanie na sytuacje, które [*nie posiadaj¹ niezmienniczoœci przesuniêcia*]{}, za³o¿enia nie wykorzystywano w dowodzie. [^51]: Amplitudy $q_{n}$ s¹ w przypadku rozk³adów ci¹g³ych zwi¹zane z $p_{n}$, które s¹ [*gêstoœciami*]{} prawdopodobieñstw. [^52]: O ile nie bêdzie to prowadzi³o do nieporozumieñ, bêdziemy pominijali s³owo “sk³adowa”. [^53]: Co jest s³uszne nawet na poziomie gêstoœci o ile tylko $P(\tilde{\Theta})$ $\in$ ${\cal S}$ nie posiada w $\Theta$ wy¿szych d¿etów ni¿ drugiego rzêdu. [^54]: W tym punkcie, dzia³a EFI w stosunku do estymowanego równania ruchu jak MNW w stosunku do estymowanego parametru dla rozk³adu znanego typu. [^55]: Wspólnotê mechaniki falowej z kwantow¹ ograniczymy do typów równañ ró¿niczkowych zagadnienia Sturm’a-Liouville’a oraz zasady nieoznaczonoœci Heisenberga. Dowód przeprowadzony w ramach mechaniki falowej w obszarze wspólnym dla obu teorii falowych, uznajemy co najwy¿ej jako przes³ankê jego s³usznoœci w mechanice kwantowej. [^56]: Wielowymiarowoœæ czasoprzestrzenna mo¿e, w kontekœcie obecnych rozwa¿añ, zmieniæ co najwy¿ej znak pojemnoœci informacyjnej. [^57]: Frieden co prawda wyprowadzi³ równie¿ mechanikê klasyczn¹ z mechaniki falowej, ale jedynie jako graniczny przypadek $\hbar \rightarrow 0$, a wartoœæ $N$ w tym wyprowadzeniu jest nieistotna [@Frieden]. [^58]: Wyj¹tkiem jest równanie Kleina-Gordona, w wyprowadzeniu którego odwo³ujemy siê jedynie do zasady wariacyjnej, natomiast struktura uk³adu jest narzucona z góry (por. (\[Q for N scalar\]).) [^59]: Równie¿ wyprowadzenie zasady strukturalnej (\[eq zero\]) czyni j¹ mniej fundamentaln¹ ni¿ w sformu³owaniu Friedena-Soffera [@Frieden], a tym co staje siê fundamentalne jest funkcja wiarygodnoœci $P(\Theta)$ próbki, czyli ³¹czna gêstoœæ prawdopodobieñstwa jej (niewidocznej dla badacza) realizacji. [^60]: Równoœæ: $$\begin{aligned} \label{tw Parsevala} \int_{\cal X} d^{4}{\bf x}\,\psi_{n}^{*}({\bf x})\,\psi_{n}({\bf x}) = \int_{\cal P} d^{4}{\bf p}\,\phi_{n}^{*}({\bf p})\,\phi_{n}({\bf p}) \; ,\end{aligned}$$ jest treœci¹ twierdzenia Parseval’a. [^61]: Patrz równanie (\[row KL dla swobodnego\]) w przypisie. [^62]: Zgodnie z uwag¹ uczynion¹ powy¿ej, skrót rezultatów metody EFI dla samych pól cechowania w elektrodynamice Maxwella znajduje siê w Dodatku \[Maxwell field\]. [^63]: [**Uwaga o g³ównej wi¹zce (w³óknistej)**]{}: Mówi¹c zwiêŸle, g³ówna wi¹zka to taka wi¹zka w³óknista (okreœlona w Rozdziale \[Uwaga o rozwinieciu funkcji w szereg Taylora\]), której w³ókno jest strukturaln¹ grup¹ symetrii $G$.\ Podsumujmy jednak ca³¹ dotychczasow¹ informacjê na temat g³ównej wi¹zki w³óknistej precyzyjnie: Maj¹c rozmaitoœæ ${\cal X}$ oraz grupê Liego $G$, g³ówna wi¹zka w³óknista $E({\cal X},G)\,$ jest rozmaitoœci¹ tak¹, ¿e:\ 1. Grupa $G$ dzia³a na $E$ w sposób ró¿niczkowalny i bez punktów sta³ych.\ 2. Przestrzeñ bazowa ${\cal X}=E/G$, tzn. ${\cal X}$ jest przestrzeni¹ ilorazow¹ $E$ wzglêdem $G$, oraz istnieje ró¿niczkowalne odwzorowanie (nazywane rzutowaniem) $\pi: E \rightarrow {\cal X}$.\ 3. Dla ka¿dej mapy $\{ U_{i} \}$ w atlasie dla ${\cal X}$, istnieje ró¿niczkowalne i odwracalne odwzorowanie $\phi_{j}: \pi^{-1}(U_{j}) \rightarrow U_{j} \times G $ zadane przez $E \rightarrow (\pi({\cal P}), f({\cal P}))$ w ka¿dym punkcie ${\cal P} \in E$, gdzie $f: \pi^{-1}(U_{j}) \rightarrow G$ spe³nia warunek $f(g \,{\cal P}) = g \, f({\cal P})$, dla ka¿dego $g \in G$.\ Obraz $\pi^{-1}$, czyli $U_{j} \times G$, jest nazywany w³óknem. Zatem ka¿de w³ókno niesie z sob¹ kopiê grupy strukturalnej $G$. [^64]: Jako, ¿e baza dla pozosta³ych pól nie jest jeszcze wybrana. [^65]: Macierze Diraca $\gamma^{\mu}$, $\mu=0,1,2,3\,$, wystêpuj¹ce w tzw. kowariantnej formie równania Diraca: $$\begin{aligned} \label{kowariantna postac r Diraca} (i \, \gamma^{\mu} D_{\mu} - m) \psi = 0 \; ,\end{aligned}$$ wyra¿aj¹ siê poprzez macierze Diraca $\beta$ oraz $\alpha^{l}$, $l=1,2,3$, w sposób nastêpuj¹cy: $\gamma^{0} = \beta$ oraz $\gamma^{l} = \beta \alpha^{l}\,$, ($\gamma_{\mu} = \sum_{\nu=0}^{3} \eta_{\mu \nu} \gamma^{\nu}$). Macierze Diraca $\vec{\alpha} \equiv \left(\alpha^{1},{\alpha^{2}},{\alpha^{3}}\right)$ oraz $\beta$ maj¹ postaæ: $$\begin{aligned} \label{m Diraca alfa beta} {\alpha^{l}}=\left({\begin{array}{cc} 0 & {\sigma^{l}}\\ {\sigma^{l}} & 0\end{array}}\right)\!,\;\; l=1,2,3, \quad\quad \beta = \left({\begin{array}{cc} \! {\mathbf 1}\! & \!0\!\\ \!0\! & \!{-{\mathbf 1}}\!\end{array}}\right)\end{aligned}$$ gdzie $\sigma^{l}=\sigma_{l}$, $\left(l=1,2,3\right)$ s¹ macierzami Pauliego, a ${\mathbf 1}$ jest macierz¹ jednostkow¹: $$\begin{aligned} \label{m Pauliego} \sigma^{1}=\left({\begin{array}{cc} \!0 & 1\!\\ \!1 & 0\!\end{array}}\right)\!\!,\;\; \sigma^{2}=\left({\begin{array}{cc} \!0\! & \!{-i}\!\\ \!{i}\! & \!0\!\end{array}}\right)\!\!,\;\; \sigma^{3}=\left({\begin{array}{cc} \!1\! & \!0\!\\ \!0\! & \!{-1}\!\end{array}}\right)\!\!,\;\; {\mathbf 1} = \left({\begin{array}{cc} \!1 & 0\!\\ \!0 & 1\!\end{array}}\right) \;\; .\end{aligned}$$ [^66]: Równanie Kleina-Gordona otrzymane z wariacji informacji (\[TPI every field jawna postac\]) ma postaæ ($\vec{\nabla}=(\partial/\partial x^{l})\,, \;l=1,2,3$): $$\begin{aligned} \label{row KL dla dowolnego N} - c^{2} \hbar^{2} \; ( \vec{\nabla} - \frac{ie\vec{A}}{c\hbar} ) \cdot ( \vec{\nabla} - \frac{ie\vec{A}}{c\hbar} ) \; \psi_{n} + \hbar^{2} (\frac{\partial}{\partial t} + \frac{ie\phi}{\hbar} )^{2} \; \psi_{n} + m^{2} \, c^{4} \, \psi_{n} = 0 \; .\end{aligned}$$ Dla pola swobodnego czteropotencja³ cechowania $A_{\mu} = (\phi, -\vec{A})$ jest równy zeru i wtedy otrzymujemy: $$\begin{aligned} \label{row KL dla swobodnego} -c^{2}\hbar^{2}\, \nabla^{2} \psi_{n} + \hbar^{2}\frac{{\partial^{2}}}{{\partial t^{2}}} \, \psi_{n} + m^{2}c^{4} \, \psi_{n} = 0 \; .\end{aligned}$$ [^67]: Jednak to przypuszczenie nale¿y udowodniæ. [^68]: Z drugiej strony, same równania master le¿¹ w innej czêœci klasycznej statystycznej estymacji, tzn. w obszarze dzia³ania teorii procesów stochastycznych [@Sobczyk_Luczka]. [^69]: W przypadku wyprowadzenia równania Boltzmanna w Rozdziale \[rozdz.energia\], po rozwi¹zaniu uk³adu równañ strukturalnego (\[rownanie strukt E\]) i wariacyjnego (\[rownanie wariacyjne E\]), otrzymuje siê postaæ (\[postac mikro Q dla E\]) na $\texttt{q\!F}_{n}(q_{n}({\bf x}))$. Jest to postaæ, która po wstawieniu do równania wariacyjnego (\[rownanie wariacyjne E\]) daje równanie generuj¹ce (\[falowe kappa 1\]) amplitudê $q_{n}({\bf x})$. Zasada wariacyjna (\[gggg\]) (gdzie $\texttt{q\!F}_{n}(q_{n}({\bf x}))$ wyznaczone w (\[postac mikro Q dla E\]) wystêpuje w $k$, (\[k dla Boltzmanna\])), da³aby równie¿ równanie generuj¹ce (\[falowe kappa 1\]). Równoœæ (\[S and K connection\]) nale¿y wiêc rozumieæ nastêpuj¹co. Znaj¹c $\mathbb{S}(q_{n}({\bf x}),\partial q_{n}({\bf x}))$, które wystêpuje po lewej stronie równania (\[S and K connection\]) jako $K$ ze wstawion¹, samospójnie wyznaczon¹ postaci¹ $\texttt{q\!F}_{n}(q_{n}({\bf x}))$ i wariuj¹c je dopiero wtedy ze wzglêdu na $q_{n}({\bf x})$, otrzymujemy to samo równanie generuj¹ce co rozwi¹zanie obu zasad strukturalnej i wariacyjnej jednoczeœnie. [^70]: Natomiast w ogólnoœci, nie mo¿na przyrównaæ samej zasady wariacyjnej metody EFI daj¹cej równanie Eulera-Lagrange’a (dla amplitudy $q_{n}({\bf x})$), w którym mo¿e byæ uwik³ana obserwowana informacja strukturalna $\texttt{q\!F}_{n}(q_{n}({\bf x}))$, do zasady najmniejszego dzia³ania $\mathbb{S}$, która daje koñcow¹ postaæ równania Eulera-Lagrange’a dla $q_{n}({\bf x})$ bez ¿adnego, bezpoœredniego œladu postaci $\texttt{q\!F}_{n}(q_{n}({\bf x}))$. [^71]: W zgodzie z ogóln¹ w³asnoœci¹ kontrakcji indeksu Minkowskiego dla czterowektora $x~\equiv~(x_{\nu})_{\nu=0}^{3}$ $=~(x_{0},x_{1},x_{2},x_{3})$: $$\sum_{\nu \mu = 0}^{3} \eta_{\nu\mu}\frac{df}{dx_{\nu}}\frac{df}{dx_{\mu}}=\left(\frac{df}{dx_{0}}\right)^{2}-\sum_{k=1}^{3} \left(\frac{df}{dx_{k}}\right)^{2} \; .$$ W [@Frieden] metryka ma postaæ Euklidesow¹, a zmienne maj¹ urojon¹ wspó³rzêdn¹ przestrzenn¹ $\left(x_{0}, i \vec{x} \right)$, por. Rozdzia³ \[Poj inform zmiennej los poloz\]. [^72]: Jak w Rozdziale \[The kinematical form of the Fisher information\], indeks $n$ przy wspó³rzêdnej pominiêto, korzystaj¹c z za³o¿enia, ¿e rozproszenie zmiennej nie zale¿y od punktu pomiarowego próby. [^73]: OdpowiedŸ na pytanie, czy rozwa¿ania termodynamiczne s¹ przyczyn¹ metryki Minkowskiego niezbêdnej w relatywistycznej teorii pola, wykracza poza obszar skryptu. Niemniej autor skryptu uwa¿a, ¿e tak siê istotnie sprawy maj¹, tzn. ¿e przestrzeñ Euklidesowa z transformacj¹ Galileusza s¹ pierwotne wobec przestrzeni Minkowskiego z transformacj¹ Lorentza. St¹d podejœcie efektywnej teorii pola Logunova [@Denisov-Logunov] do teorii grawitacji jest bli¿sze teorii pomiaru fizycznego Friedena-Soffera (któr¹ jest EFI). Nieco wiêcej na ten temat mo¿na znaleŸæ w Dodatku \[general relativity case\]. [^74]: Gdyby nie wyca³kowaæ w (\[postac I po calk czesci\]) $I$ przez czêœci, wtedy z (\[obserwowana zas strukt z P i z kappa\]) i po skorzystaniu z postaci kinematycznej pojemnoœci, zasada strukturalna na poziomie obserwowanym mia³aby postaæ: $$\begin{aligned} \label{mikroskowowa dla E} q_{n}^{' \,2}({\bf x}_{{\epsilon}}) + \frac{\kappa}{4} q_{n}^{2}({\bf x}_{{\epsilon}}) \,\texttt{q\!F}_{n}(q_{n}({\bf x}_{{\epsilon}})) = 0 \; .\end{aligned}$$ [^75]: Omówienie metody Aoki i Yoshikawy zamieszczono na koñcu obecnego rozdzia³u. [^76]: Przyjmuj¹c, ¿e $r>>1$, otrzymujemy, w podanych granicach, nastêpuj¹cy rozk³ad prawdopodobieñstwa produkcyjnoœci pracownika: $$\begin{aligned} \label{rozklad koncowy A dysktretny przybl} P\left( i \right) \approx (\frac{1}{r} + \frac{1}{2 r^2}) \, \left(e^{- \frac{i}{r}} \; + \; \frac{1}{r} \right) \;\;\;\;\; {\rm dla} \quad \;\;\; i= 1,2,...\; \;\;\;{ \rm oraz} \;\;\; a_{0} > 0 \; , \;\;\; r >> 1 \, .\end{aligned}$$ [^77]: Jeœli uk³ad opisany jest funkcj¹ falow¹ $\psi(\vec{y}, t)$, wtedy $p(\vec{y}, t) = |\psi(\vec{y}, t)|^{2}$. [^78]: Lecz bior¹c pod uwagê ogólne stwierdzenie, ¿e jeœli $P_{12}$ jest ³¹cznym rozk³adem prawdopodobieñstwa pewnych dwóch zmiennych losowych “1” oraz “2”, a ${P_{1}}$ oraz ${P_{2}}$ ich rozk³adami brzegowymi, to jeœli w ogólnoœci nie s¹ one wzglêdem siebie niezale¿ne (tzn. ³¹czne prawdopodobieñstwo nie jest iloczynem brzegowych), wtedy: $$\begin{aligned} \label{subaddytywnosc informacji} I_{F}\left({P_{12}}\right) \ge I_{F} \left({P_{1}}\right) + I_{F} \left({P_{2}}\right)\equiv \tilde{C} \, \end{aligned}$$ gdzie $\tilde{C}$ jest pojemnoœci¹ informacyjn¹ z³o¿onego uk³adu. Relacja (\[subaddytywnosc informacji\]) oznacza, ¿e jeœli wystêpuj¹ jakiekolwiek korelacje miêdzy zmiennymi to, jeœli znamy wynik doœwiadczenia dla pierwszej zmiennej to maleje iloœæ informacji koniecznej do okreœlenia wyniku doœwiadczenia dla drugiej z nich, tzn. istnienie korelacji w uk³adzie zwiêksza informacjê Fishera $I_{F}$ o parametrach charakteryzuj¹cych rozk³ad uk³adu. [^79]: W przypadku braku wspólnej przestrzeni zdarzeñ $\Omega_{AB}$ dla zmiennych losowych powiedzmy $A$ oraz $B$, nie mo¿na by zdefiniowaæ ³¹cznego rozk³adu prawdopodobieñstwa $P(A,B)$ dla tych dwóch zmiennych, pomimo istnienia ich rozk³adów brzegowych $P(A)$ oraz $P(B)$, co oznacza, ¿e nie da³o by siê dokonaæ ich ³¹cznego pomiaru. Równie¿ na ogó³, pomimo istnienia ³¹cznego rozk³adu brzegowego $P(A,B)$ dla zmiennych $A$ oraz $B$, oraz rozk³adu brzegowego $P(B,C)$ dla zmiennych $B$ oraz $C$, nie istnieje rozk³ad ³¹czny $P(A,B,C)$ dla zmiennych $A$, $B$ i $C$. Zwróæmy uwagê na fakt, ¿e w dowodzie nierównoœci Bella [@Bell; @Khrennikov] przyjmuje siê za oczywisty fakt istnienia ³¹cznego rozk³adu $P(A,B,C)$. Mo¿liwoœæ taka istnieje zawsze, gdy wspólna przestrzeñ zdarzeñ $\Omega_{ABC}$ tych trzech zmiennych losowych istnieje i jest iloczynem kartezjañskim $\Omega_A \times \Omega_B \times \Omega_C$. Z drugiej strony, nierównoœci Bell’a s¹ znane w klasycznej teorii prawdopodobieñstwa od czasów Boole’a jako test, który w przypadku ich niespe³nienia œwiadczy o niemo¿liwoœci konstrukcji ³¹cznego rozk³adu prawdopodobieñstwa. Rozumowanie to mo¿na rozszerzyæ na dowoln¹ liczbê zmiennych losowych [@Khrennikov; @Accardi].\ W pe³nym opisie rzeczywistego eksperymentu EPR-Bohm’a, a nie tylko w eksperymencie typu “gedanken”, powinny wystêpowaæ obok dwóch zmiennych losowych rzutów spinów mierzonych w analizatorach “a” oraz “b”, równie¿ dwie zmienne losowe k¹towe mierzone dla tych cz¹stek w chwili ich produkcji. [^80]: Podobnie dla $S_{a}$, wychodz¹c z $P\left(S_{a}|\vartheta\right) = C' = const. $ i postêpuj¹c analogicznie jak przy przejœciu od (\[Sb\]) do (\[cpol\]), otrzymujemy: $$\begin{aligned} \label{pol dla Sa} P\left(S_{a}|\vartheta\right) = \frac{1}{2} \; .\end{aligned}$$ [^81]: Przypomnijmy sobie analizê przeprowadzon¹ w rozdziale (\[rozdz.energia\]) dla rozk³adu energii cz¹steczki gazu. Wtedy ze wzglêdu na nieograniczony zakres argumentu amplitudy, wybraliœmy rozwi¹zanie o charakterze eksponencjalnym. [^82]: W mechanice kwantowej powiedzielibyœmy, ¿e $\psi_{ab}(\vartheta)$ oznacza amplitudê prawdopodobieñstwa zdarzenia, ¿e wartoœæ k¹ta wynosi $\vartheta$, o ile pojawi³aby siê ³¹czna konfiguracja spinów $S_{ab}$. [^83]: Jest tak w ca³ej analizie EFI nie uwzglêdniaj¹cej wp³ywu urz¹dzenia pomiarowego. [^84]: Dodatkowo dla eksperymentu EPR-Bohm’a $\widetilde{\mathbf{C}} \equiv \widetilde{\mathbf{C}}_{EPR}=0$, zgodnie z (\[postac stalej Cab\]). [^85]: Zaszumienie z aparatury Sterna-Gerlacha, o którym wspomniano powy¿ej, dla uproszczenia pomijamy. [^86]: W przypadku braku separowalnoœci I oraz Q na sumy odpowiednio pojemnoœci informacyjnych oraz informacji strukturalnych, w³aœciwych dla poduk³adów. [^87]: Na przyk³ad, czêstoœci rejestracji okreœlonych rzutów spinu $S_{a}$ w eksperymencie EPR-Bohm’a. [^88]: Na przyk³ad, k¹tem $\vartheta$ w eksperymencie EPR-Bohm’a. [^89]: Wnioski w pracy [@Frieden] s¹ nastêpuj¹ce: Przekaz \[zwi¹zanej\] informacji J do [*cz¹stki obserwowanej*]{} wzbudza wartoœæ spinu $S_{a}$ do poziomu danej w próbce, zgodnie z koñcowym prawem (\[wynikEPR\]). Oznacza to, ¿e przekaz informacji sprawia, ¿e cz¹stka [*spe³nia*]{} koñcowe prawo (\[wynikEPR\]) \[opisuj¹ce zachowanie siê\] spinu. Przyczyn¹ jest jakiœ nieznany mechanizm oddzia³ywania, byæ mo¿e pewna “bogata i z³o¿ona struktura, która mo¿e odpowiadaæ na informacjê i kierowaæ zgodnie z tym, swoim w³asnym ruchem” . W ten sposób, przekaz \[zwi¹zanej\] informacji J wi¹¿e z sob¹ stany spinowe nieobserwowanej i obserwowanej cz¹stki. A zatem J (zgodnie z wymogiem) “nadaje postaæ” w³asnoœciom spinu [*nie*]{}obserwowanej cz¹stki. [^90]: Gdyby na przyk³ad chcieæ wyznaczyæ metod¹ EFI ³¹czny rozk³ad energii i pêdu cz¹stki gazu. [^91]: Dla brzegowego rozk³adu pola cechowania wyznaczonego z ³¹cznego rozk³adu wszystkich pól. [^92]: Normalizacja czteropotencja³u $A_{\nu}$ zadana przez (\[A normalization\]) do jednoœci mog³aby nie zachodziæ [@Leonhardt]. Warunkiem koniecznym dla $q_{\nu}({\bf x})$ jest, aby niezbêdne œrednie mog³y byæ wyliczone. Porównaj tekst poni¿ej (\[observed IF\]). [^93]: Otrzymanej wczeœniej jako wynik uzgodnienia rezultatu metody EFI z równaniami Maxwella. [^94]: A nie tzw. metryki s³abego pola pochodz¹cej od liniowej czêœci zaburzenia metryki. Metryka s³abego pola ma postaæ: $\bar{h}_{\nu\mu} = h_{\nu\mu} - \frac{1}{2} \eta_{\mu \nu} h$, gdzie $h=\sum_{\mu, \nu =0}^{3} \eta^{\mu \nu} h_{\mu \nu}$ oraz $h_{\mu\nu} = g_{\mu\nu} - \eta_{\mu \nu}$, dla $|h_{\mu\nu}| << 1$. [^95]: Pojemnoœæ informacyjna (\[postac I dla p po x bez n\]) dla $N=1$ i skalarnego parametru $\theta$ wynosi: $$\begin{aligned} \label{I kinetyczny_dodatek} I_{F} = \int dx \frac{1}{p_{\theta}(x)} \left(\frac{\partial p_{\theta}(x)}{\partial x}\right)^{2} \; ,\end{aligned}$$ i jest to informacja Fishera parametru $\theta$, gdzie informacjê o parametrze $\theta$ pozostawiono w indeksie rozk³adu. Interesuj¹cy zwi¹zek informacji Fishera z entropi¹ Kullbacka-Leiblera pojawia siê na skutek zmiany rozk³adu zmiennej losowej spowodowanego nie zmian¹ parametru rozk³adu, ale zmian¹ wartoœci $x$ na $x + \Delta x$. Zast¹pmy wiêc (\[I kinetyczny\_dodatek\]) sum¹ Riemanna (\[zwiazek I oraz S\]) i wprowadŸmy wielkoœæ: $$\begin{aligned} \label{delta_dodatek} \delta_{\Delta x} \equiv \frac{p_{\theta}\left({x_{k} + \Delta x}\right)}{p_{\theta}(x_{k})} - 1 \; .\end{aligned}$$ Postêpuj¹c dalej podobnie jak przy przejœciu od (\[delta\]) do (\[I porownanie z Sn\]) (tyle, ¿e teraz rozk³ady ró¿ni¹ siê z powodu zmiany wartoœci fluktuacji $x$), otrzymujemy (\[iewf\]).

--- abstract: 'Recent work has demonstrated significant anonymity vulnerabilities in Bitcoin’s networking stack. In particular, the current mechanism for broadcasting Bitcoin transactions allows third-party observers to link transactions to the IP addresses that originated them. This lays the groundwork for low-cost, large-scale deanonymization attacks. In this work, we present [[Dandelion++]{}]{}, a first-principles defense against large-scale deanonymization attacks with near-optimal information-theoretic guarantees. [[Dandelion++]{}]{} builds upon a recent proposal called [Dandelion]{} that exhibited similar goals. However, in this paper, we highlight some simplifying assumptions made in [Dandelion]{}, and show how they can lead to serious deanonymization attacks when violated. In contrast, [[Dandelion++]{}]{} defends against stronger adversaries that are allowed to disobey protocol. [[Dandelion++]{}]{}  is lightweight, scalable, and completely interoperable with the existing Bitcoin network. We evaluate it through experiments on Bitcoin’s mainnet (i.e., the live Bitcoin network) to demonstrate its interoperability and low broadcast latency overhead.' author: - Giulia Fanti - Shaileshh Bojja Venkatakrishnan - Surya Bakshi - Bradley Denby - Shruti Bhargava - Andrew Miller - Pramod Viswanath bibliography: - 'privacy.bib' title: '[[Dandelion++]{}]{}: Lightweight Cryptocurrency Networking with Formal Anonymity Guarantees' --- [^1] &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10002950.10003648.10003671&lt;/concept\_id&gt; &lt;concept\_desc&gt;Mathematics of computing Probabilistic algorithms&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10002978.10003014.10003015&lt;/concept\_id&gt; &lt;concept\_desc&gt;Security and privacy Security protocols&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; [^1]: This work was supported in part by NSF grant CIF-1705007, Input Output Hong Kong (IOHK), Jump Trading, CME Group, and the Distributed Technologies Research Foundation

--- abstract: | We introduce two classes of multivariate log skewed distributions with normal kernel: the log canonical fundamental skew-normal (log-CFUSN) and the log unified skew-normal (log-SUN). We also discuss some properties of the log-CFUSN family of distributions. These new classes of log-skewed distributions include the log-normal and multivariate log-skew normal families as particular cases. We discuss some issues related to Bayesian inference in the log-CFUSN family of distributions, mainly we focus on how to model the prior uncertainty about the skewing parameter. Based on the stochastic representation of the log-CFUSN family, we propose a data augmentation strategy for sampling from the posterior distributions. This proposed family is used to analyze the US national monthly precipitation data. We conclude that a high dimensional skewing function lead to a better model fit. [*Keywords:*]{} skewed distributions; data augmentation; bayesian inference. [*AMS 1991 subject classification:*]{} 62H05; 62F15; 62E10 --- Marina M. de Queiroz [^1], Rosangela H. Loschi [^2], Roger W. C. Silva [^3]\ Departamento de Estatística, Universidade Federal de Minas Gerais. Introduction {#sec.intro} ============ The construction of new parametric distributions has received considerable attention in recent years. This growing interest is motivated by datasets that often present strong skewness, heavy tails, bimodality and some other characteristics that are not well fitted by the usual distributions, such as the normal, Student-$t$, log-normal, exponential and many others. The main goal is to build more flexible parametric distributions with additional parameters allowing to control such characteristics. If compared to finite mixtures of distributions [see @LiLeYe07; @CaBoPe08 for instance] or nonparametric methods [for recent surveys on Bayesian nonparametric see @MuQu04; @Wa05; @Dey98], one advantage of this approach is that, in general, more parsimonious models are obtained and, as a consequence, the inference process tends to become simpler. It is not feasible to mention all developments in this area in recent years. [@ArBe02], [@Ge04] and [@Az05] review several recent works in the area and are important sources of a detailed discussion of such distributions properties. Further advances in the area can be found in [@GeLo05], [@ArAz06], [@ArBrGe06], [@ArGoSa09], [@ElGoQu09], [@ArCoGo10], [@MaGe10], [@GoElSaBo11], [@BoGoRi11], [@RoLoAr13] and many others. The seminal paper by [@Az85] is one of the main references in this topic and has inspired many other works. [@Az85] introduced the so called skew-normal (SN) family of distributions which probability density function (pdf) is $$\label{skewnormal} f(z \mid \mu,\omega,\alpha)=\frac{2}{\omega}\phi\left(\frac{z-\mu}{\omega}\right) \Phi\left(\alpha\left(\frac{z-\mu}{\omega}\right)\right),\,\,z \in \mathbb{R},$$ where $\mu\in\mathbb{R}$ and $\omega\in\mathbb{R}^+$ are the location and scale parameters, respectively, $\alpha \in\mathbb{R}$ is the skewness parameter and $\phi$ and $\Phi$ denote, respectively, the pdf and the cumulative distribution function (cdf) of the $N(0,1)$. The family in (\[skewnormal\]) extends the normal one by introducing an extra parameter to control the asymmetry of the distribution and has the normal family as a particular subclass whenever $\alpha$ equals zero. It also preserves some nice properties of the normal family. Another extension of the univariate distribution in (\[skewnormal\]) recently appeared in [@MaBo14] which introduced the so called skew-normal alpha-power distibution. The multivariate analog of the SN distribution was introduced by [@AzDa96]. In a more general setting, [@GeLo05] introduced the class of generalized multivariate skew elliptical (GSE) distributions which pdf is $$\label{fundamental} f(\mathbf{z}|Q)=2f_k(\mathbf{z})Q(\mathbf{z}),\,\,\mathbf{z}\in\RR^k,$$ where $f_k$ is the pdf of a $k$-dimensional elliptical distribution and $Q$ is a skewing function satisfying $Q(-\mathbf{z})=1-Q(\mathbf{z})$, for all $\mathbf{z}\in \RR^k$. Many of the SN distribution properties also follow to any distribution in this class. Particularly, [@GeLo05] prove that distributions of quadratic forms in the GSE family do not depend on the skewing function $Q$. Some other properties of the GSE family, such as the joint moment generating functions of linear transformations and quadratic forms of ${{\bf Z}}$ and the conditions for their independence, can be found in [@HuSuGu13]. It should be also mentioned that the multivariate SN families of distributions defined by [@AzDa96] and [@AzCa99] and the family of skew-spherical (elliptical) distributions defined in [@BrDe01] are subclasses of (\[fundamental\]). [@AzDa96]’s family of distributions is also a subclass of the fundamental SN (FUSN) class of distributions defined by [@ArGe05]. A vector $Z^*$ has a $n$-variate canonical fundamental skew-normal (CFUSN) distribution with an $n \times m$ skewness matrix ${\mbox{\boldmath $\Delta$}}$, which will be denoted by $Z^*\sim CFUSN_{n,m}({\mbox{\boldmath $\Delta$}})$, if its density is given by $$\label{CFUSN} f_{{{\bf Z}}^*}(\mathbf{z})=2^m\phi_n(\mathbf{z})\Phi_m({\mbox{\boldmath $\Delta$}}'\mathbf{z}|{\bf{I}}_m-{\mbox{\boldmath $\Delta$}}' {\mbox{\boldmath $\Delta$}}),\,\,\,\,\mathbf{z}\in \RR^n,$$ where ${\mbox{\boldmath $\Delta$}}$ is such that $||{\mbox{\boldmath $\Delta$}}\mathbf{a}|| < 1$, for all unitary vectors $\mathbf{a}\in \RR^m$, and $||^{.}||$ denotes euclidean norm. Along this paper, we denote by $\phi_n({\mathbf y}\mid{\mathbf {\mbox{\protect\boldmath $\mu$}}},{\mathbf \Sigma})$ the p.d.f. associated with the multivariate $N_n({\mathbf {\mbox{\protect\boldmath $\mu$}}},{\mathbf \Sigma})$ distribution, and by $\Phi_n({\mathbf y}\mid{\mathbf {\mbox{\protect\boldmath $\mu$}}},{\mathbf \Sigma})$ the corresponding cumulative distribution function (c.d.f.). If $\mathbf{{\mbox{\protect\boldmath $\mu$}}}={\mathbf 0}$ (respectively $\mathbf{{\mbox{\protect\boldmath $\mu$}}}={\mathbf 0}$ and ${\mathbf \Sigma} ={\bf{I}}_n$) these functions will be denoted by $\phi_n({\mathbf y}\mid{\mathbf \Sigma})$ and $\Phi_n({\mathbf y}\mid{\mathbf \Sigma})$ (respectively $\phi_n({\mathbf y})$ and $\Phi_n({\mathbf y})$). For simplicity, $\phi({y})$ and $\Phi({y})$ will be used in the univariate case. Several classes of SN distributions were defined in the literature. An unification of these families is proposed by [@ArAz06] which define the unified skew-normal family of distribution, the so-called SUN family. A random vector $Z^* \sim SUN_{n,m}({\mbox{\protect\boldmath $\eta$}},{\mbox{\boldmath $\gamma$}}, \bar{{\mbox{\protect\boldmath $\omega$}}}, {\mbox{\protect\boldmath $\Omega$}}^*)$ if its pdf is $$\label{DeSUN} f_{{{\bf Z}}^*}(\mathbf{z})=\phi_n(\mathbf{z}- {\mbox{\protect\boldmath $\eta$}}\mid {\mbox{\protect\boldmath $\Omega$}}) \frac{\Phi_m({\mbox{\boldmath $\gamma$}}+ {\mbox{\boldmath $\Delta$}}'\bar{{\mbox{\protect\boldmath $\Omega$}}}^{-1}{\mbox{\protect\boldmath $\omega$}}^{-1}(\mathbf{z}- {\mbox{\protect\boldmath $\eta$}})|{\mbox{\boldmath $\Gamma$}}- {\mbox{\boldmath $\Delta$}}'\bar{{\mbox{\protect\boldmath $\Omega$}}}^{-1} {\mbox{\boldmath $\Delta$}})} {\Phi_m({\mbox{\boldmath $\gamma$}}\mid {\mbox{\boldmath $\Gamma$}})},\,\,\,\,\mathbf{z}\in\RR^n,$$ where the vectors ${\mbox{\protect\boldmath $\eta$}}\in \RR^n$ and ${\mbox{\boldmath $\gamma$}}\in \RR^m$, $\bar{{\mbox{\protect\boldmath $\omega$}}}$ is the vector of the diagonal elements of $\omega$, $\omega$ is a diagonal matrix formed by the standard deviations of ${\mbox{\protect\boldmath $\Omega$}}= {\mbox{\protect\boldmath $\omega$}}\bar{{\mbox{\protect\boldmath $\Omega$}}} {\mbox{\protect\boldmath $\omega$}}$, $\bar{{\mbox{\protect\boldmath $\Omega$}}}$, ${\mbox{\boldmath $\Gamma$}}$ and ${\mbox{\boldmath $\Delta$}}$ are, respectively, $n \times n$, $m \times m$ and $n \times m$ matrices such that $${\mbox{\protect\boldmath $\Omega$}}^* = \left( \begin{array}{cc} {\mbox{\boldmath $\Gamma$}}& {\mbox{\boldmath $\Delta$}}' \\ {\mbox{\boldmath $\Delta$}}& \bar{{\mbox{\protect\boldmath $\Omega$}}} \\ \end{array}\right)$$ is a correlation matrix. For another unification of multivariate skewed distributions see [@AbTo13]. In limit cases, some of these distributions concentrate their probability mass in positive (or negative) values. The half-normal distribution, for instance, is obtaind from (\[skewnormal\]) by assuming $\alpha$ equal to infinite. Because of this, such family of distributions has also been considered to model data with positive support, such as income, precipitation, pollutants concentration and so on. However, such limit distributions are not flexible enough to accommodate the diversity of shapes of positive (or negative) data. In the univariate context, Gamma, exponential and log-normal distributions are commonly used to model non-negative random variables. Less conventional analysis can be done using the log-SN and log-Skew-$t$ introduced by [@AzCaKo03] or the log-power-normal distribution introduced by [@MaBo12]. In the multivariate context, however, distributions with positive support are usually intractable, with the exception of the multivariate log-normal distribution. With the above problem in mind, [@MaGe10] built the multivariate log-skew elliptical family of distributions as follows. Denote by $El_n(\mathbf{\mu},\mathbf{\Sigma},g^{(n)})$ the family of $n$-dimensional elliptical distributions (with existing pdf) with generating function $g^{(n)}(u), u\geq 0$, defining a $n$-dimensional spherical density, a location column vector $\mathbf{\mu} \in \mathbb{R}^n$, and a $n\,$x$\,n$ positive definite dispersion matrix $\mathbf{\Sigma}$. If $\mathbf{X}\sim El_n(\mathbf{\mu},\mathbf{\Sigma},g^{(n)})$, then its pdf is $ f_{n}(\mathbf{x}; \mathbf{\mu},\mathbf{\Sigma}, g^{(n)})=|\mathbf{\Sigma}|^{-\frac{1}{2}}g^{(n)}(Q_{\mathbf x}^{\mathbf{\mu}, \mathbf{\Sigma}})$, where $Q_{\mathbf x}^{\mathbf{\mu},\mathbf{\Sigma}}=(\mathbf{x}-\mathbf{\mu})'\mathbf{\Sigma}^{-1} (\mathbf{x}-\mathbf{\mu})$, $\mathbf{x}\in\mathbb{R}^n$ [@Fang90]. Consider the class of skew elliptical distributions with pdf given by $$\label{skewelliptical} f_{SEl_{n}}(\mathbf{x})=2f_{n}(\mathbf{x}; \mathbf{\mu},\mathbf{\Sigma}, g^{(n)})F(\mathbf{\alpha'} \mathbf{\omega^{-1}}(\mathbf{x}-\mathbf{\mu});g_{Q_{\mathbf{x}}^{\mathbf{\mu},\mathbf{\Sigma}}}),\,\, \mathbf{x} \in \mathbb{R}^{n},$$ where $\mathbf{\alpha} \in \mathbb{R}^n$ is a shape parameter, $\mathbf{\omega}=diag(\mathbf{\Sigma})^{1/2}$ is a $n\,$x$\,n$ scale matrix, $f_{n}(\mathbf{x}; \mathbf{\mu},\mathbf{\Sigma}, g^{(n)})$ is the pdf of a $n$-dimensional random vector of $El_{n}(\mathbf{\mu}, \mathbf{\Sigma}, g^{(n)})$ and $F(u;g_{Q_{\mathbf x}^{\mathbf{\mu},\mathbf{\Sigma}}})$ is the cdf of the $El_{1}(0,1,g_{Q_{\mathbf x}^{\mathbf{\mu},\mathbf{\Sigma}}})$ with generating function $g_{Q_{\mathbf x}^{\mathbf{\mu},\mathbf{\Sigma}}}(u)=g^{(n+1)}(u+ Q_{\mathbf x}^{\mathbf{\mu},\mathbf{\Sigma}})/g^{(n)} (Q_{\mathbf x}^{\mu,\Sigma})$. The distribution in (\[skewelliptical\]) is denoted by $SEl_n(\mathbf{\mu},\mathbf{\Sigma}, \mathbf{\alpha},g^{(n+1)})$. Consider the transformation ${\exp(\mathbf{X})}=(\exp(X_1),\dots,\exp(X_n))$, where $\mathbf{X}\sim SEl_n(\mathbf{\mu},\mathbf{\Sigma}, \mathbf{\alpha},g^{(n+1)})$. Then, $\mathbf{X}$ has log-skew elliptical distribution denoted by $\mathbf{X} \sim LSEl_n(\mathbf{\mu},\mathbf{\Sigma}, \mathbf{\alpha},g^{(n+1)})$ with pdf $$\label{logskewelliptical} f_{LSEl_{n}}(\mathbf{x})=2\left(\prod_{i=1}^{n}{x_{i}}\right)^{-1}f_{n}(\ln(\mathbf{x}); \mathbf{\mu},\mathbf{\Sigma}, g^{(n)})F(\mathbf{\alpha'} \mathbf{\omega^{-1}}(\ln(\mathbf{x})-\mathbf{\mu});g_{Q_{\mathbf{x}}^{\mathbf{\mu},\mathbf{\Sigma}}}), \ \mathbf{x}>0.$$ It is immediate that the multivariate skew-normal [@AzDa96] and skew-t [@AzCa03] distributions are special cases of (\[skewelliptical\]). Consequently, the log-skewed class of distributions in (\[logskewelliptical\]) introduced by [@MaGe10] also defines particular classes of multivariate log-SN and log-skew-$t$ distributions and has, as a special case, the multivariate log-normal family of distributions. Our main motivation to introduce new classes of multivariate log-skewed distribution are some results that recently appeared in a paper by [@SaLoAr13]. That paper focused on the parameter interpretation in the mixed logistic regression models which is done through the so called odds ratio as in the usual logistic regression model. However, by considering the random effects, the odds ratio to compare two individuals in two different clusters becomes a random variable ($OR$) that depends on the random effects related to the two clusters under comparison [@LaPeJoEn00]. Because of this, [@LaPeJoEn00] propose to interpret the odds ratio in terms of the median of its distribution in order to quantify appropriately the heterogeneity among the different clusters. If the random effects are independent and identically distributed (iid) with $SN(\xi,\sigma^2,\lambda)$ then [@SaLoAr13] prove that the odds ratio has distribution with pdf given by $$f_{OR|{\mbox{\boldmath $\beta$}},{\mbox{\protect\boldmath $\theta$}}, {{\bf x}}}(r) = \frac{4}{r}\phi(\ln{r}|\kappa_{12},2\sigma^2) \times\Phi_2\left(\frac{\delta\ln{r}}{2\sigma}{\mbox{\protect\boldmath $\epsilon$}}| \frac{\delta\kappa_{12}}{2\sigma}{\mbox{\protect\boldmath $\epsilon$}}, {\bf{I}}_2 - \frac{\delta^{2}}{2}{\mbox{\protect\boldmath $\epsilon$}}{\mbox{\protect\boldmath $\epsilon$}}' \right), \;\;r\in\RR_{+}, \label{EqLNA11}$$ where $\kappa_{12}=({{\bf x}}'_{i_1j_1}-{{\bf x}}'_{i_2j_2}){\mbox{\boldmath $\beta$}}$, ${\mbox{\protect\boldmath $\epsilon$}}= (1, -1)'$ and $\delta = \lambda(1+ \lambda^2)^{-0.5}$. Similar distributions were also obtained under independent skew-normally distributed random effects. The univariate log-skewed distribution in (\[EqLNA11\]) does not belong to the class of distributions defined by [@MaGe10], nor to that introduced by [@AzCaKo03]. Moreover, only its median was obtained by [@SaLoAr13] but no other property of it was studied. In this paper, we introduce the multivariate log-CFUSN and log-SUN family of distributions. We explore their relationship and study some properties of the log-CFUSN family of distributions. Such classes of distributions have as subclasses the multivariate log-skew-normal family introduced by [@MaGe10], the log-SN family by [@AzCaKo03] and the family of distributions given in (\[EqLNA11\]). We also discuss some issues related to Bayesian inference in this family. To illustrate its use we analyze the USA monthly precipitation data recorded from 1895 to 2007, that is available at the National Climatic Data Center (NCDC). This paper is organized as follows. In Section \[Sec2\] we define the log-CFUSN and the log-SUN families of distributions and establish some of the probabilistic properties of the log-CFUSN family of distributions. Bayesian inference in the log-CFUSN family is discussed in Section \[Sec3\]. In Section \[SecCS\] we present some data analysis using the proposed log-CFUSN family of distributions. Finally, Section \[SecCo\] finishes the paper with a discussion and our main conclusions. Log-SUN and Log-CFUSN families of distributions {#Sec2} =============================================== Under the normal theory, the log-normal family of distributions is obtained assuming the logarithimic transformation. If a random variable $Y$ is log-normally distributed it follows that the log transformation of it, that is, $X=\ln Y$, has a normal distribution. Following this idea, in this section, we formally define the log-canonical-fundamental-skew-normal (log-CFUSN) and the log-unified-skew-normal (log-SUN) families of distributions and explore some properties of the log-CFUSN such as conditional and marginal distributions, mixed moments and stochastic representations. Let ${{\bf Z}}^{*}=(Z^{*}_1, \dots, Z^{*}_n)'$ be an $n \times 1$ random vector and consider the transformations ${\exp({\mathbf{Z}}^*)}=(\exp(Z^{*}_1), \dots,\exp(Z^{*}_n))'$ and ${\ln {\mathbf{Z}}^*}=(\ln Z^{*}_1, \dots,\ln Z^{*}_n)'$. (Log-CFUSN family of distributions) \[Def1\] Let ${{\bf Z}}^{*}$ and ${{\bf Y}}$ be $n \times 1$ random vectors such that ${{\bf Z}}^{*} = \ln {{\bf Y}}$. We say that ${{\bf Y}}$ has a log-canonical-fundamental-skew-normal distribution with $ n \times m$ skewness matrix $\mathbf{\Delta}$ denoted by $\mathbf{Y} \sim LCFUSN_{n,m}(\mathbf{\Delta})$, if $\mathbf{Z^{*}} \sim CFUSN_{n,m}(\mathbf{\Delta})$ with pdf given in (\[CFUSN\]). Thus, from definition \[Def1\], we have that $\mathbf{Y}={\exp({\mathbf{Z}}^*)}$ and using some results of probability calculus, we can prove that the pdf of the log-CFUSN family of distributions with skewness matrix $\mathbf{\Delta}$ is $$\label{LCFUSN} f_{\mathbf{Y}}(\mathbf{y})=2^{m}\left(\prod_{i=1}^{n}y_{i}\right)^{-1} \phi_{n}(\ln \mathbf{y})\Phi_{m}(\mathbf{\Delta}'\ln \mathbf{y}|{\bf{I}}_{m}-\mathbf{\Delta}'\mathbf{\Delta}), \,\,\,\,\mathbf{y} \in \mathbb{R}^{n^{+}},$$ where $\mathbf{\Delta}$ is an $n \times m$ matrix such that $||\mathbf{\Delta} \textbf{a}||<1$, for all unity vectors **a** $\in \mathbb{R}^{m}$. This distribution generalizes the multivariate log-SN distribution defined by [@MaGe10] by assuming a $m$-variate skewing function. If in (\[LCFUSN\]) we take $m=1$ and assume $\mathbf{\alpha}= ({\bf{I}}_{m}-\mathbf{\Delta}'\mathbf{\Delta})^{-\frac{1}{2}}\mathbf{\Delta}'$ we obtain the family defined by [@MaGe10] which general expression is given in $(\ref{logskewelliptical})$. If $\mathbf{\Delta}$ is a matrix with all entries equal to zero we have the multivariate log-normal distribution. Another reason to study this distribution comes from results in [@SaLoAr13] summarized in the introduction. As it can be noticed, the distribution for the odds ratio given in (\[EqLNA11\]) also belongs to the log-CFUSN family of distributions whenever the individuals under comparison have the same characteristics, that is, equal vector of covariates (${{\bf x}}^{t}_{i_1j_1}={{\bf x}}^{t}_{i_2j_2}$), and the scale parameter for the distribution of the random effects is $\sigma^2 =1$. In that case, $OR \sim LCFUSN_{1,2}(\mathbf{\Delta})$ where $ \mathbf{\Delta}= \delta {\mbox{\protect\boldmath $\epsilon$}}$. Figure \[densidade\] depicts the densities of $LCFUSN_{n,m}(\mathbf{\Delta})$ for the case $n=1$ and some values of $m$ and $\mathbf{\Delta}$. To simplify the presentation let ${\bf{1}}_{n,m}$ be the matrix of ones of order $n \times m$ and denote by ${\bf{1}}_{n}$ the column vector of ones of order $n$. Clearly the distribution allocates more mass to the tails when $m$ increases. Moreover, the densities shape becomes more flexible if compared with (\[logskewelliptical\]). ![Log-CFUSN densities $LCFUSN_{1,m}(\mathbf{\Delta})$ for different values of $m$ and $\mathbf{\Delta}=0.4 \times{\bf{1}}_{m}'$ (left) and $\mathbf{\Delta}= -0.4\times{\bf{1}}_{m}'$ (right).[]{data-label="densidade"}](figure1a.eps "fig:"){width="7cm"} ![Log-CFUSN densities $LCFUSN_{1,m}(\mathbf{\Delta})$ for different values of $m$ and $\mathbf{\Delta}=0.4 \times{\bf{1}}_{m}'$ (left) and $\mathbf{\Delta}= -0.4\times{\bf{1}}_{m}'$ (right).[]{data-label="densidade"}](figure1b.eps "fig:"){width="7cm"} In order to show the effect of $m$ in the asymmetry of the distribution, Figures \[cnivelm2\] and \[cnivelm3\] show the contour plots for the log-CFUSN densities $LCFUSN_{n,m}(\mathbf{\Delta})$ whenever $m=2$ and $3$, respectively. In both cases we assume bivariate ($n=2$) log-CFUSN densities. In Figure \[cnivelm2\] the following skewness matrices of parameters $\mathbf{\Delta}$ are assumed $\mathbf{\Delta}_1= - \mathbf{\Delta}_4 = 0.3 \times {\bf{1}}_{2,2}$, $\mathbf{\Delta}_2= - \mathbf{\Delta}_5= 0.1 \times {\bf{1}}_{2,2}$ and $\mathbf{\Delta}_3= - \mathbf{\Delta}_6 =\left(\begin{array}{cc}0.4 & 0.8 \\0.3 & 0.3\end{array}\right) $. In Figure \[cnivelm3\] the skewness matrices of parameters $\mathbf{\Delta}$ are $\mathbf{\Delta}_1= - \mathbf{\Delta}_4 = 0.3 \times {\bf{1}}_{2,3}$, $\mathbf{\Delta}_2= - \mathbf{\Delta}_5= 0.2 \times {\bf{1}}_{2,3}$, $\mathbf{\Delta}_3=0.1 \times {\bf{1}}_{2,3}$ and $\mathbf{\Delta}_6=\left(\begin{array}{ccc}-0.1 & -0.3 & -0.2 \\-0.1 & -0.3 & -0.2\end{array}\right)$. It is clear that the curves in Figures \[cnivelm2\] and \[cnivelm3\] deviate from the origin when the entries of $\mathbf{\Delta}$ are positive and curves are more concentrated around the origin when these entries are negative. Similar behavior is noted in the contour curves of the $CFUSN_{n,m}({\mbox{\boldmath $\Delta$}})$ distribution in [@ArGe05]. ![Contour plots for the log-CFUSN densities with $n=2$ and $m=3$ and $\mathbf{\Delta_1}$ (top left), $\mathbf{\Delta_2}$ (top middle), $\mathbf{\Delta_3}$ (top right), $\mathbf{\Delta_4}$ (bottom left), $\mathbf{\Delta_5}$ (bottom middle), $\mathbf{\Delta_6}$ (bottom right).[]{data-label="cnivelm3"}](figure2.eps "fig:"){width="13cm"}\ ![Contour plots for the log-CFUSN densities with $n=2$ and $m=3$ and $\mathbf{\Delta_1}$ (top left), $\mathbf{\Delta_2}$ (top middle), $\mathbf{\Delta_3}$ (top right), $\mathbf{\Delta_4}$ (bottom left), $\mathbf{\Delta_5}$ (bottom middle), $\mathbf{\Delta_6}$ (bottom right).[]{data-label="cnivelm3"}](figure3.eps "fig:"){width="13cm"}\ It must be also noticed that the log-CFUSN family of distributions is a subclass of an extended class of log-skewed distributions with normal kernel which can be built similarly from the family defined by [@ArAz06]. If we consider the SUN family of distribution in (\[DeSUN\]), we can define the log-SUN family of distibution as follows. (Log-SUN family of distributions) \[DefLSUN\] Let ${{\bf Z}}^{*}$ and ${{\bf Y}}$ be $n \times 1$ random vectors such that ${{\bf Z}}^{*} = \ln {{\bf Y}}$. We say that ${{\bf Y}}$ has a log-unified-skew-normal distribution with parameters ${\mbox{\protect\boldmath $\eta$}}$, ${\mbox{\boldmath $\gamma$}}$, $\bar{{\mbox{\protect\boldmath $\omega$}}}$ and ${\mbox{\protect\boldmath $\Omega$}}^*$ as defined in (\[DeSUN\]) denoted by $\mathbf{Y} \sim LSUN_{n,m}({\mbox{\protect\boldmath $\eta$}},{\mbox{\boldmath $\gamma$}}, \bar{{\mbox{\protect\boldmath $\omega$}}}, {\mbox{\protect\boldmath $\Omega$}}^*)$, if $Z^* \sim SUN_{n,m}({\mbox{\protect\boldmath $\eta$}},{\mbox{\boldmath $\gamma$}}, \bar{{\mbox{\protect\boldmath $\omega$}}}, {\mbox{\protect\boldmath $\Omega$}}^*)$ with pdf given in (\[DeSUN\]). It follows, as a consequence of Definition \[DefLSUN\], that the pdf of ${{\bf Y}}$ is given by $$\label{LogSUN} f_{{{\bf Y}}}(\mathbf{y})= \left(\prod_{i=1}^{n}y_{i}\right)^{-1} \phi_n(\ln{{{\bf y}}}- {\mbox{\protect\boldmath $\eta$}}\mid {\mbox{\protect\boldmath $\Omega$}}) \frac{\Phi_m( {\mbox{\boldmath $\gamma$}}+ {\mbox{\boldmath $\Delta$}}'\bar{{\mbox{\protect\boldmath $\Omega$}}}^{-1}{\mbox{\protect\boldmath $\omega$}}^{-1}(\ln{{{\bf y}}}- {\mbox{\protect\boldmath $\eta$}})|{\mbox{\boldmath $\Gamma$}}- {\mbox{\boldmath $\Delta$}}'\bar{{\mbox{\protect\boldmath $\Omega$}}}^{-1} {\mbox{\boldmath $\Delta$}})} {\Phi_m({\mbox{\boldmath $\gamma$}}\mid {\mbox{\boldmath $\Gamma$}})},\,\,\,\,$$ for $\mathbf{y}\in\RR^n_{+}.$ Particularly, if $\mathbf{Y} \sim LSUN_{n,m}(\bf{0},\bf{0}, {\bf{1}}_n, {\mbox{\protect\boldmath $\Omega$}}^*)$, where ${\bf{1}}_n$ is the column vector of ones of order $n$ and $ {\mbox{\protect\boldmath $\Omega$}}^* = \left( \begin{array}{cc} {\mbox{\protect\boldmath $I$}}_m & {\mbox{\boldmath $\Delta$}}' \\ {\mbox{\boldmath $\Delta$}}& {\mbox{\protect\boldmath $I$}}_n \\ \end{array}\right), $ it follows that $\mathbf{Y} \sim LCFUSN_{n,m}(\mathbf{\Delta})$ with pdf given in (\[LCFUSN\]). Some properties of the Log-CFUSN family of distributions -------------------------------------------------------- We now present several properties of the log-CFUSN family of distributions, among them are the mixed moments, the cdf and, marginal and conditional distributions. We also establish conditions for independence in the log-CFUSN family of distributions. Proposition \[propcfusncdf\] provides the cdf for this family. \[propcfusncdf\] If $\mathbf{Y} \sim LCFUSN_{n,m}(\mathbf{\Delta})$, then its cdf is given by $$\label{fdalcfusn} F_{\mathbf{Y}}(\mathbf{y})=2^{m}\Phi_{n+m}((\ln \mathbf{y'},\mathbf{0}')'|\mathbf{\Omega}), \;\;\mathbf{y} \in \mathbb{R}^{n^{+}}$$ where $\mathbf{\Omega}=\left( \begin{array}{cc} \mathbf{I}_{n} & -\mathbf{\Delta} \\ -\mathbf{\Delta}' & \mathbf{I}_{m} \\ \end{array}\right).$ The proof of Proposition \[propcfusncdf\] follows from Proposition 2.1 in [@ArGe05] by noticing that $P(\mathbf{Y}\leq \mathbf{y})=P(\exp({\mathbf{Z}^{*})} \leq \mathbf{y})=P(\mathbf{Z}^{*} <\ln \mathbf{y})= F_{\mathbf{Z}^{*}}(\ln \mathbf{y})$. The mixed moments of a random vector $\mathbf{Y}\sim LCFUSN_{n,m}(\mathbf{\Delta})$ can be expressed in terms of the moment generating function of a $CFUSN_{n,m}(\mathbf{\Delta})$ distribution. This can be seen in the following proposition. \[momentoslcfusn\] If $ \mathbf{Y} \sim LCFUSN_{n,m}(\mathbf{\Delta})$ and $\mathbf{t}=(t_{1}, ..., t_{d})'$, $t_{i} \in \mathbb{N}$, then the mixed moments of $ \mathbf{Y}$ are given by $$\label{mmlcfusn} E(\prod_{i=1}^{n}{Y_{i}}^{t_{i}})=2^{m}e^{(1/2)\mathbf{t}'\mathbf{t}}\Phi_{m}(\mathbf{\Delta}'\mathbf{t}).$$ The proof of Proposition \[momentoslcfusn\] follows by noticing that $E(\prod_{i=1}^{n}{Y_{i}}^{t_{i}})$ $=$ $E(\prod_{i=1}^{n}e^{{t_{i}}\ln{Y_{i}}})$ $=$ $E(e^{\sum_{i=1}^{n}{t_{i}}\ln{Y_{i}}})$ $= $ $E(e^{\mathbf{t} \ln \mathbf{Y}})$ $=$ $M_{\ln \mathbf{Y}}(\mathbf{t})$. As $ \mathbf{Y} \sim LCFUSN_{n,m}(\mathbf{\Delta})$ , we have $ \ln \mathbf{Y} \sim CFUSN_{n,m}(\mathbf{\Delta})$. The result follows from Proposition 2.3 in [@ArGe05]. Considering the result in $(\ref{mmlcfusn})$, we can calculate the moments of a random vector with distribution $LCFUSN_{n,m}(\mathbf{\Delta})$. For example, if we consider $\mathbf{Y} \sim LCFUSN_{1,m}(\mathbf{\Delta})$, we have that $$\begin{aligned} E(Y) &=& 2^{m}e^{1/2}\Phi_{m}({\mbox{\boldmath $\Delta$}})\nonumber\\ \nonumber E(Y^{2}) &=& 2^{m}e^{2}\Phi_{m}(2{\mbox{\boldmath $\Delta$}}) \\ \nonumber E(Y^{3}) &=& 2^{m}e^{9/2}\Phi_{m}(3{\mbox{\boldmath $\Delta$}}) \\ \nonumber E(Y^{4}) &=& 2^{m}e^{8}\Phi_{m}(4{\mbox{\boldmath $\Delta$}}) \nonumber.\end{aligned}$$ Considering these results it can be proved that the coefficient of asymmetry and kurtosis of $Y\sim LCFUSN_{1,m}({\mbox{\boldmath $\Delta$}})$ are given, respectively, by $$\label{asymmetrylcfusn} \gamma_{Y}= \frac{e^{3}\Phi_{m}(3{\mbox{\boldmath $\Delta$}})-2^{m}\Phi_{m}({\mbox{\boldmath $\Delta$}})(3e\Phi_{m}(2{\mbox{\boldmath $\Delta$}})-2\Phi_{m}({\mbox{\boldmath $\Delta$}}))}{2^{\frac{m}{2}}(e \Phi_{m}(2{\mbox{\boldmath $\Delta$}})-2^{m}\Phi^{2}_{m}({\mbox{\boldmath $\Delta$}}))^{3/2}},$$ and $$\label{kurtosislcfusn} \kappa_{Y}= \frac{e^{6}\Phi_{m}(4{\mbox{\boldmath $\Delta$}})-2^{m}(4e^{3}\Phi_{m}({\mbox{\boldmath $\Delta$}})\Phi_{m}(3{\mbox{\boldmath $\Delta$}})-3.2^{m+1}e\Phi^{2}_{m}({\mbox{\boldmath $\Delta$}})\Phi_{m}(2{\mbox{\boldmath $\Delta$}})+3.2^{2m}\Phi^{4}_{m}({\mbox{\boldmath $\Delta$}}))}{2^{m}(e^{2}\Phi^{2}_{m}(2{\mbox{\boldmath $\Delta$}})-2^{m+1}e\Phi_{m}(2{\mbox{\boldmath $\Delta$}})\Phi^{2}_{m}({\mbox{\boldmath $\Delta$}})+2^{2m}\Phi^{4}_{m}({\mbox{\boldmath $\Delta$}}))}.$$ Consequently, if $m=1$ and $\Delta$ is a matrix with all entries equal to zero, that is, if $Y \sim LN(0,1)$ then $\gamma_{Y}= (2+e)\sqrt{e-1}$ and $\kappa_{Y}=e^{4}+2e^{3}+3e^{2}-3$. Figure \[asykurt\] depicts the asymmetry coefficient and kurtosis for the $LCFUSN_{1,1}({\mbox{\boldmath $\Delta$}})$ distribution. Observe that ${\mbox{\boldmath $\Delta$}}=0$ corresponds to the log normal case. It is clear, at least in the case $n=m=1$, that asymmetry and kurtosis can change significantly depending on the choice of ${\mbox{\boldmath $\Delta$}}$. ![Asymmetry (left) and Kurtosis (right) for the $LCFUSN_{1,1}({\mbox{\boldmath $\Delta$}})$ distribution.[]{data-label="asykurt"}](figure4.eps "fig:"){width="12cm"}\ Table \[kurtsym\] displays the asymmetry and kurtosis coefficients of the $LCFUSN_{1,m}({\mbox{\boldmath $\Delta$}})$ as a function of $m$ and it suggests a monotonic decreasing behavior of these quantities as $m$ increases. Although the behavior of these coefficients depends on ${\mbox{\boldmath $\Delta$}}$, particularly, for ${\mbox{\boldmath $\Delta$}}=0.4\times{\bf{1}}_{m}'$ and ${\mbox{\boldmath $\Delta$}}=-0.4\times{\bf{1}}_{m}'$ the asymmetry and kurtosis coefficients of the $LCFUSN_{1,m}(\Delta)$ are both smaller than those obtained for the $LN(0,1)$ for all $m$ considered in the study. ----- -- ---------- ----------- -- ---------- ------------ -- $m$ Kurtosis Asymmetry Kurtosis Asymmetry. 1 $92.84$ $5.64$ $74.39$ $5.20$ 2 $76.30$ $5.16$ $48.38$ $4.33$ 3 $63.39$ $4.73$ $31.12$ $3.55$ 4 $53.42$ $4.36$ $19.52$ $2.84$ 5 $45.91$ $4.05$ $11.59$ $2.14$ ----- -- ---------- ----------- -- ---------- ------------ -- : Kurtosis and asymmetry for the $LCFUSN_{1,m}({\mbox{\boldmath $\Delta$}})$. []{data-label="kurtsym"} Similar to what is observed for the CFUSN family of distributions, the log-CFUSN is closed under marginalization but not under conditioning. The next result establishes that the $LCFUSN_{n,m}(\mathbf{\Delta})$ distribution is closed under marginalization. The proof of this result will be omitted. It follows immediately from Proposition 2.6 in [@ArGe05] and Definition \[Def1\]. \[marginallcfusn\] Let $\mathbf{Y}\sim LCFUSN_{n,m}(\mathbf{\Delta})$ and consider the partitions $\mathbf{Y}=\left( \mathbf{Y}_{1}', \mathbf{Y}_{2}' \right)'$ and $\mathbf{\Delta}=\left(\mathbf{\Delta}_{1}', \mathbf{\Delta}_{2}' \right)'$, where $\mathbf{Y}_{i}$ and $\mathbf{\Delta}_{i}$ has dimensions $n_{i} \times 1$ and $n_{i}\times m$, respectively, and $n_{1}+n_{2}=n$. Then, for $i=1,2$, $\mathbf{Y}_{i}\sim LCFUSN_{n_{i},m}(\mathbf{\Delta}_{i})$ with pdf given by $$\label{fdplcfusnmarg} f_{\mathbf{Y}_{i}}(\mathbf{y}_{i})=2^{m}\left(\prod_{j=1}^{n_{i}}y_{j}\right)^{-1}\phi_{n_{i}}(\ln \mathbf{y}_{i})\Phi_{m}(\mathbf{\Delta}'_{i}\ln \mathbf{y}_{i}|\mathbf{I}_{m}-\mathbf{\Delta}'_{i}\mathbf{\Delta}_{i}), \mathbf{y}_{i} \in \mathbb{R}^{n_{i}^{+}}.$$ It is also possible to derive conditions for independence under the log-CFUSN family of distributions by assuming some constraints on the partitions defined in Proposition \[marginallcfusn\]. \[indlcfsun\] Let $\mathbf{Y} \sim LCFUSN_{n,m}(\mathbf{\Delta})$ and consider the partitions $\mathbf{Y}=\left( \mathbf{Y}_{1}', \mathbf{Y}_{2}' \right)'$ and $\mathbf{\Delta}=\left(\mathbf{\Delta}_{1}', \mathbf{\Delta}_{2}' \right)'$, where $\mathbf{Y}_{i}$ and $\mathbf{\Delta}_{i}$ has dimensions $n_{i} \times 1$ and $n_{i}\times m$, respectively, and $n_{1}+n_{2}=n$. Let ${\mbox{\boldmath $\Delta$}}_i=({\mbox{\boldmath $\Delta$}}_{i,1},{\mbox{\boldmath $\Delta$}}_{i,2})$, where ${\mbox{\boldmath $\Delta$}}_{i,j}$ has dimension $n_i \times m_j$, $j=1,2$, and $m_1+m_2=m$, $m>1$. Then, under each of the conditions below on the shape matrix $\mathbf{\Delta}$, the random vectors $\mathbf{Y}_{1}$ and $\mathbf{Y}_{2}$ are independent - $\mathbf{\Delta}_{12}=\mathbf{\Delta}_{21}=\mathbf{0}$ and, in this case, $\mathbf{Y}_{i}\sim LCFUSN_{n_{i},m_{i}}(\mathbf{\Delta}_{ii}), i=1,2$;\ - $\mathbf{\Delta}_{ii}=\mathbf{0}, i=1,2$ and, in this case, $\mathbf{Y}_{1}\sim LCFUSN_{n_{1},m_{2}}(\mathbf{\Delta}_{12})$ e $\mathbf{Y}_{2}\sim LCFUSN_{n_{2},m_{1}}(\mathbf{\Delta}_{21})$. The proof of Proposition \[indlcfsun\] is straightforward from Proposition 2.7 in [@ArGe05] and thus is omitted. We now obtain the conditional distributions under the $LCFUSN_{n,m}({\mbox{\boldmath $\Delta$}})$ family. \[lcfusncond\] Let $\mathbf{Y}\sim LCFUSN_{n,m}(\mathbf{\Delta})$ and consider the partitions $\mathbf{Y}=\left( \mathbf{Y}_{1}', \mathbf{Y}_{2}' \right)'$ and $\mathbf{\Delta}=\left(\mathbf{\Delta}_{1}', \mathbf{\Delta}_{2}' \right)'$, where $\mathbf{Y}_{i}$ and $\mathbf{\Delta}_{i}$ has dimensions $n_{i} \times 1$ and $n_{i}\times m$, respectively, and $n_{1}+n_{2}=n$. Then, the conditional pdf of $\mathbf{Y}_{1}$ given $\mathbf{Y}_{2}=\mathbf{y}_{2}$, $\mathbf{y}_{2}\in \mathbb{R}^{n_{2}^{+}}$ is given by $$\label{lcfusncondeq} f_{\mathbf{Y}_{1}|\mathbf{Y}_{2}=\mathbf{y}_{2}}(\mathbf{y}_{1})=\left(\prod_{i=j}^{n_{1}}y_{j}\right)^{-1}\phi_{n_{1}}(\ln \mathbf{y}_{1}) \frac{\Phi_{m}(\mathbf{\Delta}_{1}'\ln \mathbf{y}_{1}|-\mathbf{\Delta}'_{2}\ln \mathbf{y}_{2},\mathbf{I}_{m}-\mathbf{\Delta}'\mathbf{\Delta})}{\Phi_{m}(\mathbf{\Delta}'_{2}\ln \mathbf{y}_{2}|\mathbf{I}_{m}-\mathbf{\Delta}'_{2}\mathbf{\Delta}_{2})}, \;\;\mathbf{y}_{1} \in \mathbb{R}^{n_{1}^{+}}.$$ The proof follows from results of probability calculus and by noticing that, given $\mathbf{y}_{2} \in \mathbb{R}^{n_{2}^{+}}$, we have that $\Phi_{m}(\mathbf{\Delta}'\ln{{{\bf y}}}|\mathbf{I}_{m} - \mathbf{\Delta}'\mathbf{\Delta})=$ $\Phi_{m}(\mathbf{\Delta}_{1}'\ln{{{\bf y}}}_{1}+\mathbf{\Delta}_{2}'\ln{{{\bf y}}}_{2}|\mathbf{I}_{m} - \mathbf{\Delta}'\mathbf{\Delta})=\Phi_{m}(\mathbf{\Delta}_{1}'\ln{{{\bf y}}}_{1}|-\mathbf{\Delta}_{2}'\ln{{{\bf y}}}_{2}, \mathbf{I}_{m} - \mathbf{\Delta}'\mathbf{\Delta})$. Notice that the log-CFUSN family of distribution per se is not closed under conditioning. However, if considered as a particular subclass of the log-SUN family of distribution, we notice from (\[lcfusncondeq\]) and (\[LogSUN\]) that ${\mathbf{Y}}_{1} \mid {\mathbf{Y}}_{2}={\mathbf{y}}_{2} \sim LSUN_{n,m}({\bf{0}},{\mbox{\boldmath $\Delta$}}_{2}'\ln {\mathbf{y}}_{2}, {\bf{1}}_{n_1}, {\mbox{\protect\boldmath $\Omega$}}^*)$, where $ {\mbox{\protect\boldmath $\Omega$}}^* = \left( \begin{array}{cc} {\mbox{\protect\boldmath $I$}}_m - {\mbox{\boldmath $\Delta$}}_2'{\mbox{\boldmath $\Delta$}}_2 & {\mbox{\boldmath $\Delta$}}_1' \\ {\mbox{\boldmath $\Delta$}}_1 & {\mbox{\protect\boldmath $I$}}_{n_1} \\ \end{array}\right). $ A location-scale extension of the log-CFUSN distribution -------------------------------------------------------- More flexible class of distributions are obtained if we are able to include on it location and scale parameters. Usually, this is done considering a linear transformation of a variable with the standard distribution. Assuming this principle, we introduce the location-scale extension of the $LCFUSN_{n,m}$ distribution as follows. Assume that $\mathbf{X}\sim CFUSN_{n,m}(\mathbf{\Delta})$ and define the linear transformation $\mathbf{W}={{\mbox{\protect\boldmath $\mu$}}}+ \mathbf{\Sigma}^{\frac{1}{2}}\mathbf{X}$, where ${\mbox{\protect\boldmath $\mu$}}$ is an $n \times 1$ vector and $\mathbf{\Sigma}$ is an $n \times n$ positive definite matrix. As shown by [@ArGe05], the pdf of $\mathbf{X}$ is $$\label{fdp2} f_{\mathbf{W}}(\mathbf{w})=2^{m}|\mathbf{\Sigma}|^{-1/2}\phi_{n}(\mathbf{\Sigma}^{-1/2}(\mathbf{w}-{{\mbox{\protect\boldmath $\mu$}}})) \Phi_{m}(\mathbf{\Delta}'\mathbf{\Sigma}^{-1/2}(\mathbf{w}-{{\mbox{\protect\boldmath $\mu$}}})|\mathbf{I}_{m}-\mathbf{\Delta}'\mathbf{\Delta}), \mathbf{w} \in \mathbb{R}^{n}.$$ Let us consider the transformation $\mathbf{U}= \exp({\mathbf{W}})$. By definition, $\mathbf{U}$ has a location-scale log-CFUSN distribution denoted by $\mathbf{U} \sim LCFUSN_{n,m}({{\mbox{\protect\boldmath $\mu$}}}, \mathbf{\Sigma}, \mathbf{\Delta})$ and its pdf is $$\begin{aligned} \label{fdpescalaloc} f_{\mathbf{U}}(\mathbf{u})&=&2^{m}|\mathbf{\Sigma}|^{-1/2}\left(\prod_{j=1}^{n}u_{j}\right)^{-1} \phi_{n}(\mathbf{\Sigma}^{-1/2}(\ln \mathbf{u}-{{\mbox{\protect\boldmath $\mu$}}})) \nonumber \\ & \times &\Phi_{m}(\mathbf{\Delta}'\mathbf{\Sigma}^{-1/2} (\ln \mathbf{u}-{{\mbox{\protect\boldmath $\mu$}}})|\mathbf{I}_{m}-\mathbf{\Delta}'\mathbf{\Delta}), \mathbf{u} \in \mathbb{R}^{n^{+}}.\end{aligned}$$ It is important to note that if $\Sigma = {\rm diag}\{\sigma_1^2, \dots, \sigma_n^2\}$, that is, if we are skewing an independent $n$-variate normal distribution, the distribution in (\[fdpescalaloc\]) can be obtained from the log-SUN distribution given in (\[LogSUN\]) by assuming ${\mbox{\protect\boldmath $\eta$}}= \mu$, ${\mbox{\boldmath $\gamma$}}= \bf{0}$, $\bar{{\mbox{\protect\boldmath $\omega$}}} = (\sigma_1, \dots, \sigma_n)$ and $ {\mbox{\protect\boldmath $\Omega$}}^* = \left( \begin{array}{cc} {\mbox{\protect\boldmath $I$}}_m & {\mbox{\boldmath $\Delta$}}' \\ {\mbox{\boldmath $\Delta$}}& {\mbox{\protect\boldmath $I$}}_n \\ \end{array}\right), $ that is, we have that $\mathbf{\mathbf{U}} \sim LSUN_{n,m}({\mbox{\protect\boldmath $\mu$}},\bf{0}, \bar{{\mbox{\protect\boldmath $\omega$}}}, {\mbox{\protect\boldmath $\Omega$}}^*)$. Marginal and conditional distributions in the location-scale log-CFUSN class of distributions are not easily obtainable. However, under some particular structures for $\mathbf{\Sigma}$ we can derive such results. Let $W \sim CFUSN_{n,m}(\mathbf{\mu}, \mathbf{\Sigma}, \mathbf{\Delta})$, as defined in Expression 2.11 in [@ArGe05], and consider the partitions $$\mathbf{W}=\left( \begin{array}{c} \mathbf{W}_{1} \\ \mathbf{W}_{2} \\ \end{array} \right), \mathbf{\Delta}=\left( \begin{array}{c} \mathbf{\Delta}_{1} \\ \mathbf{\Delta}_{2} \\ \end{array} \right), {\mbox{\protect\boldmath $\mu$}}=\left( \begin{array}{c} {{\mbox{\protect\boldmath $\mu$}}}_{1} \\ {{\mbox{\protect\boldmath $\mu$}}}_{2} \\ \end{array} \right),$$ where $\mathbf{W}_{i}$, ${{\mbox{\protect\boldmath $\mu$}}}_{i}$ and $\mathbf{\Delta}_{i}$ have dimensions $n_{i}\times 1$, $n_{i}\times 1$ and $n_{i}\times m$, $i=1,2$, respectively, and $n_{1}+n_{2}=n$. Suppose also that $\mathbf{\Sigma}$ is a diagonal matrix such that $$\mathbf{\Sigma} = \left( \begin{array}{cc} \mathbf{\Sigma}_{11} & \textbf{0} \\ \textbf{0} & \mathbf{\Sigma}_{22} \\ \end{array} \right),$$ where $\mathbf{\Sigma}_{ij}$ has dimension $n_{i}\times n_{j}$. Under these conditions, it follows that $\mathbf{U}_i= \exp({\mathbf{W}_{i}}) \sim LCFUSN({{\mbox{\protect\boldmath $\mu$}}}_i, \mathbf{\Sigma}_{ii}, \mathbf{\Delta}_i)$, that is the location-scale log-CFUSN family of distributions preserves closeness under marginalization. It also follows that the conditional distribution of $\mathbf{U}_{1}|\mathbf{U}_{2}=\mathbf{u}_{2}$ is given by $$\begin{aligned} \label{condcomescala1} f_{\mathbf{U}_{1}|\mathbf{U}_{2}=\mathbf{u}_{2}}(\mathbf{u}_{1})&=& \left(\prod_{j=1}^{n_{1}}u_{j}\right)^{-1} \phi_{n_{1}}(\mathbf{\Sigma}_{11}^{-1/2}(\ln \mathbf{u}_{1}-{{\mbox{\protect\boldmath $\mu$}}}_{1})) \\ &\times& \frac{\Phi_{m}(\mathbf{\Delta}_{1}'\mathbf{\Sigma}_{11}^{-1/2} (\ln \mathbf{u}_{1}-{{\mbox{\protect\boldmath $\mu$}}}_{1})|-\mathbf{\Delta}_{2}'\mathbf{\Sigma}_{22}^{-1/2}(\ln \mathbf{u}_{2}-{{\mbox{\protect\boldmath $\mu$}}}_{2}), I_{m}-\mathbf{\Delta}'\mathbf{\Delta})} {\Phi_{m}(\mathbf{\Delta}_{2}'\mathbf{\Sigma}_{22}^{-1/2}(\ln \mathbf{u}_{2}-{{\mbox{\protect\boldmath $\mu$}}}_{2})|I_{m}- \mathbf{\Delta}_{2}'\mathbf{\Delta}_{2})}, \nonumber \end{aligned}$$ $\mathbf{u}_{1} \in \mathbb{R}^{n_{1}^{+}}$ and $\mathbf{u}_{2} \in \mathbb{R}^{n_{2}^{+}}$. Stochastic representation ------------------------- Stochastic representations of skewed distributions are useful, for instance, to generate samples from those distributions more easily. They also play a very important role in inference if we are interested in apply MCMC or EM methods. A stochastic representation of the log-CFUSN family is straightforward from the marginal stochastic representation of the CFUSN family given in [@ArGe05]. Assume that ${{\bf Z}}^{*}\sim CFUSN_{n,m}({\mbox{\boldmath $\Delta$}})$, where $||{\mbox{\boldmath $\Delta$}}'{{\bf a}}||<1$ for any unitary vector ${{\bf a}}\in \RR^n$. Let ${{\bf D}}\sim N_{m}(\bf{0}, {\mbox{\protect\boldmath $I$}}_{m})$, $\mathbf{V} \sim N_{n}(\bf{0},{\mbox{\protect\boldmath $I$}}_{n})$ where ${{\bf D}}$ and $\mathbf{V}$ are independent column random vectors of order $m$ and $n$, respectively. Denote by $|{{\bf D}}|$ the vector $(|D_{1}|,...,|D_{m}|)'$. [@ArGe05] prove that the marginal representation of ${{\bf Z}}^{*}$ is $$\label{EqSR} {{\bf Z}}^{*}\buildrel d \over = {\mbox{\boldmath $\Delta$}}|{{\bf U}}|+({\mbox{\protect\boldmath $I$}}_{n}-{\mbox{\boldmath $\Delta$}}{\mbox{\boldmath $\Delta$}}')^{1/2}\mathbf{V}.$$ If ${{\bf Y}}\sim LCFUSN_{n,m}({\mbox{\boldmath $\Delta$}})$ then its marginal representation follows as a consequence of (\[EqSR\]) by noticing that ${{\bf Y}}\buildrel d \over = \exp({{{\bf Z}}^*})$ $\buildrel d \over = \exp({\mbox{\boldmath $\Delta$}}|{{\bf D}}|) \exp(({\mbox{\protect\boldmath $I$}}_{n}-{\mbox{\boldmath $\Delta$}}{\mbox{\boldmath $\Delta$}}')^{1/2}\mathbf{V})$ $\buildrel d \over = \exp({\mbox{\boldmath $\Delta$}}|{{\bf D}}|) {{\bf T}}$, where ${{\bf T}}$ has a multivariate log-normal distribution with a null location parameter and scale matrix equal to ${\mbox{\protect\boldmath $I$}}_{n}-{\mbox{\boldmath $\Delta$}}{\mbox{\boldmath $\Delta$}}'$. Some aspects of Bayesian Inference in the LCFUSN Family {#Sec3} ======================================================= Let ${{\bf Y}}_1, \dots, {{\bf Y}}_L \mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}}\buildrel iid \over\sim LCFUSN_{n,m}({\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}})$ with pdf given in (\[fdpescalaloc\]). Define the $L \times n$ matrices ${{\bf Y}}= ({{\bf Y}}_1, \dots, {{\bf Y}}_L)'$ and $\ln{{\bf Y}}= (\ln{{\bf Y}}_1, \dots, \ln{{\bf Y}}_L)'$. Therefore, it follows that the likelihood function is given by $$\begin{aligned} \label{EqVeroG1} f({{\bf Y}}\mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}}) &=& 2^{Lm} \prod_{l=1}^{L} \prod_{j=1}^{n} Y_{lj}^{-1} \phi_{L,n}(\ln {{\bf Y}}\mid {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}', {\bf{I}}_L, {\mbox{\protect\boldmath $\Sigma$}}) \\ &\times &\Phi_{Lm} ({\bf{I}}_L \otimes{\mbox{\boldmath $\Delta$}}'{\mbox{\protect\boldmath $\Sigma$}}^{-1/2} vec(\ln {{\bf Y}}) \mid {\bf{1}}_L \otimes({\mbox{\boldmath $\Delta$}}'{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}{\mbox{\protect\boldmath $\mu$}}), {\bf{V}}^* ), \nonumber\end{aligned}$$ where ${\bf{V}}^*= {\bf{I}}_{Lm} - {\bf{I}}_L \otimes {\mbox{\boldmath $\Delta$}}'{\mbox{\boldmath $\Delta$}}$ and $\mathbf{A} \otimes \mathbf{B}$ denotes the Kronecker product of $\mathbf{A}$ and $\mathbf{B}$, $vec(\cdot)$ is the operator vec and $\phi_{L,n}(\cdot \mid M ; C, V)$ denotes the pdf of a matrix-variate normal distribution where $M$ is an $L\times n$ constant vector and $C$ and $V$ are, respectively, $L \times L$ and $n \times n$ constant matrices. Observe that the likelihood function in (\[EqVeroG1\]) defines a class of matrix-variate log-CFUSN distributions. In this work, inference is done under the Bayesian paradigm. Therefore we need to specify prior distributions for all parameters. We consider $m$ as a fixed constant and also assume some usual prior distributions for the location and scale parameters. In the following proposition, we present the posterior full conditional distributions for ${\mbox{\protect\boldmath $\mu$}}$, ${\mbox{\protect\boldmath $\Sigma$}}$ and ${\mbox{\boldmath $\Delta$}}$ whenever the prior distributions for ${\mbox{\protect\boldmath $\mu$}}$, ${\mbox{\protect\boldmath $\Sigma$}}$ are, respectively, $$\begin{aligned} \left.\begin{array}{rcl} {\mbox{\protect\boldmath $\mu$}}&\sim& N_n ({\mbox{\protect\boldmath $\mu$}}_{0}, {\mbox{\protect\boldmath $\Sigma$}}_{\mu}) \\ {\mbox{\protect\boldmath $\Sigma$}}&\sim& IW(d, D), \end{array}\right\} \label{EqPriorG}\end{aligned}$$ where ${\mbox{\protect\boldmath $\mu$}}_{0} \in \RR^n$, ${\mbox{\protect\boldmath $\Sigma$}}_{\mu}$ is an $n \times n$ symmetric, positive definite matrix, $D$ is an $n \times n$ constant matrix, $d \in \RR_+$ with $d >n$, and $IW(d, D)$ denotes the inverse-Wishart distribution with parameters $d$ and $D$. A flat prior distribution for ${\mbox{\protect\boldmath $\Sigma$}}$ is obtained by setting $d$ close to zero. Let ${{\bf Y}}_1, \dots, {{\bf Y}}_L \mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}}\buildrel iid \over\sim LCFUSN_{n,m}({\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}})$. Assume that, [*a priori*]{}, the parameters ${\mbox{\protect\boldmath $\mu$}}$, ${\mbox{\protect\boldmath $\Sigma$}}$ and ${\mbox{\boldmath $\Delta$}}$ are independent and that the prior distributions for ${\mbox{\protect\boldmath $\mu$}}$, ${\mbox{\protect\boldmath $\Sigma$}}$ are given in (\[EqPriorG\]). Suppose ${\mbox{\boldmath $\Delta$}}$ has a proper prior distribution $\pi({\mbox{\boldmath $\Delta$}})$. Then, the posterior full conditional distributions for ${\mbox{\protect\boldmath $\mu$}}$, ${\mbox{\protect\boldmath $\Sigma$}}$ and ${\mbox{\boldmath $\Delta$}}$ are given, respectively, by $$\begin{aligned} \pi({\mbox{\protect\boldmath $\mu$}}\mid {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}}, {{\bf Y}}) &\propto& \phi_n({\mbox{\protect\boldmath $\mu$}}\mid {\mbox{\protect\boldmath $\Sigma$}}^* [{\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{-1} {\mbox{\protect\boldmath $\mu$}}_{0} + ({\mbox{\protect\boldmath $\Sigma$}}^{-1}\otimes {\bf{1}}_L)' vec(\ln {{\bf Y}})] \mid {\mbox{\protect\boldmath $\Sigma$}}^* ) \\ &\times & \Phi_{mL}({\bf{I}}_L \otimes[{\bf{I}}_m - {\mbox{\boldmath $\Delta$}}' {\mbox{\boldmath $\Delta$}}]^{-1/2}{\mbox{\boldmath $\Delta$}}'{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}[vec(\ln {{\bf Y}}) - {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}])\\[2mm] \pi({\mbox{\protect\boldmath $\Sigma$}}\mid {\mbox{\protect\boldmath $\mu$}},{\mbox{\boldmath $\Delta$}}, {{\bf Y}}) &\propto& {\mathcal{I}}{\mathcal{W}}_n(d+L+1, D + [\ln{{\bf Y}}- {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}']'[\ln{{\bf Y}}- {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}'])\\ & \times& \Phi_{mL}({\bf{I}}_L \otimes[{\bf{I}}_m - {\mbox{\boldmath $\Delta$}}' {\mbox{\boldmath $\Delta$}}]^{-1/2}{\mbox{\boldmath $\Delta$}}'{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}[vec(\ln {{\bf Y}}) - {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}])\\[2mm] \pi({\mbox{\boldmath $\Delta$}}\mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {{\bf Y}}) &\propto& \pi({\mbox{\boldmath $\Delta$}})\Phi_{mL}({\bf{I}}_L \otimes[{\bf{I}}_m - {\mbox{\boldmath $\Delta$}}' {\mbox{\boldmath $\Delta$}}]^{-1/2}{\mbox{\boldmath $\Delta$}}'{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}[vec(\ln {{\bf Y}}) - {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}])\\\end{aligned}$$ where ${\mbox{\protect\boldmath $\Sigma$}}^* = [L {\mbox{\protect\boldmath $\Sigma$}}^{-1} + {\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{-1}]^{-1}$ and ${\mathcal I}{\mathcal W}_n(a, A)$ denotes the pdf of the inverse-Wishart distribution with parameters $a$ and $A$. \[ProFCDG\] The proof of Proposition \[ProFCDG\] follows by mixing (\[EqVeroG1\]), (\[EqPriorG\]) and $\pi({\mbox{\boldmath $\Delta$}})$ using the Bayes’s theorem and some well-known results of matrix theory. It is noteworthy that the posterior full conditional distribution of ${\mbox{\protect\boldmath $\mu$}}$ belongs to the SUN family of distributions. The univariate case is presented in the following corollary. Denote by $IG(\alpha, \beta)$, $\alpha>0$ and $\beta>0$, the inverse-gamma distribution with $E(\sigma^{2}) = \alpha(\beta-2)^{-1}$. Let $Y_{1},..., Y_{L}\mid \mu, \sigma, {\mbox{\boldmath $\Delta$}}\buildrel iid \over\sim LCFUSN_{1,m}(\mu,\sigma^2,{\mbox{\boldmath $\Delta$}})$ and assume that, a priori, $\mu$, $\sigma$ and ${\mbox{\boldmath $\Delta$}}$ are independent and such that $\mu \sim N(\mu_0,v)$, $\sigma^{2} \sim IG(\alpha, \beta)$, where $l \in \RR$, $v$, $\alpha$ and $\beta$ are non-negative numbers, and ${\mbox{\boldmath $\Delta$}}$ has a proper prior distribution $\pi({\mbox{\boldmath $\Delta$}})$. Then, the posterior full conditional distributions for $\mu$, $\sigma$ and ${\mbox{\boldmath $\Delta$}}$ are given, respectively, by $$\begin{aligned} f(\mu \mid {{\bf y}}, \sigma^{2}, {\mbox{\boldmath $\Delta$}}) &\propto&\phi\left(\mu \Big| \frac{v^{2} {\bf{1}}_L' \ln {{\bf y}}+\mu_0\sigma^{2}}{Lv^{2}+ \sigma^{2}}, \frac{v^{2}\sigma^{2}}{Lv^{2}+\sigma^{2}}\right) \\ & \times&\Phi_{mL}(\sigma^{-1}(\mathbf{I}_{L} \otimes {\mbox{\boldmath $\Delta$}}') (\ln {{\bf y}}- \mu {\bf{1}}_L)\mid \mathbf{I}_{mL}- \mathbf{I}_{L} \otimes{\mbox{\boldmath $\Delta$}}'{\mbox{\boldmath $\Delta$}})\\ [2mm] f(\sigma^{2} \mid {{\bf x}}, \mu, {\mbox{\boldmath $\Delta$}}) &\propto& \left(\frac{1}{\sigma^{2}}\right)^{\frac{L+2\alpha+2}{2}} \exp{\left( \frac{2\beta- (\ln {{\bf y}}- \mu {\bf{1}}_L)' (\ln {{\bf y}}- \mu {\bf{1}}_L)}{2\sigma^{2}}\right)}\\ \nonumber & \times& \Phi_{mL}(\sigma^{-1}(\mathbf{I}_{L} \otimes {\mbox{\boldmath $\Delta$}}') (\ln {{\bf y}}- \mu {\bf{1}}_L)\mid \mathbf{I}_{mL}- \mathbf{I}_{L} \otimes{\mbox{\boldmath $\Delta$}}'{\mbox{\boldmath $\Delta$}})\\ [2mm] f({\mbox{\boldmath $\Delta$}}\mid {{\bf x}}, \mu, \sigma^{2}) &\propto& \pi({\mbox{\boldmath $\Delta$}})\Phi_{mL}(\sigma^{-1}(\mathbf{I}_{L} \otimes {\mbox{\boldmath $\Delta$}}') (\ln {{\bf y}}- \mu {\bf{1}}_L)\mid \mathbf{I}_{mL}- \mathbf{I}_{L} \otimes{\mbox{\boldmath $\Delta$}}'{\mbox{\boldmath $\Delta$}}),\end{aligned}$$ where $\ln {{\bf y}}= (\ln y_1, \dots, \ln y_L)'$. \[CoBA1\] This result is a straightforward consequence of Proposition \[ProFCDG\]. It follows by observing that the likelihood function of ${{\bf y}}$ is given by $$\begin{aligned} \label{vero1lcfsun} f({{\bf y}}| \mu, \sigma^{2}, {\mbox{\boldmath $\Delta$}}) &=& 2^{Lm} (2\pi\sigma^{2})^{-L/2}\left(\prod_{i=1}^{L}y_{i}\right)^{-1} \exp\left\{-\sum_{i=1}^{L} \frac{(\ln y_{i}-\mu)^{2}}{2\sigma^{2}}\right\} \\ &\times& \prod_{i=1}^{L} \Phi_{m}({\mbox{\boldmath $\Delta$}}'\sigma^{-1}(\ln y_{i}-\mu)|\mathbf{I}_{m}-{\mbox{\boldmath $\Delta$}}'{\mbox{\boldmath $\Delta$}}),\nonumber \end{aligned}$$ and that the inverse-Wishart distribution is a generalization of the multivariate inverse-gamma distribution. Since the parameter ${\mbox{\boldmath $\Delta$}}$ is an $ n \times m$ vector with $||{\mbox{\boldmath $\Delta$}}\textbf{a}||<1$, for all unitary vectors $\textbf{a} \in \mathbb{R}^{m}$, the elicitation of a prior distribution for ${\mbox{\boldmath $\Delta$}}$ becomes a hard task. To overcome this difficulty, we can assume an alternative parametrization of the model by setting ${\mbox{\boldmath $\Delta$}}= {\mbox{\boldmath $\Lambda$}}(\mathbf{I}_{m}+{\mbox{\boldmath $\Lambda$}}'{\mbox{\boldmath $\Lambda$}})^{-1/2}$ for some $n \times m$ real matrix ${\mbox{\boldmath $\Lambda$}}$. A possible prior distribution for ${\mbox{\boldmath $\Lambda$}}$ is a multivariate normal distribution. The calculation of the full conditional distributions under these choices is similar to that presented in Proposition \[ProFCDG\] and thus will be omitted. However, we remark that the posterior full conditional distributions for ${\mbox{\protect\boldmath $\mu$}}$ and ${\mbox{\boldmath $\Lambda$}}$ belong to the SUN class of distributions and a skewed inverse-Wishart distribution is the posterior full conditional distribution for ${\mbox{\protect\boldmath $\Sigma$}}$. Consequently, by considering this class of joint prior distributions for $({\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Lambda$}})$ we have conjugacy. It is notable that we are also performing a conjugate analysis for $({\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}})$ in the cases discussed in Proposition \[ProFCDG\] and Corollary \[CoBA1\]. Another way to overcome the problem is to assume ${\mbox{\boldmath $\Delta$}}= \delta {\bf{1}}_{n,m}$ where $\delta$ is a real number belonging to the interval $(-1, 1)$. By carrying this out, the model loses some flexibility. On the other hand we obtain a more parsimonious model which is still able to accommodate different degrees of asymmetry. From now on, we consider this approach and elicit a non-informative uniform prior distribution for $\delta$. Under this more parsimonious model, the posterior full conditional distributions for all parameters follow from Proposition \[ProFCDG\] and are given by $$\begin{aligned} \pi({\mbox{\protect\boldmath $\mu$}}\mid {\mbox{\protect\boldmath $\Sigma$}}, {\mbox{\boldmath $\Delta$}}, {{\bf Y}}) &\propto& \phi_n({\mbox{\protect\boldmath $\mu$}}\mid {\mbox{\protect\boldmath $\Sigma$}}^* [{\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{-1} {\mbox{\protect\boldmath $\mu$}}_{0} + ({\mbox{\protect\boldmath $\Sigma$}}^{-1}\otimes {\bf{1}}_L)' vec(\ln {{\bf Y}})] \mid {\mbox{\protect\boldmath $\Sigma$}}^* ) \\ &\times & \Phi_{mL}(I_L \otimes[I_m - \delta^2 {\bf{1}}_{m,m}]^{-1/2} \delta{\bf{1}}_{m,n}{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}[vec(\ln {{\bf Y}}) - {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}]),\\[2mm] \pi({\mbox{\protect\boldmath $\Sigma$}}\mid {\mbox{\protect\boldmath $\mu$}},{\mbox{\boldmath $\Delta$}}, {{\bf Y}}) &\propto& {\mathcal I}{\mathcal W}_n(d+L+1, D + [\ln{{\bf Y}}- {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}']'[\ln{{\bf Y}}- {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}'])\\ & \times& \Phi_{mL}(I_L \otimes[I_m - \delta^2 {\bf{1}}_{m,m}]^{-1/2} \delta{\bf{1}}_{m,n}{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}[vec(\ln {{\bf Y}}) - {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}]),\\[2mm] \pi({\mbox{\boldmath $\Delta$}}\mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {{\bf Y}}) &\propto& \Phi_{mL}(I_L \otimes[I_m - \delta^2 {\bf{1}}_{m,m}]^{-1/2} \delta{\bf{1}}_{m,n}{\mbox{\protect\boldmath $\Sigma$}}^{-1/2}[vec(\ln {{\bf Y}}) - {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}]).\\\end{aligned}$$ A difficulty encountered in inference under this family of distributions is that, independently of the model we assume (a general ${\mbox{\boldmath $\Delta$}}$, ${\mbox{\boldmath $\Delta$}}= \delta {\bf{1}}_{n,m}$ or the reparametrization ${\mbox{\boldmath $\Lambda$}}$), the skewing function for all posterior full conditional distributions is the cdf of some $mL$-variate normal distribution. Hence the computational cost for sampling of the posterior distributions tends to become very high. Data augmentation: Simplifying the computation using the Stochastic representation ----------------------------------------------------------------------------------- A strategy that greatly facilitates Bayesian inference under complex models is the data augmentation technique. It consists of including latent variables or unobserved data into the model in order to simplify the computational procedures [@DyMe01]. In the proposed model, we accomplish this by considering the stochastic representations for the CFUSN family of distributions obtained by [@ArGe05]. By applying a logarithmic transformation to the data, we can estimate the parameters of the log-CFUSN distribution via the CFUSN distribution. Formally, if we consider the marginal stochastic representation in (\[EqSR\]), the model in (\[EqVeroG1\]) can be hierarchically represented as follows. Let ${{\bf Y}}_{i} \sim LCFUSN_{n,m}({\mbox{\protect\boldmath $\mu$}},{\mbox{\protect\boldmath $\Sigma$}},{\mbox{\boldmath $\Delta$}})$ and ${{\bf Z}}_{i}=\ln {{\bf Y}}_{i} \sim CFUSN_{n,m}({\mbox{\protect\boldmath $\mu$}},{\mbox{\protect\boldmath $\Sigma$}},{\mbox{\boldmath $\Delta$}})$. Assume also that ${\mbox{\boldmath $\Delta$}}= \delta {\bf{1}}_{n,m}$, $\delta \in (-1, 1)$. Then, it follows that $$\label{eq1} {{\bf Z}}_{i}\buildrel d \over = \delta {\mbox{\protect\boldmath $\Sigma$}}^{1/2}{\bf{1}}_{n,m}|{{\bf X}}_{i}|+[{\mbox{\protect\boldmath $\Sigma$}}({\mbox{\protect\boldmath $I$}}_{n}-\delta^2{\bf{1}}_{n,n})]^{1/2}{\mbox{\protect\boldmath $V$}}_{i} +{\mbox{\protect\boldmath $\mu$}},$$ where ${{\bf X}}_{i} \sim N_{m}(\bf{0}, {\mbox{\protect\boldmath $I$}}_{m})$, ${\mbox{\protect\boldmath $V$}}_{i} \sim N_n(\bf{0}, {\mbox{\protect\boldmath $I$}}_{n})$, ${{\bf X}}_{i}$ and ${\mbox{\protect\boldmath $V$}}_{i}$ are independent random vectors and $|{{\bf X}}_{i}|=(|X_{i1}|,...,|X_{im}|)'$. As a consequence, the model in (\[EqVeroG1\]) is equivalent to $$\begin{aligned} \label{hier} {{\bf Y}}_{i} &=& \exp {{\bf Z}}_{i} \nonumber\\ {{\bf Z}}_{i}|{{\bf X}}_{i}={{\bf x}}_{i}&\sim& N_n( {\mbox{\protect\boldmath $\mu$}}+ \delta {\mbox{\protect\boldmath $\Sigma$}}^{1/2}{\bf{1}}_{n,m} |{{\bf X}}_{i}|, {\mbox{\protect\boldmath $\Sigma$}}({\bf{I}}-\delta^2{\bf{1}}_{n,n})) \nonumber\\ {{\bf X}}_{i} &\sim& N_{m}(\bf{0}, {\mbox{\protect\boldmath $I$}}_{m}),\end{aligned}$$ where ${{\bf X}}_{i}$ is a latent (unobserved) random variable. This hierarchical representation of the model is known as data augmentation strategy and great facilitates the process of sampling from the posterior distributions. Let ${{\bf Z}}= ({{\bf Z}}_1, \dots, {{\bf Z}}_L)'$ and $|{{\bf X}}|= (|{{\bf X}}_1|, \dots, |{{\bf X}}_L|)'$. Under this hierarchical representation, the likelihood for the augmented data becomes $$f({{\bf Z}}\mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, \delta, {{\bf X}}) = \phi_{L,n}({{\bf Z}}\mid {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}' + \delta {\mbox{\protect\boldmath $\Sigma$}}^{-1/2} {\bf{1}}_{n,m}|{{\bf X}}|', {\bf{I}}_L, {\mbox{\protect\boldmath $\Sigma$}}({\bf{I}}-\delta^2{\bf{1}}_{n,n})).$$ Assume the prior distributions for ${\mbox{\protect\boldmath $\mu$}}$ and ${\mbox{\protect\boldmath $\Sigma$}}$ given in (\[EqPriorG\]) and suppose that, [*a priori*]{}, $\delta \sim {\mathcal{U}}(-1,1)$. It follows that the full conditional distributions for the parameter ${\mbox{\protect\boldmath $\mu$}}$, ${\mbox{\protect\boldmath $\Sigma$}}$ and $\delta$ and for the latent variables ${{\bf X}}_i$, $i=1, \dots, L$ are, respectively, $$\begin{aligned} {\mbox{\protect\boldmath $\mu$}}\mid {\mbox{\protect\boldmath $\Sigma$}}, \delta, {{\bf Z}}, {{\bf X}}&\sim& N_n({\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{*-1} [{\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{-1}{\mbox{\protect\boldmath $\mu$}}_0 + ({\mbox{\protect\boldmath $\Sigma$}}W_{\delta})^{-1} ({{\bf Z}}' {\bf{1}}_L - \delta {\mbox{\protect\boldmath $\Sigma$}}^{-1/2}{\bf{1}}_{n,m} |{{\bf X}}|'{\bf{1}}_L )], {\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{*}), \\ f({\mbox{\protect\boldmath $\Sigma$}}\mid {\mbox{\protect\boldmath $\mu$}}, \delta, {{\bf Z}}, {{\bf X}}) &\propto& |{\mbox{\protect\boldmath $\Sigma$}}|^{-L/2} \exp \left\{\frac{-tr[(W_{\delta}{\mbox{\protect\boldmath $\Sigma$}})^{-1}({{\bf Z}}- {\mbox{\protect\boldmath $\mu$}}^* )' ({{\bf Z}}- {\mbox{\protect\boldmath $\mu$}}^* )]}{2} \right\},\\ f(\delta \mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {{\bf Z}}, {{\bf X}}) &\propto& |W_{\delta}|^{-L/2} \exp \left\{\frac{-tr[(W_{\delta}{\mbox{\protect\boldmath $\Sigma$}})^{-1}({{\bf Z}}- {\mbox{\protect\boldmath $\mu$}}^* )' ({{\bf Z}}- {\mbox{\protect\boldmath $\mu$}}^* )]}{2} \right\},\\ f({{\bf X}}_i \mid {\mbox{\protect\boldmath $\mu$}}, {\mbox{\protect\boldmath $\Sigma$}}, {{\bf Z}}, \delta{{\bf X}}_{(-i)} ) &\propto& \exp \left\{ -\frac{1}{2}\left[ |{{\bf X}}_i|'[ {\bf{I}}_m + \delta^2 {\bf{1}}_{m,n} {\mbox{\protect\boldmath $\Sigma$}}^{1/2} W_{\delta}^{-1}{\mbox{\protect\boldmath $\Sigma$}}^{-1}{\mbox{\protect\boldmath $\Sigma$}}^{1/2}{\bf{1}}_{n,m}]|{{\bf X}}_i|\right]\right\}\nonumber \\ &\times& \exp\left\{-\frac{1}{2}\left[- \delta |{{\bf X}}_i|' {\bf{1}}_{m,n} {\mbox{\protect\boldmath $\Sigma$}}^{1/2} W_{\delta}^{-1} {\mbox{\protect\boldmath $\Sigma$}}^{-1}({{\bf Z}}_i -{\mbox{\protect\boldmath $\mu$}}) \right]\right\}\nonumber \\ &\times& \exp\left\{-\frac{1}{2}\left[ - \delta({{\bf Z}}_i -{\mbox{\protect\boldmath $\mu$}})'W_{\delta}^{-1}{\mbox{\protect\boldmath $\Sigma$}}^{-1}{\mbox{\protect\boldmath $\Sigma$}}^{1/2}{\bf{1}}_{n,m}|{{\bf X}}_i|\right]\right\},\end{aligned}$$ where ${\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{*}= [{\mbox{\protect\boldmath $\Sigma$}}_{\mu}^{-1} + L [{\mbox{\protect\boldmath $\Sigma$}}W_{\delta}]^{-1}]^{-1}$, $W_{\delta}={\bf{I}}_n - \delta^2{\bf{1}}_{n,n}$ and ${\mbox{\protect\boldmath $\mu$}}^* = {\bf{1}}_L \otimes {\mbox{\protect\boldmath $\mu$}}' + \delta {\mbox{\protect\boldmath $\Sigma$}}^{-1/2} {\bf{1}}_{n,m}|{{\bf X}}|'$. Notice that by using the stochastic representation, the Gibbs sampler can be used to sample from the posterior full conditional distribution of ${\mbox{\protect\boldmath $\mu$}}$. The posterior full conditional distributions of ${\mbox{\protect\boldmath $\Sigma$}}$, $\delta$ and $ {{\bf X}}_i$, $i=1, \dots, n$, have no closed forms and thus the Metropolis-Hastings algorithm can be used. Moreover, the hierarchical representation in $(\ref{hier})$ also allows us to use the software Winbugs to obtain samples from the posterior distributions. We consider it to analyse the dataset in next section. Case Study {#SecCS} ========== In this section we analyze the USA monthly precipitation data recorded from 1895 to 2007. This dataset is available at the National Climatic Data Center (NCDC) and consists of 1.344 observations of the US precipitation index (PCL). Denote by $Y_i$ the precipitation index in the $i$th month. In order to consider the strategy for data analysis described in Section \[Sec3\], we consider the log-transformed data. Figure $\ref{histN}$ shows the histogram for the transformed data (left) and the original data (right), both of them suggesting the existence of asymmetry in the data, disclosing that the use of asymmetric distributions can be a reasonable choice to analyze it. ![Histogram of logarithm of PCP (left) and PCP (right).[]{data-label="histN"}](figure5.eps "fig:"){width="12cm"}\ Similar data was previously analyzed by [@MaGe10] using the log-skew-normal and the log-skew-$t$ distributions. If compared to the log normal distribution, these models provide a better fit to data. [@MaGe10] concluded that, due to its flexibility, the log-skew-$t$ distribution, although less parsimonious, worked better than the log-skew normal distribution in capturing the skewness and heavier tails in the data. Depending on $m$, the log-CFSUN family of distributions can be heavier tailed than the log-skew-normal distribution defined by [@MaGe10]. The main goal here is to fit models in the log-CFSUN family of distributions and evaluate if there is some gain in assuming a higher dimensional skewing function. We consider $Y_i \mid \mu, \sigma^2, {\mbox{\boldmath $\Delta$}}\sim LCFUSN_{1,m}(\mu, \sigma^2, {\mbox{\boldmath $\Delta$}})$ and assume the more parsimonious log-CFSUN family discussed in the previous section where ${\mbox{\boldmath $\Delta$}}= \delta{\bf{1}}_{m,1}$. To complete the model specification we assume flat prior distributions for all parameters setting $\mu \sim N(0,100)$, $\sigma^2 \sim IG(0.1, 0.1)$ and $\delta \sim U(-1,1)$. We provide a sensitivity analysis considering different values for $m$ ($m=1$ to $5$), which is assumed to be fixed. We name $M_i$ the model for which we assume $m=i$. Table \[TaPoSu\] shows some summaries of the posterior distributions of all parameters. The posterior means for $\mu$ and $\sigma^2$ are similar for all models and increase as $m$ increases. Also, all models point out a negative skewness in the data and the highest estimate for $\delta$ is obtained if $m=1$, that is, whenever a less dimensional skewing function is assumed. It is also noteworthy that the posterior inference about $\mu$ is less precise for models with high $m$ since the posterior variance for that parameter becomes higher as $m$ increases. The opposite is observed for $\sigma^2$ and $\delta$. The $95\%$ HPD intervals disclose strong evidence in favour of an asymmetric model with negative skewness (see also Figure \[PoDel\] that shows the posterior distribution for $\delta$ in all cases). ----- -- --------- ---------- -- --------- ---------- -- ---------- ---------- -------------------- $m$ Mean St. Dev. Mean St. Dev. Mean St. Dev. $95\%$HPD 1 $1.140$ $0.010$ $0.375$ $0.011$ $-0.947$ $0.010$ $[-0.962, -0.925]$ 2 $1.276$ $0.013$ $0.384$ $0.010$ $-0.686$ $0.005$ $[-0.694, -0.674]$ 3 $1.392$ $0.016$ $0.392$ $0.010$ $-0.570$ $0.004$ $[-0.575, -0.561]$ 4 $1.483$ $0.015$ $0.394$ $0.008$ $-0.497$ $0.003$ $[-0.499, -0.490]$ 5 $1.562$ $0.015$ $0.394$ $0.008$ $-0.446$ $0.001$ $[-0.447, -0.441]$ ----- -- --------- ---------- -- --------- ---------- -- ---------- ---------- -------------------- : Posterior summaries, Precipitation data []{data-label="TaPoSu"} \ Figure \[AplicPre\] presents the plug-in estimates of the true density for all $m$ and Table \[TaPre\] presents the posterior predictive probabilities of exceeding the data average ($2.42$), the maximum ($4.20$) and also the probability of not exceed the minimum ($0.54$). Both informations disclose that the models are comparable. Moreover, the predictive summaries show that the left tail of the posterior predictive distribution is lighter than the right one which is in agreement with the empirical distribution of the data. ![Fitted log-CFUSN densities, precipitation data.[]{data-label="AplicPre"}](figure7.eps){width="7.5cm"} $m$ $Prob>2.42$ $Prob>4.2$ $Prob<0.54$ ----- ------------- -------------------------- ------------------------- 1 $0.5068$ $ 6.4514\times 10^{-4} $ $3.2422\times 10^{-6} $ 2 $0.4952$ $ 5.2508\times 10^{-4} $ $1.7958\times 10^{-6} $ 3 $0.4920$ $ 3.1177\times 10^{-4} $ $1.3747\times 10^{-6} $ 4 $0.4907$ $ 4.4087\times 10^{-4} $ $8.4866\times 10^{-7} $ 5 $0.4909$ $ 3.1727\times 10^{-3} $ $5.7553\times 10^{-7} $ : Posterior Predictive Probabilities, Precipitation data []{data-label="TaPre"} Some measures for model comparison are presented in Table \[TaMoSe\]. Specifically, we consider the sum of the logarithm of the conditional predictive ordinate (SlnCPO) [@GeDe94; @Ge96] and the deviance information criterion (DIC) [@Spieg02; @Celeux06]. Both criteria point out the model with high dimensional skewing function ($M_5$) as the best model. It is also remarkable that the DIC presents a monotonic behaviour. The Kolmogorov-Smirnov goodness of fit test comparing the plug-in estimate and the empirical cdf is also shown in Table \[TaMoSe\]. The statistic $D_n$ and the $p$-value are calculated as in [@LiLeHs07]. The differences between the empirical and the estimated c.d.f are not significant and, differently of DIC and the SlnCPO, the $D_n$ indicates model $M_1$ as the best one. ----- -- ----------- ----------- -- ------------ ------------ $m$ $D_n$ P-value DIC SlnCPO 1 $0.02508$ $0.33978$ $-13,190$ $-0.83766$ 2 $0.02765$ $0.25874$ $-36,960$ $-0.83545$ 3 $0.03033$ $0.17621$ $-112,400$ $-0.83765$ 4 $0.03244$ $0.11524$ $-321,100$ $-0.84144$ 5 $0.03082$ $0.16208$ $-895,300$ $-0.81057$ ----- -- ----------- ----------- -- ------------ ------------ : Model selection statistics, Precipitation data []{data-label="TaMoSe"} Conclusions {#SecCo} =========== In this paper we introduced two classes of log-skewed distributions with normal kernels: the log-CFUSN and the log-SUN. We studied some properties of the log-CFUSN family of distributions such as marginal and conditional distributions, moments and stochastic representation. We also discussed some issues related to Bayesian inference in that family. Our discussion was devoted to the elicitation of a prior distribution for the skewness parameter. The main motivation for studying the log-CFUSN family of distribution in detail, and other new classes of log-skewed distributions, is the result that appeared in [@SaLoAr13] where it was shown that such family is of fundamental interest in the interpretation of the parameters in mixed logistic regression model if the random effects are skew-normally distributed. In that paper it was proved that, under skew-normality, the odds ratio has distribution in the log-CFUSN family. Analizing the USA precipitation dataset, we concluded that the use of a skewing function with higher dimension than that assumed by [@MaGe10] can bring some gain to the model fit. Acknowledgement {#acknowledgement .unnumbered} =============== The authors would like to thank the Editors and the referee for their comments and suggestions which improved the paper. We would like to express our gratitude to Professors Fredy Castellares and Reinaldo Arellano for their comments on the first draft of this work. The research of M.M. Queiroz was partially supported by CAPES (*Coordenação de Aperfeiçoamento de Pessoal de Nível Superior*) and CNPq (*Conselho Nacional de Desenvolvimento Científico e Tecnológico*) of the Ministry for Science and Technology of Brazil. R. H. Loschi would like to thank to CNPq, grants 301393/2013-3 and 306085/2009-7, for a partial allowance to her researches. The research of R.W.C. Silva was partially supported by PRPq-UFMG (Edital 12/2011). [15]{} natexlab\#1[\#1]{}url\#1[`#1`]{}urlprefix Abtahi A and Towhidi M. The new unified representation of multivariate skewed distributions. [*Statistics*]{}. 2013; [**[47]{}**]{}: 126-140. Arellano-Valle RB, Cortés MA. and Gómez HW. An extension of the epsilon-skew-normal distribution. [*Communications in Statistics - Theory and Methods*]{}. 2010; [**[39]{}**]{}: 912-922. Arellano-Valle RB, Azzalini A. On the unification of families of skew-normal distributions. [*Scandinavian Journal of Statistics*]{}. 2006; [**33**]{}: 561–574. Arellano-Valle RB, Branco MD and Genton MG. [A unified view on skewed distributions arising from selections.]{} [*The Canadian Journal of Statistics*]{}. 2006; [**[34]{}**]{}: 581-601. Arellano-Valle, RB and Genton MG. On fundamental skew distributions. [*Journal of Multivariate Analysis*]{}. 2005; [**96**]{} (1): 93-116. Arnold BC, Goméz HW and Salinas HS. On multiple constraint skewed models. [*Statistics: A Journal of Theorectical and Applied Statistics*]{}. 2009; [**[43]{}**]{} (3): 279–293. Arnold BC and Beaver JR. Skewed multivariate models related to hidden truncation. [*Test*]{}. 2002; [**[11]{}**]{}: 7–54. Azzalini A. The skew-normal distribution and related multivariate families. With discussion by Marc G. Genton and a rejoinder by the author. [*Scandinavian Journal of Statistics*]{}. 2005; [**32**]{}: 159–200. Azzalini A. A class of distributions which includes the normal ones. [*Scandinavian Journal of Statistics*]{}. 1985; [**[12]{}**]{}: 171-178. Azzalini A, Capitanio A. Distributions generated by perturbation of symmetry with enphasis on a multivariate skew $t$ distribution. [**]{}. 2003; [**65**]{}: 367-389. Azzalini A, Capitanio A. Statistical applications of the multivariate skew normal distribution. [**]{}. 1999; [**61**]{}: 579-602. Azzalini A and Dalla Valle A. [ The multivariate skew-normal distribution.]{} [*Biometrika*]{}.1996; [**83**]{}: 715-726. Azzalini A, Dal Cappello T and Kotz S. [Log-skew-normal and log-skew-t distributions as models for family income data.]{} [*Journal of Income Distribution*]{}. 2003 [**11**]{}: (3,4), 12-20. Branco MD, Dey DK. A general class of multivariate skew-elliptical distributions. [*[Journal of Multivariate Analysis]{}*]{}. 2001; [**[79]{}**]{}: 93-113. Bolfarine H, Goméz HW and Rivas L. The log-bimodal-skew-normal model: A geochemical application. [*[Journal of Chemometrics]{}*]{}. 2011; [**[25]{}**]{}(6): 329–332. Cabral CRB, Bolfarine H and Pereira JRG. Bayesian density estimation using student-$t$-normal mixtures. [*[Computational Statistics and Data Analysis]{}*]{}. 2008 [**[52]{}**]{}: 5075–5090. Celeux G, Forbes F, Robert CP and Titterington DM. Deviance information criteria for missing data models. [*Bayesian Analysis*]{}. 2006; [**1**]{}(4): 651-674. Dey DK, Müller P and Sinha D. [*[Practical Nonparametric and Semiparametric Bayesian Statistics]{}*]{}. Springer-Verlag, New York; 1998. Elal-Olivero D, Gómez HW, Quintana FA. Bayesian Modeling using a class of bimodal skew-Elliptical distributions. [*Journal of Statistical Planning and Inference*]{}. 2009; [**[139]{}**]{}(4): 1484-1492. Fang KT, Kotz S and Ng KW. [*Symmetric Multivariate and Related Distributions.*]{} Chapman and Hall, London; 1990. Gelfand AE. Model determination using sampling-based methods. In [*[Markov Chain Monte Carlo in Pratice]{}*]{}. 1996; W. R. Gilks, S. Richardson, D.J. Spiegelhalter (ed): 145–161Chapman & Hall, London. Gelfand AE and Dey DK. Bayesian model choice: Asymptotics and exact calculations. [*[Journal of the Royal Statistical Society B]{}*]{}. 1994; [**56**]{}: 501–514. Genton MG. [*Skew-elliptical distributions and their applications: A journey beyond normality*]{}. 2004; Edited Volume, Chapman & Hall / CRC, Boca Raton, FL, 416 pp. Genton MG and Loperfido N. [Generalized skew-elliptical distributions and their quadratic forms.]{} [*Annals of the Institute of Statistal Mathematics*]{}. 2005; [**57**]{}(2): 389-401. Goméz HW, Elal-Olivero D, Salinas HS and Bolfarine H. Bimodal extension based on the skew-normal distribution with application to pollen data. [*Environmetrics*]{}. 2011; [**[22]{}**]{}: 50–62. Huang WJ, Su NC, Gupta AK. A study of generalized skew-normal distribution. [*[Statistics]{}*]{}. 2013; [**[47]{}**]{}: 942–953. Larsen K, Petersen JH, Budtz-J$\o$rgensen E and Endahl L. Interpreting parameters in the logistic regression model with random effects. [*[Biometrics]{}*]{}. 2000; [**[56]{}**]{}: 909–914. Lin TI, Lee JC. and Hsieh WJ. Robust mixture modelling using the skew $t$ distribution. *Stat. Comput.*. 2007a; [**[17]{}**]{}: 81-92. Lin TI, Lee JC and Yen SY. Finite mixture modelling using the skew normal distribution. [*Statistica Sinica*]{}. 2007b; [**[17]{}**]{}: 909-927. Marchenko YV and Genton MG. Multivariate log-skew-elliptical distributions with applications to precipitation data. [*Environmetrics*]{}. 2010; [**21**]{}: 318-340. Martinez-Florez GD, Bolfarine H and Gomez HW. The log-power-normal distribution with application to air pollution. [*Environmetrics*]{}. 2012; [**25**]{}(1): 44-56. Martinez-Florez GD, Bolfarine H and Gomez HW. Skew-normal alpha power model. [*Statistics*]{}. 2014; [**48**]{}(1): 1414-1428. Müller P and Quintana FA. Nonparametric Bayesian Data Analysis. [*Statistical Science*]{}. 2004;[**[19]{}**]{}(1): 95–1109. Rocha GHMA, Loschi RH and Arellano-Valle RB. Inference in flexible families of distributions with normal kernel. [*Statistics: A Journal of Theorectical and Applied Statistics*]{}. 2013; [**47**]{}: 1184-1206. Santos CC, Loschi RH and Arellano-Valle RB. Parameter Interpretation in Skewed Logistic Regression with Random Intercept. [*Bayesian Analysis*]{}. 2013; [**8**]{}(2): 381–410. Spiegelhalter DJ, Best NG, Carlin BP and Linde AVD. Bayesian measures of model complexity and fit (with discussion). [*Journal of the Royal Statistical Society*]{}. 2002; [**64**]{}(4): 583-639. van Dyk DA and Meng XL. The art of data augmentation. [*[Journal of Computational and Graphical Statistics]{}*]{}. 2001; (discussion paper), [**[10]{}**]{}(1): 1–50. Walker S. Bayesian Nonparametric Inference. In [*Handbook of Statistics*]{}, vol.25. Elsevier B. V., New York; 2005. [^1]: Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901 - Belo Horizonte - MG, Brazil. E-mail: marinamunizdequeiroz@gmail.com. [^2]: Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901 - Belo Horizonte - MG, Brazil. E-mail: loschi@est.ufmg.br [^3]: Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901 - Belo Horizonte - MG, Brazil. E-mail: rogerwcs@est.ufmg.br (corresponding author).

--- abstract: 'Initial-boundary value problem for the generalized Zakharov-Kuznetsov equation posed on a bounded rectangle is considered. Critical and subcritical powers in nonlinearity are studied.' address: | Departamento de Matemática\ Universidade Estadual de Maringá\ 87020-900, Maringá - PR, Brazil. author: - | M. Castelli$^\ast$,  G. Doronin\ [Departamento de Matemática,\ Universidade Estadual de Maringá,\ 87020-900, Maringá - PR, Brazil. ]{} title: ' Subcritical and critical generalized Zakharov-Kuznetsov equation posed on bounded rectangles ' --- [^1] Introduction ============ We are concerned with initial-boundary value problems (IBVPs) posed on bounded rectangles located at the right half-plane $\{(x,y)\in\mathbb{R}^2:\ x>0\}$ for the generalized Zakharov-Kuznetsov [@pastor] equation $$u_t+u_x+u^{1+\delta}u_x+u_{xxx}+u_{xyy}=0,\label{mzk}$$ with $\delta \in [0,1].$ When $\delta=0,$ turns the classical Zakharov-Kuznetsov (ZK) equation [@zk], while $\delta=1$ corresponds to so-called modified Zakharov-Kuznetsov (mZK) equation [@pastor1] which is a two-dimensional analog of the well-known modified Korteweg-de Vries (mKdV) equation [@bona2] $$\label{mkdv} u_t+u_x+u^2u_x+u_{xxx}=0.$$ Notes that both ZK and mZK possess real plasma physics applications [@zk]. As far as ZK is concerned, the results on both IVP and IBVPs can be found in [@faminski; @faminski2; @farah; @pastor; @pastor2; @saut; @temam; @temam2]. For IVP to mZK, see [@pastor1]; at the same time we do not know solid results concerning IBVP to mZK. The main difference between initial and initial-boundary value problems is that IVP provides (almost immediately) good estimates in $(L^{\infty}_t;H^1_{xy})$ by the conservation laws, while IBVP does not possesses this advantage. Our work is a natural continuation of [@doronin] where with $\delta =0$ has been considered. There one can find out a more detailed background, descriptions of main features and the deployed reference list. In the present note we put forward an analysis of for $\delta \in (0,1].$ When $\delta = 1,$ the power is critical (see [@pastor; @pastor1]) and a challenge concerning the well-posedness of IBVPs appears. For one-dimensional dispersive models the critical nonlinearity has been treated in [@larkin19]. Once $\delta\in (0,1)$ the existence of a weak solution in $\left((L^{\infty}_T; L^2)\cap (L^2_T;H^1_0)\right)$ with $u_0\in L^2_{xy}$ is proved in our work via parabolic regularization. If $\delta =1,$ we apply the fixed point arguments to prove the local existence and uniqueness of solutions with more regular initial data. We also show the exponential decay of $L^2$ norm of solutions as $t\to\infty$ if $u\in (L^{\infty}_{\mathbb{R}^+} ; H^1_0),$ under domain’s size restrictions. These are the main results of the paper. Problem and notations {#problem} ===================== Let $L,B,T$ be finite positive numbers. Define $\Omega$ and $Q_T$ to be spatial and time-spatial domains $$\Omega=\{(x,y)\in\mathbb{R}^2: \ x\in(0,L),\ y\in(-B,B) \},\ \ \ Q_T=\Omega \times (0,T).$$ In $Q_T$ we consider the following IBVP: $$\begin{aligned} A&u\equiv u_t+u_x+u^{1+\delta}u_x+u_{xxx}+u_{xyy}=0,\ \ \text{in}\ Q_T; \label{2.1} \\ &u(x,-B,t)=u(x,B,t)=0,\ \ x\in(0,L),\ t>0; \label{2.2} \\ &u(0,y,t)=u(L,y,t)=u_x(L,y,t)=0,\ \ y\in(-B,B),\ t>0; \label{2.3} \\ &u(x,y,0)=u_0(x,y),\ \ (x,y)\in\Omega, \label{2.4}\end{aligned}$$ where $u_0:\Omega\to\mathbb{R}$ is a given function. Hereafter subscripts $u_x,\ u_{xy},$ etc. denote the partial derivatives, as well as $\partial_x$ or $\partial_{xy}^2$ when it is convenient. Operators $\nabla$ and $\Delta$ are the gradient and Laplacian acting over $\Omega.$ By $(\cdot,\cdot)$ and $\|\cdot\|$ we denote the inner product and the norm in $L^2(\Omega),$ and $\|\cdot\|_{H^k}$ stands for the norm in $L^2$-based Sobolev spaces. Abbreviations like $(L^s_t;L^l_{xy})$ are also used for anisotropic spaces. Existence in sub-critical case {#existence} ============================== In this section we state the existence result in sub-critical case, i.e., for $\delta\in(0,1).$ We provide a short motivation for this study at the final of the section. Sub-critical nonlinearity ------------------------- \[theorem1\] Let $\delta\in(0,1)$ and $u_0\in L^2(\Omega)$ be a given function. Then for all finite positive $B,\ L,\ T$ there exists a weak solution to - such that $$u\in L^{\infty}(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega)).$$ To prove this theorem we consider for all real $\eps>0$ the following parabolic regularization of -: $$\begin{aligned} A^{\eps}u_{\eps}&\equiv Au_{\eps}+\eps(\partial_x^4u_{\eps}+\partial_y^4u_{\eps})=0\ \ \text{in}\ Q_T;\label{3.1}\\ &u_{\eps}(x,-B,t)=u_{\eps}(x,B,t) % \notag\\& =\partial_y^2u_{\eps}(x,-B,t)=\partial^2_yu_{\eps}(x,B,t)=0,\ x\in(0,L),\ t>0;\label{3.2}\\ &u_{\eps}(0,y,t)=u_{\eps}(L,y,t) %\notag\\& =\partial_x^2u_{\eps}(0,y,t)=\partial_xu_{\eps}(L,y,t)=0,\ y\in (-B,B),\ t>0;\label{3.3}\\ &u_{\eps}(x,y,0)=u_{0}(x,y),\ (x,y)\in \Omega.\label{3.4}\end{aligned}$$ For all $\eps>0,$ - admits a unique regular solution in $Q_T$ [@lady2]. In what follows we omit the subscript $\eps$ whenever it is unambiguous. Multiplying $A^{\eps}u_{\eps}$ by $u_{\eps}$ and integrating over $Q_T,$ we have $$\label{estimate 2.1} \| u\| ^2 (t) + \int_0^t \int_{-B}^B u^2_x (0,y,\tau) \, dyd\tau + 2\epsilon \int_0^t \big( \|u_{xx}\|^2 (\tau) + \|u_{yy}\|^2 (\tau) \big)d\tau = \|u_0\|^2, \,\,t\in(0,T).$$ Multiplying $A^{\eps}u_{\eps}$ by $x u_{\eps}$, integrating over $\Omega$ with the use of the Nirenberg, Hölder and Young inequalities yields $$\begin{aligned} \label{estimate 2.3} \dfrac{d}{dt} \| \sqrt{x}u\|^2(t) +\frac{1}{2} \|\nabla u \|^2(t) + 2\|u_x\|^2(t) + 2\eps\Big( \|\sqrt{x}u_{xx}\|^2(t) + \|\sqrt{x} u_{yy}\|^2 (t)\Big) \nonumber \\ \leq \|u\|^2(t)+2\eps\int_{-B}^B u_x^2(0,y,t)\, dy + \dfrac{C(\xi,\delta)C_{\Omega}^{\frac{2}{1-\delta}}}{3+\delta}\|u\|^{\frac{4}{1-\delta}}(t).\end{aligned}$$ Integrating with respect to $t>0$ in and taking $\eps<1/2$ gives $$\begin{aligned} \label{estimate 22.3} &\| \sqrt{x}u\|^2(t)+\frac{1}{2}\int_0^{t} \|\nabla u \|^2(\tau)\, d\tau + 2\int_0^{t} \|u_x\|^2(\tau)\, d\tau + 2\eps\int_0^{t}\Big( \|\sqrt{x}u_{xx}\|^2(\tau) + \|\sqrt{x} u_{yy}\|^2 (\tau)\Big)\, d\tau \nonumber \\ &\leq \int_0^{t}\|u_0\|^2\, d\tau +\int_0^{t}\int_{-B}^B u_x^2(0,y,\tau)\, dyd\tau+\dfrac{C(\xi,\delta)C_{\Omega}^{\frac{2}{1-\delta}}}{3+\delta}\cdot\int_0^{t}\|u_0\|^{\frac{4}{1-\delta}}d\tau \nonumber\\ &\leq (T+1)\|u_0\|^2 + \dfrac{C(\xi,\delta)C_{\Omega}^{\frac{2}{1-\delta}}}{3+\delta}\cdot T \|u_0\|^{\frac{4}{1-\delta}}. %\nonumber \\\end{aligned}$$ Note that does not hold for critical case, i.e., while $\delta \to 1.$ Estimates and thus become $$\label{limitations} \begin{array}{c} \hspace{-2cm}u_{\eps} \;\;\;\; \text{is bounded in } \;\;\;\; L^{\infty}\big(0,T;L^2(\Omega)\big) ,\\ u_{\eps x}(0,y,t) \;\;\;\; \text{is bounded in } \;\;\;\; L^2\big(0,T;L^2(-B,B)\big) ,\\ \hspace{-2cm}\nabla u_{\eps}\;\;\;\; \text{is bounded in } \;\;\;\; L^2\big(0,T;L^2(\Omega)\big), \end{array}$$ where limitations do not depend on $\eps$ but depend only on $T$, $\delta$, $\Omega$ and $\|u_0\|$. Thanks to (\[limitations\]) we have boundness of $u_{\eps}^{1+\delta}u_{\eps x}$ for all $\delta \in (0,1)$. In fact, given $\delta \in (0,1)$ take $m=\frac{4}{3+\delta}$ and $ \kappa (\delta) = \frac{1+\delta}{3+\delta}.$ Then Hölder’s and Nirenberg’s inequality yield $$\begin{aligned} \label{estimativa 3.3} \|u^{1+\delta}u_x\|^m_{L^m(0,T;L^m(\Omega))} &= \int_0^T \|u^{1+\delta}u_x\|^m_{L^m(\Omega)}(t)\, dt %\nonumber \\& \leq C_{\Omega}^{4 \kappa (\delta)} \int_0^T \| \nabla u \|^{2 } (t)\|u \|^{2 \kappa (\delta)}(t) \, dt \nonumber \\ & = C_{\Omega}^{4 \kappa (\delta)} \|u \|^{2 \kappa (\delta)}_{L^{\infty}(0,T;L^2(\Omega))} \| \nabla u \|^{2 } _{L^2(0,T;L^2(\Omega))} .\end{aligned}$$ Therefore, due to (\[estimativa 3.3\]) and (\[limitations\]) we conclude that $u^{1+\delta}u_x$ is bounded in $L^m(0,T;L^m(\Omega)).$ Since $L^{\frac{4}{1-\delta}}$ is the dual space of $L^{ \frac{4}{3+\delta} } $ and $ H^{ 1 } \subset L^{\frac{4}{1-\delta}}$ in dimension 2, we have as well $$\label{estimativa 3.5} u^{1+\delta}u_x \;\;\;\text{is bounded in}\;\;\; L^{\frac{4}{3+\delta}}(0,T;H^{-1}(\Omega)).$$ Thanks to (\[limitations\]) and (\[estimativa 3.5\]) jointly with the equation, we get $$\label{estimativa 3.6} \dfrac{\partial u_{\epsilon}}{\partial t } \;\;\;\text{is bounded (independently of $\eps$) in} \;\;\;L^{\frac{4}{3+\delta}}(0,T;H^{-3}(\Omega))$$ which assures the family $u_{\eps}$ to be relatively compact in $L^2(0,T;L^2(\Omega))$. This is sufficiently to obtain the existence of $\lim u_{\eps}$ as $\eps \rightarrow 0$, using the compactness argument in the nonlinear term. The initial condition $u(x,y,0) = u_0(x,y)$ is fulfilled; indeed, due to (\[estimativa 3.6\]) $ u_{\eps}$ converges to $u$ in $C\big([0,T]; H^{-3}_{w}(\Omega)\big), $ where $H^{-3}_{w}$ is $H^{-3}$ equipped with the weak topology. By the same way, the Dirichlet condition $u=0$ onto $\partial \Omega$ is satisfied since $u_{\eps}$ converges to $u$ weakly in $L^{2}(0,T;H^{1}_0(\Omega)).$ It remains to show that $u_x(L,y,t)=0,$ which is done by the following two lemmas (cf. [@temam; @temam2]). \[lema cont\] If $u\in L^{\infty }(0,T;L^2(\Omega))\cap L^2 (0,T;H^1_0 (\Omega) )$ solves (\[2.1\]), then $$\label{eq 1 lm cont} u_x , u_{xx} \in C(0,L; V)\ \text{ with }\ V= H^{-2}\big((0,T)\times (-B,B)\big),$$ and, in particular, $$\label{eq 2 lm cont} u_x \big|_{x=0,1}, \;\; u_{xx}\big|_{x=0,1}$$ are well defined in $V$. Moreover, these traces depend continuously of $u$ in an appropriate sense. To prove this lemma, write (\[2.1\]) in the form $$\label{lm eq 1} u_{xxx} = -u_x - u_{xyy} - u^{1+\delta}u_x - u_t ,$$ and observe that $$\begin{aligned} u_t \in L^{2}(0,L;H^{-1}\big(0,T;L^2 (-B,B)\big), \\ u_{xyy} \in L^{2}(0,L;L^2\big(0,T;H^{-2} (-B,B)\big).\end{aligned}$$ Accordingly with (\[estimativa 3.5\]) and definition of $V$ in , it holds $$\label{lm eq 2} u^{1+\delta}u_x \in L^{\frac{4}{3+\delta}}\big(0,L;L^{\frac{4}{3+\delta}}((0,T)\times(-B,B))\big) \hookrightarrow L^{\frac{4}{3+\delta}}\big(0,L;V\big). %H^{-2}((0,T)\times(-B,B))\big).$$ Thus we have $$\label{lm eq 3} u_{xxx} \in L^{\frac{4}{3+\delta}}\big(0,L;V\big) %H^{-2}((0,T)\times(-B,B))\big),$$ and (\[eq 1 lm cont\]) and (\[eq 2 lm cont\]) follow. Moreover, if a sequence of functions $u_m$ satisfies (\[eq mZK\]) and $u_m \rightarrow u $ in $ L^{\infty }(0,T;L^2(\Omega))\cap L^2 (0,T;H^1_0 (\Omega) ) $ strongly, then $ u_{mx} \big|_{x=0,1}, \;\; u_{mxx}\big|_{x=0,1} $ converge to $u_x \big|_{x=0,1}, \;\; u_{xx}\big|_{x=0,1}$ in $V.$ If a convergence of $u_m$ being weak (star-weak for $L^{\infty }$,) then a convergence take place in $C(0,L; V_w)$ and $Y_w$. This is based on compactness arguments justified by (\[estimativa 3.6\]), used to prove that $u_m^{1+\delta}u_{mx} \rightarrow u^{1+\delta}u_x$. \[lema ux(L)\]Let $U$ be a reflexive Banach space and $p\geq 1$. Suppose that two function sequences $u_{\eps} ,\ g_{\eps}\in L^p(0,L;U)$ satisfy $$\label{eq traco ux(L)} \left. \begin{array}{c} %------------------------------- u_{\eps xxx} + \eps u_{\eps xxxx}= g_{\eps}, \\ %------------------------------- u_{\eps}(0)=u_{\eps}(L)=u_{\eps x}(L)=u_{\eps xx}(0)=0 , %------------------------------- \end{array} \right.$$ with $g_{\eps}$ being bounded in $L^p(0,L;U)$ as $\eps \to 0.$ Then $ u_{\eps xx} $ (consequently $ u_{\eps x}, $ and $ u_{\eps} $) is bounded in $L^{\infty}(0,L;U)$ as $\eps \rightarrow 0.$ Moreover, for a subsequence $u_{\eps} \to u $ converging (strongly or weakly) in $L^q(0,L;U),$ $1\leq q < \infty,$ it holds that $u_{\eps x}(L)$ converges to $u_x(L)$ in $U$ (at least weakly), and therefore $u_x(L)=0.$ See [@temam2] for the proof. To prove Theorem \[theorem1\], apply the above lemmas with $$g_{\eps} := - u_{\eps t} - \eps u_{\eps x} - u_{\eps xyy} - u^{1+\delta}_{\eps}u_{\eps \eps} - \eps u_{\eps yyyy} ,$$ $$U= H^{-1} (0,T ; L^2(-B,B)) + L^2 (0,T ; H^{-4}(-B,B)) + L^{\frac{4}{3+\delta}}(0,T;L^{\frac{4}{3+\delta}}(-B,B)),$$ and $$p=\frac{4}{3+\delta}.$$ The proof is completed. Motivation and explanation of the main difficulty ------------------------------------------------- Note that inclusions can be obtained also for $\delta=1$ with $\|u_0\|<1/2.$ Using embedding machinery and interpolation theory for anisotropic spaces, one could pass to the limit as $\eps \to 0$ in nonlinear term, as well. Indeed, let $\delta=1.$ Multiplying $A^{\eps}u_{\eps}=0$ by $2(1+x)u_{\eps}$ and integrating over $\Omega,$ we have $$\begin{aligned} \label{estimate 1.1} \frac{d}{dt}\left((1+x), u^2\right) (t) + \|\nabla u\|^2(t)+2\|u_x\|^2(t)+ (1-2\eps)\int_{-B}^B u^2_x (0,y,t) \, dy\\ \le \|u\|^2(t)+2\|u\|^4_{L^4(\Omega)} \le \|u\|^2(t)+2\|\nabla u\|^2(t)\|u\|^2(t).\end{aligned}$$ Bearing in mind that $\|u\|(t)\le \|u_0\|(t)<1/2$ and integrating in $t>0,$ Gronwall’s lemma gives $$u\in L^{\infty}\left(0,T;L^2(\Omega)\right) \cap L^2\left(0,T;H^1_0(\Omega)\right)$$ with both estimates independent of $\eps<1/4.$ Now we observe that $$\begin{aligned} \int_0^T \int_{\Omega} |u^3|^{\frac{4}{3}} dx dt \leq C \|u_0\|^2 \| \nabla u\|^2_{L^2_T L^2_{xy}}\end{aligned}$$ and by estimate above this implies $ u^3 \in L^{\frac{4}{3}}(Q_T). $ Since $L^{\frac{4}{3}}(\Omega) \hookrightarrow H^{-1}(\Omega),$ we conclude that $$u^2u_x = \frac{1}{3}\partial_x(u^3) \in L^{\frac{4}{3}}(0,T; H^{-2}(\Omega))$$ whence $$u_t \in L^{\frac{4}{3}}(0,T; H^{-2}(\Omega))$$ and passage to the limit as $\eps \to 0$ in nonlinear term can be justified as above. It is difficult, however, to obtain explicit estimates like with $m>1$ for $\delta=1.$ In fact, let $r,s \ge 1.$ We are going to determine conditions upon $r$ and $s$ such that $u^2 u_x $ lies in $L^r\left((0,T;L^s(\Omega)\right).$ Consider $p,q>1$ with $1/p +1/q = 1.$ Then $$\begin{aligned} \label{3.12} \|u^2u_x \|_{L^r_T L^s_{xy}}^r &=& \int_0^T \left( \int_{\Omega} u^{2s} u_x^s \, d\Omega \right) ^{\frac{r}{s}} dt \nonumber \\ %==== &\leq & \int_0^T \|u\|^{2r}_{L^{2sp}_{xy}}(t)\|u_x\|^{r}_{L^{sq}_{xy}}(t) \, dt .\end{aligned}$$ By Nirenberg’s inequality with $ %\begin{equation} \alpha = \frac{sp-1}{sp} $ one has $$\|u\|^{2r}_{L^{2sp}_{xy}}(t) \leq C \|\nabla u\|^{2r \alpha} \|u\|^{2r(1-\alpha)}.$$ Supposing $sq\leq 2,$ estimate reads $$\begin{aligned} \|u^2u_x \|_{L^r_T L^s_{xy}}^r &\leq & C \|u \|_{L^{\infty}_T L^2_{xy}}^{2r(1-\alpha)} \int \|\nabla u\|^{2r \alpha} \|u_x\|^{r}(t) \, dt \nonumber \\ &\leq & C \|u \|_{L^{\infty}_T L^2_{xy}}^{2r(1-\alpha)} C \|\nabla u \|_{L^{r(2\alpha +1)}_T L^2_{xy}}^{r(2\alpha +1)}.\end{aligned}$$ In order to gain $r(2\alpha +1) =2,$ it should be $\alpha = 1/r -1/2.$ Therefore, $ \frac{1}{sp}= \frac{3}{2} - \frac{1}{r}, $ which implies $$\begin{aligned} sq = \frac{2rs}{2(r+s) - 3rs}.\end{aligned}$$ Since $sq \leq 2,$ it follows that $\frac{2rs}{2(r+s) - 3rs} \leq 2$ which means $sr \leq \frac{ r+s}{2}.$ Observe that for $r,s>1$ this condition does not hold. The only possibility thus reads $r=s=1,$ i.e., $u^2 u_x \in L^1\left((0,T;L^1(\Omega)\right).$ The space $(L^1_t;L^1_{xy})$ is known to be difficult to deal with. For example, it is not clear even whether the condition $u_x(L,y,t)=0$ being satisfied. We leave it here only to illustrate a challenge appearing in the critical case. Local result for critical case {#local existence} ============================== Consider the following Cauchy problem in abstract form: $$\label{Eq sem. grp.} \left\{ \begin{array}{c} %------------------------------- u_t + Au= f, \;\; \\ %\text{em} \;\; \mathrm{Q}_T = \Omega\times (0,T), \;\; ,\\ %------------------------------- %\hspace{-0.5cm}u=0\;\;\text{em}\;\;\partial \Omega , \;\; u_x(L,y,t)=0 , \\ %------------------------------- \hspace{0cm}u(0)=u_0 ,\;\;%\text{em}\;\;\Omega ; %------------------------------- \end{array} \right.$$ where $f\in L^1(0,T;L^2(\Omega))$ and $A: L^2(\Omega)\to L^2(\Omega)$ defined as $A\equiv \partial_x + \Delta\partial_x $ with the domain $$D(A) = \{ u \in L^2(\Omega ) \, ; \, \Delta u_x + u_x \in L^2(\Omega) \, \text{with}\, u|_{\partial \Omega}=0 \ \text{ and }\ u_x(L,y,t) = 0, \ t\in (0,T) \},$$ endowed with its natural Hilbert norm $ \|u\|_{D(A)}(t) = \left( \|u\|^2_{L^2(\Omega)}(t) + \| \Delta u_x + u_x \|^2_{L^2(\Omega)}(t) \right)^{1/2} $ for all $t\in (0,T)$. \[res. 1 teman\] Assume $u_0\in D(A)$ and $f\in L^1_{loc}(\R^+;L^2(\Omega))$ with $f_t\in L^1_{loc}(\R^+;L^2(\Omega))$. Then problem possesses the unique solution $u(t)$ such that $$\label{res 2} u\in C([0,T]; D(A)) , \; u_t \in L^{\infty}(0,T; L^2(\Omega)) \; T>0.$$ Moreover, if $u_0\in L^2(\Omega)$ and $f \in L^1_{loc}(\R^+;L^2(\Omega)),$ then possesses a unique (mild) solution $u \in C([0,T]; L^2(\Omega)) $ given by $$\label{res 3} u(t)=S(t)u_0 + \int_0^t S(t-s)f(s)\, ds.$$ \[res. 2 teman\] Under the hypothesys of Proposition \[res. 1 teman\], the solution $u$ in (\[res 2\]) satisfies $$\label{res 2.1} u\in L^{\infty}((0,T);H^1_0(\Omega)\cap H^2(\Omega)),$$ For the proof, see [@temam2]. Furthermore, one can get (see [@kato], for instance) the estimate for strong solution (\[res 2\]): $$\label{estimativa temam/kato 1} \|u_t \| (t) \leq \|Au_0\| + \|f\|(0) + \|f_t\|_{L^1_tL^2_{xy}},$$ and $$\label{estimativa temam/kato 2} \left\| A u \right\| (t) \leq \| u_t \|(t)+ \|f\|(t).$$ Since $D(A){ \hookrightarrow } H^1_0(\Omega)\cap H^2(\Omega)$ compactly (see [@temam2] for instance), we have the estimate $$\begin{aligned} \label{estimativa temam/kato 4} \|u\|_{L^{\infty}0,T; H^1_0\cap H^2 (\Omega)}(t) \leq C \big(\|u\|_{L^{\infty}_t L^2_{xy}} + \|Au_0\| + \|f\|(0) + \|f_t\|_{L^1_tL^2_{xy}} + \|f\| _{L^{\infty}_t L^2_{xy}} \big). \\\end{aligned}$$ where $C$ depends only on $\Omega $. Next, we define $$Y_T = \{f \in L^{1}\big( 0,T; L^2(\Omega) \big) \; \text{such that} \; f_t \in L^{1} \big( 0,T; L^2(\Omega) \big) \}$$ with the norm $$\|f\|_{Y_T}= \|f\|_{ L^{1}_t L^2_{xy} } + \|f_t\|_{L^{1}_t L^2_{xy} }.$$ \[prop traço f\] If $f\in Y_T,$ then $ f \in C([0,T];L^2(\Omega))$, with the constant $C_T$ from $\|f\|_{C_t L^2_{xy}} \leq C_T \|f\|_{Y_T} $ which is proportional to $T$ and its positive powers [@evans]. Consider $X_T^0 = L^{\infty}\big(0,T; H^1_0(\Omega) \cap H^2(\Omega) \big) $ and define the Banach space $$\begin{aligned} \label{esp XT} X_T=\{u \in X_T^0:\ u_t \in L^{\infty}\big(0,T; L^2(\Omega)\big) \;\;\text{and}\;\; \nabla u_t \in L^{2}\big(0,T; L^2(\Omega)\big) \} .\end{aligned}$$ with the norm $$\begin{aligned} \label{morma esp XT} \| u \|_{X_T} = \| u \|_{L^{\infty}_T H^1_0 \cap H^2_{xy} } + \| u_t \|_{L^{\infty}_T L^2_{xy}} +\| \nabla u_t \|_{L^{2}_T L^2_{xy}} . \\\end{aligned}$$ \[Teo solu local mZK\] Let $u_0 \in D(A)$. Then there exists $T>0$ such that IBVP (\[2.1\])-(\[2.4\]) possesses a unique solution in $X_T$. The proof of the Theorem consists in three lemmas below. \[lemma1\] The function $Y_T %& \longrightarrow %& X_T;\ \ %\\ %------------------------------- f %& \mapsto %& \int_0^t S(t-s)f(s) ds$ is well defined and continuous. For the proof, note that this function maps $f$ to the solution of homogeneous linear problem with zero initial datum. Estimates (\[estimativa temam/kato 1\]) and (\[estimativa temam/kato 4\]) then give $$\label{estimativa ut 0} \| u \|_{L^{\infty}_T H^1_0 \cap H^2_{xy} } + \| u_t \|_{L^{\infty}_T L^2_{xy}} \leq C \|f\|_{Y_T},$$ where $C$ is as above. Thus, it rests to estimate the term $\| \nabla u_t \|_{L^{2}_T L^2_{xy}}$ in (\[morma esp XT\]). Differentiate the equation in (\[Eq sem. grp.\]) with respect to $t,$ multiply it by $(1+x) u_{t}$ and integrate the outcome over $\Omega.$ The result reads $$\label{estimativa ut} \dfrac{d}{dt} \left( (1+x),u_t^2\right) (t) + \|\nabla u_t \|^2(t) + 2\| u_{xt} \|^2+\int_{-B}^B u_{xt}^2(0,y,t)\,dy = \|u_t\|^2(t) + 2\int_{\Omega}(1+x) f_t u_{t} \, d\Omega.$$ Hölder’s inequality and (\[estimativa temam/kato 1\]) imply $$\begin{aligned} \label{estimativa ut 1.2} \int_0^T \|\nabla u_t \|^2(t)\, dt &\leq & T\big(\|f\|(0) + \|f_t\|_{L^1_T L^2_{xy}} \big)^2 \nonumber \\ &+& 2(1+L)\big(\|f\|(0) + \|f_t\|_{L^1_T L^2_{xy}} \big) \|f_t\|_{L^1_T L^2_{xy}} +\left( (1+x),u_t^2\right) (0) . %==============================\end{aligned}$$ Using the equation from (\[Eq sem. grp.\]) and taking in mind that $u_0 \equiv 0 $, we get $$\begin{aligned} \label{estimativa ut 1.3} u_t (x,y,0) = f(x,y,0) - Au_0= f(x,y,0)\end{aligned}$$ Inserting (\[estimativa ut 1.3\]) into (\[estimativa ut 1.2\]) provides $$\begin{aligned} \label{estimativa ut 1.7} \|\nabla u_t \|^2_{L^2_TL^2_{xy}} \leq \Big(4T K_T^2 + 4K_T(1+L) + K_T^2(1+L) \Big)\|f\|^2_{Y_T},\end{aligned}$$ where $K_T=\max \{1,C_T\}$. Therefore, estimates (\[estimativa ut 0\]) and (\[estimativa ut 1.7\]) read $$\begin{aligned} \label{estimativa ut 1.9} \|u\|_{X_T} \leq K \|f\|_{Y_T}.\end{aligned}$$ The function $$D(A) \longrightarrow X_T ;\ u_0 \mapsto S(t)u_0$$ is well defined and continuous. The proof follows the same steps as Lemma \[lemma1\], taking into account that now $f\equiv 0$. The resulting estimate is $$\begin{aligned} \label{estimativa linear XT} \|u\|_{X_T} & \leq & M \|u_0 \|_{D(A)},\end{aligned}$$ where $M$ is given by $$\begin{aligned} \label{estimativa linear M} M = 2C + 1 + \sqrt{1+L+T},\end{aligned}$$ and $C$ (which depends only on $\Omega$) is defined by continuous immersion $D(A) \hookrightarrow H^1_0(\Omega)\cap H^2(\Omega).$ \[contração\] Given $R>0$, consider the closed ball $ B_R = \{u \in X_T ; \|u\|_{X_T} \leq R \}.$ Then the operator $$\begin{aligned} \label{contraçao 0} \Phi : B_R \longrightarrow X_T ;\ %------------------------------- v \mapsto S(t)u_0 -\int_0^t S(t-s)v^2 v_x (s) \,ds \nonumber\end{aligned}$$ is the contraction. Fix $R>0$ and $u,v \in B_R.$ We have $$\begin{aligned} \Phi (v) - \Phi (u)= \int_0^t S(t-s)[u^2u_x - v^2 v_x] (s) \,ds \nonumber\end{aligned}$$ so that (\[estimativa ut 1.9\]) implies $$\begin{aligned} \label{contraçao 0.1} \| \Phi (u) - \Phi (v) \|_{X_T} \leq K \|u^2u_x - v^2v_x\|_{Y_T}.\end{aligned}$$ We study the right-hand norm in detail: $$\begin{aligned} \label{contaçao 1} \|u^2u_x - v^2v_x\|_{Y_T} &=& \|u^2u_x - v^2v_x\|_{L^{1}_T L^2_{xy}} +\left\|\big(u^2u_x\big)_t - \big(v^2v_x\big)_t\right\|_{L^{1}_Y L^2_{xy}} \nonumber \\ &=& I + J.\end{aligned}$$ First, we write $$\begin{aligned} \label{contaçao 2} I &=& \left\|(u^2 - v^2)u_x\right\|_{L^{1}_T L^2_{xy}} +\left\| v^2(u_x-v_x)\right\|_{L^{1}_T L^2_{xy}} \nonumber \\ &=& I_1 + I_2 .\end{aligned}$$ For the integral $I_1$ one has $$\begin{aligned} \label{contaçao 2.1} I_1 \leq \int_0^T \| u-v\|_{L^{6}(\Omega)} \| u+v\|_{L^{6}(\Omega)} \|u_x \|_{L^{6}(\Omega)} dt.\end{aligned}$$ Nirenberg’s inequality gives $$\begin{aligned} \label{contaçao 2.4} I_1 &\leq & T C_{\Omega} \| \nabla (u + v)\|^{\frac{2}{3}}_{L^{\infty}_T L^2_{xy}} \|u + v\|^{\frac{1}{3}}_{L^{\infty}_T L^2_{xy}} \| \nabla u_x\|^{\frac{2}{3}}_{L^{\infty}_T L^2_{xy}} \|u_x\|^{\frac{1}{3}}_{L^{\infty}_T L^2_{xy}} \| \nabla (u - v)\|^{\frac{2}{3}}_{L^{\infty}_T L^2_{xy}} \|u - v\|^{\frac{1}{3}}_{L^{\infty}_T L^2_{xy}} \nonumber \\ %============================= &= & T C_{\Omega} D^{\frac{2}{3}} \| u + v\|_{X_T} \|u\|_{X_T} \| u - v \|_{X_T} ,\end{aligned}$$ where $D$ is the Poincare’s constant from $\|w\|\leq D \| \nabla w\|.$ Since $u$ and $v$ lie in $B_R,$ we conclude $$\begin{aligned} \label{contaçao 2.5} I_1 \leq TK_0R^2 \| u - v \|_{X_T}.\end{aligned}$$ The integral $I_2$ can be treated in the similar way as $I_1$. It rests to estimate the integral $J$. $$\begin{aligned} \label{contaçao 3} J \leq \| 2u u_t(u_x - v_x) \|_{L^1_T L^2_{xy}} + \| u^2 (u_{xt} - v_{xt}) \|_{L^1_T L^2_{xy}} + \|2v_x u(u_t - v_t) \|_{L^1_T L^2_{xy}} \nonumber \\ %============================= + \|2v_x v_t (u-v) \|_{L^1_T L^2_{xy}} +\| v_{xt}(u-v)(u+v) \|_{L^1_T L^2_{xy}} \nonumber \\ %============================= = J_1 + J_2 + J_3 +J_4 + J_5.\end{aligned}$$ For $J_1$ we have $$\begin{aligned} \label{contaçao 3.1} J_1 & \leq \int_0^T \| u\|_{L^{6}(\Omega)} \| u_t\|_{L^{6}(\Omega)} \|u_x - v_x \|_{L^{6}(\Omega)} \, dt .\end{aligned}$$ Niremberg’s inequality implies $$\begin{aligned} \label{contaçao 3.2} J_1 \leq C_{\Omega} \| \nabla u\|^{\frac{2}{3}}_{L^{\infty}_T L^2_{xy}} \|u\|^{\frac{1}{3}}_{L^{\infty}_T L^2_{xy}} \| \nabla (u_x - v_x)\|^{\frac{2}{3}}_{L^{\infty}_T L^2_{xy}} \|u_x-v_x\|^{\frac{1}{3}}_{L^{\infty}_T L^2_{xy}}\|u_t\|^{\frac{1}{3}}_{L^{\infty}_T L^2_{xy}} \nonumber \\ \leq T^{\frac{2}{3}} K_2 R^2 \| u - v \|_{X_T}.\end{aligned}$$ The integrals $J_3$ and $J_4$ are analogous to $J_1$. To get bound for $J_5$ we observe that $$\begin{aligned} \label{contaçao 3.6} J_5 &=& \int_0^T \left(\int_{\Omega} v_{xt}^2 (u -v)^2 (u+v)^2 \, d\Omega\right)^{\frac{1}{2}} dt \nonumber \\ %============================== & \leq & \int_0^T \left( \sup (u-v)^2 \right)^{\frac{1}{2}} \left( \sup (u+ v)^2 \right)^{\frac{1}{2}} \|v_{xt}\|(t) \, dt \nonumber \\ %============================== &\leq & \int_0^T \big( \|u-v\|^2_{H^1_{xy}}(t) + \|u_{xy} - v_{xy}\|^2 (t) \big)^{\frac{1}{2}} \big( \|u+v\|^2_{H^1_{xy}}(t) + \|u_{xy} + v_{xy}\|^2 (t) \big)^{\frac{1}{2}} \| v_{xt}\|(t) \, dt \nonumber \\ %============================= &\leq & \big( \|u-v\|_{L^{\infty}_T H^1_{xy}} + \|u_{xy}- v_{xy}\|_{L^{\infty}_T L^2_{xy}} \big)\big( \|u+v\|_{L^{\infty}_T H^1_{xy}} + \|u_{xy}+ v_{xy}\|_{L^{\infty}_T L^2_{xy}} \big)\| v_{xt}\|_{L^{1}_T L^2_{xy}} \nonumber \\ %============================= & \leq & 4 T^{\frac{1}{2}} \| v\|_{X_T}\|u+v\|_{X_T} \|u-v\|_{X_T} \nonumber \\ %============================= & \leq & 8 T^{\frac{1}{2}} R^2 \|u-v\|_{X_T}.\end{aligned}$$ The integral $J_2$ follows like $J_5$. Thus, $$\begin{aligned} \label{contaçao 3.6} \|u^2u_x - v^2v_x\|_{Y_T} \leq K K^* T^{\frac{1}{2}}R^2 \|u-v\|_{X_T}.\end{aligned}$$ Finally, choosing $T>0$ such that $K K^* T^{\frac{1}{2}}R^2<1 ,$ we conclude that $\Phi $ is a contraction map. Lemma \[contração\] is proved. Let $u \in B_R$. If $R= 2 M\|u_0\|_{D(A)},$ then estimates (\[estimativa linear XT\]) and (\[contaçao 3.6\]) with $v \equiv 0$ assure $$\begin{aligned} \label{contaçao 3.7} \|u\|_{X_T} & \leq & \|S(t)u_0\|_{X_T} +\| \int_0^t S(t-s)u^2u_x \, ds \|_{X_T} \nonumber \\ %============================= & \leq & M\|u_0\|_{D(A)} + K K^* T^{\frac{1}{2}}R^2 \|u\|_{X_T} \nonumber \\ %============================= &\leq & \frac{R}{2} + K K^* T^{\frac{1}{2}}R^3.\end{aligned}$$ Setting $T>0$ such that $K K^* T^{\frac{1}{2}}R^3 < \frac{R}{2}, $ one get $$\begin{aligned} \label{contaçao 3.8} \|u\|_{X_T} \leq R.\end{aligned}$$ Choose $T>0$ such that $K K^* T^{\frac{1}{2}}R^2<1 $ and $ K K^* T^{\frac{1}{2}}R^3 < \frac{R}{2} .$ Then $\Phi$ is the contraction from the ball $B_R$ into itself. Therefore, the Banach fixed point theorem assures the existence of a unique element $u\in B_R$ such that $\Phi (u) = u.$ This completes the proof of Theorem \[Teo solu local mZK\]. Decay ===== Let $B,L>0$ satisfy $$\pi^2 \left[\frac{3}{L^2}+\frac{1}{4B^2}\right] - 1 :=2 A^2 >0\ \ \text{ and }\ \ \|u_0\|^2 < \dfrac{A^2}{2\pi^2\left( \frac{1}{L^2}+\frac{1}{4B^2}\right)}.$$ If there exists solution $$u\in L^{\infty}\left(0,\infty;H^1_0(\Omega)\right)$$ to -, then $$\label{decaimento 1} %------------------------------- \|u\|^2(t) \leq \left( 1+x, u^2 \right) (t) \leq e^{-\left(\frac{A^2}{(1+L)}\right)t}\left( 1+x, u^2_0 \right). %-------------------------------$$ To prove this result we will use \[prop 1 decaimento\] Let $L,B>0$ and $\omega \in H_0^1(\Omega)$. Then $$\label{decaimento 1.4} \int_0^L \int_{-B}^B \omega^2(x,y)dxdy \leq \frac{4B^2}{\pi^2}\int_0^L \int_{-B}^B \omega_y^2(x,y)dxdy,$$ and $$\label{decaimento 1.5} \int_0^L \int_{-B}^B \omega^2(x,y)dxdy \leq \frac{L^2}{\pi^2}\int_0^L \int_{-B}^B \omega_x^2(x,y)dxdy.$$ See [@doronin] for the proof. We start the proof of , multiplying by $u$ and integrating over $Q_t,$ which easily gives $$\label{decaimento 1.3} \| u\| ^2 (t) \leq \|u_0\|^2.$$ Multiplying by $(1+x)u$ and integrating over $\Omega,$ we have $$\begin{aligned} \label{decaimento 1.6} \dfrac{d}{dt} \left( 1+x, u^2 \right)(t) + \int_{-B}^B u^2_x (0,y,t) \, dy + \|\nabla u \|^2(t) + 2\|u_x\|^2(t) - \|u\|^2(t) \nonumber \\ %------------------------------- = -2\int_{\Omega}(1+ x) u (u^2u_x) \, d\Omega %\nonumber \\ %------------------------------- = \frac{1}{2}\int_{\Omega}u^4 \, d\Omega.%\nonumber \\\end{aligned}$$ For the integral $I_1 = \frac{1}{2}\int_{\Omega}u^4 = \frac{1}{2}\|u\|^4_{L^4(\Omega)}(t),$ Nirenberg’s inequality implies $$\begin{aligned} \label{decaimento 1.8} I_1 &\leq & \frac{1}{2} \big( 2^{\frac{1}{2}} \|\nabla u \|^{\frac{1}{2}}(t)\| u \|^{\frac{1}{2}} (t) \big)^4 \nonumber \\ %------------------------------- &=& 2 \|\nabla u \|^{2}(t)\| u \|^{2}(t)%\nonumber \\ %------------------------------- \leq 2 \|\nabla u \|^{2}(t)\| u_0\|^{2}(t).\end{aligned}$$ Take $$I_2 = 3\|u_x\|^2(t)+ \|u_y\|^2(t).$$ For all $\varepsilon>0$ we have $$I_2 = (3-\varepsilon)\|u_x\|^2(t)+ (1-\varepsilon)\|u_y\|^2(t) + \varepsilon\big( \|u_x\|^2(t)+ \|u_y\|^2(t) \big).$$ Lemma \[prop 1 decaimento\] jointly with (\[decaimento 1.6\]) and (\[decaimento 1.8\]) provides $$\begin{aligned} \label{decaimento 1.9} \dfrac{d}{dt} \left( 1+x, u^2 \right)(t) + \left[ \pi^2\left( \frac{3}{L^2}+\frac{1}{4B^2} \right) -1 - \varepsilon \pi^2\left( \frac{1}{L^2}+\frac{1}{4B^2} \right) \right]\|u\|^2(t) \nonumber \\ %------------------------------- + \left( \varepsilon - 2\|u_0\|^2 \right)\|\nabla u\|^2(t) \leq 0.\end{aligned}$$ Define $$2 A^2:=\pi^2 \left[\frac{3}{L^2}+\frac{1}{4B^2}\right] - 1 >0, %$$  = . $$ The result for reads $$\begin{aligned} \label{decaimento 1.10} \dfrac{d}{dt} \left( 1+x, u^2 \right)(t) + A^2\|u\|^2(t)+ \left( \varepsilon - 2\|u_0\|^2 \right)\|\nabla u\|^2(t) \leq 0.\end{aligned}$$ If $0\leq \varepsilon - 2\|u_0\|^2,$ then $$\begin{aligned} \label{decaimento 1.11} \dfrac{d}{dt} \left( 1+x, u^2 \right)(t) + \frac{A^2}{(1+L)}\left( 1+x, u^2 \right)(t) \leq 0,\end{aligned}$$ and consequently $$\label{decaimento 1.12} %------------------------------- \|u\|^2(t) \leq \left( 1+x, u^2 \right) (t) \leq e^{-\left(\frac{A^2}{(1+L)}\right)t}\left( 1+x, u^2_0 \right). %-------------------------------$$ The proof is completed. [99]{} J. L. Bona and R. W. Smith, The initial-value problem for the Korteweg-de Vries equation, Phil. Trans. Royal Soc. London Series A 278 (1975), 555–601. G. G. Doronin and N. A. Larkin, Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip, Proc. Edinb. Math. Soc. (2) 58 (2015), 661–-682. L. C. Evans, Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp. ISBN: 978-0-8218-4974-3 A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation (Russian), Differentsial’nye Uravneniya, 31 (1995), 1070–1081; Engl. transl. in: Differential Equations 31 (1995), 1002–1012. A. V. Faminskii, Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electronic Journal of Differential equations 127 (2008), 1–23. L. G. Farah, F. Linares and A. Pastor, A note on the 2D generalized Zakharov-Kuznetsov equation: Local, global, and scattering results, J. Differential Equations 253 (2012), 2558–2571. T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xxii+619 pp. ISBN: 3-540-58661-X O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, Rhode Island, 1968. F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal. 260 (2011), 1060–1085. F. Linares and A. Pastor, Well-posedness for the 2D modified Zakharov-Kuznetsov equation, J. . F. Linares, A. Pastor and J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV Soliton, Comm. Part. Diff. Equations 35 (2010), 1674–1689. F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Disc. Cont. Dynamical Systems A 24 (2009), 547–565. N. Larkin, J. Luckesi, Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals, Recommended to cite as: Larkin, N.A. & Luchesi, J. Appl Math Optim (2019). https://doi.org/10.1007/s00245-019-09579-w. J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Advances in Differential Equations 15 (2010), 1001–1031. J.-C. Saut, R. Temam and C. Wang, An initial and boundary-value problem for the Zakharov-Kuznetsov equation in a bounded domain, J. Math. Phys. 53 115612(2012). V. E. Zakharov and E. A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP 39 (1974), 285–286. [^1]: $^\ast$Partially supported by CAPES

--- abstract: 'Using adjoint representation of Lie superalgebras, we obtain the matrix form of super Jacobi and mixed super Jacobi identities of Lie super-bialgebras. By direct calculations of these identities, and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie super-bialgebras.' author: - | A. Eghbali , A. Rezaei-Aghdam F. Heidarpour\ \ title: ' Classification of two and three dimensional Lie super-bialgebras' --- **Introduction** ================ Lie bialgebras were first introduced, from the mathematical point of view, by Drinfel’d as algebraic structures and classical limit of underlying quantized enveloping algebras ([*quantum groups*]{}) [@Drin]. In particular, every deformation of a universal enveloping algebra induces a Lie bialgebra structure on the underlying Lie algebra. Conversely, it has been shown in Ref. [@Etin] that each Lie bialgebra admits quantization. So the classification of Lie bialgebras can be seen as the first step in the classification of quantum groups. Many interesting examples of Lie bialgebras based on complex semisimple Lie algebras have been given in Drinfel’d [@Drin]. A complete classification of Lie bialgebras with reduction was given in Ref. [@Del]. However, a classification of Lie bialgebras is out of reach, with similar reasons as for Lie algebra classification. In the non-semisimple case, only a bunch of low dimensional examples have been thoroughly studied . On the other hand, from the physical point of view, the theory of classical integrable systems naturally relates to the geometry and representation theory of Poisson-Lie groups and the corresponding Lie bialgebras and their classical r-matrices (see, for example, Ref. [@Kosmann]). In the same way, Lie super-bialgebras [@N.A], as the underlying symmetry algebras, play an important role in the integrable structure of $AdS/CFT$ correspondence [@Bs]. In this way, and by considering that there is a universal quantization for Lie super-bialgebras [@Geer], one can assign an important role to the classification of Lie super-bialgebras (especially low dimensional Lie super-bialgebras) from both physical and mathematical point of view. There are distinguished and nonsystematic ways for obtaining low dimensional Lie super-bialgebras (see, for example, Refs. ). In this paper, using the adjoint representation of Lie superalgebras, we present a systematic way for obtaining and classifying low dimensional Lie super-bialgebras. We apply this method to the classification of two and three dimensional Lie super-bialgebras. The paper is organized as follows. In section two, we give basic definitions and notations that are used throughout the paper. The systematic way for classification of Lie super-bialgebras by using matrix form of super Jacobi $(sJ)$ and mixed super Jacobi $(msJ)$ identities of Lie super-bialgebras is described in section three. A list of two and three dimensional Lie superalgebras of Ref. [@B] is offered in section 4. The automorphism Lie supergroups of these Lie superalgebras are also presented in section 4 . Then, using the method mentioned in section two, we classify all $48$ two and three dimensional Lie super-bialgebras in section five. The details of calculations are explained, using an example. The concluding section discusses some remarks. Certain properties of tensors and supermatrices are given in appendix A. Solutions of $(sJ-msJ)$ identities and their isomorphism matrices are give in appendix B. **Basic definitions and notations** ==================================== $\; \; \; \;$In this paper we use DeWitt notation for supervector spaces, supermatrices, etc [@D]. Some of definitions and related notations are given in appendix A. [*Definition*]{}: A [*Lie superalgebra*]{} ${\bf g}$ is a graded vector space ${\bf g}={\bf g}_B \oplus {\bf g}_F$ with gradings; $grade({\bf g}_B)=0,\; grade({\bf g}_F)=1$; such that Lie bracket satisfies the super antisymmetric and super Jacobi identities, i.e., in a graded basis $\{X_i\}$ of ${\bf g}$ if we put [@F1] $$[X_i , X_j] = {f^k}_{ij} X_k,$$ then $$(-1)^{i(j+k)}{f^m}_{jl}{f^l}_{ki} + {f^m}_{il}{f^l}_{jk} + (-1)^{k(i+j)}{f^m}_{kl}{f^l}_{ij}=0,$$ so that $${f^k}_{ij}=-(-1)^{ij}{f^k}_{ji}.$$ Note that, in the standard basis, ${f^B}_{BB}$ and ${f^F}_{BF}$ are real c-numbers and ${f^B}_{FF}$ are pure imaginary c-numbers and other components of structure constants ${f^i}_{jk}$ are zero [@D], i.e. we have $${f^k}_{ij}=0, \hspace{10mm} if \hspace{5mm} grade(i) + grade(j)\neq grade(k)\hspace{3mm} (mod 2).$$ Let ${\bf g}$ be a finite-dimensional Lie superalgebra and ${\bf g}^\ast$ be its dual vector space and let $( .~, ~. )$ be the canonical pairing on ${\bf g}^\ast \oplus {\bf g}$. [*Definition*]{}: A [*Lie super-bialgebra*]{} structure on a Lie superalgebra ${\bf g}$ is a super skew-symmetric linear map $\delta : {\bf g }\longrightarrow {\bf g}\otimes{\bf g}$ ([*the super cocommutator*]{}) so that [@N.A] 1) $\delta$ is a super one-cocycle, i.e. $$\delta([X,Y])=(ad_X\otimes I+I\otimes ad_X)\delta(Y)-(-1)^{|X||Y|}(ad_Y\otimes I+I\otimes ad_Y)\delta(X) \qquad \forall X,Y\in {\bf g},$$         where $|X|(|Y|)$ indicates the grading of $X(Y)$; 2\) the dual map ${^t}{\delta}:{\bf g}^\ast\otimes {\bf g}^\ast \to {\bf g}^\ast$ is a Lie superbracket on ${\bf g}^\ast$, i.e., $$(\xi\otimes\eta , \delta(X)) = ({^t}{\delta}(\xi\otimes\eta) , X) = ([\xi,\eta]_\ast , X) \qquad \forall X\in {\bf g} ;\;\, \xi,\eta\in{\bf g}^\ast.$$ The Lie super-bialgebra defined in this way will be indicated by $({\bf g},{\bf g}^\ast)$ or $({\bf g},\delta)$. [*Proposition*]{}: If there exists an automorphism $A$ of ${\bf g}$ such that $$\delta^\prime = (A\otimes A)\circ\delta\circ A^{-1},$$ then the super one-cocycles $\delta$ and $\delta^\prime$ of the Lie superalgebra $\bf g$ are [*equivalent*]{} [@F2]. In this case the two Lie super-bialgebras $({\bf g},\delta)$ and $({\bf g},\delta^\prime)$ are equivalent (as in the bosonic case [@RHR]). [*Definition*]{}: A [*Manin*]{} super triple [@N.A] is a triple of Lie superalgebras $(\cal{D} , {\bf g} , {\bf \tilde{g}})$ together with a non-degenerate ad-invariant super symmetric bilinear form $<.~, ~. >$ on $\cal{D}$ such that 1)$\;$${\bf g}$ and ${\bf \tilde{g}}$ are Lie sub-superalgebras of $\cal{D}$, 2)$\;$ $\cal{D} = {\bf g}\oplus{\bf \tilde{g}}$ as a supervector space, 3)$\;$ ${\bf g}$ and ${\bf \tilde{g}}$ are isotropic with respect to $< .~, ~. >$, i.e. $$<X_i , X_j> = <\tilde{X}^i , \tilde{X}^j> = 0, \hspace{10mm} {\delta_i}^j=<X_i , \tilde{X}^j> = (-1)^{ij}<\tilde{X}^j, X_i>=(-1)^{ij}{\delta^j}_i,$$ where $\{X_i\}$ and $\{\tilde{X}^i\}$ are basis of Lie superalgebras ${\bf g}$ and ${\bf \tilde{g}}$, respectively. Note that in the above relation ${\delta^j}_i$ is the ordinary delta function. There is a one-to-one correspondence between Lie super-bialgebra $({\bf g},{\bf g}^\ast)$ and Manin super triple $(\cal{D} , {\bf g} , {\bf \tilde{g}})$ with $ {\bf g}^\ast={\bf \tilde{g}}$ [@N.A]. If we choose the structure constants of Lie superalgebras ${\bf g}$ and ${\bf \tilde{g}}$ as $$[X_i , X_j] = {f^k}_{ij} X_k,\hspace{20mm} [\tilde{X}^i ,\tilde{ X}^j] ={{\tilde{f}}^{ij}}_{\; \; \: k} {\tilde{X}^k}, \\$$ then ad-invariance of the bilinear form $<.~ ,~. >$ on $\cal{D} = {\bf g}\oplus{\bf \tilde{g}}$ implies that $$[X_i , \tilde{X}^j] =(-1)^j{\tilde{f}^{jk}}_{\; \; \; \:i} X_k +(-1)^i {f^j}_{ki} \tilde{X}^k.$$ Clearly, using Eqs. (6) and (9) we have $$\delta(X_i) = (-1)^{jk}{\tilde{f}^{jk}}_{\; \; \; \:i} X_j \otimes X_k,$$ note that the appearance of $(-1)^{jk}$ in this relation is due to the definition of natural inner product between ${\bf g}\otimes {\bf g}$ and ${\bf g}^\ast \otimes {\bf g}^\ast $ as $ (\tilde{X}^i \otimes \tilde{ X}^j ,X_k \otimes X_l)=(-1)^{jk}{\delta^i}_k {\delta^j}_l$. As a result, if we apply this relation in the super one-cocycle condition (5), super Jacobi identities (2) for the dual Lie superalgabra and the following mixed super Jacobi identities are obtained $${f^m}_{jk}{\tilde{f}^{il}}_{\; \; \; \; m}= {f^i}_{mk}{\tilde{f}^{ml}}_{\; \; \; \; \; j} + {f^l}_{jm}{\tilde{f}^{im}}_{\; \; \; \; \; k}+ (-1)^{jl} {f^i}_{jm}{\tilde{f}^{ml}}_{\; \; \; \; \; k}+ (-1)^{ik} {f^l}_{mk}{\tilde{f}^{im}}_{\; \; \; \; \; j}.$$ This relation can also be obtained from super Jacobi identity of $\cal{D}$. **Calculation of Lie super-bialgebras using adjoint representation** ===================================================================== $\; \; \; \;$As discussed in the introduction, by use of super Jacobi and mixed super Jacobi identities we classify two and three dimensional Lie super-bialgebras. This method was first used to obtain three dimensional Lie bialgebras in Ref. [@JR]. Because of tensorial form of super Jacobi and mixed super Jacobi identities, working with them is not so easy and we suggest writing these equations as matrix forms using the following adjoint representations for Lie superalgebras ${\bf g}$ and ${\bf \tilde{g}}$ $$({\tilde{\cal X}}^i)^j_{\; \;k} = -{\tilde{f}^{ij}}_{\; \; \; k}, \hspace{10mm} ({\cal Y}^i)_{ \;jk} = -{f^i}_{jk}.$$ Then the matrix forms of super Jacobi identities (2) for dual Lie superalgebra $\tilde {\bf g}$ and mixed super Jacobi identities (12) become as follows, respectively: $$({\tilde{\cal X}}^i)^j_{\; \;k}{\tilde{\cal X}}^k - {\tilde{\cal X}}^j {\tilde{\cal X}}^i + (-1)^{ij}{\tilde{\cal X}}^i {\tilde{\cal X}}^j = 0,$$ $${({\tilde{\cal X}}^i)}^j_{\; \;l}\;{\cal Y}^l =-(-1)^{k} ({\tilde{\cal X}}^{st})^{j}\; {\cal Y}^i + {\cal Y}^j{\tilde{\cal X}}^i - (-1)^{ij}{\cal Y}^i {\tilde{\cal X}}^j + (-1)^{k+ij} ({\tilde{\cal X}}^{st})^{i} \;{\cal Y}^j.$$ Note that in the above relations we use the right indices for the matrix elements and index $k$ represents the column of matrix ${\tilde{\cal X}}^{st}$. Having the structure constants of the Lie superalgebra ${\bf g}$, we solve the matrix Eqs. (14) and (15) in order to obtain the structure constants of the dual Lie superalgebras ${\bf {\tilde g}}$ so that $({\bf g}, {\bf {\tilde g}})$ is Lie super-bialgebra. We assume that, we have classification and tabulation of the Lie superalgebras ${\bf g}$ (for example two and three dimensional Lie superalgebras for this paper). Here we explain the method in general and one can apply it for classification of other low dimensional Lie super-bialgebras [@RE]. We fulfill this work in the following three steps. Note that in solving Eqs. (14), (15), (17) and (21) we use the MAPLE program. [*Step 1:*]{}  [*Solutions of super Jacobi and mixed super Jacobi identities and determination of Lie superalgebras $\bf g'$ which are isomorphic with dual solutions*]{} With the solution of matrix Eqs. (14) and (15) for obtaining matrices ${\tilde{\cal X}}^i$, some structure constants of ${\bf {\tilde g}}$ are obtained to be zero, some unknown and some obtained in terms of each other. In order to know whether ${\bf {\tilde g}}$ is one of the Lie superalgebras of table or isomorphic to them, we must use the following isomorphic relation between obtained Lie superalgebras ${\bf {\tilde g}}$ and one of the Lie superalgebras of table, e.g., $\bf g'$. Applying the transformation (37) for a change of basis ${\bf {\tilde g}}$ we have $$\tilde{X}^{'\;i}=C^i_{\;\;j}\tilde{X}^j,\hspace{20mm} [\tilde{X}^{'\;i} ,\tilde{X}^{'\;j}] ={\tilde{f}^{'\;ij}}_{\; \; k} \tilde{X}^{'\;k},$$ then the following matrix equations for isomorphism $$(-1)^{i(j+l)}\; C\;(C^i_{\;\;k}\;\tilde{\cal X}^k_{\tilde{(\bf g)}})={\cal X}^{i}_{(\bf g')}\;C,$$ are obtained, where the indices $j$ and $l$ correspond to the row and column of $C$ in the left hand side of (17). In the above matrix equations ${\cal X}^{i}_{(\bf g')}$ are adjoint matrices of known Lie superalgebra $\bf g'$ of the classification table. Solving (17) with the condition $sdet C\neq 0$ we obtain some extra conditions on ${{\tilde{f}^{kl}}_{(\bf {\tilde g})}}\hspace{1mm}_m$’s that were obtained from (14) and (15). [*Step 2:*]{}  [*Obtaining general form of the transformation matrices $B:{\bf g}'\longrightarrow {\bf g}'.i$; such that $({\bf g}, {\bf g}'.i)$ are Lie super-bialgebras*]{} As the second step we transform Lie super-bialgebra $({\bf g}, {\bf {\tilde g}})$ (where in the Lie superalgebra $\bf {\tilde g}$ we impose extra conditions obtained in the step one) to Lie super bialgebra $({\bf g}, {\bf g}'.i)$ (where ${\bf g}'.i$ is isomorphic as Lie superalgebra to ${\bf g}'$) with an automorphism of the Lie superalgebra ${\bf g}$. As the inner product (8) is invariant we have $A^{-st}:{\bf {\tilde g}}\longrightarrow {\bf g}'.i$, $$X'_i=(-1)^k A_i^{\;\;k} X_k,\hspace{10mm}\tilde{X}^{'\;j}=(A^{-st})^j_{\;\;l}\tilde{X}^l,\hspace{10mm}<X'_i , \tilde{X}^{'\;j}> = \delta_i^{\;\;j},$$ where $A^{-st}$ is superinverse supertranspose of every matrix $A\in Aut(\bf g)$ \[$Aut(\bf g)$ is the automorphism supergroup of the Lie superalgebra $\bf g$\]. Thus we have the following transformation relation for the map $A^{-st}$: $$(-1)^{k(j+l)} (A^{-st})^i_{\;\;k}{{\tilde{f}^{kl}}_{(\bf {\tilde g})}}\hspace{1mm}_m (A^{-st})^j_{\;\;l} = {{{{f}}^{ij}}_{(\bf {g}'.i)}}\hspace{0.5mm}_n (A^{-st})^n_{\;\;m}.\\$$ Now, for obtaining Lie super-bialgebras $({\bf g}, {\bf g}'.i)$, we must find Lie superalgebras ${\bf g}'.i$ or transformations\ $B:{\bf g}'\longrightarrow {\bf g}'.i$, such that $$(-1)^{k(j+l)} B^i_{\;\;k}{{{f}^{kl}}_{(\bf { g}')}}\hspace{0.5mm}_m B^j_{\;\;l} = {{{{f}}^{ij}}_ {(\bf {g}'.i)}}\hspace{0.5mm}_n B^n_{\;\;m}.\\$$ For this purpose, it is enough to omit ${{{{f}}^{ij}}_ {(\bf {g}'.i)}}\hspace{1mm}_n $ between (19) and (20). Then we will have the following matrix equation for $B$: $$(A^{-st})^i_{\;\;m}\tilde{\cal X}^{st\;m}_{\tilde{(\bf g)}}A^{-1} = (-1)^{i(l+j)}(B^{st} A)^{-1}(B^i_{\;\;k}{{\cal X}^{st\;k}}_{({\bf g}')}) B^{st},\\$$ where indices $l$ and $j$ are row and column of $B^{st}$ in the right hand side of the above relation. Note that for $A^{-st}$ and $A^{-1}$ in $(21)$ we use row and column indices on the right hand sides. Now by solving (21) we obtain the general form of matrix $B$ with the condition $sdetB \neq 0$. In solving (21) one can obtain conditions on elements of matrix $A$, yet we must only consider those conditions under which we have $sdet A\neq 0$ and matrices $A$, $B$ and $A^{-st}$ have the general transformation matrix form (38). [*Step 3:*]{}  [*Obtaining and classificating the nonequivalent Lie super-bialgebras*]{} Having solved (21), we obtain the general form of the matrix $B$ so that its elements are written in terms of the elements of matrices $A$, $C$ and structure constants ${{\tilde{f}^{ij}}_{(\bf {\tilde g})}}\hspace{0.5mm}_k $. Now with substituting $B$ in $(20)$, we obtain structure constants ${{{{f}}^{ij}}_ {(\bf {g}'.i)}}\hspace{0.5mm}_n $ of the Lie superalgebra ${\bf g}'.i$ in terms of elements of matrices $A$ and $C$ and some ${{\tilde{f}^{ij}}_{(\bf {\tilde g})}}\hspace{1mm}_k $. Then we check whether it is possible to equalize the structure constants ${{{{f}}^{ij}}_ {(\bf {g}'.i)}}\hspace{0.5mm}_n$ with each other and with $\pm1$ or not so as to remark $sdet B\neq0$, $sdet A\neq0$, and $sdet C\neq0$. In this way, we obtain isomorphism matrices $B_1$, $B_2$,.... As $B_i^{st}$ must be in the form of transformation matrices (38), we obtain conditions on the matrices $B_i$. The reason is that if $({\bf g}, {\bf g}'.i)$ is Lie super-bialgebra, then $( {\bf g}',{\bf g}.i)$ will be Lie super-bialgebras with $B^{st}_i:{\bf g}\longrightarrow {\bf g}.i$. Note that in obtaining $B_i$s we impose the condition $B{B_i}^{-1}\in Aut^{st}(\bf g)$ \[$Aut^{st}(\bf g)$ is the supertranspose of $Aut(\bf g)$\]; if this condition is not satisfied then we cannot impose it on the structure constants because $B$ and $B_i$ are not equivalent (see bellow). Now using isomorphism matrices $B_1$, $B_2$, etc, we can obtain Lie super-bialgebras $({\bf g}, {\bf g}'.i)$, $({\bf g}, {\bf g}'.ii)$, etc. On the other hand, there is the question: which of these Lie super-bialgebras are equivalent? In order to answer this question, we use the matrix form of the relation (7). Consider the two Lie super-bialgebras $({\bf g}, {\bf g}'.i)$, $({\bf g}, {\bf g}'.ii)$; then using [@F3] $$A(X_i)=(-1)^j A_i^{\;\;j} X_j,$$ relation (7) will have the following matrix form: $$(-1)^{i(j+l)} A^{st} ((A^{st})^i_{\;\;k}{{\cal X}_{({\bf g}'.i)}}^k) = {{{\cal X}}_{({\bf g}'.ii)}}^i A^{st}.\\$$ On the other hand, the transformation matrix between ${\bf g}'.i$ and ${\bf g}'.ii$ is $B_2B_1^{-1}$ if $B_1:{\bf g}'\longrightarrow {\bf g}'.i$ and $B_2:{\bf g}'\longrightarrow {\bf g}'.ii$; then we have $$(-1)^{i(j+l)} (B_2B_1^{-1}) ((B_2B_1^{-1})^i_{\;\;k}{{\cal X}_{({\bf g}'.i)}}^k) = {{{\cal X}}_{({\bf g}'.ii)}}^i (B_2B_1^{-1}).\\$$ A comparison of $(24)$ with $(23)$ reveals that if $B_2B_1^{-1}\in A^{st}$ holds, then the Lie super-bialgebras $({\bf g}, {\bf g}'.i)$ and $({\bf g}, {\bf g}'.ii)$ are equivalent. In this way, we obtain nonequivalent class of $B_i$s and we consider only one element of this class. Thus, we obtain and classify all Lie super-bialgebras. In the next section, we apply this formulation to two and three dimensional Lie superalgebras. **Two and three dimensional Lie superalgebras and their automorphism supergroups** =================================================================================== In this section, we use the classification of two and three dimensional Lie superalgebras listed in Ref. [@B]. In this classification, Lie superalgebras are divided into two types: trivial and nontrivial Lie superalgebras for which the commutations of fermion-fermion is zero or nonzero, respectively (as we use DeWitt notation here, the structure constant $C^B_{FF}$ must be pure imaginary). The results have been presented in tables $1$ and $2$. As the tables $(m, n-m)$ indicate, the Lie superalgebras have $m, \{X_1,...,X_m\}$ bosonic and $n-m, \{X_{m+1},...,X_n\}$ fermionic generators. :\ --------------------------------------------------------------------------------------------------------------------------------------- [Type ]{} [${\bf g}$ ]{} [Bosonic basis]{} [Fermionic basis]{} [Non-zero commutation relations]{} [Comments]{} ------------- ---------------- ------------------- --------------------- ------------------------------------------ ------------------- [$(1,1)$]{} [$B$]{} [$X_1$]{} [$X_2$]{} [$[X_1,X_2]=X_2$]{} [$(2,1)$]{} [${C^1_p}$]{} [$X_1,X_2$]{} [$X_3$]{} [$[X_1,X_2]=X_2,\; [X_1,X_3]=pX_3 $]{} [$p\neq0$ ]{} [$ C^2_p$]{} [$X_1$]{} [$X_2,X_3$]{} [$[X_1,X_2]=X_2, \;[X_1,X_3]=pX_3$ ]{} [$0<|p|\leq 1$]{} [$C^3$]{} [$X_1$]{} [$X_2,X_3$]{} [$[X_1,X_3]=X_2 [Nilpotent ]{} $]{} [$(1,2)$]{} [$C^4$]{} [$X_1$]{} [$X_2,X_3$]{} [$[X_1,X_2]=X_2,\; [X_1,X_3]=X_2+X_3$]{} [$C^5_p$]{} [$X_1$]{} [$X_2,X_3$]{} [$[X_1,X_2]=pX_2-X_3,\; [$p\geq0 $]{} [X_1,X_3]=X_2+pX_3$ ]{} --------------------------------------------------------------------------------------------------------------------------------------- : ([@F4]).\ ------------------------------------------------------------------------------------------------------------------------------------------------------- [Type ]{} [${\bf g}$ ]{} [Bosonic basis]{} [Fermionic basis]{} [Non-zero (anti)commutation relations]{} [Comments]{} ------------- -------------------------- ------------------- --------------------- --------------------------------------------------- ---------------- [$(1,1)$]{} [$(A_{1,1}+A)$]{} [$X_1$]{} [$X_2$]{} [$\{X_2,X_2\}=iX_1$]{} [$(2,1)$]{} [$C^1_{\frac{1}{2}}$ ]{} [$X_1,X_2$]{} [$X_3$]{} [$[X_1,X_2]=X_2,\; [X_1,X_3]=\frac{1}{2}X_3,\;\{X_3,X_3\}=iX_2 $ ]{} [$(A_{1,1}+2A)^1$]{} [$X_1$]{} [$X_2,X_3$]{} [$\{X_2,X_2\}=iX_1,\; \{X_3,X_3\}=iX_1 $]{} [Nilpotent]{} [$(A_{1,1}+2A)^2$]{} [$X_1$]{} [$X_2,X_3$]{} [$\{X_2,X_2\}=iX_1, [Nilpotent ]{} \;\{X_3,X_3\}=-iX_1 $]{} ------------------------------------------------------------------------------------------------------------------------------------------------------- One can check whether super Jacobi identities for the above Lie superalgebras are satisfied. As mentioned in section 3 for obtaining dual Lie superalgebras we need automorphism supergroups of Lie superalgebras. In order to calculate the automorphism supergroups of two and three dimensional Lie superalgebras, we use the following transformation: $${X'_i}=(-1)^j\;A_i^{\;\;j} X_j,\hspace{20mm} [X'_i ,X'_j] =f^k_{\; \; ij} X'_k, \\$$ thus we have the following matrix equation for elements of automorphism supergroups: $$(-1)^{ij+mk} A {\cal Y}^k A^{st} = {\cal Y}^e A_e^{\;\;k},\\$$ where index $j$ is the column of matrix $A$ and indices $i$ and $m$ are row and column of $A^{st}$ in the left hand side. Using (26) with the condition $sdet A\neq 0$ and imposing the condition that $A$ must be in the form (38), we obtain table $3$ for automorphism supergroups.\ [ ]{}: [Automorphism supergroups of the two and three dimensional Lie superalgebras.]{}\ [ l l l l p[30mm]{} ]{} & [Automorphism supergroups]{}&       [Comments]{}\ &[$\left(\begin{array}{cc} 1 & 0 \\ 0 & b \\ \end{array} \right)$]{}&       [$b \in \Re-\{0\}$]{}\ [$ (A_{1,1}+A)$]{} & [$\left(\begin{array}{cc} a^2 & 0 \\ 0 & a \\ \end{array} \right)$ ]{}&       [$a\in\Re-\{0\}$]{}\ [$ C^1_p$]{} & [$\left( \begin{array}{ccc} 1 & a & 0 \\ 0 & c & 0 \\ 0 & 0 & d \end{array} \right)$]{}&       [$p \in \Re-\{0\},\;\;\;c,d\in\Re-\{0\}$, $a\in\Re$ ]{}\ [$ C^1_\frac{1}{2}$]{} &[$\left(\begin{array}{ccc} 1 & a & 0 \\ 0 & b^2 & 0 \\ 0 & 0 & b \end{array} \right)$]{}&       [$b\in\Re-\{0\}$ , $a\in\Re$ ]{}\ [$C^2_1$]{} &[$\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & b \\ 0 & a & d \end{array} \right)$]{}&       [$ab- cd\neq 0$ ]{}\ [$ C^2_p$ ]{} & [$\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & 0 \\ 0 & 0 & d \end{array} \right)$]{}&       [$c,d\in\Re-\{0\},\;\;p \in [-1,1]-\{0\}$]{}\ [$ C^3$]{} &[$\left( \begin{array}{ccc} a & 0 & 0 \\ 0 & ad & 0 \\ 0 & e & d \end{array} \right)$]{} &       [$a,d\in\Re-\{0\}$, $e\in\Re$]{}\ [$ C^4$]{} & [$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & 0 \\ 0 & d & c \end{array} \right)$]{} &        [$c\in\Re-\{0\},\;\;d\in\Re$]{}\ [$ C^5_p$ ]{} &[$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & -d \\ 0 & d & c \end{array} \right)$]{} &       [$p\geq 0,\;\;c\in\Re-\{0\}$ or $d\in\Re-\{0\}$]{}\ [$(A_{1,1}+2A)^1$ ]{} & [$\left(\begin{array}{ccc} b^2+c^2 & 0 & 0 \\ 0 & b & -c \\ 0 & c & b \end{array} \right)$]{} &       [$b\in\Re-\{0\}\verb""$]{} [or]{} [$c\in\Re-\{0\}\verb""$]{}\ [$ (A_{1,1}+2A)^2$]{} &[$\left(\begin{array}{ccc} b^2-c^2 & 0 & 0 \\ 0 & b & c \\ 0 & c & b \end{array} \right)$]{}&        [$b\in\Re-\{0\}\verb""$]{} [or]{} [$c\in\Re-\{0\}\verb""$]{}\ **Two and three dimensional Lie super-bialgebras** =================================================== Using the methods discussed in section $3$, we can classify two and three dimensional Lie super-bialgebras. We have applied MAPLE $9$ and obtained $48$ Lie super-bialgebras. These have been listed in the following tables 4-6: :\ [l l l l p[5mm]{} ]{} & &\ [$(A_{1,1}+A)$]{} &&\ &\ [$B$]{}& &&\ & & &\ :\ [l l l l p[0.15mm]{} ]{} & [$\tilde{\bf g}$]{} &[Non-zero (anti) commutation relations of $\tilde{\bf g}$]{}& [Comments]{}\ &[$I_{(2 , 1)} $]{}& &&\ [$C^1_p$]{}&&&\ &[$C^1_{-p}.i$ ]{}&[$[{\tilde X}^1,{\tilde X}^2]={\tilde X}^1,\;\;\;[{\tilde X}^2,{\tilde X}^3]=p{\tilde X}^3$]{} &[$p\in\Re-\{0\}$]{}\ &[$I_{(2 , 1)} $]{}&\ &[$C^1_{p}.i{_{|_{p=-\frac{1}{2}}}}$]{}&[$[{\tilde X}^1,{\tilde X}^2]= {\tilde X}^1,\;\;\;[{\tilde X}^2,{\tilde X}^3]=\frac{1}{2} {\tilde X}^3$]{}&\ &[$C^1_{p}.ii{_{|_{p=-\frac{1}{2}}}}$]{}&[$[{\tilde X}^1,{\tilde X}^2]=-{\tilde X}^1,\;\;\;[{\tilde X}^2,{\tilde X}^3]=-\frac{1}{2} {\tilde X}^3$]{}&\ [$C^1_{\frac{1}{2}}$]{}&\ &[$C^1_{\frac{1}{2}}.i$ ]{}&[$[{\tilde X}^1,{\tilde X}^2]={\tilde X}^1,\;\;\;[{\tilde X}^2,{\tilde X}^3]=-\frac{1}{2} {\tilde X}^3,\;\;\; \{{\tilde X}^3,{\tilde X}^3\}=i{\tilde X}^1$]{}&\ &[$C^1_{\frac{1}{2}}.ii$ ]{}&[$[{\tilde X}^1,{\tilde X}^2]=-{\tilde X}^1,\;\;\;[{\tilde X}^2,{\tilde X}^3]=\frac{1}{2} {\tilde X}^3,\;\;\; \{{\tilde X}^3,{\tilde X}^3\}=-i{\tilde X}^1$]{}&\ &[${{C^1_{\frac{1}{2},k}}}$]{}&[$[{\tilde X}^1,{\tilde X}^2]=k{\tilde X}^2,\;[{\tilde X}^1,{\tilde X}^3]=\frac{k}{2}{\tilde X}^3,\;\{{\tilde X}^3,{\tilde X}^3\}=ik{\tilde X}^2$]{}&[$k\in {\Re-\{0\}}$]{}\ :\ [l l l p[0.5mm]{} ]{} & [$\tilde{\bf g}$]{} &[Comments ]{}\ &[$I_{(1 , 2)} $]{}&\ [$C^2_1$]{}&[$(A_{1,1}+2A)^1_{\epsilon,0,\epsilon}$ ]{} & &\ &[$(A_{1,1}+2A)^2_{\epsilon,0,-\epsilon}$ ]{}& &\ &[$I_{(1 , 2)} $]{}&\ [$C^2_p$]{}& [$(A_{1,1}+2A)^1_{\epsilon , k,\epsilon }$]{} &[$ -1<k<1$]{}&\ [$p \in [-1,1)-\{0\}$ ]{}&[$(A_{1,1}+2A)^2_{0 ,1,0}\;,\;\;(A_{1,1}+2A)^2_{\epsilon,1,0}\;,\;\;(A_{1,1}+2A)^2_{0,1,\epsilon}\;,\;\; (A_{1,1}+2A)^2_{\epsilon ,k,-\epsilon }$]{} &[$k\in\Re$]{}&\ & [$I_{(1 , 2)} $]{}&\ [$C^3$]{}&[$(A_{1,1}+2A)^1_{\epsilon , 0,\epsilon }$]{} &&\ &[$(A_{1,1}+2A)^2_{0 ,\epsilon,0 }\;,\;\;(A_{1,1}+2A)^2_{\epsilon ,0,-\epsilon}$]{} &&\ &[$I_{(1 , 2)} $]{}&\ [$C^4$]{}&[$(A_{1,1}+2A)^1_{k , 0,1}\;,\;\;(A_{1,1}+2A)^1_{s , 0,-1}$]{} &[$0<k,\;\;s<0$]{}&\ &[$(A_{1,1}+2A)^2_{0 ,\epsilon,0 }\;,\;\;(A_{1,1}+2A)^2_{k ,0,1}\;,\;\;(A_{1,1}+2A)^2_{s,0,-1}$]{} &[$k<0,\;\;0<s$]{}\ & [$I_{(1 , 2)} $]{}&\ [$C^5_p$]{}&[$(A_{1,1}+2A)^1_{k , 0,1}\;,\;\;(A_{1,1}+2A)^1_{s , 0,-1}$]{} &[$0<k,\;\;s<0$]{}&\ [$p\geq 0$]{}&[$(A_{1,1}+2A)^2_{k ,0,1}\;,\;\;(A_{1,1}+2A)^2_{s,0,-1}$]{} &[$ k<0,\;\;0<s$]{}\ [$(A_{1,1}+2A)^1$]{}&[$I_{(1 , 2)} $]{}&\ [$(A_{1,1}+2A)^2$]{}& [$I_{(1 , 2)} $]{}&\ For three dimensional dual Lie superalgebras $(A_{1,1}+2A)^1_{\alpha ,\beta,\gamma }\;$ and $\; (A_{1,1}+2A)^2_{\alpha ,\beta,\gamma }$ which are isomorphic with $(A_{1,1}+2A)^1$ and $(A_{1,1}+2A)^2$, respectively, we have the following commutation relations: $$\{{\tilde X}^2,{\tilde X}^2\}=i\alpha {\tilde X}^1 ,\quad \{{\tilde X}^2,{\tilde X}^3\}=i \beta {\tilde X}^1 ,\quad \{{\tilde X}^3,{\tilde X}^3\}=i\gamma {\tilde X}^1,\qquad \alpha, \beta, \gamma \in\Re.$$ Note that these Lie superalgebras are non isomorphic and they differ in the bound of their parameters. Meanwhile note that for every Lie superalgebra of tables 1 and 2 there are trivial solutions for Eqs. (14) and (15) with ${\tilde f}^{ij}_{\;\;k}=0$. (i.e. the dual Lie superalgebra is Abelian) we denote these dual Lie superalgebras with $I_{(m,n)}$. We see that for two and three dimensional Lie super-bialgebras (with two fermions), trivial Lie superalgebras are only dual to nontrivial one. The solution of super Jacobi and mixed super Jacobi identities and related isomorphism matrices $C$ (in step 1) are listed in Appendix $B$. Here for explanation of steps 1-3 of section 3 we give an example. **An example** -------------- What follows is an explanation of the details of calculations for obtaining dual Lie superalgebras of Lie super-bialgebras $(C^4 , (A_{1,1}+2A)^1_{k , 0,1})$ and $(C^4 , (A_{1,1}+2A)^1_{s , 0,-1})$. As mentioned in Appendix $B$, the solution of super Jacobi and mixed super Jacobi identities has the following form: $${{\tilde{f}}^{22}}_{\; \; \: 1}=i\alpha,\qquad {{\tilde{f}}^{33}}_{\; \; \; 1}=i\beta ,\qquad {{\tilde{f}}^{23}}_{\; \; \; 1}=i\gamma,\;\;\;\;\forall \alpha, \beta, \gamma \in \Re. \hspace{3cm}$$ This solution is isomorphic with the Lie superalgebra $(A_{1,1}+2A)^1$ with the following matrix: $$C_1=\left( \begin{tabular}{ccc} $ c_{11}$& 0 & 0 \\ $ c_{21}$ & $c_{22}$ & $ c_{23}$ \\ $c_{31}$ & 0 & $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ where $c_{11}= -ic^2_{33} {{\tilde{f}}^{33}}_{\; \; \; 1},\; c_{23}= -c_{22} \frac{{{\tilde{f}}^{23}}_{\; \; \; 1}}{{{\tilde{f}}^{33}}_{\; \; \; 1}},\; {{\tilde{f}}^{22}}_{\; \; \;1}=(\frac{c^2_{23}+c^2_{33}}{c^2_{22}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{22}$, $c_{33}\in\Re-\{0\}$. By imposing that $C_1$ must be the transformation matrix $(38)$, we have $c_{31}=c_{21}=0$.\ Now with the help of automorphism supergroup of $C^4$, the solution of $(21)$ for the matrix $B$ will be $$B=\left( \begin{tabular}{ccc} $ \frac{c^2(b^2_{32}+b^2_{33})}{\beta}$& 0 & 0 \\ $ 0$ & $-\frac{b_{33}c_{33}}{c_{22}}$ & $ \frac{b_{32}c_{33}}{c_{22}}$\\ $0$ & $b_{32}$ & $b_{33} $ \\ \end{tabular} \right),\;\;\;\beta=-i {{\tilde{f}}^{33}}_{\; \; \; 1}.\hspace{2cm}$$ Now using $(20)$, we obtain the following commutation relations for the dual Lie superalgebra ${(A_{1,1}+2A)^1}{'}$: $$\{{\tilde X}^2,{\tilde X}^2\}=ia^{'}{\tilde X}^1,\qquad \{{\tilde X}^3,{\tilde X}^3\}=ib^{'}{\tilde X}^1,\;\;\;\;\;\;a^{'}=\frac{\beta c^2_{33}}{c^2 c^2_{22}},\;\;b^{'}=\frac{\beta}{c^2},$$ such that $a^{'}$ and $b^{'}$ have the same sings.\ One cannot choose $a^{'}=b^{'}$ because in this case we have $$B_1=\left( \begin{tabular}{ccc} $\frac{{c}^2({b^{'}}^2_{32}+{b^{'}}^2_{33})}{\beta}$& 0 & 0 \\ $ 0$ & $-b^{'}_{33}$ & $b^{'}_{32}$ \\ $0$ &$b^{'}_{32}$ & $b^{'}_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ so as $BB^{-1}_1\notin A^{st}(C^4)$. Now by choosing $b^{'}=1$ i.e., $\beta=c^2$ we have $$B_2=\left( \begin{tabular}{ccc} ${b^{''}}^2_{32}+{b^{''}}^2_{33}$& 0 & 0 \\ $ 0$ & $-\frac{c^{''}_{33}}{c^{''}_{22}}b^{''}_{33}$ & $\frac{c^{''}_{33}}{c^{''}_{22}}b^{''}_{32}$ \\ $0$ & $b^{''}_{32}$ & $b^{''}_{33} $ \\ \end{tabular} \right). \hspace{2cm}$$ such that $BB^{-1}_2\in A^{st}(C^4)$, this means that we can choose $b^{'}=1$ and in this case, we have the following dual Lie superalgebras ($a^{'}=\frac{c^2_{33}}{c^2_{22}}=k>0$):\ $$(A_{1,1}+2A)^1_{k , 0,1}:\hspace{1cm} \quad \{{\tilde X}^2,{\tilde X}^2\}=ik{\tilde X}^1,\qquad \{{\tilde X}^3,{\tilde X}^3\}=i{\tilde X}^1,\;\;k>0. \hspace{1cm}$$ In the same way by choosing $b^{'}=-1$ i.e., $\beta=-c^2$ such that $a^{'}\neq b^{'}$ we have $$B_3=\left( \begin{tabular}{ccc} $-({b^{'''}}^2_{32}+{b^{'''}}^2_{33})$& 0 & 0 \\ $ 0$ & $-\frac{c^{'''}_{33}}{c^{'''}_{22}}b^{'''}_{33}$ & $\frac{c^{'''}_{33}}{c^{'''}_{22}}b^{'''}_{32}$ \\ $0$ & $b^{'''}_{32}$ & $b^{'''}_{33} $ \\ \end{tabular} \right). \hspace{2cm}$$ so as $BB^{-1}_3\in A^{st}(C^4)$, this means that we can choose $b^{'}=-1$ and in this case, we have the following dual Lie superalgebras ($a^{'}=-\frac{c^2_{33}}{c^2_{22}}=s<0$):\ $$(A_{1,1}+2A)^1_{s , 0,-1}:\hspace{1cm} \{{\tilde X}^2,{\tilde X}^2\}=is{\tilde X}^1,\qquad \{{\tilde X}^3,{\tilde X}^3\}=-i{\tilde X}^1,\;\;s<0. \hspace{1cm}$$ Note that as $B_2B^{-1}_3\notin A^{st}(C^4)$, this means that the Lie super-bialgebras $(C^4 , (A_{1,1}+2A)^1_{k , 0,1})$ and $(C^4 , (A_{1,1}+2A)^1_{s , 0,-1})$ are not equivalent. **Conclusion** ============== We have presented a new method for obtaining and classifying low dimensional [*Lie super-bialgebras*]{} which can also be applied for classification of low dimensional [*Lie bialgebras*]{}. We have classified all of the $48$ two and three dimensional Lie super-bialgebras [@F5]. Using this classification one can construct Poisson-Lie T-dual sigma models [@K.S1] on the low dimensional Lie supergroups [@ER] (as Ref. [@JR] for Lie groups). Moreover, one can determinate the coboundary type of these Lie super-bialgebras and their classical r-matrices and Poisson brackets for their Poisson-Lie supergroups [@ER1]. Determination of doubles and their isomorphism and investigation of Poisson-Lie plurality [@Von] are open problems for further investigation.\ [**Acknowledgments**]{} We would like to thank S. Moghadassi and F. Darabi for carefully reading the manuscript and useful comments. [**Appendix A: Some properties of matrices and tensors in supervector spaces**]{} In this appendix, we will review some basic properties of tensors and matrices in Ref. [@D].\ 1. We consider the standard basis for the supervector spaces so that in writing the basis as a column matrix, we first present the bosonic base, then the fermionic one. The transformation of standard basis and its dual bases can be written as follows: $${e'}_i=(-1)^j{K_i}\;^j e_j ,\hspace{10mm}{e'}^i={{K^{-st}}^i}_j\; e^j,$$ where the transformation matrix $K$ has the following block diagonal representation [@D] $$K=\left( \begin{tabular}{c|c} A & C \\ \hline D & B \\ \end{tabular} \right),$$ where $A,B$ and $C$ are real submatrices and $D$ is pure imaginary submatrix [@D]. Here we consider the matrix and tensors having a form with all upper and lower indices written in the right hand side. 2\. The transformation properties of upper and lower right indices to the left one for general tensors are as follows: $$^iT_{jl...}^{\;k}=T_{jl...}^{ik},\qquad _jT^{ik}_{l...}=(-1)^j\;T_{jl...}^{ik}.$$ 3. For supertransposition we have $${{L}^{st\;i}}_j=(-1)^{ij}\;{L_j}^{\;i},\qquad {L^{st}_{\;i}}^{\;j}=(-1)^{ij}\;{L^i}_{\;j},$$ $$M^{st}_{\;ij}=(-1)^{ij}\;M_{\;ji},\qquad M^{st\;ij}=(-1)^{ij}\;M^{\;ji}.$$ 4. For superdeterminant we have $$sdet\left( \begin{tabular}{c|c} A & C \\ \hline D & B \\ \end{tabular} \right)=det{(A-CB^{-1}D)}(det B)^{-1},$$ when $det B\neq0$ and $$sdet\left( \begin{tabular}{c|c} A & C \\ \hline D & B \\ \end{tabular} \right)=(det{(B-DA^{-1}C)})^{-1}\;(det A),$$ when $det A\neq0$. For the inverse of matrix we have $${\footnotesize \left( \begin{tabular}{c|c} A & C \\ \hline D & B \\ \end{tabular} \right)^{-1}=\left( \begin{tabular}{c|c} $(1_m-A^{-1}C B^{-1}D)^{-1}A^{-1}$& $-(1_m-A^{-1}CB^{-1}D)^{-1}A^{-1}CB^{-1}$ \\ \hline $-(1_n-B^{-1}DA^{-1}C)^{-1}B^{-1}DA^{-1}$ & $(1_n-B^{-1}DA^{-1}C)^{-1}B^{-1}$ \end{tabular} \right),}$$ where $det A,det B\neq0$ and $m$,$n$ are dimensions of submatrices $A$ and $B$, respectively. [**Appendix B: Solutions $(sJ-msJ)$ for dual Lie superalgebras and isomorphism matrices**]{} This appendix includes solutions of super Jacobi and mixed super Jacobi identities $(sJ-msJ)$ for dual Lie superalgebras and isomorphism matrices $C$ which relate these solutions to other Lie superalgebras. 1\. Solution of $(sJ-msJ)$ for dual Lie superalgebras of $B$ is $${{\tilde{f}}^{22}}_{\; \;\:1}=i\alpha,\qquad{{\tilde{f}}^{12}}_{\; \; \: 1}=0,$$ where $\alpha\in\Re$.\ Isomorphism matrix between these solutions and $(A_{1,1}+A)$ is as follows: $$C=\left( \begin{tabular}{cc} $ -ic^2_{22} {{\tilde{f}}^{22}}_{\; \; \: 1} $ & 0\\ $ c_{21}$ & $c_{22}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions $c_{22}\in\Re-\{0\}$ and imposing that $C$ must be the transformation matrix, we have $c_{21}=0$. 2\. Solutions of $(sJ-msJ)$ for dual Lie superalgebras of $C^1_p\;(p\in\Re-\{0\})$ are [l l l l p[2mm]{} ]{} $i)$&${{\tilde{f}}^{12}}_{\; \; \: 1}=\alpha,\;\;{{\tilde{f}}^{23}}_{\; \; \: 3}=p\alpha,$&$\;\;\;\;\;\;\;\;p\in\Re-\{0\},\;\;\;\alpha \in\Re,$&\ $ii)$&${{\tilde{f}}^{12}}_{\; \; \: 1}={{\tilde{f}}^{23}}_{\; \; \: 3}=\beta,$&$\;\;\;\;\;\;\;\;p=1,\;\;\;\beta \in\Re,$&\ $iii)$ &${{\tilde{f}}^{33}}_{\; \; \: 1}=i\gamma,$ &$\;\;\;\;\;\;\;\;p\in\Re-\{0\},\;\;\;\gamma \in\Re,$&\ $iv)$ &${{\tilde{f}}^{33}}_{\; \; \: 1}=i\lambda,\;\;{{\tilde{f}}^{33}}_{\; \; \: 2}=i\eta$ &$\;\;\;\;\;\;\;\;p=\frac{1}{2},\;\;\;\lambda, \eta \in\Re,$&\ $v)$ &${{\tilde{f}}^{12}}_{\; \; \: 1}=\mu,\;\;{{\tilde{f}}^{23}}_{\; \; \: 3}=-\frac{\mu}{2},\;\;{{\tilde{f}}^{33}}_{\; \; \: 1}=i\nu$ &$\;\;\;\;\;\;\;\;p=-\frac{1}{2},\;\;\;\mu, \nu \in\Re.$&\ For solution $(i)$ we have isomorphism matrix between these solutions and ${C^1_p}$ as follows: $$C=\left( \begin{tabular}{ccc} $ c_{11}$ & $-\frac{1}{{{\tilde{f}}^{12}}_{\; \; \: 1}}$& $ c_{13}$ \\ $ c_{21}$ & 0 & 0 \\ 0 & 0 & $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{23}}_{\; \; \:3}=p {{\tilde{f}}^{12}}_{\; \; \: 1}$ and $c_{21}$, $c_{33}\in\Re-\{0\}$, $c_{11}\in\Re$, as well as imposing that $C$ must be transformation matrix we have $c_{13}=0$. Solution $(ii)$ is a special case of solution $(i)$. For solution $(iii)$ and $(iv)$ $sdet C=0$. Solution $(v)$ is investigated for $(C^1_\frac{1}{2} , C^1_p)$ in the following step.\ 3. Solutions of $(sJ-msJ)$ for dual Lie superalgebras of $C^1_\frac{1}{2}$ are [l l l l p[2mm]{} ]{} $i)$&${{\tilde{f}}^{23}}_{\; \; \: 3}=\alpha,\;\;\;{{\tilde{f}}^{12}}_{\; \; \: 1}=2\alpha,\;\;\;\;\alpha \in\Re,$&&\ $ii)$&${{\tilde{f}}^{33}}_{\; \; \: 1}=i\beta,\;\;\;{{\tilde{f}}^{33}}_{\; \; \: 2}=i\gamma,\;\;\;{{\tilde{f}}^{23}}_{\; \; \: 3}=\frac{\beta}{2},\;\;\;{{\tilde{f}}^{13}}_{\; \; \: 3}=\frac{-\gamma}{2},\;\;\;{{\tilde{f}}^{12}}_{\; \; \: 1}=-\beta,\;\;\;{{\tilde{f}}^{12}}_{\; \; \: 2}=-\gamma,\;$&$\beta, \gamma \in\Re.$&\ For solution $(i)$ we have isomorphism matrix $C$ between $(C^1_\frac{1}{2} , C^1_p)$ as follows: $$C=\left(\begin {tabular}{ccc} $ c_{11}$ & $-\frac{1}{2{{\tilde{f}}^{23}}_{\; \; \; 3}}$& $ c_{13}$ \\ $ c_{21}$ & 0 & 0 \\ 0 & 0 & $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{12}}_{\; \; \;3}=2 {{\tilde{f}}^{23}}_{\; \; \; 3}$, $c_{21}$, $c_{33}\in\Re-\{0\}$, $c_{11}\in\Re$ and $p=-\frac{1}{2}$, as well as imposing that $C$ must be the transformation matrix, we have $c_{13}=0$.\ For solution $(ii)$ we have isomorphism matrices $C_1$ and $C_2$ between $(C^1_\frac{1}{2} , C^1_\frac{1}{2})$ as follows: $$C_1=\left( \begin{tabular}{ccc} $ c_{11}$ & $\frac{i}{{{\tilde{f}}^{33}}_{\; \; \; 1}}$& $ 0 $ \\ $ -ic^2_{33} {{\tilde{f}}^{33}}_{\; \; \; 1}$ & 0 & 0 \\ 0 & 0 & $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{33}}_{\; \; \:2}= {{\tilde{f}}^{13}}_{\; \; \; 3}={{\tilde{f}}^{12}}_{\; \; \; 2}=0$, $c_{33}\in\Re-\{0\}$ and $c_{11}\in\Re$\ $$C_2=\left( \begin{tabular}{ccc} $ c_{12}\frac{{{\tilde{f}}^{33}}_{\; \; \;1}}{{{\tilde{f}}^{33}}_{\; \; \;2}} -\frac{i}{{{\tilde{f}}^{33}}_{\; \; \; 2}}$ & $c_{12}$& $ 0 $ \\ $ -ic^2_{33} {{\tilde{f}}^{33}}_{\; \; \; 1}$ &$-ic^2_{33} {{\tilde{f}}^{33}}_{\; \; \; 2}$& 0 \\ 0 & 0 & $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions $c_{33}\in\Re-\{0\}$ and $c_{12}\in\Re$.\ 4. Solution of $(sJ-msJ)$ for dual Lie superalgebras of $C^2_p$, $C^3$, $C^4$ and $C^5_p$ is $${{\tilde{f}}^{22}}_{\; \; \: 1}=i\alpha,\qquad {{\tilde{f}}^{33}}_{\; \; \; 1}=i\beta ,\qquad {{\tilde{f}}^{23}}_{\; \; \; 1}=i\gamma, \hspace{4cm}$$ where $\alpha$,$\beta$ and $\gamma \in\Re$. Furthermore for $C^3$ we have another solution as follows: $${{\tilde{f}}^{22}}_{\; \; \: 1}=i \alpha,\qquad \alpha \in\Re,$$ where for this solution we have $sdet C=0$, therefore we omit it.\ 4.1. For the above solution, we have the following isomorphism matrices $C_1$, $C_2$ and $C_3$ between $C^2_p$, $C^3$, $C^4$ and $C^5_p$ with $(A_{1,1}+2A)^1$: $$C_1=\left( \begin{tabular}{ccc} $ -ic^2_{33} {{\tilde{f}}^{33}}_{\; \; \; 1}$& 0 & 0 \\ $ c_{21}$ & $c_{22}$ & $ -c_{22}\frac{{{\tilde{f}}^{23}}_{\; \; \;1}}{{{\tilde{f}}^{33}}_{\; \; \;1}}$ \\ $c_{31}$ & 0 & $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\; \; \;1}=(\frac{c^2_{23}+c^2_{33}}{c^2_{22}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{22}$, $c_{33}\in\Re-\{0\}$, as well as imposing that $C_1$ must be the transformation matrix, we have $c_{31}=c_{21}=0$,\ $$C_2=\left( \begin{tabular}{ccc} $ -ic^2_{23} {{\tilde{f}}^{33}}_{\; \; \; 1}$& 0 & 0 \\ $ c_{21}$ & $0$ & $c_{23} $ \\ $c_{31}$ & $-c_{33}\frac{{{\tilde{f}}^{33}}_{\; \; \;1}}{{{\tilde{f}}^{23}}_{\; \; \;1}}$& $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\; \; \;1}=(\frac{c^2_{23}+c^2_{33}}{c^2_{32}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{23}$, $c_{33}\in\Re-\{0\}$, as well as imposing that $C_2$ must be transformation matrix, we have $c_{31}=c_{21}=0$,\ $$C_3=\left( \begin{tabular}{ccc} $c_{11}$& 0 & 0 \\ $ c_{21}$ & $c_{22}$ & $c_{23} $ \\ $c_{31}$ & $c_{32}$& $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\; \; \;1}=(\frac{c^2_{23}+c^2_{33}}{c^2_{22}+c^2_{32}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$, ${{\tilde{f}}^{23}}_{\; \; \;1}=-(\frac{c_{33}c_{32}+c_{22}c_{23}}{c^2_{22}+c^2_{32}}){{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{11}\in\Re-\{0\}, \;\;c_{22},c_{23},c_{32},$\ $c_{33}\in\Re$ with the conditions $c^2_{23}+c^2_{33}\neq 0$, $c^2_{22}+c^2_{32}\neq 0$, as well as imposing that $C_3$ must be the transformation matrix, we have $c_{31}=c_{21}=0$.\ 4.2. For the above solution we have isomorphism matrices $C_4$, $C_5$, $C_6$, $C_7$ and $C_8$ between $C^2_p$, $C^3$, $C^4$ and $C^5_p$ with $(A_{1,1}+2A)^2$ as follows: $$C_4=\left( \begin{tabular}{ccc} $ -ic_{32}(c_{23}-c_{33} ) {{\tilde{f}}^{23}}_{\; \; \; 1}$& 0 &0\\ $ c_{21}$ & $c_{32}$ & $ c_{23}$ \\ $c_{31}$ & $c_{32}$ & $ c_{33} $ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\;\;\;1} =-(\frac{c_{23}+c_{33}}{c_{32}}) {{\tilde{f}}^{23}}_{\; \; \; 1}$, ${{\tilde{f}}^{33}}_{\; \; \; 1}=0$, $c_{23} \neq c_{33}$, $c_{32}\in\Re-\{0\}$ and $c_{23},c_{33}\in\Re$, as well as imposing that $C_4$ must be the transformation matrix, we have $c_{31}=c_{21}=0$,\ $$C_5=\left( \begin{tabular}{ccc} $ ic_{32}(c_{23}+c_{33} ) {{\tilde{f}}^{23}}_{\; \; \; 1}$& 0 &0\\ $ c_{21}$ & $-c_{32}$ & $ c_{23}$ \\ $c_{31}$ & $c_{32}$ & $ c_{33} $ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\;\;\;1} =(\frac{c_{23}-c_{33}}{c_{32}}) {{\tilde{f}}^{23}}_{\; \; \; 1}$, ${{\tilde{f}}^{33}}_{\; \; \; 1}=0$, $c_{23} \neq -c_{33}$, $c_{32}\in\Re-\{0\}$ and $c_{23},c_{33}\in\Re$, as well as imposing that $C_5$ must be the transformation matrix, we have $c_{31}=c_{21}=0$,\ $$C_6=\left( \begin{tabular}{ccc} $ -ic^2_{23}{{\tilde{f}}^{33}}_{\; \; \; 1}$& 0 &0\\ $ c_{21}$ & $0$ & $ c_{23}$ \\ $c_{31}$ & $-c_{33}\frac{{{\tilde{f}}^{33}}_{\; \; \; 1}} {{{\tilde{f}}^{23}}_{\; \; \; 1}}$ & $ c_{33} $ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\;\;\;1} =-(\frac{c^2_{23}-c^2_{33}}{c^2_{32}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{23}$, $c_{33}\in\Re-\{0\}$, as well as imposing that $C_6$ must be the transformation matrix, we have $c_{31}=c_{21}=0$,\ $$C_7=\left( \begin{tabular}{ccc} $ ic^2_{33}{{\tilde{f}}^{33}}_{\; \; \; 1}$& 0 &0\\ $ c_{21}$ & $c_{22}$ & $ -c_{22}\frac{{{\tilde{f}}^{23}}_{\; \; \; 1}} {{{\tilde{f}}^{33}}_{\; \; \; 1}}$ \\ $c_{31}$ & $0$ & $ c_{33} $ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\;\;\;1} =(\frac{c^2_{23}-c^2_{33}}{c^2_{22}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{22}$, $c_{33}\in\Re-\{0\}$, as well as imposing that $C_7$ must be the transformation matrix, we have $c_{31}=c_{21}=0$,\ $$C_8=\left( \begin{tabular}{ccc} $c_{11}$& 0 & 0 \\ $ c_{21}$ & $c_{22}$ & $c_{23} $ \\ $c_{31}$ & $c_{32}$& $ c_{33}$ \\ \end{tabular} \right), \hspace{2cm}$$ with the conditions ${{\tilde{f}}^{22}}_{\; \; \;1}=(\frac{c^2_{33}-c^2_{23}}{c^2_{32}-c^2_{22}}) {{\tilde{f}}^{33}}_{\; \; \; 1}$, ${{\tilde{f}}^{23}}_{\; \; \;1}=(\frac{c_{22}c_{23}-c_{32}c_{33}}{c^2_{32}-c^2_{22}}){{\tilde{f}}^{33}}_{\; \; \; 1}$ and $c_{11}\in\Re-\{0\}, \;\;c_{22},c_{23},c_{32},$\ $c_{33}\in\Re$ with the conditions $c^2_{33}-c^2_{23}\neq 0, c^2_{32}-c^2_{22}\neq 0$, as well as imposing that $C_8$ must be the transformation matrix, we have $c_{31}=c_{21}=0$.\ [99]{} V. G. Drinfel’d, *Hamiltonian Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation*, Sov. Math. Dokl. **27** (1983) 68-71. P. Etingof, D. Kazhdan, *Quantization of Lie bialgebras,* I, Selecta Math. **2** (1996) no. 1, 1-41.\ P. Etingof, D. Kazhdan, *Quantization of Lie bialgebras*, II. Selecta Math. **4** (1998) no. 2, 213-231. P. Delorme, *Sur les triples de Manin pour les complexes,* Jour. Alg. **246** (2001) 97-174. M.A. Jafarizadeh and A. Rezaei-Aghdam, *Poisson-Lie T-duality and Bianchi type algebras*, *Phys. Lett. B* **458** (1999) 477-490, `[arXiv:hep-th/9903152]`. X. Gomez, *Classification of three-dimensional Lie bialgebras*, J. Math. Phys. **41** (2000) 4939. L. Hlavaty and L. Snobl, *Classification of 6-dimensional Manin triples*, `[arXiv:math-ph/0202209]`. A. Rezaei-Aghdam, M. Hemmati, A. R. Rastkar, *classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups*, *J. Phys. A:Math.Gen.* **38** (2005) 3981-3994, `[arXiv:math-ph/0412092]`. Y. Kosmann-Schwarzbach, *Lie bialgebras, Poisson-Lie groups and dressing transformations integrability of nonlinear Systems*: Proc.(Pondicherry) ed Y Kosmann-Schwarzbach, B Grammaticos and K M Tamizhmani (Berlin: Springer, 1996) pp 104-70. N. Andruskiewitsch, *Lie Superbialgebras and Poisson-Lie supergroups*, Abh. Math. Sem. Univ. Hamburg [**63**]{} (1993) 147-163. N. Beisert and E. Spill, *The classical r-matrices of AdS/CFT and its Lie bialgebras structure*, `[arXiv:0708.1762 [hep-th]]`. N. Geer, *Etingof-Kazdan quantization of Lie super bialgebras*, `[arXiv:math.QA/0409563]`. C. Juszczak and J. T. Sobczyk, *Classification of low dimentional Lie super-bialgebras,* *J. Math. Phys.* **39**, (1998) 4982-4992. C. Juszczak, *Classification of $osp(2|2)$ Lie super-bialgebras,* `[arXiv:math.QA/9906101]`. G. Karaali, *A New Lie Bialgebra Structure on sl(2,1)*, *Contemp. Math.* **413** (2006), pp.101-122. N. Backhouse, *A classification of four-dimensional Lie superalgebras*, *J. Math. Phys.* **19** (1978) 2400-2402. B. DeWitt, *Supermanifolds,* Cambridge University Press 1992. *Note that the bracket of one boson with one boson or one fermion is usual commutator but for one fermion with one fermion is anticommutator. Furthermore we identify grading of indices by the same indices in the power of (-1), for example $grading(i)\equiv i$; this is the notation that DeWitt applied in his book (Ref. [@D]).* *For the proof of this proposition it is enough to use (11) and $\delta^\prime(X_i) = (-1)^{jk}{{\tilde{f}}'^{jk}}_{\; \; \; \:i} X_j \otimes X_k$ and (22) in (5), then if super Jacobi and mixed super Jacobi identities \[(2) and (12)\] for ${\tilde{f}^{jk}}_{\; \; \; \:i}$ are satisfied then they satisfy for ${{\tilde{f}}'^{jk}}_{\; \; \; \:i}$ as well and vise versa.* A. Eghbali and A. Rezaei-Aghdam, *Classification of four-dimensional Lie super-bialgebras of the type $(2 , 2)$*, work in progress. *Note that the effect of $A$ is linear and even i.e. $$A(a_i^{\;\;j}X_j)=a_i^{\;\;j} A(X_j).$$* *Note that the superalgebra $A$ is one dimensional Abelian Lie superalgebras with one fermionic generator where Lie superalgebra $A_{1,1}$ is its bosonization.* *Note that in this classification one can omit the $i=\sqrt{-1}$ from commutation relations and to obtain Lie super-bialgebras in the nonstandard basis.* C. Klimčik and P. Ševera, *Dual non-Abelian duality and the Drinfeld double*, *Phys. Lett.* **B351** (1995) 445-462, `[arXiv:hep-th/9502122]`.\ C. Klimčik, *Poisson-Lie $T$-duality*, *Nucl. Phys. Proc. Suppl.* **46** ( 1996) [116-121]{}, `[arXiv:hep-th/9509095]`. A. Eghbali, A. Rezaei-Aghdam, *Poisson-Lie T-dual sigma models on supermanifolds*, JHEP 2009, [**094**]{} (2009), `[arXiv:0901.1592v3 [hep-th]]`. A. Eghbali and A. Rezaei-Aghdam, *Classical $r$-matrices of two and three dimensional Lie super-bialgebras and their Poisson-Lie supergroups*, `[ arXiv:0908.2182 [math-ph]]`. R. Von Unge, *Poisson-Lie T-plurality*, JHEP, 2002, [**014**]{} (2002) (preprint `[arXiv:hep-th/0205245]`).

--- abstract: 'In this letter we complement previous studies on exclusive vector meson photoproduction in hadronic collisions presenting a comprehensive analysis of the $t$ - spectrum measured in exclusive $\rho$ and $J/\Psi$ photoproduction in $pA$ collisions at the LHC. We compute the differential cross sections considering two phenomenological models for the gluon saturation effects and present predictions for $pPb$ and $pCa$ collisions. Moreover, we compare our predictions with the recent preliminary CMS data for the exclusive $\rho$ photoproduction. We demonstrate that the gluon saturation models are able to describe the CMS data at small - $t$. On the other hand, the models underestimate the few data point at large – $t$. Our results indicate that future measurements of the large – $t$ region can be useful to probe the presence or absence of a dip in the $t$ – spectrum and discriminate between the different approaches to the gluon saturation effects.' author: - 'V. P. Gonçalves $^{1}$, F. S. Navarra $^{2}$ and D. Spiering $^{2}$' title: 'Exclusive $\rho$ and $J/\Psi$ photoproduction in ultraperipheral $pA$ collisions: Predictions of the gluon saturation models for the momentum transfer distributions' --- During the last years the study of photon – induced interactions at hadronic colliders has been strongly motivated by the possibility of constraining the dynamics of the strong interactions at large energies (For a recent review see Ref. [@review_forward]). One of most promising observables is the exclusive vector meson photoproduction cross section [@vicbert; @vicmag], which is driven by the gluon content of the target (proton or nucleus) and is strongly sensitive to non-linear effects (parton saturation). Such expectation has motivated the analysis of exclusive $\rho$, $\phi$, $J/\Psi$, $\Psi(2S)$ and $\Upsilon$ photoproduction in $pp$, $pA$ and $AA$ collisions at RHIC and LHC energies considering different theoretical approaches for the treatment of the QCD dynamics and for the vector meson wave function (See, e.g., Refs. [@vicmag_varios; @bruno; @schafer; @guzey; @jones; @run2; @armesto; @contreras]). In particular, the recent study performed in Ref. [@run2] indicated that a global analysis of the experimental data for the rapidity distributions of all these different final states will be necessary to discriminate between the distinct theoretical approaches. On the other hand, the results presented in Refs. [@armesto; @nos_tdist] indicate that the study of the squared momentum transfer ($t$) distributions is an important alternative to probe the QCD dynamics at high energies. These distributions are expected to provide information about the spatial distribution of the gluons in the hadron and about fluctuations of the color fields (See e.g. Ref. [@heike]). In Ref. [@nos_tdist] we have presented predictions for the $t$ - spectrum measured in the exclusive vector meson photoproduction considering $pp$ and $PbPb$ collisions at the LHC. Our goal is this letter is twofold. First, to complement that study and present, for the first time, predictions for the momentum transfer distributions measured in exclusive $\rho$ and $J/\Psi$ photoproduction in $pPb$ collisions considering two phenomenological models for the treatment of the gluon saturation effects. Second, to present a comparison of gluon saturation predictions with the recent (preliminary) CMS data on exclusive $\rho$ photoproduction in ultraperipheral $pPb$ collisions at $\sqrt{s_{NN}} = 5.02$ TeV [@cms_prel]. Such comparison is also performed here for the first time. As we will demonstrate in what follows, our results indicate that the analysis of the $t$ – spectrum can be useful to discriminate between the different approaches to gluon saturation effects. Moreover, we will show that these models are able to describe the CMS data at small – $t$ but underestimate the few data points at large – $t$. Initially, let’s present a brief review of the formalism used in our calculations. The exclusive vector meson photoproduction in $pA$ collisions is dominated by photon - proton interactions, since the nuclear photon flux is enhanced by the square of the nuclear charge ($Z$) [@upc]. The process is represented in Fig. \[fig:diagrama\]. The final state will be characterized by two intact hadrons ($A$ and $p$) and two rapidity gaps, i.e. the outgoing particles ($A$, $V = \rho, \, J/\Psi$ and $p$) are separated by a large region in rapidity in which there is no additional hadronic activity observed in the detector. The differential cross section can be expressed as follows $$\begin{aligned} \frac{d\sigma \,\left[A + p \rightarrow A \otimes V \otimes p\right]}{dY\,dt} = n_A(\omega) \, \cdot \, \frac{d\sigma}{dt}(\gamma p \rightarrow V \otimes p)\,\,\,, \label{dsigdy}\end{aligned}$$ where the rapidity ($Y$) of the vector meson in the final state is determined by the photon energy $\omega$ in the collider frame and by the mass $M_{V}$ of the vector meson \[$ Y \propto \ln \, ( \omega/M_{V})$\]. Moreover, $d\sigma/dt$ is the differential cross section of the $\gamma p \rightarrow V \otimes p$ process, with the symbol $\otimes$ representing the presence of a rapidity gap in the final state. Furthermore, $n_A(\omega)$ denotes the equivalent photon spectrum of the relativistic incident nucleus. As in our previous studies [@run2; @nos_tdist] we will assume a point – like form factor for the nucleus, which implies that [@upc] $$\begin{aligned} n_{A}(\omega) = \frac{2Z^{2}\alpha_{em}}{\pi } \left[ \xi K_{0}(\xi) K_{1}(\xi) -\frac{\xi^{2}}{2} \left( K_{1}^{2}(\xi) - K_{0}^{2}(\xi) \right ) \right] , \end{aligned}$$ where $ \xi = \omega \left( R_{A} + R_{p} \right) / \gamma_{L}$, with $\gamma_L$ being the Lorentz factor. The differential cross section for the $\gamma p \rightarrow V \otimes p$ process is given by $$\begin{aligned} \frac{d\sigma}{dt} & = & \frac{1}{16\pi} |{\cal{A}}^{\gamma p \rightarrow V p }(x, \Delta)|^2\,\,, \label{dsigdt}\end{aligned}$$ where ${\cal{A}}$ is the amplitude for producing an exclusive vector meson diffractively. In the color dipole formalism [@nik], this amplitude can be factorized in terms of the fluctuation of the virtual photon into a $q \bar{q}$ color dipole, the dipole-hadron scattering by a color singlet exchange (denoted $\pom$ in Fig. \[fig:diagrama\]) and the recombination into the vector meson $V$. Consequently, the amplitude can be expressed as follows $$\begin{aligned} {\cal A}^{\gamma p \rightarrow V p }({x},\Delta) = i \int dz \, d^2{\mbox{\boldmath $r$}}\, d^2{\mbox{\boldmath $b$}}\, e^{-i[{\mbox{\boldmath $b$}}-(1-z){\mbox{\boldmath $r$}}].{\mbox{\boldmath $\Delta$}}} \,\, (\Psi^{V*}\Psi) \,\,2 {\cal{N}}^p({x},{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}) \,\,, \label{amp}\end{aligned}$$ where $(\Psi^{V*}\Psi)$ denotes the wave function overlap between the photon and vector meson wave functions, $\Delta = - \sqrt{t}$ is the momentum transfer and ${\mbox{\boldmath $b$}}$ is the impact parameter of the dipole relative to the proton target. Moreover, the variables ${\mbox{\boldmath $r$}}$ and $z$ are the dipole transverse pair separation and the momentum fraction of the photon carried by a quark (an antiquark carries then $1-z$), respectively. As in Ref. [@nos_tdist], in what follows we will consider the Boosted Gaussian model [@KT; @KMW] for the overlap function. The function $ {\cal N}^p (x, {\mbox{\boldmath $r$}}, {\mbox{\boldmath $b$}})$ is the forward dipole-proton scattering amplitude (for a dipole at impact parameter ${\mbox{\boldmath $b$}}$) which encodes all the information about the hadronic scattering. It depends on the $\gamma h$ center - of - mass reaction energy, $W = [2 \omega \sqrt{s_{NN}}]^{1/2}$, through the variable $ x = M^2_V/W^2$. One of the main open questions in QCD is the treatment of its high energy regime, where non – linear (gluon saturation) effects are expected to contribute [@hdqcd]. Currently, the bCGC and IP-Sat models, which are based on different assumptions for the treatment of the gluon saturation effects, describe with success the high precision HERA data for inclusive and exclusive processes. In the impact parameter Color Glass Condensate (bCGC) model [@KMW] the dipole - proton scattering amplitude is given by $$\begin{aligned} \mathcal{N}^p(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}) = \left\{ \begin{array}{ll} {\mathcal N}_0\, \left(\frac{ r \, Q_s(b)}{2}\right)^{2\left(\gamma_s + \frac{\ln (2/r \, Q_s(b))}{\kappa \,\lambda \,Y}\right)} & \mbox{$r Q_s(b) \le 2$} \\ 1 - e^{-A\,\ln^2\,(B \, r \, Q_s(b))} & \mbox{$r Q_s(b) > 2$} \,\,, \end{array} \right. \label{eq:bcgc}\end{aligned}$$ with $\kappa = \chi''(\gamma_s)/\chi'(\gamma_s)$, where $\chi$ is the LO BFKL characteristic function. The coefficients $A$ and $B$ are determined uniquely from the condition that $\mathcal{N}^p(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}})$, and its derivative with respect to $r\,Q_s(b)$, are continuous at $r\,Q_s(b)=2$. The impact parameter dependence of the proton saturation scale $Q_s(b)$ is given by: $$Q_s(b)\equiv Q_s(x,b)=\left(\frac{x_0}{x}\right)^{\frac{\lambda}{2}}\; \left[\exp\left(-\frac{{b}^2}{2B_{\rm CGC}}\right)\right]^{\frac{1}{2\gamma_s}}, \label{newqs}$$ with the parameter $B_{\rm CGC}$ being obtained by a fit of the $t$-dependence of exclusive $J/\psi$ photoproduction. The factors $\mathcal{N}_0$ and $\gamma_s$ were taken to be free. In what follows we consider the set of parameters obtained in Ref. [@amir] by fitting the recent HERA data on the reduced $ep$ cross sections: $\gamma_s = 0.6599$, $\kappa = 9.9$, $B_{CGC} = 5.5$ GeV$^{-2}$, $\mathcal{N}_0 = 0.3358$, $x_0 = 0.00105$ and $\lambda = 0.2063$. In the bCGC model, the saturation regime, where $r Q_s(b) > 2$, is described by the Levin - Tuchin law [@levin_tuchin] and the linear one by the BFKL dynamics near of the saturation line. On the other hand, in the IP-Sat model [@ipsat2; @ipsat3], ${\cal N}^p$ has an eikonalized form and depends on a gluon distribution evolved via DGLAP equation, being given by $$\begin{aligned} {\cal N}^p(x,\mbox{\textbf{\textit{r}}},\mbox{\textbf{\textit{b}}}) = 1 - \exp \left[ \frac{\pi^{2}r^{2}}{N_{c}} \alpha_{s}(\mu^{2}) \,\,xg\left(x, \frac{4}{r^{2}} + \mu_{0}^{2}\right)\,\, T_{G}(b) \right] , \label{ipsat}\end{aligned}$$ with a Gaussian profile $$\begin{aligned} T_{G}(b) = \frac{1}{2\pi B_{G}} \exp\left(-\frac{b^{2}}{2B_{G}} \right) .\end{aligned}$$ The initial gluon distribution evaluated at $\mu_{0}^{2}$ is taken to be $ xg(x,\mu_{0}^{2}) = A_{g}x^{-\lambda_{g}} (1-x)^{5.6}$. In this work we assume the parameters obtained in Ref. [@ipsat4]. One have that as in the bCGC model, the IP-Sat predicts the saturation of $ {\cal N}^p$ at high energies and/ot large dipoles, but the approach to this regime is not described by the Levin - Tuchin law. Moreover, in contrast to the bCGC model, the IP-Sat takes into account the effects associated to the DGLAP evolution, which are expected to be important in the description of the small dipoles. Consequently, both models are based on different assumptions for the linear and non - linear regimes. As pointed above, the current high precision HERA data are not able to discriminate between these models. In what follows we analyze the possibility of constraining the models of gluon saturation effects in exclusive vector meson photoproduction at $pA$ collisions. \ Let’s consider the exclusive $\rho$ and $J/\Psi$ photoproduction in $pCa$ and $pPb$ collisions at the LHC energies. Our main focus will be on the transverse momentum distributions, which are expected to be studied considering the higher statistics of Run 2 and 3 [@review_forward]. However, firstly let us analyse the impact of the gluon saturation effects on the rapidity distributions at a fixed value of the momentum transfer $t$. We will estimate Eq. (\[dsigdy\]) for $t = t_{min}$, with $t_{min} = - m_N^2 M_V^4/W^4$. In Fig. \[Fig:rap\_pA\] we present our predictions for the rapidity distributions considering the exclusive $\rho$ (upper panels) and $J/\Psi$ (lower panels) photoproduction in $pCa$ and $pPb$ collisions. We observe that the difference between the bCGC and IP-Sat is larger for $\rho$ production, with the IP-Sat predictions being smaller than the bCGC ones. On the other hand, the IP-Sat model predicts larger values of the rapidity distribution when the $J/\Psi$ production is considered. These results are expected, since the bCGC and IP-Sat models assume different behavior for the linear and non - linear regimes. In the $\rho$ case, the process is dominated by the contribution of large dipole sizes, which are expected to be strongly suppressed by the gluon saturation effects. On the other hand, $J/\Psi$ production is dominated by small dipoles, i. e. the cross section is expected to be mainly determined by the linear regime of the QCD dynamics. The main difference between the predictions for $pCa$ and $pPb$ collisions is the normalization of the distributions. This result is also expected, since the distribution is calculated by the product of the photon flux and the photon - proton cross section \[See Eq. (\[dsigdy\])\], with $n_A$ being proportional to $Z^2$. The rapidity and transverse momentum dependencies are determined by the $\gamma p \rightarrow V p$ cross section, which is the same for $pCa$ and $pPb$ collisions. Consequently, in what follows, we will only present our predictions for the $t$– distributions in $pPb$ collisions. -- -- -- -- -- -- Let us now analyze the predictions of the different gluon saturation models for the transverse momentum distributions considering $pPb$ collisions at $\sqrt{s_{NN}} = 8.16$ TeV and assuming three different fixed values for the vector meson rapidity ($Y = 0$, 2 and 4). Our results for the exclusive $\rho$ and $J/\Psi$ photoproduction are presented in the upper and lower panels of Fig. \[Fig:dsigdt\_tdist\], respectively. We observe that the bCGC and IP-Sat predictions are similar at small - $|t|$, but differ at larger values. The position of the dip is dependent on the description of the gluon saturation effects, with the bCGC model predicting the dip at smaller values of $|t|$, independently of the produced vector meson. Moreover, we see that the position of the dip is displaced at smaller $|t|$ with the growth of the rapidity and the number of dips predicted for the $\rho$ production in the range $|t| \le 4$ GeV$^2$ is larger than for the $J/\Psi$ case. These results indicate that the study of the $t$ - distribution in the range $0.75 \le |t| \le 1.5$ GeV$^2$ ($2.0 \le |t| \le 3.0$ GeV$^2$) for the case of $\rho$ ($J/\Psi$) production can be useful to contrain the description of the gluon saturation effects. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Transverse momentum distributions for the exclusive $\rho$ photoproduction at different center - of - mass energies of the $\gamma p $ system. Preliminary data from the CMS Collaboration [@cms_prel].[]{data-label="Fig:dsigdt_ep"}](dsdt_rho_gp_W36_boosted.eps "fig:") ![Transverse momentum distributions for the exclusive $\rho$ photoproduction at different center - of - mass energies of the $\gamma p $ system. Preliminary data from the CMS Collaboration [@cms_prel].[]{data-label="Fig:dsigdt_ep"}](dsdt_rho_gp_W59_boosted.eps "fig:") ![Transverse momentum distributions for the exclusive $\rho$ photoproduction at different center - of - mass energies of the $\gamma p $ system. Preliminary data from the CMS Collaboration [@cms_prel].[]{data-label="Fig:dsigdt_ep"}](dsdt_rho_gp_W108_boosted.eps "fig:") ![Transverse momentum distributions for the exclusive $\rho$ photoproduction at different center - of - mass energies of the $\gamma p $ system. Preliminary data from the CMS Collaboration [@cms_prel].[]{data-label="Fig:dsigdt_ep"}](dsdt_rho_gp_W176_boosted.eps "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The exclusive $\rho$ photoproduction in ultraperipheral $pPb$ collisions at $\sqrt{s_{NN}} = 5.02$ TeV has been studied by the CMS Collaboration. In particular, they released, very recently, the first (preliminary) data [@cms_prel] for the $t$ – distributions of the $\gamma p \rightarrow \rho p$ process at different center – of – mass energies of the $\gamma p$ system. Assuming that the nuclear photon flux is well known and that there is a direct relation between the rapidity $Y$ of the vector meson and the $\gamma p$ center - of - mass energy ($W$), they measured $d\sigma/dt$ for different rapidity bins and, consequently, for averaged values of $W$. A comparison between our predictions and these preliminary data is presented in Fig. \[Fig:dsigdt\_ep\]. The bCGC and IP-Sat models describe quite well the distributions for $|t| \le 0.4$ GeV$^2$. On the other hand, at larger values of $|t|$, where the number of experimenal points is smaller and the uncertainty is larger, the predictions of the gluon saturation models underestimate the data, with the IP-Sat predictions being closer to the data. It is important to emphasize that the discrepancy starts to occur exactly in the region where the presence of dips becomes important and the $t$ - distribution can no longer be described by an exponential with a fixed slope. If confirmed, these data can be a first indication that the model of the spatial distribution of gluons in the proton (present in the bCGC and IP-Sat models) should be improved in the study of gluon saturation effects. Certainly, more data on exclusive vector meson photoproduction will be very useful to improve our understanding of the QCD dynamics at high energies. Finally, let’s summarize our main results and conclusions. In this letter we have investigated the exclusive $\rho$ and $J/\Psi$ photoproduction in $pA$ collisions at the LHC motivated by the expectation that this process may allow us to constrain the description of the QCD dynamics at high energies. Differently from $pp$ and $AA$ collisions, in $pA$ collisions the rapidity of the vector meson allows to unambiguously determine the $\gamma p$ center - of - mass energy and, consequently, to probe the QCD dynamics at the given value of the Bjorken – $x$ variable. We have considered $\rho$ and $J/\Psi$ production, which mainly probes the non - linear and linear QCD regimes, respectively, and presented the bCGC and IP-Sat predictions for the rapidity and transverse momentum distributions. These two models, even though describing the available HERA data, are based on different assumptions for the gluon saturation effects. We demonstrated that their predictions for the $t$ – spectra are similar at small values of $|t|$ but differ at large - $|t|$, with the position of the dip being model dependent. A comparison of our predictions with the very recent (preliminary) CMS data has been presented for the first time, with the data at small - $|t|$ being quite well described by both gluon saturation models. However, the large - $|t|$ data are underestimated by the models. This can be a first indication that the description of the spatial distribution of the gluons in the proton should be improved. These results indicate that the experimental analysis of the transverse momentum distribution is useful to discriminate between different approaches for the QCD dynamics as well to improve our description of the gluon saturation effects. VPG acknowledges useful discussions with J. Cepila, J. G. Contreras, J. D. Tapia - Takaki and W. Schafer. VPG is grateful to the members of the Department of Physics and Astronomy of the University of Kansas by the warm hospitality during the initial phase of this study. This work was partially financed by the Brazilian funding agencies CNPq, FAPERGS, FAPESP and INCT-FNA (process number 464898/2014-5). [99]{} K. Akiba [*et al.*]{} \[LHC Forward Physics Working Group Collaboration\], J. Phys. G [**43**]{}, 110201 (2016) V. P. Goncalves and C. A. Bertulani, Phys. Rev. C [**65**]{}, 054905 (2002). V. P. Goncalves and M. V. T. Machado, Eur. Phys. J.  C [**40**]{}, 519 (2005). V. P. Goncalves and M. V. T. Machado, Phys. Rev.  C [**73**]{}, 044902 (2006); Phys. Rev.  D [**77**]{}, 014037 (2008); Phys. Rev. C [**84**]{}, 011902 (2011) V. P. Goncalves, B. D. Moreira and F. S. Navarra, Phys. Rev. C [**90**]{}, 015203 (2014); Phys. Lett. B [**742**]{}, 172 (2015); Phys. Rev. D [**95**]{}, no. 5, 054011 (2017) W. Schafer and A. Szczurek, Phys. Rev. D [**76**]{}, 094014 (2007); A. Rybarska, W. Schafer and A. Szczurek, Phys. Lett. B [**668**]{}, 126 (2008); A. Cisek, W. Schafer and A. Szczurek, Phys. Rev. C [**86**]{}, 014905 (2012); A. Cisek, W. Schafer and A. Szczurek, JHEP [**1504**]{}, 159 (2015) L. Frankfurt, V. Guzey, M. Strikman and M. Zhalov, JHEP [**0308**]{}, 043 (2003); V. Guzey and M. Zhalov, JHEP [**1310**]{}, 207 (2013); JHEP [**1402**]{}, 046 (2014). S. P. Jones, A. D. Martin, M. G. Ryskin and T. Teubner, JHEP [**1311**]{}, 085 (2013); Eur. Phys. J. C [**76**]{}, no. 11, 633 (2016) V. P. Goncalves, M. V. T. Machado, B. D. Moreira, F. S. Navarra and G. S. dos Santos, Phys. Rev. D [**96**]{}, no. 9, 094027 (2017) N. Armesto and A. H. Rezaeian, Phys. Rev. D [**90**]{}, no. 5, 054003 (2014) J. Cepila, J. G. Contreras and J. D. Tapia Takaki, Phys. Lett. B [**766**]{}, 186 (2017); J. Cepila, J. G. Contreras and M. Krelina, Phys. Rev. C [**97**]{}, no. 2, 024901 (2018); J. Cepila, J. G. Contreras, M. Krelina and J. D. Tapia Takaki, Nucl. Phys. B [**934**]{}, 330 (2018) V. P. Goncalves, F. S. Navarra and D. Spiering, Phys. Lett. B [**768**]{}, 299 (2017) H. Mantysaari and B. Schenke, Phys. Rev. Lett.  [**117**]{}, no. 5, 052301 (2016); Phys. Rev. D [**94**]{}, 034042 (2016) CMS Collaboration \[CMS Collaboration\], “Exclusive $\rho(770)^{0}$ photoproduction in ultra-peripheral pPb collisions at $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$ with the CMS experiment,” CMS-PAS-FSQ-16-007. C.A. Bertulani and G. Baur, Phys. Rep. [**163**]{}, 299 (1988); G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, Y. Kharlov, Phys. Rep. [**364**]{}, 359 (2002); V. P. Goncalves and M. V. T. Machado, Mod. Phys. Lett. A [**19**]{}, 2525 (2004); C. A. Bertulani, S. R. Klein and J. Nystrand, Ann. Rev. Nucl. Part. Sci. [**55**]{}, 271 (2005); K. Hencken [*et al.*]{}, Phys. Rept.  [**458**]{}, 1 (2008); J. G. Contreras and J. D. Tapia Takaki, Int. J. Mod. Phys. A [**30**]{}, 1542012 (2015). N. N. Nikolaev, B. G. Zakharov, Phys. Lett. B [**332**]{}, 184 (1994); Z. Phys. C [**64**]{}, 631 (1994). H. Kowalski and D. Teaney, Phys. Rev. D [**68**]{}, 114005 (2003) H. Kowalski, L. Motyka and G. Watt, Phys. Rev.  D [**74**]{}, 074016 (2006) F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, Ann. Rev. Nucl. Part. Sci.  [**60**]{}, 463 (2010); H. Weigert, Prog. Part. Nucl. Phys.  [**55**]{}, 461 (2005); J. Jalilian-Marian and Y. V. Kovchegov, Prog. Part. Nucl. Phys.  [**56**]{}, 104 (2006). A. H. Rezaeian and I. Schmidt, Phys. Rev. D [**88**]{}, 074016 (2013) E. Levin and K. Tuchin, Nucl. Phys. A [**691**]{}, 779 (2001) H. Kowalski and D. Teaney, Phys. Rev. D [**68**]{}, 114005 (2003). H. Kowalski, T. Lappi and R. Venugopalan, Phys. Rev. Lett.  [**100**]{}, 022303 (2008). A. H. Rezaeian, M. Siddikov, M. Van de Klundert and R. Venugopalan, Phys. Rev. D [**87**]{}, 034002 (2013).

--- abstract: 'We report a record-size polariton condensate of a fraction of a millimeter. This macroscopically occupied state of macrosopic size is not constrained to the excitation spot and is free from the usual complications brought by high-energy reservoir excitons, which strongly alter the physics of polaritons, including their mobility, energy distribution and particle interactions. The density of this trap-free condensate is lower than , reducing the phase noise induced by the interaction energy. Experimental findings are backed up by numerical simulations using a hydrodynamic model which takes into account both the polariton expansion and the phonon-assisted relaxation towards the lowest energy state. These results propel polariton condensates at the fundamental level set by their cold-atomic counterparts by getting rid of several solid-state difficulties, while still retaining their unique driven/dissipative features.' author: - Dario Ballarini - Davide Caputo - Carlos Sánchez Muñoz - Milena De  Giorgi - Lorenzo Dominici - 'Marzena H. Szymańska' - Kenneth West - 'Loren N. Pfeiffer' - Giuseppe Gigli - 'Fabrice P. Laussy' - Daniele Sanvitto title: Formation of a macroscopically extended polariton condensate without an exciton reservoir --- Under suitable conditions, light-matter interaction can be strong enough to drive the coherent exchange of energy between photonic and electronic modes [@Hopfield1958]. This is the paradigm of microcavity exciton-polaritons: quasi-particles created by the strong coupling between the photonic mode of a microcavity and the excitonic transition of semiconductor quantum wells [@Weisbuch1992]. Polaritons manifest their composite nature with a combination of photonic and excitonic properties [@Dominici2014]. Thanks to their photonic component, polaritons can ballistically propagate in the plane of the microcavity with velocities up to a few percent of the speed of light [@Freixanet2000]. On the other hand, the exciton component results in strong optical nonlinearities and induces an energy renormalization of the polariton dispersion at high densities [@Senellart1999]. This energy shift can be much larger than the linewidth and is at the foundation of most polaritonic effects and applications [@Baas2004b; @Amo2010; @Ballarini2013]. As bosonic quasiparticles, polaritons experiment final-state stimulated scattering, which results, above a density threshold, in a laser-like emission without population inversion, a collective phenomenon that is explained in the framework of Bose-Einstein condensation [@Baumberg2000; @Deng2002; @Kasprzak2006]. A unique feature of polariton condensates is their driven/dissipative nature, in which the steady state is reached through a dynamical balance of pumping and dissipation. Polariton condensate have been experimentally observed in different materials, both inorganic [@Richard2005; @Balili2007; @Christopoulos2007; @Guillet2011] and organic semiconductors [@Daskalakis2014; @Plumhof2014], and thanks to their light mass, condensation is achieved also at room temperature [@Baumberg2008]. However, differently from their atomic counterpart, these condensates suffer from dephasing and density fluctuations induced by the interactions with the exciton reservoir, effectively resulting in multimode condensates [@Kasprzak2008; @Love2008; @Kim2016]. The exciton reservoir also acts as a trapping mechanism, if the polariton lifetime is too short, confining the condensation process within the region of the excitation spot [@Tassone1997; @Wouters2008b; @Roumpos2010; @yama1]. Moreover, polariton condensation is often localized in potential minima caused by imperfections of the sample structure, yielding to a fragmentation of the phase coherence [@Wouters2008; @Baas2008; @Lagoudakis2011; @Thunert2016; @Daskalakis2015]. All these aspects of polariton condensates are not welcomed as they blur the fundamental character of the phenomenon by disrupting it with technical impediments. These become obstacles for prospective applications with polaritons, such as investigating out-of-equilibrium phase transitions or to implement simulation and related devices [@Berloff2016]. On the other hand, confinement of polaritons in one dimensional structures provides a striking evidence of the mechanism of expulsion and acceleration of polariton condensates far from the exciton reservoir [@Wertz2010; @Wouters2010b; @Wouters2012; @Anton2013b]. In two-dimensional (2D) structures, interferences from scattering potentials, effects of laser-induced confinement and the wedge in the microcavity thickness, add additional difficulties in achieving extended and uniform condensates, even in samples with long polariton lifetimes [@Christmann2012; @Cristofolini2013; @Gao2016; @Sun2016]. In this work, we use a high quality 2D microcavity without spatial inhomogeneities, and observe the formation of a condensate that extends much beyond the laser spot region. Photoluminescence measurements are employed to demonstrate the expansion of polaritons from the excitation spot and the subsequent relaxation into the lowest energy level at the bottom of the polariton dispersion. The extended condensate is formed thanks to two main ingredients: the high homogeneity of the sample, which avoids localization effects, and the long radiative lifetime ($\sim\SI{100}{\pico\second}$), that allows the propagating polariton to relax into the ground state. Remarkably, condensation occurs at low densities ($\sim \SI{0.1}{polariton\per\micro\meter\squared}$), without the presence of the exciton reservoir, and covers an area of more than $\SI{0.03}{\milli\meter\squared}$. The energy-resolved spatial profiles obtained by solving a theoretical model, which combines the hydrodynamics of the expanding polaritons with their energy relaxation, reproduce the experimental results and confirm that, above a threshold power, phonon-mediated scattering into the lowest energy mode is effective in forming an extended 2D polariton condensate. The sample used in this study is a high quality-factor $3/2$ $\lambda$ GaAs/AlGaAs planar cavity containing 12 GaAs quantum wells placed at three anti-node positions of the electric field. The front (back) mirror consists of 34 (40) pair of AlAs/Al$_{0.2}$Ga$_{0.8}$As layers. The Rabi splitting is of and it is excited close to the zero detuning condition. Photoluminescence measurements are performed under non-resonant excitation with a low-noise, narrow-linewidth Ti:sapphire laser with stabilized output frequency to reduce the fluctuations in the exciton reservoir. The laser is focused on the sample in a spot with a Gaussian intensity profile of FWHM$=\SI{20}{\micro\meter}$. The sample emission is collected and imaged on the entrance slit of a streak camera coupled to a spectrometer in order to measure the time-, energy- and space-resolved polariton dynamics. ![**(a)** Energy versus real-space polariton emission along one direction, starting from the center of the pumping spot, displaying the expanding polaritons and the bottom condensate. The white-dashed line indicates the Gaussian potential induced by the excitonic repulsive interactions under the laser spot. **(b)** Energy-resolved reciprocal-space emission measured above the condensation threshold. The energy scale in the vertical axes is the same as in (a). **(c)** Two-dimensional spatial emission map of the bottom condensate. []{data-label="fig:1"}](emission-figure1-WF.pdf){width="48.00000%"} Under nonresonant excitation, a high density of excitons accumulate within the region of the pumping spot, inducing a blueshift of the polariton energy proportional to their repulsive interaction strength. Outside the optically pumped area, the density of uncoupled excitons decreases quickly in space, due to the small exciton mobility (2–5 microns), and the polariton energy recovers the linear regime. The Gaussian profile of the exciting beam is roughly reproduced by the potential landscape, evidenced in Fig. \[fig:1\](a) by a dashed, white line. In Fig. \[fig:1\](a), the emitted intensity is energy- (vertical axis) and spatially- (horizontal axis) resolved along one direction passing through the center of the excitation spot. The high-energy polaritons sitting at the top ( above the bottom energy) and formed at the center of the laser spot, expands radially outwards with a large in-plane wavevector ($k=\SI{2.2}{\per\micro\meter}$), as can be seen in the cross-section of the lower polariton dispersion (LPB, energy distribution in momentum space) shown in Fig. \[fig:1\](b). The macroscopic occupation of the lowest energy mode (k=0), visible at the bottom of the LPB in Fig. \[fig:1\](b), corresponds to the condensation outside of the spot region in Fig. \[fig:1\](a). At the same time, also lower energy states along the whole dispersion ($k\leq\SI{2.2}{\per\micro\meter}$) are occupied and expand. The high spatial homogeneity of the sample grants a uniform expansion of the polariton gas, as shown in the two-dimensional, energy-filtered space map shown in Fig. \[fig:1\](c). In order to study the polariton dynamics, the steady state, populated through the continuous wave (CW) pump laser, is perturbed by focusing an additional pulsed beam on top of the CW laser (both lasers are tuned to nonresonantly excite the system at the first minimum of the mirrors’ stop band). The repetition rate of the pulsed beam is slow enough to allow the system to recover its steady state condition before the arrival of the next pulse [@note:note1]. The evolution of the additional polaritons injected by the pulse is recorded with a time resolution of [@note:note2]. ![**(a)** Time dependence of the polariton emission intensity at a given energy, $E=\SI{2.6}{\milli\electronvolt}$, perturbed by a pulse, as a function of the distance from the excitation spot. At this energy, polaritons propagate with a constant velocity of . **(b)** The propagation speeds extracted as in panel **(a)** for the whole energy dispersion (blue dots) are compared to the polariton velocities as calculated from the LPB (green line). Faster expansions than expected are observed for energies below $E=\SI{1.7}{\milli\electronvolt}$.[]{data-label="fig:2"}](speedPanel.pdf){width="48.00000%"} By extracting the space-time images at different energies, as shown in Fig. \[fig:2\](a) for $E=\SI{2.6}{\milli\electronvolt}$, the expansion velocity as a function of energy can be estimated from the slope of the emission intensity. These velocities are compared in Fig. \[fig:2\](b) to the group velocities calculated from the LPB dispersion and the difference is indicated by the red-filling region, showing that the effect of relaxation from higher energy states becomes considerable at lower energies. An independent estimation of the group velocity can be obtained under CW pumping only: here, we use the expanding polariton fluid to perform an optical tomography of a spatial region where a single natural defect, injected in the DBR heterostructure during the growing process, breaks the spatial homogeneity of the sample. The pumping spot, the expanding fluid and the defect are clearly visible in Fig. \[fig:3\](a), which shows the $(x,y)$ image of the emission at $E=\SI{0.5}{\milli\electronvolt}$. The region around the defect is magnified and filtered at different energies, showing increasing flow speeds from Fig. \[fig:3\](b) to Fig. \[fig:3\](d). The velocities extracted from the interference pattern (shown in Fig. \[fig:3\](f)) match perfectly with the group velocities of the fluid calculated from the polariton dispersion (black line in Fig. \[fig:3\](e)) as $v_{g}=\frac{1}{\hbar} \frac{\partial E_{LPB}(k)}{\partial k} $. This shows unambiguously that the rate at which the polariton population distributes in space is the sum of two rates: the ballistic propagation and the filling rate due to relaxation processes. The second one becomes important for states close to the bottom of the LPB, and helps in reaching distances longer than expected from the bare propagation of low-speed polaritons. ![ **(a)** Two-Dimensional spatial emission with propagation against a natural defect ($E=\SI{0.5}{\milli\electronvolt}$). **(b-c-d)** Particular cases of intensity oscillations against the defect with the characteristic angle in the defect shadow for different propagation velocities (E=, , in panels **(b-c-d)**, respectively). **(e)** Energies versus wavevector extracted from the polariton dispersion (black line) and from the periodical spatial oscillations. Polariton density as a function of energy (red line). **(f)** Intensity cross section of the emission ($E=\SI{1.1}{\milli\electronvolt}$) along the white line in panel **(d)**. []{data-label="fig:3"}](Fig4.pdf){width="48.00000%"} This is the first time that such a behavior (fluid passing a defect) is recorded for a continuum of states, suggesting that, at higher densities, a new class of experiments on polariton superfluidity could be performed on expanding clouds free from reservoir artifacts and involving macroscopic distances. Remarkably, the polariton density outside of the pumped region, at the bottom of the LPB, manifests a nonlinear increase as a function of the pumping power for extremely low density values, as shown in Fig. \[fig:4\](a). At the same time, the formation of the bottom condensate is marked by a narrowing of the linewidth, as shown in Fig. \[fig:4\](b). The low density of the extended condensate dwindles the effects of polariton-polariton interaction, reducing the intrinsic dephasing of the condensate [@Kim2016] and making this configuration appealing both for applications and for future investigations on phase transition dynamics in polariton systems. In Fig. \[fig:4\](c), the ratio between the density in the lowest energy state and the whole expanding cloud is compared at different excitation powers, showing a nonlinear increase at the condensation threshold. ![ **(a)** Polariton density at a distance of from the excitation spot as a function of energy for different pumping power. **(b)** Reciprocal-space linewidth as a function of power (same horizontal scale as in **(c)**). **(c)** Population ratio between the lowest energy state and the whole continuum of states at $\SI{40}{\micro\meter}$ from the spot region, showing the nonlinear increase above the condensation threshold. []{data-label="fig:4"}](density-energy-figure.pdf){width="48.00000%"} A theory joining hydrodynamics and relaxation of an expanding and relaxing condensate can be developed by combining the mean field description of the dynamics given by the Gross-Pitaevskii equation (GPE) [@Wouters2008b; @Carusotto13a] with the rate equations that account for stimulated scattering due to the interaction with a phonon bath [@Doan2005; @Wouters2010b]. In order to obtain a simplified differential equation describing the steady-state polariton distribution in energy and space, we adapt the recent approach that merges both components of the dynamics at a level of the description that involves only the polariton density [@Bobrovska2016]. We study the dynamics outside of the pumping spot, without the presence of an exciton reservoir and with low polariton densities, allowing to ignore the nonlinear term, so that the GPE reads ${i\hbar \partial_t \psi(r)= \left(E_0 - \frac{\hbar^2}{2m}\nabla^2 - \frac{i \hbar}{2}\gamma \right)\psi(r)}$ (where $E_0$ is the energy corresponding to the bottom of the lower branch, $\gamma$ is the decay rate and we have assumed cylindrical symmetry). A continuity equation, relating spatial and temporal fluxes, is derived from the GPE for each polariton energy through spectral expansion: $$\left.\frac{\partial}{\partial t}n_\omega \right|_\mathrm{exp} = -\nabla\cdot\left(n_\omega \mathbf{v_\omega}\right) - \gamma n_\omega \label{eq:main-hidrodynamics}$$ where $\mathbf{v_\omega}$ is the velocity field of the condensate [@note:note3]. On the other hand, a rate equation is written for phonon-mediated stimulated scattering, coupling the different spectral modes: $$\left.\frac{\partial}{\partial t} n_\omega \right|_\mathrm{relax}= -\int \left( W_{\omega,\omega'} + W_{\omega',\omega}\right) n_\omega n_{\omega'}d\omega' \label{eq:main-rate-equation}$$ where $W_{\omega,\omega'}$ is the scattering rate from a state with energy $\omega$ to another with energy $\omega'$. Combining the two, and taking the steady state solution, we derive a differential equation for the polariton density in energy and space, ${\left.\partial_t n_\omega(r)\right|_\mathrm{relax} + \left.\partial_t n_\omega(r)\right|_\mathrm{exp}=0}$, that takes into account the relaxation and radiative lifetime. The equation is numerically solved outside of the pumping spot, with $\sigma$ the spot radius and $r=0$ the center of the spot, assuming as boundary initial condition for each energy the experimentally obtained polariton density at $r=\sigma$, and then solving for $r>\sigma$. The results of the calculation for two different conditions, with and without the presence of a phonon bath, are shown in Fig. \[fig:5\](a) and Fig. \[fig:5\](b) respectively. While Fig. \[fig:5\](a) reproduces the formation of an extended polariton condensate at the bottom of the dispersion, in Fig. \[fig:5\](b) the small polariton velocity dominates the dynamics close to $k=0$ and precludes the observation of an extended 2D polariton condensate. ![Comparison between two numerical simulations, with and without relaxation. This result shows that relaxation is needed to account for the main feature observed, that is, the formation of an extended lower condensate. The extension of the lower condensate cannot be given by any kind of expansion due to its small velocity.[]{data-label="fig:5"}](fig5ab.pdf){width="48.00000%"} In conclusion, the formation under non-resonant pumping of an extended 2D polariton condensate at the bottom of the LPB has been experimentally demonstrated. The polariton energy under the laser spot is blues-shifted with respect to the lowest energy outside of the laser spot. This produces an expanding flow of polaritons at the energy of the blue-shift, but also a continuum of states along the dispersion, which are formed through relaxation and which expand with decreasing velocities for decreasing energies. Thanks to the high quality of the sample, we are able to observe the relaxation until the bottom of the LPB. Above a threshold density, a condensate forms at $k=0$ at the lowest available energy, covering a spatial region larger then $\SI{0.03}{\milli\meter\squared}$ around the pumping spot. These results are well reproduced by a simple theoretical model that captures the physics of expansion and relaxation of higher energy polaritons. Our findings provide the closest realization so far of a textbook Bose–Einstein condensate in a solid-state matrix, with macroscopic sizes and dilute, tunable densities. Such systems should allow to truly take advantage of polariton condensates for fundamental research of out-of-equilibrium quantum dynamics. Acknowledgments {#acknowledgments .unnumbered} --------------- Funding from the POLAFLOW ERC Starting Grant is acknowledged. [45]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , , , , ****, (). , , , , , , , , , , , ****, (). , , , , ****, (). , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , ****, (). , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , , ****, (). , , , ****, (). , , , , , (). , , , , , , , , , , , ****, (). , , , ****, (). , ****, (). , , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , (). . . , ****, (). , , , , ****, (). , ****, (). . [ @column@grid ]{} **Formation of a macroscopically extended polariton condensate without an exciton reservoir\ Supplemental Material** Polariton Lifetime {#polariton-lifetime .unnumbered} ------------------ In order to determine the polariton lifetime, time-resolved measurements are performed using a Ti:sapphire laser delivering 100 fs pulses with a 82 MHz repetition rate. This beam resonantly injects polariton inside the cavity by tuning the frequency and angle of incidence of the laser in order to match the LPB close to the bottom energy, $E_{k=0}=772.85$ nm, but sligthly shifted in angle to allow the filtering out of the reflected light. In this way, the bottleneck effect is limited and the polariton lifetime can be extracted from the polariton emission. This is visible from the exponential decay shown in Fig. \[fig:1-3\]. From the exponential decay in time of the population it is possible to extract a lifetime of $100$ ps. ![**(a)** Time resolved emission taken in a small bound of $k=0$ and with the laser pump at $E_{k=0}=771$ nm. **(b)** Cross Section of the time resolved decay emission in the region within the dashed, red lines in panel **(a)**. The exponential decay fitting gives a decay of 100 ps (red line).[]{data-label="fig:1-3"}](pulsedTR_Emission.pdf){width="48.00000%"} Time-Resolved Dynamics {#time-resolved-dynamics .unnumbered} ---------------------- In Fig. \[fig:2S\], four snapshots are shown with the emission intensity as a function of energy (vertical axes) and space (horizontal axes) at different delay times (42 ps, 66 ps, 77 ps and 104 ps after the arrival of the non-resonant pulse in Fig. \[fig:2S\](a), Fig. \[fig:2S\](b), Fig. \[fig:2S\](c) and Fig. \[fig:2S\](d), respectively). Polaritons at high energy quickly expand and relax into lower energy states. The expanding flow is apparent in the space-energy emission map shown in Fig.\[fig:2S\]b, and it is more effective for the lower energy states (Fig.\[fig:2S\](c)-(d)). ![Energy vs real space at different times shown by the labels. In the figure, the photoluminescence intensity is displayed in a false-color linear scale.[]{data-label="fig:2S"}](emissionPanelDiffTime.pdf){width="50.00000%"} An accumulation points, induced by the pulse spot width being slightly larger than the CW one, allows the observation of a characteristic “whiplash” of relaxation in the space-energy map. Eventually, at longer times, polaritons relax into the bottom of the LPB, as shown in Fig.\[fig:2S\](d). Theoretical description {#theoretical-description .unnumbered} ----------------------- We consider a mean field description of the dynamics of the condensate wave function $\psi(\mathbf r)$ given by the Gross-Pitaevskii equation [@Wouters2008bS] $$\begin{aligned} i\hbar \frac{\partial \psi(\mathbf r)}{\partial t} &= \left\{E_0 - \frac{\hbar^2}{2m}\nabla_\mathbf{r}^2 + \frac{i \hbar}{2}\left[R[n_R(\mathbf{r})]-\gamma\right] + V(\mathbf r) \right. \nonumber \\ &\left. \vphantom{\frac{\hbar^2}{2m}} + \hbar g|\psi(\mathbf r)|^2 \right\}\psi(\mathbf r). \label{eq:full-GP}\end{aligned}$$ Here, $R[n_R(\mathbf r)]$ describes a coupling to the excitonic reservoir $n_R(\mathbf r)$, that has its own dynamics: $$\dot{n}_R(\mathbf r) = P(\mathbf r) - \gamma_R n_R(\mathbf r) - R[n_R(\mathbf r)]|\psi(\mathbf r)|^2$$ In this work we are interested on the expansion and relaxation dynamics outside the excitation spot. The excitonic interaction with the reservoir produces a repulsive potential that can be described by $V(\mathbf r)$ and that tends to expel the polaritons from the spot where they have been created: we will study the dynamics of these polaritons outside the spot and, due to the low densities, ignore the nonlinearity. Therefore, in the our region of interest, $|\mathbf r|>\sigma$ ($\sigma$ being the size of the spot, i.e., the region where $n_R(\mathbf r)$ and $V(\mathbf r)$ are zero) the hydrodynamical description of the polariton flow is given by: $$i\hbar \frac{\partial \psi(\mathbf r)}{\partial t} = \left\{E_0 - \frac{\hbar^2}{2m}\nabla_\mathbf{r}^2 - \frac{i \hbar}{2}\gamma \right\}\psi(\mathbf r). \label{eq:outside-GP}$$ Since the mechanism that creates these polaritons take place for $|r|<\sigma$, the polaritons created by the pumping terms in Eq. \[eq:full-GP\] must be accounted by an appropriate boundary condition in Eq. \[eq:outside-GP\]. Let’s consider now a time-dependent spectral expansion of the wavefunction as follows: $$\psi(\mathbf r, t) = \sum_\omega \psi_\omega(\mathbf r,t) e^{i\omega t}$$ where we assume that $ \psi_\omega(\mathbf r,t) $ evolves in time much slower than the oscillations given by $e^{i \omega t}$. Next, we write $\psi_\omega(\mathbf r,t)$ in terms of density and phase: $$\psi_\omega(\mathbf r,t) = \sqrt{n_\omega(\mathbf r,t)}e^{i \phi_\omega(\mathbf r, t)}$$ By plugging this ansatz in Eq.  and taking the imaginary part, we find: $$\begin{aligned} &i\hbar \sum_\omega e^{i[\omega t + \phi_\omega]}\left\{\frac{\partial}{\partial t}\sqrt{n_\omega}+ i (\omega + \frac{\partial}{\partial t}\phi_\omega) \sqrt{n_\omega}\right\} \nonumber \\ &= -\frac{\hbar^2}{2m}\sum_\omega e^{i[\omega t + \phi_\omega]} \left[ \nabla^2(\sqrt{n_\omega}) + 2i\nabla\phi_\omega \nabla(\sqrt{n_\omega}) \right. \nonumber\\ & +\left. i \sqrt{n_\omega} \nabla^2\phi_\omega - (\nabla \phi_\omega)^2\sqrt{n_\omega} \right]- i\frac{\hbar}{2}\gamma \sum_\omega e^{i[\omega t + \phi_\omega]}\sqrt{n_\omega}\end{aligned}$$ Now, we equate the terms in the sum with the same $\omega$ and take the imaginary part of this equation to get: $$\frac{\partial}{\partial t}\sqrt{n_\omega} = -\frac{\hbar^2}{2m}\left(2i\nabla\phi_\omega \nabla(\sqrt{n_\omega})+i \sqrt{n_\omega} \nabla^2\phi_\omega\right) -i\frac{\hbar}{2}\gamma \sqrt{n_\omega}$$ which we can rewrite as an energy-resolved continuity equation: $$\left.\frac{\partial}{\partial t}n_\omega \right|_\mathrm{exp} = -\nabla\left(n_\omega \mathbf{v_\omega}\right) - \gamma n_\omega \label{eq:hidrodynamics}$$ where $$\mathbf{v_\omega} = \frac{\hbar}{m}\nabla\phi_\omega.$$ The suffix in the partial derivative accounts for the fact that this equation only describes the hydrodynamics of expansion of the polariton fluid. We now describe the variation in $n_\omega$ due to the relaxation to lower energy modes mediated by phonon scattering. A rate equation is usually written in $k$ space, and if we take into account only stimulated scattering, it reads as [@Doan2005S]: $$\left.\frac{\partial}{\partial t} n_\mathbf{k} \right|_\mathrm{relax}= -\sum_{\mathbf{k'}} \left[ W_{\mathbf{k},\mathbf{k'}} n_{\mathbf{k}}n_{\mathbf{k'}} + (\mathbf{k} \leftrightarrow \mathbf{k'})\right] \label{eq:rate-equation}$$ If the scattering is mediated by a phonon bath, Fermi’s golden rule gives the following transition rate in $k$ space: $$\begin{aligned} W_{\mathbf{k},\mathbf{k'}} &\approx \frac{L_z (\chi_k \chi_{k'} \Delta\tilde E_{k,k'})^2}{\hbar \rho V u^2 q_z}B^2(q_z)|D_e - D_h|\times\nonumber\\& |n_{\mathrm{ph}}(\omega_{k'}-\omega_k)|\theta(\Delta\tilde E_{k,k'}-|\mathbf{k}-\mathbf{k'}|)\end{aligned}$$ where $u$ is the longitudinal sound velocity, $L_z$ is the quantum well width, $V$ is the crystal volume, $\rho$ is the mass density of the solid, ${\Delta\tilde E_{\mathbf{k},\mathbf{k'}} = |\omega_\mathbf{k'}-\omega_\mathbf{k}|/\hbar u}$, $q_z$ is, for a given in plane momentum change $\mathbf{k}-\mathbf{k'}$, the momentum in the $z$ direction that must be taken by the phonons for the scattering process to conserve energy ${q_z = \sqrt{\Delta\tilde E_{k,k'}^2-|\mathbf{k}-\mathbf{k'}|^2}}$, $|n_\mathrm{ph}(\omega)|$ is the absolute of the phonon density and $B(q)$ is given by: $$B(q) = \frac{8\pi^2}{L_z q(4\pi^2 - L_z^2 q^2)}\sin\left(\frac{L_z q}{2}\right).$$ Since the experiment is performed on a polar symmetry, we can assume that $n_\mathbf{k}$ depends only on the module of $\mathbf k$: $$n_\mathbf{k} = n_k$$ and we can perform the sum in $\mathbf{k'}$ in going to the continuum limit $\sum_\mathbf{k'} \rightarrow \frac{S}{(2\pi)^2}\int dk_x dk_y = \frac{S}{(2\pi)^2}\int k \, dk\, d\theta$ $$\left.\frac{\partial}{\partial t} n_k \right|_\mathrm{relax}= -\frac{S}{(2\pi)^2} \int k'\left[ W_{\mathbf{k},\mathbf{k'}} n_k n_{k'} + (\mathbf{k} \leftrightarrow \mathbf{k'})\right]dk'\,d\theta \label{eq:rate-equation-polar}$$ and writing: $$W_{k,k'} = \int_0^{2\pi} d\theta \,W_{\mathbf{k},\mathbf{k'}}$$ where we integrated the angular dependence of the scattering rate, entering in the expression of $W_{\mathbf{k},\mathbf{k'}}$ from $q_z = \sqrt{\Delta \tilde E_{k,k'}^2-k^2-{k'}^2+2k k' \cos(\theta)}$, we get: $$\left.\frac{\partial}{\partial t} n_k \right|_\mathrm{relax}= -\frac{S}{(2\pi)^2} \int k'\left( W_{k,k'} +W_{k',k} \right) n_k n_{k'} dk' \label{eq:rate-equation-polar-simp}$$ Since the energy of the polaritons depends only on the modulus of the momentum, $\omega_k \approx \hbar^2 k^2/2m_{LP}$, we can write $n_k$ as a function of energy $n_\omega$, and using $d\omega = \hbar^2 k \, dk/2 m_{LP}$, write: $$\left.\frac{\partial}{\partial t} n_\omega \right|_\mathrm{relax}= -\int \left( W_{\omega,\omega'} + W_{\omega',\omega}\right) n_\omega n_{\omega'}d\omega' \label{eq:rate-equation-polar-simp2}$$ where $W_{\omega,\omega'} = 2m_{LP}S/(\hbar^2 4\pi^2) W_{k(\omega),k'(\omega')}$. Finally, we write a joint equation for the time evolution of $n_\omega$ by joining both the hydrodynamics of expansion of Eq. : $$\begin{aligned} \frac{d n_\omega}{dt} &= \left.\frac{\partial}{\partial t} n_\omega \right|_\mathrm{exp} +\left.\frac{\partial}{\partial t} n_\omega \right|_\mathrm{relax} = \nonumber \\ & -\frac{1}{r}\frac{\partial}{\partial r}\left(r n_\omega v_\omega \right) - \gamma n_\omega \nonumber\\ & - \int \left( W_{\omega,\omega'} + W_{\omega',\omega}\right) n_\omega n_{\omega'}d\omega' \label{eq:final-equation}\end{aligned}$$ where we wrote the velocity as a vector field with radial component only $\mathbf{v_\omega} = v_\omega \mathbf{u_r}$. Our next approximation will be to assume that the velocity will be given by ${v_\omega \approx (1/\hbar) \partial \omega_k /\partial k }$. To study the spatial profile resolved in energies, we will look now for steady state solutions of equation setting the time derivative to zero and getting the following integro-differential equation in real space: $$\begin{aligned} \frac{\partial n_\omega(r)}{\partial r} &= - \frac{n_\omega(r)}{r}+ \frac{1}{v_\omega}\left[ \vphantom{\frac{2 m_{LP} S}{\hbar^2(2\pi)^2}}-\gamma n_\omega(r) \right.\nonumber \\ &\left. -\int \left( W_{\omega,\omega'} + W_{\omega',\omega}\right) n_\omega n_{\omega'}d\omega' \right]\end{aligned}$$ This we solve numerically by writing the integral again as a discrete sum: $$\begin{aligned} \frac{\partial n_\omega(r)}{\partial r} &= - \frac{n_\omega(r)}{r}+ \frac{1}{v_\omega}\left[ \vphantom{\frac{2 m_{LP} S}{\hbar^2(2\pi)^2}}-\gamma n_\omega(r) \right.\nonumber \\ &\left. - \sum_{\omega'} \left( \tilde W_{\omega,\omega'} + \tilde W_{\omega',\omega}\right) n_\omega n_{\omega'} \right] \label{eq:numerical}\end{aligned}$$ with $$\tilde W_{\omega,\omega'} = \kappa \frac{(\chi_\omega \chi_{\omega'} \Delta\tilde E_{\omega,\omega'})^2}{|e^{\beta (\omega'-\omega)}-1|}\int \frac{B(q_z)^2}{q_z}d\theta$$ where $\Delta \tilde E_{\omega,\omega'} = |\omega'-\omega|/(\hbar u)$, and $$\kappa = \frac{2 m_{LP} S}{\hbar^2(2\pi)^2}\frac{L_z}{\hbar \rho V u^2}|D_e - D_h|\Delta\omega'$$ [11]{} T. D. Doan, H.T. Cao, D.B.T. Thoai and H. Haug, Phys. Rev. B **72**, 085301, (2005). M. Wouters, I. Carusotto and C. Ciuti, Phys. Rev. B **77**, 115340 (2008).

--- abstract: 'New tunneling data are reported in underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8-\delta}$ using superconductor-insulator-superconductor break junctions. Energy gaps, $\Delta$, of 51$\pm$2, 54$\pm$2 and 57$\pm$3 meV are observed for three crystals with T$_{c}$=77, 74, and 70 K respectively. These energy gaps are nearly three times larger than for overdoped crystals with similar T$_{c}$. Detailed examination of tunneling spectra over a wide doping range from underdoped to overdoped, including the Josephson $I_{c}R_{n}$ product, indicate that these energy gaps are predominantly of superconducting origin.' address: | $^{1}$Science and Technology Center for Superconductivity,\ Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439\ $^{2}$Department of Applied Physics, Science University of Tokyo, Kagurazaka 1-3, Shinjuku-ku, Tokyo, 162-8601 Japan\ $^{3}$Illinois Institute of Technology, Chicago, Illinois 60616\ $^{4}$Department of Physics, Izmir Institute of Technology, TR-35210 Izmir, Turkey\ $^{5}$Naval Research Laboratory, Washington, D.C. 20375 author: - | N. Miyakawa$^{1,2}$, J. F.  Zasadzinski$^{1,3}$, L. Ozyuzer$^{1,4}$, P. Guptasarma,$^{1}$ , D.  G. Hinks$^{1}$,\ C. Kendziora $^{5}$ and K. E. Gray $^{1}$ title: | Predominantly Superconducting Origin of Large Energy Gaps in\ Underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8-\delta}$ from Tunneling Spectroscopy --- Efforts to understand the mechanism of pairing in high-T$_{c}$ superconducting (HTS) cuprates are currently focused on the unusual doping dependences of superconducting and normal state properties. In particular, underdoped HTS compounds have exhibited pseudogap phenomena above T$_{c}$ in both spin and charge excitations.[@1] Recently, tunneling[@2] studies on Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8-\delta}$ (Bi2212) in the superconducting state have shown a remarkable effect whereby the energy gap exhibits a strong, monotonic dependence on doping, increasing substantially in the underdoped phase even as T$_{c}$ decreases. It has been pointed out [@3] that the smooth dependence on doping may nevertheless originate from a quasiparticle gap that evolves from superconducting character in the overdoped phase to another type (e.g. charge density wave, spin density wave, etc.) in the underdoped phase. While the measured tunneling gap vs.  doping is consistent with other probes[@2] including angle resolved photoemission (ARPES)[@4; @5; @6] and Raman[@7], it is at odds with some measurements that support a superconducting order parameter scaling with T$_{c}$.[@8] Thus a critical question is whether this relatively large energy gap originates entirely from superconducting pairing or has a contribution from some other electronic effect. Here we address the nature of the gap measured by tunneling and report new data in very underdoped Bi2212 by superconductor-insulator-superconductor (SIS) break junctions. Energy gaps, $\Delta$, of 51$\pm$2, 54$\pm$2 and 57$\pm$3 meV are observed for three underdoped crystals with T$_{c}$=77, 74, and 70 K respectively, extending the previously reported trend[@2] further into the underdoped regime. Detailed examination of the tunneling spectra over a wide doping range, including the Josephson $I_{c}R_{n}$ product, show that these energy gaps are predominantly of superconducting origin. Historically, tunneling studies have been relied upon to examine the magnitude of the superconducting energy gap in both conventional and HTS.[@9] In particular, SIS junctions provide an accurate measure of 2$\Delta$ from the peak in tunneling conductance which is only weakly affected by thermal smearing or quasiparticle scattering.[@2] However, the large magnitudes of energy gaps observed here lead to such extraordinarily large values of $2 \Delta $/kT$_{c}$ (as high as 20) that it is necessary to examine carefully the entire tunneling spectrum to clarify the physical origin of these energy gaps. Most theoretical models of HTS stress the importance of electronic correlations such as spin density waves[@10] and its precursors[@11], or charge density waves in the underdoped phase[@3; @12] which might give rise to momentum-dependent quasiparticle excitation gaps, $\Delta _{c}$([ **k**]{}) in addition to those arising from superconductivity, $\Delta _{s}$([**k**]{}). In these “two-gap” scenarios the energy gaps often add in quadrature[@3; @12] such that the total energy gap $\Delta$= $(\Delta _{s}^{2} + \Delta _{c}^{2})^{1/2}$. Since these other correlation gaps are often used to explain the pseudogap above T$_{c}$, our investigation here has a bearing on this issue as well. We argue first that if two distinct gaps exist, (1) they should have very different doping and temperature dependences; and (2) it is unlikely that $\Delta _{s}$([**k**]{}) and $\Delta _{c}$([**k**]{}) will have identical momentum dependences. Thus the quasiparticle density of states (DOS) should exhibit distinct features corresponding to each gap. We observe no evidence of a second gap feature and it will be shown that the shape of the gap region spectrum smoothly evolves with doping, with features changing mainly in energy scale. We also examine a property of SIS junctions that depends solely on superconductivity, the Josephson current. A statistical summary of the Josephson $I_{c}R_{n}$ products of over 40 SIS junctions is presented and it is shown that the largest values (both average and maximum) are all found in underdoped crystals, where the measured quasiparticle gap is the largest. This links the measured quasiparticle gap to a purely superconducting energy scale, the Josephson strength. Thus we are forced to conclude that over the range of doping studied (from 70 K underdoped to 62 K overdoped) the measured quasiparticle gap appears to be due predominantly to superconductivity. We grew high quality single crystals using a slightly modified floating-zone process as described elsewhere.[@2] This yields an optimal T$_{c}$ onset of 95 K and the doping is varied through the oxygen concentration. The 70 K underdoped crystal was prepared by a different procedure (see ref. [@13]) and there is good agreement among the differently processed samples. Both SIS break junctions and SIN (superconductor-insulator-normal metal) junctions were prepared on freshly cleaved surfaces by a point contact technique with Au tip.[@2; @14; @15] Tunneling spectra and gap values in the SIN junctions[@16] are consistent with those presented here but in this paper we focus on the SIS junctions. In Fig. 1 is shown the dI/dV vs. V for an SIS junction on the most underdoped crystal with T$_{c}$ = 70 K. The conductance data have been normalized by a constant which is the conductance at V= 340 mV. The shape of the conductance is similar to that found on optimally doped crystals[@2] exhibiting sharp conductance peaks (eV$_{p}$ = 2$\Delta$), subgap conductance and pronounced dip features at eV= 3$\Delta$. At zero bias there is a small Josephson current in the I(V) curve (inset of Fig.  1) which shows up as a conductance peak. The dashed line of Fig.  1 is a fit using a weighted, momentum averaged $d$-wave density of states (DOS).[@17] The fit is good except for the obvious discrepancies at the dips, and the weighting factor indicates that there is preferential tunneling along the ($\pi$,0) point, the maximum of the $d$-wave gap. This analysis gives $\Delta$ = 60 meV for the maximum $d$-wave gap and $\Gamma$ = 6 meV where $\Gamma$ is a quasiparticle scattering rate. The data of Fig. 1 can also be adequately fit with a smeared BCS DOS leading to $\Delta$=57 meV, which is exactly half of conductance peak voltage, V$_{p}$. Thus far, reproducibility on this crystal is limited to four separate SIS junctions, but in each case a well-defined energy gap is found with $\Delta$= 57$\pm$3 meV. The large energy gaps found here extends the previously reported trend \[2\] further into the underdoped regime and leads to a value of 2$\Delta$/kT$_{c}$= 20. In Fig. 2 we show representative SIS tunneling spectra for a wide doping range from overdoped with a T$_{c}$ = 62 K to underdoped with a T$_{c}$ = 70 K. The Josephson peak has been removed for clarity. Four of the curves in Fig.  2 have been published previously[@2; @14] but are included to display the trend with doping. To compare the spectra, which have widely different gap values, we rescale the voltage axis by V$_{p}$/2. In this way the voltage scale is approximately in units of $\Delta$/e. The bottom three curves of Fig.  2 are new results of this study and in the cases of the other underdoped crystals (T$_{c}$ = 74 K, 77 K) the curves are representative of many different junctions formed on each crystal. Well-defined gap structure was reproducibly observed with energy gaps, $\Delta$, of 51$\pm$2, and 54$\pm$2 for the crystals with T$_{c}$=77 and 74 K respectively. What is observed in Fig. 2 is a smooth evolution of the spectra from overdoped to underdoped, with a single gap feature that grows monotonically as the doping level is reduced. All of the curves exhibit subgap conductance, which as shown in Fig. 1, can be attributed to a $d$-wave DOS. A notable feature in the SIS junctions of Fig. 2 is the pronounced dip structure[@14] which remains at eV$\sim$3$\Delta$ over the entire range of this study, i.e. $\Delta$= 15 meV - 60 meV. In none of the curves is there seen any evidence of a second peak in the conductance which would be a clear indication of another gap in the quasiparticle spectrum. Rather, what is most striking is that the entire gap region spectrum has nearly the same shape over the entire doping range, and all that is changing is the energy scale of the spectral features. Thus we find no evidence that the nature of the energy gap is changing over the doping range. The $I_{c}R_{n}$ value for the junction in Fig. 1 is about 2 mV as obtained from the I(V) curve, but values as large as 14 mV have been observed for one of the other junctions of this same crystal. Here $R_{n}$ is estimated from the high bias conductance which is relatively constant as shown in Fig. 2. Large values of $I_{c}R_{n}$ (15 mV-25 mV) were previously reported for an 83 K underdoped sample.[@2] Table I shows the average and maximum $I_{c}R_{n}$ values for over 40 SIS junctions on Bi2212 for a variety of doping levels. We find that the average $I_{c}R_{n}$ increases with decreased doping and that the three highest values among all the junctions are found in underdoped samples, consistent with the large quasiparticle gaps observed. Although the statistical distribution is still rough at present, the trend indicates that the quasiparticle gap is linked to the Josephson strength, $I_{c}R_{n}$, a purely superconducting energy scale. The temperature dependence of tunneling conductance was measured in some cases and in Fig. 3 are shown the results for another junction on the T$_{c}$ =77 K underdoped crystal. The high bias junction resistance is $\sim$ 11 k$\Omega$ and all of the data have been normalized by this constant value. Here the Josephson peak at zero bias is left in. As clearly seen in this figure, the superconducting gap peak, V$_{p}$, changes very little up to 50 K, but for T $>$ 60 K the magnitude of the superconducting gap starts decreasing and states at Fermi level start filling in. The quasiparticle peak coming from superconducting gap and the zero-bias peak coming from Josephson current continuously disappear near the bulk T$_{c}$. This is important because it means that the local T$_{c}$ of the junction is essentially the same as the bulk T$_{c}$. The decrease of the gap magnitude with temperature is also seen directly in the raw data for an 83 K underdoped and a 95 K optimal doped crystal[@2] and is consistent with other SIS break junctions on Bi2212.[@18] To attempt a more quantitative analysis, the superconducting gap, $\Delta$(T), and quasiparticle scattering rate, $\Gamma$(T), as a function of temperature have been estimated by fitting the data in Fig.  3 to a simple model for SIS junctions[@2; @9] which uses a smeared BCS DOS to describe the Bi2212. As discussed earlier in the examination of Fig.  1 this analysis might lead to a minor discrepancy in the magnitude of the gap when compared to a $d$-wave model, but the simpler smeared BCS function makes the analysis much easier. The results of this procedure are shown in Fig. 4. The principal result is that the gap magnitude decreases significantly as T increases near T$_{c}$. For T$>$72 K the conductance data are so smeared out that no accurate values for $\Delta$ and $\Gamma$ can be obtained. The decrease in gap magnitude near T$_{c}$ is in disagreement with the interpretation of recent STM experiments on underdoped Bi2212 where the raw data seem to suggest a T-independent gap.[@19] We note, however, that $\Delta$(T) cannot be inferred directly in SIN junctions due to importance of fermi factors in the tunneling conductance. We estimate the maximum Josephson current $I_{c}^{*}$ from the measured conductance peak at zero bias and use the junction resistance $R_{n}$ at high bias to plot the temperature dependence of the Josephson strength, $I_{c}^{*}R_{n}$, normalized to the value obtained at T = 4.2 K. The normalized Josephson strength plotted in Fig. 4 is approximate and it is used only to estimate the T$_{c}$ of the junction which can be seen is very close to the measured bulk T$_{c}$ of the crystal (77 K). We thus can say that the large energy gaps in our underdoped crystals correspond to the measured bulk T$_{c}$ and are not a consequence of some local deviation in stoichiometry. Above the junction T$_{c}$ the quasiparticle gap feature has essentially disappeared and only a very weak depression in the conductance at zero bias remains. This behavior is consistent with our previous T-dependent SIS data[@2] on an 83 K underdoped crystal and a 95 K optimally doped crystal as well as other SIS data in the literature.[@18] We thus find no clear evidence of a pseudogap above T$_{c}$ in the SIS tunneling data, again in contrast to the strong gap feature observed in recent STM experiments.[@19] One possible explanation is that the T-dependent pseudogap is highly anisotropic in k-space as is indicated in ARPES.[@1; @6] Since the SIS tunneling probes a momentum averaged DOS as indicated by the strong subgap conductance in Figs. 1 and 2, then the effect of a highly anisotropic pseudogap on these junctions may give just a weak depression in conductance as is found. We now summarize the doping dependence of the tunneling data. The shape of the SIS quasiparticle spectra are qualitatively the same for all doping levels, exhibiting a single gap feature at eV=2$\Delta$ that evolves smoothly with doping. The full gap-region spectrum scales with $\Delta$ indicating that the character of the energy gap is the same over the doping range studied. Josephson currents are found in all the SIS junctions and in some cases the magnitudes of the $I_{c}R_{n}$ product are very large, nearly 25% of $\Delta$/e. Furthermore, the three highest $I_{c}R_{n}$ values among more than 40 SIS junctions were all found in underdoped samples where the quasiparticle gap is the largest. These results indicate that the quasiparticle gaps are predominantly due to superconductivity and we find no evidence of another gap, $\Delta _{c}$([**k**]{}), at low temperatures. The trend of $\Delta$ vs.  doping closely follows the doping dependence of T$^{*}$, the pseudogap temperature for Bi2212.[@20] Thus the large gaps in the underdoped region suggests that significant superconducting pairing correlations exist at temperatures between T$_{c}$ and T\* and that T$_{c}$ is the temperature where long-range phase coherence sets in.[@1; @21; @22] The temperature dependence of the superconducting gap, $\Delta$(T), in underdoped crystals may also be providing a subtle clue about the nature of the superconducting fluctuations above T$_{c}$. While $\Delta$ does not close at T$_{c}$, there is nevertheless a significant decrease in its magnitude that is seen even in the raw data. This argues against a picture of tightly bound, pre-formed bosons below T\*, which would be expected to have a T-independent gap near T$_{c}$. Rather the data are more consistent with intermediate coupling regime scenarios for the pairing fluctuations which persist up to T\*.[@1; @22] This work was partially supported by U.S. Department of Energy, Division of Basic Energy Science-Material Science under contract No. W-31-109-ENG-38, and the National Science Foundation, Office of Science and Technology Centers under contract No. DMR 91-20000. M. Randeria, cond-mat/9710223. N. Miyakawa, P.  Guptasarma, J. F.  Zasadzinski, D. G.  Hinks, K. E.  Gray, Phys.  Rev.  Lett.  [**80**]{}, 157 (1998). R.  S.  Markiewicz, C. Kusko, M. T. Vaughn, J. Supercond. (in press). J. M.  Harris, Z.-X.  Shen, P.  J.  White, D. S.  Marshall, M. C.  Schabel, J. N.  Eckstein and I.  Bozovic, Phys.  Rev.  B [**54**]{}, 15665 (1996). A. G. Loeser, Z.-X. Shen, D. S. Dessau, D. S. Marshall, C. H. Park, P.  Fournier, A.  Kapitulnik, Science [**273**]{}, 325 (1996). H. Ding et al., Nature [**382**]{}, 51 (1996); M. R. Norman, et al., Nature [**392**]{}, 157 (1998). G.  Blumberg et al, Proc.  SNS 1998 (cond-mat/9711073). X. K. Chen et al., Phys. Rev. B [**48**]{}, 10530 (1993); C.  Panagopoulos et al., Phys. Rev. B [**57**]{}, 13422 (1998). E. L. Wolf, [*Principles of Electron Tunneling Spectroscopy*]{} (Oxford Univ. Press, New York, 1985). A. P. Kampf, J. R. Schrieffer, Phys. Rev. B [**42**]{}, 7967 (1990). H. Kotani, K. Ueda, Phys.  Rev.  Lett.  [**80**]{}, 5619 (1998). T. Dahm, D. Manske, L. Tewordt, Phys. Rev. B [**56**]{}, R11419 (1997). C. Kendziora, R. J. Kelley, E. Skelton, M.  Onellion, Physica C [**257**]{}, 74 (1996). Y. DeWilde, N. Miyakawa, P. Guptasama, M. Iavarone, L. Ozyuzer, J. F. Zasadzinski, P. Romano, D. G. Hinks, C. Kendziora, G. W. Crabtree and K.E. Gray, Phys. Rev. Lett. [**80**]{}, 153 (1998). J. F. Zasadzinski, L. Ozyuzer, Z. Yusof, J. Chen, K. E. Gray, M. Mogilevsky, D. G. Hinks, J. L. Cobb, J. T. Markert, in [*Spectroscopic Studies of High T$_{c}$ Cuprates*]{}, ed. I. Bozovic, D. van der Marel, Proc. SPIE [**2696**]{}, 338 (1996) (Bellingham). N. Miyakawa, J. F. Zasadzinski, Y. DeWilde, L. Ozyuzer, P.  Guptasarma, D. G. Hinks, C. Kendziora, G. W. Crabtree, K. E.  Gray (unpublished). H. Hancotte, D. N. Davydov, R. Deltour, A. G. M.  Jansen, P. Wyder, Physica C [**280**]{}, 71 (1997). D.  Mandrus, J.  Hartge, C.  Kendziora, L.  Mihaly, L.  Forro, Europhys.  Lett., [**22**]{}, 199 (1993); D. Mandrus et al., Nature (London) [**351**]{}, 460 (1991). Ch.  Renner, B.  Revaz, J.-Y Genoud, K.  Kadowaki, and O.  Fischer, Phys.  Rev.  Lett.  [**80**]{}, 149 (1998). M. Oda, K. Hoya, R. Kubota, C. Manabe, N. Momono, T. Nakano, M. Ido, Physica C [**281**]{}, 135 (1997). V.J. Emery, S.A. Kivelson, Nature [**374**]{}, 434 (1995). Q. Chen, I. Kosztin, B. Janko, K. Levin, cond-mat/9807414. Fig.  1 Differential conductance, dI/dV, for SIS break junction on an underdoped single crystal of Bi2212 with T$_{c}$=70 K. The inset shows a Josephson current at zero bias. Fig.  2 Normalized SIS tunneling conductances of Bi2212 with various doping levels from underdoped to overdoped. The voltage axis has been rescaled in units of $\Delta$/e. Fig.  3 Temperature dependence of SIS tunneling conductance on an underdoped Bi2212 (T$_{c}$=77 K) break junction. For clarity, each conductance has been normalized by its value at 200 mV and (except for the 5 K curve) is offset vertically. Fig.  4 Temperature dependence of superconducting gap $\Delta$(T) (circle), quasiparticle scattering rate $\Gamma$(T) (triangle) and normalized Josephson strength, $I_{c}^{*}R_{n}$ (square) (see text in details), where the normalization of $I_{c}^{*}R_{n}$(T) has been done by $I_{c}^{*}R_{n}$(4.2 K). The full curve represents the BCS curve of superconducting gap $\Delta$(T).

--- abstract: 'In this paper, motivated by recent important works due to Frank-Lewin-Lieb-Seiringer [@FLLS] and Frank-Sabin [@frank-sabin-1], we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates in certain sense. An application of the inequality shows the well-posedness to the periodic Hartree equation describing the infinitely many quantum particles with the power type interaction.' address: 'Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan' author: - Shohei Nakamura title: The orthonormal Strichartz inequality on torus --- Introduction and Main results ============================= The classical Strichartz inequality for the free Schrödinger propagator $e^{it\Delta}$ may be stated that for any space dimension $d\geq1$ and any admissible pair $p,q\in[1,\infty]$, namely $\frac 2p +\frac dq =d$ and $(p,q,d)\neq (1,\infty,2)$, $$\big\| | e^{it\Delta} f |^2\big \|_{L^p_tL^q_x(\mathbb{R}^{d+1})} \lesssim 1$$ holds as long as $\|f\|_{L^2(\mathbb{R}^d)}=1$ where the notation $\lesssim$ denotes the inequality with some implicit constant, for example, $A\lesssim B$ means an inequality $A\leq CB$ holds for some constant $C>0$. Such inequality is first observed by Strichartz in [@Strichartz] and later extended to mixed norm setting and applied for nonlinear Schrödinger equations, for example [@GinibreVelo; @GuoPeng; @KeelTao; @Tutumi; @Yajima]. To explain the problem we address in, let us overview two topics concerning the classical Strichartz inequality, the first one is the generalization of the Strichartz inequality involving the orthonormal system and the second one is the theory for the nonlinear periodic Schrödinger equation, especially the Strichartz inequality on torus. Orthonormal Strichartz inequality on $\mathbb{R}^d$ --------------------------------------------------- Recently, the classical Strichartz inequality has been generalized to the orthonormal setting by Frank-Lewin-Lieb-Seiringer [@FLLS] and Frank-Sabin [@frank-sabin-1]. Let us recall what the orthonormal Strichartz inequality is and their results. For the admissible pair $p,q$ and suitable $\alpha\in[1,\infty]$, we consider the inequality $$\label{e:ONS-Rd} \bigg\| \sum_j\lambda_j |e^{it\Delta}f_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{R}^{d+1})} \lesssim \|\lambda\|_{\ell^\alpha}$$ for all $\lambda=(\lambda_j)_j\in\ell^\alpha$ and all orthonormal system $(f_j)_j$ in $L^2(\mathbb{R}^d)$. Clearly, the case $\alpha=1$ follows from the triangle inequality and the classical Strichartz inequality without any making use of the orthonormal hypothesis. So, in view of the inclusion relation of $\ell^\alpha$ space, the problem we are interested in is to find the largest $\alpha=\alpha(p,q)$ for which the inequality holds given the admissible pair $p,q$. It is convenient to introduce some notations to overview the known results, see Figure \[f:points\]: $$A= \big(\frac{d-1}{d+1},\frac{d}{d+1}\big),\quad B=(1,0),\quad C=\big(\frac{d-2}{d},1\big).$$ When $d=1$, $A=C=(0,\frac12)$. For two points $X,Y\in [0,1]^2$, we use a notation $(X,Y)$ to represent the open line combining $X,Y$. Similarly, we define $[X,Y]$, $(X,Y]$ and $[X,Y)$. (0,2.2) node (yaxis) \[left\] [$1/p$]{} |- (2.2,0) node (xaxis) \[below\] [$1/q$]{}; (0, 0) rectangle (2, 2); at (0,0) [$O$]{}; at (1.2,2) [$C$]{}; (4/3,5/3)–(2,0); \(C) at (1.2,2); (C) circle\[radius=0.2mm\]; \(B) at (2,0); (B) circle\[radius=0.2mm\]; (Frank-Sabin) at (4/3,5/3); (Frank-Sabin) circle\[radius=0.2mm\]; at (4/3,5/3) [$A$]{}; at (2,.1) [$B$]{}; (0,1)–(2,2); (1.2,2)–(1.2,0); at (1.2,0) [$\frac{d-2}{d}$]{}; at (0,2) [1]{}; at (0,1) [$\frac12$]{}; at (0,0) [0]{}; at (2,0) [1]{}; (0,1)–(1,1)–(2,1); (4/3,5/3)–(1.2,2); (4/3,0)–(4/3,5/3); at (4/3,0) [$\frac{d-1}{d+1}$]{}; (0,5/3)–(4/3,5/3); at (0,5/3) [$\frac{d}{d+1}$]{}; (0,0)–(2,0); \[t:ONS-Rd\] Let $d\geq1$. If $(\frac1q,\frac1p) \in (A,B]$, then holds for any $\lambda=(\lambda_j)_j\in\ell^{\alpha}$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{R}^d)$ whenever $\alpha\leq\frac{2q}{q+1}$. Moreover, this is sharp in the sense that the inequality fails if $\alpha>\frac{2q}{q+1}$. While this theorem gives the answer to the problem on $(A,B]$, namely $\alpha=\frac{2q}{q+1}$ is the best possible, this theorem does not cover all admissible exponents and the problem on $[A,C]$ is still open regardless of recent contributions [@BHLNS; @FLLS; @frank-sabin-2]. As far as we are aware, the following are the best known results on $[A,C]$. \[t:ONS-Rd-beyond\] Let $d\geq1$. 1. (Critical point) On the point $(\frac1q,\frac1p)=A$, the estimate with $\alpha=\frac{2q}{q+1}=p=\frac{d+1}{d}$ fails. 2. On the region $(\frac1q,\frac1p)\in (A,C)$, the estimate holds as long as $\alpha < p$ and this is sharp up to $\varepsilon$-loss in the sense that fails if $\alpha > p$. Moreover, the weak type estimate $$\bigg\| \sum_j \lambda_j |e^{it\Delta}f_j|^2 \bigg\|_{L^{p,\infty}_tL^q_x(\mathbb{R}^{d+1})} \lesssim \|\lambda\|_{\ell^p}$$ also holds true for any $\lambda=(\lambda_j)_j\in\ell^p$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{R}^d)$ where $L^{p,\infty}_t$ is the weak $L^p$-space. 3. (Keel-Tao endpoint) On the point $(\frac1q,\frac1p)=C$, the estimate holds with $\alpha=1$ and this is sharp in the sense that fails if $\alpha>1$. From this theorem, one may notice that the point $A$ plays a critical role in the sense that the sharp exponent is $\alpha = \frac{2q}{q+1}$ on the lower region and the expected sharp exponent is $\alpha=p$ on the upper region. Such generalization involving the orthonormal system is strongly motivated by the theory for the many body quantum mechanics and it is important to find the sharp sequence exponent $\alpha$ as in Theorem \[t:ONS-Rd\] in this context. The first initiative work of such generalization goes back to the famous work due to Lieb-Thirring [@Lieb-Thirring-1] where the Gagliardo-Nirenberg-Sobolev inequality was generalized to the orthonormal inequality, so-called Lieb-Thirring’s inequality. Importantly, the sharp orthonormal inequality played a crucial role to prove the stability of matter [@LiebBAMS; @Lieb-Thirring-1], see also [@sabin-2]. It is also notable that the sharp orthonormal Strichartz inequality as in Theorem \[t:ONS-Rd\] was employed crucially to establish well-posedness and scattering theory for the certain Hartree equation in [@CHP-1; @CHP-2; @LewinSabinWP; @LewinSabinScatt; @sabin]. One functional Strichartz inequality on torus --------------------------------------------- There is another theory regarding the classical Strichartz inequality, namely the nonlinear periodic PDE problem. In [@Bourgain-restrictiontori] Bourgain studied the nonlinear periodic Schrödinger equation on torus $\mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d$ and established the well-posedness theory. One crucial feature of the equation on $\mathbb{T}^d$ is that the dispersion of the solution is weaker than the solution of the equation on $\mathbb{R}^d$ since $\mathbb{T}^d$ is compact and hence, new difficulty occurs to established the well-posedness theory. A decisive tool to study the nonlinear periodic Schrödinger equation is the Strichartz inequality on torus which can be stated as follows: Let $d\geq1$ and $p_* = \frac{d+2}{d}$. Then for arbitrary small $\varepsilon>0$, there exists $C_\varepsilon>0$ such that for any $N>1$ and any $f\in L^2(\mathbb{T}^d)$ whose Fourier support is contained in $[-N,N]^d$, $$\label{e:strichartz-tori} \big\| |e^{it\Delta}f|^2 \big\|_{L^{p_*}_{x,t}(\mathbb{T}^{d+1})} \leq C_\varepsilon N^\varepsilon \|f\|_{L^2(\mathbb{T}^d)}$$ holds. Remark that the $N^\varepsilon$-loss in is not removable. Historically, in [@Bourgain-restrictiontori], Bourgain proved when $d=1,2$ via number theoretical argument so-called Hardy-Littlewood circle method and conjectured that holds for any $d\geq3$. After some improvements were obtained in [@BourgainDemeter-tori-1; @BourgainDemeter-tori-2], this conjecture was finally solved positively by the celebrated work due to Bourgain-Demeter [@BourgainDemeter-decoupling] where they employed deep theory from Harmonic analysis so-called decoupling theorem. Moreover, it was also observed that the inequality still holds replacing the torus by more general irrational torus. For further discussion and the theory on the irrational torus including survey, see [@CatoireWang; @Demirbas; @GuoOhWang; @KenigPonceVega; @Nahmod; @Staffilani; @Vega]. It is notable that in [@BurqGerardTzvetkov], Burq-Gérard-Tzvetkov studied the nonlinear Schrödinger equation on the compact manifold. In this paper, we employ their idea used to establish the Strichartz inequality on the compact manifold. Further improvement were obtained in their continued works [@BurqGerardTzvetkov-2; @BurqGerardTzvetkov-3] where they employed bilinear and multilinear approach. For the study of the Hartree equation on compact manifold, see the work of Gérard-Pierfelice [@GerardPierfelice]. Main results ------------ With these two topics concerning the classical Strichartz inequality in mind, it is natural to investigate the nonlinear periodic equation in the framework of orthonormal systems. So, our main aim in this paper is to establish the sharp orthonormal Strichartz inequality on torus and apply it to the periodic Hartree equation for the density matrices of infinite trace. More precisely, our first main goal is to determine the largest $\alpha$ for which the inequality $$\label{e:ONS-Td} \bigg\| \sum_{j} \lambda_j |e^{it\Delta} P_{\leq N} f_j|^2 \bigg\|_{L_{t}^{p}L^q_x(\mathbb{T}^{d+1})} \leq C_\rho N^\rho \|\lambda\|_{\ell^\alpha}$$ holds for any $N>1$, any $\lambda=(\lambda_j)_j\in\ell^\alpha$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{T}^d)$, given a parameter $\rho>0$ and admissible pair $p,q$. Here, the operator $P_{\leq N}$ denotes the frequency cut-off operator which is defined by $P_{\leq N}\phi= ( \1_{[-N,N]^d} \hat{\phi} )^{\vee}$, where $(\hat{\phi}(n))_n$ is the Fourier coefficient of $\phi$ and ${}^\vee$ is its inverse. When $p=q=p_*$, again applying the triangle inequality and , we can prove for any small $\varepsilon$, $$\label{e:ONS-one} \bigg\| \sum_{j} \lambda_j |e^{it\Delta} P_{\leq N} f_j|^2 \bigg\|_{L_{x,t}^{p_*}(\mathbb{T}^{d+1})} \leq C_\varepsilon N^\varepsilon \|\lambda\|_{\ell^\1}.$$ Our first observation is that if we define $\alpha(\rho)$ for each $\rho>0$ by $$\label{e:alpha} \frac{1}{\alpha(\rho)} = 1 - \frac\rho d,$$ then $\alpha\leq \alpha(\rho)$ is necessary for the inequality , we will see this in Lemma \[l:necessary\] by testing the inequality with a simple example. So, in the orthonormal framework, the sharp exponent $\alpha$ for the inequality should be related to the power $\rho$ and more interestingly, we can easily see that $\alpha(\rho)\to1$ as $\rho\to0$. This reveals that the trivial estimate is almost sharp when $\varepsilon\to0$. In other words, to make $\alpha$ strictly bigger than one, we need to lose the factor $N$ with certain power. Our first result is the following. \[t:ONS-pure\] Let $d\geq1$ and $\rho\in(0,\frac{1}{p_*}]$. Then $$\label{e:pure-ONS} \bigg\| \sum_{j} \lambda_j |e^{it\Delta} P_{\leq N} f_j|^2 \bigg\|_{L_{t,x}^{p_*}(\mathbb{T}^{d+1})} \leq C_\rho N^\rho \|\lambda\|_{\ell^\alpha}$$ holds for any $N>1$, any $\lambda\in\ell^{\alpha}$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{T}^d)$ whenever $\alpha < \alpha(\rho)$. Moreover, this is sharp up to $\varepsilon$-loss in the sense that fails if $\alpha>\alpha(\rho)$. Remark that the possibility of with the expected exponent $\alpha=\alpha(\rho)$ remains open except the case $\rho=\frac1p$. Theorem \[t:ONS-pure\] is a consequence of the following more general mixed norm orthonormal Strichartz inequality via the complex interpolation with . Note that one can easily check that $\alpha(1/p)=\frac{2q}{q+1}$ holds if $\frac2p+\frac dq=d$. \[t:ONS-mix\] Let $d\geq1$ and $(\frac1q,\frac1p) \in (A,B]$. Then for any $N>1$, any $\lambda\in\ell^{\alpha}$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{T}^d)$, $$\label{e:ONS-mix} \bigg\| \sum_{j} \lambda_j |e^{it\Delta} P_{\leq N} f_j|^2 \bigg\|_{L_{t}^{p}L^q_x(\mathbb{T}^{d+1})} \leq C N^\frac1p \|\lambda\|_{\ell^{\alpha}}$$ holds true whenever $\alpha \leq \frac{2q}{q+1}$. Moreover, this is sharp in the sense that fails if $\alpha > \frac{2q}{q+1}$. Recall that the exponent $\alpha(1/p)=\frac{2q}{q+1}$ has already appeared in Theorem \[t:ONS-Rd\] as the sharp exponent for the orthonormal Strichartz inequality on $\mathbb{R}^d$. Furthermore, the range $(A,B]$ also corresponds to the range of Theorem \[t:ONS-Rd\]. So, we may find some connections between the orthonormal Strichartz inequality on $\mathbb{R}^d$ and the one on $\mathbb{T}^d$ with the case $\rho=\frac1p$. It is natural to ask further what happens in the region $[A,C]$. In view of the similarity between the $\mathbb{R}^d$ case and the $\mathbb{T}^d$ case with $\rho=\frac1p$ and Theorem \[t:ONS-Rd-beyond\], one may expect some different phenomena on $[A,C]$. Especially, recall that at the point $(\frac1q,\frac1p)=A$, the inequality on $\mathbb{R}^d$: $$\bigg\| \sum_{j} \lambda_j |e^{it\Delta} f_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{R}^{d+1})} \lesssim \|\lambda\|_{\ell^{\frac{2q}{q+1}}}$$ fails. In spite of such similarity and the failure, we interestingly have a positive result at the point $A$ for $\mathbb{T}^d$ case at least when $d=1$. Recall that when $d=1$, exponents are $A=C=(0,\frac12)$ and $\alpha(1/p)=\frac{2q}{q+1}=2$. \[t:endpoint-1d\] Let $(\frac1q,\frac1p)=A=(0,\frac12)$. Then for any $N>1$, $\lambda\in\ell^\alpha$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{T})$, $$\label{e:ONS-d1} \bigg\| \sum_j \lambda_j |e^{it\Delta}P_{\leq N}f_j|^2 \bigg\|_{L^2_tL^\infty_x(\mathbb{T}^{1+1})} \leq C N^{\frac12} \|\lambda\|_{\ell^{\alpha}}$$ holds true whenever $\alpha\leq 2$. Moreover, this is sharp in the sense that fails if $\alpha>2$. We emphasize that to prove the endpoint estimate Theorem \[t:endpoint-1d\] we follow the spirit of the Hardy-Littlewood circle method via Frank-Sabin’s $TT^*$ argument in Schatten space. This is possible since the right-hand side of becomes $\ell^2$ when $d=1$ and $(\frac1q,\frac1p) =A= (0,\frac12)$. We will make use of the speciality of $\ell^2$. The problem on the region $[A,C]$ for $d\geq2$ remains open although we will give one observation in Theorem \[t:ONS-beyond\]. There are some possibility to extend Theorem \[t:ONS-mix\] to more general compact manifold as Burq-Gérard-Tzvetkov did from view the point of our proof of Theorem \[t:ONS-mix\]. However, we will not go to such direction here. As an application of the above orthonormal Strichartz inequalities, we consider $M$ couple of nonlinear periodic Hartree equations which describes the dynamics of $M$ fermions interacting via a power type potential $w_a(x)=|x|^{-a}$ for certain $0<a<d$ $$\label{e:Hartree-0} \left\{ \begin{array}{ll} i\partial_tu_1&=(-\Delta+w_{a}\ast\rho)u_1, \quad u_1|_{t=0} = f_1 \\ &\ \vdots \\ i\partial_tu_M&=(-\Delta+w_{a}\ast\rho)u_M, \quad u_M|_{t=0} = f_M , \end{array} \right.$$ where $(x,t) \in \mathbb{T}^d\times\mathbb{R}$, $(f_j)_{j=1}^M$ is an orthonormal system in $L^2(\mathbb{T}^d)$ and $\rho$ is a density function defined by $\rho(x,t) = \sum_{j=1}^M |u_j(x,t)|^2 $. Remark that the solution $(u_j(t))_{j=1}^M$ continues to be an orthonormal system in $L^2(\mathbb{T}^d)$ for each $t>0$. Our main interest is the case $M\to\infty$ and hence, we naturally arrive at the operator valued equivalent formulation of as follows: $$\label{e:Hartree} \left\{ \begin{array}{ll} i\partial_t\gamma=[-\Delta+w_{a}\ast\rho_\gamma,\gamma], \quad (x,t)\in \mathbb{T}^d \times \mathbb{R} \\ \gamma|_{t=0}=\gamma_0. \end{array} \right.$$ Here $\gamma_0,\gamma=\gamma(t)$ are bounded and self-adjoint operators on $L^2(\mathbb{T}^d)$, $[A,B]$ is a commutator of two operators $A$ and $B$ and $\rho_\gamma:\mathbb{T}^d\to\mathbb{C}$ is given by $\rho_\gamma(x) = \gamma(x,x)$ where $\gamma(\cdot,\cdot)$ denotes the integral kernel of the operator $\gamma$. There are several context for this equation on $\mathbb{R}^d$ when $\gamma_0$ is in the trace class [@BovePratoFano1; @BovePratoFano2; @Chadam] and more importantly Lewin-Sabin [@LewinSabinWP; @LewinSabinScatt] and Chen-Hong-Pavlović [@CHP-1; @CHP-2] study the equation when $\gamma$ is not in the trace class. We will obtain the $\mathbb{T}^d$ counterpart of the (local) well-posedness result due to Frank-Sabin [@frank-sabin-1 Theorem 14]. To state our result concerning to the equation , let us introduce more notions. For $\alpha\in[1,\infty)$, $\mathcal{C}^\alpha=\mathcal{C}^\alpha(L^2(\mathbb{T}^d))$ denotes the Schatten space based on $L^2(\mathbb{T}^d)$ which is the space of all compact operators $A$ on $L^2(\mathbb{T}^d)$ such that ${\rm Tr}|A|^\alpha<\infty$, where $|A|=\sqrt{A^*A}$, and its norm is defined by $\|A\|_{\mathcal{C}^\alpha}=({\rm Tr}|A|^\alpha)^\frac1\alpha$. If $\alpha=\infty$, we define $\|A\|_{\mathcal{C}^\infty} = \|A\|_{L^2\to L^2}$. Also, we use the Sobolev type Schatten space $\mathcal{C}^{\alpha,s}=\mathcal{C}^{\alpha,s}(L^2(\mathbb{T}^d))$, $s\in\mathbb{R}$, introduced in [@CHP-1; @CHP-2] whose norm is defined by $$\|\gamma\|_{\mathcal{C}^{\alpha,s}(L^2(\mathbb{T}^d))} = \| \langle D\rangle^s \gamma \langle D \rangle^s \|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^d))},$$ where $\langle D\rangle^s$ is the inhomogeneous derivative, $\langle D\rangle^s\phi = ((1+|n|^2)^\frac s2 \hat{\phi})^\vee$. \[t:wellposed\] Let $d\geq1$. Suppose $(\frac1q,\frac1p)\in (A,B)$, $\frac{1}{2p}<s$ and $0<a<\frac{3}{2p}$. 1. (Local well-posedness) For any $\gamma_0\in\mathcal{C}^{\frac{2q}{q+1},s}(L^2(\mathbb{T}^d))$, there exist   $T=T( \|\gamma_0\|_{\mathcal{C}^{\frac{2q}{q+1},s}(L^2(\mathbb{T}^d))} ,s,a) > 0$ and $\gamma \in C^0_t([0,T]; \mathcal{C}^{\frac{2q}{q+1},s}(L^2(\mathbb{T}^d)))$ satisfying on $[0,T]\times \mathbb{T}^d$ and $\rho_\gamma\in L^p_tL^q_x([0,T]\times\mathbb{T}^d)$. 2. (Almost global well-posedness) For each $T>0$, we have small $R_T=R_T(a,s)>0$ such that if $\|\gamma_0\|_{\mathcal{C}^{\frac{2q}{q+1},s}(L^2(\mathbb{T}^d))} \leq R_T$, then there exists a solution $\gamma \in C^0_t([0,T]; \mathcal{C}^{\frac{2q}{q+1},s}(L^2(\mathbb{T}^d)))$ satisfying on $[0,T]\times \mathbb{T}^d$ and $\rho_\gamma\in L^p_tL^q_x([0,T]\times\mathbb{T}^d)$. Note that if $d=3$ and $(\frac1q,\frac1p)\in(A,B)$ is sufficiently close to $A$, we may choose $a=1$ which is the most meaningful case from view point of physical motivation in Theorem \[t:wellposed\]. In fact, the condition $\frac32 \cdot \frac{d}{d+1} = \frac32\cdot\frac{3}{3+1}>1$ holds and hence $\frac{3}{2p}>1$ holds if $\frac1p$ is sufficiently close to $\frac{d}{d+1} = \frac{3}{3+1}$ which means $(\frac1q,\frac1p)$ is sufficiently close to $A$, recall $A=(\frac{d-1}{d+1},\frac{d}{d+1})$. So, this exhibits one importance of extending the orthonormal Strichartz inequality up to near the point $A$. To have more range of $a$, we need to establish the orthonormal Strichartz inequality on the beyond region $[A,C]$ as in Theorems \[t:endpoint-1d\] and \[t:ONS-beyond\]. Also, in such case, namely $(\frac1q,\frac1p)$ close to $A$, the gain of the Schatten exponent $\alpha=\frac{2q}{q+1}$ is close to $\frac{d+1}{d}$ which is the largest number among $\{\frac{2q}{q+1}:(\frac1q,\frac1p)\in[A,B]\}$. This paper is organized as follows. In Section 2, we give a few definitions and recall the duality principle. In Section 3, we prove orthonormal Strichartz inequality Theorems \[t:ONS-pure\], \[t:ONS-mix\] and \[t:endpoint-1d\]. In Section 4, we prove the well-posedness result, Theorem \[t:wellposed\]. In Section 5, we give one observation concerning to the orthonormal Strichartz inequality on the beyond region $[A,C]$ where we will show the almost sharp inequality at $A$ even when $d\geq2$. Preliminaries ============= In this section, we provide further definitions and recall the duality principle due to Frank-Sabin [@frank-sabin-1]. For $s\in\mathbb{R}$ and $p\in[1,\infty]$, we use $B^s_{p,\infty}=B^s_{p,\infty}(\mathbb{T}^d)$ to denote the Besov space on $\mathbb{T}^d$ whose norm is defined by $$\|f\|_{B^s_{p,\infty}(\mathbb{T}^d)} = \sup_{k\in \mathbb{N}\cup\{0\}} 2^{ks} \|P_k f\|_{L^p(\mathbb{T}^d)}.$$ Here, $P_k$ is the frequency cutoff operator, $P_k\phi(x)=(\varphi_k\hat{\phi})^\vee$ for $k\in \mathbb{N}\cup\{0\}$ where $\{\varphi_k\}_{k=0}^\infty$ is the partition of unity, namely $\varphi_k$ is a smooth function whose support is contained in $\{|\xi|\sim 2^k\}$ when $k\geq1$ and $\varphi_0$ is a smooth function whose support is contained in $\{|\xi|\leq2\}$ such that $\sum_{k=0}^\infty \varphi_k = 1$. See [@Triebel] for the details of this function space. It is notable that for $a\in(0,d)$, $w_a(x)=|x|^{-a} \in B^s_{p,\infty}(\mathbb{T}^d)$ if and only if $a \leq \frac dp - s$ holds. We will use this to show Theorem \[t:wellposed\] in Section 4. In the sequel, we sometimes abbreviate $\mathbb{T}^d$ and use $L^2$ instead of $L^2(\mathbb{T}^d)$ for example. It is reasonable to reformulate the inequality in terms of the Fourier extension operator. Let us introduce the notation $S_{d,N}=\mathbb{Z}^d\cap [-N,N]^d$ and define the Fourier extension operator $\mathcal{E}_N$ by $$\mathcal{E}_N a(x,t)=\sum_{n\in S_{d,N}}a_ne^{2\pi i(x\cdot n+t|n|^2)},\quad (x,t)\in\mathbb{T}^{d+1},$$ for $a=(a_n)_n\in\ell^2$. Then its dual operator $\mathcal{E}_N^*$ (Fourier restriction operator) is given by $$\mathcal{E}_N^*F(n)= \int_{\mathbb{T}^{d+1}}F(x,t)e^{-2\pi i(x\cdot n+t|n|^2)}\, \mathrm{d}x\mathrm{d}t $$ if $n\in S_{d,N}$ and $\mathcal{E}_N^*F(n)=0$ if $n\notin S_{d,N}$. Here, the dual operator of $\mathcal{E}_N$ means that for any $a\in\ell^2$ and any $F\in L^2(\mathbb{T}^{d+1})$, $$\langle \mathcal{E}_Na,F\rangle_{L^2_{x,t}(\mathbb{T}^{d+1})}=\langle a,\mathcal{E}_N^*F\rangle_{\ell^2_n}$$ holds. Also, it is notable that from few calculations the operator $\mathcal{E}_N\mathcal{E}_N^*$ is given by $$\begin{aligned} \mathcal{E}_N\mathcal{E}_N^*F(x,t) &= \int_{\mathbb{T}}e^{i(t-t')\Delta}[F(\cdot,t')](x)\, \mathrm{d}t'\\ &= \int_{\mathbb{T}}\sum_{n\in S_{d,N}}\widehat{F(\cdot,t')}(n)e^{2\pi i(x\cdot n+(t-t')|n|^2)}\, \mathrm{d}t',\end{aligned}$$ and hence if we write $$K_N(x,t)=\sum_{n\in S_{d,N}}e^{2\pi i(x\cdot n+t|n|^2)},$$ then we have $$\label{e:convolution} \mathcal{E}_N\mathcal{E}_N^*F(x,t) =K_N\ast F(x,t)= \int_{\mathbb{T}^{d+1}}K_N(x-x',t-t')F(x',t')\, \mathrm{d}x'\mathrm{d}t'.$$ Using these notations, the inequality can be reformulated as follows. The inequality holds for any $N>1$, any $\lambda\in\ell^\alpha$ and any orthonormal system $(f_j)_j$ in $L^2(\mathbb{T}^d)$ if and only if $$\label{e:ONS-extension} \bigg\| \sum_j\lambda_j|\mathcal{E}_N a_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_{\rho} N^\rho \|\lambda\|_{\ell^\alpha}$$ holds for any $N>1$, $\lambda\in \ell^\alpha$ and any orthonormal system $(a_j)_j$ in $\ell^2$. This is because if we let $a_j = \hat{f_j}$, then the orthonormality of $(f_j)_j$ in $L^2(\mathbb{T}^d)$ is equivalent to the one of $(a_j)_j$ in $\ell^2$ and $e^{it\Delta}f_j = \mathcal{E}_N a_j$. From now on, we will mainly consider the inequality of the form . All our results concerning to the orthonormal inequality would be shown in terms of the Schatten spaces. In fact, thanks to the duality principle due to Frank-Sabin [@frank-sabin-1], the orthonormal inequality we will prove can be rephrased as follows. \[l:duality\] The inequality is equivalent to $$\label{e:FS-dual} \big\| W_1\mathcal{E}_N\mathcal{E}_N^* W_2 \big\|_{\mathcal{C}^{\alpha'}(L^2(\mathbb{T}^{d+1}))} \leq C_\rho N^\rho \|W_1\|_{L^{2p'}_tL^{2q'}_x(\mathbb{T}^{d+1})} \|W_2\|_{L^{2p'}_tL^{2q'}_x(\mathbb{T}^{d+1})}$$ for all $W_1,W_2 \in L^{2p'}_tL^{2q'}_x(\mathbb{T}^{d+1})$. Proof of Theorems \[t:ONS-pure\], \[t:ONS-mix\] and \[t:endpoint-1d\] ===================================================================== The necessity of $\alpha\leq \alpha(\rho)$ ------------------------------------------ First, we prove the necessity $\alpha\leq\alpha(\rho)$ for the inequality by testing a simple example. \[l:necessary\] Let $d\geq1$ and $p,q,\alpha\in[1,\infty]$ be arbitrary. Suppose or equivalently with some $\rho>0$ holds for any $N>1$, any $\lambda\in\ell^\alpha$ and any orthonormal system $(a_j)_j$ in $\ell^2$. Then it must be $\alpha\leq\alpha(\rho)$. Let $a_j=\1_{\{j\}}$ for each $j\in\mathbb{Z}^d$ and $\lambda_j=\1_{S_{d,N}}(j)$. Notice that if $j\in S_{d,N}$, then $$| \mathcal{E}_N a_j(x) | = \Big| \sum_{n\in S_{d,N}} e^{2\pi i(x\cdot n + t|n|^2)} \1_{\{j\}}(n) \Big| = 1,$$ which implies $$\bigg\| \sum_j \lambda_j |\mathcal{E}_N a_j|^2 \bigg\|_{ L^p_tL^q_x(\mathbb{T}^{d+1}) } = \sharp S_{d,N} \sim N^d.$$ On the other hand, the right-hand side of is $$N^\rho \|\lambda\|_{\ell^\alpha} = N^\rho ( \sharp S_{d,N} )^\frac1\alpha \sim N^\rho N^\frac d\alpha.$$ So, applying reveals $ N^d \lesssim N^\rho N^\frac d\alpha$, which gives $d \leq \rho + \frac d\alpha$ as $N\to\infty$. As we mentioned in Section 1, $\alpha(\rho) = \frac{2q}{q+1}$ when $\rho=\frac1p$ and $\frac2p+\frac dq=d$. Hence, Lemma \[l:necessary\] shows the sharpness part of Theorems \[t:ONS-mix\] and \[t:endpoint-1d\]. Proof of Theorems \[t:ONS-pure\] and \[t:ONS-mix\] -------------------------------------------------- Let us prove Theorem \[t:ONS-mix\]. Once we prove Theorem \[t:ONS-mix\], then Theorem \[t:ONS-pure\] follows from the complex interpolation between Theorem \[t:ONS-mix\] and . In this subsection, we use the notation $I_N=[-\frac{1}{2N},\frac{1}{2N}]$. The key point is the following dispersive estimate observed in Kenig-Ponce-Vega [@KenigPonceVega]. \[l:dispersive\] It holds that $$\bigg|\sum_{n=-N}^Ne^{2\pi i(xn+t|n|^2)}\bigg|\leq C|t|^{-\frac12}$$ for any $(x,t)\in \mathbb{T} \times [-N^{-1},N^{-1}]$. From Lemma \[l:dispersive\], we clearly have $$\label{e:dispersive} \bigg|\sum_{n\in S_{d,N}}e^{2\pi i(x\cdot n+t|n|^2)}\bigg|\leq C|t|^{-\frac d2}$$ for any $(x,t)\in\mathbb{T}^d\times [-N^{-1},N^{-1}]$. Using this with Stein’s analytic interpolation, we prove the following proposition. See Vega [@Vega] for the one functional counterpart. \[p:ONS-Td\] Let $d\geq1$ and suppose $(\frac1q,\frac1p) \in (A,B]$. Then for any $N>1$, any $\lambda\in\ell^\frac{2q}{q+1}$ and any orthonormal system $(a_j)_j$ in $\ell^2$, $$\label{e:ShortTime} \bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^d\times I_N)} \leq C\|\lambda\|_{\ell^\frac{2q}{q+1}}.$$ Thanks to the duality principle, Lemma \[l:duality\], to prove the desired estimate for all $(\frac1q,\frac1p)\in(A,B]$, it suffices to to show $$\label{e:dual-ShortTime} \big\| W_1 \1_{I_N} \mathcal{E}_N \mathcal{E}_N^*[ \1_{I_N} W_2 ] \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))} \lesssim \|W_1\|_{L^\beta_tL^\alpha_x(\mathbb{T}^{d+1})} \|W_2\|_{L^\beta_tL^\alpha_x(\mathbb{T}^{d+1})}$$ for all $\alpha,\beta\geq1$ such that $\frac2\beta + \frac d\alpha = 1$ and $0 \leq \frac1\alpha < \frac{1}{d+1}$. Moreover, it is enough to show on $\frac{1}{d+2}\leq \frac1\alpha <\frac{1}{d+1}$ since we trivially have when $\alpha=\infty$ from the Plancherel theorem. Define for $\varepsilon>0$, $T_{N,\varepsilon}=K_{N,\varepsilon}\ast$ where $K_{N,\varepsilon}(x,t)=\1_{\varepsilon<|t|<N^{-1}}K_N(x,t)$. Once we have $$\label{0927-1} \big\|W_1 \1_{I_N} T_{N,\varepsilon}[ \1_{I_N}W_2] \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))} \leq C \|W_1\|_{L^\beta_tL^\alpha_x(\mathbb{T}^{d+1})}\|W_2\|_{L^\beta_tL^\alpha_x(\mathbb{T}^{d+1})}$$ for some $C$ independent of $\varepsilon$, then follows by taking $\varepsilon\to0$. To do Stein’s analytic complex interpolation, we further define for $z\in\mathbb{C}$ with ${\rm Re}z\in[-1,\frac{d}{2}]$, $$K_{N,\varepsilon}^z(x,t) =t^zK_{N,\varepsilon}(x,t)$$ and $T_{N,\varepsilon}^z = K^z_{N,\varepsilon}\ast$. From , we have for $(x,t)\in \mathbb{T}^d\times I_N$ $$|K_{N,\varepsilon}^z(x,t)| \leq C |t|^{{\rm Re}z-\frac d2}.$$ This involving the Hardy-Littlewood-Sobolev inequality reveals that $$\begin{aligned} &\big\| W_1 \1_{I_N} T_{N,\varepsilon}^z[ \1_{I_N} W_2] \big\|_{\mathcal{C}^2(L^2(\mathbb{T}^{d+1}))}^2\\ =& \int_{(x,t)\in \mathbb{T}^d \times I_N} \int_{(x',t')\in \mathbb{T}^d \times I_N} |W_1(x,t)K_{N,\varepsilon}^z(x-x',t-t')W_2(x',t')|^2\, \mathrm{d}x\mathrm{d}t\mathrm{d}x'\mathrm{d}t'\\ \leq& C \big\| \|W_1\|_{L^2_x(\mathbb{T}^d)}^2\big\|_{L^{\tilde{u}}_t(\mathbb{T})} \big\| \|W_2\|_{L^2_x(\mathbb{T}^d)}^2\big\|_{L^{\tilde{u}}_t(\mathbb{T})},\end{aligned}$$ where $2{\rm Re}z-d\in(-1,0]$ and $\frac{2}{\tilde{u}}+( d-2{\rm Re}z)=2$. If we write $2\tilde{u}=u$, then $\frac1u\in(\frac14,\frac12]$ and we have $$\big\| W_1 \1_{I_N} T_{N,\varepsilon}^z[ \1_{I_N} W_2] \big\|_{\mathcal{C}^2(L^2(\mathbb{T}^{d+1}))} \leq C \|W_1\|_{L^u_tL^2_x(\mathbb{T}^{d+1})} \|W_2\|_{L^u_tL^2_x(\mathbb{T}^{d+1})},$$ provided $\frac1u= \frac12+\frac12({\rm Re}z-\frac d2),{\rm Re}z\in(\frac{d-1}{2},\frac d2]$. On the other hand, we claim that for ${\rm Re}z=-1$, $T^{z}_{N,\varepsilon}:L^2_{x,t}(\mathbb{T}^d \times I_N)\to L^2_{x,t}(\mathbb{T}^d \times I_N)$ holds with some constant depending only on $d$ and ${\rm Im}z$ exponentially. In fact, from Plancherel’s theorem, we have for each $t\in\mathbb{T}$, $$\begin{aligned} \big\| T^z_{N,\varepsilon} F(\cdot,t) \big\|_{L^2_x}^2 &= \sum_{m\in S_{d,N}} \bigg| \int_{\varepsilon<|t'|<N^{-1}} t'^{-1+i{\rm Im}z} e^{-2\pi i (t-t') |m|^2} \mathcal{F}_x [ F(\cdot,t-t') ](m)\, \mathrm{d}t' \bigg|^2 \\ &= \sum_{m\in S_{d,N}} \bigg| \int_{\varepsilon<|t'|<N^{-1}} t'^{-1+i{\rm Im}z} G_m(t-t')\, \mathrm{d}t' \bigg|^2, \end{aligned}$$ where $G_m(s) = e^{-2\pi i s |m|^2} \mathcal{F}_x [ F(\cdot,s) ](m)$. So, if we further define $H_{N,\varepsilon}^z: G(t) \mapsto \int_{\varepsilon<|t'|<N^{-1}} t'^{-1+i{\rm Im}z} G(t-t')\, \mathrm{d}t'$, then $$\big\| T^z_{N,\varepsilon} F \big\|_{L^2_{x,t}}^2 = \sum_{m\in S_{d,N}} \| H^z_{N,\varepsilon} G_m \|_{L^2_t}^2.$$ Therefore, once we have the bound $H^z_{N,\varepsilon}: L^2 \to L^2$ with some constant depending only on ${\rm Im}z$ exponentially, then we obtain the desired bound $T^{z}_{N,\varepsilon}:L^2_{x,t}(\mathbb{T}^d \times I_N)\to L^2_{x,t}(\mathbb{T}^d \times I_N)$. Indeed, the bound $H^z_{N,\varepsilon}: L^2 \to L^2$ holds true since the operator $H^z_{N,\varepsilon}$ is just Hilbert transform up to $i{\rm Im}z$. For further detail, see Vega [@Vega]. Hence, using $T^{z}_{N,\varepsilon}:L^2_{x,t}(\mathbb{T}^d \times I_N)\to L^2_{x,t}(\mathbb{T}^d \times I_N)$, we obtain for ${\rm Re}z=-1$, $$\big\| W_1 \1_{I_N} T_{N,\varepsilon}^z[ \1_{I_N} W_2] \big\|_{\mathcal{C}^\infty(L^2(\mathbb{T}^{d+1}))} \leq C({\rm Im}z) \|W_1\|_{L^\infty_tL^\infty_x(\mathbb{T}^{d+1})} \|W_2\|_{L^\infty_tL^\infty_x(\mathbb{T}^{d+1})}.$$ Applying Stein’s analytic interpolation, holds as long as $$\frac 2\beta+\frac d\alpha=1,\quad \frac{1}{d+2}\leq \frac{1}{\alpha}<\frac{1}{d+1}.$$ Once we have , then the same inequality replacing $I_N$ by an arbitrary interval $I$ whose length is $N^{-1}$ holds true: $$\label{e:anyinterval} \bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^d\times I)} \leq C\|\lambda\|_{\ell^\frac{2q}{q+1}}$$ where the constant $C$ is independent of $I$. In fact, if we denote the center of the interval $I$ by $c(I)$, then changing variables give $$\bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^d\times I)} = \bigg\| \sum_j \lambda_j |\mathcal{E}_Nb_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^d\times I_N)}$$ where $b_j(n) = a_j(n)e^{-2\pi i c(I)|n|^2}$. Since $(b_j)_j$ is orthonormal in $\ell^2$ if $(a_j)_j$ is orthonormal, reveals the desired inequality. From this observation, we may prove Theorem \[t:ONS-mix\]. We have a covering $\mathbb{T}=\bigcup_{i=1}^NI_i$ where $\{I_i\}_{i=1}^N$ is the collection of disjoint intervals whose length is $N^{-1}$ and decompose $$\bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})}^p = \sum_{i=1}^N \bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^d\times I_i)}^p.$$ Applying , we obtain . Proof of Theorem \[t:endpoint-1d\] ---------------------------------- One notices that $\frac{2q}{q+1}=2$ holds if $(\frac1q,\frac1p)=(0,\frac12)$ and this is a key point for the proof of Theorem \[t:endpoint-1d\]. So, the desired inequality is equivalent to $$\label{e:critical-1d} \bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2\bigg\|_{L^2_tL^\infty_x(\mathbb{T}^{2})} \lesssim N^\frac12 \|\lambda\|_{\ell^2}.$$ From Lemma \[l:duality\], is equivalent to $$\label{e:goal-1d} \| W_1 \mathcal{E}_N\mathcal{E}_N^* W_2 \|_{\mathcal{C}^2(L^2(\mathbb{T}^{2}))} \lesssim N^\frac12 \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})} \|W_2\|_{L^4_tL^2_x(\mathbb{T}^{2})}.$$ Recalling , we see that the left-hand side of turns into $$\begin{aligned} &\big\| W_1 \mathcal{E}_N\mathcal{E}_N^* W_2 \big\|_{\mathcal{C}^2(L^2(\mathbb{T}^{2}))}^2\\ = & \int_{\mathbb{T}^{2}} \int_{\mathbb{T}^{2}} |W_1(x,t) K_N(x-x',t-t') W_2(x',t')|^2\, \mathrm{d}t\mathrm{d}x\mathrm{d}t'\mathrm{d}x'.\end{aligned}$$ Now, we expand $|K_N(x-x',t-t')|^2$ as follows. $$|K_N(x-x',t-t')|^2 = \sum_{n_1,n_2=-N}^N e^{2\pi i [ (x-x')(n_1-n_2) + (t-t')(|n_1|^2 - |n_2|^2) ]}.$$ If we write $|W_i|^2 = \psi_i$, then $$\begin{aligned} &\| W_1 \mathcal{E}_N\mathcal{E}_N^* W_2 \|_{\mathcal{C}^2(L^2(\mathbb{T}^{2}))}^2 = {\rm I} + {\rm II}, \end{aligned}$$ where I is the case when $n_1=n_2$: $${\rm I} = \sum_{n=-N}^N \int_{\mathbb{T}^{2}} \int_{\mathbb{T}^{2}} \psi_1(x,t) \psi_2(x',t')\, \mathrm{d}t\mathrm{d}x\mathrm{d}t'\mathrm{d}x',$$ and II is the case when $n_1\neq n_2$: $${\rm II} = \sum_{n_1\neq n_2} \int_{\mathbb{T}^{2}} \int_{\mathbb{T}^{2}} \psi_1(x,t) e^{2\pi i [ (x-x')(n_1-n_2) + (t-t')(|n_1|^2 - |n_2|^2) ]} \psi_2(x',t')\, \mathrm{d}t\mathrm{d}x\mathrm{d}t'\mathrm{d}x'.$$ We first handle II. Rewrite $$\begin{aligned} &\sum_{n_1\neq n_2} e^{2\pi i [ (x-x')(n_1-n_2) + (t-t')(|n_1|^2 - |n_2|^2) ]} \\ =& \sum_{\substack{m_1=-2N,\\ m_1\neq 0}}^{2N} \sum_{m_2=-N^2}^{N^2} e^{2\pi i [ (x-x')m_1 + (t-t')m_2 ]} \sum_{\substack{n_1\neq n_2: \\ n_1-n_2=m_1, |n_1|^2-|n_2|^2=m_2}} 1\\ =& \sum_{\substack{m_1=-2N,\\ m_1\neq 0}}^{2N} \sum_{m_2=-N^2}^{N^2} e^{2\pi i [ (x-x')m_1 + (t-t')m_2 ]} \1_{S_{2,N}}\Big(2^{-1}(m_1+\frac{m_2}{m_1}), 2^{-1}(-m_1+\frac{m_2}{m_1}) \Big),\end{aligned}$$ since the number of $(n_1,n_2)$ satisfying the condition $n_1\neq n_2$, $n_1-n_2=m_1$ and $|n_1|^2-|n_2|^2=m_2$ for fixed $m_1\neq0,m_2$ is at most one. For the sake of simplicity, we write $m_2 \in M_{N}(m_1)$ if $m_2 \in [-N^2,N^2]$ and $2^{-1}(m_1+\frac{m_2}{m_1}),2^{-1}(-m_1+\frac{m_2}{m_1})\in \mathbb{Z}\cap [-N,N]$. From this observation, $$\begin{aligned} {\rm II} &= \sum_{\substack{m_1=-2N,\\ m_1\neq 0}}^{2N} \sum_{m_2\in M_N(m_1)} \int_{\mathbb{T}^{2}} \int_{\mathbb{T}^{2}} \psi_1(x,t) e^{2\pi i [ (x-x')m_1 + (t-t')m_2 ]} \psi_2(x',t')\, \mathrm{d}t\mathrm{d}x\mathrm{d}t'\mathrm{d}x'\\ &= \sum_{\substack{m_1=-2N,\\ m_1\neq 0}}^{2N} \sum_{m_2\in M_N(m_1)} \overline{\widehat{\psi_1}(m_1,m_2)} \cdot \widehat{\psi_2}(m_1,m_2) \\ &\leq \sum_{\substack{m_1=-2N,\\ m_1\neq 0}}^{2N} \Big( \sum_{m_2\in \mathbb{Z}} |\widehat{\psi_1}(m_1,m_2)|^2 \Big)^\frac12 \Big(\sum_{m_2\in \mathbb{Z}} |\widehat{\psi_2}(m_1,m_2)|^2 \Big)^\frac12. \end{aligned}$$ If we use the notation $\mathcal{F}_x \psi_1 (m_1,t) = \int_{\mathbb{T}} e^{-2\pi i xm_1} \psi_1(x,t)\, \mathrm{d}x$, then we clearly have $\widehat{\psi_1}(m_1,m_2) = \mathcal{F}_t[\mathcal{F}_x\psi_1 (m_1, \cdot)](m_2) $. Applying the Plancherel and the Hausdorff-Young which states that $\mathcal{F}_x: L^1(\mathbb{T}) \to \ell^\infty$, $$\begin{aligned} \Big( \sum_{m_2\in \mathbb{Z}} |\widehat{\psi_1}(m_1,m_2)|^2 \Big)^\frac12 &= \Big( \int_{\mathbb{T}} |\mathcal{F}_x\psi_1 (m_1, t)|^2\, \mathrm{d}t \Big)^\frac12 \\ &\leq \bigg( \int_{\mathbb{T}} \Big(\int_{\mathbb{T}}|\psi_1 (x, t)|\, \mathrm{d}x\Big)^2\, \mathrm{d}t \bigg)^\frac12. \end{aligned}$$ Putting together with $\psi_i = |W_i|^2$, we see $${\rm II} \leq 4N \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})}^2 \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})}^2.$$ On the other hand, for $I$, we easily have from Hölder, $$I \leq 2N \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})}^2 \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})}^2.$$ In total, $$\big\| W_1 \mathcal{E}_N\mathcal{E}_N^* W_2 \big\|_{\mathcal{C}^2(L^2(\mathbb{T}^{2}))}^2 \leq 6N \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})}^2 \|W_1\|_{L^4_tL^2_x(\mathbb{T}^{2})}^2,$$ which implies . The well-posedness of the Hartree equation =========================================== In this section, we prove Theorem \[t:wellposed\] applying our orthonormal Strichartz inequalities. We obtained the orthonormal inequality in the form of in the previous sections. By the same proof, it is also possible to replace $P_{\leq N}$ by $P_k$ for any $k\in\mathbb{N}\cup\{0\}$. For example, Theorem \[t:ONS-pure\] can be rephrased by for any $k\in\mathbb{N}\cup\{0\}$, $$\bigg\| \sum_{j}\lambda_j |e^{it\Delta}P_{k} f_j|^2 \bigg\|_{L^{p}_{t}L^q_x(\mathbb{T}^{d+1})} \leq C_\rho 2^{k\rho} \|\lambda\|_{\ell^\alpha}.$$ Keeping this in mind, we give a more general result which can be derived by assuming $$\label{e:generalONS} \bigg\| \sum_{j}\lambda_j |e^{it\Delta}P_{k} f_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_\rho 2^{k\rho} \|\lambda\|_{\ell^\alpha},\quad (k\in\mathbb{N}\cup\{0\}).$$ \[p:generalwellposed\] Suppose for some $p,q,\alpha\in[1,\infty]$ and some $\rho>0$. Let $s>\frac\rho2$ and $w\in B^s_{q',\infty}$. 1. For any $\gamma_0\in\mathcal{C}^{\alpha,s}(L^2)$ with $R:=\|\gamma_0\|_{\mathcal{C}^{\alpha,s}(L^2)}<\infty$, there exists $T=T(R, \|w\|_{B^s_{q',\infty}} ) > 0$ and $\gamma\in C^0_t([0,T];\mathcal{C}^{\alpha,s}(L^2))$ satisfying on $[0,T]\times\mathbb{T}^d$ and $\rho_\gamma\in L^p_tL^q_x([0,T]\times\mathbb{T}^d)$. 2. For each $T>0$, we have $R_T=R_T(\| w \|_{B^s_{q',\infty}} )$ such that if $\|\gamma_0\|_{\mathcal{C}^{\alpha,s}(L^2)} \leq R_T$, then there exists a solution $\gamma \in C^0_t([0,T]; \mathcal{C}^{\alpha,s}(L^2))$ satisfying on $[0,T]\times \mathbb{T}^d$ and $\rho_\gamma\in L^p_tL^q_x([0,T]\times\mathbb{T}^d)$. Once we have Proposition \[p:generalwellposed\], then it suffices to combine this with Theorem \[t:ONS-mix\] to have Theorem \[t:wellposed\]. In fact, using Proposition \[p:generalwellposed\] with $(\frac1q,\frac1p)\in(A,B)$, $\rho=\frac1p$, $w=w_a$ and $\alpha=\frac{2q}{q+1}$, we obtain Theorem \[t:wellposed\] since the assumption of Proposition \[p:generalwellposed\] can be ensured by Theorem \[t:ONS-mix\] and $w_a\in B^s_{q',\infty}$ holds if $a\leq \frac{d}{q'}-s=\frac2p-s$. So, from now on, we prove Proposition \[p:generalwellposed\] following the argument due to Frank-Sabin [@frank-sabin-1 Theorem 14] with few twists. Our ingredient is the part of the control of the nonlinearity where we employ the estimate involving the Besov space $B^s_{q',\infty}$. As a direct corollary of , we have for any $\varepsilon>0$, any $\lambda\in\ell^\alpha$ and any orthonormal system $(f_j)_j$ in $L^2$, $$\label{e:globalONS} \bigg\| \sum_{j}\lambda_j |e^{it\Delta}\langle D \rangle^{-\frac\rho2 -\varepsilon} f_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_{\rho,\varepsilon} \|\lambda\|_{\ell^\alpha}.$$ In fact, using the vector-valued version of the Littlewood-Paley theorem (for example, Lemma 1 in [@sabin-2]) and , we obtain $$\begin{aligned} &\bigg\|\sum_{j}\lambda_j|e^{it\Delta}\langle D\rangle^{-(\frac\rho2+\varepsilon)}f_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})}\\ \lesssim & \bigg\|\sum_{j}\lambda_j|e^{it\Delta}P_{0}f_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} + \bigg\|\sum_{k=1}^\infty\sum_{j}\lambda_j|2^{-k(\frac\rho2+\varepsilon)}e^{it\Delta}P_kf_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})}\\ \lesssim & \|\lambda\|_{\ell^{\alpha}} + \sum_{k=1}^\infty2^{-k(\rho+2\varepsilon)} \bigg\|\sum_j\lambda_j |e^{it\Delta}P_{k}f_j|^2\bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \lesssim_\varepsilon \|\lambda\|_{\ell^{\alpha}},\end{aligned}$$ as we desired. In the sequel, we denote $s=\frac\rho2+\varepsilon$. Before going to the next step, let us recall about the density function, although we do not give the complete treatment of the density function of $\gamma$ here. We refer to [@FLLS] for further detail. A concrete example of our interest is $\rho_{e^{it\Delta} \langle D \rangle^{-s}\gamma_0 \langle D \rangle^{-s} e^{it\Delta}} (x) = \sum_j \lambda_j|e^{it\Delta} \langle D \rangle^{-s} f_j(x)|^2$ where $\gamma_0=\sum_j\lambda_j |f_j\rangle\langle f_j|$, $(f_j)_j$ is the orthonormal system in $L^2(\mathbb{T}^d)$. Then the density function $\rho_{e^{it\Delta} \langle D \rangle^{-s}\gamma_0 \langle D \rangle^{-s} e^{it\Delta}} (x)$ satisfies $$\label{e:density-prop} \int_{\mathbb{T}^d} \rho_{e^{it\Delta} \langle D \rangle^{-s}\gamma_0 \langle D \rangle^{-s} e^{it\Delta}} (x) V(x)\, \mathrm{d}x = {\rm Tr}_{L^2(\mathbb{T}^d)} (\gamma_0 e^{-it\Delta}\langle D \rangle^{-s} V \langle D \rangle^{-s} e^{it\Delta})$$ for any nice function $V:\mathbb{T}^d\to [0,\infty)$. From the definition, it is clear that is equivalent to $$\label{e:density} \left\|\rho_{e^{it\Delta} \langle D \rangle^{-s} \gamma_0 \langle D \rangle^{-s} e^{-it\Delta}}\right\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_{\rho,\varepsilon} \|\gamma_0\|_{\mathcal{C}^\alpha(L^2)}, \quad \gamma_0\in\mathcal{C}^\alpha(L^2).$$ \[p:inhomONS\] 1. The orthonormal Strichartz inequality or is equivalent to for any $V\in L^{p'}_tL^{q'}_x(\mathbb{T}^{d+1})$, $$\label{e:dual-ons} \bigg\| \int_{\mathbb{T}}e^{-it\Delta}\langle D \rangle^{-s}V(x,t) \langle D \rangle^{-s}e^{it\Delta}\, \mathrm{d}t \bigg\|_{\mathcal{C}^{\alpha'}(L^2)} \leq C_{\rho,\varepsilon} \|V\|_{L^{p'}_tL^{q'}_x(\mathbb{T}^{d+1})}.$$ 2. (Inhomogeneous estimate) Let $R(t'):L^2\to L^2$ be self-adjoint for each $t'\in\mathbb{T}$ and define $$\gamma(t)=\int_0^te^{i(t-t')\Delta}R(t')e^{i(t'-t)\Delta}\, \mathrm{d}t',\quad (t\in\mathbb{T}).$$ Suppose one of , and holds true. Then $$\label{e:inhom} \|\rho_{\langle D \rangle^{-s} \gamma(t) \langle D \rangle^{-s}} \|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_{\rho,\varepsilon} \bigg\|\int_{\mathbb{T}}e^{-is\Delta}|R(s)|e^{is\Delta}\, \mathrm{d}s\bigg\|_{\mathcal{C}^\alpha(L^2)}.$$ Since the proof of this proposition is almost the same as in [@FLLS; @frank-sabin-1], we omit details and give key steps. To show , in view of the duality, we have only to show $$\label{e:1214-1} \bigg|{\rm Tr}_{L^2}\bigg( \gamma_0 \int_{\mathbb{T}}e^{-it\Delta}\langle D \rangle^{-s}V(x,t) \langle D \rangle^{-s}e^{it\Delta}\, \mathrm{d}t \bigg)\bigg| \lesssim \|V\|_{L^{p'}_tL^{q'}_x(\mathbb{T}^{d+1})}$$ for any $\gamma_0: \|\gamma_0 \|_{\mathcal{C}^\alpha(L^2)}=1$ which follows from the combination of $\eqref{e:density-prop}$ and . To show , we notice from the duality and the property of the density function that for some non-negative function $V=V(x,t)$ such that $\|V\|_{L^{p'}_tL^{q'}_x(\mathbb{T}^{d+1})} = 1$, $$\begin{aligned} &\|\rho_{\langle D \rangle^{-s} \gamma(t) \langle D \rangle^{-s}} \|_{L^p_tL^q_x(\mathbb{T}^{d+1})} = \int_{\mathbb{T}} {\rm Tr}_{L^2} ( \gamma(t) \langle D\rangle^{-s} V(t) \langle D\rangle^{-s} )\, \mathrm{d}t\\ \leq& \bigg\| \int_{\mathbb{T}} e^{-it\Delta} \langle D\rangle^{-s} V(t) \langle D\rangle^{-s} e^{it\Delta}\, \mathrm{d}t \bigg\|_{\mathcal{C}^{\alpha'}(L^2)} \bigg\| \int_{\mathbb{T}} e^{-it'\Delta} |R(t')| e^{it'\Delta}\, \mathrm{d}t' \bigg\|_{\mathcal{C}^{\alpha}(L^2)}, \end{aligned}$$ where we used the fact that $|{\rm Tr}_{L^2}(AB)| \leq {\rm Tr}_{L^2} (|A| |B|)$ for self-adjoint opeartors $A,B$. So, applying , we obtain . Note that from Duhamel’s principle the solution of the inhomogeneous equation $$\label{e:inhomHartree} \left\{ \begin{array}{ll} i\partial_t\gamma=[-\Delta,\gamma] + R(t), \quad (x,t)\in \mathbb{T}^d \times \mathbb{R} \\ \gamma|_{t=0}=\gamma_0, \end{array} \right.$$ can be written by $$e^{it\Delta}\gamma_0e^{-it\Delta} -i\int_0^t e^{i(t-t')\Delta}R(t')e^{i(t'-t)\Delta}\, \mathrm{d}t'.$$ So, the inequality is an estimate of the inhomogeneous term. Remark that and can be generalize: for any $T>0$, $$\label{e:density-T} \left\|\rho_{e^{it\Delta} \langle D \rangle^{-s} \gamma_0 \langle D \rangle^{-s} e^{-it\Delta}}\right\|_{L^p_tL^q_x([0,T]\times\mathbb{T}^{d})} \leq C_{\rho,\varepsilon} T^{1/p} \|\gamma_0\|_{\mathcal{C}^\alpha(L^2)},$$ and $$\label{e:inhom-T} \|\rho_{\langle D \rangle^{-s} \gamma(t) \langle D \rangle^{-s}} \|_{L^p_tL^q_x([0,T]\times\mathbb{T}^{d})} \leq C_{\rho,\varepsilon} T^{1/p} \bigg\|\int_{\mathbb{T}}e^{-is\Delta}|R(s)|e^{is\Delta}\, \mathrm{d}s\bigg\|_{\mathcal{C}^\alpha(L^2)}.$$ Now, we prove Proposition \[p:generalwellposed\] using Proposition \[p:inhomONS\]. First we prove the local well-posedness Proposition \[p:generalwellposed\]-(1). Let us write $\|\gamma_0\|_{\mathcal{C}^{\alpha,s}(L^2)} = R <\infty$ and take $T=T(R, \|w\|_{B^s_{q',\infty}})\leq1$ to be chosen later. To capture the solution by employing the fixed point theorem, define the space $X$ by $$\begin{aligned} X_T= \{ (\gamma,\rho)\in C^0_t([0,T];\mathcal{C}^{\alpha,s}(L^2))\times L^p_tL^q_x([0,T]\times\mathbb{T}^d): \| (\gamma, \rho) \|_{X_T} \leq C^*R\},\end{aligned}$$ where $$\| (\gamma,\rho) \|_{X_T} := \|\gamma\|_{C^0_t([0,T];\mathcal{C}^{\alpha,s}(L^2))}+\|\rho\|_{L^p_tL^q_x([0,T]\times\mathbb{T}^d)}$$ and $C^*$ is chosen so that $C^*>\max{(10,10C_{\rho,\varepsilon})}$. Next, define the contraction map $\Phi$. First, define $$\Phi_1(\gamma,\rho)(t) = e^{it\Delta}\gamma_0e^{-it\Delta} -i\int_0^t e^{i(t-t')\Delta}[w_a\ast \rho(t'),\gamma(t')]e^{i(t'-t)\Delta}\, \mathrm{d}t'$$ and $$\Phi(\gamma,\rho) = (\Phi_1(\gamma,\rho),\rho[\Phi_1(\gamma,\rho)]).$$ Here, we used the notation $\rho[\gamma]=\rho_\gamma$. In this formulation, is equivalent to $(\gamma,\rho_\gamma) = \Phi(\gamma,\rho_\gamma)$. We now claim that for any $T>0$ and any small $\delta>0$, $$\label{e:contract-1} \|\Phi_1(\gamma,\rho)\|_{C^0_t([0,T];\mathcal{C}^{\alpha,s}(L^2))} \leq R + C_{s,\delta}T^{1/p'}\|w\|_{{B}^{s+\delta}_{q',\infty}}(C^*R)^2$$ and recalling $C_{\rho,\varepsilon}$ is the constant of the orthonormal Strichartz inequality , $$\label{e:contract-2} \|\rho[\Phi_1(\gamma,\rho)]\|_{L^p_tL^q_x([0,T]\times\mathbb{T}^d)} \leq C_{\rho,\varepsilon}T^{1/p}\big\{ R+C_{s,\delta}T^{1/p'}\|w\|_{{B}^{s+\delta}_{q',\infty}}(C^*R)^2\big\}.$$ Once these claims are proved, then choosing $T\leq1$ small enough so that $$C_{s,\delta}C_{\rho,\varepsilon}T^{1/p'}\|w\|_{{B}^{s+\delta}_{q',\infty}}(C^*R)^2\leq \frac{C^*R}{4},$$ we see that $\Phi(\gamma,\rho)\in X_T$ for $(\gamma,\rho)\in X_T$ (precisely speaking, $T$ depends on $\|w\|_{{B}^{s+\delta}_{q',\infty}}$, not $\|w\|_{{B}^{s}_{q',\infty}}$, but this is harmless since $s = \rho + \varepsilon$ and $\varepsilon,\delta$ are arbitrary small). Similarly, we can show that $\Phi$ is a contraction mapping. So, we find a solution to the Hartree equation on $[0,T]$. Let us prove . To evaluate $\|\Phi_1(\gamma,\rho)\|_{C^0_t([0,T];\mathcal{C}^{\alpha,s}(L^2))}$, fix any $t\in[0,T]$ and calculate $$\begin{aligned} &\|\Phi_1(\gamma,\rho)(t)\|_{\mathcal{C}^{\alpha,s}(L^2)}\\ \leq& \|e^{it\Delta}\gamma_0e^{-it\Delta}\|_{\mathcal{C}^{\alpha,s}(L^2)} + \int_0^T \big\|e^{i(t-t')\Delta}[w\ast \rho(t'),\gamma(t')]e^{i(t'-t)\Delta}\big\|_{\mathcal{C}^{\alpha,s}(L^2)}\, \mathrm{d}t'.\end{aligned}$$ The first term is easy to handle since if $(f_j)_j$ is orthonormal in $L^2$, then $(e^{it\Delta}f_j)_j$ is as well for each $t$: $$\|e^{it\Delta}\gamma_0e^{-it\Delta}\|_{\mathcal{C}^{\alpha,s}(L^2)}= \|\gamma_0\|_{\mathcal{C}^{\alpha,s}(L^2)}= R.$$ For the second term, we use the Hölder inequality for Schatten spaces to have $$\begin{aligned} &\big\|e^{i(t-t')\Delta}[w\ast \rho(t'),\gamma(t')]e^{i(t'-t)\Delta}\big\|_{\mathcal{C}^{\alpha,s}(L^2)}\\ \leq& \big\{ \| \langle D \rangle^s w\ast \rho(t') \langle D \rangle^{-s} \|_{\mathcal{C}^\infty(L^2)} + \| \langle D \rangle^{-s} w\ast \rho(t') \langle D \rangle^{s} \|_{\mathcal{C}^\infty(L^2)} \big\} \| \gamma(t') \|_{\mathcal{C}^{\alpha,s}(L^2)} $$ The estimate we employ to evaluate the above nonlinear term is the following (see Corollary on p. 205 in [@Triebel] where the inequality was proved for $\mathbb{R}^d$ case, but the same proof is applicable for $\mathbb{T}^d$ case) $$\label{eq:1012-8} \|f\cdot g\|_{{H}^{r}} \leq C_{s,\delta} \|f\|_{{B}^{|r|+\delta}_{\infty,\infty}} \|g\|_{{H}^{r}},$$ where $r\in\mathbb{R}$ and $\delta>0$ are arbitrary. From this estimate and Young’s inequality, $$\begin{aligned} \| \langle D \rangle^s w\ast \rho(t') \langle D \rangle^{-s} \|_{\mathcal{C}^\infty(L^2)} \leq C_{s,\delta} \|w\ast \rho(t')\|_{{B}^{s+\delta}_{\infty,\infty}} \leq C_{s,\delta} \|w\|_{{B}^{s+\delta}_{q',\infty}}\|\rho(t')\|_{L^q_x}.\end{aligned}$$ Similarly, $$\begin{aligned} \| \langle D \rangle^{-s} w\ast \rho(t') \langle D \rangle^{s} \|_{\mathcal{C}^\infty(L^2)} \leq C_{-s,\delta} \|w\|_{{B}^{s+\delta}_{q',\infty}}\|\rho(t')\|_{L^q_x}.\end{aligned}$$ In total, from $(\gamma,\rho)\in X_T$, we estimate the second term by $$\begin{aligned} \int_0^T \left\|e^{i(t-t')\Delta}[w\ast \rho(t'),\gamma(t')]e^{i(t'-t)\Delta}\right\|_{\mathcal{C}^{\alpha,s}(L^2)}\, \mathrm{d}t' \leq C_{s,\delta}'\|w\|_{{B}^{s+\delta}_{q',\infty}}T^{1/p'}(C^{*}R)^2.\end{aligned}$$ where $C_{s,\delta}'=C_{s,\delta} + C_{-s,\delta}$ which shows . To show , we employ homogeneous and inhomogeneous orthonormal Strichartz estimates and to have $$\begin{aligned} &T^{-1/p}C_{\rho,\varepsilon}^{-1} \|\rho[\Phi_1(\gamma,\rho)]\|_{L^p_tL^q_x([0,T]\times\mathbb{T}^d)}\\ \leq& \| \langle D \rangle^s \gamma_0 \langle D \rangle^s \|_{\mathcal{C}^\alpha(L^2)} + \bigg\|\int_{0}^Te^{-it'\Delta} \langle D \rangle^s |[w_a\ast \rho(t'),\gamma(t')]| \langle D \rangle^s e^{it'\Delta}\, \mathrm{d}t'\bigg\|_{\mathcal{C}^\alpha(L^2)}.\end{aligned}$$ For the first term, $\| \langle D \rangle^s \gamma_0 \langle D \rangle^s \|_{\mathcal{C}^\alpha(L^2)} = R$. For the second term, we may employ the same argument as and we see . Let us show proposition \[p:generalwellposed\]-(2). In this case, we first fix an arbitrary $T>0$. The key estimates are and which have been already proved. These two estimates yield that $$\| \Phi (\gamma,\rho) \|_{X_T} \leq (1+C_{\rho,\varepsilon}T^{1/p}) \big( \| \gamma_0 \|_{\mathcal{C}^{\alpha,s}(L^2)} + C_{s,\delta} T^{1/p'}\|w\|_{B^{s+\delta}_{q',\infty}} \| (\gamma, \rho) \|_{X_T}^2 \big).$$ With this in mind, we choose $R_T = R_T(\| w \|_{B^s_{q',\infty}})$ small enough (precisely speaking, $R_T$ depends on $\| w \|_{B^{s+\delta}_{q',\infty}}$, not $\| w \|_{B^s_{q',\infty}}$, but again this is harmless) so that we can find $M>0$ such that for any $y\in [0, M]$, it holds $$(1+C_{\rho,\varepsilon}T^{1/p}) \big( \| \gamma_0 \|_{\mathcal{C}^{\alpha,s}(L^2)} + C_{s,\delta} T^{1/p'} \|w\|_{B^{s+\delta}_{q',\infty}} y^2 \big) \leq M$$ as long as $ \| \gamma_0 \|_{\mathcal{C}^{\alpha,s}(L^2)} \leq R_T$. So, if we define the space $X_{T,M}$ by $$X_{T,M} := \{ (\gamma,\rho)\in X_T: \| (\gamma, \rho) \|_{X_T} \leq M \},$$ then we see that $\Phi : X_{T,M} \to X_{T,M}$. By choosing $R_T$ smaller further, we can also show that $\Phi$ is a contraction map on $X_{T,M}$ by the similar way and hence from the fixed point theorem we find a solution $\gamma \in C^0_t([0,T]; \mathcal{C}^{\alpha,s}(L^2))$ satisfying $\rho_\gamma \in L^p_tL^q_x([0,T]\times\mathbb{T}^d)$. On the beyond region $[A,C]$ ============================ In this final Section, we give one observation on the beyond region $[A,C]$ when $d\geq2$ and this at least gives almost sharp inequality with $\varepsilon$-loss at the point $A$. \[t:ONS-beyond\] Let $d\geq2$, $N>1$ and $(a_j)_j$ be any orthonormal system in $\ell^2$. 1. \[item:1\] On $(\frac1q,\frac1p)=A$, $$\label{e:ONS-A} \bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_\varepsilon N^{\frac1p + \varepsilon} \|\lambda\|_{\ell^{\alpha(1/p)}}$$ holds true for any $\lambda\in\ell^{\alpha(1/p)}$ and arbitrary small $\varepsilon>0$. Moreover, this is sharp up to $\varepsilon$. 2. \[item:2\] On $(\frac1q,\frac1p)=C$, $$\label{e:ONS-C} \bigg\| \sum_j \lambda_j |\mathcal{E}_Na_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_\varepsilon N^{\frac1p + \frac1d + \varepsilon} \|\lambda\|_{\ell^{\alpha(1/p)}}$$ holds true for any $\lambda\in\ell^{\alpha(1/p)}$ and arbitrary small $\varepsilon>0$. We will show Theorem \[t:ONS-beyond\] in a more general form: for any $(\frac1q,\frac1p)\in[A,C]$, $$\label{e:beyond} \bigg\| \sum_j \lambda_j |\mathcal{E}_N a_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d+1})} \leq C_\varepsilon N^{\frac12(d-1-\frac{d+1}{q}) + \frac1p + \varepsilon} \|\lambda\|_{\ell^\frac{2q}{q+1}}.$$ Note that while gives an almost sharp estimate at $A$ up to $\varepsilon$, seems not sharp because of the factor $N^\frac{1}{d}$. If we recall the argument which we used to prove Theorem \[t:ONS-mix\], then it suffices to show $$\bigg\| \sum_j \lambda_j |\mathcal{E}_N a_j|^2 \bigg\|_{L^p_tL^q_x(\mathbb{T}^{d}\times I_N)} \leq C_\varepsilon N^{\frac12(d-1-\frac{d+1}{q}) + \varepsilon} \|\lambda\|_{\ell^\frac{2q}{q+1}}.$$ Moreover, in view of Lemma \[l:duality\], this inequality follows from $$\label{e:dual-beyond} \big\| W_1^N \mathcal{E}_N \mathcal{E}_N^* W_2^N \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))} \lesssim N^{\frac{d+1-\alpha}{\alpha}+\varepsilon} \|W_1\|_{L^\beta_t L^\alpha_x(\mathbb{T}^{d+1})} \|W_1\|_{L^\beta_t L^\alpha_x(\mathbb{T}^{d+1})}$$ for $\frac 2\beta +\frac d\alpha =1$ and $d\leq \alpha \leq d+1$ where $W_i^N:= \1_{I_N}(t) W_i $. To this end, we decompose the operator $\mathcal{E}_N\mathcal{E}_N^* = K_N \ast$ as follows: for $(x,t)\in\mathbb{T}^d\times I_N$, $$\begin{aligned} \mathcal{E}_N\mathcal{E}_N^* W^N(x,t) &= \sum_{j=-\infty}^{{\rm log}_2 (N^{-1})} \int_{\mathbb{T}^d}\int_{2^{j-1} \leq |t-t'| < 2^{j}} K_N(x-x',t-t') W(x',t')\, \mathrm{d}x'\mathrm{d}t' \\ &= \sum_{j=-\infty}^{{\rm log}_2 (N^{-1})} T_{N,j} W(x,t), \end{aligned}$$ where $T_{N,j}=K_{N,j}\ast$ and $K_{N,j}=K_N\1_{2^{j-1}\leq |t| <2^j}$. Hereafter we evaluate each term $\big\| W_1^N T_{N,j} W_2^N \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))}$ . We claim that for any $\sigma\in [2,\infty]$ and any parameters $\mu\in[0,1]$, $\rho\geq4$, $$\begin{aligned} \label{e:local-dual} &\big\| W_1^N T_{N,j} W_2^N \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))} \nonumber \\ \lesssim & 2^{j[(\frac12-\frac d2 (1-\mu))\frac 2\alpha+1-\frac 2\alpha]} N^{(d\mu - 2(\frac14-\frac1\rho))\frac 2\alpha} \| W_1 \|_{L^{\frac{\rho\alpha}{2}}_tL^\alpha_x(\mathbb{T}^{d+1})} \| W_2 \|_{L^{\frac{\rho\alpha}{2}}_tL^\alpha_x(\mathbb{T}^{d+1})}\end{aligned}$$ To see this, we consider two cases $\alpha=2$ and $\alpha=\infty$. When $\alpha=2$, we employ the kernel estimate: for $(x,t) \in \mathbb{T}^d\times I_{N}$, $$|K_{N,j}(x,t)| \lesssim \min{(|t|^{-\frac{d}{2}}, N^d)} \leq |t|^{-\frac d2 (1-\mu)} N^{d\mu} \, \quad (\mu\in [0,1]).$$ From this estimate, Young’s inequality and Hölder’s inequality, $$\begin{aligned} &\big\| W_1^N T_{N,j} W_2^N \big\|_{\mathcal{C}^2(L^2(\mathbb{T}^{d+1}))}^2 \\ \lesssim& N^{2d\mu} \int_{|t-t'|\sim 2^j} \| W^N_1(\cdot,t) \|_{L^2_x(\mathbb{T}^d)}^2 |t-t'|^{-d(1-\mu)} \|W^N_2(\cdot,t') \|_{L^2_x(\mathbb{T}^d)}^2\, \mathrm{d}t\mathrm{d}t'\\ \lesssim& N^{2d\mu} 2^{j(1-d(1-\mu))} N^{ -4(\frac14-\frac1\rho) } \|W_1\|_{L^\rho_tL^2_x(\mathbb{T}^{d+1})}^2 \|W_2\|_{L^\rho_tL^2_x(\mathbb{T}^{d+1})}^2 \end{aligned}$$ holds for any $\rho\geq4$. On the other hand, when $\alpha=\infty$, we see from Plancherel’s theorem that for any $F\in L^2(\mathbb{T}^{d+1})$ $$\begin{aligned} \big\| W_1^N T_{N,j} [W_2^N F] \big\|_{L^2(\mathbb{T}^{d+1}))} &\lesssim 2^j \|W_1\|_{L^\infty_tL^\infty_x(\mathbb{T}^{d+1})} \| W_2 \|_{L^\infty_tL^\infty_x(\mathbb{T}^{d+1})} \| F \|_{L^2(\mathbb{T}^{d+1})}, \end{aligned}$$ since we have for any $(n,n_{d+1}) \in \mathbb{Z}^{d+1}$, $$|\mathcal{F}_{x,t} K_{N,j}(n,n_{d+1})| \lesssim 2^j.$$ Interpolating these two estimates, we obtain . To sum up each estimate , we need to $$(\frac12-\frac d2 (1-\mu))\frac 2\alpha+1-\frac 2\alpha >0$$ or equivalently, $ \mu>\frac{d+1-\alpha}{d} $ which gives the restriction of $\mu$. Under this restriction, we can sum up and obtain $$\begin{aligned} &\big\| W_1^N \mathcal{E}_N \mathcal{E}_N^* W_2^N \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))} \\ \lesssim & N^{(d\mu - 2(\frac14-\frac1\rho))\frac 2\alpha} N^{-(1-\frac d\alpha(1-\theta) -\frac1\alpha)} \| W_1 \|_{L^{\frac{\rho\alpha}{2}}_tL^\alpha_x(\mathbb{T}^{d+1})} \| W_2 \|_{L^{\frac{\rho\alpha}{2}}_tL^\alpha_x(\mathbb{T}^{d+1})}. \end{aligned}$$ The parameter $\rho\geq4$ is determined to establish the scaling condition $ 2\cdot\frac{2}{\rho\alpha} + \frac d\alpha =1$ which means $\frac1\rho = \frac{\alpha - d}{4}$. From this and a few computations we learn $\alpha$ is restriced to $d\leq\alpha\leq d+1$. Then we finally have $$\big\| W_1^N \mathcal{E}_N \mathcal{E}_N^* W_2^N \big\|_{\mathcal{C}^\alpha(L^2(\mathbb{T}^{d+1}))} \lesssim N^{\frac d\alpha \mu} \| W_1 \|_{\beta,\alpha} \| W_2 \|_{\beta,\alpha},$$ for any $\alpha \in [d,d+1]$, $\mu \in (\frac{d+1-\alpha}{d},1]$ and $\frac 2\beta +\frac d\alpha =1$. In particular, taking $\mu = \frac{d+1-\alpha}{d} + \varepsilon$, we arrive at . This work was supported by Grant-in-Aid for JSPS Research Fellow no. 17J01766. This work grows out the collaboration with Professors Neal Bez, Younghun Hong, Sanghyuk Lee and Yoshihiro Sawano [@BHLNS]. The author thank Neal Bez for introducing me to this problem, Sanghyuk Lee for sharing very useful insight, Younghun Hong for giving me many comments from view point of PDE perspective and Yoshihiro Sawano for making the paper nicer. [MMMMM]{} N. Bez, Y. Hong, S. Lee, S. Nakamura, Y. Sawano, *On the Strichartz estimates for orthonormal systems of initial data with regularity*, arXiv:1708.05588. J. Bourgain, *Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations*, Geom. Funct. Anal. **3** (1993), no. 2, 107–156. J. Bourgain, C. Demeter, *Improved estimates for the discrete Fourier restriction to the higher dimensional sphere*, Illinois J. Math. **57** (2013), no. 1, 213–227. J. Bourgain, C. Demeter, *New bounds for the discrete Fourier restriction to the sphere in four and five dimensions*, Internat. Math. Res. Notices (2015), no. 11, 3150–3184. J. Bourgain, C. Demeter, *The proof of the $\ell^2$ Decoupling Conjecture*, Annals of Math. **182** (2015), no. 1, 351–389. N. Burq, P. Gérard, N. Tzvetkov, *Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds*, Amer. J. of Math. **126** (2004), no. 3, 569–605. N. Burq, P. Gérard, N. Tzvetkov, *Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces*, Invent. Math. **159** (2005), 187–223. N. Burq, P. Gérard, N. Tzvetkov, *Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations*, Ann. Sci. École Norm. Sup. **38** (2005), 255–301. A. Bove, G. Da Prato, G. Fano, *An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction*, Commun. Math. Phys. **37** (1974), 183–191. A. Bove, G. Da Prato, G. Fano, *On the Hartree-Fock time-dependent problem*, Commun. Math. Phys. **49** (1976), 25–33. F. Catoire, W.-M. Wang, *Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori*, Commun. Pure Appl. Anal. **9** (2010), 483–491. J. M. Chadam, *The time-dependent Hartree-Fock equations with Coulomb two-body interaction*, Commun. Math. Phys. **46** (1976), 99–104. T. Chen, Y. Hong, N. Pavlović, *Global well-posedness of the NLS system for infinitely many fermions*, Arch. Ration. Mech. Anal. **224** (2017), 91–123. T. Chen, Y. Hong, N. Pavlović, *On the scattering problem for infinitely many fermions in dimension $d\geq3$ at positive temperature*, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire. S. Demirbas, *Local well-posedness for 2-d Schrödinger equation on irrational tori and bounds onSobolev norms*, Commun. Pure Appl. Anal. **16** (2017), 1517–1530. R. Frank, M. Lewin, E. Lieb, R. Seiringer, *Strichartz inequality for orthonormal functions*, J. Eur. Math. Soc. **16** (2014), 1507–1526. R. Frank, J. Sabin, *Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates*, to appear in Amer. J. of Math. R. Frank, J. Sabin, *The Stein-Tomas inequality in trace ideals*, Séminaire Laurent Schwartz – EPD et applications (2015-2016), Exp. No. XV, 12 pp., 2016. P. Gérard, V. Pierfelice, *Nonlinear Schrödinger equation on four-dimensional compact manifolds*, Bull. Soc. math. France. **138** (2010), 119–151. J. Ginibre, G. Velo, *Smoothing properties and retarded estimates for some dispersive evolution equations*, Comm. Math. Phys. **123** (1989), 535–573. Z. Guo, T. Oh, Y. Wang, *Strichartz estimates for Schödinger equations on irrational tori*, Proc. London Math. Soc. **109** (2014), 975–1013. Z. Guo, L. Peng, *Endpoint Strichartz estimate for the kinetic transport equation in one dimension*, C. R. Math. Acad. Sci. Paris **345** (2007), 253–256. M. Keel, T. Tao *Endpoint Strichartz estimates*, Amer. J. Math. **120** (1998), 955–980. C. E. Kenig, G. Ponce, L. Vega, *Oscillatory integrals and regularity of dispersive equations*, Indiana Univ. Math. J. **40** (1991), 33–69. M. Lewin, J. Sabin, *The Hartree equation for infinitely many particles. I. Well-posedness theory*, Comm. Math. Phys. **334** (2015), 117–170. M. Lewin, J. Sabin, *The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D*, Analysis & PDE **7** (2014), 1339–1363. E. H. Lieb, *The stability of matter: from atoms to stars*, Bull. Amer. Math. Soc. **22** (1990), 1–49. E. H. Lieb, W. Thirring, *Bound on kinetic energy of fermions which proves stability of matter*, Phys. Rev. Lett. **35** (1975), 687–689. A. R. Nahmod, *The nonlinear Schrödinger equation on tori: integrating Harmonic analysis, Geometry, and Probability*, Bull. Amer. Math. Soc. **53** (2016), 57–91. J. Sabin, *The Hartree equation for infinite quantum systems*, Journées équations aux dérivées partielles, (2014), Exp. No. 8. 18p. J. Sabin, *Littlewood–Paley decomposition of operator densities and application to a new proof of the Lieb–Thirring inqeuality*, Math. Phys. Anal. Geom. **19** (2016), no. 2, Art. 11, 11 pp. R. S. Strichartz, *Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations*, Duke Math. J. **44** (1977), 705–774. G. Staffilani, *Dispersive equations and their role beyond PDE*, http://math.mit.edu/ gigliola/AMS-Bulletin.pdf. H. Triebel, *Theory of function spaces II*, Birkhäuser, Basel, 1992. Y. Tsutsumi, *$L^2$-solutions for nonlinear Schrödinger Equations and nonlinear groups*, Funkcialaj Ekvacioj **30** (1987), 115–125. L. Vega, *Restriction Theorems and the Schrödinger multiplier on the torus*, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990), IMA Vol. Math. Appl., vol. 42, Springer-Verlag, New York, 1992, pp. 199–211. K. Yajima, *Existence of solutions for Schrödinger evolution equations*, Comm. Math. Phys., **110** (1987), 415–426.

--- abstract: 'We study the dynamics of skyrmions under spin-transfer torque in Dzyaloshinskii-Moriya materials with easy-axis anisotropy. In particular, we study the motion of a topological skyrmion with skyrmion number ${Q}=1$ and a non-topological skyrmionium with ${Q}=0$ using their linear momentum, virial relations, and numerical simulations. The non-topological ${Q}=0$ skyrmionium is accelerated in the direction of the current flow and it either reaches a steady state with constant velocity, or it is elongated to infinity. The steady-state velocity is given by a balance between current and dissipation and has an upper limit. In contrast, the topological ${Q}=1$ skyrmion converges to a steady-state with constant velocity at an angle to the current flow. When the spin current stops the ${Q}=1$ skyrmion is spontaneously pinned whereas the ${Q}=0$ skyrmionium continues propagation. Exact solutions for the propagating skyrmionium are identified as solutions of equations given numerically in a previous work. Further exact results for propagating skyrmions are given in the case of the pure exchange model. The traveling solutions provide arguments that a spin polarized current will cause rigid motion of a skyrmion or a skyrmionium.' author: - Stavros Komineas - Nikos Papanicolaou title: 'Skyrmion dynamics in chiral ferromagnets under spin-transfer torque' --- Introduction {#sec:intro} ============ Soliton structures are found in ferromagnets and they can be considered as an encoding of magnetic information which is robust both under temperature and external probes. Stable topological solitons with the structure of a skyrmion had been predicted in the presence of the Dzyaloshinskii-Moriya (DM) interaction [@BogdanovYablonskii_JETP1989; @BogdanovHubert_JMMM1994] and they were observed in recent years as isolated structures [@RommingHanneken_Science2013; @RommingKubetzka_PRL2015] or forming lattices [@MuhlbauerBinz_Science2009; @YuOnose_Nature2010; @YuKanazawa_NatMat2011]. In the presence of easy-axis anisotropy there are topological skyrmions with skyrmion number ${Q}=1$ as well as non-topological ${Q}=0$ solitons ($2\pi$ vortices) [@BogdanovHubert_JMMM1999; @LeonovRoessler_EPJ2013]. Skyrmions could be the stable and robust entities that are needed for the technology of recording and transferring information, currently mainly obtained in magnetic media using domain walls [@AllwoodXiong_Science2005]. The propagation of magnetic information is done most conveniently by the injection of electrical spin-polarized current. Single skyrmions and skyrmion lattices can be set in motion suggesting a promising technique for the manipulation of magnetic information [@EverschorGarst_PRB2012; @FertCros_NatNano2013; @SampaioCros_NatNano2013; @NagaosaTokura_NatNano2013; @IwasakiMochizuki_NatComms2013; @IwasakiMochizuki_NatNano2013; @SchutteIwasaki_PRB2014]. Propagation of skyrmions by spin current or the related spin-Hall effect may be a promising strategy for the implementation of racetrack memories [@TomaselloMartinez_SciRep2014]. The existence of two species of skyrmions (${Q}= 0$ and ${Q}\neq 0$) has allowed theoretical predictions for dramatically different dynamical behaviors [@KomineasPapanicolaou_PRB2015]. We show here that spin torque accelerates a ${Q}= 0$ skyrmionium and we describe the process theoretically, while the study is complemented by numerical simulations. The skyrmionium may reach a steady state or it may absorb energy from the current and expand without limit. A skyrmionium propagating even when the external probe is switched off can be obtained. The situation is contrasted to the more well-studied case of a ${Q}= 1$ skyrmion under spin torque. The outline of the paper is the following. Sec. \[sec:model\] gives a description of the Landau-Lifshitz model including damping and spin-transfer torques and provides the main theoretical tools. Sec. \[sec:traveling\_skyrmionium\] is on the dynamics of a ${Q}=0$ skyrmionium and Sec. \[sec:traveling\_skyrmion\] is on the dynamics of a ${Q}=1$ skyrmion. Sec. \[sec:conclusions\] contains our concluding remarks. An Appendix gives analytical results on the pure exchange model. Magnetization dynamics under spin-transfer torque {#sec:model} ================================================= We assume a thin ferromagnetic film with a Dzyaloshinksii-Moriya (DM) interaction. Let $\bm{M}(x,y,t)$ be the magnetization vector with $M_s$ the saturation magnetization and define the normalised magnetization $\bm{m} = \bm{M}/M_s$, so that $\bm{m}^2=1$. The conservative Landau-Lifshitz (LL) equation for the statics and dynamics of the magnetization is $$\label{eq:LL} {\partial}_t \bm{m} = -\bm{m}\times\bm{f}$$ and is valid in the absence of damping and external probes. We consider an effective field $\bm{f}$ which includes an exchange interaction with constant $A$, an easy-axis anisotropy perpendicular to the $(x_1,x_2)$-plane of the film with constant ${K}$, and a DM interaction with constant ${D}$ [@BogdanovHubert_JMMM1994]. If the energy is ${W}$ then the effective field $\bm{f}=-\delta{W}/\delta\bm{m}$ is $$\begin{aligned} \label{eq:effective_field} \mathbf{f} & = \Delta\mathbf{m} + {\kappa}\, m_3 {\mathbf{\hat{e}}_3}\\ & - 2{\lambda}\,\left[ {\partial}_2 m_3\, {\mathbf{\hat{e}}_1}- {\partial}_1 m_3\,{\mathbf{\hat{e}}_2}+ ({\partial}_1 m_2 - {\partial}_2 m_1)\,{\mathbf{\hat{e}}_3}\right] \notag\end{aligned}$$ where we have used ${\ell_{\rm D}}= 2A/|D|$ as the unit of length. The parameter $$\label{eq:anisotropy} {\kappa}\equiv \frac{{K}}{{K}_0},\qquad {K}_0 = \frac{D^2}{4A}$$ is the rationalized (dimensionless) anisotropy constant and ${\lambda}= D/|D| = \pm 1$ will be referred to as the chirality. We choose chirality ${\lambda}=1$ in all of our numerical calculations, while ${\kappa}$ is taken to be positive (easy-axis anisotropy). We have not included the demagnetizing field in Eq.  because it does not affect skyrmion configurations in a qualitatively significant way [@BegChernyshenko_arXiv2014]; it introduces a dependence of the skyrmion size on the film thickness [@KiselevBogdanov_JPD2011]. The time variable $t$ in Eq.  is measured in units of $\tau_0=2A M_s/(\gamma {D}^2)$ where $\gamma$ is the gyromagnetic ratio. When a spin-polarized current is flowing in the plane of the film, say, in the $x_1$ direction, and we also include damping effects, the magnetization obeys the Landau-Lifshitz-Gilbert equation with two additional terms for the spin-transfer torque [@Slonczewski_JMMM1996; @Berger_PRB1996; @ZhangLi_PRL2004]: $$\label{eq:llg_stt_ip} ({\partial}_t + {u}\, {\partial}_1)\bm{m} = -\bm{m}\times\bm{f} + \bm{m}\times \left( \alpha{\partial}_t + {\beta}u\, {\partial}_1 \right) \bm{m}.$$ The dissipation constant is $\alpha$ while ${u}$ is the effective spin velocity parallel to the spin current and ${\beta}$ is the non-adiabatic spin-transfer torque parameter. In the absence of spin torque, that is, for ${u}=0$ the ground state of the model is the [*spiral state*]{} for sufficiently small anisotropy. For ${\kappa}> {\kappa}_c = \pi^2/4\approx 2.4674$ the ground state is either of the two uniform ferromagnetic states $\bm{m}=(0,0,\pm 1)$. In the latter case, skyrmions are excited states and they are classified by the skyrmion number defined as $$\label{eq:skyrmion_number} {Q}= \frac{1}{4\pi} \int {q}\, d^2x,\qquad {q}= \frac{1}{2} {\epsilon_{\mu\nu}}\mathbf{m}\cdot({\partial}_\nu\mathbf{m}\times {\partial}_\mu\mathbf{m}),$$ where ${q}$ is called the [*topological density*]{}. The skyrmion number ${Q}$ is integer-valued (${Q}=0,\pm 1,\pm 2,\ldots$) for all magnetic configurations such that $\mathbf{m}=(0,0,\pm 1)$ at spatial infinity. For definiteness we will assume $\mathbf{m}=(0,0,1)$ in all our calculations. Axially symmetric skyrmion configurations are conveniently described in terms of the standard spherical parametrisation given by $$m_1 = \sin\Theta \cos\Phi,\quad m_2 = \sin\Theta \sin\Phi,\quad m_3 = \cos\Theta$$ with the ansatz $$\label{eq:axially_symmetric} \Theta = \theta(\rho),\qquad \Phi = \phi + \pi/2,$$ where $(\rho,\phi)$ are polar coordinates. Solving Eq.  with boundary conditions $\theta(\rho=0)=\pi$ and $\theta(\rho\to\infty)=0$, leads to a static skyrmion with ${Q}=1$ shown in Fig. \[fig:skyrmion\_vecs\]. If the boundary conditions are $\theta(\rho=0)=2\pi, \theta(\rho\to\infty)=0$ a ${Q}=0$ configuration is found [@BogdanovHubert_JMMM1999] which has been called a “skyrmionium” [@FinazziSavoini_PRL2013; @KomineasPapanicolaou_PRB2015] and is shown in Fig. \[fig:skyrmionium\_vecs\]. ![The axially symmetric (${Q}=1$) skyrmion represented through the projection $(m_1,m_2)$ of the magnetization vector on the plane. It is calculated as a static solution of Eq.  for anisotropy ${\kappa}=3$.[]{data-label="fig:skyrmion_vecs"}](skyrmion_vecs){width="6truecm"} ![The axially symmetric (${Q}=0$) skyrmionium represented through the projection $(m_1,m_2)$ of the magnetization vector on the plane It is calculated as a static solution of Eq.  for anisotropy ${\kappa}=3$.[]{data-label="fig:skyrmionium_vecs"}](skyrmionium_vecs){width="6truecm"} For the conservative LL Eq.  it can be proved that the moments of the topological density defined by $$\label{eq:topological_moments} I_\mu = \int x_\mu{q}\, d^2x,\qquad \mu=1,2$$ are conserved quantities [@PapanicolaouTomaras_NPB1991; @KomineasPapanicolaou_PhysD1996]. The conservation laws hold in the case of an infinite film. For a magnetization field obeying an equation of the form ${\partial}_t \bm{m}= -\bm{m}\times\bm{g}$ the time derivative of the topological density is $$\label{eq:vorticity_time_derivative} \dot{{q}} = -{\epsilon_{\mu\nu}}\,{\partial}_\mu (\bm{g}\cdot{\partial}_\nu\bm{m})$$ as can be found by a straightforward calculation. For $\bm{g}=\bm{f}$ it can be shown that $\dot{I}_\mu=0$ and the ingredients of the proof for the specific $\bm{f}$ of Eq.  are given in Ref. [@KomineasPapanicolaou_PRB2015]. Eq.  including damping and spin torques can be written in the form $$\begin{aligned} & {\partial}_t \bm{m} = -\bm{m}\times\bm{g} \\ & \bm{g} = \frac{1}{1+\alpha^2} \left[ \bm{f} + \alpha\,\bm{m}\times\bm{f} - ({\beta}-\alpha) {u}\, {\partial}_1\bm{m} - \alpha({\beta}-\alpha) {u}\,\bm{m}\times{\partial}_1\bm{m} \right] \notag\end{aligned}$$ and the time derivatives for $I_\mu$ can be found by using Eq. . The moments $I_\mu$ are no longer conserved due to the damping and spin torque terms and we find $$\begin{aligned} \label{eq:virial1_stt_inplane} (1+\alpha^2)\dot{I}_1 & = - ({\beta}-\alpha){u}\, d_{12} + \alpha\, D_2 + (1+\alpha{\beta}) {u}\, (4\pi{Q}) \notag \\ (1+\alpha^2) \dot{I}_2 & = ({\beta}-\alpha){u}\, d_{11} - \alpha\, D_1\end{aligned}$$ where we have used the notation $$\label{eq:d_munu} \begin{split} d_{\mu\nu} & = \int ({\partial}_\mu\bm{m}\cdot{\partial}_\nu\bm{m})\, d^2x,\qquad \qquad \mu,\nu=1,2 \\ D_\mu & = \int (\bm{m}\times \bm{f} ) \cdot{\partial}_\mu\bm{m}\, d^2x. \end{split}$$ Eqs.  give an explicit result since they may be applied for any magnetic configuration. Let us consider a skyrmion which is initially static within the conservative LL Eq.  and we suddenly apply an electrical current according to Eq. . This will start to move and the overall motion is given by Eqs. . As a next step we will assume that it will eventually reach a steady-state with velocity $\bm{v}=(v_1, v_2)$. We may then write the traveling wave ansatz for the magnetization $$\begin{aligned} \label{eq:traveling_wave} & \bm{m}(x_1,x_2,t) = \bm{m}_0(\xi_1,\xi_2;v_1,v_2) \\ & \xi_1 \equiv x_1-v_1 t,\quad \xi_2=x_2-v_2 t \notag\end{aligned}$$ so that ${\partial}_t\bm{m} = -v_\nu {\partial}_\nu\bm{m}$ with $\nu=1,2$. Inserting this in Eq.  we obtain $$\label{eq:llg_stt_ip_steadystate} {u}\,{\partial}_1\bm{m} - v_\nu {\partial}_\nu\bm{m} = -\bm{m}\times \bm{f} + \bm{m}\times \left( {\beta}\, {u}\,{\partial}_1\bm{m} - \alpha v_\nu\, {\partial}_\nu\bm{m} \right).$$ We will assume that ${\partial}_1, {\partial}_2$ in the above equation denote derivatives with respect to $\xi_1, \xi_2$. We may now take the cross product of both sides in Eq.  with ${\partial}_\mu\bm{m}$ for $\mu=1$ or $2$ and then contract with $\bm{m}$. In the result, the term containing the effective field $\bm{f}$ (i.e., the term due to the conservative part of the equation) is written as a total derivative [@PapanicolaouTomaras_NPB1991; @KomineasPapanicolaou_PhysD1996]: $$\label{eq:sigma_divergence} -\bm{f}\cdot {\partial}_\mu \bm{m} = {\partial}_\nu \sigma_{\mu\nu}$$ where the explicit form of the tensor $\sigma_{\mu\nu}$ for the effective field is given in Ref. [@KomineasPapanicolaou_PRB2015]. Upon integrating over all space the total derivative vanishes and we obtain the pair of virial relations [@HeLiZhang_PRB2006; @EverschorGarst_PRB2011] $$\label{eq:virial} \begin{split} & ( -4\pi{Q}+ \alpha d_{21} ) v_1 + \alpha d_{22} v_2 = {\beta}{u}\, d_{21} - {u}\, (4\pi{Q}) \\ & \alpha d_{11} v_1 + (4\pi{Q}+ \alpha d_{12} ) v_2 = {\beta}{u}\, d_{11}. \end{split}$$ Some general conclusions can be drawn from Eqs.  and . The dynamics and the steady-state velocity $(v_1,v_2)$ can be obtained in particular cases as will be discussed in the next sections for the cases of a skyrmionium and a skyrmion. A traveling ${Q}=0$ skyrmionium {#sec:traveling_skyrmionium} =============================== Let us consider the ${Q}=0$ skyrmionium of Fig. \[fig:skyrmionium\_vecs\] which is a static solution of Eq.  and we suddenly apply a spin polarized current. In order to follow how the initial skyrmionium will be accelerated we will follow its [*linear momentum*]{} which is defined via the conserved $I_\mu$ of Eq.  as [@PapanicolaouTomaras_NPB1991; @KomineasPapanicolaou_PhysD1996] $$\label{eq:linear_momentum} P_\mu = {\epsilon_{\mu\nu}}I_\nu,\qquad \mu, \nu = 1\; \hbox{or}\; 2.$$ To be sure, the above quantities have the meaning of a linear momentum within the hamiltonian equations . We will though extend their use in the full model . The time derivatives of the components of the linear momentum are given by Eqs.  applied for ${Q}=0$: $$\label{eq:virial1_stt_inplane_Q0} \begin{split} (1+\alpha^2) \dot{P}_1 & = ({\beta}-\alpha){u}\, d_{11} - \alpha\, D_1 \\ (1+\alpha^2)\dot{P}_2 & = ({\beta}-\alpha){u}\, d_{12} - \alpha\, D_2. \end{split}$$ These relations give a simple result when we apply them for the initial skyrmionium which is a static solution of Eq.  (thus $D_\mu=0$) and is axially symmetric (thus $d_{12}=0$). We obtain $$\label{eq:virial1_stt_inplane_Q0_t0} \dot{P}_1 = \frac{{\beta}-\alpha}{1+\alpha^2}{u}\, d_{11} \qquad \dot{P}_2 = 0.$$ The skyrmionium is accelerated acquiring a linear momentum component along the $x_1$-direction only. The acceleration is zero when ${\beta}= \alpha$ and this point will be clarified in the following when we present solutions of Eq. . If the skyrmionium eventually reaches a traveling steady state then the virial relations apply. For ${Q}=0$ they reduce to a simple form and give the velocity of the steady state: $$\label{eq:virial_Q0} \begin{pmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{pmatrix} \begin{pmatrix} \alpha v_1 - {\beta}{u}\\ \alpha v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \Rightarrow \begin{cases} v_1 & = \frac{{\beta}}{\alpha}{u}\\ v_2 & = 0 \end{cases}$$ provided $\det (d_{\mu\nu}) \neq 0$. Therefore, the skyrmionium in steady state moves in the direction of the current. The presence of dissipation is crucial as there is apparently no steady state for $\alpha=0$. The acceleration process along the axis of the current described by Eqs.  is compatible with the steady-state velocity in Eq. . We can find exact traveling solutions for a skyrmionium under spin-transfer torque. In order to show how this can be achieved, we start by considering the special case ${\beta}=\alpha$, for which Eq.  becomes $$\label{eq:llg_stt_ip_special} ({\partial}_t + {u}\, {\partial}_1)\bm{m} = -\bm{m}\times\bm{f} + \alpha\,\bm{m}\times \left( {\partial}_t + {u}\, {\partial}_1 \right) \bm{m}.$$ We look for traveling wave solutions of the form and we choose $v_1={u},\, v_2 = 0$. We have ${\partial}_t\bm{m} = -u{\partial}_1\bm{m}$, which is used to reduce Eq.  to $\bm{m}\times\bm{f}=0$. Thus, if we choose $\bm{n}(x_1,x_2)$ to be a static solution of Eq.  then $\bm{m}(x_1,x_2,t)=\bm{n}(\xi_1,x_2)$ gives a configuration which satisfies Eq.  and is a traveling solution with velocity $(v_1, v_2)=({u},0)$. In conclusion, the static skyrmionium solution of the conservative LL Eq.  (shown in Fig. \[fig:skyrmionium\_vecs\]) is a traveling solution of the full equation for the case ${\beta}=\alpha$. That also explains the vanishing acceleration in Eqs. . This mathematical result has the following physical content. If we apply spin polarized current to an initially static skyrmionium this is expected to be set into rigid motion without significant deformations. The same can also be argued for the traveling skyrmion in the next Section. Let us now generalize the above for ${\beta}\neq \alpha$. The traveling wave ansatz with the choice $(v_1,v_2)=(v,0)$ is used to reduce Eq.  to $$\label{eq:llg_stt_ip_traveling} ({u}- v){\partial}_1\bm{m} = -\bm{m}\times\bm{f} + ({\beta}{u}- \alpha v)\,\bm{m}\times {\partial}_1 \bm{m}.$$ We now choose $v={\beta}{u}/\alpha$ and the equation is further reduced to $$\label{eq:LL_steadystate} v_0\,{\partial}_1\bm{m} = \bm{m}\times\bm{f},\qquad v_0 = \frac{{\beta}-\alpha}{\alpha}\,{u}.$$ This equation is identical to that for a steady-state traveling with a velocity $v_0$ within the conservative LL Eq. . Such states were numerically calculated and studied in Ref. [@KomineasPapanicolaou_PRB2015]. A family of traveling skyrmioniums were found with velocities up to a critical velocity $v_c \approx 0.102$. In conclusion, if we denote the solutions of Eq.  by $\bm{n}(x_1,x_2;v_0)$ then the form $\bm{m}(x_1,x_2,t)=\bm{n}(\xi_1,x_2;v_0)$ with $\xi_1=x_1-vt$ is a traveling wave solution of Eq.  with velocity $v={\beta}{u}/\alpha$. The condition $v_0 < v_c$ for the skyrmionium configuration satisfying Eq. , becomes in the presence of spin-transfer torque $$\label{eq:velocity_condition} v < u + v_c.$$ ![The velocity components $(v_1, v_2)=(\dot{{X}}_1,\dot{{X}}_2)$ for a skyrmionium under spin torque with parameter values . The expected velocity at the steady state is $(v_1, v_2) = (0.167, 0)$.[]{data-label="fig:Q0_velocity"}](Q0_velocity){width="6truecm"} We have conducted a numerical simulation using the parameter set $$\label{eq:parameter_set0} \alpha = 0.06,\qquad {u}= 0.1,\qquad {\beta}= 0.1.$$ We use as an initial condition the skyrmionium of Fig. \[fig:skyrmionium\_vecs\] with ${\kappa}=3$ and apply the spin current. The expected velocity in the steady-state is given by Eq.  and is $v_1=0.167,\; v_2=0$. The prediction is confirmed by the numerical simulation. We define the position of the skyrmionium as $({X}_1,{X}_2)$ with $$\label{eq:moments_magnetization} {X}_\mu = \frac{\int x_\mu(1-m_3)\, d^2x}{\int (1-m_3)\, d^2x}.$$ Fig. \[fig:Q0\_velocity\] shows the velocity $(v_1, v_2)=(\dot{{X}}_1,\dot{{X}}_2)$ as a function of time. The skyrmionium is accelerated and the change of linear momentum at $t=0$ verifies the prediction $\dot{P}_1 = 0.34$ calculated by Eq.  when we use the numerically calculated value $d_{11}=57.1$. The velocity at $t=0$ is $v_1(t=0) \approx {u}= 0.1$ and it increases to $v_1(t=100) = 0.16$ at the end of this simulation. The component $v_2$ acquires some small value at the initial stages of the simulation and it later goes to zero. The results have been confirmed also by a simulation in a moving frame, running for times longer than those shown Fig. \[fig:Q0\_velocity\], where it is seen that $(v_1,v_2)$ converge to the expected values at the steady-state. ![Contour plots of $m_3$ for a skyrmionium under spin torque with parameter values . (Left) The initial condition, at $t=0$, is a static skyrmionium solution of Eq. . (Right) The skyrmionium at $t=40$ when it has been accelerated. The inner part has moved down relative to the outer part. The contour levels plotted are $m_3=0.9,0.6,0.3,0.0$ (solid lines) and $m_3=-0.3,-0.6,-0.9$ (dashed lines). []{data-label="fig:Q0_stt"}](Q0_t000 "fig:"){width="4truecm"} ![Contour plots of $m_3$ for a skyrmionium under spin torque with parameter values . (Left) The initial condition, at $t=0$, is a static skyrmionium solution of Eq. . (Right) The skyrmionium at $t=40$ when it has been accelerated. The inner part has moved down relative to the outer part. The contour levels plotted are $m_3=0.9,0.6,0.3,0.0$ (solid lines) and $m_3=-0.3,-0.6,-0.9$ (dashed lines). []{data-label="fig:Q0_stt"}](Q0_t040 "fig:"){width="4truecm"} Fig. \[fig:Q0\_stt\] shows two snapshots of the skyrmionium under spin torque. At the initial time $t=0$ we have the axially symmetric static solution of Eq. . The accelerated skyrmionium at $t=40$ has velocity $v_1 = 0.143$ and has lost axial symmetry: its central part has moved lower. It is very similar to the propagating skyrmionium studied in Ref. [@KomineasPapanicolaou_PRB2015]. We conclude that the application of spin current is a method to obtain a propagating skyrmionium in a steady state. We note that the skyrmionium continues to travel at its acquired velocity when the spin currents is switched off, irrespectively of whether a steady state was reached or not. This is in stark contrast to the dynamics of a skyrmion or to ordinary domain wall dynamics. Let us now consider a second set of parameter values $$\label{eq:parameter_set1} \alpha = 0.04,\qquad {u}= 0.1,\qquad {\beta}= 0.1.$$ which gives a velocity for the skyrmionium $v={\beta}{u}/\alpha=0.25$ violating condition . In this case the skyrmionium is accelarated until its velocity approaches the limiting value ${u}+ v_c$ while the configuration becomes elongated along the $x_2$ axis and eventually reaches the boundaries of our numerical grid. Presumably, the process would continue until the skyrmionium configuratiom is destroyed or until it turns to a domain wall extending to infinity in the $x_2$ direction. A traveling ${Q}=1$ skyrmion {#sec:traveling_skyrmion} ============================ Let us now consider topologically nontrivial solutions (${Q}\neq 0$) such as the ${Q}=1$ skyrmion of Fig. \[fig:skyrmion\_vecs\] which is a static solution of Eq. , and suddenly apply a spin polarized current. In order to follow the skyrmion as it moves we will follow the coordinates of its guiding center $(R_1, R_2)$ defined as the normalized moments in Eq. : $$\label{eq:guiding_center} R_\mu = \frac{I_\mu}{4\pi{Q}} = \frac{1}{4\pi{Q}} \int x_\mu{q}\, d^2x.$$ They give a measure of the position of a skyrmion and are conserved quantities as explained in connection with Eqs. , . The instantaneous velocity $(\dot{R}_1, \dot{R}_2)$ is given through Eqs. . For the initial axially symmetric skyrmion solution of Eq.  we have $D_\mu=0$ and $d_{12}=0,\; d_{11}=d_{22}= {W_{\rm ex}}$, where ${W_{\rm ex}}$ is the exchange energy. Eq.  gives $$\label{eq:Rdot_static} \dot{R}_1 = {u}\frac{1+\alpha{\beta}}{1+\alpha^2} = {u}+ \frac{\alpha}{\bar{d}}\dot{R}_2\,, \qquad \dot{R}_2 = {u}\frac{({\beta}-\alpha)\, \bar{d}}{1+\alpha^2}$$ where we denoted $\bar{d}=d_{11}/(4\pi{Q})=d_{22}/(4\pi{Q})$. Thus, the skyrmion will initially have a velocity component in the direction of the current flow while its velocity component perpendicular to it depends on the sign of ${\beta}-\alpha$. If we now assume that the skyrmion will eventually reach a propagating steady state then its velocity will satisfy the virial relations . For ${Q}\neq 0$ we define $\bar{d}_{\mu\nu} = d_{\mu\nu}/(4\pi{Q})$ and write the virial relations as $$\label{eq:virial_velocity_topological} \begin{split} & ( 1 - \alpha \bar{d}_{12} ) v_1 - \alpha \bar{d}_{22} v_2 = (1 - {\beta}\bar{d}_{12}){u}\\ & \alpha \bar{d}_{11} v_1 + (1 + \alpha \bar{d}_{12} ) v_2 = {\beta}{u}\, \bar{d}_{11}. \end{split}$$ These imply that, in general, both components of the velocity are nonzero, $v_1, v_2 \neq 0$, therefore a propagating skyrmion will move at an angle to the flow of the spin current. Unlike in the case of a skyrmionium, dissipation is not necessary in order to obtain a steady-state in the case of the skyrmion and Eqs.  give for $\alpha=0$ the velocity $$v_1={u}- {\beta}{u}\,\bar{d}_{12},\qquad v_2= {\beta}{u}\,\bar{d}_{11}.$$ In order to obtain more detailed information on the propagating skyrmion configuration we study the case ${\beta}=\alpha$ since it emerges again as a special case as seen in Eq. . In a steady state Eq.  gives for the skyrmion a velocity $(v_1,v_2)=({u},0)$ colinear with the spin current. When we substitute this in Eq.  we obtain the static LL equation $\bm{m}\times\bm{f}=0$. Thus the argument employed in Sec. \[sec:traveling\_skyrmionium\] for a skyrmionium also applies for a skyrmion: a static skyrmion solution of the conservative LL Eq. , which we denote $\bm{n}(x_1,x_2)$, is a traveling solution of the full equation with $\bm{m}(x_1,x_2,t)=\bm{n}(\xi_1,x_2)$ and $\xi_1=x_1-{u}t$. It is well-known that there are no traveling skyrmion solutions of Eq. , i.e., there are no ${Q}\neq 0$ skyrmion solutions of Eq. , and this can be rigorously established [@PapanicolaouTomaras_NPB1991]. Therefore, the arguments about traveling skyrmionium solutions following Eq.  cannot be applied in the case of skyrmions. For ${\beta}\neq \alpha$ a traveling skyrmion should have $v_2 \neq 0$ as shown by Eq. , that is, the skyrmion travels at an angle with respect to the direction of the flow of the spin current. In the case ${\beta}\neq \alpha$ we could not find exact traveling wave solutions of Eq. , for the effective field . Exact results are though indeed obtained for the pure exchange model in Appendix \[sec:exchange\_model\]. For small deviations from the simple case ${\beta}=\alpha$ and $(v_1,v_2)=({u},0)$ we may assume that the skyrmion retains approximately axial symmetry, and thus we have a diagonal $\bar{d}_{\mu\nu}=\bar{d}\,\delta_{\mu\nu}$ where $\bar{d}$ is a constant. Eqs.  have now a relatively simple solution: $$\label{eq:virial_velocity_topological_simple} v_1 = u\frac{1 + \alpha{\beta}\bar{d}^2}{1 + (\alpha \bar{d})^2} = {u}+ \alpha \bar{d}\,v_2, \quad v_2 = u\frac{({\beta}-\alpha)\,\bar{d}}{1 + (\alpha \bar{d})^2}.$$ For the pure exchange model $\bar{d}=1$ (see Appendix \[sec:exchange\_model\]) while for other models such as in Eq.  we have $\bar{d}>1$ calculated by substituting the static skyrmion solution of Eq.  in Eq. . Eq.  gives the so-called mobility relation, i.e., a linear relation between the velocity and the current. ![Simulation results for the velocity components $(v_1, v_2)=\dot{R}_1, \dot{R}_2)$ as a function of time $t$ for a skyrmion under spin torque with parameter values .[]{data-label="fig:Q1_velocity"}](Q1_velocity){width="6truecm"} In order to check the theoretical predictions we have conducted a numerical simulation using the parameter set $$\label{eq:parameter_set1b} \alpha = 0.2,\qquad {u}= 0.1,\qquad {\beta}= 0.5$$ with large values for $\alpha,{\beta}$ and ${\beta}-\alpha$. We use as an initial condition the skyrmion of Fig. \[fig:skyrmion\_vecs\] with ${\kappa}=3$ and apply the spin current. The numerically found velocity for the skyrmion is shown in Fig. \[fig:Q1\_velocity\]. We initially have $\dot{R}_1 = 0.106,\; \dot{R}_2 = 0.044$; the velocity presents oscillations and eventually converges to constant values $(v_1, v_2) = (0.1140, 0.0436)$. The expected initial velocity is found from Eqs.  and it is $\dot{R}_1 = 0.106,\; \dot{R}_2 = 0.046$, where we have used the numerically calculated value for the exchange energy ${W_{\rm ex}}=20.08 \Rightarrow \bar{d} = 1.598$. The final velocity is in excellent agreement with the velocity at a steady state predicted by Eq.  which gives $(v_1, v_2) = (0.1139, 0.0435)$, where we assume that the initial skyrmion profile is not significantly changed. The skyrmion configuration is indeed not visibly distorted during the simulation compared to the initial axially symmetric skyrmion. We have also conducted a numerical simulation using the parameter set and the results are again in excellent agreement with the predictions of Eqs.  and . We finally note that the skyrmion is pinned at its final position when the spin current is switched off. This is contrasted with the dynamics of a skyrmionium which travels at a constant velocity also in the absence of external forces (and damping). Concluding remarks {#sec:conclusions} ================== We have studied the dynamics under spin torque of non-topological $({Q}= 0)$ and topological $({Q}\neq 0)$ skyrmions in films of Dzyaloshinskii-Moriya materials with easy-axis anisotropy. Analytical results are obtained using the equations for the linear momentum (for ${Q}= 0$) or the guiding centre (for ${Q}\neq 0$), and virial relations. The study is complemented by a set of numerical simulations. Furthermore, we obtain exact solutions of the Landau-Lifshitz equation including spin torques, given in Eq. , for some particular cases. The traveling solutions obtained provide arguments that a spin polarized current will cause rigid motion of a skyrmion or a skyrmionium. This result is applicable not only for the solitons studied in the present work but also for other models, e.g., for driven motion of magnetic bubbles. The ${Q}=0$ skyrmionium is accelerated by the spin torque and it continues moving after switching off the current. This Newtonian dynamics was also observed in the case of the skyrmionium under an external field gradient [@KomineasPapanicolaou_PRB2015]. It is dramatically different than the more well-studied dynamics of a skyrmion [@FertCros_NatNano2013; @IwasakiMochizuki_NatComms2013; @IwasakiMochizuki_NatNano2013] which is spontaneously pinned in the absence of external torques. On the other hand, the skyrmionium motion presents some similarities with the skyrmion motion in a stripe geometry [@IwasakiMochizuki_NatNano2013]. Our methods can be also applied to simpler one-dimensional models (wires) provided static domain wall solutions of Eq.  exist. The pure exchange model {#sec:exchange_model} ======================= If we set $\bm{f}=\Delta\bm{m}$ we have the so-called pure exchange model and we will present analytic results which elucidate the discussion in Sec. \[sec:traveling\_skyrmion\]. In the pure exchange model the Bogomolnyi relations [@BelavinPolyakov_JETP1975; @Rajaraman] $$\label{eq:Bogomolnyi_BP} {\partial}_1\bm{m} = \bm{m}\times{\partial}_2\bm{m},\qquad {\partial}_2\bm{m} = -\bm{m}\times{\partial}_1\bm{m},$$ which contain only first order derivatives, are sufficient for obtaining static solutions of Eq. , i.e., solutions for $\bm{m}\times\bm{f}=0$. A large class of ${Q}\neq 0$ skyrmion solutions can be found by solving . Of those, the axially symmetric ${Q}= 1$ skyrmion configuration of the form will be denoted $\bm{m}=\bm{n}(x_1,x_2)$ and reads $$\label{eq:axially_symmetric_Q1} n_1 = -\frac{2a x_2}{\rho^2+a^2},\quad n_2 = \frac{2a x_1}{\rho^2+a^2},\quad n_3 = \frac{\rho^2-a^2}{\rho^2+a^2},$$ where $a$ is a arbitrary positive constant giving the skyrmion radius and $\rho^2=x_1^2+x_2^2$. Configuration is similar in its gross features to that shown in Fig. \[fig:skyrmion\_vecs\]. We turn to Eq.  for a traveling steady state with velocity $(v_1, v_2)$ and we require $$\label{eq:velocity_topological_bogomolny} \begin{cases} & {u}- v_1 = -\alpha v_2 \\ & v_2 = {\beta}{u}- \alpha v_1 \end{cases} \Rightarrow \begin{cases} v_1 & = \frac{1+\alpha{\beta}}{1+\alpha^2}\, {u}\\ v_2 & = \frac{{\beta}-\alpha}{1+\alpha^2}\, {u}. \end{cases}$$ Then, Eq.  simplifies to $$\label{eq:llg_stt_ip_steadystate_exchangemodel} \left( \alpha v_2\,{\partial}_1 + v_2 {\partial}_2 \right)\bm{m} = \bm{m}\times\bm{f} + \bm{m}\times \left( \alpha v_2{\partial}_2 - v_2\,{\partial}_1 \right) \bm{m}.$$ Under the Bogomolnyi relations the terms on the left-hand-side cancel with the last terms on the right-hand-side which originate in the damping and the non-adiabatic spin torque. Since the same Bogomolnyi relations are sufficient for the vanishing of the first term on the right-hand-side, we conclude that relations are sufficient conditions for solutions of Eq. . Therefore, any solution $\bm{m}=\bm{n}(x_1,x_2)$ of Eqs. , such as the skyrmion in Eq. , is a traveling solution $\bm{m}(x_1,x_2,t)=\bm{n}(\xi_1,\xi_2)$ of the full Eq.  with spin current and damping, with a velocity given by Eq. . Regarding calculations presented in Sec. \[sec:traveling\_skyrmion\] it is useful to note that skyrmion solutions with ${Q}\neq 0$ which satisfy have exchange energy ${W_{\rm ex}}=4\pi{Q}$ [@Rajaraman]. Thus, for axially symmetric skyrmions, such as , the tensor $d_{\mu\nu}$, defined in Eq. , is diagonal with $\bar{d}_{\mu\nu} = \delta_{\mu\nu}$. We then see that the velocity in Eq.  coincides with that given in Eq.  applied for $\bar{d}=1$. [10]{} A. N. Bogdanov and D. A. Yablonskii, Zh. Eksp. Teor. Fiz. [**95**]{}, 178 (1989). A. N. Bogdanov and A. Hubert, JMMM [**138**]{}, 255 (1994). N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Science [**341**]{}, 636 (2013). N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, and R. Wiesendanger, Phys. Rev. Lett. [**114**]{}, 177203 (2015). S. Mühlbauer, F. Binz, B. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Science [**323**]{}, 915 (2009). X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature [**465**]{}, 901 (2010). X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Isiwata, Y. Matsui, and Y. Tokura, Nature Materials [**10**]{}, 106 (2011). A. N. Bogdanov and A. Hubert, JMMM [**195**]{}, 182 (1999). A. Leonov, U. K. Rößler, and M. Mostovoy, EPJ Web of Conferences [ **75**]{}, 05002 (2013). D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, , and R. P. Cowburn, Science [**309**]{}, 1688 (2005). K. Everschor, M. Garst, B. Binz, F. Jonietz, S. Mühlbauer, C. Pfleiderer, and A. Rosch, Phys. Rev. B [**86**]{}, 054432 (2012). A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnology [**8**]{}, 152 (2013). J. Sampaio, C. Vincent, S. Rohart, A. Thiaville, and A. Fert, Nature Nanotechnology [**8**]{}, 839 (2013). N. Nagaosa and Y. Tokura, Nature Nanotechnology [**8**]{}, 899 (2013). J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature Communications [**4**]{}, 1463 (2013). J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature Nanotechnology [**8**]{}, 742 (2013). C. Schütte, J. Iwasaki, A. Rosch, and N. Nagaosa, Phys. Rev. B [**90**]{}, 174434 (2014). R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G. Finocchio, Sci. Rep. [**4**]{}, 6784 (2014). S. Komineas and N. Papanicolaou, Phys. Rev. B [**92**]{}, 064412 (2015). M. Beg, D. Chernyshenko, M.-A. Bisotti, W. Wang, M. Albert, R. L. Stamps, and H. Fangohr, arXiv:1312.7665 (2014). N. S. Kiselev, A. N. Bogdanov, R. Schäfer, and U. K. Rößler, J. Phys. D [**44**]{}, 392001 (2011). J. C. Slonczewski, J. Magn. Magn. Mat. [**159**]{}, L1 (1996). L. Berger, Phys. Rev. B [**54**]{}, 9353 (1996). S. Zhang and Z. Li, Phys. Rev. Lett. [**93**]{}, 127204 (2004). M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A. Itoh, L. Duò, A. Kirilyuk, T. Rasing, and M. Ezawa, Phys. Rev. Lett. [**110**]{}, 177205 (2013). N. Papanicolaou and T. N. Tomaras, Nucl. Phys. B [**360**]{}, 425 (1991). S. Komineas and N. Papanicolaou, Physica D [**99**]{}, 81 (1996). J. He, Z. Li, and S. Zhang, Phys. Rev. B [**73**]{}, 184408 (2006). K. Everschor, M. Garst, R. A. Duine, and A. Rosch, Phys. Rev. B [**84**]{}, 064401 (2011). A. A. Belavin and A. M. Polyakov, JETP Lett. [**22**]{}, 245 (1975). R. Rajaraman, [*Solitons and Instantons*]{} (North Holland, Amsterdam, 1982).

Events containing two photons plus large missing transverse energy ($\gamma\gamma \not \!\! E_T$) represent an important signature for some classes of supersymmetric models [@xer]. Models that predict the existence of light neutralinos [@lit:neu] can give rise to this kind of event when the next to lightest neutralino decays $\tilde{\chi}_2^0 \to \tilde{\chi}_1^0 \gamma$, where $\tilde{\chi}_1^0$ is the lightest supersymmetric particle (LSP). When a light gravitino is present [@lit:gra], like in models with gauge–mediated low energy supersymmetry breaking [@gau:med], the lightest neutralino is unstable and decays via $\tilde{\chi}_1^0 \to \tilde{G} \gamma$, which also yield an event topology with two photons together with missing energy, since the gravitino ($\tilde{G}$) escapes undetected. DØ Collaboration have reported a recent search for diphoton events with large missing transverse energy in $p\bar{p}$ collisions at $\sqrt{s} = 1.8$ TeV [@prl:old; @prl:new; @eno]. Their analysis indicates a good agreement with the expectations from the Standard Model (SM). In this way, DØ Collaboration were able to set limits on the production cross section $\sigma(p\bar{p} \to \gamma\gamma \not \!\! E_T + X)$, and consequently, to establish an exclusion region in the supersymmetry parameter space and lower bounds on the masses of the lightest chargino and neutralino. In this work, we point out that the experimental search for $\gamma\gamma \not \!\! E_T$ events is also able to constraint new physics in the bosonic sector of the SM. For instance, associated Higgs–$Z$ boson production, with the subsequent decay of the Higgs into two photons and the $Z$ going to neutrinos, can yield this signature. In the SM, the decay width $H \to \gamma \gamma$ is very small since it occurs just at one–loop level [@h:gg]. However, the existence of new interactions can enhance this width in a significant way. We can describe the deviations of the SM predictions for the couplings in the bosonic sector via effective Lagrangians [@classical; @linear; @drghm; @hisz]. The new couplings among light states are described by anomalous effective operators representing residual interactions, after the heavy degrees of freedom are integrated out. A complete set of eleven $C$ and $P$ conserving and $SU_L(2) \times U_Y(1)$ invariant operators can be found in Refs. [@linear; @drghm; @hisz]. The dimension–6 operators that alter the $HVV$ couplings, like $HWW$, $HZZ$, $H\gamma\gamma$ and $HZ\gamma$, can be written in terms of the Higgs doublet ($\Phi$) as $$\begin{aligned} {\cal L}_{\text{eff}} &=& f_{WW} \Phi^{\dagger} \hat{W}_{\mu \nu} \hat{W}^{\mu \nu} \Phi + f_{BB} \Phi^{\dagger} \hat{B}_{\mu \nu} \hat{B}^{\mu \nu} \Phi \\ \label{lag} &+& f_W (D_{\mu} \Phi)^{\dagger} \hat{W}^{\mu \nu} (D_{\nu} \Phi) + f_B (D_{\mu} \Phi)^{\dagger} \hat{B}^{\mu \nu} (D_{\nu} \Phi) \nonumber\end{aligned}$$ where $\hat{B}_{\mu\nu} = i (g'/2) B_{\mu \nu}$, and $\hat{W}_{\mu \nu} = i (g/2) \sigma^a W^a_{\mu \nu}$, with $B_{\mu \nu}$ and $ W^a_{\mu \nu}$ being the field strength tensors of the $U(1)$ and $SU(2)$ gauge fields respectively. Other possible operators like $\Phi^{\dagger}\hat{B}_{\mu \nu}\hat{W}^{\mu \nu}\Phi$ (not “blind” operators) contribute to gauge–boson two–point functions at tree level and are strongly constrained. The first two operators appearing in Eq.(\[lag\]) do not modify the $WW\gamma$ and $WWZ$ tree–point couplings, while the operators ${\cal O}_{W}$ and ${\cal O}_{B}$ generate both Higgs–vector boson and self–vector–bosons anomalous couplings. Therefore, the linearly realized effective Lagrangians relate the modifications in the Higgs couplings to those in the vector boson vertex [@linear; @drghm; @hisz; @hsz]. It is important to notice that the coefficient of the operators ${\cal O}_{WW}$ and ${\cal O}_{BB}$ cannot be constrained by the $W^+W^-$ production at LEP2, since they do not generate anomalous triple gauge boson couplings. They can only be studied in processes involving the Higgs boson in electron–positron [@hsz; @epem; @ggg:bbg] or hadronic collisions [@prl]. We examine here the production of anomalously coupled Higgs boson at Fermilab Tevatron $p\bar{p}$ collider. In particular, we concentrate on the signature $\gamma\gamma \not \!\! E_T$ which can originate from the reactions, $$\begin{aligned} p \bar{p} & \to & Z (\to \nu \bar{\nu}) + H (\to \gamma \gamma) + X \nonumber \\ p \bar{p} & \to & W (\to \ell \nu) + H (\to \gamma \gamma) + X \label{zw} \end{aligned}$$ where in the latter case the charged lepton ($\ell = e, \mu$) escapes undetected. We have computed the cross sections (\[zw\]) taking into account all electroweak subprocess $q \bar{q}^\prime \to \nu \bar{\nu} (\ell \nu) \gamma \gamma$, with $\ell = e, \mu$. The anomalous contributions coming from the Lagrangian (\[lag\]) and the interference with the SM diagrams were consistently included via modified Helas [@helas] subroutines. For the proton structure functions, we have employed the MRS (G) set [@mrs] at the scale $Q^2 = \hat{s}$. In order to compare our predictions with the data collected by the DØ Collaboration, we have applied the same cuts of Ref.[@prl:new]. We required that one photon has transverse energy $E_T^{\gamma_1} > 20$ GeV and the other $E_T^{\gamma_2} > 12$ GeV, each of them with pseudorapidity in the range $|\eta^\gamma| < 1.2$ or $1.5 < |\eta^\gamma| < 2.0$. We further required that $\not \!\!E_T > 25$ GeV. For the $\ell \nu \gamma \gamma$ final state, we imposed that the charged lepton is outside the covered region of the electromagnetic calorimeter and it escapes undetected ($|\eta_{e}|> 2$ or $1.1<|\eta_e|<1.5 $, $ |\eta_{\mu}|> 1$). After these cuts we find that 80% to 90% of the signal comes from associated Higgs–$Z$ production while 10% to 20% arrises from Higgs–$W$. We also include in our analysis the particle identification and trigger efficiencies which vary from 40% to 70% per photon [@fermilab]. We estimate the total effect of these efficiencies to be 35%. The main sources of background to this reaction [@prl:new] arise from SM processes containing multijets, direct photon, $W + \gamma$, $W + j$, $Z \to ee$ and $Z \to \tau\tau \to ee$ where photons are misidentified and/or the missing energy is mismeasured. The DØ Collaboration estimate the contribution of all these backgrounds to yield $2.3 \pm 0.9$ events. DØ Collaboration has observed 2 events that have passed the above cuts in their data sample of $106.3 \pm 5.6$ pb$^{-1}$. The invariant mass of the photon pair in these events are $50.4$, and $264.3$ GeV [@eno]. In our analysis, we search for Higgs boson with mass in the range $70 < M_H \lesssim 2 M_W$, since after the $W^+W^-$ threshold is reached the diphoton branching ratio of Higgs is quite reduced. Since no event with two–photon invariant mass in the range $70 < M_{\gamma\gamma} \lesssim 2 M_W$ were observed, a $95\%$ CL in the determination of the anomalous coefficient $f_i$, $i=WW, BB, W, B$ of Eq. (\[lag\]) is attained requiring 3 events coming only from the anomalous contributions. In Fig. \[fig:1\], we present the exclusion region in the $f_{WW}\times f_{BB}$ plane, when we assume that just these two coefficients are different from zero. The clear (dark) shadow represents the excluded region, at $95\%$ CL, for $M_H = 80 \, (140)$ GeV. We have used and integrated luminosity of 100 pb$^{-1}$ . Since the anomalous contribution to $H \to \gamma\gamma$ width becomes zero for $f_{WW} = - f_{BB}$ a very loose bound is obtained near this axis. We should also notice that the reactions (\[zw\]) are more sensible to $f_{WW}$, while the dependence on $f_{BB}$ is very weak. In Fig.\[fig:2\], we show the $f_{WW}$ values that can be excluded as a function of the Higgs boson mass at $64$ % ($95\%$) CL. When we assume that all the coefficients of the Lagrangian (\[lag\]) have the same magnitude the $H \to \gamma\gamma$ coupling becomes related to the triple vector boson coupling, $WW\gamma$. Therefore, the limits obtained from Higgs production, with the subsequent decay into two photons, is able to generate an indirect bound on $\Delta \kappa_\gamma$ [@linear; @drghm; @hisz; @hsz; @prl]. In Fig. \[fig:3\], we compare our indirect limit on $\Delta \kappa_\gamma$ with the experimental limit of DØ Collaboration from gauge boson pair production [@fermilab] for $f \equiv f_{WW} = f_{BB} = f_{W} = f_{B}$ (light shadow) and $f \equiv f_{WW} = f_{BB} = -f_{W} = -f_{B}$ (dark shadow). We also display the expected bounds at the upgraded Tevatron (Run II) and at TeV33, assuming 1 and 10 fb$^{-1}$ of integrated luminosity, respectively [@tevatron], and the limit that will be possible to extract from LEP II, operating at 190 GeV with an integrated luminosity of 500 fb$^{-1}$ [@lep]. We can see that, for $M_H \lesssim 170 \, (140)$ GeV, the limit that can be established at 95% CL from our analysis based on the present Tevatron luminosity is tighter than the present limit coming from gauge boson production. If the result from the recent global fit to LEP, SLD, $p\bar{p}$, and low energy data that favors a Higgs boson with mass $M_H = 127 \stackrel{+127}{\scriptstyle{-72}}$ GeV [@ewwg] is not substantially modified by the presence of the new operators, our indirect limit on $\Delta\kappa_\gamma$ applies for the most favoured Higgs masses. In conclusion, we have shown how to extract important information on anomalous Higgs boson coupling from the analysis of $\gamma\gamma \not \!\! E_T$ events in $p\bar{p}$ collisions. In particular, we were able to establish limits on the coefficients of general effective operators that give rise to the coupling $H\gamma\gamma$. Since linearly realized effective Lagrangians relate the modifications in the Higgs couplings to the ones involving vector boson self–interaction, one can extract indirect limits on the anomalous $WW\gamma$ coupling that are competitive with the bounds from direct searches in gauge boson production at present and future collider experiments. The authors would like to thank O. J. P. Éboli for suggesting this work. We also want to thank Sarah Eno for providing us with information on the DØ data on two photons plus missing transverse energy data. M. C. G–G is grateful to the Instituto de Física Teórica for its kind hospitality. This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), by DGICYT under grant PB95-1077, by CICYT under grant AEN96–1718, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). For a recent review on the phenomenological aspects of supersymmetry see for instance: X. Tata, in the Proceedings of the IX Jorge André Swieca Summer School: Particles and Fields, São Paulo, Brazil, report UH-511-872-97, and hep-ph/9706307. S. Ambrosanio [*et al.*]{}, Phys. Rev.Lett.  [**76**]{}, 3498 (1996); [*idem*]{} Phys. Rev. D[**55**]{} 1372 (1997). S. Ambrosanio [*et al.*]{}, Phys. Rev. D[**54**]{}, 5395 (1996); J. Kim [*et al.*]{}, ACT-11-97, and hep-ph/9707331 P. Fayet, Phys. Lett. [**B70**]{}, 461 (1977); S. Dimopoulos [*et al.*]{}, Phys. Rev. Lett.  [**76**]{}, 3494 (1996); H. Baer, M. Brhlik, C. Chen, and X.  Tata, Phys.Rev. D[**55**]{} 4463 (1997). S. Abachi [*et al.*]{}, DØ Collaboration, Phys. Rev. Lett. [**78**]{}, 2070 (1997). B. Abbott [*et al.*]{}, DØ Collaboration, FERMILAB-PUB-97-273-E, and hep-ex/9708005. See also the DØ Collaboration public Web page: `http://www-d0.fnal.gov/public/new/analyses/gauge` `/welcome.html`. J. Ellis, M. K. Gaillard, D. V. Nanopoulos, Nucl.  Phys. [**B106**]{}, 292 (1976); M. A. Shifman, A. I.Vainshtein, M.  B. Voloshin, V. I. Zakharov, Sov. J. Nucl.Phys. [**30**]{}, 711 (1979). K. Hagiwara, H. Hikasa, R. D. Peccei and D. Zeppenfeld, Nucl. Phys. [**B282**]{}, 253 (1987). C. J. C. Burguess and H. J. Schnitzer, Nucl. Phys. [**B228**]{}, 464 (1983); C. N. Leung, S. T. Love and S. Rao, Z. Phys. [**31**]{}, 433 (1986); W. Buchmüller and D. Wyler, Nucl. Phys. [**B268**]{}, 621 (1986). A. De Rujula, M. B. Gavela, P. Hernandez and E. Masso, Nucl. Phys. [**B384**]{}, 3 (1992); A. De Rujula, M. B. Gavela, O. Pene and F. J. Vegas, Nucl. Phys. [**B357**]{}, 311 (1991). K. Hagiwara, S. Ishihara, R. Szalapski and D.Zeppenfeld, Phys. Lett. [**B283**]{}, 353 (1992); [*idem*]{}, Phys. Rev. [**D48**]{}, 2182 (1993); K. Hagiwara, T. Hatsukano, S. Ishihara and R. Szalapski, Nucl. Phys. [**B496**]{}, 66 (1997). K. Hagiwara, R. Szalapski and D. Zeppenfeld, Phys. Lett. [**B318**]{}, 155 (1993). K. Hagiwara, and M. L. Stong, Z. Phys. [**C62**]{}, 99 (1994); B. Grzadkowski, and J. Wudka, Phys. Lett., 49 (1995); G. J. Gounaris, J. Layssac and F. M.Renard, Z. Phys. [**C65**]{}, 245 (1995); G. J. Gounaris, F.M. Renard and N. D. Vlachos, Nucl. Phys. [**B459**]{}, 51 (1996). S. M. Lietti, S. F. Novaes and R.Rosenfeld, Phys. Rev. [**D54**]{}, 3266 (1996); F. de Campos, S. M. Lietti, S. F. Novaes and R. Rosenfeld, Phys. Lett., 93 (1996); [*idem*]{}, Phys. Rev. [**D56**]{}, 4384 (1997); S. M. Lietti and S. F. Novaes, hep–ph/9708443, Phys. Lett. (in press). F. de Campos, M. C. Gonzalez–Garcia and S. F.Novaes, Phys. Rev. Lett. (in press). H. Murayama, I. Watanabe and K. Hagiwara, KEK report 91-11 (unpublished). A. D. Martin, W. J. Stirling, R. G. Roberts Phys. Lett. [**B354**]{}, 155 (1995). B. Abbott [*et al.*]{}, DØ Collaboration, Phys.  Rev. Lett. [**79**]{}, 1441 (1997). D. Amidei [*et al.*]{}, [*Future Electroweak Physics at the Fermilab Tevatron: Report of the TeV–2000 Study Group*]{}, preprint FERMILAB-PUB-96-082 (1996). T. Barklow [*et al.*]{}, Summary of the Snowmass Subgroup on Anomalous Gauge Boson Couplings, to appear in the [*Proceedings of the 1996 DPF/DPB Summer Study on New Directions in High-Energy Physics*]{}, June 25 — July 12 (1996), Snowmass, CO, USA, and hep-ph/9611454. The LEP Collaborations ALEPH, DELPHI, L3, OPAL, and the LEP Electroweak Working Group, contributions to the 28th International Conference on High-energy Physics (ICHEP 96), Warsaw, Poland, 1996, report CERN-PPE/96-183 (1996); [*idem*]{}, contributions to the 1997 Winter Conferences, LEPEWWG/97–01 (1997).

--- abstract: 'We study the stability region of the topological superfluid phase in a trapped two-dimensional polarized Fermi gas with spin-orbit coupling and across a BCS-BEC crossover. Due to the competition between polarization, pairing interaction and spin-orbit coupling, the Fermi gas typically phase separates in the trap. Employing a mean field approach that guarantees the ground state solution, we systematically study the structure of the phase separation and investigate in detail the optimal parameter region for the preparation of the topologically non-trivial superfluid phase. We then calculate the momentum space density distribution of the topological superfluid state and demonstrate that the existence of the phase leaves a unique signature in the trap integrated momentum space density distribution which can survive the time-of-flight imaging process.' author: - Jing Zhou - Wei Zhang - Wei Yi title: 'Topological superfluid in a trapped two-dimensional polarized Fermi gas with spin-orbit coupling' --- Introduction ============ The study of non-Abelian topological order has attracted a great amount of interest recently, due to the potential applications in fault-tolerant quantum computation [@tqc1; @tqc2]. In addition to systems with intrinsic chiral $p$-wave pairing order, e.g. fractional quantum hall systems [@fqh0; @fqh1; @fqh2], chiral $p$-wave superconductors [@pwavesc; @rad1], $p$-wave superfluidity in ultracold fermions [@sade1; @pwavesf1; @pwavesf2; @rad2] etc., it has been shown that a topologically non-trivial superfluid phase that supports non-Abelian excitations can be induced from an underlying $s$-wave superfluidity. An example is the semiconductor/superconductor heterostructures with spin-orbit coupling (SOC), $s$-wave pairing superfluidity and an external Zeeman field [@tsfsolid1; @tsfsolid2; @tsfsolid3; @tsfsolid4; @soc1]. With the rapidly developing toolbox available for the quantum control of ultracold atomic systems, the elements above can now be implemented experimentally in ultracold Fermi gases. Importantly, the spin-orbit coupling in ultracold Bose gases has been made possible by the recent experimental achievement of synthetic gauge field in ultracold atoms and has generated considerable amount of theoretical interests [@gauge1; @gauge2; @wucongjun; @zhaibec]. On the other hand, the pairing superfluidity in an ultracold Fermi gas and the quantum phase transition in a polarized Fermi gas have been investigated extensively during the past decade [@expf1; @expf2; @expf3; @stoofreview; @creview; @pmit; @price; @preview]. With clean environment and highly tunable parameters, ultracold Fermi gases may serve as an ideal platform for the observation of topological superfluidity and for the study of the interesting physics therein. In particular, it has been suggested that the Majorana zero modes, which have eluded experimental observation for decades, may be detected in an ultracold atomic system with s-wave interactions [@maj; @pwave; @zhang; @sato]. Spin-orbit coupled Fermi gas has been under intensive theoretical study recently [@zhang; @lewenstein; @soc2; @soc3; @chuanwei; @soc4; @soc6; @iskin; @melo; @thermo; @2d1; @2d2; @sade2; @2d3; @2dbkt; @puhan]. For an unpolarized Fermi gas near a wide Feshbach resonance, the SOC has been found to result in a BCS-BEC type crossover even on the BCS side of the resonance [@soc3; @soc4]. Furthermore, it has been suggested that the topological superfluid (TSF) phase can be stabilized in a spin polarized Fermi gas in the presence of SOC [@zhang; @sato; @lewenstein; @chuanwei; @iskin; @thermo]. For a polarized Fermi gas without SOC, phase separation takes place near a wide Feshbach resonance in a uniform gas, due to the competition between population imbalance and pairing interactions [@preview; @trapwy; @sade3; @parish]. With the introduction of SOC, the phase separation develops a rich structure involving the topologically non-trivial superfluid state [@iskin; @thermo]. For a polarized Fermi gas in an external trap, which is always the case in experiments, the various phases naturally separate in real space, with different phases occurring at different places in the trapping potential [@expf1; @expf2; @trapwy]. The important questions here are whether the topological superfluid is stable in a trapping potential and what the detailed structure of the phases in a trap is. The phase structure involving the topologically non-trivial superfluid phase in a trapped three-dimensional (3D) polarized Fermi gas with SOC has been examined previously, where it has been found that two distinct types of topological superfluid may be stabilized [@iskin; @thermo]. In this work, we focus on the phase structure of a spin-orbit coupled two-dimensional (2D) polarized Fermi gas near a wide Feshbach resonance in a trapping potential. We study the system using a BCS-type mean field theory at zero temperature. Due to the existence of metastable or unstable solutions of the gap equation that are typical in the presence of population imbalance, we directly minimize the thermodynamic potential [@preview; @thermo]. In contrast to the 3D case where there are two distinct topologically non-trivial phases with either two or four gapless points in the quasi-particle excitation spectrum, in 2D we find that there is only one topologically non-trivial superfluid phase which is always protected by an excitation gap away from its phase boundary against the conventional superfluid state (SF). This agrees with the previous calculations [@tsfsolid1; @soc1]. As has been shown in Ref. [@tsfsolid1], when a vortex is created in this phase, a Majorana zero mode can be found at the center of it. Due to the competition between population imbalance and pairing, the various phases appear at different locations in an external trapping potential. We then map out the phase diagrams illustrating the structure of the phase separation in typical trapping potentials across a wide Feshbach resonance under the local density approximation (LDA). From our zero temperature mean field calculations, we investigate the parameter regions for the existence of a stable topological superfluid phase (TSF) in the trap. To characterize the properties of different phases, we calculate the momentum space density distribution for the various phases. Notably, the momentum space density distribution for the minority spin features a dip near the origin in the topological superfluid state, which is unique among the phases that we consider. We further demonstrate that for appropriate parameters such that the center of the trap is occupied by the TSF state, this signature dip of the momentum distribution can survive the process of time-of-flight imaging and thus may serve as an unambiguous signal for the TSF. Note that the aim of the current work is to provide a qualitatively correct general picture of the phase structure in a trap, as mean field theory is not quantitatively reliable near a wide Feshbach resonance in 2D due to large fluctuations. We have also not considered the possibility of pairing states with non-zero center of mass momentum in our mean field theory. The paper is organized as follows: in Sec. II, we write down the model Hamiltonian and outline the mean field theory that we adopt; in Sec. III, we analyze the minima of the thermodynamic potential and discuss the competition between the various phases; we then map out the phase diagram of a homogeneous system using the chemical potential and SOC strength as parameters in Sec. IV, which provides us with valuable information regarding the phase separation in a potential trap; in Sec. V, we derive a set of universal dimensionless number equations, with which we study the real space distribution of particle density and pairing gap in a trap; we investigate the momentum distribution of the phases in Sec. VI, and identify the signature of the TSF state in the momentum distribution; finally, we summarize in Sec. VII. Formalism ========= We first consider the model Hamiltonian for a uniform 2D polarized Fermi gas with Rashba spin-orbit coupling near a wide Feshbach resonance. The Hamiltonian can be expressed as a sum of three parts: the unperturbed Hamiltonian $H_0$, the SOC Hamiltonian $H_{\text{soc}}$ and the interaction Hamiltonian $H_{\text{int}}$ [@sato; @lewenstein; @chuanwei]: $$\begin{aligned} &H-\sum_{\sigma}\mu_{\sigma}N_{\sigma}=H_0+H_{\text{soc}}+H_{\text{int}}\nonumber\\ &=\sum_{\mathbf{k},\sigma}(\epsilon_{\mathbf{k}}-\mu_{\sigma})a^{\dag}_{\mathbf{k},\sigma}a_{\mathbf{k},\sigma} +\sum_{\mathbf{k}}\alpha k\left(e^{-i\varphi_{\mathbf{k}}}a^{\dag}_{\mathbf{k},\uparrow}a_{\mathbf{k},\downarrow} + {\rm H.C.} \right)\nonumber\\ &+\frac{U}{\cal V}\sum_{\mathbf{k},\mathbf{k}'}a^{\dag}_{\mathbf{k},\uparrow}a^{\dag}_{-\mathbf{k},\downarrow}a_{-\mathbf{k}',\downarrow}a_{\mathbf{k}',\uparrow}, \label{OrgH}\end{aligned}$$ where the kinetic energy $\epsilon_{\mathbf{k}}=\hbar^2k^2/(2m)$, $\mu_{\sigma}$ is the chemical potential for atoms with spin $\sigma=\{\uparrow,\downarrow\}$, $N_{\sigma}$ denotes the total number of particles with spin $\sigma$, $a_{\mathbf{k},\sigma}$($a^{\dag}_{\mathbf{k},\sigma}$) annihilates (creates) a fermion with momentum $\mathbf{k}$ and spin $\sigma$, ${\cal V}$ is the quantization area in 2D, and H.C. stands for Hermitian conjugate. The Rashba spin-orbit coupling strength $\alpha$ can be tuned via parameters of the gauge-field generating lasers [@gauge2], while $\varphi_{\mathbf{k}}=\arg{\left(k_x+ik_y\right)}$. In writing the interaction Hamiltonian $H_{\text{int}}$, we assume an $s$-wave contact interaction between the two fermion species, with the bare interaction rate $U$ renormalized following the standard relation in two dimensions [@2drenorm]: $$\frac{1}{U}=-\frac{1}{\cal V}\sum_{\mathbf{k}}\frac{1}{2\epsilon_{\mathbf{k}}+E_b}.$$ Here, $E_b > 0$ is the binding energy of the two-body bound state in two dimensions without SOC. By tuning through a Feshbach resonance from a high-field BCS side, $E_b$ increases from zero and becomes large in the low-field BEC limit. Therefore, we use the variation of $E_b$ to represent the BCS-BEC crossover in the following discussions. One should notice that the $E_b$ we use in this manuscript is not the binding energy of the two-body bound state in the presence of SOC. The non-interacting Hamiltonian $H_0+H_{\text{soc}}$ can be diagonalized in the helicity basis: $$\begin{aligned} a_{\mathbf{k},\uparrow}&=\frac{1}{\sqrt{2}}e^{-i\varphi_{\mathbf{k}}}\left(a_{\mathbf{k},+}+a_{\mathbf{k},-}\right),\\ a_{\mathbf{k},\downarrow}&=\frac{1}{\sqrt{2}}\left(a_{\mathbf{k},+}-a_{\mathbf{k},-}\right),\end{aligned}$$ where $a_{\mathbf{k},\pm}$ ($a^{\dag}_{\mathbf{k},\pm}$) are the annihilation (creation) operators for the dressed spin states with different helicities ($\pm$). Under this basis, the interaction Hamiltonian can be written as $$\begin{aligned} H_{\text{int}}&=\frac{U}{4}\sum_{\mathbf{k},\mathbf{k}'}e^{i\varphi_{\mathbf{k}}} \left(a^{\dag}_{\mathbf{k},+}a^{\dag}_{-\mathbf{k},+}-a^{\dag}_{\mathbf{k},-}a^{\dag}_{-\mathbf{k},-}\right)\nonumber\\ &\times e^{-i\varphi_{\mathbf{k}'}} \left(a_{-\mathbf{k}',+}a_{\mathbf{k}',+}-a_{-\mathbf{k}',-}a_{\mathbf{k}',-}\right).\end{aligned}$$ Taking the pairing mean field $$\begin{aligned} \Delta&=\frac{U}{2}\sum_{\mathbf{k}}\left\langle e^{-i\varphi_{\mathbf{k}}}\left(a_{-\mathbf{k},+}a_{\mathbf{k},+}-a_{-\mathbf{k},-}a_{\mathbf{k},-}\right)\right\rangle\nonumber\\ &=U\sum_{\mathbf{k}}\left\langle a_{-\mathbf{k},\downarrow}a_{\mathbf{k},\uparrow}\right\rangle,\end{aligned}$$ the mean field Hamiltonian becomes $$\begin{aligned} &H_m-\sum_{\sigma}\mu_{\sigma}N_{\sigma}=\sum_{\mathbf{k},\lambda=\pm}\xi_{\lambda}a^{\dag}_{\mathbf{k},\lambda}a_{\mathbf{k},\lambda} \nonumber\\ &+\sum_{\mathbf{k}}\left[\frac{\Delta^{\ast}}{2}e^{-i\varphi_{\mathbf{k}}}\left(a_{-\mathbf{k},+}a_{\mathbf{k},+}-a_{-\mathbf{k},-}a_{\mathbf{k},-}\right)+h.c.\right]\nonumber\\ &-\frac{h}{2}\sum_{\mathbf{k}}\left(a^{\dag}_{\mathbf{k},+}a_{\mathbf{k},-}+h.c.\right)-{\cal V}\frac{|\Delta|^2}{U}, \label{meanH}\end{aligned}$$ where we have defined the chemical potentials $\mu=(\mu_{\uparrow}+\mu_{\downarrow})/2$ and $h=\mu_{\uparrow}-\mu_{\downarrow}$; and $\xi_{\pm}=\xi_{\mathbf{k}}\pm\alpha k$ with $\xi_{\mathbf{k}}=\epsilon_{\mathbf{k}}-\mu$. The mean field Hamiltonian is quadratic and can be diagonalized in the helicity basis: $\left\{a_{\mathbf{k},+},a_{-\mathbf{k},+}^{\dag},a_{\mathbf{k},-},a_{-\mathbf{k},-}^{\dag}\right\}^T$: $$\begin{aligned} &H_m-\sum_{\sigma}\mu_{\sigma}N_{\sigma} = \sum_{\mathbf{k},\lambda=\pm} E_{\mathbf{k},\lambda} \alpha^{\dag}_{\mathbf{k},\lambda}\alpha_{\mathbf{k},\lambda} \nonumber\\ & +\frac{1}{2}\sum_{\mathbf{k},\lambda=\pm}(\xi_{\lambda}-E_{\mathbf{k},\lambda})-\frac{|\Delta|^2}{U}.\end{aligned}$$ Here, $\alpha_{\mathbf{k},\sigma}$ ($\alpha^{\dag}_{\mathbf{k},\sigma}$) is the annihilation (creation) operator for the quasi-particles. The quasi-particle excitation spectra take the form $$\label{disp} E_{\mathbf{k},\pm}=\sqrt{\xi_{\mathbf{k}}^2+\alpha^2k^2+|\Delta|^2+\frac{h^2}{4} \pm 2E_0},$$ where $E_0=\sqrt{(h^2/4+\alpha^2k^2)\xi_{\mathbf{k}}^2 + h^2 |\Delta|^2 / 4}$. From this dispersion relation, we see that for finite pairing gap and SOC strengths, the only possible gapless point in 2D lies in the $E_{\mathbf{k},-}$ branch at $k=0$. This takes place when $h/2=\sqrt{\mu^2+\Delta^2}$. As the chemical potential imbalance $h$ increases across this point, the excitation gap first vanishes and then opens up again, and the system enters a topologically non-trivial superfluid phase [@soc1; @chuanwei]. This is in contrast to the 3D case, where two distinct topologically non-trivial phases exist, with two or four gapless points in the quasi-particle excitation spectrum [@iskin; @thermo]. In this manuscript, we consider only the zero temperature case and write down the thermodynamic potential as $$\begin{aligned} \label{thermoeqn} \Omega&=-\left.\frac{1}{\beta}\ln\text{tr}\left[e^{-\beta(H_m-\sum_{\sigma}\mu_{\sigma} N_{\sigma})}\right]\right|_{T\rightarrow 0}\nonumber\\ &=\frac{1}{2}\sum_{\mathbf{k},\lambda=\pm}\left(\xi_{\lambda}-E_{\mathbf{k},\lambda}\right)-{\cal V}\frac{|\Delta|^2}{U},\end{aligned}$$ where $\beta=1/k_BT$ and $k_B$ is the Boltzmann constant. Considering the extrema condition of the thermodynamic potential $\partial\Omega / \partial \Delta=0$ and the number constraints $n_{\sigma}= (- 1/ {\cal V}) \partial \Omega / \partial \mu_{\sigma}$, we get the gap and the number equations, respectively: $$\begin{aligned} &\Delta\sum_{\mathbf{k}}\left[\frac{1}{4E_{\mathbf{k},+}}\left(1+\frac{h^2}{4E_0}\right) +\frac{1}{4E_{\mathbf{k},-}}\left(1-\frac{h^2}{4E_0}\right)\right]+\frac{\Delta}{U}=0, \label{gapeqn}\\ & n_{\sigma}=\frac{1}{\cal V} \sum_{\mathbf{k}}\left[1-\frac{\xi_{\mathbf{k}}+\delta_{\sigma}\frac{h}{2}}{2E_{\mathbf{k},+}} - \frac{\xi_{\mathbf{k}}+\delta_{\sigma}\frac{h}{2}}{2E_{\mathbf{k},-}} + \frac{\xi_{\mathbf{k}}\left(\frac{h^2}{4}+\alpha^2k^2\right) + \delta_{\sigma}\frac{h}{2}(\xi_{\mathbf{k}}^2 +\Delta^2)}{2E_0} \left(\frac{1}{E_{\mathbf{k},-}}-\frac{1}{E_{\mathbf{k},+}}\right)\right], \label{numbereqn}\end{aligned}$$ where $\delta_{\uparrow}=-\delta_{\downarrow}=-1$, and we have taken $\Delta$ to be real for simplicity. The ground state of the system at zero temperature is given by the global minimum of the thermodynamic potential in Eq. (\[thermoeqn\]) under the number constraints Eq. (\[numbereqn\]). For a uniform gas, one has to consider explicitly the possibility of phase separation and introduce a mixing coefficient in order to get the correct ground state. In an external trapping potential, the various phases naturally separate in real space [@preview; @parish; @thermo]. Next, we focus on the phase separation in the presence of an external trapping potential $V(\mathbf{r})$, due to its experimental relevance. Assuming the potential to be slowly varying and taking the local density approximation (LDA), we can write the chemical potentials at each spatial location $\mathbf{r}$ as: $\mu_{\uparrow}(\mathbf{r})=\mu_{\mathbf{r}} + h/2$, $\mu_{\downarrow}(\mathbf{r})=\mu_{\mathbf{r}} - h/2$, and $\mu_{\mathbf{r}}=\mu-V(\mathbf{r})$, where the chemical potential at trap center $\mu$ and the chemical potential imbalance $h$ are related to the total particle number $N=N_{\uparrow}+N_{\downarrow}$ and the polarization $P=\left(N_{\uparrow}-N_{\downarrow}\right)/N$. The total particle number for each spin species can be determined from a trap integration: $N_{\sigma}=\int d^{2}\mathbf{r} n_{\sigma}(\mathbf{r})$, where the local density $n_{\sigma}(\mathbf{r})$ can be calculated from Eq. (\[numbereqn\]) with $\mu$ replaced by $\mu_{\mathbf{r}}$, and with the local pairing order parameter $\Delta(\mathbf{r})$ determined from the global minimum of the thermodynamic potential at each spatial location $\mathbf{r}$. Without loss of generality, we assume $N_{\uparrow}>N_{\downarrow}$ throughout this work such that $h$ and $P$ are both positive. ![(Color online) Typical shapes of thermodynamic potential as a function of the pairing order parameter $\Delta$ with various SOC strength $\alpha k_{h}/h$ (left column) and chemical potential $\mu/h$ (right column). The two-body binding energy is chosen as $E_b/h = 0.5$. The parameters of the subplots are: (a)$\mu/h=0.2,\alpha k_{h}/h=0.1$, (b)$\mu/h=0.2, \alpha k_{h}/h=0.35$, (c)$\mu/h=0.2, \alpha k_{h}/h=0.6$, (d)$\mu/h=0.2, \alpha k_{h}/h=0.8$, (e)$\mu/h=0.4, \alpha k_{h}/h=0.3$, (f)$\mu/h=0.24, \alpha k_{h}/h=0.3$, (g)$\mu/h=0.1, \alpha k_{h}/h=0.3$, (h)$\mu/h=-0.2, \alpha k_{h}/h=0.3$. The chemical potential difference $h$ is taken to be the energy unit, while the unit of momentum $k_h$ is defined as $\hbar^2 k_h^2/(2m)=h$.[]{data-label="Thermo_po"}](new4_thermo){width="8.5cm"} Thermodynamic potential in the presence of SOC ============================================== ![(Color online) Dependence of the pairing gap on SOC strength. (a) Evolution of the gap equation with increasing SOC strengths. Form the upmost curve to the lowest: $\alpha k_h/h=0.8, 0.5, 0.3, 0.2$, respectively. (inset) Enlarged view. $G$ is related to the derivative of the thermodynamic potential with respect to $\Delta$, as defined in the text. (b) Scaling of the SOC-induced pairing order with SOC strength. (inset) Semi-logarithm plot of the scaling relation. For both plots, $E_b/h=0.5$, $\mu/h=0.1$.[]{data-label="gapillustrate"}](addfigure.pdf){width="8.5cm"} In Fig. \[Thermo\_po\], we show the behavior of the thermodynamic potential for a set of typical parameters. Similar to the case of a polarized Fermi gas without SOC, the competition between polarization and pairing leads to a double-well structure in the thermodynamic potential in certain parameter regions \[c.f. Fig. \[Thermo\_po\](a)(b)(e)(f)(g)\]. As a consequence, two distinct gapped phases can be present in the phase diagram, which are separated by a quantum phase transition. Specifically, if one of the two gapped states is a conventional superfluid with $h/2<\sqrt{\mu^2+\Delta^2}$, while the other a topological superfluid with $h/2>\sqrt{\mu^2+\Delta^2}$, there must be a first order phase transition between SF and TSF phases as the parameters are tuned so that the ground state of the system changes from one local minimum to the other. If both of the pairing orders are conventional superfluid with the same symmetry, there can only be a first-order-like crossover [@soc1]. Due to the non-monotonic behavior of the thermodynamic potential, the solution of the gap equation may also correspond to metastable or unstable states, in addition to the ground state. To avoid this complication, we directly minimize the thermodynamic potential to ensure that the ground state is reached. In the absence of SOC, when the polarization becomes large enough, a phase transition occurs and brings the system from a superfluid phase to a normal phase [@preview]. However, an arbitrarily small SOC will change this picture and introduces novel type of phases and phase transitions to the system. In fact, when $\Delta=0$, a singularity exists in the integrand of the gap equation (\[gapeqn\]) over considerably large parameter regions. So long as this singularity exists, the gap equation [*always*]{} has at least one finite solution regardless of the SOC strength $\alpha$ and chemical potential combinations $(\mu, h)$. In order to understand this picture, we show in Fig. \[gapillustrate\](a) the behavior of the function $G \equiv (-1/2\Delta) \partial \Omega/\partial \Delta$, which is proportional to the left-hand side of the gap equation (\[gapeqn\]). In the presence of the singularity, for arbitrary SOC strength, the function $G$ is always diverging as $\Delta \to 0$, and tends to large negative values as $\Delta \to \infty$. Therefore, there is at least one solution to the gap equation under these conditions, indicating the presence of gapped phases, as has been pointed out previously in Ref [@soc1]. Further analysis shows that one of the gapped phases is the global minimum of the thermodynamic potential. This observation shows that superfluidity can survive arbitrary polarization, provided that an SOC is introduced. In Fig. \[gapillustrate\](b), we present the pairing gap $\Delta$ of the ground state as a function of SOC strength $\alpha$. The numerical result suggests that the pairing gap decreases super-exponentially as $\alpha$ approaches zero. The singularity responsible for the divergence of $G$ comes from the terms proportional to $E_{\mathbf{k},-}^{-1}$, which diverges at $\Delta=0$. As $E_{\mathbf{k},-}=||\xi_{\mathbf{k}}|-\sqrt{\alpha^2k^2+h^2/4}|$ at $\Delta=0$, the solutions of the equation $|\xi_{\mathbf{k}}|=\sqrt{\alpha^2k^2+h^2/4}$ give the singularity points in momentum space. It is easy to see that the equation above does not have real-valued solutions under the conditions $\mu<-(\alpha^4+h^2)/(4\alpha^2)$ or $\mu<\min(-h/2,-\alpha^2/2)$. This suggests that the ground states corresponding to Fig. \[Thermo\_po\](a)(g)(h) are superfluid phases with small but finite pairing gap. ![(Color online) The phase diagram in the $\alpha$-$\mu$ plane with the binding energy $E_{b}/h=0.5$. The first order phase transition is shown in red solid curve while the second order phase transition in dash-dotted black curve. The thin dashed curve in the TSF region marks the $\Delta/h=10^{-3}$ threshold. The chemical potential difference $h$ is taken to be the energy unit, while the unit of momentum $k_h$ is defined through $\hbar^2k_h^2/(2m)=h$.[]{data-label="Eb0.5"}](Eb.pdf){width="8cm"} ![(Color online) The phase diagram in the $\alpha$-$\mu$ plane with various binding energies $E_{b}$. (a)$E_{b}/h=0.2$, (b)$E_{b}/h=0.65$, (c)$E_{b}/h=0.95$, (d)$E_{b}/h=1.2$. The thin dashed curves in (a-c) mark the $\Delta/h=10^{-3}$ threshold. The chemical potential $h$ is taken to be the energy unit, while the unit of momentum $k_h$ is defined through $\hbar^2k_h^2/(2m)=h$.[]{data-label="Eb_phase"}](new_Eb_phase){width="8.5cm"} {width="12cm"} Phase diagram in the $\alpha$-$\mu$ plane ========================================== From the discussions in the previous section, we see that the presence of SOC can lead to a rich structure of phases in a trapping potential. As a first step to understand the spatial distribution of the various phases in the trap, we consider in this section a homogeneous system and investigate the phase diagram as a function of $(\alpha,\mu)$ for given $E_b$ and $h$. Under LDA while assuming both spin species experience the same harmonic potential, a downward vertical line in such a phase diagram represents a trajectory from a trap center to its edge, with the chemical potential at the trap center fixed by that at the starting point of the line. To this end, we only need to minimize the thermodynamic potential in Eq. (\[thermoeqn\]) for given SOC strength $\alpha$ and chemical potential difference $h$ while sweeping the chemical potential $\mu$. In Fig. \[Eb0.5\], we show a typical phase diagram in the $\alpha$-$\mu$ plane for $E_b/h=0.5$. Notice that there is only one topologically non-trivial superfluid phase, which is clearly different from the 3D case as discussed before. The TSF phase is separated from the conventional superfluid phase by two kinds of phase boundaries. The solid curve in Fig. \[Eb0.5\] represents a first order phase boundary, along which the states corresponding to the two local minima of the double well structure in the thermodynamic potential are degenerate in energy. Compared to the 3D case, the first order phase boundary is dramatically extended. The other kind of TSF-SF phase boundary is of second order, given by $h/2=\sqrt{\mu^2+\Delta^2}$, along which the pairing gap remains finite and the excitation gap vanishes. To determine the phase boundary for $\Delta=0$, we need to examine the existence of divergence in the gap equation as $\Delta$ approaches zero. As discussed in the previous section, the singularities go away when $\mu<- (\alpha^4+h^2)/(4\alpha^2)$ or $\mu<\min(-h/2,-\alpha^2/2)$. The phase boundary of the superfluid phases with $\Delta=0$ can be calculated by solving the gap equation in these parameter regions. We note that the maximum value of the chemical potential satisfying these relations is $-h/2$, below which the chemical potential of both spin species $\mu_{\sigma}$ become negative. Hence there will not be a phase boundary between a superfluid phase (SF or TSF) and a normal phase. Instead, only phase boundaries between a superfluid state and vacuum (VAC) exist. For the calculations above, we always check the thermodynamic potential to ensure that states along the phase boundary with $\Delta=0$ represent ground state solutions. According to the phase diagram in Fig. \[Eb0.5\], the stability region for the TSF phase appears to be significant. Yet this can be misleading for experimental detection. In fact, the size of the pairing gap in the TSF phase with small SOC strength is typically vanishingly small. This can be seen from the dashed curve traversing the TSF phase in Fig. \[Eb0.5\], which is solved from the gap equation by setting $\Delta/h=10^{-3}$. To the left of the curve, the pairing gap $\Delta/h<10^{-3}$ and decreases exponentially fast as $\alpha$ approaches zero. The order parameter $\Delta$ only becomes significant when $\alpha$ is further increased toward the phase boundary between TSF and SF. Given the fluctuations in 2D systems at finite temperatures, experimental observation of the TSF phase is only possible to the right of the dashed curve and with reasonably large pairing gap $\Delta$. Figure \[Eb0.5\] also provides information regarding the structure of the phase separation in a trapping potential. When the SOC is small, the Fermi gas will phase separate into two regions, a conventional superfluid core surrounded by a TSF phase with large spin polarization and vanishingly small pairing order. The phase boundary between them is of first order. As the SOC increases, the local minima in the thermodynamic potential corresponding to the TSF and the SF states move closer as the pairing gap of TSF state increases. The two local minima merge at a critical end point beyond which the double-well structure in the thermodynamic potential disappears and the phase boundary between TSF and SF becomes second order. Further increasing the SOC, there may be a parameter window where the TSF phase appears as a ring structure in the trap. Finally, when the SOC is large enough, phase separation no longer occurs and the trap is filled with a superfluid of rashbons. We have also calculated the $\alpha$-$\mu$ phase diagram for a homogeneous system with different bound state energies $E_b$ (see Fig. \[Eb\_phase\]). Toward the BCS side \[Fig. \[Eb\_phase\](a)\], the stability region of the TSF phase increases while the first order phase boundary between TSF and SF no longer exists. For small SOC and large chemical potential, there may exist two different SF phases at the trap center, separated by a first-order-like boundary. In this case as the symmetries of the two SF phases are the same, the boundary is merely a crossover. On the phase diagram, this first-order-like crossover boundary ends at a critical end point where the two potential wells in the double well structure of the thermodynamic potential merge. Immediately below this first order crossover boundary and with small SOC, an SF phase with vanishingly small order parameter and small polarization appears where the chemical potentials of both spin species are positive. This corresponds to an SOC induced SF phase out of a normal phase with two spin species without SOC. Toward BEC side \[Fig. \[Eb\_phase\](b-d)\], the stability region of the TSF phase becomes smaller and eventually disappears from the phase diagram. The trap is then occupied by superfluid of rashbons. ![ (Color online) The phase structure appearing in a harmonic trapping potential with the parameters $E_{b}/E_F=0.5$ and $\alpha k_{F}/E_F=0.75$. Here, the total polarization $P$ is a trapped integrated result as calculated from Eq. (\[no2\]), and $\tilde{r}=r/R$ is the dimensionless distance from the trap center. Notice that the trap is filled with TSF phase as $P$ is above a critical value. All phase boundaries shown in this plot are of second order. First order phase boundaries show up at smaller SOC strengths and/or smaller $E_b$. The units of energy $E_F$ and of length $R$ are defined in the text.[]{data-label="prd"}](prdiagram){width="8cm"} Phase separation in a trap ========================== Next, we adopt the LDA and explicitly include the trapping potential in our calculation. To make our calculation universal and applicable to systems with any total particle number, we derive a dimensionless form following Ref. [@trapwy]. We take the unit of energy to be the Fermi energy ($E_F$) at the center of a 2D axially symmetric trap for N non-interacting fermions with equal population for the two spin species, with $E_F=\hbar\omega\sqrt{N}$ and $\omega$ is the trapping frequency. The harmonic trapping potential in the dimensionless form can be expressed as $V(\mathbf{r})/E_F=r^2/R^2=\tilde{r}^2$, where $R=\sqrt{E_F/m}$ is the Thomas-Fermi radius in two dimensions [@footnote1]. The number equation in the dimensionless form then becomes $$\begin{aligned} &1=4\int d^2\tilde{\mathbf{r}}[\tilde{n}_{\uparrow}(\tilde{\mathbf{r}})+\tilde{n}_{\downarrow}(\tilde{\mathbf{r}})], \label{no1}\\ &P=4\int d^2\tilde{\mathbf{r}}[\tilde{n}_{\uparrow}(\tilde{\mathbf{r}})-\tilde{n}_{\downarrow}(\tilde{\mathbf{r}}))]\label{no2}\end{aligned}$$ with dimensionless number density $\tilde{n}_{\sigma}=n_{\sigma}/n_0$. Here, $n_{\sigma}$ is the number density given by the number equation Eq. (\[numbereqn\]) at position $\mathbf{r}$. The Fermi momentum $k_F$ is defined as $E_F=\hbar^2k_F^2/(2m)$, and $n_0=k_F^2/(2\pi)$. It is obvious that the properties of the system only depend on the dimensionless parameters $\{E_b/E_F, \alpha k_F/E_F, P\}$. Solving the dimensionless equations above, we get the typical phase structure in a trapping potential with a various sets of parameters. Note that for simplicity, we first choose an appropriate chemical potential difference, e.g. $h=E_F$, and solve for the chemical potential $\mu$ at the center of the trap from Eq. (\[no1\]) with fixed SOC strength $\alpha$ and $E_b$. The polarization $P$ can then be calculated from Eq. (\[no2\]). The resulting shell structures are shown in Fig. \[shell\]. The topological superfluid phase typically appears toward the edge of the trap or as a ring between two SF phases, in agreement with our previous discussions based on the phase diagram of Fig. \[Eb0.5\]. Notably, there are parameter regions where the TSF state can occupy the entire trap. This is shown in the right-most column in Fig. \[shell\], and corresponds to a vertical line in the $\alpha$-$\mu$ plane phase diagram with the starting point in the TSF phase. To better understand the phase structures in a trap, we plot in Fig. \[prd\] the zero temperature phase boundaries in a harmonic trapping potential in the $P$-$\tilde{r}$ plane, where $P$ is the trap-integrated total polarization calculated from Eq. (\[no2\]), and $\tilde{r}=r/R$ is the dimensionless distance from the trap center. Notice that the TSF phase occupies the entire trap only when the total polarization exceeds a critical value ($\sim 0.69$ for the chosen parameters in Fig. \[prd\]). As we will show in the next section, this regime provides an ideal setup for the detection of the TSF state in a trapped gas. When $P$ decreases from this value, the conventional SF phase will emerge from the trap center, gradually extend to the trap edge, and eventually occupy the entire trap for small polarization case. ![(Color online) The density distribution in momentum space. (a) TSF with small gap: $Eb/h=0.5$, $\mu/h=0$, $\alpha k_{f}/h=0.1$; (b) TSF with larger gap: $Eb/h=0.5$, $\mu/h=0$, $\alpha k_{f}/h=0.45$; (c) SF: $Eb/h=0.5$, $\mu/h=0$, $\alpha k_{f}/h=0.8$, (d) SF with small gap: $Eb/h=1.2$, $\mu/h=0$, $\alpha k_{f}/h=0.1$, (e) SF: $Eb/h=1.2$, $\mu/h=0$, $\alpha k_{f}/h=0.45$, (f) SF: $Eb/h=1.2$, $\mu/h=0$, $\alpha k_{f}/h=0.8$. The chemical potential $h$ is taken to be the energy unit, the unit of momentum $k_h$ is defined through $\hbar^2k_h^2/(2m)=h$, and the unit of density is defined as $n_h=k_h^2/(2\pi)$.[]{data-label="nk"}](re3_nk){width="8.5cm"} Momentum distribution and the signature of the topological superfluid phase =========================================================================== To characterize the properties of the various phases in the phase diagram, we calculate their respective momentum distribution (see Fig. \[nk\]), which is given by the summand in the number equations (\[numbereqn\]). In Fig. \[nk\], we show the momentum distribution of a homogeneous system with various parameters. In the first row of Fig. \[nk\], the binding energy is set as $E_b/h = 0.5$ with increasing SOC strength. In the second row, a similar evolution with $\alpha$ is shown but with a binding energy $E_b/h = 1.2$, more toward the BEC regime. It is apparent that the momentum distribution in a TSF phase \[c.f. Fig. \[nk\](a-b)\] is drastically different from that in an SF phase. In particular, the momentum distribution of the minority spin in the TSF phase features a dip near zero momentum, which can be explained by the observation that $n_{\mathbf{k},\downarrow}=0$ at $k=0$, where $n_{\mathbf{k},\downarrow}$ is the summand in the corresponding number equation (\[numbereqn\]). As this dip in the momentum distribution is unique to the TSF phase, one may think of using it as a signature for the experimental detection of the TSF phase. To measure the momentum distribution experimentally, a commonly used practice is the time-of-flight imaging technique, which involves a ballistic expansion of the gas after suddenly switching off the trapping potential. As there are typically several different phases in the trapping potential, the observed momentum distribution is usually a trap-integrated distribution which includes the contribution from all the phases in the trap. In this case, the signal of the topological superfluid is washed out and cannot be detected. ![(Color online) (left column)The trap-integrated density distribution in momentum space $n_{k}$. (right column) The number density distribution for both spin up (black solid curves) and spin down (dashed red curves) atoms in a trapping potential. The insets show the distribution of the order parameter $\Delta(\tilde{\mathbf{r}})$. The dashed line in (d) illustrates the TSF-SF phase boundary. The parameters are: (a)(b)$E_{b}/E_F=0.2$, $\alpha k_{F}/E_F=1.25$, $P=0.82$; (c)(d) $E_{b}/E_F=0.2$, $\alpha k_{F}/E_F=1$, $P=0.65$. The energy unit $E_F$ is defined in the text and the unit in momentum space $k_F$ is related to the unit of energy as $\hbar^2k_F^2/(2m)=E_F$. The unit of density is defined through $n_0=k_F^2/(2\pi)$.[]{data-label="trap_nk"}](trap_nk5){width="8.5cm"} One possible way to overcome this difficulty is to prepare the system in an appropriate parameter region such that the center of the trap is filled with the TSF state. An example of this is demonstrated in Fig. \[trap\_nk\](b). The corresponding trap-integrated momentum distribution is shown in Fig. \[trap\_nk\](a), where the signature of the TSF state apparently survives the trap integration. In comparison, we show in Fig. \[trap\_nk\](c)(d) similar calculations for the case with an SF core surrounded by the TSF phase. As is clear from Fig. \[trap\_nk\](c), the signature dip for the TSF state can no longer be observed in the trap-integrated momentum distribution. This suggests that the existence of the dip can serve as a signature for the existence of the TSF phase if the momentum distribution of the minority spin species can be detected. Summary ======= We have developed a mean field theory to characterize the phases of a trapped 2D polarized Fermi gas with SOC near a wide Feshbach resonance. Under LDA, we have calculated in detail the structure of phase separation of the pairing gap in a trapping potential with various parameters. Compared to the 3D case, we find dramatically increased first order phase boundary between the SF core and the TSF phase that surrounds it, which makes it observable in experiments from the density distributions of the spin species. We then develop a universal scheme for the characterization of a trapped gas. The resulting phase and density distributions are therefore independent of the trapping geometry and the total particle number, and are determined by a set of dimensionless parameters. We explicitly calculate the density and momentum distribution of the gas in a trapping potential. Importantly, we find a parameter region where the trap is occupied by the TSF phase only. In this regime, the characteristic signature of the TSF state in the momentum distribution can survive the trap integration, rendering the signal detectable in a time-of-flight imaging process. We would like to thank Chuanwei Zhang for helpful discussions. This work is supported by NFRP (2011CB921200, 2011CBA00200), NNSF (60921091), NSFC (10904172, 11105134), the Fundamental Research Funds for the Central Universities (WK2470000001, WK2470000006), and the Research Funds of Renmin University of China (10XNL016). WZ would also like to thank the NCET program for support. [99]{} A. Kitaev, Ann. Phys. (N.Y.) [**303**]{}, 2 (2003). C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. [**80**]{}, 1083 (2008). G. Moore and N. Read, Nucl. Phys. B [**360**]{}, 362 (1991). N. Read and D. Green, Phys. Rev. B [**61**]{}, 10267 (2000). S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett. [**94**]{}, 166802 (2005). S. Das Sarma, C. Nayak, and S. Tewari, Phys. Rev. B [**73**]{}, 220502(R) (2006). V. Gurarie and L. Radzihovsky, Phys. Rev. B [**75**]{}, 212509 (2007). S. S. Botelho and C. A. R. Sá de Melo, J. Low Temp. Phys. [**140**]{}, 409 (2005). V. Gurarie, L. Radzihovsky, and A. V. Andreev, Phys. Rev. Lett. [**94**]{}, 230403 (2005). S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and P. Zoller, Phys. Rev. Lett. [**98**]{}, 010506 (2007). V. Gurarie, L. Radzihovsky, Ann. Phys. [**322**]{}, 2 (2007). J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. [**104**]{}, 040502 (2010). J. Alicea, Phys. Rev. B [**81**]{}. 125318 (2010). Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. [**105**]{}, 177002 (2010). L. Mao, J. Shi, Q. Niu, and C. Zhang, Phys. Rev. Lett. [**106**]{}, 157003 (2011). S. Tewari, T. D. Stanescu, J. D. Sau, and S. Das Sarma, New J. Phys. [**13**]{}, 065004 (2011). Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. [**102**]{}, 130401 (2009). Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature (London) [**471**]{}, 83 (2011). C.-J. Wu, I. M. Shem, and X.-F. Zhou, Chin. Phys. Lett. [**28**]{}, 097102 (2011). C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett. [**105**]{}, 160403 (2010). C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. [**92**]{}, 040403 (2004). M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. [**92**]{}, 120401 (2004). M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Phys. Rev. Lett. [**92**]{}, 120403 (2004). R. A. Duine and H. T. C. Stoof, Phys. Rep. [**396**]{}, 115 (2004). Q.-J. Chen, J. Stajic, S. Tan, and K. Levin, Phys. Rep. [**412**]{}, 1 (2005). M. W. Zwierlein, A. Schirotzek, C. Schunck, and W. Ketterle, Science [**311**]{}, 492 (2006). G. B. Patridge, W. Li, R. Kamar, Y. Liao, and R. G. Hulet, Science [**311**]{}, 503 (2006). D. E Sheehy and L. Radzihovsky, Ann. Phys. [**322**]{}, 1790 (2007). E. Majorana, Nuovo Cimento [**5**]{}, 171 (1937). S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and P. Zoller, Phys. Rev. Lett. [**98**]{}, 010506 (2007). C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. [**101**]{}, 160401 (2008). M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett. [**103**]{}, 020401 (2009). A. Kubasiak, P. Massignan, and M. Lewenstein, Euro. Phys. Lett. [**92**]{}, 46004 (2010). J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B [**83**]{}, 094515 (2011). J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys. Rev. B [**84**]{}, 014512 (2011). M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. [**107**]{}, 195303 (2011). Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. [**107**]{}, 195305 (2011). H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett. [**107**]{}, 195304 (2011). M. Iskin and A. L. Subasi, Phys. Rev. Lett. [**107**]{}, 050402 (2011). L. Han and C. A. R. Sá de Melo, arXiv:1106.3613. W. Yi and G.-C. Guo, Phys. Rev. A [**84**]{}, 031608(R) (2011). G. Chen, M. Gong, and C. Zhang, arXiv:1107.2627. L. Dell’Anna, G. Mazzarella, and L. Salasnich, Phys. Rev. A [**84**]{}, 033633 (2011). K. Seo, L. Han, and C. A. R. Sá de Melo, arXiv:1108.4068. B. Huang and S. Wan, arXiv:1109.3970. L. He and X.-G. Huang, arXiv:1109.5577. L. Jiang, X.-J. Liu, H. Hu, and H. Pu, arXiv:1110.0805. W. Yi and L.-M. Duan, Phys. Rev. A [**74**]{}, 013610 (2006). M. Iskin and C. A. R. Sá de Melo, Phys. Rev. Lett. [**97**]{}, 100404 (2006). M. M. Parish, F. M. Marchetti, A. Lamacraft and B. D. Simons, Nat. Phys. [**3**]{}, 124 (2007). M. Randeria, J.-M. Duan, and L.-Y. Shieh, Phys. Rev. Lett. [**62**]{}, 981 (1989). For simplicity, in this work we assume the external trapping potential to be symmetric with respect to the origin. However, the universal approach for the trapped gas can be extended to anisotropic cases so long as the trapping potential remains harmonic. This can be achieved by applying different scalings along different spatial directions. See Ref. [@trapwy] for details.

--- abstract: | Within the last few years, a countless number of blockchain systems have emerged on the market, each one claiming to revolutionize the way of distributed transaction processing in one way or the other. Many blockchain features, such as byzantine fault tolerance (BFT), are indeed valuable additions in modern environments. However, despite all the hype around the technology, many of the challenges that blockchain systems have to face are fundamental transaction management problems. These are largely shared with traditional database systems, which have been around for decades already. These similarities become especially visible for systems, that blur the lines between blockchain systems and classical database systems. A great example of this is Hyperledger Fabric, an open-source permissioned blockchain system under development by IBM. By having a relaxed view on BFT, the transaction pipeline of Fabric highly resembles the workflow of classical distributed databases systems. This raises two questions: (1) Which conceptual similarities and differences do actually exist between a system such as Fabric and a classical distributed database system? (2) Is it possible to improve on the performance of Fabric by transitioning technology from the database world to blockchains and thus blurring the lines between these two types of systems even further? To tackle these questions, we first explore Fabric from the perspective of database research, where we observe weaknesses in the transaction pipeline. We then solve these issues by transitioning well-understood database concepts to Fabric, namely transaction reordering as well as early transaction abort. Our experimental evaluation shows that our improved version Fabric++ significantly increases the throughput of successful transactions over the vanilla version by up to a factor of $3$x. author: - | Ankur SharmaFelix Martin SchuhknechtDivya AgrawalJens Dittrich\ \ Big Data Analytics Group\ Saarland Informatics Campus\ <https://bigdata.uni-saarland.de> bibliography: - 'bibliography\_sigmod.bib' title: 'How to Databasify a Blockchain: the Case of Hyperledger Fabric' --- Introduction {#sec:introduction} ============ Blockchains are one of the hottest topics in modern distributed transaction processing. However, from the perspective of database research, one could raise the question: what makes these systems so special over classical distributed databases, that have been out there for a long time already? The answer lies in byzantine fault tolerance: while classical distributed database systems require a trusted set of participants, blockchain systems are able to deal with a certain amount of *maliciously* behaving nodes. This feature opens lots of new application fields such as transactions between organizations, that do not fully trust each other. Unfortunately, ensuring BFT over all nodes of the network also heavily complicates transaction processing. If any node of the network is considered to be potentially malicious, a complex *consensus mechanism* is required to ensure the integrity of the system. This consensus mechanism assures, that a transaction can only commit, if a majority of the network agrees to it. Of course, the expensiveness of the consensus mechanism has also been observed by the blockchain engineers. Therefore, some systems trade BFT with performance by simply assuming certain parties of the network to be trustworthy. A great example for this is Hyperledger Fabric [@fabric], a popular open-source blockchain system introduced by IBM. In terms of BFT, it differs from other major players such as Bitcoin or Ethereum in two additional assumptions: First, Fabric assumes that the *ordering service*, which globally orders all transactions that go through the system, is trustworthy. Second, it allows for the forming of so called *organizations*. Within an organization, it is assumed that all peers trust each other. These two assumptions heavily simplify transaction processing. First of all, no complex consensus mechanism, such as PBFT [@pbft], is necessary. Second, the trust within an organization allows for a distribution of work within it and enables parallelism, as not every peer in the organization has to execute every transaction. With this relaxed view on BFT in mind, can we actually still consider Fabric a true blockchain system? A trustworthy ordering service, which globally arranges and schedules transactions, is a component that is present in classical distributed database systems as well. Further, the concept of an organization, in which all peers trust each other, is also present in distributed databases in its extreme form: all peers belong to a single organization. Besides of that, other core requirements of transaction management, such as ensuring transaction isolation or managing the data in a store, are essentially present one-to-one in both blockchains and database systems. At the example of Fabric, it becomes obvious that conceptually the lines between blockchain systems and distributed database systems are rather blurry. We believe this blurriness should be seen as a chance for the database community: Due to all these conceptual similarities, it becomes possible to transition well-understood database technology to the world of blockchains, significantly enhancing this new technology. The question remains which similarities can be exploited to transition database technology to Fabric and by how much can we improve on the state-of-the-art? To tackle this problem, we perform the following steps in this work: 1. To have a basis for the discussion, we first inspect the transaction flow of Hyperledger Fabric in the latest version 1.2 from a conceptual perspective. Fabric will serve as our case-study for the rest of the paper on how to “databasify” a blockchain system (Section \[sec:fabric\]). 2. Based on the analysis of the transaction flow in Fabric, we then inspect its components, that show the highest resemblance with those of database systems. We identify weaknesses in the implementation of Fabric of these components and describe, how database technology can be utilized to counter them. (Section \[sec:sidebyside\]). 3. We transition database technology to the transaction pipeline of Fabric. Precisely, we first improve on *the ordering of transactions*. By default, the system orders transactions arbitrarily after simulation, leading to unnecessary serialization conflicts. To counter this problem, we introduce an advanced *transaction reordering mechanism*, which aims at reducing the number of serialization conflicts between transactions within a block. This mechanism significantly increases the number of valid transactions, that make it through the system and therefore the overall throughput (Section \[ssec:transaction\_reodering\]). 4. Next, we advance the *abort of transactions*. By default, Fabric checks whether a transaction is valid right before the commit. This late abort unnecessarily penalizes the system by processing transactions, that have no chance to commit. To tackle this issue, we introduce the concept of *early abort* to various stages of the pipeline. We identify invalid transactions as early as possible and abort them, assuring that the pipeline is not throttled by transactions that have no chance to commit eventually. A requirement for this concept is a *fine-grained concurrency control mechanism*, by which we extend Fabric as well (Section \[ssec:early\_abort\]). These modifications significantly extend the vanilla Fabric, turning it into what we call Fabric++. 5. We perform an extensive experimental evaluation of the optimizations of Fabric++ under a custom blockchain benchmark simulating an asset transfer scenario. In total, we evaluate the transactional throughput under $108$ different configurations of Fabric and Fabric++ and show that we are able to significantly boost the performance over the vanilla version. Additionally, we vary the number of channels and clients and show, that our optimizations also have a positive impact on the scaling capabilities of the system (Section \[sec:ea\]). Hyperledger Fabric {#sec:fabric} ================== Before diving into the conceptual similarities and differences between Fabric and distributed database systems, we have to understand the workflow of Fabric. Let us describe in the following section how it behaves in the latest version 1.2. High-level Workflow ------------------- At its core, Fabric follows a *simulate-order-validate-commit* workflow, as shown in Figure \[figs:fabric\_workflow\_highlevel\]: (1) In the **simulation phase**, a client submits a transaction proposal to a specified subset of the peers, called the endorsement peers or *endorsers*. The endorsers simulate the effects of the transaction proposal against a local copy of their current state. Interestingly, none of the writes become durable in the current state at this point. If the endorsers endorse the transaction proposal, an actual transaction is formed from the execution result, that is then sent to the ordering service (via the client). (2) In the **ordering phase**, the ordering service establishes a global order among all received transactions and distributes the ordered transactions at the granularity of blocks to all peers of the network. (3) In the **validation phase**, all peers individually validate the transactions within the received blocks in terms of endorsement policy and serializability. (4) In the **commit phase**, the blocks are appended to the local ledger and the changes made by the valid transactions are applied to the current state. Following these four phases assures that each honest peer stores the same transaction sequence. ![High-level workflow of Fabric.[]{data-label="figs:fabric_workflow_highlevel"}](fabric_highlevel.pdf){width="12cm"} Architecture ------------ Fabric is a *permissioned* blockchain system, meaning all peers of the network are known at any point in time. Peers are grouped into *organizations*, which typically host them. For example, two companies trading with each other could each host an organization of $10$ machines, forming a network of $20$ peers. Within an organization, all peers trust each other. Each peer runs a local instance of Fabric. This instance includes a copy of the *ledger*, containing the ordered sequence of all transactions that went through all four phases. This includes both valid and invalid transactions. Apart from the ledger, each peer also contains the *current state* in form of a state database. The current state can be seen as an optimization of the ledger: while the ledger simply contains the sequence of all processed transactions, the current state represents the state after the *application* of all *valid* transactions in the ledger to the initial state. Fabric implements the current state in form of a versioned key-value-store. For every key in the store, a pair of value and version-number is kept, where the version-number[^1] counts the number of changes that already happened to the value of this key. Apart from the peers, which play an important role both in the simulation phase and the validation phase, there is a separate instance called the *ordering service*, which is the core component of the ordering phase and assumed to be trustworthy. Although it can be composed out of multiple machines for fault tolerance, it is a central service responsible for establishing a global order among all transactions. Running Example --------------- With the basic components of the architecture in mind, let us now discuss how transactions flow through the system. To do so, we present a simple running example in Figure \[figs:fabric\_workflow\_execute\] (simulation phase), Figure \[figs:fabric\_workflow\_order\] (ordering phase), and Figure \[figs:fabric\_workflow\_validate\] (validation and commit phase), where two organizations $A$ and $B$ want to transfer money between each other. Each organization contributes two peers to the network. The balances of the organizations are stored by two variables $BalA$ and $BalB$, where $BalA$ stores the value $100$ in its current version $v_3$ and $BalB$ stores $50$ in version $v_2$. We can also see that the ledger already contains six transactions $T_1$ to $T_6$, where the four transactions $T_1$, $T_2$, $T_4$, and $T_6$ were valid ones and lead to the current state. The transactions $T_3$ and $T_5$ were invalid transactions. They are still stored in the ledger, although they did not pass the validation phase. Simulation Phase {#ssec:simulation} ---------------- Transaction processing starts with the *simulation* phase in Figure \[figs:fabric\_workflow\_execute\]. In **step** , a client proposes a *transaction proposal* (or short *proposal*) to the system. In our example, the proposal intends to transfer the amount of $30$ from $BalA$ to $BalB$. The two involved operations and are expressed in a smart contract[^2], an arbitrary program, that is bound to the proposal. Additionally to the smart contract, an endorsement policy must be specified. It determines which and/or how many peers have to endorse the proposal. In our example of money transfer between two organizations, a reasonable endorsement policy is to request endorsement from one peer of each organization — like two lawyers, preserving and defending the individual rights of their clients. ![Simulation phase.[]{data-label="figs:fabric_workflow_execute"}](fabric_workflow_execute.pdf){width="12cm"} Therefore, in **step** , the proposal is sent to the two endorsement peers $A1$ and $B1$ according to the policy. These two peers now individually simulate the smart contract (, ), that is bound to the proposal, against their local current state. Note that, as the name suggests, the simulation of the smart contract against the current state does *not change* the current state in any way. Instead, each endorsement peer builds an auxiliary read set $RS$ and a write set $WS$ during the simulation to keep track of all accesses that happen. In our case of money transfer over the amount of $30$, the smart contract first reads the two current balances $BalA$ and $BalB$ along with their current version-numbers. Second, the smart contract updates the two balances according to the transferred amount, resulting in the new balances $BalA=70$ and $BalB=80$. Overall, this builds the following read and write set: $$RS=\{(BalA, v_3), (BalB, v_2)\} \hspace{0.2cm}WS=\{ BalA=70, BalB=80 \}$$ In this sense, the simulation of the smart contract is actually only a monitoring of the execution effects. The reason for performing only a simulation is that in this phase, we can not be sure yet whether this transaction will be allowed to commit eventually – this check will be performed later in the validation phase. After the simulation of the smart contract on all endorsement peers, in **step** , the endorsement peers return their individually computed read and write sets to the client, that sent the transaction proposal. Additionally, they return a signature of their simulation, that will be relevant in the validation phase in Section \[ssec:validation\]. If all read sets and write sets match[^3], in **step** , the actual *transaction* (called $T_7$ in the following) is formed from the results of the endorsement. This transaction $T_7$ now contains the effects of the execution in form of the read and write set as well as all signatures and can be passed on to the ordering service. Ordering Phase {#ssec:ordering} -------------- As mentioned, the central component of the *ordering* phase is the ordering service, that we visualize in Figure \[figs:fabric\_workflow\_order\]. It receives all transactions, that made it through the simulation phase. Consequently, it receives in **step**  our transaction $T_7$, that we followed through the simulation phase in Section \[ssec:simulation\]. In **step** , we assume that it also receives two other transactions $T_8$ and $T_9$, that were endorsed in parallel to $T_7$. ![Ordering phase.[]{data-label="figs:fabric_workflow_order"}](fabric_workflow_order.pdf){width="12cm"} The ordering service now has the sole purpose of establishing a global order among the transactions. It treats the transactions in a black box fashion and does not inspect the transaction semantics, such as the read and write set, in any way. By default, it essentially arranges the transactions in the order in which they arrive, resulting in what we call for the rest of the paper the *arrival order*. In **step** , the ordering service now outputs the ordered stream of transactions in form of *blocks*, containing a certain number of transactions. Outputting whole blocks instead of individual transactions reduces the pressure on the network, as less communication overhead is produced. Finally, the generated block is distributed to *all* four peers of the network to start the validation phase. Note that there is no guarantee that all peers receive a block at the same time, as the distribution happens partially from ordering service to peers directly as shown in **step**  and partially between the peers using a gossip protocol as shown in **step** . However, the service assures that all peers receive the blocks in the same order. Validation and Commit Phase {#ssec:validation} --------------------------- When a block arrives at a peer, the *validation* phase starts, visualized in Figure \[figs:fabric\_workflow\_validate\] for peer $A1$. The three remaining peers execute the same validation process. Overall, the validation phase has two purposes. ### Endorsement Policy Evaluation {#sssec:endorsement_policy_eval} The first purpose is to validate the transactions in the block with respect to the *endorsement policy*. For example, it is possible that a malicious transaction was generated by a malicious client and a malicious peer in conspiracy to take advantage of the money transfer. Let us assume that transaction $T_8$ is such a malicious transaction and that the malicious client, which proposed $T_8$, works together with peer $A2$, which is also malicious. Instead of using the legit write set $WS_{B2}=\{BalA=30, BalB=120\}$ from B2, the client creates a proposal with the write set $WS_{A2}=\{BalA=100, BalB=120\}$, that it received from its collaborator A2. How is this transaction $T_8$ now detected in the validation phase? The key to this lies in the signatures $Sig_{A2}$ and $Sig_{B2}$, that the endorsement peers generate at the end of the simulation phase. The signature is computed over the read and write set, the executed smart contract, and the used endorsement policy. The client receives these cryptographically secure signatures and must pack them into the transaction along with the read and write set. The peers that validate the transaction recompute the signatures of all endorsement peers, that were responsible for transaction $T_8$ and compare the signatures with the received ones $Sig_{A2}$ and $Sig_{B2}$. In our example, in **step** , the peers detect that the signature of the honest peer $Sig_{B2}$ does not match to the one they computed from the received write set and thus, would classify $T_8$ as invalid. $T_7$ and $T_9$, the remaining transactions in the block, are evaluated in parallel. Their signatures match the ones computed from the read and write set and therefore, these transactions are valid with respect to the endorsement policy. ![Validation and Commit Phase.[]{data-label="figs:fabric_workflow_validate"}](fabric_workflow_validate.pdf){width="12cm"} ### Serializability Conflict Check {#sssec:serializability_conflict_check} The second purpose of the validation is to analyze the transactions with respect to *serializability conflicts*, that can arise from the order of transactions. For every transaction, it must be checked whether the version-numbers of all keys in the read set match the version-numbers in the current state. Only if this is the case, a transaction operates on an up-to-date state. Considering our example, let us perform the serializability conflict check for the received block. $T_8$ is already marked as invalid as it did not pass the endorsement policy evaluation, so it is not checked again. $T_7$ passed the endorsement policy evaluation and is now tested for serialization conflicts in **step** . Its read set is $RS=\{(BalA, v_3), (BalB, v_2)\}$. The version numbers of $BalA$ and $BalB$ in the read set match the ones of the current state and therefore, $T_7$ is marked as valid. As a consequence, in **step** , the write set of $T_7$, namely $WS=\{ BalA=70, BalB=80 \}$ is written to the current state. This changes the current state to $BalA=(70,v_4)$ and $BalB=(80,v_3)$. Note that the version-numbers of the modified variables are incremented. The next transaction to be checked is $T_9$ in **step** . Let us assume it also performs a money transfer and has the following read and write set: $$RS=\{(BalA, v_3), (BalB, v_2)\} \hspace{0.2cm} WS=\{ BalA=0, BalB=150 \}$$ This transaction will not pass the conflict check, as it read $BalA$ in version $v_3$ and $BalB$ in version $v_2$, while the current state already contains $BalA$ in version $v_4$ and $BalB$ in version $v_3$. Therefore, it operated on outdated data and is marked as invalid. As a consequence, its write set is not applied to the current state and simply discarded. Finally, after validating all transactions of the block, in **step**  the entire block is appended to the ledger along with the information about which transactions are valid or invalid. Blurred Lines: Fabric vs Distributed Database Systems {#sec:sidebyside} ===================================================== As we now have an understanding of the workflow of Fabric, we are able to discuss its architecture in relation to distributed database systems. In particular, we are interested in aspects of Fabric, that are (a) conceptually shared with distributed database systems, but (b) have potential for the application of database technology. The Importance of Transaction Order {#ssec:importance_of_order} ----------------------------------- The first component we look at is the ordering mechanism. Such a component is also present in any distributed database system with transaction semantics and therefore a great candidate for transitioning database technology to Fabric. As described in Section \[ssec:ordering\], Fabric relies on a single trustworthy ordering service for ordering transactions. Since Fabric simulates the smart contracts bound to proposals *before* performing the ordering, the order actually has an influence on the number of serialization conflicts between transactions. Again, this is a property shared with any parallel database system, that separates transaction execution from transaction commit. In ordering transactions, various different strategies are possible: The simplest option is to arbitrarily order them, for instance in the order in which they arrive. While this arrival order is fast to establish, it can lead to serialization conflicts, that are *potentially unnecessary*. These conflicts increase the number of invalid transactions, which must be resubmitted by the client. Unfortunately, the vanilla Fabric follows exactly this naive strategy. This is caused by the design decision that the ordering service is not supposed to inspect the transaction semantics, such as the read and write set, in any way. Instead, it simply leaves the transactions in the order in which they arrive. This strategy can be problematic, as the example in Table \[table:naive\_order\] shows. In this example, four transactions are scheduled in the order in which they arrive, namely $T_1 \Rightarrow T_2 \Rightarrow T_3 \Rightarrow T_4$, where $T_1$ updates the key $k_1$ from version $v_1$ to $v_2$. Since the transactions $T_2$, $T_3$, and $T_4$ each read $k_1$ in version $v_1$ during their simulations, they have no chance to commit, as they operated on an outdated version of the value of $k_1$. They will be identified as invalid in the validation phase and the corresponding transaction proposals must be resubmitted by the client, resulting in a new round of simulation, ordering, and validation. [ L[3cm]{} | R[3cm]{} | R[3cm]{} | R[3cm]{} ]{} **Transaction** & **Read Set** & **Write Set** & **Is Valid?**\ 1. $T_1$ & — & &\ 2. $T_2$ & $, (k_2, v_1)$ & $(k_2, v_1 \rightarrow v_2)$ &\ 3. $T_3$ & $, (k_3, v_1)$ & $(k_3, v_1 \rightarrow v_2)$ &\ 4. $T_4$ & $, (k_3, v_1)$ & $(k_4, v_1 \rightarrow v_2)$ &\ \[table:naive\_order\] Interestingly, for the four transactions from the previous example, there exists an order that is conflict free. In the schedule $T_4 \Rightarrow T_2 \Rightarrow T_3 \Rightarrow T_1$, as shown in Table \[table:smart\_order\], all four transactions are valid, as their read and write sets do not conflict with each other in this order. [ L[3cm]{} | R[3cm]{} | R[3cm]{} | R[3cm]{} ]{} **Transaction** & **Read Set** & **Write Set** & **Is Valid?**\ 1. $T_4$ & $(k_1, v_1)$$, (k_3, v_1)$ & $(k_4, v_1 \rightarrow v_2)$ &\ 2. $T_2$ & $(k_1, v_1)$$, (k_2, v_1)$ & $(k_2, v_1 \rightarrow v_2)$ &\ 3. $T_3$ & $(k_1, v_1)$$, (k_3, v_1)$ & $(k_3, v_1 \rightarrow v_2)$ &\ 4. $T_1$ & — & $(k_1, v_1 \rightarrow v_2)$ &\ \[table:smart\_order\] This example shows that the vanilla orderer of Fabric misses a chance of removing *unnecessary* serialization conflicts. While this problem is new to the blockchain domain, as blockchains typically offer only a serial execution of transactions, within the database community, this problem is actually well known. There exist reordering mechanisms which aim at minimizing the number of serialization conflicts via a reordering of transactions [@reordering; @reordering2; @abort_dependency_detection]. However, in a database system, it is typically avoided to buffer a large number of incoming transactions before processing as low latency is mandatory. Thus, reordering is not always an option in such a setup. Fortunately, as blockchain systems buffer the incoming transactions anyways to group them into blocks, this gives us the opportunity to apply sophisticated transaction reordering mechanisms without introducing significant overhead. We will add such a transaction reordering mechanism to Fabric in Section \[ssec:transaction\_reodering\], which significantly enhances the number of valid transactions, that make it through the system. On the Lifetime of Transactions ------------------------------- The second aspect we look at from a database perspective tackles the lifetime of transactions within the pipeline. In Fabric, every transaction that goes through the system is either classified as valid or as invalid with respect to the validation criteria. In the vanilla version, this classification happens in the validation phase right before the commit phase. A severe downside of this form of *late abort* is that a transaction, that violated the validation criteria already in an earlier phase, is still processed and distributed across all peers. This penalizes the whole system with unnecessary work, throttling the performance of valid transactions. Besides, this concept also delays the abort notification to the client. We have to distinguish in which phase a violation happens. First, a violation can occur already in the simulation phase, in form of so called *cross-block conflicts*, meaning a transaction from a later block, which is currently in the simulation phase, conflicts with a valid transaction from an earlier block. Second, a violation can occur as well as in the ordering phase, in form of *within-block conflicts* between conflicting transactions in a single block. Let us look at these two scenarios in isolation in Section \[sssec:early\_abort\_simulation\] and Section \[sssec:early\_abort\_ordering\], respectively. ### Violation in the simulation phase (cross-block conflicts) {#sssec:early_abort_simulation} To understand the problem in the simulation phase, let us look at the following situation and how the vanilla version of Fabric handles it. Let us assume there are four transactions $T_1$, $T_2$, $T_3$, and $T_4$ that are currently in the ordering phase and that end up in a block of size four, which is shipped to all peers for validation. Before the validation of that block starts within a peer $P$, the smart contract of a transaction proposal $T_5$ starts its simulation in $P$. To do so, it acquires a read lock[^4] on the *entire* current state. While the simulation is running, the block has to wait for the validation, as it has to acquire an *exclusive* write lock on the current state. The problem in this situation is: if $T_1$, $T_2$, $T_3$, or $T_4$ write the value of a key, that is read by $T_5$, then $T_5$ simulates on stale data. Therefore, in the moment of the read, the transaction becomes virtually invalid. Still, in the vanilla version of Fabric, this stale read is not detected before the validation phase of $T_5$. Thus, $T_5$ would continue its simulation and go through the ordering phase, just to be invalidated in the very end. ### Violation in the ordering phase (within-block conflicts) {#sssec:early_abort_ordering} Apart from conflicts across blocks, there can be conflicts between transactions within a block. These conflicts appear after putting the transactions into a particular order in the ordering phase. For instance, the example from Table \[table:naive\_order\] in Section \[ssec:importance\_of\_order\] showed a schedule, where the three transactions $T_2$, $T_3$, and $T_4$ individually conflict with the previously scheduled transaction $T_1$ of the same block. Unfortunately, these conflicts are not detected within the orderer of the vanilla version of Fabric. The block containing $T_2$, $T_3$, and $T_4$ would be distributed across all peers of the network for validation, although $3/4$ of transactions within the block are virtually invalid. As before, this originates from the design decision that the ordering service does not inspect transaction semantics. The mentioned situations show that Fabric misses several chances to abort transactions right at the time of violation. In contrast to that, database systems are typically very eager in aborting transactions [@early_abort_paper], as it decreases network traffic and saves computing resources. This concept of “cleaning” the pipeline as early as possible is called *early abort* in the context of databases, which apply this concept in various flavors. For instance, besides of the early abort of transactions, that violate certain criteria, database systems eliminate records from the query result set as early as possible by pushing down selection and projection operations in the query plan. To overcome the mentioned problems, we will apply the concept of early abort at several stages of the transaction processing pipeline of Fabric. By this, we assure to utilize the available resources with meaningful work to the extend. We will detail this in Section \[ssec:early\_abort\]. Fabric++ {#sec:databasify} ======== We have outlined the problems of Fabric and how they relate to key problems known in the context of database systems. Let us now see precisely how we counter them. First, in Section \[ssec:transaction\_reodering\], we introduce a transaction reordering mechanism, that aims at minimizing the number of unnecessary within-block conflicts. Second, in Section \[ssec:early\_abort\], we introduce early transaction abort to several stages of the Fabric pipeline. This also involves the introduction of a fine-grained concurrency control mechanism. Transaction Reordering {#ssec:transaction_reodering} ---------------------- When reordering a set of transactions $S$, multiple challenges must be faced. First, we have to identify which transactions of $S$ actually conflict with each other with respect to the actions they perform. Precisely, we have a conflict between two transactions $T_i$ and $T_j$ (denoted as $T_i \nrightarrow T_j$), if $T_i$ writes to a key that is read by $T_j$. In this case, $T_i$ must be ordered *after* $T_j$ (denoted as $T_j \Rightarrow T_i$) to make the schedule serializable, as otherwise, the read of $T_j$ would be outdated. Unfortunately, the problem is typically more complex as *cycles of conflicts* can occur, such that simple reordering can not resolve the problem. For example, if we have the cycle of conflicts $T_i \nrightarrow T_j \nrightarrow T_k \nrightarrow T_i$, there is no order of these three transactions that is serializable. Therefore, before reordering transactions, our mechanism must actually first remove certain transactions of $S$ to form a subset $S' \subseteq S$, from which a serializable schedule can be generated. Of course, a goal must be to remove as few transactions as possible. Finally, after computing $S'$, we can derive a concrete serializable schedule from the transactions in $S'$. On a high-level, we have to carry out the steps as shown in the pseudo-code of Algorithm \[alg:ordering\] to create a serializable schedule for a set of transactions $S$. func reordering(Transaction[] S) { // Step 1: Inspect the read/write set of all transactions and build a conflict graph. (*\label{l:buildConflictGraph}*)Graph cg = buildConflictGraph(S) // Step 2: Within cg, we have to identify all occurring cycles. Divide cg into // strongly connected subgraphs using Tarjans algorithm [2] in divideIntoSubgraphs(). (*\label{l:divideIntoSubgraphs}*)Graph[] cg_subgraphs = divideIntoSubgraphs(cg) // Each strongly connected subgraph of cg with more than one node must contain at // least one cycle. We identify the cycles within the subgraphs using Johnsons // algorithm [1] in getAllCycles(). (*\label{l:step3start}*)Cycle[] cycles = emptyList() foreach subgraph in cg_subgraphs: if(subgraph.numNodes() > 1): (*\label{l:step3end}*)cycles.add(subgraph.getAllCycles()) // Step 3: To remove the cycles in cg, we have to remove conflicting transactions from // S. For each transaction of S, we count in how many cycles it occurs. MaxHeap transactions_in_cycles = emptyMaxHeap() foreach Cycle c in cycles: foreach Transaction t in c: if transactions_in_cycles.contains(t) transactions_in_cycles[t]++ else transactions_in_cycles[t] = 1 // Step 4: Let us define S(*'*) as S. We now greedily remove the transaction from S(*'*) that // occurs in most cycles, until all cycles have been resolved. (*\label{l:step4start}*)Transaction[] S(*'*) = S while not cycles.empty(): Transaction t = transactions_in_cycles.popMax() S(*'*).remove(t) foreach Cycle c in cycles: if c.contains(t): c.remove(t) (*\label{l:step4end}*)cycles.remove(c) foreach Transaction t(*'*) in c: transactions_in_cycles[t(*'*)]-- // Step 5: From S(*'*) we have to form the actual serializable schedule. We start by // building the (cycle-free) conflict graph of S(*'*). (*\label{l:buildConflictGraph2}*)Graph cg(*'*) = buildConflictGraph(S(*'*)) // Compute schedule. We start at some node of the graph, that hasn't been visited yet. Transactions[] order = emptyList() Node startNode = cg(*'*).getNextNode() while order.length() < cg(*'*).numNodes(): addNode = true (*\label{l:searchSourceStart}*)if startNode.alreadyScheduled(): startNode = cg(*'*).getNextNode() continue // Traverse upwards to find a source foreach Node parentNode in startNode.parents(): if not parentNode.alreadyScheduled(): startNode = parentNode addNode = false (*\label{l:searchSourceEnd}*)break // A source has been found, so schedule it and traverse downwards. (*\label{l:scheduleStart}*)if addNode: startNode.scheduled() order.append(startNode) foreach Node childNode in startNode.children(): if not childNode.alreadyScheduled(): startNode = childNode (*\label{l:scheduleEnd}*)break // we invert the order to get the actual schedule return order.invert() } [ C[4cm]{} || C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{} C[0.6cm]{}]{} &\ **Transactions** & $K_0$ & $K_1$ & $K_2$ & $K_3 $ & $K_4$ & $K_5$ & $K_6$ & $K_7$ & $K_8$ & $K_9$\ $T_0$ & **1** & **1** & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\ $T_1$ & 0 & 0 & 0 & **1** & **1** & **1** & 0 & 0 & 0 & 0\ $T_2$ & 0 & 0 & 0 & 0 & 0 & 0 & **1** & **1** & 0 & 0\ $T_3$ & 0 & 0 & **1** & 0 & 0 & 0 & 0 & 0 & **1** & 0\ $T_4$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & **1**\ $T_5$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\ \ &\ **Transactions** & $K_0$ & $K_1$ & $K_2$ & $K_3 $ & $K_4$ & $K_5$ & $K_6$ & $K_7$ & $K_8$ & $K_9$\ $T_0$ & 0 & 0 & **1** & 0 & 0 & 0 & 0 & 0 & 0 & 0\ $T_1$ & **1** & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\ $T_2$ & 0 & 0 & 0 & **1** & 0 & 0 & 0 & 0 & 0 & **1**\ $T_3$ & 0 & **1** & 0 & 0 & **1** & 0 & 0 & 0 & 0 & 0\ $T_4$ & 0 & 0 & 0 & 0 & 0 & **1** & **1** & 0 & **1** & 0\ $T_5$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & **1** & 0 & 0\ \[table:conflictkeys\] ### Example To understand the principle and to discuss some of the implementation details, let us go through a concrete example. Let us assume we have a set $S$ of six transactions $T_0$ to $T_5$ to consider for reordering. These six transactions have read and write sets as shown in Table \[table:conflictkeys\]. In total, they access ten unique keys $K_0$ to $K_9$. **Step (1)**: Based on this information, we now have to generate the conflict graph of the transactions as done by the function [[`buildConflictGraph()`]{}]{} in line \[l:buildConflictGraph\] of Algorithm \[alg:ordering\]. To do so in an efficient way, we interpret the rows of Table \[table:conflictkeys\] as bit-vectors of length $10$. Let us refer to them as $vec_r(T_i)$ for the reading accesses and $vec_w(T_i)$ for the writing accesses of a transaction $T_i$. For each transaction $T_i$, we now perform a bitwise `&`-operation between $vec_r(T_i)$ and $vec_w(T_j)$ for all $j \neq i$. If the result of an `&`-operation is not $0$, we have identified a read-write conflict and create an edge in the conflict graph between the corresponding transactions. For example, for $T_0$ we have the reading accesses $vec_r(T_0) = 1100000000$ The bitwise `&`-operation with $vec_w(T_1) = 1000000000$ leads to $ vec_r(T_0) \texttt{ \& } vec_w(T_1) = 1000000000$, which is not $0$. This means $T_1$ writes a key that $T_0$ is reading and thus, we put a corresponding edge in the conflict graph. As a result, we obtain the conflict graph $C(S)$ of our six transactions as shown in Figure \[figs:fabric++\_ordering\]. ![Conflict graph $C(S)$ of the transactions in $S$.[]{data-label="figs:fabric++_ordering"}](fabric++_ordering.pdf){width="6cm"} Note that this algorithm has quadratic complexity on the number of transactions. Still, we apply it as the number of transactions to consider is very small in practice due to the limitation by the block size and therefore, the overhead is negligible. **Step (2)**: To identify the cycles, we apply Tarjan’s algorithm [@Tarjan72] in the function [[`divideIntoSubgraphs()`]{}]{} in line \[l:divideIntoSubgraphs\] to identify all strongly connected subgraphs. In general, this can be done in linear time in $\mathcal{O}(N+E)$ over the number of nodes $N$ and number of edges $E$ and results in the three subgraphs as shown in Figure \[figs:fabric++\_ordering\_subgraphs\]. Using Johnson’s algorithm [@Johnson75], we then identify all cycles within the strongly connected subgraphs. Again, this step can be done in linear time in $\mathcal{O}((N+E) \cdot (C+1))$, where $C$ is the number of cycles. Therefore, if there are no cycles in the subgraphs, the overhead of this step is very small. ![The three strongly connected subgraphs of the conflict graph of Figure \[figs:fabric++\_ordering\].[]{data-label="figs:fabric++_ordering_subgraphs"}](fabric++_ordering_subgraph.pdf){width="6cm"} We identify that the first subgraph (colored in green) consisting of $T_0$, $T_1$, and $T_3$ contains the two cycles  and $c_2=T_0 \nrightarrow T_3 \nrightarrow T_1 \nrightarrow T_0$. The second subgraph (colored in red) consisting of $T_2$ and $T_4$ contains the cycle $c_3=T_2 \nrightarrow T_4 \nrightarrow T_2$. The third subgraph (colored in yellow) contains only one node and is thus cycle-free. **Step (3)**: From this information, we can build a table denoting for every transaction in which cycle it appears as shown in the lines \[l:step3start\] to \[l:step3end\] of Algorithm \[alg:ordering\]. Table \[table:fabric++\_ordering\_cycles\] visualizes the result for our example. If a transaction $T_i$ is part of a cycle $c_j$, the corresponding cell is set to $1$, otherwise $0$. The last row of the table sums up for every transaction in how many cycles it is contained in total. [ C[1.5cm]{} || C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} ]{} **Cycle** & $T_0$ & $T_1$ & $T_2$ & $T_3 $ & $T_4$ & $T_5$\ $c_1$ & 1 & 0 & 0 & 1 & 0 & 0\ $c_2$ & 1 & 1 & 0 & 1 & 0 & 0\ $c_3$ & 0 & 0 & 1 & 0 & 1 & 0\ $\sum$ & 2 & 1 & 1 & 2 & 1 & 0\ \[table:fabric++\_ordering\_cycles\] **Step (4)**: We now iteratively remove transactions, that participate in cycles, starting from the ones that appear in most cycles. The lines \[l:step4start\] to \[l:step4end\] of Algorithm \[alg:ordering\] show the corresponding pseudo-code. As we can see, $T_0$ and $T_3$ both appear in two cycles, so we take care of them first. If we can choose between two transactions, such as $T_0$ and $T_3$, we pick the one with the smaller subscript. This assures that our algorithm is deterministic. We remove $T_0$, which clears all cycles in which $T_0$ appears, namely $c_1$ and $c_2$. The sum is updated accordingly, as we can see in Table \[table:fabric++\_ordering\_cycles2\]. [ C[1.5cm]{} || C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} ]{} **Cycle** & & $T_1$ & $T_2$ & $T_3 $ & $T_4$ & $T_5$\ & & & & & &\ & & & & & &\ $c_3$ & 0 & 0 & 1 & 0 & 1 & 0\ $\sum$ & 0 & 0 & 1 & 0 & 1 & 0\ \[table:fabric++\_ordering\_cycles2\] The transactions $T_2$ and $T_4$ remain with a participation in cycle $c_3$ each. We remove $T_2$ which clears $c_3$ and thereby the last cycle. This results in the state of Table \[table:fabric++\_ordering\_cycles3\]. [ C[1.5cm]{} || C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} | C[0.9cm]{} ]{} **Cycle** & & $T_1$ & & $T_3 $ & $T_4$ & $T_5$\ & & & & & &\ & & & & & &\ & & & & & &\ $\sum$ & 0 & 0 & 0 & 0 & 0 & 0\ \[table:fabric++\_ordering\_cycles3\] From this we now know that from the set $S'=\{T_1,T_3,T_4,T_5\}$ we can generate a serializable schedule, leading to the cycle-free conflict graph $C(S')$ (line \[l:buildConflictGraph2\]) as shown in Figure \[figs:fabric++\_ordering\_cyclefree\]. **Step (5)**: Generating the final schedule is essentially a repetitive execution of two parts until all nodes are scheduled: (a) the locating of the source node in the current subgraph (lines \[l:searchSourceStart\] to \[l:searchSourceEnd\]) and (b) the scheduling of all nodes that reachable from that source (lines \[l:scheduleStart\] to \[l:scheduleEnd\]). ![The cycle-free conflict graph $C(S')$, containing only the transactions $T_1$, $T_3$, $T_4$, and $T_5$.[]{data-label="figs:fabric++_ordering_cyclefree"}](fabric++_ordering_cyclefree.pdf){width="6cm"} We start part (a) at the node of $C(S')$ representing the transaction with the smallest subscript, namely $T_1$. From this starting node, we have to find a source node, as sources have to be scheduled last. $T_1$ has two parents, namely $T_3$ and $T_4$, so it not a source. We follow the edge to $T_3$, which has not been visited yet but is also not a source, as it has $T_4$ as a parent as well. We follow the edge to $T_4$, which has not been visited yet and which is a source. Therefore, we can schedule $T_4$ safely at the last position in our schedule, to which we refer to as *position $4$*. Now, part (b) starts as all nodes that are reachable from $T_4$ must be scheduled before it. $T_4$ has two children, namely $T_1$ and $T_3$. We follow the edge to $T_1$, which has not been scheduled yet. However, as $T_1$ has an incoming edge from $T_3$, we also can not directly schedule it. First, we visit $T_3$ and identify that it has a parent in form of $T_4$, the source at which we started. With this information, we know that $T_3$ must be scheduled at position $3$ and $T_1$ must be scheduled at position $2$. This ends part (b), as all reachable nodes have been scheduled. Next, we restart at the only remaining node $T_5$. As $T_5$ is not only a source but also a sink, we can schedule it instantly at position $1$. This results in the final schedule $T_5 \Rightarrow T_1 \Rightarrow T_3 \Rightarrow T_4$, which is returned to the orderer. Please note that our reordering mechanism is not guaranteed to abort a minimal number of transactions, as this would be a NP-hard problem. However, it offers a very lightweight way to generate a serializable schedule with a small number of aborts. ### Batch Cutting In the context of transaction reordering, we have to discuss and extend a mechanism within the ordering service, that we omitted for simplicity in the description of Fabric in Section \[sec:fabric\], namely *batch cutting*. When the ordering service receives the transactions in form of a constant stream, it decides based on multiple criteria when to “cut” a batch of transactions to finalize it and to form the block. In the vanilla version, a batch is cut as soon as one of the following three conditions hold: (a) The batch contains a certain number of transactions. (b) The batch has reached a certain size in terms of bytes. (c) A certain amount of time has passed since the first transaction of this batch was received. In Fabric++, we extend these criteria by one additional condition. We also cut the batch, if (d) the transactions within the batch access a certain number of unique keys. This condition ensures that the runtime of our reordering mechanism, in particular the time of step (1), remains bounded. ### Micro-Benchmark ![Workload 1: Varying the number of conflicts.[]{data-label="figs:ordering_benchmark1"}](rotate-eps-converted-to.pdf){width="10cm"} To analyze the effectiveness of our reordering mechanism, we first evaluate it in a stand-alone micro-benchmark in isolation of Fabric. For a given sequence of input transactions we compute the number of valid transactions for this particular sequence (called “arrival order” in the following plots) as well as for the sequence that is generated by our reordering mechanism (called “reordered” in the following plots). Additionally, we measure the time to compute the reordered schedule. In Figure \[figs:ordering\_benchmark1\], we test a workload pattern with varying number of conflicts. For the interested reader, we provide a second micro-benchmark in the Appendix \[sec:ordering\_benchmark2\] on the effect of varying the length of the cycles (Figure \[figs:ordering\_benchmark2\]) and see how well our reordering mechanism performs in comparison to the naive arrival order. ### Micro-Benchmark 1: Interleave reads and writes to vary the number of conflicts The first input sequence we test consists of two equal sized sub sequences, where one subsequence contains only transactions that perform writes (colored in ) and the other sequence only transactions that read (colored in ). Each transaction performs only one operation (either read or write). Neither two writes nor two reads happen to the same key. For the example of $n=6$ transactions, we start with the following sequence $S_1$: $$S_1 = T[\textcolor{red}{w(k_1)}], T[\textcolor{red}{w(k_2)}],T[\textcolor{red}{w(k_3)}], T[\textcolor{blue}{r(k_1)}], T[\textcolor{blue}{r(k_2)}], T[\textcolor{blue}{r(k_3)}]$$ To generate $S_i$, we move the last transaction of $S_{i-1}$ to the front, leading to the following sequences $S_2$, $S_3$, and $S_4$. $$S_2 = T[\textcolor{blue}{r(k_3)}], T[\textcolor{red}{w(k_1)}], T[\textcolor{red}{w(k_2)}],T[\textcolor{red}{w(k_3)}], T[\textcolor{blue}{r(k_1)}], T[\textcolor{blue}{r(k_2)}]$$ $$S_3 = T[\textcolor{blue}{r(k_2)}], T[\textcolor{blue}{r(k_3)}], T[\textcolor{red}{w(k_1)}], T[\textcolor{red}{w(k_2)}],T[\textcolor{red}{w(k_3)}], T[\textcolor{blue}{r(k_1)}]$$ $$S_4 = T[\textcolor{blue}{r(k_1)}], T[\textcolor{blue}{r(k_2)}], T[\textcolor{blue}{r(k_3)}], T[\textcolor{red}{w(k_1)}], T[\textcolor{red}{w(k_2)}],T[\textcolor{red}{w(k_3)}]$$ The more writing transactions happen before the corresponding reading transactions, the more conflicts happen. We want to find out whether our reordering mechanism can solve this problem. Figure \[figs:ordering\_benchmark1\] shows the results for $n=1024$ transactions. As we can see, our reordering mechanism is able to reorder the transactions for every input sequence in a way such that all transactions are valid. In contrast to that, the arrival order suffers under a lot of invalid reading transactions, if writing transactions happen before. We can also see that our reordering mechanism is computationally cheap: it takes only around 1 to 2 ms to rearrange the transactions on a Macbook Pro with Intel Core i7 running at 3.1 GHz. Early Transaction Abort using Advanced Concurrency Control {#ssec:early_abort} ---------------------------------------------------------- The reordering mechanism previously described not only tries to minimize the number of unnecessary aborts, it also enables a form of *early abort*. Transactions, that are removed from $S$ because of their participation in conflict cycles can be aborted already in the ordering phase instead of later on the validation phase. This assures that less transactions are distributed across the network. In the following, we want to push this concept of aborting transactions as early as possible in the pipeline to the limits. Additionally to early aborting transactions that occur in conflict cycles, we can integrate two more applications of early abort, as we will describe in Section \[ssec:early\_abort\_simulation\] and Section \[ssec:early\_abort\_ordering\]. The first one is happening already in the simulation phase. Let us see in the following how this works. ### Early Abort in the Simulation Phase {#ssec:early_abort_simulation} To realize early abort in the simulation phase, we first have to extend Fabric by a more fine-grained concurrency control mechanism, that allows for the parallel execution of simulation and validation phase within a peer. With such a mechanism at hand, we have the chance of identifying stale reads *during* the simulation already. To understand the concept, let us consider the example from Section \[sssec:early\_abort\_simulation\] again. With a fine-grained concurrency control mechanism, the block containing $T_1$, $T_2$, $T_3$, and $T_4$ would not have to pend for validation while the smart contract bound to the proposal $T_5$ is simulating. Instead, the four transactions would apply their updates in an atomic fashion *while* $T_5$ is simulating. As a consequence of this design, for every read $T_5$ performs, we can check whether the read value is still up-to-date. As soon as we detect a stale read, we can abort the simulation of the transaction proposal. Additionally, we directly notify the corresponding client about the abort, such that it can resubmit the proposal without delay. Let us discuss in the following, how exactly our fine-grained concurrency control mechanism works and how we realize it in Fabric++. In the context of modern database systems, advanced concurrency control mechanisms are well established [@hyper; @fsfekete; @movcc; @gconemvcc; @gctwomvcc; @anker]. Instead of locking the entire store, these techniques typically perform a fine-grained locking on the record level or even at the level of individual cells/values. As there is conceptually no difference between the store of a database system and the store used within the Fabric peers, similar techniques can be applied here. ![Parallelization with early abort using our fine-grained concurrency control.[]{data-label="figs:fabric++_cc"}](fabric++_cc.pdf){width="14cm"} As discussed in Section \[sec:fabric\], Fabric implements its current state in form of a key-value store, which maps each individual key to a pair of value and version-number. The version-number is actually composed of the ID of the transaction, that performed the update, as well as the ID of the block that contains the transaction. In the original version of Fabric, the sole purpose of the version-numbers is to identify stale reads. In the validation phase, for every transaction we check whether the version-number of the read value still matches the one in the current state. We can go one step further and exploit the available version-numbers to implement a lock-free concurrency control mechanism protecting the current state. To do so, in Fabric++, we first remove the read-write lock, that was unnecessarily sequentializing simulation and validation phase. The version-number, that is maintained with each value, is sufficient to ensure the same transaction isolation semantics as the vanilla version. As no lock is acquired anymore, we need a mechanism to ensure that updates performed by the validation phase are not seen by simulation phases running in parallel. To achieve this behavior, during simulation, we have to inspect the version-number of every read value and test whether it is still up-to-date. Figure \[figs:fabric++\_cc\] visualizes this concept using a concrete example. At the start of the simulation phase, we first identify the [[`block-ID`]{}]{} of the last block that made it into the ledger. Let us refer to this [[`block-ID`]{}]{} as the [[`last-block-ID`]{}]{}. In our example, [[`last-block-ID = 4`]{}]{}. During the simulation of a smart contract bound to a transaction proposal $T_{exec}$, no read must encounter a version-number containing a [[`block-ID`]{}]{} higher than the [[`last-block-ID`]{}]{}. If it does see a higher [[`block-ID`]{}]{} it means that during the simulation phase, a validated transaction $T_{valid}$ in the validation phase modified a value in the read set of $T_{exec}$ and thus, the read set is outdated. In our example, the read of $balA=70$ in the simulation phase happens *before* the update of $balA$ to $50$ in the validation phase. This is reflected by the version-number of $balA$, namely [[`block-ID = 4`]{}]{}. Therefore, this read is up-to-date and the simulation continues. In contrast to that, the read of $balB$ happens *after* the update of $balB$ to $100$ in the validation phase. This is reflected by the version-number of $balB$, namely [[`block-ID = 5`]{}]{}. As [[`5`]{}]{} is higher than the [[`last-block-ID = 4`]{}]{}, we can directly classify $T_{exec}$ as invalid, as the transaction will not have a chance to pass the validation phase later on. Please note that the overall correctness of our lock-free mechanism is ensured by the atomic updates of the version-numbers. ### Early Abort in the Ordering Phase {#ssec:early_abort_ordering} In addition to the early abort in the simulation phase, as explained in Section \[ssec:early\_abort\_simulation\], we can transition a similar concept also to the ordering phase. As Fabric performs commits at the granularity of whole blocks, two transactions within the same block, that read the same key, must read the same version of that key. For example, let us consider two transactions $T_6$ and $T_7$, where $T_6$ is ordered before $T_7$ within the same block ($T_6 \Rightarrow T_7$). If $T_6$ read version $v_1$ of a key $k$ and $T_7$ read version $v_2$ of $k$ in their respective simulations, then $T_7$ is invalid. Such a version mismatch can happen, if between the simulations of $T_6$ and $T_7$ a change to the value of $k$ was committed by a valid transaction from a previous block. Therefore, as soon as we detect a version mismatch between transactions within the same block, we can early abort the latter transaction. Again, this strategy assures that only those transactions end up in a block, that have a realistic chance of commit. Experimental Evaluation {#sec:ea} ======================= In the previous section, we have extended and modified core components of Fabric in several ways, turning it into Fabric++. It is now time to evaluate the modifications in terms of effectiveness. Primarily, we are interested in the throughput of valid/successful and invalid/failed transactions, that make it through the system. Secondarily, we are interested on the influence of certain system configurations and the workload characteristics on the system. Setup ----- Before starting with the actual experiments, let us discuss the setup. Our cluster consists of six identical servers, that are located within the same rack and connected via gigabit-ethernet. Four machines serve as peers, one machine runs the ordering service, and one machine serves as the client, which fires transaction proposals. Each server consists of two quad-core Intel Xeon CPU E5-2407 (SandyBridge architecture) running at $2.2$ GHz with $32$KB of L1 cache, $256$KB of L2 cache, and $10$MB of a shared L3 cache. $24$GB of DDR3 ram are attached to each of the two NUMA regions. The operating system used is a 64-bit Arch Linux with kernel version $4.17$. Fabric is set up to use LevelDB as the current state database. Benchmark Framework and Workload -------------------------------- In the database community, there exist numerous established benchmarks that can be used to test and to compare systems, such as TPC-C [@tpcc], TPC-H [@tpch], or YCSB [@ycsb]. Unfortunately, since blockchains are still a relatively young field, there exist only very few benchmarks with standardized workloads. At first, we looked at the Caliper [@caliper] benchmarking suite which seemed like a natural candidate, as it is part of the Hyperledger project just like Fabric. It is compatible to Fabric $1.2$, but comes with a few limitations: First, the framework provides only sample smart contracts and not a real benchmarking workload. Second, for certain metrics such as transactions per second or latency, it remains unspecified how they are actually measured. Third, it supports only a single channel. Apart from these limitations, other researcher have experienced incorrect behavior of Caliper in form of events, that were not properly registered. [@gauge_changes]. As a consequence, they released a fork of Caliper named Gauge [@gauge] that claims to resolve these problems. Unfortunately, Gauge is not compatible with version $1.2$ of Fabric right now. Next, we looked at Blockbench, which originates from a survey paper [@BlockchainSurvey] on blockchain systems. While Blockbench actually provides some benchmarking workloads such as YCSB, again, it lacks the support for Fabric $1.2$ and would need significant changes to make it compatible. As a consequence of this journey, we decided to build our own benchmarking framework and to introduce a highly customizable workload. This allows us to fire transaction proposals at a specified rate from multiple clients in multiple channels. Our benchmark setup looks as follows: Initially, we create a certain number of accounts (10,000 accounts throughout this section, 20,000 accounts in Appendix \[sec:extended\_throughput\]), each represented by a randomly generated account balance. Our workload is formed of a single smart contract, that reads and writes an adjustable number of account balances, simulating a typical asset transfer scenario between accounts. Among the accounts, there exist a certain number of hot accounts, that are involved in transactions more frequently than the remaining ones. By varying the number of read and write accesses per transaction, the probability of picking hot accounts, and the number of hot accounts, we are able to generate a wide range of different workloads. In a single run, we fire a constant stream of transaction proposals, that are bound to our smart contract, for a certain amount of time at a certain firing rate. In the following, we test numerous different system and workload configurations to identify the impact of the system. In the individual experiments, we will detail the chosen configuration. Transactional Throughput {#ssec:throughput} ------------------------ We start our experimental evaluation by testing Fabric and Fabric++ under probably the most important criterium for a transaction processing system, namely the throughput of transactions. We differentiate between successful and failed transactions: a good system should try to maximize the number of successful transactions while keeping the number of failed transactions as small as possible. [ L[12cm]{} || L[3.3cm]{} ]{} **Experiment Parameters** & **Values**\ Fired transaction proposals per second per client & 512\ Duration in which transaction proposals are fired & 90 sec\ Number of channels & 1\ Number of clients per channel & 4\ \ **System Parameters** & **Values**\ Maximum time to form a block & 1 sec\ Maximum number of keys accessed per block& 16384\ Maximum size per block & 2MB\ Maximum number of transactions per block (BS) & 256, 512, 1024\ \ **Workload Parameters** & **Values**\ Number of account balances & 10000\ Number of read & written balances per transaction (RW) & 4, 8\ Probability for picking a hot account for reading (HR) & 10%, 20%, 40%\ Probability for picking a hot account for writing (HW) & 5%, 10%\ Number of hot account balances (HSS) & 1%, 2%, 4%\ \[table:config\] To measure this property, we fire a constant stream of transaction proposals for $90$ seconds into a single channel using four clients. Each client fires at a rate of $512$ proposals per second. This firing rate is sufficient to fully sustain the system in our setup. Table \[table:config\] shows the detailed configuration. To identify their impact on the throughput, we vary five important parameters: the maximum number of transactions per block (BS), the number of read balances and written balances per transaction (RW), the probability for picking a hot account for reading (HR) respectively for writing (HW), as well as the number of hot account balances (HSS). In total, we evaluate $108$ different configurations in this experiment. Figure \[figs:configs\] shows the results. First and foremost, we vary the maximum number of transaction per block (BS), as it has a large impact on the transaction processing in general and the ordering in particular. The results for Fabric and Fabric++ for BS=256 are presented in first and the second row, for BS=512 in the third and fourth row, and for BS=1024 in the fifth and sixth row, respectively. Along the columns, we vary the remaining four parameters RW, HR, HW, and HSS in a total of 36 configurations. For a single run, we show the transactional throughput () that was achieved for each second of the $90$ second run. This throughput is additionally split into successful transactions () and failed transactions (). To interpret the results, let us look at Figure \[figs:configs\] as a whole. We can see that Fabric++ significantly increases the throughput of successful transactions over Fabric for essentially all tested configurations. For vanilla Fabric, we can observe that under configurations accessing many accounts (RW=8), the number of failed transactions per second is actually significantly higher than the throughput of successful transactions. This problem is highly reduced by Fabric++, where the successful throughput is at least on par with the failed throughput, or even dominates it. The largest improvement of Fabric++ over Fabric in terms of successful transactions we observe is around factor 3x for the configuration BS=1024, RW=8, HR=40%, HW=10%, HSS=1%, which we also show in a zoomed-in version in Figure \[figs:zoomin\] of the appendix. We also observe a significant decrease in the throughput of the successful transactions with the increase in the hotness of the transactions. For large block-sizes (BS $\in \{512, 1024\}$), each block ($b_i$) roughly updates every key in the hotset and a large fraction of coldset. This forces most of the transactions in block ($b_{i+1}$) to abort because of read-write conflicts. So, we observe a pattern of blocks committing with alternating highly-successful and highly-failed transactions. In Fabric, most of the transactions are aborted due to this inter-block conflicts. In addition to this, due to a large block size, Fabric creates a large amount of within-block conflicts, which results in a large fraction of the total number of processed transactions to abort. In Fabric++, we observe a similar alternating behavior in terms of cross-block conflicts. However, since Fabric++ reorders the transactions within the block to remove the within-block conflicts, the number of successful transactions remain on-par with the number of failed transactions. We observe that the strength of Fabric++ lies in contended workload, where the hotness has temporal behavior. If, due to temporal behavior, hot reads, and updates end up in a same block, Fabric++ can possibly optimize the order of transactions to extract a largest set of transactions that have a chance to commit. In contrast to Fabric++, Fabric will behave similarly for temporal and non-temporal hotness in the workloads, forcing a large fraction of transactions to abort, even though they could commit. Apart from the overall comparison of Fabric and Fabric++, we can analyze the influence of the parameters on the system. A larger block size generally results in a higher throughput. In the case of Fabric++, a larger block size also increases the reordering possibilities of our mechanism. Besides, we can see that a higher number of accesses per transactions results in more failed transactions. Optimization Breakdown ---------------------- In Section \[ssec:throughput\], we measured the throughput of Fabric++ with both optimizations activated. Let us now see at a sample configuration, how much the individual optimizations of reordering and early abort contribute to the improvement. Figure \[figs:breakdown\] shows the improvement breakdown for the configuration BS=1024, RW=8, HR=40%, HW=10%, HSS=1% in comparison to standard Fabric. While Fabric achieves only a throughput of around $100$ successful transactions per second, activating one of our two optimization techniques alone improves this to around $150$ transactions per second. In comparison to that, activating both techniques at the same time results in the highest throughput of successful transactions with around $220$ transactions per second. This shows nicely how both techniques work together: Transactions, that are already early aborted in the simulation phase do not end up in a block in the ordering phase. As a consequence, only transactions, that have a realistic chance of being successful, are considered in the reordering process. Scaling Channels and Clients ---------------------------- So far, in all experiments we used four clients to fire transactions into a single channel. Let us now vary the number of channels as well as the number of clients per channel to see the effect on the throughput. Again, we use the configuration BS=1024, RW=8, HR=40%, HW=10%, HSS=1% and evaluate the average throughput of successful transactions for Fabric and Fabric++. First, we vary the number of channels in Figure \[figs:scaling\_channels\] from $1$ to $8$. Per channel, we use $2$ clients to fire transaction proposals. We can see that when going from $1$ channel to $4$ channels, the throughput of both Fabric and Fabric++ significantly increases. Obviously, the additional mechanisms of Fabric++ do not harm the scaling with the number of channels. Only when using $8$ channels, the throughput decreases again for both Fabric and Fabric++. This is simply the case because individual channels start competing for resources. This also increases the number of failed transactions: Scaling from $1$ to $8$ channels increases the number of failed transactions from $213$ TPS to $837$ TPS for Fabric and from $81$ TPS to $704$ TPS for Fabric++. Due to the competition for resources, individual simulations phase take longer and increase the chance of working on stale data. After varying the number of channels, let us now vary the number of clients per channel in Figure \[figs:scaling\_clients\]. We test $1$, $2$, $4$, and $8$ clients, where all clients fire their transaction proposals into a single channel. Here, the picture is a slightly different to the behavior when scaling channels. The throughput of Fabric increases very gently with the number of clients, and we see an improvement from around $60$ to $105$ successful transactions per seconds when going from $1$ to $8$ clients. For Fabric++, we see the highest throughput with around $205$ successful transactions per second already for $2$ clients. For $8$ clients, the throughput drops by around factor $2$ to the throughput of Fabric, clearly showing that the firing clients also compete for resources. This is also visible in an increase in failed transactions when going from $1$ to $8$ clients per channel, which increase from $86$ TPS to $928$ TPS for Fabric and from $20$ TPS to $841$ TPS for Fabric++. Conclusion {#sec:conclusion} ========== In this work, we identified strong similarities of the transaction pipeline of contemporary blockchain systems at the case of Hyperledger Fabric and distributed database systems in general. We analyzed these similarities in detail and exploited them to transition mature techniques from the context of database systems to Fabric, namely transaction reordering to remove serialization conflicts as well as early abort of transactions, that have no chance to commit. In an extended experimental evaluation, where we tested $108$ different configurations of workload and system, we showed that this improved version Fabric++ outperforms the vanilla Fabric in terms of throughput of successful transactions by up to factor $3$x, while keeping the scaling capabilities intact. Acknowledgments {#acknowledgments .unnumbered} ---------------- We would like to thank Immanuel Haffner for helping us in setting up the Fabric cluster, running benchmarks, as well as profiling the internals of the system. Throughput Timeline =================== Figure \[figs:zoomin\] presents the detailed zoom-in of the run for configuration BS=1024, RW=8, HR=40%, HW=10%, HSS=1% for Fabric (Figure \[figs:zoomin\_baseline\]) and Fabric++ (Figure \[figs:zoomin\_master\]). We can see that the throughput remains very stable over the run of $90$ seconds. In the beginning, there is a small ramp-up phase visible, which is actually very interesting. For Fabric, the throughput of successful transactions directly starts very low with only $100$ transactions per second. In contrast to that, for Fabric++, the initial throughput of successful transactions almost reaches $500$ transactions per second with the number of failed transactions per second being $0$. This shows that for the first block, our reordering mechanism manages to completely resolve all intra-block conflicts. After that, inter-block conflicts can arise which increase the number of failed transactions in any case. Ordering Service Micro-Benchmark 2: Vary the length of cycles {#sec:ordering_benchmark2} ============================================================= In the following experiment, we want to analyze the impact of cycles on the arrival order and on our reordering mechanism. To do so, we again form a sequence of $n$ transactions, that contains $n/t$ cycles of size $t$ transactions of the form $$T[\textcolor{blue}{r(k_0)}, \textcolor{red}{w(k_0)}], T[\textcolor{blue}{r(k_0)}, \textcolor{red}{w(k_1)}], T[\textcolor{blue}{r(k_1)}, \textcolor{red}{w(k_2)}], T[\textcolor{blue}{r(k_2)}, \textcolor{red}{w(k_0)}]$$ Again, we want to identify how many transactions are valid under the arrival order and when using our reordering mechanism. Figure \[figs:ordering\_benchmark2\] shows the results for $1024$ transactions. For the arrival order, only half of transactions are valid, no matter of the cycle length. This is because aborting every second transaction breaks the cycles. In comparison to that, our reordering mechanism is able to achieve a high number of valid transactions, if the cycles are sufficiently long respectively, there are not too many cycles to cancel. Of course, our algorithm becomes more expensive with the length of the cycles to break. However, since extremely long cycles are very unlikely to occur in reality, the runtime of our mechanism will in general remain low in the ordering phase, as we will see in the full fledged evaluation later on. ![Workload 2: Varying the length of the cycles.[]{data-label="figs:ordering_benchmark2"}](cycles-eps-converted-to.pdf){width="8.5cm"} Extended Throughput Evaluation {#sec:extended_throughput} ============================== Additionally to 10,000 account balances, as used in the previous experimental evaluation, we test Fabric and Fabric++ as well under 20,000 account balances and 2 read and write accesses per transaction (RW=2). We can observe that for RW=2, the number of successful transactions is significantly higher than the number of failed transactions due to less conflicts than for RW=4 or RW=8. [^1]: The version number is actually composed of transaction-ID and the block-ID, see Section \[ssec:early\_abort\_simulation\] for details. [^2]: Smart contracts are typically called *chaincodes* in Fabric. However, as they do not conceptually differ from smart contracts in blockchain systems such as Ethereum, we stick to this term throughout the paper. [^3]: They might not match due to non-determinism in the smart contract or due to malicious behavior of the endorsement peer(s). [^4]: The read lock can be shared by multiple simulation phases, as they do not modify the current state.

--- abstract: 'We construct breather and rogue wave solutions of a variable coefficient nonlinear Schrödinger equation with an external linear potential. This generalized model describes the nonlinear wave propagation in an inhomogeneous plasma/medium. We derive several localized solutions including Ma breather, Akhmediev breather, two-breather and rogue wave solutions of this model and show how the inhomogeneity of space modifies the shape and orientation of these localized structures. We also depict the trajectories of the inhomogeneous rogue wave. Our results may be useful for controlling plasmonic energy along the plasma surface.' address: | Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University,\ Tiruchirappalli - 620 024, Tamil Nadu, India author: - 'K. Manikandan and M. Senthilvelan' title: On the characterization of breather and rogue wave solutions of an inhomogeneous nonlinear Schrödinger equation --- Introduction ============ The study of nonlinear wave propagation in dispersive and inhomogeneous media is of great interest. It has a wide range of applications such as radio waves in the ionosphere [@budd], waves in the ocean [@osbore], optical pulses in glass fibers [@hase], laser radiation in plasma [@tewari] and so on. The propagation of a general dimensionless nonlinear wave envelope in a weakly inhomogeneous plasma/medium obeys the following variable coefficients nonlinear Schrödinger equation (NLS) equation [@chen:liu; @bala; @serkin], namely $$i\psi_t+\frac{1}{2}\psi_{xx}+h(t)\vert \psi\vert ^2\psi+M(x,t)\psi=0, \label{wav:eqn}$$ where $\psi(x,t)$ represents the complex envelope of the physical field, $x$ is the longitudinal variable, $t$ is the transverse variable and subscripts denote partial derivative with respect to that variable, $h(t)$ and $M(x,t)$ refer the nonlinearity and inhomogeneity management parameters respectively. The study of inhomogeneous NLS (INLS) equation with longitudinally and transversely varying inhomogeneity is of contemporary interest in several branches of physics [@atre:pani; @solli; @yan; @zhong]. In this paper, we choose the inhomogeneity of the linear potential $M(x,t)$ to be $2kx$, so that Eq. (\[wav:eqn\]) takes the following form [@chen:liu; @bala] $$i\psi_t+\frac{1}{2}\psi_{xx}+h(t)\vert \psi\vert ^2\psi+2kx\psi=0, \label{aa1}$$ where $k$ is the inhomogeneity parameter. During the past two decades several investigations have been made on to study how the inhomogeneous medium affects the propagation of solitary waves [@chen:liu; @bala; @belm; @haseg]. However only fewer works have been devoted to analyze how the inhomogeneity of space and time affects the other localized solutions like breather and rogue waves (RWs) [@dai; @yang; @xfwu; @wang; @khaw; @shuk]. In this work we intend to concentrate on this particular aspect. Breather is a localized solution with temporally or spatially periodic structure and appears to be the internal oscillations and bound states of nonlinear wave packets [@mande]. A RW is a wave which is localized in both space and time and appears from nowhere and disappears without a trace [@karif]. A wave is classified under this category when its wave height (distance from trough to crest) reaches a value which is at least twice the significant wave height [@kharif; @osborn]. Even though it was first observed in arbitrary depth of ocean, the phenomenon is now shown to appear in diverse areas of physics [@blud:kono; @kibler; @chab:hoff; @bailung; @shat; @ya; @mosl]. Certain kinds of exact solutions of NLS equation such as Peregrine soliton [@pere; @akmv:anki], time periodic breather or Ma soliton (MS) [@ma] and space periodic breather or Akhmediev breather (AB) [@eleon; @kadz] have been considered to describe the possible mechanism for the formation of RWs. Motivated by the contemporary development in the investigation of the breather and RW solutions of NLS type equations, in this paper, we construct Ma breather, AB, two-breathers, first-, second-, and third-order RW solutions with and without free parameters through the similarity transformation method. We also investigate in-detail the impact of inhomogeneity on these localized solutions. To construct exact solutions of (\[aa1\]) one may look for a transformation that can map the given equation to the standard NLS equation. The most popular way of transforming INLS equation to the NLS equation is the similarity transformation method [@serkin; @belm; @susl; @pono]. The necessary similarity transformation can be obtained by considering a generalized transformation $\psi(x,t)$=$\frac{1}{\sqrt{\mu(t)}}\phi(\xi,\tau)\exp[i (\alpha(t)x^2+\delta(t)x+\kappa(t))]$, where $\xi=\beta(t)x+\epsilon(t)$, $\tau=\gamma(t)$ and $\mu(t)$, $\alpha(t)$, $\beta(t)$, $\gamma(t)$, $\delta(t)$, $\epsilon(t)$ and $\kappa(t)$ are arbitrary functions of $t$, which can map Eq. (\[aa1\]) to the NLS equation $$i\phi_{\tau}+\frac{1}{2}\phi_{\xi\xi}+ \vert \phi\vert ^2\phi=0. \label{a3}$$ Substituting this transformation in (\[aa1\]) and eliminating $\phi_{\tau}$ by Eq. (\[a3\]) one can obtain a set of differential equations for these unknown arbitrary functions. Solving them we can find these arbitrary functions which in turn fix the exact form. As a result $$\begin{aligned} \label{a16} \psi(x,t)&=&\frac{\phi(\xi,\tau)}{\sqrt{\mu_0(1+2\alpha_0 t)}}\exp\left[i \left(\frac{\alpha_0}{1+2\alpha_0 t}x^2 +\left(kt+\frac{\delta_0+kt}{1+2\alpha_0 t}\right)x \nonumber \right.\right. \\ && \left.\left. +\kappa_0-\frac{k^2t^3}{6}-\frac{t(\delta_0+kt)^2}{2(1+2\alpha_0 t)}\right)\right],\end{aligned}$$ where $\phi(\xi,\tau)$ is the solution of NLS equation and provided $h(t)$ to be $\displaystyle{\frac{h_0 \mu_0 \beta_0^2}{1+2\alpha_0 t}}$. With this choice of $h(t)$ we have an integrable INLS equation of the form $$\begin{aligned} \label{a17} i\psi_t+\frac{1}{2}\psi_{xx}+\frac{\mu_0 \beta_0^2}{1+2\alpha_0 t}\vert \psi\vert ^2\psi+2kx\psi=0,\end{aligned}$$ where we have taken $h_0=1$ without loss of generality. In the above $h_0$, $\mu_0$, $\alpha_0$, $\beta_0$, $\gamma_0$, $\delta_0$, $\epsilon_0$ and $\kappa_0$ are integration constants. Since (\[a17\]) can be mapped to the standard NLS equation we can generate certain new localized structures including breather and rogue wave (RW) solutions and study how these localized solutions are affected by the inhomogeneity parameter. The paper is organized as follows. In Sec. 2, we construct AB, Ma and two-breather solutions to INLS equation (\[a17\]) and study their characteristics in detail. In Sec. 3, we construct RW solutions without and with free parameters and investigate how these RW structures get modified in the plane wave background. In Sec. 4, we examine certain characteristics of RW, namely the evolution of its peak and width and depict the trajectories of inhomogeneous RW. Finally, in section 5, we present a summary of our results and conclusions. Characteristics of breathers ============================ To begin with we construct first- and second-order breather solutions of (\[a17\]). The breather solution of NLS equation is given by [@eleon] $$\label{a18} \phi_1(\xi,\tau)=\left[\frac{m^2 \cosh(d \tau_s) + 2 i m v \sinh(d \tau_s)}{2 (\cosh(d \tau_s) - v \cos(m (\xi - \xi_1)))}-1\right]e^{i\tau},$$ where $\tau_s$=$\tau - \tau_1$, the parameters $m$ and $v$ are expressed in terms of a complex eigenvalue (say $l$), that is $m=2\sqrt{1+l^2}$ and $v$ = $Im(l$), and $\xi_1$ and $\tau_1$ serve as coordinate shifts from the origin. The parameter $d(=m v)$ in (\[a18\]) is the growth rate of modulation instability. Substituting the breather solution (\[a18\]) in (\[a16\]) we can capture the breather solution of (\[a17\]). When $v$ lies between $0$ and $1$ and $m$ is real, one can obtain the AB solution which is periodic in $x$ and localized in $t$. On the other hand when $v>1$ and $m$ is imaginary, Eq. (\[a16\]) provides Ma breather solution which is periodic in $t$ and localized in $x$. ![(a) AB profile for $l=0.5 I$, (b) Ma breather profile for $l=1.2 I$, (c) and (d) are their corresponding contour plots. The parameters are $\beta_0=2.0$, $\mu_0=2.0$, $\alpha_0=0.01$, $\gamma_0=1$, $\delta_0=0.01$, $\epsilon_0=1$ and $\kappa_0=1$ and $k=0.01$.[]{data-label="fig1"}](inho-br.pdf){width="0.85\linewidth"} In Fig. \[fig1\] we display the evolution of AB and Ma breather wave solutions of (\[a17\]) for $k=0.01$. Fig. \[fig1\](a) represents the evolution of an AB for the eigenvalue $l=0.5 i$ and Fig. \[fig1\](b) corresponds to Ma breather for the eigenvalue $l=1.2 i$. The corresponding contour plots are given in the second row. When we increase the strength of the inhomogeneity parameter ($k$) to $2$, the AB gets stretched in space along the positive $x$ direction which is demonstrated in Fig. \[fig1a\](a). For $k=-2$ the stretching occurs in the reverse direction as shown in Fig. \[fig1a\](b). As for as the Ma breather is concerned, if we $k$ is increase it bends in the plane wave background. The corresponding structure is given in Fig. \[fig1a\](c) for $k=0.5$. The Ma breather gets curved in the reverse direction for negative value of $k$ which is not displayed here. The second row in Fig. \[fig1a\] represents their corresponding contour plots which clearly illustrate how the inhomogeneity parameter influences the breather structures. ![(a) AB for $k=2.0$, (b) AB for $k=-2.0$, (c) Ma breather for $k=0.5$, (d), (e) and (f) are their corresponding contour plots. The other parameters are same as in Fig. \[fig1\].[]{data-label="fig1a"}](inho-br1.pdf){width="0.99\linewidth"} We proceed to construct two-breather solutions of (\[a17\]) and analyze how these solutions get distorted by the inhomogeneity of space. The two-breather solution of NLS equation is given by [@kadz], $$\label{a18a} \phi_2(\xi,\tau)=\left[(-1)^j+\frac{G_2(\xi,\tau)+i H_2(\xi,\tau)}{D_2(\xi,\tau)}\right]\exp{(i\tau)},$$ where $G_2$, $H_2$, and $D_2$ are given by $$\begin{aligned} G_2 &=& -(k_1^2-k_2^2) \left[\frac{k_1^2\delta_2}{k_2}\cosh(\delta_1\tau_{s1})\cos(k_2\xi_{s2})-\frac{k_2^2\delta_1}{k_1}\cosh(\delta_2\tau_{s2})\cos(k_1 \xi_{s1}) \right. \nonumber \\ && \left. - (k_1^2-k_2^2)\cosh(\delta_1\tau_{s1})\cosh(\delta_2\tau_{s2})\right], \nonumber \\ \label{a19} H_2&=& -2 (k_1^2 - k_2^2) \left[\frac{\delta_1 \delta_2}{k_2} \sinh(\delta_1 \tau_{s1}) -\cos(k_2 \xi_{s2}) - \frac{\delta_1 \delta_2}{k_1}\sinh(\delta_2\tau_{s2}) \cos(k_1 \xi_{s1}) \right. \nonumber \\ && \left. -\delta_1 \sinh(\delta_1 \tau_{s1})\cosh(\delta_2 \tau_{s2}) + \delta_2 \sinh(\delta_2 \tau_{s2})\cosh(\delta_1 \tau_{s1})\right], \\ D_2 &=& 2 (k_1^2 + k_2^2) \frac{\delta_1 \delta_2}{k_1 k_2} \cos(k_1\xi_{s1})\cos(k_2 \xi_{s2}) + 4 \delta_1 \delta_2 (\sin(k_1 \xi_{s1}) \sin(k_2 \xi_{s2}) \nonumber \\ && + \sinh(\delta_1 \tau_{s1}\sinh(\delta_2 \tau_{s2}) - (2 k_1^2 - k_1^2 k_2^2 + 2 k_2^2) \cosh(\delta_1 \tau_{s1})\cosh(\delta_2 \tau_{s2}) \nonumber \\ && - 2 (k_1^2 - k_2^2) \left(\frac{\delta_1}{k_1}\cos(k_1 \xi_{s1})\cosh(\delta_2 \tau_{s2})- \frac{\delta_2}{k_2}\cos(k_2 \xi_{s2}) \cosh(\delta_1 \tau_{s1})\right), \nonumber\end{aligned}$$ where the modulation frequencies, $k_j=2\sqrt{1+l_j^2}$, $j=1,2$, are described by the (imaginary) eigenvalues $l_j$. In Eq. (\[a19\]), $\xi_j, \tau_j$, $j=1,2$, represents the shifted point of origin, $\delta_j(=k_j\sqrt{4-k_j^2}/2)$ is the instability growth rate of each component and $\xi_{sj}=\xi-\xi_j$ and $\tau_{sj}=\tau-\tau_j$ are shifted variables. ![(a) Two AB profile for $l_1=0.5i$ and $l_2=0.7i$ with $\tau_1=5$ and $\tau_2=-5$, (b) Two AB without time shifts, (c) the intersection of AB-Ma breathers for $l_1=0.5i$ and $l_2=1.2i$, (d) Two Ma breather for $l_1=1.1i$ and $l_2=1.2i$ with $\xi_1=3$ and $\xi_2=-3$, (e) Two Ma breather without space shifts and (f) Two Ma breather for $l_1=1.1i$ and $l_2=1.11i$. The other parameters are same as in Fig. \[fig1\].[]{data-label="fig2"}](inho-br2.pdf){width="0.99\linewidth"} ![(a) Two AB for $k=1.1$, (b) the intersection of AB-Ma breathers for $k=0.5$, (c) two Ma breather for $k=0.5$, (d), (e) and (f) are their corresponding contour plots.[]{data-label="fig2a"}](inho-br2b.pdf){width="0.99\linewidth"} With two purely imaginary eigenvalues, $l_j$, $j=1,2$, the solution (\[a18a\]) is capable of describing a variety of possible second-order breather structures. The solution includes ABs, Ma solitons and the intersection of AB and Ma breathers in certain combination of eigenvalues. For example, when both the eigenvalues $Im(l_j)$, $j=1,2$, lie between $0$ and $1$, we obtain the ABs. On the other hand when both of them are greater than one ($Im(l_j)>1$) we obtain the Ma breathers and the mixed possibility, that is one of the eigenvalues is less than one and the other eigenvalue is greater than one, we obtain the intersection of AB and Ma breathers. Inserting (\[a18a\]) in (\[a16\]) we obtain the general two-breather solution of the INLS Eq. (\[a17\]). Fig. \[fig2\] displays the evolution of two-breather solution of (\[a17\]) for $k=0.01$ with the imaginary eigenvalues. To obtain the ABs profile from (\[a18a\]) we restrict both the eigenvalues $Im(l_1)$ and $Im(l_2)$ to be less than 1 ($l_1=0.5i$ and $l_2=0.7i$). One AB developing with a time delay after another is shown in Fig. \[fig2\](a) and without the time delay is given in Fig. \[fig2\](b). When we change the eigenvalues to $l_1=0.5i$ and $l_2=1.2i$, the AB intersects with Ma breather which is demonstrated in Fig. \[fig2\](c). When both the eigenvalues $Im(l_1)$ and $Im(l_2)$ are greater than 1, say for example $l_1=1.1i$ and $l_2=1.2i$, we obtain two Ma breather solutions from (\[a16\]). Similarly the developing of the Ma breather with and without spatial delay is shown in Figs. \[fig2\](d) and \[fig2\](e) respectively. We also observe that the distance between the Ma breathers increases when we set both the eigenvalues to be nearly equal, say for example $l_1=1.1i$ and $l_2=1.11i$. It is shown in Fig. \[fig2\](f). When the strength of the inhomogeneity parameter $k$ is raised to $1.1$, both the ABs are obliquely stretched in $x$ plane which is demonstrated in Fig. \[fig2a\](a). When $k=0.5$, the intersection of AB and Ma breather and both the Ma breathers develop a bending structure in the plane wave background. They are illustrated in Figs. \[fig2a\](b) and \[fig2a\](c) respectively. When $k$ is negative, ABs get stretched in the reverse direction. As we expect AB-Ma/both the Ma breathers develop a bending structure in the negative $x$ direction which is not presented here. Recently breathers have been realized experimentally in optical fibers [@kibler] and plasmas [@bailung]. The results presented here will also be realized in the experimental context of nonlinear wave propagation in an inhomogeneous plasma. Characteristics of rogue waves ============================== Next we move on to investigate the RW solution of (\[a17\]). The RW solution can be obtained from the AB/Ma breather solutions as a limiting case [@chab:hoff]. The RW solution has the following basic structure: $$\label{a22} \phi_j(\xi,\tau)=\left[(-1)^j+\frac{G_j+i\tau H_j}{D_j}\right]\exp{(i\tau)}, \;\;\; j=1,2,...N,$$ where $G_j,H_j$ and $D_j$ are polynomials in $\xi$ and $\tau$. The first-order $(j=1)$ RW solution is given by [@akmv:anki] $$\label{a23} \phi_1(\xi,\tau)=\left(1-\frac{4(1+2i\tau)}{1+4\xi^2+4\tau^2}\right)\exp{(i\tau)},$$ and the second-order $(j=2)$ RW solution is given by $$\label{a24} \phi_2(\xi,\tau)=\left[1+\frac{G_2+i\tau H_2}{D_2}\right]\exp{(i\tau)},$$ where $$\begin{aligned} G_2&=&\frac{3}{8}-3\xi^2-2\xi^4-9\tau^2-10\tau^4-12\xi^2\tau^2, \nonumber \\ H_2&=&\frac{15}{4}+6\xi^2-4\xi^4-2\tau^2-4\tau^4-8\xi^2\tau^2, \nonumber \\ D_2 & = &\frac{1}{8}\left(\frac{3}{4}+9\xi^2+4\xi^4+\frac{16}{3}\xi^6+33\tau^2+36\tau^4 \right. \nonumber \\ & & \left.+\frac{16}{3}\tau^6-24\xi^2\tau^2+16\xi^4\tau^2+16\xi^2\tau^4\right). \nonumber\end{aligned}$$ ![(a) First-order RW, (b) second-order RW and (c) third-order RW for $k=0.1$. The other parameters are same as in Fig. \[fig1\].[]{data-label="fig3"}](inho-rog.pdf){width="0.99\linewidth"} When $j=3$ in Eq. (\[a22\]) we get the third-order $(j=3)$ RW solution [@akmv:anki] where $G_3$, $H_3$ and $D_3$ are polynomials in $\xi$ and $\tau$. Since the explicit expression of third-order RW solution is very lengthy we have not given the underlying expression here and only a graphical analysis of the third-order RW solution is presented. Substituting (\[a23\]) and (\[a24\]) in (\[a16\]) one can get first and second-order RW solutions of the INLS Eq. (\[a17\]). We fix the constants $\mu_0$ and $\beta_0$ to be $2.0$ and display the RW solution of (\[a17\]) for two different values of the inhomogeneous parameter, say $k=0.1$ and $k=2$, in Figs. \[fig3\] and \[fig3a\]. In Fig. \[fig3\], we have given the solution plots of (a) first-, (b) second- and (c) third-order RW solutions for $k=0.1$. These waves are localized both in space and time thus revealing the characteristic feature of RWs. When we increase the strength of inhomogeneity parameter $k$ to $2$, we observe that the wave crests get stretched in space and the RW structures get distorted in the plane which is demonstrated in Fig. \[fig3a\]. The variations can be seen more clearly in their respective contour plots which are given in the second row in Fig. \[fig3a\]. Here also when $k$ is negative the RWs get stretched in the reverse direction. ![(a) First-order RW, (b) second-order RW, (c) third-order RW for $k=2$, (d), (e) and (f) are their corresponding contour plots.[]{data-label="fig3a"}](inho-roga.pdf){width="0.99\linewidth"} Very recently it has been shown that one can also introduce certain free parameters in the RW solutions and by varying these free parameters one can extract certain patterns exhibited by these RWs [@ankie]. Motivated by this, in the following, we consider the second- and third-order RW solutions with suitable free parameters and analyze how the RW patterns change with respect to these free parameters for a particular value of inhomogeneity parameter. To begin with we confine our attention to the second-order RW solution. In this case, we have the following modified expressions for $G_2$, $H_2$ and $D_2$, that is $$\begin{aligned} G_2&=& 12(3-16\xi^4-24\xi^2(4\tau^2+1)-48 l\xi-80\tau^4-72\tau^2-48 m\tau), \nonumber \\ H_2&=& 24 \left[\tau(15-16\xi^4+24\xi^2-48 l\xi-8(1-4\xi^2)\tau^2-16\tau^4)+6m(1-4\tau^2+4\xi^2)\right], \nonumber \\ D_2 & = & 64\xi^6+48\xi^4(4\tau^2+1)+12\xi^2(3-4\tau^2)^2+64\tau^6+432\tau^4+396\tau^2+9 \nonumber \\ & & +48m(18m+\tau(9+4\tau^2-12\xi^2))+48l(18l+\xi(3+12\tau^2-4\xi^2)). \nonumber\end{aligned}$$ ![RW triplets for $k=0.1$. Parameters (a) $l=20$ and $m=30$, (b) $l=60$ and $m=100$ and (c) $l=120$ and $m=20$.[]{data-label="fig4"}](inho-trip.pdf){width="0.99\linewidth"} ![(a), (b) and (c) RW triplets when $k=1.5$, (d), (e) and (f) are their corresponding contour plots.[]{data-label="fig4a"}](inho-tripa.pdf){width="0.99\linewidth"} This RW solution contains two free parameters, namely $l$ and $m$. When $l= m=0$, this solution coincides with the one given earlier (vide Eq. (\[a24\])) which contains one largest crest and four subcrests with two deepest troughs. When $l$ and $m$ are not equal to 0, the second-order RW splits into three first-order RWs. These waves emerge in a triangular fashion (a triplet pattern). The parameters $l$ and $m$ describe the relative positions of the first-order RWs in the triplet. The three first-order RWs form a triangular pattern with $120$ degrees of angular separation between them [@ankie]. We observe this triangular pattern for small values of $l$ and large values of $m$. On the other hand when $l$ is large and $m$ is small the peaks in the triplet move to new positions in the triangle. In Fig. \[fig4\] we display the triplet pattern for $k=0.1$. The formation of triangular pattern is shown in Fig. \[fig4\](a) for $l=20$ and $m=30$. When we increase the value of $l$ and $m$ to $60$ and $100$ respectively the distance between the peaks in the triplet increases but amplitude of none of the peaks changes which can be seen in Fig. \[fig4\](b). At $l=120$ and $m=20$, the three peaks take new positions which is shown in Fig. \[fig4\](c). When we increase the value of $k$ to $1.5$ the triplet RWs get stretched in space with a curved structure in the plane wave background which is illustrated in Fig. \[fig4a\]. When $k$ is negative the triplet RWs get stretched in the reverse direction. ![RW sextet for $k=0.1$. Parameters (a) $l=10$ and $m=10$, (b) $l=100$ and $m=10$ and (c) $l=-100$ and $m=-10$.[]{data-label="fig5"}](inho-sixt.pdf){width="0.99\linewidth"} ![(a), (b) and (c) RW sextet when $k=1.5$, (d), (e) and (f) are their corresponding contour plots.[]{data-label="fig5a"}](inho-sixta.pdf){width="0.99\linewidth"} We then move on to investigate the structure of third-order RW solution with four free parameters, namely $l,m,g$ and $e$. The third-order RW solution with four free parameters is much lengthier than the one without free parameters and so we leave the expression and analyze the results only graphically. Here also we investigate the solution based on the free parameters. When $l= m= g= e=0$, we have the classical third-order RW solution which is shown in Fig. \[fig3\](c). It has one largest crest and six subcrests with two deepest troughs. For non-zero values, the third-order RW splits into six separated first-order RWs. When we increase the value of free parameters, the six first-order RWs take new positions. However, the maximum amplitude of each one of the peaks remains the same even in the new orientation. For small values of $l$ and $m$ and large values of $g$ and $e$ the RWs form a ring structure which contains six peaks. Figs. \[fig5\] and \[fig5a\] display the evolution of third-order RW solution for non-zero values of $l,m,g$ and $e$ for two different values of the inhomogeneous parameter, $k=0.1$ and $k=1.5$ respectively. When $l= m=10$ and $g= e=500$, we obtain a ring structure with six separated first-order RWs which is shown in Fig. \[fig5\](a). Among the 6 peaks 5 of them assemble on a circle and the sixth one appears in the middle of this circle. When we increase the value of $l$ to $100$ and keep the other parameters to be $m=10$ and $g= e=500$ the first-order RWs re-assemble in a triangular form which is shown in Fig. \[fig5\](b). When we fix the values of $l$ and $m$ to be $-100$ and $-10$ respectively and $g$ = $e=500$ the triangular pattern still persists but the orientation of the triangle changes, which is illustrated in Fig. \[fig5\](c). When we increase the value of $k$ to $1.5$, six first-order RWs get stretched in space with a curved structure in the plane wave background which is shown in Fig. \[fig5a\]. When $k$ is negative the six first-order RWs get stretched in the reverse direction. Recent experimental observation of RWs in nonlinear fiber optics [@kibler] and in a water tank experiment [@chab:hoff] and in plasma [@bailung] has open up possibilities to study their characteristics in detail. Our results, as discussed here, could be useful for controlling highly energetic pulses in plasmas. Trajectories of inhomogeneous rogue wave ======================================== In this section, we investigate certain characteristics of RW, namely the evolution of its peak, the distance between the valleys’, that is width of the RW and the trajectory of the RW analytically. The trajectory of RW can be described by the motion of the hump and valleys’ center location [@ling]. Substituting (\[a23\]) in (\[a16\]) one can obtain first-order RW solution of the INLS Eq. (\[a17\])and we calculate the expression of the motion of its hump’s center as $$\label{xhump} x_h=-\frac{\epsilon(t)}{\beta(t)},$$ and the motions of the two valleys’ center is ![The trajectory of the first-order RW when (a) $k=0.1$, (b) $k=2.0$, (c) the evolution of the RW’s hump and (d) the evolution of the RW’s valleys. The other parameters are same as in Fig. \[fig1\].[]{data-label="fig6"}](traj.pdf){width="0.99\linewidth"} $$\label{xval} x_v=\frac{\pm \left(\sqrt{3\beta(t)^2(1+4 \gamma(t)^2)}\right) - 2 \beta(t) \epsilon(t)}{2 \beta(t)^2},$$ where $x_h$ and $x_v$ denote the trajectory of hump and valleys of RW respectively. The trajectory of the RW’s hump is shown by orange line and the trajectories of the two valleys look like an “X” shape, as shown by the thick and dashed lines in Fig. \[fig6\](a). The corresponding evolution plot is shown in Fig. \[fig3\](a). When the inhomogeneity parameter $k=2$, the trajectory of the RW is depicted in Fig. \[fig6\](b) and its corresponding evolution diagram is shown in Fig. \[fig3a\](a). Furthermore, we can define the width of the RW as the distance between the two valleys’ centers [@ling]. Its evolution is $$\label{width} W(t)_{rw}=\frac{\sqrt{3\beta(t)^2(1+\gamma(t)^2)}}{\beta(t)^2}.$$ Substituting (\[xhump\]) and (\[xval\]) into the expression of (\[a16\]), we can calculate the expressions for the evolution of the RW’s hump and valleys as shown in Figs. \[fig6\](c) and (d) respectively. Conclusion ========== In this work, we have considered a variable coefficient NLS equation with an external linear potential that describes the nonlinear wave propagation in an inhomogeneous plasma/medium. We have constructed several localized solutions for the INLS equation, including Ma breather, AB, two-breathers, first-, second- and third-order RW solutions by mapping it to the NLS equation. Our aim was to investigate in-detail the impact of inhomogeneity on all these rational solutions. When the inhomogeneity parameter $k$ is small, say $0.01$, we have noticed that the breather and RW solutions retain their shapes. For $k=1.5$, the shape of AB gets stretched and the Ma breather bends in the plane wave background. For negative $k$ values the same effects occur but in the reverse direction. When we increase the strength of inhomogeneity parameter in two-breather solutions, both the ABs obliquely stretches in space whereas the two Ma breathers and the intersection of AB and Ma breathers get curved in the plane wave background. Similar effects were also observed for the RW solutions. For example, when we increase the value of $k$ in the INLS equation, the RW solution get stretched in space or curved in the plane wave background. We have constructed the second- and third-order RW solutions with certain free parameters and also analyzed how they are modified by the inhomogeneity parameter. We have derived the second-order RW solution with two free parameters. For small values of these free parameters we have the triangular pattern of three separated first-order RWs and when we increase the value of free parameters the distance between the peaks in the triplet increases and these peaks occupy a different position. Finally, we have derived the third-order RW solution with four free parameters. By varying these free parameters we have obtained two different patterns of six first-order RWs, namely (i) the ring structure and (ii) the triangular structure. We have also investigated the impact of inhomogeneity on these solutions as well. The nonlinear structures, as reported here, may be useful for controlling plasmonic energy along the plasma surface. Our results provide the many possibilities to manipulate RWs both theoretically as well as experimentally in their relative fields such as fluids, nonlinear optics and Bose-Einstein condensates. Acknowledgements {#acknowledgements .unnumbered} ================ KM thanks the University Grants Commission (UGC-RFSMS), Government of India, for providing a Research Fellowship. The work of MS forms part of a research project sponsored by NBHM, Government of India. References {#references .unnumbered} ========== [90]{} K.G. Budden, Radio Wave in the Ionosphere, Cambridge University Press, London, 1961. A.R. Osborne, Nonlinear Ocean Waves, Academic Press, New York, 2009. A. Hasegawa, Opt. Lett. [**5**]{} (1980) 416-417. D.P. Tewari, R.R. Sharma, J. Phys. D: Appl. Phys. [**12**]{} (1979) 1019. H.H. Chen, C.S. Liu, Phys. Rev. Lett. [**37**]{} (1976) 693; H.H. Chen, C.S. Liu, Phys. Fluids [**21**]{} (1978) 377. R. Balakrishnan, Phys. Rev. A [**32**]{} (1985) 1144. V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. [**98**]{} (2007) 074102. R. Atre, R.K. Panigrahi, G.S. Agarwal, Phys. Rev. E [**73**]{} (2006) 056611. D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature [**450**]{} (2007) 1054. Z.Y. Yan, Phy. Lett. A [**374**]{} (2010) 672. W.P. Zhong, R.H. Xie, M. Belic, N. Petrovic, G. Chen, Phys. Rev. A [**78**]{} (2008) 023821. J. Belmonte-Beitia, V.M. Perez-Garcia, V. Vekslerchik, V.V. Konotop, Phys. Rev. Lett. [**100**]{} (2008) 164102. A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers, Springer-Verlag, Berlin, 2003. C.Q. Dai, C.L. Zheng, H.P. Zhu, Eur. Phys. J. D [**66**]{}, (2012) 112. Z. Yang, W. Zhong, M.R. Belie, Phys. Scr. [**86**]{}, (2012) 015402. X.F. Wu, G.S. Hua, Z.Y. Ma, Commun. Nonlinear. Sci. Numer. Simulat. [**18**]{} (2013) 3325. Y.Y. Wang, J.S. He, Y.S. Li, Commun. Theor. Phys. [**56**]{} (2011) 995. U. Al Khawaja, M. Taki, Phys. Lett. A [**377**]{} (2013) 2944. W.M. Moslem, P.K. Shukla, B. Eliasson, Euro. Phys. Lett. [**96**]{} (2011) 25002. D. Mandelik, H.S. Eisenberg, Y. Silberberg, R. Morandotti, J.S Aitchison, Phys. Rev. Lett. [**90**]{} (2003) 253902. C. Kharif, E. Pelinovsky, Eur. J. Mech. B/Fluids [**22**]{} (2003) 603. C. Kharif, E. Pelinovsky, A. Slungyaev, Rogue waves in the Ocean, Springer, Heidelberg, 2009. A.R. Osborne, M. Onorato, M. Serio, Phys. Lett. A [**275**]{} (2000) 386. Y.V. Bludov, V.V. Konotop, N. Akhmediv, Phys. Rev. A [**80**]{} (2009) 033610. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J. M. Dudley, Nat. Phys. [**6**]{} (2010) 790. A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Phys. Rev. Lett. [**106**]{} (2011) 204502. H. Bailung, S.K. Sharma, Y. Nakamura, Phys. Rev. Lett. [**107**]{}, (2011) 255005. M. Shats, H. Punzmann, H. Xia, Phys. Rev. Lett. [**104**]{}, (2010) 104503. Z.Y. Yan, Commun. Theor. Phys. [**54**]{}, (2010) 947. W.M. Moslem, P.K. Shukla, B. Eliasson, Europhys. Lett. [**96**]{} (2011) 25002. D.H. Pergrine, J. Austral. Math. Soc. Ser. B [**25**]{} (1983) 16. N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E [**80**]{} (2009) 026601. Y.C. Ma, Stud. Appl. Math. [**60**]{} (1979) 43. N. Akhmediev, V.M. Eleonskii, N.E. Kulagin, Zh. Eksperimentalnoii i Teoreticheskoii Fiziki [**89**]{} (1985) 1542 \[Sov. Phys. JETP [**62**]{}, (1985) 894\] D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E [**85**]{}, (2012) 066601. S.K. Suslov, Proc. Ame. Mat. Soc. [**140**]{} (2012) 3067. S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. [**97**]{} (2006) 013901. A. Ankiewicz, D.J. Kedziora, N. Akhmediev, Phy. Lett. A [**375**]{} (2011) 2782-2785. L. Ling, L.C. Zhao, Phys. Rev. E [**88**]{} (2013) 043201.

e the fermionic degrees of freedom of the gravitino field are very restricted, we have found two bosonic quantum physical states, namely the wormhole and the Hartle-Hawking state. From the point of view of perturbation theory, it seems that the gravitational and gravitino modes that are allowed to be excited in a supersymmetric Bianchi-IX model contribute in such a way to forbid any physical solutions of the quantum constraints. This suggests that in a complete perturbation expansion we would have to conclude that the full theory of N=1 supergravity with a non-zero cosmological constant should have no physical states.\ mplete perturbation expansion we would have to conclude that the full theory of N=1 supergravity with a non-zero cosmological constant should have no physical states.\ 1 .5 mssymb =cmbx10 scaled2 ‘@=11 \#1[@y to]{} \#1[@y to]{} ‘@=12 =msxm10 =msym10 \#1[0= -.025em0-0 0.05em0-0 -.025em.0433em0 ]{} \#1[0= -.12em0-0 .12em0-00]{} \#1\#2 ${\left(} \def$[)]{} $${\left[} \def$$[\]]{} = ‘@=12 **Quantization of Bianchi Models in N=1 Supergravity** **with a Cosmological Constant[^1][[Lecture presented at the International School-Seminar “Multidimensional Gravity and Cosmology”, Yaroslavl, Russia, June 20 – 26, 1994; to be published by World Scientific, ed. V. Melnikov]{}]{}** .1 true in A.D.Y. Cheng, P.D. D’Eath and $\underline{{\rm P.R.L.V. Moniz}}$[^2] .1 true in Department of Applied Mathematics and Theoretical Physics University of Cambridge, Silver Street,Cambridge CB3 9EW, UK .2 true in **ABSTRACT** .1 true in [We study the quantization of some cosmological models within the theory of N=1 supergravity with a positive cosmological constant. We find, by imposing the supersymmetry and Lorentz constraints, that there are [no]{} physical states in the models we have considered. For the k=1 Friedmann-Robertson-Walker model, where the fermionic degrees of freedom of the gravitino field are very restricted, we have found two bosonic quantum physical states, namely the wormhole and the Hartle-Hawking state. From the point of view of perturbation theory, it seems that the gravitational and gravitino modes that are allowed to be excited in a supersymmetric Bianchi-IX model contribute in such a way to forbid any physical solutions of the quantum constraints. This suggests that in a complete perturbation expansion we would have to conclude that the [full]{} theory of N=1 supergravity with a non-zero cosmological constant should have [no]{} physical states.]{} 1 PACs numbers: 04.60.+ $n$, 04.65.+ $e$, 98.80. $Hw$ 6 pt 1 .5 mssymb =cmbx10 scaled2 ‘@=11 \#1[@y to]{} \#1[@y to]{} ‘@=12 =msxm10 =msym10 \#1[0= -.025em0-0 0.05em0-0 -.025em.0433em0 ]{} \#1[0= -.12em0-0 .12em0-00]{} \#1\#2 ${\left(} \def$[)]{} $${\left[} \def$$[\]]{} = [**I. Introduction**]{} Recently a number of quantum cosmological models have been studied in which the action is that of supergravity, with possible additional coupling to supermatter \[1-11,14,15,18,26-29,31-34,38,41\]. In addition, a review on this rather fascinating subject is under preparation \[12\]. It is sufficient, in finding a physical state, to solve the Lorentz and supersymmetry constraints of the theory \[13,14\]. Because of the anti-commutation relations $ \[ S_A,~\wti S_{A'} \]_+ \sim \cH_{A A'} $, the supersymmetry constraints $ S_A \Psi = 0,~\ol S_{A'} \Psi = 0 $ on a physical wave function $ \Psi $ imply the Hamiltonian constraint $ \cH_{A A'} \Psi = 0 $ \[13,14\]. In the case of the Bianchi-I model in $ N = 1 $ supergravity with no cosmological constant ($ \Lam = 0 $) \[8\], the quantum states are in the bosonic and filled fermionic sectors and are of the form $\exp (-\half h^{-\half})$, where $ h = \det h_{i j} $ is the determinant of the three-metric. In the case of Bianchi IX with $ \Lam = 0 $, there are two states, of the form $ \exp ( \pm I / \hbar ) $ where $ I $ is a certain Euclidean action, one in the empty and one in the filled fermionic sector \[9,15\]. When the usual choice of spinors constant in the standard basis is made for the gravitino field, the bosonic state $ \exp ( - I / \hbar ) $ is the wormhole state \[9,16\]. With a different choice, one obtains the Hartle–Hawking state \[15,17\]. Similar states were found for $ N = 1 $ supergravity in the more general Bianchi models of class $ A $ \[10\]. \[Supersymmetry (as well as other considerations) forbids mini-superspace models of class $ B $.\] It is of interest to extend these results, by studying more general locally supersymmetric actions, initially in Bianchi models. Possibly the simplest such generalization is the addition of a cosmological constant in $ N = 1 $ supergravity \[19\]. On the one hand, the appearance of a cosmological constant term in some supergravity models is a consequence of the coupling to matter. In particular, when one gauges internal SO(2) or SO(3) symmetries in N=2, N=3 [*extended*]{} supergravities \[20\] (coupling the spin-1 fields in a electromagnetic way to the gravitino) one needs at the same time a cosmological constant and a mass-like term for the gravitino in the action \[20–24\]. Such interesting connection between spin-1 fields and a cosmological constant has led to the suggestion that the electromagnetism might be due to a De Sitter space-time curvature within a supergravity context \[20\]. Furthermore, the action for our model presented in eq. (2.1) can be obtained from the O(2)-gauge extended supergravity model when one eliminates from the lagrangian the spin ($ {3 \over 2} $,1)-multiplet while keeping a non-zero gauge coupling constant \[25\]. But, on the other hand, our model can also be derived as an extension of pure N=1 supergravity \[19\]. Further, little has been written about extensions of pure supergravity to include $R^2$ terms, etc. The main idea in this extension is based on the fact that as the presence of a non zero cosmological constant induces a constant curvature of space-time independent of matter, the symmetry properties of such spaces will be in correspondence with the De Sitter group rather than the Poincaré group. The new action is determined by the prescription that quantities such as the covariant derivative and curvature terms which are characteristic of the Poincaré group should be replaced in the [*field equations*]{} by new ones which are charactristic of the De Sitter group. Following this procedure, one concludes that even in N=1 supergravity one needs a mass-like term for the spin-$ {3 \over 2} $ field if one adds a non-zero cosmological constant even though there are no spin-1 fields. Using the triad ADM canonical formulation we shall see that there are no physical quantum states in the cases of Bianchi type-I and IX (diagonal) models \[26,27\]. The calculations are described in Sec. II. We also treat briefly in Sec. III the spherical $ k = + 1 $ Friedmann model, and find that there is a two-parameter family of solutions of the quantum constraints with a $ \Lam $-term. Nevertheless, as will be seen, the Bianchi-IX model provides a better guide to the generic result, since more spin-$ {3 \over 2} $ modes are available to be excited in the Bianchi-IX model, while the form of the fermionic fields needed for supersymmetry in the $ k = + 1 $ Friedmann model is very restrictive \[6\]. In Sec. IV we briefly comment on how other different approaches from the one presented in the previous sections allows one to extract some similar results \[4,5,28,29\], \[30-34\]. Sec. V contains the Conclusion. [**II. QUANTUM STATES FOR THE BIANCHI MODELS WITH A $ \Lam $-TERM**]{} Using two-component spinors \[6,14\], the action \[19\] is $$S = \int d^4 x \[ \eqalign { {}~& \( 2 \kap^2 \)^{- 1} \( \det e \)~\( R - 3 g^2 \) \cr + &{1 \over 2} \eps^{\mu \nu \rho \sig} \( \ol \psi^{A'}_{~~\mu} e_{A A' \nu} D_\rho \psi^A_{~~\sig} + H.c. \) \cr - &{1 \over 2} g (\det e)~\( \psi^A_{~~\mu} e_{A B'}^{~~~~\mu} e_B^{~~B' \nu} \psi^B_{~~\nu} + H.c. \) \cr } \]~. \eqno (2.1)$$ Here the tetrad is $ e^a_{~\mu} $ or equivalently $ e^{A A'}_{~~~~\mu} $. The gravitino field $ \(\psi^A_{~~\mu}, \ol \psi^{A'}_{~~\mu} \) $ is an odd (anti-commuting) Grassmann quantity. The scalar curvature $ R $ and the covariant derivative $ D_\rho $ include torsion. We define $ \kap^2 = 8 \pi $. Here $ g $ is a constant, and the cosmological constant is $ \Lam = {3 \over 2} g^2 $. There are two possible approaches to the quantization of this model. One possibility is to substitute the Bianchi Ansatz, e.g., $$d s^2 = \( N^j N_j - N^2 \) d t^2 + 2 N_i d t~d x^i + h_{i j} d x^i d x^j~, \eqno (2.2)$$ for the geometry $ e^{A A'}_{~~~~\mu} $, where all metric components are functions of time, and gravitino field $ \( \psi^A_{~~\mu}, \ol \psi^{A'}_{~~\mu} \) $ into the action (2.1). The components $ \psi^A_{~~\mu} e^{B B' \mu} $ and $ \ol \psi^{A'}_{~~\mu} e^{B B' \mu} $ are required to be spatially constant with respect to the standard triad \[35\] on the Bianchi hypersurfaces. One finds that, in order for the form of the Ansatz to be left invariant by one-dimensional local supersymmetry transformations, possibly corrected by coordinate and Lorentz transformations \[6\], one must study the general non-diagonal Bianchi model \[35\]. The reduced action could then be computed, leading to the Hamiltonian standard form $$H = \wti N \cH + \rrho_A S^A + \wti S^{A'} \ti \rrho_{A'} + M_{A B} J^{A B} + \wti M_{A' B'} \wti J^{A' B'}~. \eqno (2.3)$$ Hence $ \cH $ is the generator of local time translations, $ S^A $ and $ \wti S^{A'} $ are the generators of local supersymmetry transformations, and $ J^{A B} $ and $ \wti J^{A' B'} $ are the generators of local Lorentz transformations; they are formed from the basic dynamical variables $ \( e^{A A'}_{~~~~i} , \psi^A_{~~i} , \wti \psi^{A'}_{~~i} \) $. \[At this point it is natural in the classical theory to free $ \wti \psi^{A'}_{~~i} $ from being the hermitian conjugate of $ \psi^A_{~~i} $.\] The quantities $ \wti N, \rrho_A, \ti \rrho_{A'}, M_{A B} $ and $ \wti M_{A' B'} $ are Lagrange multipliers. The supersymmetry and Lorentz constraints are imposed on physical wave functions but they would be complicated because of the number of parameters needed to describe the off-diagonal model. The other alternative, taken here, is to apply the supersymmetry constraints of the general theory at a Bianchi geometry \[9\]. This is valid since the supersymmetry constraints are of first order in bosonic derivatives, and give expressions such as $ \de \Psi / \de h_{i m} (x) $ in terms of known quantities and $ \Psi $. These equations can be evaluated in the case of, e.g., a diagonal Bianchi-IX geometry, parametrized by three radii $ A,~B,~C $. One multiplies (e.g.) by $ \de h_{i m} (x) = \pt h_{i m} / \pt A $ and integrates $ \int d^3 x (~~) $ to obtain an equation for $ \pt \Psi / \pt A $ in terms of known quantities. The need to consider off-diagonal metrics is thereby avoided. In general, it is only necessary to solve the quantum constraints $$S^A \Psi = 0~,\bar S^{A'} \Psi = 0~, J^{AB} \Psi = 0~, \bar J^{A' B'} \Psi = 0~, \eqno (2.4)$$ for a physical state $ \Psi $, since the anti-commutator of $ S^A $ and $ \bar S^{A'} $ includes $ \cH^{AA'} $, so that Eq. (2.4) implies also $ \cH^{AA'} \Psi = 0 $. The wave function can (e.g.) be taken as $ \Psi \( e^{A A'}_{~~~~i}, \psi^A_{~~i} \) $ or $ \wti \Psi \( e^{A A'}_{~~~~i}, \wti \psi^{A'}_{~~i} \) $. These representations are related by a fermionic Fourier transform \[5, 9\]. Classically, the supersymmetry constraints are $$S_A = g h^\half e_A^{~~A' i} n_{B A'} \psi^B_{~~i} + \eps^{i j k} e_{A B' i}~^{3 s} D_j\ol \psi^{B'}_{~~k} {}~- \half i \kap^2 p_{A A'}^{~~~~i} \wti \psi^{A'}_{~~i} {}~, \eqno (2.5)$$ $$\ol S_{A'} = g h^{1 \over 2} e^{A~~~i}_{~~A'} n_{A B'} \ol \psi^{B'}_{~~i} + \eps^{i j k} e_{A A' i}~^{3 s} D_j \psi^A_{~~k} + {1 \over 2} i \kap^2 \psi^A_{~~i} p_{A A'}^{~~~~i}~. \eqno (2.6)$$ Here $ n^{A A'} $ is the spinor version of the unit future-pointing normal $ n^\mu $ to the constant $ t $ surface. It is a function of the $ e^{A A'}_{~~~~i} $, defined by $$n^{A A'} e_{A A' i} = 0~, \ \ \ \ n^{A A'} n_{A A'} = 1~. \eqno (2.7)$$ In Eq. (2.5),(2.6), $ p_{A A'}^{~~~~i} $ is the momentum conjugate to $ e^{A A'}_{~~~~i} $. The expression $ ^{3 s}D_j $ denotes the three-dimensional covariant derivative without torsion. Since the components of $ \psi^A_{~~k} $ are taken to be constant in the Bianchi basis, one can replace $ ^{3s}D_j \psi^A_{~~k} $ by $ \om^A_{~~B j} \psi^B_{~~k} $, where $ \om^A_{~~B j} $ gives the torsion-free connection \[14\]. Quantum-mechanically, in the representation $ \Psi \( e^{A A'}_{~~~~i}, \psi^A_{~~i} \) $, one represents \[9,14\] $$p_{A A'}^{~~~~i} \lrta - i \hbar {\de \over \de e^{A A'}_{~~~~i}} + \half \eps^{i j k} \psi_{A j} \bar \psi_{A' k}~, \eqno (2.8)$$ where $$\bar \psi^{A'}_{~~i} \lrta - i \hbar D^{A A'}_{~~~~j i} h^{1 \over 2} {\pt \over \pt \psi^A_{~~j}}~, \eqno (2.9)$$ where $ \pt / \pt \psi^A_{~~j} $ denotes left differentiation \[13\], and $$D^{A A'}_{~~~~j i} = - 2 i h^{- \half} e^{A B'}_{~~~~i} e_{B B' j} n^{B A'}~. \eqno (2.10)$$ We have made the replacement $ \de \Psi / \de \psi^B_{~~j} \longrightarrow h^{1 \over 2} \pt \Psi / \pt \psi^B_{~~j} $. This replacement is important when considering space-time manifolds whose spatial sections are compact. The $ h^{1 \over 2} $ factor ensures that each term has the correct weight in the equations, namely when one takes a variation of a (compact) Bianchi geometry, multiplying by $ \de / \de h_{i j} $ and integrating over the three-geometry (see eq. (2.21),(2.23), (2.24)). One can check, e.g., that this replacement gives the correct supersymmetry constraints in the $ k = + 1 $ Friedmann model, where the model was quantized using the alternative approach via a supersymmetric Ansatz \[6\]. The corresponding quantum constraints read, with the help of \[14\], $$\eqalignno { \ol S_{A'} \Psi &= - i \hbar g h^{1 \over 2} e^{A~~~i}_{~~A'} n_{A B'} D^{B B'}_{~~~~j i} \( h^{1 \over 2} {\pt \Psi \over \pt \psi^B_{~~j}} \) \cr {}~&+ \eps^{i j k} e_{A A' i} \om^A_{~~B j} \psi^B_{~~k} \Psi - {1 \over 2} \hbar \kap^2 \psi^A_{~~i} {\de \Psi \over \de e^{A A'}_{~~~~i}} = 0~, &(2.11) \cr S_A \Psi &= g h^{1 \over 2} e_A^{~~A' i} n_{B A'} \psi^B_{~~i} \Psi - i \hbar \om_{A~~i}^{~~B} \( h^{1 \over 2} {\pt \Psi \over \pt \psi^B_{~~i}} \) \cr {}~&+ {1 \over 2} i \hbar^2 \kap^2 D^{B A'}_{~~~~j i} \( h^{1 \over 2} {\pt \over \pt \psi^B_{~~j}} \)~{\de \Psi \over \de e^{A A'}_{~~~~i}} = 0~. &(2.12) \cr }$$ The constraints $ J^{A B} \Psi = 0,~~\bar J^{A' B'} \Psi = 0 $ imply that $ \Psi \( e^{A A'}_{~~~~i}, \psi^A_{~~i} \) $ is a Lorentz-invariant function. One solves them by taking expressions in which all spinor indices have been contracted together. As described in \[5\], it is reasonable also to consider only wave functions $ \Psi $ which are spatial scalars, where all spatial indices $ i, j, \ldots $ have also been contracted together. To specify this, note the decomposition \[11\] of $ \psi^A_{~~B B'} = e_{B B'}^{~~~~i} \psi^A_{~~i} $: $$\psi_{A B B'} = - 2 n^C_{~~B'} \rga_{A B C} + {2 \over 3} \( \beta_A n_{B B'} + \beta_B n_{A B'} \) - 2 \eps_{A B} n^C_{~~B'} \beta_C~, \eqno (2.13)$$ where $ \rga_{A B C} = \rga_{(A B C)} $ is totally symmetric and $ \eps_{A B} $ is the alternating spinor. The general Lorentz-invariant wave function is a polynomial of sixth degree in Grassmann variables: $$\eqalignno { \Psi \( e^{A A'}_{~~~~i},~\psi^A_{~~i} \) &= \Psi_0 \( h_{i j} \) + \( \beta_A \beta^A \) \Psi_{21} \( h_{i j} \) + \( \rga_{A B C} \rga^{A B C} \) \Psi_{22} \( h_{i j} \) \cr {}~&+ \( \beta_A \beta^A \)~\( \rga_{B C D} \rga^{B C D} \) \Psi_{41} \( h_{i j} \) + \( \rga_{A B C} \rga^{A B C} \)^2 \Psi_{42} \( h_{i j} \) \cr {}~&+ \( \beta_A \beta^A \)~\( \rga_{B C D} \rga^{B C D} \)^2 \Psi_6 \( h_{i j} \)~. &(2.14) \cr }$$ Any other Lorentz-invariant fermionic polynomials can be written in terms of these. Note that, for example, the term $ \( \beta^A \rga_{A B C} \)^2 = \beta^A \rga_{A B C} \beta^D \rga_D^{~~B C} $ can be rewritten, using the anti-commutation of the $ \beta $’s and $ \rga $’s, as $${\rm const.}~\beta^E \beta_E \eps^{A D} \rga_{A B C} \rga_D^{~~B C} = {\rm const.}~ \( \beta_E \beta^E \)~\(\rga_{A B C} \rga^{A B C} \).~\eqno(2.15)$$ Similarly, any quartic in $ \rga_{A B C} $ can be rewritten as a multiple of $ \( \rga_{A B C} \rga^{A B C} \)^2 $. Since there are only four independent components of $ \rga_{A B C} = \rga_{(A B C)} $, only one independent quartic can be made from $ \rga_{A B C} $, and it is sufficient to check that $ \( \rga_{A B C} \rga^{A B C} \)^2 $ is non-zero. Now $ \rga_{A B C} \rga^{A B C} = 2 \rga_{000} \rga_{111} - 6 \rga_{100} \rga_{011} $. Hence $ \( \rga_{A B C} \rga^{A B C} \)^2 $ includes a non-zero quartic term const. $\rga_{000} \rga_{100} \rga_{110} \rga_{111} $. Unlike the case of $ N = 1 $ supergravity \[5\], here the nonzero $ g $ (or $ \Lam $) implies that there is coupling between different fermionic levels. We now proceed to solve the supersymmetry and Lorentz constraints for the case of a diagonal Bianchi-IX \[29\], whose three-metric is given in terms of the three radii $ A, B, C $ by $$h_{i j} = A^2 E^1_{~i} E^1_{~j} + B^2 E^2_{~i} E^2_{~j} + C^2 E^3_{~i} E^3_{~j}~, \eqno (2.16)$$ where $ E^1_{~i}, E^2_{~i}, E^3_{~i} $ are a basis of unit left-invariant one-forms on the three-sphere \[35\]. In the calculation, we shall repeatedly need the expression: $$\eqalignno { \om_{A B i} n^A_{~~B'} e^{B B' j} &= {i \over 4} \( {C \over A B} + {B \over C A} - {A \over B C} \) E^1_{~i} E^{1 j} \cr {}~&+ {i \over 4} \( {A \over B C} + {C \over A B} - {B \over C A} \) E^2_{~i} E^{2 j} \cr {}~&+ {i \over 4} \( {B \over C A} + {A \over B C} - {C \over A B} \) E^3_{~i} E^{3 j} &(2.17) \cr }$$ This can be derived from the expressions for $ \om^{A B}_{~~~i} $ given in \[9,13\]. First consider the $ \ol S_{A'} \Psi = 0 $ constraint at the level $ \psi^1 $ in powers of fermions. One obtains $${3 \over 16} \hbar g h^{1 \over 2} e_{B A'}^{~~~~i} \psi^B_{~~i} \Psi_{21} + \eps^{j k i} e_{A A' j} \om^A_{~~B k} \psi^B_{~~i} \Psi_0 + \hbar \kap^2 e_{B A' j} \psi^B_{~~i} {\de \Psi_0 \over \de h_{i j}} = 0~. \eqno (2.18)$$ Since this holds for all $ \psi^B_{~~i} $, one can conclude $${3 \over 16} \hbar g h^{1 \over 2} e_{B A'}^{~~~~i} \Psi_{21} + \eps^{j k i} e_{A A' j} \om^A_{~~B k} \Psi_0 + \hbar \kap^2 e_{B A' j} {\de \Psi_0 \over \de h_{i j}} = 0~. \eqno (2.19)$$ Now multiply this equation by $ e^{B A' m} $, giving $$- {3 \over 16} \hbar g h^{i m} h^{1 \over 2} \Psi_{21} + \eps^{j k i} e_{A A' j} e^{B A' m} \om^A_{~~B k} \Psi_0 - \hbar \kap^2 {\de \Psi_0 \over \de h_{i m}} = 0~. \eqno (2.20)$$ The second term can be simplified using \[6\] $$e_{A A' j} e^{B A'}_{~~~~m} = - { 1 \over 2} h_{j m} \eps_A^{~~B} + i \eps_{j m n} h^{1 \over 2} n_{A A'} e^{B A' n}~. \eqno (2.21)$$ One then notes, as above, that by taking a variation among the Bianchi-IX metrics, such as $$\de h_{i j} = {\pt h_{i j} \over \pt A} = 2 A E^1_{~i} E^1_{~j}~, \eqno (2.22)$$ multiplying by $ \de \Psi_0 / \de h_{i j} $ and integrating over the three-geometry, one obtains $ \pt \Psi_0 / \pt A $. Putting this information together one obtains the constraint $$\hbar \kap^2 {\pt \Psi_0 \over \pt A} + 1 6 \pi^2 A \Psi_0 + 6 \pi^2 \hbar g B C \Psi_{21} = 0~, \eqno (2.23)$$ and two others given by cyclic permutation of $ A B C $. Next we consider the $ S_A \Psi = 0 $ constraint at order $ \psi^1 $. One uses the relations $ \pt \( \beta_A \beta^A \) / \pt \psi^B_{~~i} = - n_A^{~~B'} e_{B B'}^{~~~~i} \beta^A $ and $ \pt \( \rga_{A D C} \rga^{A D C} \) / \pt \psi^B_{~~i} = - 2 \rga_{B D C}~ n^{C C'} e^{D~~~~~i}_{~~~C'} $, and writes out $ \beta^A $ and $ \rga_{B D C} $ in terms of $ e^{E E'}_{~~~~j} $ and $ \psi^E_{~~j} $. Proceeding by analogy with the previous calculation above, one again ‘divides out’ by $ \psi^B_{~~j} $. One replaces the free spinor indices $ A B $ by the spatial index $ n $ on multiplying by $ n^A_{~~D'} e^{B D' n} $, then multiplying by different choices $ \de h_{i m} = \pt h_{i m} / \pt A $ etc. and integrating over the manifold, one finds the constraints $$\eqalignno { &{1 \over 16} \hbar^2 \kap^2 A^{- 1} \( A {\pt \Psi_{21} \over \pt A} + B {\pt \Psi_{21} \over \pt B} + C {\pt \Psi_{21} \over \pt C} \) \cr - &{1 \over 3} \hbar \kap^2 \[ 3 {\pt \Psi_{22} \over \pt A} - A^{- 1} \( A {\pt \Psi_{22} \over \pt A} + B {\pt \Psi_{22} \over \pt B} + C {\pt \Psi_{22} \over \pt C} \) \] \cr &- 16 \pi^2 g B C \Psi_0 - \pi^2 \hbar B C \( {A \over B C} + {B \over C A} + {C \over A B} \) \Psi_{21} \cr &+ {1 \over 3} \( 16 \pi^2 \) \hbar B C \( {2 A \over B C} - {B \over C A} - {C \over A B} \) \Psi_{22} = 0~. &(2.24) \cr }$$ and two more equations given by cyclic permutation of $ A B C $. Now consider the $ \ol S_{A'} \Psi = 0 $ constraint at order $ \psi^3 $. It will turn out that we need go no further than this. The constraint can be written as $$\eqalignno { &{1 \over 2} \hbar g h^{1 \over 2} e^B_{~~A' j} n_C^{~~B'} e_{B B'}^{~~~~j} \beta_C \( \rga_{D E F} \rga^{D E F} \) \Psi_{41} \cr &+ \eps^{i j k} e_{A A' i} \om^A_{~~B j} \psi^B_{~~k} \[ \( \beta_C \beta^C \) \Psi_{21} + \( \rga_{C D E} \rga^{C D E} \) \Psi_{22} \] \cr &- {1 \over 2} \hbar^2 \kap^2 \psi^A_{~~i} \[ \( \beta_C \beta^C \) {\de \Psi_{21} \over \de e^{A A'}_{~~~~i}} + \( \rga_{C D E} \rga^{C D E} \) {\de \Psi_{22} \over \de e^{A A'}_{~~~~i}} \] = 0~. &(2.25) \cr }$$ The terms $ \psi^B_{~~k} $ and $ \psi^A_{~~i} $ in the last two lines can be rewritten in terms of $ \beta_A $ and $ \rga_{F G H} $, using Eq. (2.13). Then one can set separately to zero the coefficient of $ \beta^C \( \rga_{D E F} \rga^{D E F} \) $, the symmetrized coefficient of $ \rga_{D E F} \( \beta_C \beta^C \) $ and the symmetrized coefficient of $ \rga_{F G H} \( \rga_{C D E} \right . $ $ \left . \rga^{C D E} \) $. These three equations give $${3 \over 4} \hbar g h^{1 \over 2} n^C_{~~A'} \Psi_{41} - {8 \over 3} \eps^{i j k} e_{A A' i} \om^A_{~~B j} n^B_{~~C'} e^{C C'}_{~~~~k} \Psi_{22} + {4 \over 3} \hbar \kap^2 n^A_{~~B'} e^{C B'}_{~~~~i} {\de \Psi_{22} \over \de e^{A A'}_{~~~~i}} = 0~, \eqno (2.26)$$ $$2 \eps^{i j k} e_{A A' i} \om^A_{~~B j} n^D_{~~B'} e^{C B'}_{~~~~k} \Psi_{21} - \hbar \kap^2 n^D_{~~B'} e^{C B'}_{~~~~i} {\de \Psi_{21} \over \de e^{B A'}_{~~~~i}}$$ $$+ \( B C D \to C D B \) + \( B C D \to D B C \) = 0~, \eqno (2.27)$$ and Eq. (2.27) with $ \Psi_{21} $ replaced by $ \Psi_{22} $. Contracting Eq. (2.26) with $ n_C^{~~A'} $ and integrating over the three-surface gives $${3 \over 4} \( 16 \pi^2 \) \hbar g A B C \Psi_{41} + {2 \over 3} \( 16 \pi^2 \)~ \( A^2 + B^2 + C^2 \) \Psi_{22}$$ $$+ {2 \over 3} \hbar \kap^2 \( A {\pt \Psi_{22} \over \pt A} + B {\pt \Psi_{22} \over \pt B} + C {\pt \Psi_{22} \over \pt C} \) = 0~. \eqno (2.28)$$ Contracting Eq. (2.27) with $ e^{B A' \ell} n_{C C'} e_D^{~~C' N} $, multiplying by $ \de h_{\ell n} = \pt h_{\ell n} / \pt A $ and integrating gives $$3 \hbar \kap^2 {\pt \Psi_{21} \over \pt A} - \hbar \kap^2 A^{- 1} \( A { \pt \Psi_{21} \over \pt A} + B {\pt \Psi_{21} \over \pt B} + C {\pt \Psi_{21} \over \pt C} \)$$ $$- 16 \pi^2 B C \( {C \over A B} + {B \over C A} - 2 {A \over B C} \) \Psi_{21} = 0~, \eqno (2.29)$$ and two more equations given by permuting $ A B C $ cyclically. The equation (2.29) also holds with $ \Psi_{21} $ replaced by $ \Psi_{22} $. There is a duality between wave functions $ \Psi \( e^{A A'}_{~~~~i}, \psi^A_{~~i} \) $ and wave functions $ \wti \Psi \( e^{A A'}_{~~~~i} \right . $, $ \left . \wti \psi^{A'}_{~~i} \) $, given by a fermionic Fourier transform \[14\]. The $ S_A $ and $ \ol S_{A'} $ operators interchange rôles under this transformation, and the rôles of $ \Psi_0 $ and $ \Psi_6,~\Psi_{21} $ and $ \Psi_{42} $, and $ \Psi_{22} $ and $ \Psi_{41} $ are interchanged. We shall proceed by showing that $ \Psi_{22},~\Psi_{21} $ and $ \Psi_0 $ must vanish for $ g \ne 0 $ (or $ \Lam \ne 0 $), and hence by the duality the entire wave function must be zero. Consider first the equation (2.29) and its permutations for $ \Psi_{21} $ and $ \Psi_{22} $. One can check that these are equivalent to $$\hbar \kap^2 \( A {\pt \Psi_{21} \over \pt A} - B {\pt \Psi_{21} \over \pt B} \) = 16 \pi^2 \( B^2 - A^2 \) \Psi_{21} \eqno (2.30)$$ and cyclic permutations. One can then integrate Eq. (2.30) along a characteristic $ A B = $ const., $ C = $ const., using the parametric description $ A = w_1 e^\tau $, $ B = w_2 e^{- \tau} $, to obtain $$\Psi_{21} = h_1 (A B, C) \exp \[ - {8 \pi^2 \over \hbar \kap^2} ~ \( A^2 + B^2 \) \]~. \eqno (2.31)$$ Replacing $A,B$ for $B,C$ in Eq. (2.30) gives the solution $$\Psi_{21} = h_2 (B C, A) \exp \[ - {8 \pi^2 \over \hbar \kap^2} ~\( B^2 + C^2 \) \]~. \eqno (2.33)$$ Eqs. (2.31) and (2.33) are only consistent if $ \Psi_{21} $ has the form $$\Psi_{21} = F (A B C) \exp \[ - {8 \pi^2 \over \hbar \kap^2} ~\( A^2 + B^2 + C^2 \) \]~. \eqno (2.34)$$ Similarly $$\Psi_{22} = G (A B C) \exp \[ - {8 \pi^2 \over \hbar \kap^2} ~\( A^2 + B^2 + C^2 \) \]~. \eqno (2.35)$$ Substituting Eqs. (2.34),(2.35) into Eq. (2.24), one obtains $$\eqalignno { &16 \pi^2 g \Psi_0 = - 2 \pi^2 \hbar (A B C)^{- 1} \( A^2 + B^2 + C^2 \)~(\exp) F \cr &+ {3 \over 16} \hbar^2 \kap^2 (\exp) F^{\prime} + {2 \over 3} \( 16 \pi^2 \) \hbar (A B C)^{- 1} \( 2 A^2 - B^2 - C^2 \)~(\exp) G &(2.36) \cr }$$ and cyclically, where $$\exp = \exp \[ - {8 \pi^2 \over \hbar^2 \kap^2}~\( A^2 + B^2 + C^2 \) \]~. \eqno (2.37)$$ Now $ \Psi_0 $ should be invariant under permutations of $ A, B, C $. Hence $ G = 0 $. I.e. $$\Psi_{22} = 0~. \eqno (2.38)$$ The equation (2.36) and its cyclic permutations, with $ \Psi_{22} = 0 $, must be solved consistently with Eq. (2.23) and its cyclic permutations. Eliminating $ \Psi_0 $, one finds $$\eqalignno { &{3 \hbar^3 \kap^4 \over 16 \( 16 \pi^2 g \)} F'' - {\hbar^2 \kap^2 \over 8 g}~{\( A^2 + B^2 + C^2 \) \over A B C} F' \cr &+ 6 \pi^2 \hbar g F - {\hbar^2 \kap^2 \over 4 g}~{1 \over B^2 C^2} F + {\hbar^2 \kap^2 \over 8 g}~{\( A^2 + B^2 + C^2 \) \over (A B C)^2} F = 0~, &(2.39) \cr }$$ and cyclic permutations. Since $ F = F (A B C) $ is invariant under permutations, the $ (B C)^{- 2} F $ term and its permutations imply $ F = 0 $. Thus $$\Psi_{21} = 0~. \eqno (2.40)$$ Hence, using Eq. (2.36), $$\Psi_0 = 0~.$$ Then we can argue using the duality mentioned earlier, to conclude that $$\Psi_{41} = \Psi_{42} = \Psi_6 = 0~. \eqno (2.41)$$ Hence there are no physical quantum states obeying the constraint equations in the diagonal Bianchi-IX model. This result will be discussed further in Sec. V. The same conclusion can be reached for the case of a (non-diagonal) Bianchi type I model. Following ref. \[26\], one can use the averaged ordering \[5\], with $$\eqalignno { p_{A A'}^{~~~~i} \bar \psi^{A'}_{~~i} &\lrta \half \( \bar \psi^{A'}_{~~i} p_{A A'}^{~~~~i} + p_{A A'}^{~~~~i} \bar \psi^{A'}_{~~i} \)~, \cr \psi^A_{~~i} p_{A A'}^{~~~~i} &\lrta \half \( \psi^A_{~~i} p_{A A'}^{~~~~i} + p_{A A'}^{~~~~i} \psi^A_{~~i} \) ~. &(2.42) \cr }$$ With this ordering, there is a certain symmetry between the operators $ S_A $ and $ \bar S_{A'} $ as viewed in the two representations $ \Psi \( e^{A A'}_{~~~~i}, \psi^A_{~~i} \) $ and $ \wti \Psi \( e^{A A'}_{~~~~i}, \wti \psi^{A'}_{~~i} \) $, provided one changes $ g \to - g $. However, the final result is not an artefact of the symmetric factor ordering (2.42) used here. One can repeat the calculations using a general factor ordering $$\eqalignno { p_{A A'}^{~~~~i} \bar \psi^{A'}_{~~i} &\lrta \( \half + s \) \bar \psi^{A'}_{~~i} p_{A A'}^{~~~~i} + \( \half - s \) p_{A A'}^{~~~~i} \bar \psi^{A'}_{~~i}~, \cr \psi^A_{~~i} p_{A A'}^{~~~~i} &\lrta \( \half - s \) \psi^A_{~~i} p_{A A'}^{~~~~i} + \( \half + s \) p_{A A'}^{~~~~i} \psi^A_{~~i}~, &(2.43) \cr }$$ to reach the same conclusion. Let us begin with the constraint $ \bar S_{A'} \Psi = 0 $ at order $ \psi^1 $. This gives $$\eqalignno { \biggl ( - {3 \over 4} \hbar g \Psi_{2 1} & + {4 \over 3} \hbar \kap^2 h^{\half} h_{i j} {\pt \Psi_0 \over \pt h_{i j}} - \hbar \kap^2 \Psi_0 \biggr ) n_{A A'} \beta^A \cr {}~& - \half e_{B B' i} n_C^{~~B'} e_{A A' j} \rga^{A B C} {\pt \Psi_0 \over \pt h_{i j}} = 0~, &(2.44) \cr }$$ for all $ \beta^A $ and $ \rga^{A B C} $. Take the symmetrized coefficient of $\rga^{A B C} = \rga^{(A B C)}$ and contract it with $ e^{A A'}_{~~~~k} n^{B C'} e^C_{~~C'\ell} $ to get $$\(3 h_{i k} h_{j \ell} - h_{k \ell} h_{i j} \) {\pt \Psi_0 \over \pt h_{i j}} = 0~. \eqno (2.45)$$ Since the ${{\pt ~h}\over{\pt~h_{ij}}} = h~h^{ij}$, the general solution of (2.45) may be taken in the form $$\Psi_0 = A~f(u),~u\equiv B~h^m, \eqno (2.46)$$ where $A, B, m$ are constants. Taking now the $\beta^A $ part of Eq.(2.44) and using Eq. (2.46) we get $$- {3 \over 4} \hbar g \Psi_{2 1} + 4 \hbar \kap^2 A f^{'} B m h^{m+\half} - \hbar \kap^2 A f = 0~,\eqno (2.47)$$ where $(')$ denores derivation with respect to the $u$ variable. Then, one gets as solutions $$\Psi_{2 1} = {4 \over 3} g^{-1} A \kap^2 (4 f^{'} B m h^{m+\half} - f)~. \eqno (2.48)$$ The constraint $ S_A \Psi = 0 $ at order $ \psi^1 $ can be shown to yield $$2 g h^\half \Psi_0 \beta_A - {1 \over 4} \hbar^2 \kap^2 h^{-\half} h_{ij} {\pt \Psi_{2 1} \over \pt h_{i j}}\beta_A - {8 \over 3} \hbar^2 \kap^2 h^{-\half} \Psi_{2 2} \beta_A$$ $$- {9 \over 8} \hbar^2 \kap^2 \Psi_{2 1} \beta_A - 4 e_{A A' k} D^{B A'}_{~~~~j i} n_E^{~~C'} e_{D C'}^{~~~~j} \eps_{B C} h^{m+\half} {\pt \Psi_{2 2} \over \pt h_{i k}} \rga^{C D E} \eqno (2.49)$$ for all $ \beta^A $ and $ \rga^{C D E} $. From the $ \rga^{C D E} $ part of Eq.(2.49) one gets an equation for $\Psi_{2 2}$ like Eq.(2.45) and then the solutions $\Psi_{2 2}$ are of the form of Eq.(2.48). Now let us consider the $ \bar S_{A'} \Psi = 0 $ at order $ \psi^3 $. This reads $$\hbar g e^B_{~~A' j} e_{B C'}^{~~~~j} n_C^{~~C'} \Psi_{41} \beta^C \rga_{D E F} \rga^{D E F} - \hbar \kap^2 h^{\half} {8\over 3} n^B_{~~C'} e^{E C'}_{~~~~k} e_{B A' j} {\pt \Psi_{22} \over \pt h_{j k}} \beta_E \rga_{A C D} \rga^{A C D} +$$ $$2 \hbar \kap^2 h^{\half} n^{E}_{~C'} e^{FC'}_{~~k} e_{BA'j} \left( {\pt \Psi_{21} \over \pt h_{j k}} \beta^A\beta_A \rga^{B}_{~FE} + {\pt \Psi_{22} \over \pt h_{j k}} \rga^{B}_{~FE}\rga_{A C D} \rga^{A C D}\right)$$ $$-\half \hbar \kap^2\left( \half n_{AA'}\rga^{ADP}\rga_{FDP}\beta^F - {4\over 3} n^P_{~A'} \beta^{D}\rga_{FDP}\beta^F\right) \Psi_{2 1}$$ $$- \half \hbar \kap^2 \Psi_{22} \left [ \matrix { {8 \over 3} n_{C A'} \beta_B \rga^B_{~~D A} \rga^{A C D} \cr - 2 n^E_{~~A'} \beta_A \rga_{B D A} \rga^{A B D} \cr + 4 n^E_{~~A'} \rga^B_{~~D A} \rga^{A C D} \rga_{C B E} \cr } \right ]~= 0~, \eqno (2.50)$$ for all $ \beta^A $ and $ \rga^{B C D} $. Taking the part $$-4 \hbar \kap^2 h^{\half} n^{E}_{~C'} e^{FC'}_{~~k} e_{BA'j} {\pt \Psi_{22} \over \pt h_{j k}} \rga^{B}_{~FE}\rga_{A C D} \rga^{A C D} - 2 \hbar \kap^2 n^E_{~~A'} \rga^B_{~~D A} \rga^{A C D} \rga_{C B E} \Psi_{22}=0~,$$ and substituting the solution for $\Psi_{2 2}$ of the form of Eq.(2.46) we get that the term in $\rga^{B}_{~FE}\rga_{A C D} \rga^{A C D}$ is zero and so $\Psi_{2 2} =0$. Using this result back into Eq. (2.49) together with Eq. (2.46) and (2.49) we obtain the following equation for $f$: $$\eqalignno { 2 g A h f & + {3 \over 4} \hbar \kap^4 g^{-1} A f - 2 \hbar \kap^4 g^{-1} A B m h^{m + \half} f^{'} \cr {}~& -4 \hbar \kap^4 g^{-1} A B m^2 h^{m} f^{'} - 4 \hbar \kap^4 A B^2 m^2 h^{2m + \half} f^{''} =0~.~,& (2.51)\cr }$$ Expanding $f, f^{'}$ and $f^{''}$ in a power series with respect to the $u$-variable and substituting back into Eq.(2.51) one concludes that for the constraint to be satisfied at any moment of time one needs that $A=0$, this to say that $\Psi_0 = \Psi _{2 1} =0$. Using the duality mentioned earlier, we again argue to conclude that $ \Psi_{41} = \Psi_{42} = \Psi_6 = 0.$ Hence, there are [*no*]{} physical states in the Bianchi - I case either. [**III THE $ k = + 1 $ FRIEDMANN MODEL WITH $ \Lam $-TERM**]{} The $ k = + 1 $ Friedmann model without a $ \Lam $ term has been discussed in \[2,6\]. There are two linearly independent physical quantum states. One is bosonic and corresponds to the wormhole state \[16\], the other is at quadratic order in fermions. The Hartle–Hawking state \[17\] is also found \[15\]. In the Friedmann model with $ \Lam $ term, the coupling between the different fermionic levels ‘mixes up’ this pattern \[4\]. In the Friedmann model, the wave function has the form \[6\] $$\Psi = \Psi_0 (A) + \( \beta_C \beta^C \) \Psi_2 (A)~. \eqno (3.1)$$ As part of the Ansatz of \[6\], one requires $ \psi^A_{~~i} = e^{A A'}_{~~~~i} \wti \psi_{A'} $ and $ \wti \psi^A_{~~i} = e^{A A'}_{~~~~i} \psi_A $; this is in order that the form of the one-dimensional Ansatz should be preserved under one-dimensional local supersymmetry, suitably modified by local coordinate and Lorentz transformations. Thus the gravitino field is truncated to spin $ {1 \over 2} $. Note that $ \beta^A = {3 \over 4} n^{A A'} \wti \psi_{A'} $. One then proceeds as in Sec. II to derive the consequences of the $ \ol S_{A'} \Psi = 0 $ and $ S_A \Psi = 0 $ constraints at level $ \psi^1 $, by writing down the general expression for a constraint and then evaluating it at a Friedmann geometry. Note that it is not equivalent to set $ A = B = C $ in Eqs. (2.23) and (2.24); the coefficients in the constraint equations are different. One then obtains $$\hbar \kap^2 {d \Psi_0 \over d A} + 4 8 \pi^2 A \Psi_0 + 1 8 \pi^2 \hbar g A^2 \Psi_2 = 0 \eqno (3.2)$$ and $$\hbar^2 \kap^2 {d \Psi_2 \over d A} - 4 8 \pi^2 \hbar A \Psi_2 - 2 5 6 \pi^2 g A^2 \Psi_0 = 0~. \eqno (3.3)$$ These give second-order equations, for example $$A {d^2 \Psi_0 \over d A^2} - 2 {d \Psi_0 \over d A} + \[ - {4 8 \pi^2 \over \hbar \kap^2} A - {(48)^2 \pi^4 \over \hbar^2 \kap^4} A^3 + {9 \times 512 \pi^4 g^2 \over \hbar^2 \kap^4} A^5 \] \Psi_0 = 0~. \eqno (3.4)$$ This has a regular singular point at $ A = 0 $, with indices $ \lam = 0 $ and 3. There are two independent solutions, of the form $$\eqalignno { \Psi_0 &= a_0 + a_2 A^2 + a_4 A^4 + \ldots~, \cr \Psi_0 &= A^3 \( b_0 + b_2 A^2 + b_4 A^4 + \ldots \) ~, &(3.5) \cr }$$ convergent for all $ A $. They obey complicated recurrence relations, where (e.g.) $ a_6 $ is related to $ a_4,~a_2 $ and $ a_0 $. One can look for asymptotic solutions of the type $ \Psi_0 \sim \( B_0 + \hbar B_1 + \hbar^2 B_2 + \ldots \) \exp $ $ ( - I / \hbar ) $, and finds $$I = \pm {\pi^2 \over g^2} \( 1 - 2 g^2 A^2 \)^{3 \over 2}~, \eqno (3.6)$$ for $ 2 g^2 A^2 < 1 $. The minus sign in $ I $ corresponds to taking the action of the classical Riemannian solution filling in smoothly inside the three-sphere, namely a portion of the four-sphere $ S^4 $ of constant positive curvature. This gives the Hartle–Hawking state \[17\]. For $ A^2 > \( 1 / 2 g^2 \) $, the Riemannian solution joins onto the Lorentzian solution \[36\] $$\Psi \sim \cos \left \{ \hbar^{- 1} \[ {\pi^2 \( 2 g^2 A^2 - 1 \)^{3 \over 2} \over g^2} - {\pi \over 4} \] \right \}~, \eqno (3.7)$$ which describes de Sitter space-time. [**IV OTHER APPROACHES: $\sigma-$MODEL SUPERSYMMETRIC EXTENSION AND ASHTEKAR CANONICAL QUANTIZATION**]{} In this section we describe briefly other approaches to study the quantization of cosmological models with supersymmetry, which alllows one to extract similar conclusions. The $\sigma-$model supersymmetric extension in quantum cosmology has been developed by R. Graham and collaborators \[4,5,28,29\]. As is well known, the geometrodynamics of the Bianchi models reduce, formally, to the Hamiltonian dynamics of a particle with coordinates $q^{\ti{\mu}}$ in a three or two dimensional potential $V(q^{\ti{\mu}})$ \[35\]. In this approach, quantum models are constructed by coupling additional fermionic degrees of freedom to the purely gravitational ones (the minisuperspace vielbein) in such a way that the coupled system acquires a larger symmetry, namely local supersymmetry. The supersymmetric extension of a particle motion in a potential well is treated by supersymmetric quantum mechanics \[28\]. The case of dynamics on a curved manifold with metric $ds^2 = \ti{g}_{\ti{\mu}\ti{\nu}} d q^{\ti{\mu}} d q^{\ti{\nu}}$, where $\ti{g}_{\ti{\mu}\ti{\nu}}$ is the minisuperspace metric, has been studied in the N=2 supersymmetric model $\sigma-$model \[28\]. Supersymmetry than requires that the potential $V(q^{\ti{\mu}})$ must be derivable from an underlying superpotential $\Phi(q^{\ti{\mu}})$ as $$V(q^{\ti{\mu}}) = \half \ti{g}_{\ti{\mu}\ti{\nu}} {\pt\Phi \over {\pt q^{\ti{\mu}}}} {\pt\Phi \over{ \pt q^{\ti{\nu}}}}~. \eqno (4.1)$$ Such conditions were verified for the Bianchi type I, II, VII, VIII, IX, Kantowski-Sachs, Taub and Taub-Nut models. It is important to note that this  extension of Hamiltonian dynamics of a particle only leads to a N=2 . By contrast, from (1+3) dimensional N=1 supergravity a dimensional reduction allows one to obtain a (1+0)-dimensional theory with N=4 . (The extension of R. Graham’s approach to N=4 is non-trivial but such an extension has been provided for the case of a Bianchi type-IX without matter \[29\]). One may consider the N=2  as a [*sub*]{}symmetry of the larger N=4   obtained from supergravity and an attempt to clarify that connection explicitly would be interesting. In particular \[28\], while in the N=2 supersymmetry  extension it is possible to construct solutions in all fermionic sectors, this is different from N=4 supersymmetry  minisuperspace models, in which an additional internal rotational symmetry inherited from the Lorentz invariance of supergravity rules out all states except those in the empty and filled fermionic sectors. It is also curious to mention, even if not yet clear, that if the minisuperspace metric is Wick-rotated (i.e., the scale factor is complexified) this leads to the same restrictions of physical states as the requirement of all the symmetries included in the N=4  models. The application of the $\sigma-$model supersymmetry  extension programme to a general non-diagonal Bianchi type-IX model with a cosmological constant term is given in refs. \[5,28\]. The complete Hamiltonian and the classical constraints of the model are then derived. The system is quantized [*\` a la*]{} Dirac, replacing brackets by commutators or anti-commutators and the canonical momenta by appropriatr derivatives with respect to the canonical coordinates. However, this  framework was then applied to the particular case of a closed Friedmann-Robertson-Walker case. The wave function of the Universe can be written as $$|\Psi\rangle = (\Psi_0 + \Psi_a c_a^{\dagger} + \Psi_{a\Lambda} c_{\Lambda}^{\dagger} c_a^{\dagger} + \Psi_{\Lambda} c_{\Lambda}^{\dagger} )|0\rangle~, \eqno (4.2)$$ where $c_a^{\dagger},c_{\Lambda}^{\dagger}$ stand for a set of creation operators that replace the fermionic partners in a Fock state representation; $|0\rangle $. From the N=2  constraints R. Graham obtains that the admissable solution is of the form $$\Psi_0 = \Psi_{a\Lambda} =0 ~,\eqno(4.3)$$ $$\Psi_{a(\Lambda)}^{WBK} = \sqrt{(-)1 + \left[1-{{2\Lambda}\over{9\pi^2}}6\pi a^2\right]^{-\half} } \exp\left(-\half \int_0^{6\pi a^2} du \sqrt{ 1-{{2\Lambda}\over{9\pi^2}}u}\right) ~,\eqno(4.4)$$ which in the limit $a\rightarrow 0$ gives $\Psi_a \rightarrow 0;~\Psi_{\Lambda}\rightarrow \exp(-\Phi)$ where $\Phi$ is the superpotential previously described and $\Phi = 3\pi a^2~.$ For $6\pi a^2 > \left({{2\Lambda}\over{9\pi^2}}\right)^{-1}$ the exponent in is oscillatory. These results point tothe wave function as being in the the initial state found by Vilenkin \[37\], apart from the appearance of additional Grassmann variables. Now, let us make some brief comments about canonical quantization of supergravity using Ashtekar variables. Recently Ashtekar has presented a new formulation of Einstein gravity. One of the remarkable features of this formalism is that the constraints of gravity are simple polynomials of the canonical variables. So the constraints are more manageable than the ones expressed in the metric representation. One may hope that this feature persists in the canonical quantization of N = 1 supergravity. This turns out to be true: the supersymmetry constraints are again simple polynomials of the canonical variables. The Ashtekar formalism of the N = 1 supergravity has been formulated by Jacobson \[31\]. Capovilla and Guven \[38\] have successfully carried out the quantization of all the Bianchi type A models using this formalism. They obtained similar results to D’Eath \[9\]. Recently Capovilla and Obregon \[33\], Sano and Shiraishi \[32\] have also studied the quantization of the N = 1 supergravity with a cosmological constant. In \[32\] they found a semi-classical solution in the full theory. It has the form of exponential of the N =1 supersymmetric extension of the Chern-Simons functional. They applied this semi-classical wavefuction to the FRW universe using WKB approximation. The general line element is $$ds^{2} = -d \tau^{2} + {\sigma \over 8} d^{2} \Omega~, \eqno (4.5)$$ They have considered four cases: $ \tau :{\rm real},~ \sigma < 0;~ $ $ \tau : {\rm imaginary},~\sigma > 0;~ $ $ \tau : {\rm real},~ \sigma > 0 ;~$ $ \tau : {\rm imaginary },~\sigma < 0. $  They try to find the classical solution of $\sigma$ under these 4 cases. In the first case, the universe has the form of an Euclidean hyperbola. In case 2, the universe is a 4-sphere. In case 3, it is an open universe. In the last case, there is no solution. The reality condition on the gravitino is the Majorana condition. Their solution does not satisfy this condition in general. To obtain the real solution, they must transform the solutions by the transformations corresponding to the symmetries in the theory. They did not solve this problem in \[32\] and will consider it on another occasion. In \[33\], the quantization of the class A Bianchi Models was studied. In the following, we describe briefly the work of Capovilla and Obregón. The canonical variables are $$\tilde{\sigma}^{i}_{AB}, A_{i}^{~AB}, \psi_{iA}, \tilde{\pi}^{iA} {}~, \eqno (4.6)$$ where $ A_{i}^{~AB}$ is a self-dual connection, $ \tilde{\sigma}^{i}_{AB}$ is a densitized SU(2) soldering form where $$\tilde{\sigma}^{i}_{AB} = i h^\half n^{AA'} e^{iB}_{~~A'}~, \eqno (4.7)$$ and $det(q) q^{ij} = \tilde{\sigma}^{iAB} \tilde{\sigma}^{j}_{~AB}$. Here $\tilde{\pi}^{iA}$ is the momentum conjugate to $ \psi_{iA}$. In the class A Bianchi Models, a triad of basis vectors satifsy $$[ X_{a} , X_{b} ]^{i} = C_{ab}^{~~~c} X^{i}_{c}~, \eqno (4.8)$$ where $ C_{ab}^{~~~c} = \epsilon_{abc} M^{cd}$ denote the structure constant and $M^{ab} = M^{ba}$. The Jacobson phase space variables may be expanded with respect to the triad vectors and their duals $ \chi_{i}^{a} $ $$A_{iA}^{~~B} = A_{aA}^{~~B} \chi_{i}^{a}~, \eqno (4.9a)$$ $$\tilde{\sigma}^{iAB} = det(\chi) \sigma^{aAB} X^{i}_{a}~, \eqno (4.9b)$$ $$\psi_{i}^{~A} = \psi_{a}^{~A} \chi_{i}^{a}~, \eqno (4.9c)$$ $$\tilde{\pi}^{iA} = det(\chi) \pi^{aA} X^{i}_{a}~, \eqno (4.9d)$$ The fundamental Poisson brackets are given by $$\{\sigma^{aAB}, A_{bCD}\} = {i\over\sqrt{2}} \delta^{b}_{a}\delta_{(C}^{~~A} \delta_{D)}^{~~B}~, \eqno (4.10)$$ $$\{\pi^{aA} , \psi_{bB}\} = {i\over\sqrt{2}} \delta^{a}_{b} \delta_{B}^{~A} {}~, \eqno (4.11)$$ Because of the use of the self-dual connection as a field variable, we need to impose the [*reality condition*]{} \[39\] for the construction of the inner product of the physical wave function. However, we will not need it in here because one finds that the only physical wave function is the trivial one. In the quantum theory we get from the triad representation $$A_{a}^{~AB} = {1\over\sqrt{2}} { {\pt }\over {\pt \sig^{a}_{~AB}}} ~, \eqno (4.12)$$ $$\pi^{a}_{~A} = {1\over\sqrt{2}} { {\pt }\over {\pt \psi_{a}^{~A}}} ~, \eqno (4.13)$$ Using this ansatz, one obtains $$S^{A} = \half {{\pt^{2}}\over{\pt \sig^{a}_{AB} \pt \psi_{aA}^{~~~B} }} + i2\sqrt{2}m \left(\sigma^{a} \psi_{a}\right)^{A}~, \eqno (4.14)$$ $$\bar{S}^{A} = - \half \epsilon_{abd} M^{dc} (\sigma^{a}\sigma^{b} \psi_{c})^{A} + {1\over\sqrt{2}}\left(\sigma^{[a}\sigma^{b]}{{\pt \psi_{b}}\over{\pt\sig^{a}}}\right)^{A} -i2mh^{-\half}\left(\sigma_{a}{\pt\over{\pt\psi_{aA}}}\right)^{A}~, \eqno (4.15)$$ where $\Lambda = -4m^{2}$ The last terms in $S^{A},\bar{S}^{A}$ are the contributions from the cosmological constant. The supersymmetry constraints here are simpler than those in the metric formulation. Using the same decompositon of the gravitino field and the same ansatz of the wavefunction, one again obtains a set of equations. The only solution which satisfies this set of equations is the trivial solution: there are no physical states. The simple polynomial form of the constraints suggests that the study of the full theory with a cosmological constant term might be easier in this formalism. [**V CONCLUSION**]{} There are no physical quantum states for $ N = 1 $ supergravity with a $ \Lam $-term in the Bianchi models. The physical states found in Sec. III for the $ k = + 1 $ Friedmann model, where the degrees of freedom carried by the gravitino field are $ \beta_A $, disappear when the further fermionic degrees of freedom $ \rga_{A B C} $ of the Bianchi-IX model are included. One could also study this from the point of view of perturbation theory about the $ k = + 1 $ Friedmann model. As well as the usual gravitational harmonics \[40\], gravitino harmonics can be used \[41\]. For example, the Bianchi-IX model with radii $ A, B, C $ close together describes a particular type of ‘gravitational wave’ distortion of the Friedmann model; similarly for the $ \rga_{A B C} $ of the Bianchi-IX model, which describes a particular ‘gravitino wave’ distortion. Quite generally, in perturbation theory \[40,42\] one expects to find a wave function which is a product of the background wave function $ \Psi (A) $ times an infinite product of wave functions $ \psi_n $ (perturbations) where $ n $ labels the harmonics. And one further expects that the perturbation wave function corresponding to the Bianchi-IX modes must be zero, by a perturbative version of the argument of Sec. II. \[It will be interesting to investigate this.\] Hence the complete perturbative wave function should be zero; then physical states would be forbidden for a generic model of the gravitational and gravitino fields with $ \Lam $-term. This suggests that the full theory of $ N = 1 $ supergravity with a non-zero $ \Lam $-term should have [*no*]{} physical states. [**ACKNOWLEDGEMENTS**]{} The authors are very grateful to P.K. Townsend, H. Kodama for discussions and helpful suggestions. A.D.Y.C. thanks the Croucher Foundation of Hong Kong for financial support. P.R.L.V.M. is very grateful to Prof. V.N. Melnikov and to the Organizing Committee of the International School-Seminar “Multidimensional Gravity and Cosmology” for all their kind assistance. P.R.L.V.M. also gratefully acknowledges the support of a Human Capital and Mobility grant from the European Union (Program ERB4001GT930714). .2 true in [**REFERENCES**]{} by 4em = -4em \[1\] A. Maciás, O. Obregón and M.P. Ryan, Class. Quantum Grav. [**4**]{}, 1477 (1987). \[2\] P.D. D’Eath and D.I. Hughes, Phys. Lett. [**214**]{}B, 498 (1988). \[3\] R. Graham, Phys. Rev. Lett. [**67**]{}, 1381 (1991). \[4\] R. Graham, Phys. Lett. [**277**]{}B, 393 (1992). \[5\] R. Graham and J. Bene, Phys. Lett. [**302**]{}B, 183 (1993). \[6\] P.D. D’Eath and D.I. Hughes, Nucl. Phys. B [**378**]{}, 381 (1992). \[7\] L.J. Alty, P.D. D’Eath and H.F. Dowker, Phys. Rev. D [**46**]{}, 4402 (1992). \[8\] P.D. D’Eath, S.W. Hawking and O. Obregón, Phys. Lett. [**300**]{}B, 44 (1993). \[9\] P.D. D’Eath, Phys. Rev. D [**48**]{}, 713 (1993). \[10\] M. Asano, M. Tanimoto and N. Yoshino, Phys. Lett. [**314**]{}B, 303 (1993). \[11\] A.D.Y. Cheng, P.D. D’Eath and P.R.L.V. Moniz, [*Canonical Formulation of N=1 Supergravity with Supermatter*]{} DAMTP R94/13, submitted to Physical Review D;[*Quantization of a Friedmann Model with Supermatter in N=1 Supergravity*]{} DAMTP R94/21, gr-qc 9406048. \[12\] P. D. D’Eath and P. V. Moniz, [*Supersymmetric Quantum Cosmology*]{}, work in preparation. \[13\] C. Teitelboim, Phys. Rev. Lett. [**38**]{}, 1106 (1977). \[14\] P.D. D’Eath, Phys. Rev. D [**29**]{}, 2199 (1984). \[15\] R. Graham and H. Luckock, Sidney Univ. Maths. Report. 93-06, gr-qc 9311004. \[16\] S.W. Hawking and D.N. Page, Phys. Rev. D [**42**]{}, 2655 (1990). \[17\] J.B. Hartle and S.W. Hawking, Phys. Rev. D [**28**]{}, 2960 (1983). \[18\] P.D. D’Eath, Phys. Lett. B [**321**]{}, 368 (1994). \[19\] P.K. Townsend, Phys, Rev. D [**15**]{}, 2802 (1977). \[20\] P. van Nieuwenhuizen, Phys. Rep. [**68**]{}, 189 (1981). \[21\] A. Das and D.Z. Freedman, Nucl. Phys. B[**120**]{}, 271 (1979). \[22\] E.S. Fradkin and M.A. Vasiliev, Lebedev Institute preprint No.197 (1976). \[23\] D.Z. Freedman, Phys. Rev. Lett. [**38**]{}, 105 (1977). \[24\] S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. [**66**]{}B, 35 (1977). \[25\] S. Deser and B. Zumino, CERN report 1977 \[26\] P.D. D’Eath, Phys. Lett. B[**320**]{}, 20 (1994). \[27\] A.D.Y. Cheng, P.D. D’Eath and P.R.L.V. Moniz, Phys. Rev. D[**49**]{} (1994) 5246. \[28\] R. Graham, Phys. Rev. D[**48**]{}, 1602 (1993). \[29\] R. Graham and J. Bene, Phys. Rev. D[**49**]{}, p.799 (1994). \[30\] H. Kodama, Phys. Rev. D[**42**]{}, 2548 (1990) \[31\] T. Jacobson, Class. Quantum Grav. [**5**]{}, 923, (1988). \[32\] T. Sano and J. Shiraishi, Nucl. Phys. B[**410**]{}, 423, (1993). \[33\] R. Capovilla and O. Obregon, CIEA-GR-9402, gr-qc:9402043. \[34\] H.-J. Matschull, DESY 94-037, gr-qc:9403034. \[35\] M.P. Ryan and L.L. Shepley, [*Homogeneous Relativistic Cosmologies*]{} (Princeton University Press, Princeton, 1975). \[36\] J.B. Hartle, in [*High Energy Physics 1985*]{}, ed. M.J. Bowick and F. Gürsey (World Scientific, Singapore, 1986). \[37\] A. Vilenkin, Phys. Rev. D[**30**]{}, 509, (1984); Nucl. Phys. B[**252**]{}, 141 (1985). \[38\] R Capovilla and J Guven, CIEA-GR-9401, gr-qc: 9402025. \[39\] A. Ashtekar,[*Lectures on Non-Perturbative Canonical Gravity*]{}, Singapore, Singapore: World Scientific (1991) (Advanced series in astrophysics and cosmology, 6). \[40\] J.J. Halliwell and S.W. Hawking, Phys. Rev. D [**31**]{}, 1777 (1985). \[41\] D.I. Hughes, Ph.D. thesis, University of Cambridge (1990), unpublished. \[42\] P.D. D’Eath and J.J. Halliwell, Phys. Rev. D [**35**]{}, 1100 (1987).     [^1]: $\dagger$ [^2]: [e-mail address: prlvm10@amtp.cam.ac.uk]{}

--- abstract: 'We present novel online mechanisms for traffic intersection auctions in which users bid for priority service. We assume that users at the front of their lane are requested to declare their delay cost, i.e. value of time, and that users are serviced in decreasing order of declared delay cost. Since users are expected to arrive dynamically at traffic intersections, static pricing approaches may fail to estimate user expected waiting time accurately, and lead to non-strategyproof payments. To address this gap, we propose two Markov chain models to determine the expected waiting time of participants in the auction. Both models take into account the probability of future arrivals at the intersection. In a first model, we assume that the probability of future arrivals is uniform across lanes of the intersection. This queue-based model only tracks the number of lower- and higher-bidding users on access lanes, and the number of empty lanes. The uniformness assumption is relaxed in a second, lane-based model which accounts for lane-specific user arrival probabilities at the expense of an extended state space. We then design a mechanism to determine incentive-compatible payments in the dynamic sense. The resulting online mechanisms maximize social welfare in the long run. Numerical experiments on a four-lane traffic intersection are reported and compared to a static incentive-compatible mechanism. Our findings show that static incentive-compatible mechanisms may lead users to misreport their delay costs. In turn, the proposed online mechanisms are shown to be incentive-compatible in the dynamic sense and computationally efficient.' address: - 'School of Civil and Environmental Engineering, UNSW Sydney, NSW, 2052, Australia' - 'Univ. Gustave Eiffel, Univ. Lyon, IFSTTAR, ENTPE, LICIT UMR\_T9401, Lyon, F-69675, France' - 'Department of Civil, Environmental, and Geo-Engineering, University of Minnesota, Minneapolis, MN 55455, United States' author: - David Rey - 'Michael W. Levin' - 'Vinayak V. Dixit' bibliography: - 'DR.bib' title: 'Online Incentive-Compatible Mechanisms for Traffic Intersection Auctions' --- Auctions/bidding; incentive-compatibility; online mechanism design; Markov chain; traffic intersection Introduction ============ Traffic intersections are major bottlenecks of urban transport networks. Signalized traffic intersections are typically controlled so as to improve throughput, minimize vehicle delays or reduce emissions. While these design criteria have merits they are oblivious to users’ preferences. To address this limitation, auction-based mechanisms have emerged as promising alternatives to traditional traffic intersection control approaches [@schepperle2007agent; @vasirani2012market; @carlino2013auction]. In auction-based mechanisms, users are assumed to be able to declare their preferences, e.g. value of time, to an intersection manager so as to obtain services commensurate to their need. The intersection can be viewed as a server which role is to process users’ service requests. In this context, the intersection manager acts as a controller which decides users’ service sequence and users’ payments. In this paper, we propose novel online auction-based traffic intersection mechanisms that allows users in the front of their lanes to bid for intersection access. We show that the proposed online mechanisms are incentive-compatible, i.e. that users cannot achieve higher utility by bidding untruthfully. Vehicles bid for individual vehicle access to the intersection, as opposed to actuating traffic signal phases which serve multiple vehicles. Such individualized control is compatible with, for example, individual-vehicle signal lighting systems (such as ramp meters) or autonomous intersection management which controls individual vehicle trajectories [e.g. @dresner2004multiagent; @levin2017conflict]. The resulting online mechanisms are strategyproof and thus incentivize users to declare their true value of time, which is critical to promote auction-based mechanisms in urban transport networks. Intersection auctions have previously been explored in the context of autonomous intersection management due to the natural relationship between auction-based intersection controls and autonomous intersection management, which both decide intersection access for individual connected vehicles. We first discuss the literature on autonomous intersection management before discussing pricing mechanisms in queueing systems and their relevance to traffic intersection auctions. We then position our paper with respect to the field and highlight our contributions. Autonomous intersection management ---------------------------------- Due to the real-time computing needs of auction-based traffic intersection mechanisms, such approaches are typically conceived in the context of autonomous intersection management (AIM), as proposed in the seminal work of @dresner2004multiagent [@dresner2008multiagent]. In this paradigm, traffic control is assumed to be signal-free and users are assumed to be able reserve space-time trajectories through intersections. Several works have built on and extended the AIM protocol to richer configurations. @fajardo2011automated and @li2013modeling proposed AIM protocols based on a First-Come-First-Served (FCFS) policies where vehicles are prioritized based on their arrival time at the intersection. @de2015autonomous and @altche2016analysis developed microscopic vehicle trajectory optimization formulations to coordinate vehicles through intersections. In these formulations, the intersection manager decides vehicles’ service time and speed while time is discretized. @zhang2016optimal [@zhang2017decentralized] and @malikopoulos2018decentralized proposed decentralized approaches for traffic control based on FCFS conditions and considered energy consumption as well as vehicle separation and throughput in their formulations. @mirheli2019consensus developed a consensus-based approach for cooperative trajectory planning at signal-free intersections and shows that near-optimal solution can be achieved in competitive time. @wu2019dcl designed a multi-agent Markov decision process for cooperative trajectory planning in AIM. @levin2017conflict proposed a conflict point model that obviates the need to discretize the intersection space using tiles. This conflict point model was then adapted by @rey2019blue to accommodate both legacy and automated vehicles with signalized and signal-free traffic phases; and more recently by @chen2020stability to account for pedestrians movements. Pricing in queueing systems --------------------------- Traffic intersections can be viewed as queueing systems where users wait for service, i.e. intersection access, to pursue their journey. In this regard, auction mechanisms in traffic intersections are analogous to pricing mechanisms in queueing systems, which have been extensively studied over the past few decades. @naor1969regulation was the first to study the management of queues via user payments: he considered a queueing system where arriving users faced the choice of joining a queue with a fixed payment or refrain from joining the queue. He showed that user payments could often lead to solutions maximizing social welfare. @dolan1978incentive examined a queueing system where users are serviced in a sequence determined by their declared delay costs with priority given to higher delay costs. He proposed a dynamic mechanism to determine incentive-compatible user payments. @mendelson1990optimal proposed mechanisms for optimal incentive-compatibility pricing for the M/M/1 queue. The authors considered several user classes that differ by their delay costs and identify optimality conditions for net value maximization under elastic demand. In a similar context, @bradford1996pricing studied incentive-compatibility in the context of multiserver queues. Further details on equilibrium behavior in queueing systems can be found in a review by @hassin2003queue. @kittsteiner2005priority studied a priority queueing system where users have varying service (processing) times and where queue lengths are unobservable. Users are allowed to bid for service and are privately informed of their processing time. In this incomplete information context, they show that the shortest processing time (SPT) discipline is approximately efficient if the curvature of the cost function is relatively low. Traffic intersection auctions ----------------------------- Traffic intersection auctions have received a growing attention over the past few years, notably with the advent of connected and automated vehicle technology. [@schepperle2007agent] proposed a subsidy-based mechanism for slot allocation which aim to balance vehicle waiting time. First-In First-Out (FIFO) constraints are accounted for but incentive-compatibility is not guaranteed. The “effect of starvation” and its impact on fairness are discussed. [@vasirani2012market] developed a policy based on combinatorial auctions for the allocation of reservations at traffic intersections. The auction winner pays a price that is exactly the bid that was submitted, which is not incentive-compatible. [@carlino2013auction] proposed several types of auctions where users can bid on phases or reservations (one vehicle at a time). All drivers can participate in the auction but the candidates only include drivers at the front of their lane. Winners split the cost of the second-highest bid with proportional payment, which yields a static incentive-compatible mechanism and the strategyproofness of the dynamic case is not guaranteed. [@sayin2018information] proposed an auction protocol and mechanism in which all vehicles within range of the intersection manager communicate their value of time, and the intersection manager assigns reservations to those vehicles via a static auction. The proposed mechanism is incentive-compatible in the static sense, i.e. assuming all participants in the auction are known at the time payments are determined. [@censi2019today] introduced a credit-based auction mechanism, i.e. a karma system in which agents pay other agents karma for priority. Agents with a low priority today have an incentive to lose bids so they can achieve higher priority tomorrow by acquiring more karma. The authors attempt to calculate a Nash equilibria of user bids, and they show that a centralized strategy can be more unfair than some of the Nash equilibria identified. Recently, [@lin2020pay] proposed a mechanism for pricing intersection priority based on transferable utility games. In each game, players have the possibility to trade time among themselves and “winners pay losers to gain priority”. The authors provide empirical evidence that their approach is robust against adversarial user behavior, but no formal proof of incentive-compatibility is presented. Our contributions ----------------- As highlighted in our review of the literature, existing payment mechanisms for traffic intersection auctions are either only incentive-compatible for static auctions, or dynamic but not incentive-compatible. Static auctions do not account for future arrivals when determining user payments. As we will show in our numerical experiments, static payment mechanisms may incentivize users to misreport their preferences by bidding lower than their true delay costs and obtaining a higher utility, thus potentially compromising the acceptability of the corresponding mechanisms. Since users are expected to arrive dynamically at traffic intersections, online incentive-compatible mechanisms are required to ensure truthful user behavior in the long run. We make the following contributions to the field. We propose two online payment mechanisms for traffic intersection auctions. The two proposed mechanisms use Markov Chains (MC) to model the expected waiting time of users in the system but differ in the way the expected waiting time of users is determined. First, a queue-based model which assumes uniform arrival rates across the intersection is proposed and shown to be computationally inexpensive. This approach is then extended to a lane-based model which is able to capture non-uniform arrival rates across the lanes of the intersection. We show that both online mechanisms are incentive-compatible in the dynamic sense. We conduct numerical experiments to explore the behavior of the proposed online mechanisms and quantify the benefits of both queue-based and lane-based mechanisms. Finally, our simulations highlight the limitations of static incentive-compatible payment mechanisms compared to the proposed dynamic approaches. We introduce the traffic intersection auction in Section \[auction\] and present the proposed online mechanisms in Section \[mechanisms\]. Numerical experiments are reported in Section \[num\] and conclusions are summarized in Section \[con\]. Dynamic traffic intersection auction framework {#auction} ============================================== We consider a traffic intersection with a finite number of access lanes. We consider discrete time and we assume that at each time period, users arrive on access lanes with a known probability. Upon arriving at the front of their lane, users are requested to declare their delay cost which will be used as their bid in a combinatorial auction. We assume that users seek to minimize a generalized cost function which is a linear combination of their expected waiting time and their payment to the intersection manager. We assume that users are serviced sequentially by a single server, i.e. the traffic intersection and that service is nonpreemptive, i.e. cannot be interrupted once started. This does not prevent several users to simultaneously traverse the intersection. Instead, the proposed traffic intersection auction only aims to determine the sequence in which users are to be serviced and the payments they should be charged based on their declared preferences. The goal of the intersection manager is to maximize social welfare which is defined as the total generalized user cost. To achieve optimal social welfare, we seek to determine incentive-compatible user payments to ensure the truthful declaration of their delay costs. To ensure truthful user behavior, traffic intersection lanes are serviced in order of decreasing declared user delay costs. The expected waiting time of users is a function of the declared delay costs and the state of the system. In turn, the payment of users is to be determined by the intersection manager so as to maximize social welfare and ensure truthfulness. To present the proposed online mechanism for determining incentive-compatible payments, we consider the case of a user $i$ arriving at the front of its lane and declaring a delay cost of ${v_i}$. Let $Q$ be the number of access lanes of the intersection. We denote ${\overline{q}_i}$ (resp. ${\underline{q}_i}$) the number of lanes which are occupied by users with greater (resp. lower) delay cost than $i$. Further, we denote ${q_{\emptyset}}$ the number of empty lanes. The following equation holds for any user $i$ at any time period: $$Q = {\underline{q}_i}+ {\overline{q}_i}+ {q_{\emptyset}}+ 1. \label{eq:Q}$$ Accordingly, we can define the pricing queue of the proposed dynamic traffic intersection auction. In a traffic intersection with $Q$ access lanes, the *pricing queue* is a dynamic set of at most $Q$ users which represents the users at the front of their lane at a given time period. The pricing queue consists of the set of users which have no users in front of them in their access lane to the intersection. In the proposed traffic intersection auction, only users in the pricing queue are participants. This implies that users who are behind other users are not eligible to participate until they reach the front of their lane. This auction design aims to obviates blocking effects induced by non-overtaking conditions in access lanes of the intersection. Under such typical, non-overtaking conditions users waiting behind other users in their lane cannot bypass users in front of them, thus preventing them to be serviced before reaching the front of their lane. In the proposed approach, only users in the pricing queue participate in the auction. While this dynamic auction framework restricts the number of participants to at most $Q$ users, every user queueing in the intersection will eventually reach the front of its lane and participate in a single instance of the auction. This is in contrast to approaches in the literature where auction “losers” may participate in several auctions. We refer readers to @carlino2013auction who presented pricing mechanisms for traffic intersection auctions in which blocked users are allowed to “vote” for their lane. The objective of the proposed traffic intersection auction is to determine incentive-compatible user payments upon users joining the pricing queue. Let ${W_i({v_i})}$ be the expected waiting time of user $i$ declaring a delay cost of ${v_i}$. Let ${v_i^\star}$ be the true delay cost of user $i$, the expected waiting time cost of user $i$ declaring a delay cost of ${v_i}$ is ${v_i^\star}{W_i({v_i})}$. Let ${P_i({v_i})}$ be the payment of user $i$ declaring ${v_i}$, which is to be determined by the intersection manager. We denote ${C_i({v_i})}$ the generalized cost for user $i$ and define the user objective function as: The user objective function of user $i$ is: $$\min_{{v_i}} {C_i({v_i})}= \min_{{v_i}} ({v_i^\star}{W_i({v_i})}+ {P_i({v_i})}). \label{eq:obj}$$ Following the approach of [@dolan1978incentive], we will show that setting the payment ${P_i({v_i})}$ equal to the expected marginal delay cost user $i$ imposes to other users is incentive-compatible, i.e. the user objective is minimal for ${v_i}= {v_i^\star}$. We assume that the intersection manager has knowledge of the probability distribution of users’ valuation (delay costs), and of the arrival probability of users on access lanes of the intersection. These assumptions are reasonable since one can assume that the intersection manager can observe user behavior (arrival rate and valuations) and learn these probabilities over time. Let ${F({v_i})}$ be the cumulative distribution function representing the probability that a user declares a delay cost $v \leq {v_i}$. Further, let ${\underline{v}}$ and ${\overline{v}}$ be lower and upper bounds on users’ delay costs. For conciseness, we hereby use the term *lower-bidding user* (resp. *higher-bidding user*) to refer to a user with a declared delay cost lower (resp. higher) than that of user $i$. We next introduce the proposed online mechanisms for traffic intersection auctions. Online incentive-compatible mechanisms {#mechanisms} ====================================== We propose two online mechanisms to determine the expected waiting time and payment of users. The proposed mechanisms differ in the way the expected waiting time of users is determined. We first consider a queue-based model which only tracks the number of users bidding higher or lower than the reference user (Section \[queue\]). We next extend this model to a lane-based approach which tracks users’ arrival lane (Section \[lane\]). A generic payment mechanism is proposed to determine incentive-compatible payments for both queue- and lane-based models which result in two online mechanisms (Section \[payment\]). The queue-based mechanism assumes that the arrival probability of users is uniform across lanes of the intersection and provides a relatively low-dimensional mathematical framework. This uniformness assumption is relaxed in the lane-based mechanism which can accommodate non-uniform lane arrival probabilities at the expense of an extended state space. Queue-based model {#queue} ----------------- We use a MC to determine the expected waiting time of user $i$ under different system states. The state of the queuing system w.r.t. user $i$ is represented by the number of lanes occupied by lower bidding vehicles lanes ${\underline{q}_i}$ the number of empty lanes ${q_{\emptyset}}$. We denote ${\bm{q}_i}$ the vector ${\bm{q}_i}= ({\underline{q}_i},{q_{\emptyset}})$ characterizing the state of the system w.r.t. user $i$. User $i$ is serviced when all lanes are either occupied by a lower-bidding user or empty. Observe that once a lane is occupied by a lower-bidding user w.r.t. user $i$, it remains in this state until $i$ is serviced. Hence, from the perspective of user $i$, the state space of the MC can be characterized by the inequality ${\underline{q}_i}+{q_{\emptyset}}\leq Q-1$. This leads to the following characterization of the state space. In the queue-based model, the state space of the MC for user $i$ declaring a delay cost of ${v_i}$ in a pricing queue of size $Q$ is: $${\mathcal{S}_i^q}=\left\{{\bm{q}_i}=({\underline{q}_i},{q_{\emptyset}}) \in \mathbb{N}^2: {\underline{q}_i}+ {q_{\emptyset}}\leq Q-1 \right\}.$$ For each state ${\bm{q}_i}\in {\mathcal{S}_i^q}$, we are interested in determining the cost-to-go from this state through the MC. If ${\underline{q}_i}+ {q_{\emptyset}}= Q-1$, then yields ${\overline{q}_i}= 0$ which is equivalent to require that the pricing queue does not contain any user with declared delay costs higher than that of $i$. Hence, if ${\underline{q}_i}+ {q_{\emptyset}}= Q-1$, then the MC converges and user $i$ is serviced. Hence, to determine the state transition probabilities from state ${\bm{q}_i}$ to state ${\bm{q}_i}'$, we assume that ${\underline{q}_i}+{q_{\emptyset}}<Q-1$. To calculate the expected waiting time of user $i$ in the queue-based model, we enumerate system states with varying number of lower-bidding and empty lanes w.r.t. user $i$. We assume that users’ delay cost can be represented by a continuous random variable with a known distribution. Further, we assume that the arrival of users on intersection lanes can be represented by a boolean random variable. Let $\xi \in {\mathbb{R}}^Q \times 2^Q$ represent the uncertain data in the system. Let $f_q :{\mathcal{S}_i^q}\times {\mathbb{R}}^Q \times 2^Q\rightarrow {\mathcal{S}_i^q}$ be the transition function of the MC which maps the current state and the uncertain data to the next state. We denote ${g_q({\bm{q}_i})}$ the one-step-cost of the MC representing the service time of the system in state ${\bm{q}_i}$ and we assume that the one-step-cost is deterministic. This assumption is plausible since users’ behavior is assumed to be deterministic, i.e. users cannot change their declared delay cost over time. Let ${W_q({v_i},{\bm{q}_i})}$ be the expected waiting time of user $i$ declaring ${v_i}$ for state ${\bm{q}_i}$, which represents the cost-to-go of the MC representing the state evolution in the queue-based model. Let ${\mathcal{T}_i^q}= \{{\bm{q}_i}\in {\mathcal{S}_i^q}: {q_{\emptyset}}+ {\underline{q}_i}= Q - 1\}$ be the set of terminal states in the queue-based mechanism. The expected waiting time of user $i$ declaring ${v_i}$ for state $q$ can be determined as: $${W_q({v_i},{\bm{q}_i})}=\begin{cases} 0 & \text{if } {\bm{q}_i}\in {\mathcal{T}_i^q}, \\ {\mathbb{E}\left[{W_q\left({v_i}, f_q\left({\bm{q}_i}, \xi\right)\right)}\right]} + {g_q({\bm{q}_i})}& \text{otherwise}. \end{cases} \label{eq:wq}$$ Note that the term ${\mathbb{E}\left[{W_q\left({v_i}, f_q\left({\bm{q}_i}, \xi\right)\right)}\right]}$ represents the expected value of the expected waiting time ${W_q\left({v_i}, f_q\left({\bm{q}_i}, \xi\right)\right)}$ after the transition function $f_q$. If ${\bm{q}_i}$ is not a terminal state, i.e. ${\underline{q}_i}+{q_{\emptyset}}< Q-1$, the expected waiting time of user $i$ can be calculated using the probability ${\mathrm{Pr}\left[{\bm{q}_i}'|{\bm{q}_i}\right]}$ of transitioning from state ${\bm{q}_i}$ to state ${\bm{q}_i}'$. The expected waiting time of user $i$ declaring ${v_i}$ is: $${\mathbb{E}\left[{W_q\left({v_i}, f_q\left({\bm{q}_i}, \xi\right)\right)}\right]} = \sum\limits_{{\bm{q}_i}' \in {\mathcal{S}_i^q}} {\mathrm{Pr}\left[{\bm{q}_i}'|{\bm{q}_i}\right]} {W_q({v_i},{\bm{q}_i}')}. \label{eq:ewt}$$ Combining Eqs. and yields the following system of linear equations: $$\begin{aligned} {W_q({v_i},{\bm{q}_i})}&= 0 \quad &\forall {\bm{q}_i}\in {\mathcal{T}_i^q}, \\ {W_q({v_i},{\bm{q}_i})}(1 - {\mathrm{Pr}\left[{\bm{q}_i}|{\bm{q}_i}\right]}) - \sum_{{\bm{q}_i}' \in {\mathcal{S}_i^q}: {\bm{q}_i}' \neq {\bm{q}_i}} {W_q({v_i},{\bm{q}_i}')}{\mathrm{Pr}\left[{\bm{q}_i}'|{\bm{q}_i}\right]} &= {g_q({\bm{q}_i})}\quad &\forall {\bm{q}_i}\in {\mathcal{S}_i^q}\setminus {\mathcal{T}_i^q}. \end{aligned}$$ \[eq:wq2\] Let $p$ be the probability that a user arrives on a lane of the intersection at the next time period. Recall that ${F({v_i})}$ is the cumulative distribution function of users bids. On each lane, three events may occur with the following probabilities: i) a lane can become or remain empty with probability $(1-p)$; ii) a lower-bidding user can arrive with probability $p{F({v_i})}$; and iii) a higher-bidding user can arrive with probability $p(1-{F({v_i})})$. Observe that there are $\binom{{q_{\emptyset}}+1}{{q_{\emptyset}}'}$ ways to choose ${q_{\emptyset}}'$ empty lanes among ${q_{\emptyset}}+1$ lanes. There are ${\underline{q}_i}' - {\underline{q}_i}$ lanes which become occupied by lower-bidding users and there are $\binom{{q_{\emptyset}}+1-{q_{\emptyset}}'}{{\underline{q}_i}'-{\underline{q}_i}}$ ways to choose ${\underline{q}_i}'-{\underline{q}_i}$ of the remaining ${q_{\emptyset}}+1-{q_{\emptyset}}'$ non-empty lanes to become occupied by a lower-bidding user. Accordingly, the transition probability from state $({\underline{q}_i},{q_{\emptyset}})$ to state $({\underline{q}_i}',{q_{\emptyset}}')$ is: $${\mathrm{Pr}\left[{\bm{q}_i}'|{\bm{q}_i}\right]}= \binom{{q_{\emptyset}}+1}{{q_{\emptyset}}'}(1-p)^{{q_{\emptyset}}'} \binom{{q_{\emptyset}}+1-{q_{\emptyset}}'}{{\underline{q}_i}'-{\underline{q}_i}} (p{F({v_i})})^{{\underline{q}_i}'-{\underline{q}_i}} (p(1-{F({v_i})}))^{{q_{\emptyset}}+1-{q_{\emptyset}}'- {\underline{q}_i}'+{\underline{q}_i}}. \label{eq:prob}$$ The expected waiting time defined in Eq. is a recursive function which can be calculated by solving a series of systems of linear equations where each equation corresponds to a possible state ${\bm{q}_i}\in {\mathcal{S}_i^q}$. Since all states such that ${\underline{q}_i}+{q_{\emptyset}}=Q-1$ are terminal, their corresponding expected waiting time is null and these variables can be eliminated from the system of equations. Hence, we need only to consider the set of pairs of nonnegative integers $({\underline{q}_i},{q_{\emptyset}})$ such that ${\underline{q}_i}+ {q_{\emptyset}}\leq Q-2$. Observe that each variable ${\underline{q}_i}$ or ${q_{\emptyset}}$ can take $Q-2+1$ values, thus the number of equations in the system is $\sum_{k=0}^{Q-1} k = \binom{Q}{2}$. Further, observe that ${\underline{q}_i}$ can never decrease while ${q_{\emptyset}}$ can increase or decrease. Hence, for any state ${\bm{q}_i}=({\underline{q}_i},{q_{\emptyset}})$, we can calculate ${W_q({v_i},{\bm{q}_i})}$ in a dynamic programming fashion starting with the maximum number of lanes, i.e. $k=Q-1$, and iterating downwards until $k={\underline{q}_i}$. At each iteration $k$, we solve the system of $Q-k$ linear equations with $Q-k$ variables corresponding to states ${\bm{q}_i}=(k,l)$ for $0\leq l \leq Q-1-k$. We illustrate the queue-based model in Example \[ex:queue\]. \[ex:queue\] Consider an intersection with $Q=4$ access lanes. The state space ${\mathcal{S}_i^q}$ can be represented as in Table \[tab:ss\] where the bottom row corresponds to terminal states. --------- --------- --------- --------- --------- --------- --------- $(0,0)$ $(1,0)$ $(0,1)$ $(2,0)$ $(1,1)$ $(0,2)$ $(3,0)$ $(2,1)$ $(1,2)$ $(0,3)$ --------- --------- --------- --------- --------- --------- --------- : State space ${\mathcal{S}_i^q}$ for an intersection with $Q=4$.[]{data-label="tab:ss"} The reachable sets denoted ${\mathcal{S}_i^q}({\underline{q}_i},{q_{\emptyset}})$ of the six non-terminal states of this MC are: $$\begin{aligned} {\mathcal{S}_i^q}(0,0) &= \{(0,0),(0,1),(1,0)\} \\ {\mathcal{S}_i^q}(0,1) &= \{(0,0),(0,1),(0,2),(1,0),(1,1),(2,0)\} \\ {\mathcal{S}_i^q}(0,2) &= \{(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(2,0),(2,1),(3,0)\} \\ {\mathcal{S}_i^q}(1,0) &= \{(1,0),(1,1),(2,0)\} \\ {\mathcal{S}_i^q}(1,1) &= \{(1,0),(1,1),(1,2),(2,0),(2,1),(3,0)\} \\ {\mathcal{S}_i^q}(2,0) &= \{(2,0),(2,1),(3,0)\} $$ We now illustrate the behavior of the queue-based model when determining the expected waiting time for the state ${\bm{q}_i}=(1,1)$. 1. At the first iteration, $k=Q-1=3$ and there are $Q-k=1$ equation corresponding to variable $W_q({v_i},(3,0))$, which is trivial since $(3,0)$ is a terminal state, i.e. $W_q({v_i},(3,0)) = 0$. 2. At the second iteration, $k=Q-2=2$ and there are $Q-k=2$ equations corresponding to variables $W_q({v_i},(2,0))$ and $W_q({v_i},(2,1))$, i.e.: $$\begin{aligned} W_q({v_i},(2,0)) \left(1 - {\mathrm{Pr}\left[(2,0)|(2,0)\right]}\right) - W_q({v_i},(2,1)), {\mathrm{Pr}\left[(2,1)|(2,0)\right]}&\\ - W_q({v_i},(3,0)) {\mathrm{Pr}\left[(3,0)|(2,0)\right]} &= g_q({v_i},(2,0)) \\ W_q({v_i},(2,1)) &= 0.\end{aligned}$$ Since $W_q({v_i},(3,0)) = 0$ was determined at the previous iteration, the above system collapses to: $$W_q({v_i},(2,0)) = \frac{g_q({v_i},(2,0))}{\left(1 - {\mathrm{Pr}\left[(2,0)|(2,0)\right]}\right)}.$$ 3. At the third iteration, $k=Q-3=1$ and there are $Q-k=3$ equations corresponding to variables $W_q({v_i},(1,0))$, $W_q({v_i},(1,1))$ and $W({v_i},1,2)$ i.e.: $$\begin{aligned} W_q({v_i},(1,0)) \left(1 - {\mathrm{Pr}\left[(1,0)|(1,0)\right]}\right) - W_q({v_i},(1,1)){\mathrm{Pr}\left[(1,1)|(1,0)\right]} & \\ - W_q({v_i},(2,0)){\mathrm{Pr}\left[(2,0)|(1,0)\right]} &= g_q({v_i},(1,0)), \\ W_q({v_i},(1,1)) \left(1 - {\mathrm{Pr}\left[(1,1)|(1,1)\right]}\right) - W_q({v_i},(1,0)){\mathrm{Pr}\left[(1,0)|(1,1)\right]} & \\ - W_q({v_i},(1,2)){\mathrm{Pr}\left[(1,2)|(1,1)\right]} - W_q({v_i},(2,0)){\mathrm{Pr}\left[(2,0)|(1,1)\right]} & \\ - W_q({v_i},(2,1)){\mathrm{Pr}\left[(2,1)|(1,1)\right]} - W_q({v_i},(3,0)){\mathrm{Pr}\left[(3,0)|(1,1)\right]}&= g_q({v_i},(1,1)), \\ W_q({v_i},1,2) &= 0. \end{aligned}$$ Note that $W_q({v_i},(3,0))$, $W_q({v_i},(2,1))$ and $W_q({v_i},(2,0))$ were determined at previous iterations. Since $W_q({v_i},1,2) = 0$, the above system collapses to a system of two equations in the variables $W_q({v_i},(1,0))$ and $W_q({v_i},(1,1))$, the latter being the desired expected waiting time corresponding to state ${\bm{q}_i}= (1,1)$. Lane-based model {#lane} ---------------- We now consider an alternative approach where lane arrival probabilities can be non-uniform across lanes of the intersection. We abuse notation and denote $p_j$ the probability of a user arriving on lane $j$ at the next time period. Let ${\mathcal{L}_i}= \{{\sigma_{\emptyset}},{\sigma_{\underline{v}_i}},{\sigma_{\overline{v}_i}}\}$ be the set of possible lane-based states with regards to user $i$ bidding ${v_i}$; where ${\sigma_{\emptyset}}$ represents an empty lane, ${\sigma_{\underline{v}_i}}$ the arrival of a user bidding lower than ${v_i}$ and ${\sigma_{\overline{v}_i}}$ the arrival of a user bidding higher than ${v_i}$. Let ${z_{ij}}\in {\mathcal{L}_i}$ be the state of lane $j$ w.r.t. user $i$. We denote ${\bm{z}_i}= (z_{i,1}, \ldots, z_{i,Q-1})$ the state wr.r.t. to user $i$ in the lane-based model. In the lane-based model, the state space of the MC for user $i$ declaring a delay cost of ${v_i}$ in a pricing queue of size $Q$ is $${\mathcal{S}_i^z}= \left\{{\bm{z}_i}= (z_{i,j_1},\ldots,z_{i,j_{Q-1}}) \in {\mathcal{L}_i}^{Q-1}\right\}.$$ In the lane-based model, the number of possible states w.r.t. to user $i$ in an intersection with $Q$ lanes is $|{\mathcal{L}_i}|^{Q-1}$, hence a four-lane intersection has $3^3 = 27$ possible states. Let $f_z :{\mathcal{S}_i^z}\times {\mathbb{R}}^Q \times 2^Q\rightarrow {\mathcal{S}_i^z}$ be the transition function of the MC and let ${g_z({\bm{z}_i})}$ be the one-step cost in the lane-based model. Let ${\mathcal{T}_i^z}\subset {\mathcal{S}_i^z}$ be the set of terminal states, i.e. ${\mathcal{T}_i^z}= \{{\bm{z}_i}\in {\mathcal{S}_i^z}: {z_{ij}}= {\sigma_{\emptyset}}\vee {z_{ij}}= {\sigma_{\underline{v}_i}}, j=1,\ldots,Q-1\}$. The expected waiting time ${W_z({v_i},{\bm{z}_i})}$ of user $i$ declaring ${v_i}$ for state ${\bm{z}_i}$ in the lane-based model is thus: $${W_z({v_i},{\bm{z}_i})}= \begin{cases} 0 & \text{if } {\bm{z}_i}\in {\mathcal{T}_i^z},\\ {\mathbb{E}\left[{W_z\left({v_i}, f_z\left({\bm{z}_i}, \xi\right)\right)}\right]} + {g_z({\bm{z}_i})}& \text{otherwise}, \end{cases} \label{eq:wzi}$$ with: $${\mathbb{E}\left[{W_z\left({v_i}, f_z\left({\bm{z}_i}, \xi\right)\right)}\right]} = \sum_{{\bm{z}'_i}\in {\mathcal{S}_i^z}} {\mathrm{Pr}\left[{\bm{z}'_i}|{\bm{z}_i}\right]} {W_z({v_i},{\bm{z}'_i})}. \label{eq:wfzi}$$ This leads to the following system of linear equations: $$\begin{aligned} {W_z({v_i},{\bm{z}_i})}&= 0 \quad &\forall {\bm{z}_i}\in {\mathcal{T}_i^z}, \\ {W_z({v_i},{\bm{z}_i})}(1 - {\mathrm{Pr}\left[{\bm{z}_i}|{\bm{z}_i}\right]}) - \sum_{{\bm{z}'_i}\in {\mathcal{S}_i^z}: {\bm{z}'_i}\neq {\bm{z}_i}} {W_z({v_i},{\bm{z}'_i})}{\mathrm{Pr}\left[{\bm{z}'_i}|{\bm{z}_i}\right]} &= {g_z({\bm{z}_i})}\quad &\forall {\bm{z}_i}\in {\mathcal{S}_i^z}\setminus {\mathcal{T}_i^z}.\end{aligned}$$ \[eq:WTZ\] Unlike the systems of equations , the system does not admit an intuitive decomposition which could be used to solve the system of equations via a recursive algorithm. Thus, in our numerical experiments, is solved in a single step using traditional linear algebra codes. To determine transition probabilities in the lane-based mechanism, it is necessary to track which user will be serviced at the next time period. For any non-terminal state, there is at least one higher-bidder in the pricing queue and the highest-bidder in the queue will be serviced next. Let ${\mathcal{V}_{>i}}$ be the set of higher-bidders, i.e. ${\mathcal{V}_{>i}}= \{j_1, j_2, \ldots, j_{{\overline{q}_i}}\}$ with ${v_i}< v_{j_1} < v_{j_2} < \ldots < v_{j_{{\overline{q}_i}}}$. Since the bids of future arrivals are unknown, the probability distribution that a higher-bidder is the highest-bidder is uniform. Let ${z_{ij}}(k)$ be the state of lane $j$ with regards to user $i$ assuming lane $k$ is the moving lane, i.e. the lane occupied with the highest-bidder in the pricing queue. Let ${\mathrm{Pr}\left[{z'_{ij}}|{z_{ij}}(k)\right]}$ be the transition probability of lane $j$ with regards to user $i$ from state ${z_{ij}}(k)$ to state ${z'_{ij}}$. Since the arrival process of a lane is assumed to be independent of that of other lanes, the transition probability from state ${\bm{z}_i}$ to ${\bm{z}'_i}$ denoted ${\mathrm{Pr}\left[{\bm{z}'_i}|{\bm{z}_i}\right]}$ is $${\mathrm{Pr}\left[{\bm{z}'_i}|{\bm{z}_i}\right]} = {\mathrm{Pr}\left[(z'_{i,1}, \ldots, z'_{i,Q-1})|(z_{i,1}, \ldots, z_{i,Q-1})\right]} = \frac{1}{{\overline{q}_i}} \sum_{k \in {\mathcal{V}_{>i}}} \prod_{j=1}^{Q-1} {\mathrm{Pr}\left[{z'_{ij}}|{z_{ij}}(k)\right]}. \label{eq:dprob}$$ In the lane-based mechanism, transition probabilities are function of current lane state. If lane $j$ is empty or if lane $j$ is occupied by the highest bidder in the pricing queue, then we say that this lane is open. Otherwise, lane $j$ is either occupied by a lower-bidding user or a higher-bidding user who will not be serviced next and we say that his lane is closed. Let ${\mathcal{V}_{-i}}$ be the set of users in the pricing queue other than user $i$. Let ${\mathcal{O}_{i,j}}\subseteq {\mathcal{L}_i}$ and ${\mathcal{C}_{i,j}}\subseteq {\mathcal{L}_i}$ be the set of open and closed states for lane $j$ with regards to user $i$, respectively: $$\begin{aligned} {\mathcal{O}_{i,j}}= \begin{cases} \{{\sigma_{\emptyset}},{\sigma_{\overline{v}_i}}\} &\text{ if } j \in \operatorname*{arg\,max}\{v_k : k \in {\mathcal{V}_{-i}}\}, \\ \{{\sigma_{\emptyset}}\} &\text { otherwise}, \end{cases}\\ {\mathcal{C}_{i,j}}= \begin{cases} \{{\sigma_{\underline{v}_i}}\} &\text{ if } j \in \operatorname*{arg\,max}\{v_k : k \in {\mathcal{V}_{-i}}\}, \\ \{{\sigma_{\underline{v}_i}},{\sigma_{\overline{v}_i}}\} &\text { otherwise}. \end{cases}\end{aligned}$$ For any lane $j$ in state $\sigma \in {\mathcal{O}_{i,j}}$, we have the following transition probabilities: $$\begin{aligned} {\mathrm{Pr}\left[{\sigma_{\emptyset}}|\sigma\right]} &= 1-p_j \quad &&\forall j \in \{1,\ldots,Q-1\}, \forall \sigma \in {\mathcal{O}_{i,j}},\\ {\mathrm{Pr}\left[{\sigma_{\underline{v}_i}}|\sigma\right]} &= p_j{F({v_i})}\quad &&\forall j \in \{1,\ldots,Q-1\}, \forall \sigma \in {\mathcal{O}_{i,j}},\\ {\mathrm{Pr}\left[{\sigma_{\overline{v}_i}}|\sigma\right]} &= p_j(1 - {F({v_i})}) \quad &&\forall j \in \{1,\ldots,Q-1\}, \forall \sigma \in {\mathcal{O}_{i,j}}.\end{aligned}$$ Otherwise if lane $j$ is in state $\sigma \in {\mathcal{C}_{i,j}}$, this lane remains in its current state with probability one, since the corresponding user cannot be serviced at the next time period. $$\begin{aligned} {\mathrm{Pr}\left[\sigma'|\sigma\right]} &= \begin{cases} 1 \quad \text{ if } \sigma' = \sigma,\\ 0 \quad \text{ otherwise}, \end{cases} \quad &&\forall j \in \{1,\ldots,Q-1\}, \forall \sigma \in {\mathcal{C}_{i,j}}.\end{aligned}$$ Using these lane-based transition probabilities, we can determine intersection-based transition probabilities via Eq. . The queue-based model can be viewed as a special case of the lane-based model. Let $\bm{1}_{{z_{ij}}=\sigma}$ be the indicator function taking value 1 if ${z_{ij}}=\sigma$ and 0 otherwise. We first define the mapping between the state spaces of both queue- and lane-based models. Let $h_i : {\mathcal{S}_i^z}\rightarrow {\mathcal{S}_i^q}$ be a function mapping the state space of the lane-based model to the state space of the queue-based model: $h_i : {\bm{z}_i}\mapsto {\bm{q}_i}= h_i({\bm{z}_i})$ with ${\underline{q}_i}= \sum_{j=1}^{Q-1} \bm{1}_{{z_{ij}}= {\sigma_{\underline{v}_i}}}$ and ${q_{\emptyset}}= \sum_{j=1}^{Q-1} \bm{1}_{{z_{ij}}= {\sigma_{\emptyset}}}$. The proposition below highlights the correspondence between queue- and lane-based models. If lane arrival probabilities are uniform, i.e. $p = p_j$ for all $j=1,\ldots Q-1$, and if the one-step-costs of the queue- and lane-based models are such that $g_z({\bm{z}_i}) = g_q({\bm{q}_i})$ for all states ${\bm{z}_i}\in {\mathcal{S}_i^z}$ and ${\bm{q}_i}\in {\mathcal{S}_i^q}$ such that ${\bm{q}_i}= h_i({\bm{z}_i})$, then ${W_z({v_i},{\bm{z}_i})}= W_q({v_i},{\bm{q}_i})$ for all such states. The probability of state ${\bm{q}_i}$ occuring is: $${\mathrm{Pr}\left[{\bm{q}_i}\right]} = (1-p)^{{q_{\emptyset}}} p^{{\underline{q}_i}+{\overline{q}_i}} {F({v_i})}^{{\underline{q}_i}} (1-{F({v_i})})^{{\overline{q}_i}}.$$ The probabilities of occurence for lane-specific states ${z_{ij}}$ are: $${\mathrm{Pr}\left[{z_{ij}}\right]} = \begin{cases} 1 - p_j &\text{ if } {z_{ij}}= {\sigma_{\emptyset}}, \\ p_j {F({v_i})}&\text{ if } {z_{ij}}= {\sigma_{\underline{v}_i}}, \\ p_j (1-{F({v_i})}) &\text{ if } {z_{ij}}= {\sigma_{\overline{v}_i}}. \end{cases}$$ Hence, if lane arrival probabilities are uniform, i.e. $p = p_j$, then $${\mathrm{Pr}\left[{\bm{z}_i}\right]} = \prod_{j=1}^{Q-1} {\mathrm{Pr}\left[{z_{ij}}\right]} = {\mathrm{Pr}\left[{\bm{q}_i}\right]} \text{ with } {\bm{q}_i}= h_i({\bm{z}_i}).$$ Accordingly, if lane arrival probabilities are uniform, transition probabilities given by and are equal for any states ${\bm{q}_i},{\bm{q}_i}' \in {\mathcal{S}_i^q}$ and ${\bm{z}_i},{\bm{z}'_i}\in {\mathcal{S}_i^z}$ such that ${\bm{q}_i}= h_i({\bm{z}_i})$ and ${\bm{q}_i}' = h_i({\bm{z}'_i})$, i.e. ${\mathrm{Pr}\left[{\bm{z}'_i}|{\bm{z}_i}\right]} = {\mathrm{Pr}\left[h_i({\bm{z}'_i}) | h_i({\bm{z}_i})\right]}$. Hence, if the one-step-costs are such that $g_z({\bm{z}_i}) = g_q(h_i({\bm{z}_i}))$, the solutions of the systems of equations and verify ${W_z({v_i},{\bm{z}_i})}= W_q({v_i},{\bm{q}_i})$ for any pair of states ${\bm{z}_i}\in {\mathcal{S}_i^z}$ and ${\bm{q}_i}\in {\mathcal{S}_i^q}$ such that ${\bm{q}_i}= h_i({\bm{z}_i})$. \[propmap\] Proposition \[propmap\] establishes a correspondence between the proposed queue- and lane-based models for determining the expected waiting times of users and shows that under uniform lane arrival probabilities and one-step-costs, both models are equivalent. We next illustrate the lane-based model and the mapping ${\bm{q}_i}= h_i({\bm{z}_i})$. \[ex:lane\] We consider an intersection with $Q=4$ lanes. To determine the expected waiting of a state ${\bm{z}_i}\in {\mathcal{S}_i^z}$ requires solving the systems of equations with $|{\mathcal{T}_i^z}|=2^{Q-1} = 8$ terminal states and $|{\mathcal{S}_i^z}| - |{\mathcal{T}_i^z}| = 3^{Q-1} - 8 = 19$ non-terminal states. As in Example \[ex:queue\], we examine the queue-based state ${\bm{q}_i}= (1,1)$. The set of lane-based states ${\bm{z}_i}\in {\mathcal{S}_i^z}$ which are antecedents of ${\bm{q}_i}$ via the mapping ${\bm{q}_i}= h_i({\bm{z}_i})$ is, denoted ${\mathcal{S}_i^z}(h_i^{-1}((1,1)))$ is: $${\mathcal{S}_i^z}(h_i^{-1}((1,1))) = \{({\sigma_{\emptyset}},{\sigma_{\underline{v}_i}},{\sigma_{\overline{v}_i}}), ({\sigma_{\underline{v}_i}},{\sigma_{\emptyset}},{\sigma_{\overline{v}_i}}), ({\sigma_{\underline{v}_i}},{\sigma_{\overline{v}_i}},{\sigma_{\emptyset}}), ({\sigma_{\emptyset}},{\sigma_{\overline{v}_i}},{\sigma_{\underline{v}_i}}), ({\sigma_{\overline{v}_i}},{\sigma_{\emptyset}},{\sigma_{\underline{v}_i}}), ({\sigma_{\overline{v}_i}},{\sigma_{\underline{v}_i}},{\sigma_{\emptyset}})\}$$ According to Proposition \[propmap\], if $p_j = p$ for all lanes $j$ of the intersection and if $g_z({\bm{z}_i}) = g_q(h_i({\bm{z}_i}))$ for all states ${\bm{z}_i}$, then $W_q({v_i},(1,1)) = W_z({v_i},{\bm{z}_i})$ for all ${\bm{z}_i}\in {\mathcal{S}_i^z}((1,1))$. We next present a payment mechanism that can be used with both of these models to determine incentive-compatible payments. Payment mechanism {#payment} ----------------- The proposed payment mechanism can be used with both queue- and lane-based models for determining the expected waiting time of users. Hence, in this section, we simply denote ${W_i({v_i})}$ the expected waiting time of user $i$ bidding ${v_i}$. This expected waiting time can be replaced by ${W_q({v_i},{\bm{q}_i})}$ for the queue-based model, or by ${W_z({v_i},{\bm{z}_i})}$ for the lane-based model. To determine incentive-compatible user payments, we calculate the expected marginal cost user $i$ imposes on other users. For this, we first examine the period of time over which an extra user in the lane of user $i$ is expected to have an impact on other users. The busy period w.r.t. to user $i$ is the expected time required to clear the queues if $i$ declares a minimal delay cost, i.e. ${v_i}= {\underline{v}}$. Observe that if ${v_i}={\underline{v}}$, then ${\underline{q}_i}=0$, since no user can bid lower than ${\underline{v}}$ i.e. $F({\underline{v}})=0$. We measure the impact of user $i$ onto other users throughout the remaining busy period, i.e. the duration between the start of service of user $i$ if $i$ declares ${v_i}$, and the duration between the start of service of $i$ if $i$ had declared a minimal delay cost. Hence, the remaining busy period w.r.t. user $i$ bidding ${v_i}$ denoted ${\Delta W_i({v_i})}$ can be defined as: $${\Delta W_i({v_i})}\equiv {W_i({\underline{v}})}- {W_i({v_i})}. \label{eq:rmb1}$$ We can split the remaining busy period w.r.t. user $i$ into two components: the expected portion of ${\Delta W_i({v_i})}$ which affects users who arrived *before* user $i$; and the expected portion of ${\Delta W_i({v_i})}$ which affects users who are expected to arrive *after* user $i$. We denote ${B_i({v_i})}$ and ${A_i({v_i})}$ these quantities, respectively, and impose: $${\Delta W_i({v_i})}= {B_i({v_i})}+ {A_i({v_i})}. \label{eq:rmb2}$$ To determine the marginal delay caused by user $i$ onto users already in the pricing queue, we need only to consider those ${\underline{q}_i}$ users which bid lower than $i$ since others users are not delayed by user $i$. Let ${\mathcal{V}_{<i}}$ be the set of all users bidding lower than $i$ ordered by increasing declared delay costs, i.e. ${\mathcal{V}_{<i}}= \{j_1, j_2, \ldots, j_{{\underline{q}_i}}\}$ with $v_{j_1} < v_{j_2} < \ldots < v_{j_{{\underline{q}_i}}} < {v_i}$. To determine the expected portion of the remaining busy period w.r.t. user $i$ which affects user $j \in {\mathcal{V}_{<i}}$, we take the perspective of an extra user bidding ${v_j}$ in the lane of user $i$. The expected period of time during which this extra user affects user $j$ is the difference between the waiting time of the extra user if the extra user is bypassed by user $j$ and that if the extra user bypasses user $j$ in the pricing queue. We abuse notation and denote $W_i({v_j}: {z_{ij}}= {\sigma_{\overline{v}_i}})$ and $W_i({v_j}: {z_{ij}}= {\sigma_{\underline{v}_i}})$ these waiting times, respectively. We define ${B_i({v_i})}$ as: $${B_i({v_i})}\equiv \sum_{k = 1}^{{\underline{q}_i}} W_i(v_{j_k} : z_{ij_k} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_k} : z_{ij_k} = {\sigma_{\underline{v}_i}}). \label{eq:b}$$ Accordingly, the expected marginal delay cost that user $i$ imposes on users who arrived before her is: $${MB_i({v_i})}= \sum_{k = 1}^{{\underline{q}_i}} \big(W_i(v_{j_k} : z_{ij_k} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_k} : z_{ij_k} = {\sigma_{\underline{v}_i}}))\big) v_{j_k}. \label{eq:mb}$$ We next show that the period of time ${B_i({v_i})}$ as defined via Eq. cannot exceed the remaining busy period ${\Delta W_i({v_i})}$ as given by Eq. . For any user $i$ bidding ${v_i}$, ${B_i({v_i})}\leq {\Delta W_i({v_i})}$. $$\begin{aligned} {B_i({v_i})}=& \sum_{k = 1}^{{\underline{q}_i}} W_i(v_{j_k} : z_{ij_k} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_k} : z_{ij_k} = {\sigma_{\underline{v}_i}}) \nonumber \\ =& W_i(v_{j_1} : z_{ij_1} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_1} : z_{ij_1} = {\sigma_{\underline{v}_i}}) + W_i(v_{j_2} : z_{ij_2} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_2} : z_{ij_2} = {\sigma_{\underline{v}_i}}) \nonumber \\ &+ \ldots + W_i(v_{j_{{\underline{q}_i}}} : z_{ij_{{\underline{q}_i}}} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_{{\underline{q}_i}}} : z_{ij_{{\underline{q}_i}}} = {\sigma_{\underline{v}_i}}) \label{eq:bexpand}\end{aligned}$$ Recall that $v_{j_k} < v_{j_{k+1}} < {v_i}$ for any $k = 1 \ldots {\underline{q}_i}-1$. Observe that the state corresponding to the extra user bidding $v_{j_k}$ and bypassing user $j_k$ is equivalent to the state of the extra user bidding $v_{j_{k+1}}$ and being bypassed by user $j_{k+1}$, i.e. these states have identical number and location of lower/higher-bidders and empty lanes w.r.t. to the perspective of the extra user. Since $v_{j_k} < v_{j_{k+1}}$, the waiting time of the extra user bidding $v_{j_{k+1}}$ and bypassing user $v_{j_{k+1}}$ is greater than the waiting time of the extra user bidding $v_{j_k}$ and being bypassed by user $v_{j_k}$, i.e. $$W_i(v_{j_{k+1}} : z_{ij_{k+1}} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_k} : z_{ij_k} = {\sigma_{\underline{v}_i}}) \leq 0 \qquad \forall k = 1 \ldots {\underline{q}_i}-1.$$ Omitting these negative terms from Eq. yields: $${B_i({v_i})}\leq W_i(v_{j_{1}} : z_{ij_{1}} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_{{\underline{q}_i}}} : z_{ij_{{\underline{q}_i}}} = {\sigma_{\underline{v}_i}}) \label{eq:bvileq}$$ The state corresponding to the extra user bidding $v_{j_{{\underline{q}_i}}}$ and bypassing user $j_{{\underline{q}_i}}$ is equivalent to the state from the perspective of user $i$. Thus, since $v_{j_{{\underline{q}_i}}} < {v_i}$, $W_i(v_{j_{{\underline{q}_i}}} : z_{ij_{{\underline{q}_i}}} = {\sigma_{\underline{v}_i}}) > {W_i({v_i})}$. Finally, ${W_i({\underline{v}})}$ is an upper bound on the waiting time for any user on the lane of user $i$, thus $W_i(v_{j_{1}} : z_{ij_{1}} = {\sigma_{\overline{v}_i}}) < {W_i({\underline{v}})}$. Substituting in Eq. gives $${B_i({v_i})}\leq {W_i({\underline{v}})}- {W_i({v_i})}= {\Delta W_i({v_i})}.$$ \[rmbb\] Proposition \[rmbb\] together with Eq. ensure that the expected portion of the remaining busy period corresponding to future arrivals is nonnegative, i.e. ${\Delta W_i({v_i})}- {B_i({v_i})}= {A_i({v_i})}\geq 0$. To determine the expected marginal delay cost that user $i$ imposes on future arrivals, denoted ${MA_i({v_i})}$, we use the result of [@dolan1978incentive] who observed that this marginal delay cost can be calculated by integrating $\frac{d{A_i(v)}}{dv} v$ over the bid range $[{\underline{v}},{v_i})$. Further, he observed that the state is constant for any bid comprised between two consecutive bids of users in the pricing queue. Thus, for any segment corresponding to a pair of consecutive bids in the sequence $\{{\underline{v}},v_{j_1}, \ldots, v_{j_{{\underline{q}_i}-1}},v_{j_{{\underline{q}_i}}},{v_i}\}$, the expected marginal delay cost imposed by user $i$ onto users bidding in this segment can be calculated by integrating $\frac{d{A_i(v)}}{dv} v$ over the corresponding domain. Specifically, let $v_{j_0} = {\underline{v}}$ and let $v_{j_{{\underline{q}_i}+1}} = {v_i}$. Let $D(k) = (v_{j_k},v_{j_{k+1}})$ be the domain corresponding to the bid segment $(v_{j_m},v_{j_n})$ for $m < n$ such that $j_m, j_n \in {\mathcal{V}_{<i}}$. Accordingly, the expected marginal delay cost imposed by user $i$ on future arrivals can be determined as: $${MA_i({v_i})}= \sum_{k=0}^{{\underline{q}_i}}\int_{D(k)} \frac{d {A_i(v)}}{dv} v dv = \sum_{k=0}^{{\underline{q}_i}}\int_{D(k)} \frac{-dW_i(v)}{dv} v dv. \label{eq:ma}$$ The expected marginal delay cost imposed by user $i$ declaring ${v_i}$ on other users is thus: $${MC_i({v_i})}= {MB_i({v_i})}+ {MA_i({v_i})}. \label{eq:mc}$$ We next give the main result. If the user objective function is $\min_{{v_i}} {v_i^\star}{W_i({v_i})}+ {P_i({v_i})}$ and users are serviced in order of decreasing declared delay costs, the payment ${P_i({v_i})}= {MC_i({v_i})}$ is incentive-compatible. The proof follows the same logic as that of [@dolan1978incentive]. We next recall its main steps and detail elements which are specific to our mechanism. Consider a pricing queue of size $Q$ and a user with true delay cost ${v_i^\star}$ declaring ${v_i^\star}+ \delta$ with $\delta > 0$. Increasing the declared delay cost may reduce waiting time in two ways: i) bypassing a user in the queue, and, ii) having fewer future arrivals bypass the bidder. For i) the reduction in expected waiting time is $B_i({v_i^\star}+ \delta) - B_i({v_i^\star}) = W_i(v_{j_k} : z_{ij_k} = {\sigma_{\overline{v}_i}}) - W_i(v_{j_k} : z_{ij_k} = {\sigma_{\underline{v}_i}}) \geq 0$ for some user $k$ such that $v_{j_k} \in [{v_i^\star},{v_i^\star}+\delta)$. This reduction is valued ${v_i^\star}(B({v_i^\star}+ \delta) - B({v_i^\star}))$ but the added cost is $v_{j_k}(B({v_i^\star}+ \delta) - B({v_i^\star}))$ with $v_{j_k} \geq {v_i^\star}$. For ii) the reduction in expected waiting time is $A_i({v_i^\star}+ \delta) - A_i({v_i^\star})$ which is valued ${v_i^\star}(A_i({v_i^\star}+ \delta) - A_i({v_i^\star}))$. However, the induced expected marginal cost is $MA_i({v_i^\star}+ \delta) - MA_i({v_i^\star}) = \int_{{v_i^\star}}^{{v_i^\star}+ \delta} \frac{d{A_i(v)}}{dv} v dv \geq \int_{{v_i^\star}}^{{v_i^\star}+ \delta} \frac{d{A_i(v)}}{dv} {v_i^\star}dv = {v_i^\star}(A_i({v_i^\star}+ \delta) - A_i({v_i^\star}))$. A similar reasoning can be applied for users declaring a delay cost lower than their true delay cost. \[theo\] Theorem \[theo\] proves that the expected marginal delay cost as defined by Eq. represents an incentive-compatible user payment in the dynamic sense, i.e. by taking into account the expected impact of users on future arrivals. This provides a basis to implement the proposed queue- and lane-based mechanisms in an online fashion. Upon reaching the front of their lane, users are asked to declare their delay cost ${v_i}$ and receive a corresponding payment ${P_i({v_i})}$ which ensures strategyproof user behavior in the long run. The proposed queue- and lane-based traffic intersection mechanisms are computationally efficient. In terms of computational resources, both mechanisms require the solution of Eqs. and to determine user payments. Eq. requires the solution of multiple systems of linear equations ( or depending on the mechanism selected) which can be accomplished using standard linear algebra codes. To compute Eq. , standard numerical integration and differentiation techniques can be used in combination with codes for systems of linear equations. As will be shown in our numerical experiments, both mechanisms have low average computation time per user and can thus be expected to be executed in real-time to determine incentive-compatible payments. Numerical experiments {#num} ===================== [lr]{} [0.5]{} ![Relative user generalized cost $({C_i({v_i})}- C_i(v_i^\star))/C_i(v_i^\star)$. Results of a simulation with 1 million users with an arrival probability of $p=0.25$. Figure \[fig:queue\] shows relative user generalized costs using the queue-based online mechanism and Figure \[fig:static\] shows relative user generalized costs obtained using the static mechanism.\[fig:qs\]](E025-S025-W025-N025_heatmap_queue "fig:"){width="0.9\linewidth"} [0.5]{} ![Relative user generalized cost $({C_i({v_i})}- C_i(v_i^\star))/C_i(v_i^\star)$. Results of a simulation with 1 million users with an arrival probability of $p=0.25$. Figure \[fig:queue\] shows relative user generalized costs using the queue-based online mechanism and Figure \[fig:static\] shows relative user generalized costs obtained using the static mechanism.\[fig:qs\]](E025-S025-W025-N025_heatmap_static "fig:"){width="0.9\linewidth"} In this section, we conduct numerical simulations of the proposed online traffic intersection mechanisms to explore their behavior. We implement the proposed online mechanisms by simulating the arrival and departure of users at an intersection with $Q=4$ access lanes using a discrete time process. At every time period, random trials are conducted for each empty lane to simulate the stochastic arrival process of users. For all our experiments, we use a unit one-step cost, i.e. ${g_q({\bm{q}_i})}= {g_z({\bm{z}_i})}= 1$ for all states ${\bm{q}_i}\in {\mathcal{S}_i^q}$ and ${\bm{z}_i}\in {\mathcal{S}_i^z}$. Users’ delay costs are randomly drawn from a uniform distribution $\mathcal{U}(5,10)$, i.e. ${\underline{v}}=5$ \$/hour and ${\overline{v}}=10$ \$/hour. These values are chosen based on a study on value of time for automated vehicles [@neeraj2018]. In each experiment, we simulate the service of one million users and report the average user behavior by segmenting users’ delay cost. Specifically, we segment the delay cost range $[\$5, \$10]$ into 30 uniform bins and group all serviced users throughout the simulation into these bins. We then report average waiting times, payments and generalized user costs based on the average quantities obtained for each of these 30 bins. The simulations are implemented in Python and the linear systems of equations and are solved using Numpy’s linear algebra solver on a Windows 10 machine with 8 Gb of RAM and a CPU of 2.7 Ghz. Under this configuration, the average computation time per user for the queue-based mechanism is 0.007 s and that of the lane-based mechanism is 0.126 s, which highlights the computational efficiency of the proposed online mechanisms. Comparison of dynamic and static mechanisms ------------------------------------------- ![Average experienced waiting time of users based on their delay cost. Results of five simulations with 1 million users each with a varying arrival probability $p$.\[fig:awtqs\]](AWT_p.pdf){width="0.45\linewidth"} To quantify the value of using a dynamic auction compared to a static auction, we implement the static mechanism outlined by @dolan1978incentive for priority queueing systems. This static mechanism is a Vickrey-Clarke-Groves auction which is known to be incentive-compatible in the static sense, i.e. all participating users are assumed to be known when determining payments [@vickrey1961counterspeculation; @clarke1971multipart; @groves1975incentives]. The static mechanism utilizes the same priority queueing model than that of the proposed online mechanism, i.e. users in the pricing queue are sorted by decreasing declared delay costs, but differ in the determination of waiting time and user payments. In the static mechanism, the waiting time of user $i$ is only function of the position of user $i$ in the pricing queue, e.g. if user $i$ is third in the pricing queue, her waiting time is the sum of the two one-step costs for transitioning from the current state to the state where user $i$ served. The payment of user $i$ is sum of user valuations over lower-bidders in the pricing queue. This payment is incentive-compatible in the static sense [@dolan1978incentive]. The static mechanism is compared to the queue-based model. We first set the expected arrival rate of users at the intersection to 1 so as to simulate a stable regime. Since the arrival rate is uniform across all $Q=4$ lanes, this corresponds to an arrival probability of $p=0.25$. The results are reported in Figure \[fig:qs\] which depicts the relative difference of ${C_i({v_i})}$ to $C_i(v_i^\star)$ in percentage, which represents the difference between the generalized cost of user $i$ declaring a delay cost ${v_i}$ and the generalized cost of user declaring its true delay cost $v_i^\star$. Strictly positive values indicate that declaring ${v_i}$ results in a higher cost than declaring the user true delay cost ${v_i^\star}$, whereas negative values indicate that misreporting delay cost ${v_i}$ achieves a lower cost than truthful reporting. For clarity, relative user generalized cost values are bounded within $\pm 10\%$, i.e. if $({C_i({v_i})}- C_i(v_i^\star))/C_i(v_i^\star)$ is greater (resp. lower) than $10\%$ (resp. $-10\%$), this value is shown as $10\%$ (resp. $-10\%$). [0.49]{} ![Results of simulations with 1 million users with a varying arrival probability of $p$. Figure \[fig:diffwtqueue\] shows user payments obtained using the queue-based online mechanism and Figure \[fig:diffwtstatic\] shows user payments obtained using the static mechanism.\[fig:diffwt\]](AWT-EWT_queue.pdf "fig:"){width="0.95\linewidth"} [0.45]{} ![Results of simulations with 1 million users with a varying arrival probability of $p$. Figure \[fig:diffwtqueue\] shows user payments obtained using the queue-based online mechanism and Figure \[fig:diffwtstatic\] shows user payments obtained using the static mechanism.\[fig:diffwt\]](AWT-EWT_static.pdf "fig:"){width="0.95\linewidth"} [0.45]{} ![Results of simulations with 1 million users with a varying arrival probability of $p$. Figure \[fig:payqueue\] shows user payments obtained using the queue-based online mechanism and Figure \[fig:paystatic\] shows user payments obtained using the static mechanism.\[fig:payqs\]](PAY_p_queue.pdf "fig:"){width="0.95\linewidth"} [0.45]{} ![Results of simulations with 1 million users with a varying arrival probability of $p$. Figure \[fig:payqueue\] shows user payments obtained using the queue-based online mechanism and Figure \[fig:paystatic\] shows user payments obtained using the static mechanism.\[fig:payqs\]](PAY_p_static.pdf "fig:"){width="0.95\linewidth"} The relative user generalized cost obtained via the online queue-based mechanism is shown to be, on average, incentive-compatible almost everywhere, as highlighted by positive relative generalized user costs. In turn, using the static mechanism, a user misreporting her delay cost by declaring a higher delay cost can achieve a lower generalized cost than truthfully reporting her delay cost, as highlighted by the top-right negative relative user generalized costs. This highlights the benefits of online mechanisms over static approaches for traffic intersection auctions. We next examine the impact of varying the arrival probability around the stable regime. We run simulations from $p = 0.15$, which corresponds to an expected arrival rate of 0.6, to $p = 0.35$, which corresponds to an expected arrival rate of 1.4, in steps of $0.05$. The results of these simulations are reported in Figures \[fig:awtqs\]–Figure \[fig:payqs\]. Figure \[fig:awtqs\] depicts the average experienced waiting time of users based on their delay cost for varying arrival probability $p$. Since in both dynamic and static mechanisms, users are prioritized based on their declared delay cost, the experienced, i.e. simulated, waiting time is identical in both mechanisms. As expected, experienced waiting times decrease with the delay cost. The decrease rate is near-linear for delay costs greater than \$7/hour. For lower delay costs and high arrival probabilities, the decrease rate is exponential, which highlights the potentially long waiting times incurred by low-bidding users in a congested regime, i.e. $p > 0.25$. The decrease rate of experienced waiting time is lower in stable regimes, i.e. $p \leq 0.25$ and exhibits a more linear decay rate. To quantify the accuracy of the proposed MC models for estimating expected waiting times, we report the difference of experienced minus expected waiting times for the queue-based and the static mechanisms in Figure \[fig:diffwt\]. For clarity y-axis scales are different in Figure \[fig:diffwtqueue\] and Figure \[fig:diffwtstatic\] representing the queue-based and static mechanisms, respectively. Figure \[fig:diffwtqueue\] shows that the average expected waiting times obtained via the MC queue-based model are very close to the average experienced waiting times, with maximum deviations approximately 0.05 time units reported for low-bidding users. In contrast, Figure \[fig:diffwtstatic\] shows that the static mechanism almost systematically underestimates the average experienced waiting times. Further, the loss of quality increases with the arrival probability at the intersection. These trends are consequences of the static mechanism not accounting for the probability of future arrivals. Figure \[fig:payqs\] shows the average payments of users under both queue-based and static mechanisms. User payments increase with delay cost and are comparatively larger in the queue-based mechanism (Figure \[fig:payqueue\]) than in the static mechanism (Figure \[fig:paystatic\]). This emphasizes the potential impact of future arrivals which are neglected in static mechanisms. The queue-based mechanism also exhibits more concave-shaped payments with regards to user delay cost compared to the static mechanism which is more linear. We also find that the curvature of the payment curves increase with the arrival probability, highlighting the impact of the demand regime onto the payment mechanisms. Comparison of queue- and lane-based mechanisms {#queuelane} ---------------------------------------------- [0.7]{} ![Relative user generalized cost $({C_i({v_i})}- C_i(v_i^\star))/C_i(v_i^\star)$. Results of a simulation with 1 million users with lane-based arrival probabilities of $0.50$, $0.25$, $0.15$ and $0.10$. Figure \[fig:heatmaplq\] shows relative user generalized costs obtained using the queue-based online mechanism and Figure \[fig:heatmapll\] shows relative user generalized costs obtained using the lane-based mechanism.\[fig:heatmaplane\]](E050-S025-W015-N010_heatmap_lane_queue.pdf "fig:"){width="1.0\linewidth"} \ [0.7]{} ![Relative user generalized cost $({C_i({v_i})}- C_i(v_i^\star))/C_i(v_i^\star)$. Results of a simulation with 1 million users with lane-based arrival probabilities of $0.50$, $0.25$, $0.15$ and $0.10$. Figure \[fig:heatmaplq\] shows relative user generalized costs obtained using the queue-based online mechanism and Figure \[fig:heatmapll\] shows relative user generalized costs obtained using the lane-based mechanism.\[fig:heatmaplane\]](E050-S025-W015-N010_heatmap_lane_lane.pdf "fig:"){width="1.0\linewidth"} [cc]{} [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:awtlane\]](AWT_lane.pdf "fig:"){width="1.0\linewidth"} & [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:awtlane\]](AWT_lane_E025-S025-W025-N025.pdf "fig:"){width="1.0\linewidth"} \ [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:awtlane\]](AWT_lane_E035-S015-W035-N015.pdf "fig:"){width="1.0\linewidth"} & [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:awtlane\]](AWT_lane_E050-S025-W015-N010.pdf "fig:"){width="1.0\linewidth"} \ [cc]{} [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:paylane\]](PAY_lane_lane.pdf "fig:"){width="1.0\linewidth"} & [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:paylane\]](PAY_lane_E025-S025-W025-N025.pdf "fig:"){width="1.0\linewidth"} \ [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:paylane\]](PAY_lane_E035-S015-W035-N015.pdf "fig:"){width="1.0\linewidth"} & [0.35]{} ![Results of three simulations of the lane-based mechanism with varying lane-based arrival probabilities $p_j$, $j\in\{\text{E,S,W,N}\}$. Each simulation consists of 1 million users.\[fig:paylane\]](PAY_lane_E050-S025-W015-N010.pdf "fig:"){width="1.0\linewidth"} \ To quantify the trade-offs between queue- and lane-based mechanisms, we implement both mechanisms for a fully asymmetric case. We use cardinal directions, i.e. East (E), South (S), West (W) and North (N) to refer to all $Q=4$ lanes of the intersection and assign lane arrival probabilities of $0.50$, $0.25$, $0.15$ and $0.10$, respectively. For the queue-based mechanism, we use the average lane arrival probability, i.e. $p = \sum_{j=1}^{Q} p_j = 0.25$. We report lane-based heatmaps of the relative difference in terms of relative user generalized cost $({C_i({v_i})}- C_i(v_i^\star))/C_i(v_i^\star)$ for both queue- and lane-based mechanisms in Figure \[fig:heatmaplane\]. In Figure \[fig:heatmaplq\], the outcome of these simulations highlight that using the average lane arrival probability in the queue-based mechanism fails to yields incentive-compatible outcome for the East, West and North lanes which have a lane arrival probability different that $p=0.25$. Only users on the South lane with $p_{S} = 0.25 = p$ receive incentive-compatible payments. In turn, Figure \[fig:heatmapll\] illustrates that the lane-based mechanism is incentive-compatible for each lane of the intersection. Non-uniform lane arrival probabilities -------------------------------------- We next further examine the behavior of the lane-based mechanism in the context of non-uniform lane arrival probabilities. We compare three configurations which correspond to a stable regime. The first configuration consists of uniform lane arrival probabilities of $p_j = 0.25$ for all lane $j \in \{\text{E,S,W,N}\}$. The second configuration consists of an arrival probability of $p_j = 0.35$ for East and West lanes, and $p_j = 0.15$ for South and North lanes. The third configuration consists of fully asymmetric lane arrival probabilities identical to those used in Section \[queuelane\]. These three configurations are denoted Ewww-Sxxx-Wyyy-Nzzz where the three digits next to the lane direction represent the associated probability with two decimal values, e.g. E025 correspond to $p_E=0.25$. For each of the three configurations tested, we simulate the arrival of 1 million users at an intersection. Figures \[fig:awtlane\] and \[fig:paylane\] depict the average experienced waiting time and average user payment, respectively, in the three lane arrival probabilities configurations. In both figures, the top left sub-figure (\[fig:awtavglanes\] and \[fig:payavglanes\]) shows the average behavior over all four lanes of the intersection, whereas each of the other three sub-figures focus on one of the three configuration tested and shows lane-specific behavior. We find that the uniform configuration (E025-S025-W025-N025) leads to higher average waiting times and user payments compared the other two configurations tested. As expected, the uniform configuration yields a uniform behavior all lanes. In contrast, non-uniform configurations lead to lane-specific waiting times and payments. The outcome of the partially asymmetric configuration (E035-S015-W035-N015) reveals that payments for the lower arrival probability lanes (South and North) are only marginally higher than that of the uniform configuration (E025-S025-W025-N025), whereas payments for the higher arrival probability lanes (East and West) are substantially lower. Results for the fully asymmetrical configuration (E050-S025-W015-N010) show that waiting times are higher on low arrival probability lanes and decrease with the arrival probability. Conversely, user payments are lower on high arrival probability lanes and increase for on less demanded lanes. This can be explained by observing that users on low arrival probability lanes have a higher chance of delaying more users since those are expected to arrive more frequently on higher arrival probability lanes. Conclusions {#con} =========== We next summarize the findings of this study before discussing its limitations and outlining potential extensions and future research directions. Summary of findings ------------------- In this paper, we have presented novel online mechanisms for traffic intersection auctions. We assume that users can declare their delay cost privately to the intersection manager and focus on determining incentive-compatible payments for users at the front of their lane. We also assume that the intersection manager has knowledge of the distribution of users’ delay cost. The proposed mechanisms are designed to minimize user’s generalized cost which is defined as a linear combination of expected waiting time and user payment. We introduced two Markov chain models to determine users’ expected waiting time, and presented a payment mechanism that can be implemented with both models. We showed that the proposed online traffic intersection mechanisms are incentive-compatible in the dynamic sense and thus maximize social welfare. We conducted numerical experiments on a four-lane intersection to explore the behavior of the proposed online mechanisms and compared their performance to that of a static incentive-compatible mechanism. Our findings highlight that static mechanisms, i.e. which do not account for future arrivals, may not yield incentive-compatible in dynamic sense. That is, static incentive-compatible mechanisms may fail to ensure truthful user behavior by incentivizing users to misreport their delay cost in the long run. We also provide empirical evidence that the proposed online mechanisms are computationally efficient and could be used to manage traffic operations in real-time. Finally, we quantified the trade-offs between the queue- and lane-based mechanisms and find that the lane-based mechanism can accommodate lane-specific arrival probabilities in competitive time. Limitations and future research directions ------------------------------------------ We next discuss the assumptions and mechanism design choices made in this research. The proposed online mechanisms focus on determining incentive-compatible payments for users at the front of their lane queues. Since all users queueing on a lane of a traffic intersection are eventually going to reach the front of their lanes, all users will eventually join the pricing queue and be required to declare their delay cost so as to obtain their corresponding payment. However, the proposed mechanisms do not provide the option for “blocked” users, i.e. users queueing behind another user on their lane, to “vote” for the user at the front of their lane queues. Although this is a limitation of the proposed online mechanisms, since blocked users are assumed to be unaware of other users’ delay cost, it appears challenging to ensure that they could reap benefits from bidding for their lane leader. The geometry of traffic intersections is only implicitly accounted for within the proposed online mechanisms. Specifically, users’ service times are captured within the one-step-costs of the proposed Markov chain models used to determine the expected waiting time of users. Yet, in our numerical experiments, we have assumed that all service times are deterministic and uniform. The uniformness assumption could be relaxed by substituting the unit one-step costs with movement-specific service times. However, relaxing this assumption would also require the introduction of a detailed model of users’ travel time within the intersection which is function of other users’ route choice and service time. This is expected to require an extension of the state space in the proposed lane-based Markov chain model to a movement-based model. Further, the simulation of realistic traffic movements within the intersection is expected to require traffic control model that provides the flexibility of servicing in signal-free context. Such models have been widely explored in the connected and autonomous vehicles’ literature and we leave the extension of the proposed online mechanisms to realistic traffic models for future works. Finally, the deterministic service time assumption could be relaxed at the expense of introducing more complex modeling of users’ trajectory interactions within the intersection which is beyond the scope of this research. The proposed research may also be extended by considering batches or coalitions of users. This may lead to improved social benefits if, for instance, platoons of vehicles are priced to traverse the intersection in a effective manner. Such market-driven approaches could be combined with recent efforts on platoon coordination at traffic intersections [@jiang2006platoon; @lioris2017platoons; @monteil2018mathcal; @niroumand2020joint]. The proposed online mechanisms could be revisited under the light of fairness. As observed by @schepperle2007agent “starvation” effects may occur if intersection access is blocked by low delay cost users or dominated by high delay cost users on specific lanes. Future research is thus needed to advance the development of market-driven mechanisms that are able to combine dynamic incentive-compatibility with equity factors. Finally, methods on load balancing in parallel queueing systems as proposed by @down2006dynamic could be used to develop lane assignment policies to assign users to small queue-length lanes so as to improve the social welfare of the system.

--- abstract: 'Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W''_F$. The local Langlands correspondence(LLC) establishes the reciprocity maps $\textrm{Rec}_{n,F}: \mathcal{A}_n(F)\longrightarrow \mathcal{G}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $\epsilon$-factors. This is also expected to be true under parallel compositions with a complex analytic representations of $\textrm{GL}_n(\mathbb{C})$. J.W. Cogdell, F. Shahidi, and T.-L. Tsai proved the equality of the symmetric and exterior square L- and $\epsilon$-factors \[7\] in 2017. But the twisted symmetric and exterior square L- and $\epsilon$-factor are new and very different from the untwisted case. In this paper we will define the twisted symmetric square L- and $\gamma$-factors using $\textrm{GSpin}_{2n+1}$, and establish the equality of the corresponding L- and $\epsilon$-factors. We will first reduce the problem to the analytic stability of their $\gamma$-factors for supercuspidal representations, then prove the supercuspidal stability by establishing general asymptotic expansions of partial Bessel function following the ideas in \[7\].' address: 'Department of Mathematics, Purdue University, West Lafayette, IN 47907' author: - Dongming She date: 'Sept. 2019' title: 'LOCAL LANGLANDS CORRESPONDENCE FOR THE TWISTED EXTERIOR AND SYMMETRIC SQUARE $\epsilon$-FACTORS OF $\textrm{GL}_n$' --- INTRODUCTION ============ The local Langlands Correspondence(LLC) for $\textrm{GL}_n$ has been proved by G. Laumon, M. Rapoport, and U. Stuhler for function fields (1993, \[14\]), by G. Henniart (2000, \[12\]) and also by M. Harris and R. Taylor (2001, \[10\]), and later by P. Scholze (2010, \[15\]) using a different approach for p-adic fields. Let $\rho$ be an n-dimensional Frobenius semisimple representation of the local Weil-Deligne group $W'_F$, and $\pi=\pi(\rho)$ be its corresponding irreducible admissible representation of $\textrm{GL}_n(F)$, then one expects the equality of their L- and $\epsilon$-factors: $$\epsilon(s,\rho,\psi)=\epsilon(s,\pi(\rho),\psi),$$ $$L(s,\rho)=L(s,\pi(\rho)),$$ where the local arithmetic $\epsilon$-factor $\epsilon(s,\rho, \psi)$ is defined by P. Deligne in \[9\], in which he showed that the global $\epsilon$-factors admit a factorization into a product of local ones. Here $L(s,\rho)$ is the local Artin L-factor and $\psi$ is a non-trivial additive character of $F$. The local analytic $\epsilon(s,\pi(\rho),\psi)$ and $L(s,\pi(\rho))$ are defined by Langlands-Shahidi method first for generic representations, then for tempered representations and finally using Langlands classification for all irreducible admissible representations of $\textrm{GL}_n(F)$. If $r$ is a continuous representation of $\textrm{GL}_n(\mathbb{C})$, then one can define the local Artin L- and $\epsilon$-factors $L(s,r\circ\rho,\psi)$ and $\epsilon(s,r\circ \rho, \psi)$. Therefore a natural question is to see if the following equalities hold: $$L(s,r\circ\rho)=L(s,\pi,r),$$ $$\epsilon(s,r\circ\rho,\psi)=\epsilon(s,\pi,r,\psi),$$ as long as the factors on the analytic side are defined. One has the following relationship for analytic $\epsilon$-, $\gamma$-, and L-factors: $$\epsilon(s,\pi,r,\psi)=\frac{\gamma(s,\pi,r,\psi)L(s,\pi,r)}{L(1-s,\tilde{\pi},r)}.$$ On the arithmetic side, one can naturally define $$\gamma(s,r\circ\rho,\psi)=\frac{\epsilon(s,r\circ\rho,\psi)L(1-s, r\circ \rho^{\vee})}{L(s, r\circ\rho)}.$$ So the equalities of $\epsilon$- and L-factors are equivalent to the equalities of $\gamma$- and L-factors. One method to prove equalities like this was first introduced by J.W. Cogdell, F. Shahidi, and T.-L. Tsai \[7\] in 2017, for the case where $r=\wedge^2$ and $\textrm{Sym}^2$. The proof uses a globalization method and certain reductions, and relies on two main results called the arithmetic stability and analytic stability of $\gamma$-factors respectively. The former was introduced and proved by P. Deligne in \[9\], the later for the case $r=\wedge^2$ (and by symmetry also $r=\textrm{Sym}^2$) was proved in \[7\]. The authors used the group $H=\textrm{GSp}_{2n}$ and its maximal self-associate Levi subgroup $M_H\simeq \textrm{GL}_n\times \textrm{GL}_1$ to construct the analytic factors for $r=\wedge^2$, using the fact that the adjoint representation $r$ of ${^LM}_H$ on $^L\mathfrak{n}_H=\textrm{Lie}(^LN_H)$ decomposes as $r=r_1\oplus r_2$, where $r_1$ is isomorphic to the standard representation of $\textrm{GL}_n(F)$ and $r_2=\wedge^2$. As a consequence the problem was reduced to establishing the stability of Shahidi local coefficients, which can be written as the Mellin transform of certain partial Bessel functions \[19\] under some conditions. The partial Bessel functions defined on the relevant part of the big Bruhat cells have nice asymptotic behaviors. Their asymptotic expansions can be written as a sum of two parts. The first part depends only on the central character of $\pi(\rho)$, and the second part is a uniformly smooth function on certain torus, which becomes zero after a highly ramified twist. In this paper we will define the twisted symmetric and exterior square $\gamma$- and L-factors of $\text{GL}_n(F)$, and prove the following result: Let $F$ be a non-archimedean local field, $\rho$ be an n-dimensional $\Phi$-semisimple Weil-Delinge representation of $W_F'$, $\pi=\pi(\rho)$ be the corresponding irreducible admissible representation of $G=\text{GL}_n(F)$ attached to $\rho$ under the local Langlands correspondence. Let $\text{Sym}^2$ and $\wedge^2$ denote the symmetric and exterior square representations of $^LG=\textrm{GL}_n(\mathbb{C})$, fix a character $\eta: F^{\times}\rightarrow \mathbb{C}^{\times}.$ Let $\epsilon(s, \pi, \textrm{Sym}^2\otimes\eta,\psi)$ and $\epsilon(s, \pi, \wedge^2\otimes\eta,\psi)$ be the twisted symmetric and exterior square local analytic $\epsilon$-factors, and $\epsilon(s, \textrm{Sym}^2\rho\otimes\eta,\psi)$, $\epsilon(s, \wedge^2\otimes\eta,\psi)$ their corresponding local arithmetic $\epsilon$-factors. Then $$\epsilon(s, \textrm{Sym}^2\rho\otimes\eta,\psi)=\epsilon(s, \pi, \textrm{Sym}^2\otimes\eta,\psi);$$ $$\epsilon(s, \wedge^2\rho\otimes\eta,\psi)=\epsilon(s, \pi, \wedge^2\otimes\eta,\psi);$$ and $$L(s, \textrm{Sym}^2\rho\otimes\eta)=L(s, \pi, \textrm{Sym}^2\otimes\eta);$$ $$L(s, \wedge^2\rho\otimes\eta)=L(s, \pi, \wedge^2\otimes\eta)).$$ We will show the equalies of their $\gamma$- and L-factors. First, the $\gamma$-factors $\gamma(s, \pi,\textrm{Sym}^2\otimes\eta,\psi)$ and $\gamma(s, \pi,\wedge^2\otimes\eta,\psi)$, once constructed, will have to satisfy the symmetry $$\gamma(s, (\pi\times\pi)\times \eta,\psi)=\gamma(s,\pi,\wedge^2\otimes \eta,\psi)\gamma(s,\pi,Sym^2\otimes\eta,\psi),$$ $$\gamma(s,(\rho\otimes\rho)\otimes\eta,\psi)=\gamma(s,\wedge^2\rho\otimes\eta,\psi)\gamma(s,Sym^2\rho\otimes\eta,\psi).$$ As the LLC preserves $L$- and $\epsilon$-factors of pairs, and is compatible with twisting by characters, it suffices to prove Theorem 1.1 only for the twisted symmetric square $\gamma$-factors. We will use Langlands-Shahidi method for odd $\textrm{GSpin}$ groups to produce the twisted symmetric square $\gamma$-factors. The reason is that when $n$ is odd, the maximal parabolic subgroups in $\textrm{GSpin}_{2n}$ that produce the twisted exterior square $\gamma$-factors, are not self-associate, although their unipotent radicals have relatively simpler structures. Hence Theorem 6.2 of \[19\], which we will use to write the local coefficient as the Mellin transform of partial Bessel functions, can not be applied in this situation. TWISTED SYMMETRIC SQUARE L- AND $\gamma$-FACTORS ================================================ We will construct the twisted symmetric square $\gamma$- and L-factors of $\textrm{GL}_n$ using the group $H=\textrm{GSpin}_{2n+1}$. It is a reductive group of type $\textrm{B}_n$ with derived group $\textrm{Spin}_{2n+1}$, which is the simply connected double cover of $\textrm{SO}_{2n+1}$. By Proposition 2.1 of \[2\], the root datum of $H$ can be given as: $$X=\mathbb{Z}e_0\oplus\mathbb{Z}e_1\oplus\cdots\oplus \mathbb{Z}e_n,$$ $$X^{\vee}=\mathbb{Z}e^*_0\oplus \mathbb{Z}e_1^*\oplus\cdots\oplus\mathbb{Z}e_n^*,$$ $$\Delta=\{\alpha_1=e_1-e_2,\alpha_2=e_2-e_3,\cdots,\alpha_{n-1}=e_{n-1}-e_n,\alpha_n=e_n\}$$ $$\Delta^{\vee}=\{\alpha_1^{\vee}=e_1^*-e_2^*,\alpha_2^{\vee}=e^*_2-e_3^*,\cdots,\alpha^{\vee}_{n-1}=e^*_{n-1}-e_n^*,\alpha_n^{\vee}=2e_n^*-e_0^*\}.$$ Take the self-associate parabolic subgroup $P_H$ of $H$ with Levi decomposition $P_H=M_HN_H$, where $M_H=M_{\theta}$, $\theta=\Delta-\{\alpha_n\}$. Then $M_H\simeq \textrm{GL}_n\times \textrm{GL}_1$ (Theorem 2.7, \[1\]). Let $\psi$ be a non-trivial additive character of $F$, and $(\pi, V)$ be an irreducible $\psi$-generic representation of $\textrm{GL}_n(F)$. Let $\eta:F^{\times}\rightarrow \mathbb{C}^{\times}$ be a character of $F^{\times}$. We lift $\pi$ to a $\psi$-generic representation $\sigma$ of $M_H(F)$, being trivial on the $\textrm{GL}_1$-component. Define a generic representation $\sigma_{\eta}:M_H(F)\simeq \textrm{GL}_n(F)\times \textrm{GL}_1(F)\longrightarrow \textrm{GL}(V)$ by $\sigma_{\eta}(m(g,a))v= \eta^{-1}(a)\pi(g)v.$ Denote the L-group of $H$ by $^LH$, similarly we can define $^LM_H$ and $^LN_H$. We have $^LH \simeq GSp_{2n}(\mathbb{C})=\{h\in GL_{2n}(\mathbb{C}): {^th} J h=\phi (h) J \ \ \textrm{for}\ \ \textrm{some} \ \ \phi(h)\in F^\times \}$, where $$J=\begin{bmatrix} \ \ & J' \\ -{^tJ}' & \ \ \\ \end{bmatrix}, J'=\begin{bmatrix} \ \ & \ \ & \ \ & 1 \\ \ \ & \ \ & -1 \ \ \\ \ \ & \iddots & \ \ & \ \ \\ (-1)^{n-1} & \ \ & \ \ & \ \ \\ \end{bmatrix},$$ and $\phi: H\rightarrow \mathbb{C}^\times$ is the similitude character of $H$. Therefore we have $$\leftidx{^L}M_H=\{m=m(g,a_0)=\begin{bmatrix} g & \ \ \\ \ \ & a_0 J'{^tg^{-1}J'^{-1}}\\ \end{bmatrix}: g\in \textrm{GL}_n(\mathbb{C}), a_0\in \mathbb{C}^{\times})\}$$$$\simeq \textrm{GL}_n(\mathbb{C})\times \textrm{GL}_1(\mathbb{C}).$$ Let $^L\mathfrak{n}_H=\textrm{Lie}(\leftidx{^L}N_H)$. The adjoint action $r: \leftidx{^L}M_H\longrightarrow \textrm{GL}(^L\mathfrak{n}_H)$ is irreducible (Appendix A, ($\textrm{B}_{n,ii}$), \[17\]). Then by Langlands-Shahidi method (Theorem 3.1 in \[16\] or Theorem 8.3.2 in \[17\]), the local $\gamma$-factor $\gamma(s, \sigma_{\eta}, r,\psi)$ is well-defined. $\sigma_\eta$ is unramified if both $\pi$ and $\eta$ are. Fix a uniformizer $\varpi$ of $F$, then the semisimple conjugacy class $c(\pi)$ attached to $\pi$ is given by $c(\pi)=\textrm{diag}\{\chi_1(\varpi),\cdots,\chi_n(\varpi)\}$, where $\chi_1,\cdots,\chi_n$ are $n$ unramified characters of $F^\times$. Therefore the semisimple conjugacy class attached to $\sigma$ is given by $$c(\sigma)=\textrm{diag}\{\chi_1(\varpi),\cdots,\chi_n(\varpi),\chi_n(\varpi)^{-1},\cdots,\chi_1(\varpi)^{-1}\}.$$ On the other hand, $c(\eta)=\textrm{diag}\{1,\cdots,1,\eta(\varpi)^{-1},\cdots,\eta(\varpi)^{-1}\}$, so $$c(\sigma_\eta)=c(\sigma)c(\eta)=\textrm{diag}\{\chi_1(\varpi),\cdots,\chi_n(\varpi),\eta(\varpi)^{-1}\chi_n(\varpi)^{-1},\cdots,\eta(\varpi)^{-1}\chi_1(\varpi)^{-1}\}.$$ It follows that $$L(s, \sigma_{\eta}, r)=\det(1-r(c(\sigma_\eta)q_F^{-s}))^{-1}=\prod_{1\leq i\leq j\leq n}(1-(\chi_i\chi_j\eta)(\varpi)q_F^{-s})^{-1}$$ which is what we usually referred as the unramified twisted symmetric square local L-factor for $\textrm{GL}_n$ (section 1, \[20\]). We can use Langlands-Shahidi method to first define the twisted symmetric square L-factor for $\pi$ being tempered, and use Langlands classification and multiplicativity to define for any irreducible admissible representation $\pi$ of $\textrm{GL}_n(F)$ that $L(s, \pi, \textrm{Sym}^2\otimes\eta)=L(s, \sigma_{\eta},r)$ and $\gamma(s, \pi, \textrm{Sym}^2\otimes \eta,\psi)=\gamma(s, \sigma_{\eta}, r,\psi).$ STABLE EQUALITY =============== Suppose $\rho$ is mapped to $\pi=\pi(\rho)$ under the local Langlands correspondence. The character $\eta: F^\times\longrightarrow \mathbb{C}^\times$can be viewed as a character of the local Weil group $W_F$ by $ W_F\twoheadrightarrow W_F^{ab}\simeq F^{\times}\rightarrow\mathbb{C}^{\times}$ through the local Artin map $\textrm{Art}_F^{-1}: W_F^{ab}\simeq F^{\times}$. We still denote it by $\eta$. On the other hand, $\rho$ and $\eta$ define a homomorphism $$\rho_{\eta}:W_F\longrightarrow {^LM}_H\simeq \textrm{GL}_n(\mathbb{C})\times \textrm{GL}_1(\mathbb{C})$$ by $\rho_{\eta}(w)=(\rho(w), \eta^{-1}(w)).$ It is easy to see that $r\circ \rho_{\eta}\simeq \textrm{Sym}^2\rho\otimes \eta.$ Now Let $\chi:F^\times\rightarrow \mathbb{C}^\times$ be a continuous character of $F^\times$, viewed as a character of $\textrm{GL}_n(F)$ through the determinant. Similar to $\eta$ we can also view $\chi$ as a character of $W_F$. $\rho$ and $\chi$ determine a homomorphism $$\rho\otimes\chi:W_F\longrightarrow \textrm{GL}_n(\mathbb{C})$$ by $w\mapsto \chi(w)\rho(w)$. Consequently we also have $$(\rho\otimes \chi)_{\eta}: W_F\longrightarrow {^LM}_H\simeq \textrm{GL}_n(\mathbb{C})\times \textrm{GL}_1(\mathbb{C})$$ defined by $(\rho\otimes \chi)_{\eta}(w)=((\rho\otimes \chi)(w),\eta^{-1}(w))=(\chi(w)\rho(w), \eta^{-1}(w))$. We can see that $r\circ (\rho\otimes \chi)_{\eta}\simeq \textrm{Sym}^2(\rho\otimes \chi)\otimes \eta$. Therefore on the arithmetic side we have $L(s, \textrm{Sym}^2(\rho\otimes \chi)\otimes\eta)=L(s, r\circ (\rho\otimes \chi)_{\eta})$ and $\gamma(s,\textrm{Sym}^2(\rho\otimes\chi)\otimes\eta,\psi)=\gamma(s,r\circ (\rho\otimes \chi)_{\eta}, \psi)$. We aim to prove the following proposition in this section. (**Stable Equality**) Let $F$ be a p-adic field of characteristic zero, $\eta$ a fixed character of $F^{\times}$, and $\rho$ be an $n$-dimensional continuous irreducible representation of $W_F$. Then for every sufficiently highly ramified character $\chi$ of $F^{\times}$, we have $$\gamma(s,\textrm{Sym}^2(\rho\otimes \chi)\otimes \eta,\psi)=\gamma(s,\pi\otimes\chi, \textrm{Sym}^2\otimes \eta, \psi),$$ where $\pi=\pi(\rho)\in Irr(\textrm{GL}_n(F))$ is the irreducible admissible representation attached to $\rho$ under the local Langlands correspondence. We will prove Proposition 3.1 by induction on $n$. It is important to point out that the induction hypothesis will be used in the proof of Proposition 3.2 using a global-to-local argument. We will first establish the proposition for a fixed irreducible representation $\rho_0$ of $W_F$(Proposition 3.2), then use both the arithmetic and analytic stability of $\gamma$-factors (Proposition 3.3 & 3.4) on the two sides to deform the equality for the fixed representation to obtain the result of Proposition 3.1 for all $n$-dimensional representations $\rho$. We begin with the first step: (**Stable Equality at a base point**) Let $F$ be a p-adic field, fix a character $\eta$ of $F^{\times}$. Given a character $\omega_0$ of $F^{\times}$, there exists an irreducible n-dimensional representation $\rho_0$ of $W_F$ with $\det\rho_0$ corresponding to $\omega_0$ by local class field theory, such that for all characters $\chi$ of $F^{\times}$, we have $$\gamma(s,\textrm{Sym}^2(\rho_0\otimes \chi)\otimes \eta,\psi)=\gamma(s,\pi(\rho_0)\otimes\chi, \textrm{Sym}^2\otimes \eta, \psi),$$ This is essentially the same as the proof of Proposition 3.2 in \[7\]. Using the globalization method provided by Lemma 3.1 in \[7\], we see that there exists a number field $\mathbb{F}$ and an irreducible continuous n-dimensional representation $\Sigma$ of the global Weil group $W_{\mathbb{F}}$, such that if $\Sigma_v=\Sigma\vert_{W_{\mathbb{F}_v}}$, then there is a place $v_0$ of $\mathbb{F}$ such that $\mathbb{F}_{v_0}=F$, $\det \Sigma_{v_0}$ corresponds to $\omega_0$ by local class field theory. Moreover, $\Sigma_{v_0}$ is irreducible, $\Sigma_v$ is reducible for all $v<\infty$ with $v\neq v_0$, and $\Pi=\pi(\Sigma):=\otimes_v \pi(\Sigma_v)$ is a cuspidal automorphic representation of $\textrm{GL}_n(\mathbb{A}_{\mathbb{F}})$. Therefore all the local components $\Pi_v$ are generic. Let $\Psi=\otimes_v \Psi_v$ be a nontrivial additive character of $\mathbb{F}\backslash \mathbb{A}_{\mathbb{F}}$ so that $\Psi_{v_0}=\psi$, the nontrivial additive character which defines the generic character of $U_n(F)$. We also take $\tau:\mathbb{F}^{\times}\backslash \mathbb{A}^{\times}_{\mathbb{F}}\rightarrow \mathbb{C}^{\times}$ to be a Hecke character with $\tau_{v_0}=\eta$. Outside a finite set of places $S$ containing $v_0$ and the infinite places, $\Pi_v$, $\tau_v$ and $\Psi_v$ are all unramified. Take $\xi:\mathbb{F}^{\times}\backslash \mathbb{A}^{\times}_{\mathbb{F}}\rightarrow \mathbb{C}^{\times}$ a Hecke character such that $\xi_{v_0}=\chi$, it is easy to see that globally we have $\pi(\Sigma\otimes\xi)_{\tau}=(\Pi\otimes\xi)_{\tau}$. Similar to the local case the global $L$-functions are given by $L(s,\textrm{Sym}^2(\Sigma\otimes\xi)\otimes \tau)=L(s, r\circ (\Sigma\otimes \xi)_{\tau})$ and $L(s,\Pi\otimes \xi, \textrm{Sym}^2\otimes \tau)=L(s,(\Pi\otimes\xi)_{\tau}, r).$ Now we apply the global functional equations for the Artin L-functions in general as given in \[9\], and the twisted symmetric square L-function for the automorphic side through Langlands-Shahidi method as in \[17\], and do some simple calculation on the unramified places, we will be able to match the the product of L-factors at those places. We obtain the equality of the product of local $\gamma$-factors at those “bad” places. Since by \[18\] we know that the arithmetic and the analytic factors defined by the Langlands-Shahidi method always agree at all Archimedean places \[18\], we are left with the product of $\gamma$-factors of a finite set of places at which the local component $\Sigma_v$ are all reducible, and a fixed place $v_0$. Let $\Sigma_v=\Sigma_{v,1}\oplus\cdots\oplus \Sigma_{v,r_v}$ be the decomposition of $\Sigma_v$ into irreducibles. We will prove the equality $\gamma(s, \textrm{Sym}^2((\Sigma_{v,1}\oplus\cdots\oplus \Sigma_{v,r_v})\otimes \xi_v)\otimes \tau_v, \Psi_v)=\gamma(s, \textrm{Ind}(\Pi_{v,1}\otimes\cdots\otimes \Pi_{v,r_v})\otimes\xi_v, \textrm{Sym}^2\otimes\tau_v, \Psi_v)$, by induction on $r_v$. Since $\Sigma_v$ is reducible, $r_v\ge 2$. When $r_v=2$ we have $$\gamma(s, \textrm{Sym}^2((\Sigma_{v,1}\oplus\Sigma_{v,2})\otimes \xi_v)\otimes \tau_v, \Psi_v)$$$$=\gamma(s, \textrm{Sym}^2(\Sigma_{v,1}\otimes\xi_v)\otimes\tau_v, \Psi_v)\gamma(s, \textrm{Sym}^2(\Sigma_{v,2}\otimes \xi_v)\otimes\tau_v,\Psi_v)$$$$\cdot\gamma(s,((\Sigma_{v,1}\otimes \xi_v)\otimes(\Sigma_{v,2}\otimes \xi_v))\otimes\tau_v,\Psi_v)$$ $$=\gamma(s, \Pi_1\otimes \xi_v, \textrm{Sym}^2\otimes \tau_v,\Psi_v)\gamma(s, \Pi_2\otimes \xi_v, \textrm{Sym}^2\otimes \tau_v,\Psi_v)$$$$\cdot\gamma(s,((\Pi_{v,1}\otimes \xi_v)\times(\Pi_{v,2}\otimes \xi_v))\otimes\tau_v,\Psi_v)$$ $$=\gamma(s, \textrm{Ind}(\Pi_{v,1}\otimes\Pi_{v,2})\otimes\xi_v, \textrm{Sym}^2\otimes \tau_v,\Psi_v).$$ Here the first equality is the additivity of the arithmetic $\gamma$-factors, the second equality follows from our induction hypothesis of Proposition 3.1 on the dimension $n$ of $\rho$, and the fact the LLC preserves the local $\gamma$-factors in pairs. The last equality is a consequence of the multiplicativity of the analytic $\gamma$-factors. Indeed, recall that the adjoint action $r:{^LM_H}\simeq \textrm{GL}_n(\mathbb{C})\times \textrm{GL}_1(\mathbb{C})\longrightarrow \textrm{GL}({^L\mathfrak{n}}_H)$ is irreducible. ${^LM_H}=\{m=m(g,a_0)=\begin{bmatrix} g & \ \ \\ \ \ & a_0J'{^tg^{-1}J'^{-1}}\\ \end{bmatrix} g\in \textrm{GL}_n(\mathbb{C}), a_0\in \textrm{GL}_1(\mathbb{C})\}$ and ${^L\mathfrak{n}_H}=\{\begin{bmatrix} 0 & X \\ 0 & 0 \\ \end{bmatrix}: J'{^tX}J'=X\}.$ Let $Y=XJ'^{-1}$ then $J'{^tX}J'=X\Leftrightarrow {^tY}=Y.$ Denote $\mathfrak{n}(Y)=\begin{bmatrix} 0 & X \\ 0 & 0\\ \end{bmatrix}=\begin{bmatrix} 0 & YJ'^{-1}\\ 0 & 0 \\ \end{bmatrix}$. Then an easy calculation show that $r(m(g,a_0))\mathfrak{n}(Y)=\mathfrak{n}(a_0gY{^tg}J').$ Let $\theta_1\subset \theta\subset \Delta$ be the subset of simple roots which gives the Levi subgroup $M_{\theta_1}\simeq \textrm{GL}_{n_1}\times \textrm{GL}_{n_2}\times \textrm{GL}_1$ with $n=n_1+n_2$, therefore ${^L}M_{\theta_1}\simeq \textrm{GL}_{n_1}(\mathbb{C})\times \textrm{GL}_{n_2}(\mathbb{C})\times \textrm{GL}_1(\mathbb{C})$. Write $Y=\begin{bmatrix} Y_1 & Y_2 \\ Y_3 &Y_4\\ \end{bmatrix}$, then $^tY=Y$ is equivalent to say that $^tY_1=Y_1$, $Y_3={^tY}_2$ and ${^tY}_4=Y_4$. According to the inductive construction of local $\gamma$-factors through Langlands-Shahidi method, we need to decompose the restriction of the adjoint action $r$ on ${^LM_{\theta_1}}$ on $^L\mathfrak{n}_H$ into a direct sum of irreducible subrepresentations (Theorem 8.3.2 of \[17\]). In our case each of them contributes to a local $\gamma$-factor. The restriction gives that $$r(m(\begin{bmatrix} g_1 & \ \ \\ \ \ & g_2 \\ \end{bmatrix}), a_0)(\mathfrak{n}(Y))=\mathfrak{n}(a_0\begin{bmatrix} g_1 & \ \ \\ \ \ & g_2 \\ \end{bmatrix}\begin{bmatrix} Y_1 & Y_2 \\ {^tY}_2 & Y_4 \\ \end{bmatrix}\begin{bmatrix} {^tg}_1 & \ \ \\ \ \ & {^tg}_2 \\ \end{bmatrix}J')$$$$=\mathfrak{n}(\begin{bmatrix} a_0g_1 Y_2{^tg}_2 J_{n_2}' & a_0g_1Y_1{^tg}_1 J_{n_1}' \\ a_0 g_2Y_4 {^tg}_2 J_{n_2}' & a_0 g_2 {^tY}_2 {^tg_1} J_{n_1}' \\ \end{bmatrix},$$ where $J'=\begin{bmatrix} \ \ & J_{n_1}' \\ J_{n_2}' & \ \ \\ \end{bmatrix}$ with $J_{n_i}'$ the same type of matrix as $J'$ of size $n_i$. Now let’s get back to our setting. For $v\in S$, non-archmedean and $v\neq v_0$, $\Pi_1$ and $\Pi_2$ are irreducible admissible representations of $\textrm{GL}_{n_1}(\mathbb{F}_v)$ and $\textrm{GL}_{n_2}(\mathbb{F}_v)$ respectively. $\tau_v$ is a fixed character of $\mathbb{F}_v^{\times}$, and $\xi_v$ is a character of $\mathbb{F}_v^{\times}$. Notice that here $Y_2$ is a free matrix of size $n_1\times n_2$, so the two diagonal blocks above give an irreducible subrepresentation. It is isomorphic to the tensor product $\Pi_1$ and $\Pi_2$, twisted by a character $\tau_v$ which is given by the $a_0$-component in the above expression. Therefore it contributes to the twisted Rankin-Selberg local $\gamma$-factor $\gamma(s, (\Pi_1\times \Pi_2)\otimes \tau_v, \Psi_v)$. If we take $\Pi_i\otimes \xi_v$ instead of $\Pi_i$, we obtain the twisted Rankin-Selberg $\gamma$-factor $\gamma(s, ((\Pi_1\otimes \xi_v)\times (\Pi_2\otimes \xi_v))\otimes \tau_v, \Psi_v)$. Moreover, notice that ${^tY}_1=Y_1$ and ${^tY_4}=Y_4$, and the form of each of the rest blocks shows that each of them is isomorphic to the adjoint action of $^LM_i$ on $^L\mathfrak{n_i}$, where $M_i$ is the same type of Siegel Levi inside $\textrm{GSpin}_{2n_i+1}$. Therefore they are both irreducible, and they contribute to the twisted symmetric square local $\gamma$-factors $\gamma(s, \Pi_i, \textrm{Sym}^2\otimes \tau_v, \Psi_v)$, $i=1,2$. Again take $\Pi_v\otimes \xi_v$ instead of $\Pi_v$, we obtain the two $\gamma$-factors $\gamma(s, \Pi_1\otimes \xi_v, \textrm{Sym}^2\otimes \tau_v, \Psi_v)$ and $\gamma(s, \Pi_2\otimes \xi_v, \textrm{Sym}^2\otimes \tau_v, \Psi_v)$. Therefore by the multiplicativity of the local analytic $\gamma$-factors, we obtain that $$\gamma(s, \textrm{Ind}(\Pi_{v,1}\otimes\Pi_{v,2})\otimes\xi_v, \textrm{Sym}^2\otimes \tau_v,\Psi_v)$$ $$=\gamma(s, \Pi_1\otimes \xi_v, \textrm{Sym}^2\otimes \tau_v,\Psi_v)\gamma(s, \Pi_2\otimes \xi_v, \textrm{Sym}^2\otimes \tau_v,\Psi_v)$$$$\cdot\gamma(s,((\Pi_{v,1}\otimes \xi_v)\times(\Pi_{v,2}\otimes \xi_v))\otimes\tau_v,\Psi_v).$$ This establishes the last equality. The general case follows from the case $r_v=2$ by induction on $r_v$. Hence from the global functional equations we are left with $\gamma(s, \textrm{Sym}^2(\rho_0\otimes \chi)\otimes \eta, \psi)=\gamma(s, \pi(\rho_0)\otimes\chi, \textrm{Sym}^2\otimes \eta, \psi)$. To prove Proposition 3.1, besides Proposition 3.2, we also need both the arithmetic and analytic stability for $\gamma$-factors. We will explain as follows. On the arithmetic side, P. Deligne showed the existence and uniqueness of the local $\epsilon$-factors on page 535-547 in \[9\]. For $V$ a finite dimensional complex representation of the local Weil group, $\chi$ is sufficiently ramified character of $F^{\times}$, the arithmetic $\epsilon$-factor attached to $V\otimes \chi$ depends only on $\det(V)$ and $\dim(V)$. Apply this to the case when $V\simeq \textrm{Sym}^2\rho\otimes \eta$ where $\rho$ is an irreducible n-dimensional representation of $W_F$, and $\eta$ is a character of $F^{\times}$ viewed as a character of $W_F$ as before. Also notice that $L(s, V\otimes \chi)=1$ for $\chi$ sufficiently ramified, we obtain: (**Arithmetic Stability for the twisted symmetric square $\gamma$-factors**) Let $\rho_1$ and $\rho_2$ be two continuous n-dimensional representations of $W_F$ with $\det(\rho_1)=\det(\rho_2)$, $\eta$ be a fixed character of $F^{\times}$. Then for all sufficiently ramified characters $\chi$ of $F^{\times}$ we have $$\gamma(s, \textrm{Sym}^2(\rho_1\otimes \chi)\otimes \eta, \psi)=\gamma(s, \textrm{Sym}^2(\rho_2\otimes \chi)\otimes\eta, \psi).$$ On the analytic side, $\pi=\pi(\rho)$ is supercusipidal when $\rho$ is irreducible, therefore analogously we should have: (**Supercuspidal Stability for the twisted symmetric square $\gamma$-factors**) Let $\pi_1$ and $\pi_2$ be two supercusipidal representations of $\textrm{GL}_n(F)$ with $\omega_{\pi_1}=\omega_{\pi_2}$, and $\eta$ is a fixed character of $F^{\times}$. Then for all sufficiently ramified characters $\chi$ of $F^{\times}$, whose degree of ramification depends only on $\pi_1$ and $\pi_2$, identified as characters of $\textrm{GL}_n(F)$ through the determinant, we have $$\gamma(s, \pi_1\otimes \chi, \textrm{Sym}^2\otimes\eta, \psi)=\gamma(s,\pi_2\otimes\chi, \textrm{Sym}^2\otimes\eta, \psi).$$ This is the main result of this paper and will be established in the rest sections. With Proposition 3.2, 3.3, and 3.4, we are ready to prove Proposition 3.1. (Proof of Proposition 3.1) We will do induction on the dimension $n$ with the help of a globalization method provided as on page 2061-2065 in \[7\]. When $n=1$ we obtain that both sides equal to 1, and there is nothing to prove. For $n=2$, one could either follow \[8\] directly, or instead we show $\gamma(s,\wedge^2(\rho\otimes \chi)\otimes \eta, \psi)=\gamma(s, \pi\otimes \chi, \wedge^2\otimes \eta, \psi).$ These $\gamma$-factors are in general defined again through Langlands-Shahidi method by the adjoint action of $^LM$ on ${^L\mathfrak{n}}$ where $M$ is the maximal Levi isomorphic to $\textrm{GL}_n\times \textrm{GSpin}_0\simeq \textrm{GL}_n\times \textrm{GL}_1$ inside $\textrm{GSpin}_{2n}$(Theorem 2.7 \[1\]). Notice that in this case $\wedge^2\rho\otimes \eta=\det(\rho)\otimes \eta$. On the other hand, it is not hard to see that $\gamma(s,\pi, \wedge^2\otimes \eta, \psi)=\gamma(s, \omega_{\pi}\times \eta,\psi)$, where $\omega_{\pi}$ is the central character of $\pi$, and the right hand side is the $\gamma$-factor attached to the Rankin-Selberg L-function $L(s,\omega_{\pi}\times \eta).$ Since we know that $\det \rho\leftrightarrow \omega_{\pi}$ under the local Langlands correspondence, and tensor product of representations on the arithmetic side corresponds to Rankin-Selberg convolutions on the analytic side, so $\det\rho\otimes \eta\leftrightarrow \omega_{\pi}\times \eta$. Moreover, since LLC is compatible with twisting by characters, we see that the stable equality is true for the twisted exterior square $\gamma$-factors when $n=2$, and for this case we don’t even need to assume $\chi$ is highly ramified. Now apply the equalities $$\gamma(s, (\pi\times\pi)\times \eta,\psi)=\gamma(s,\pi,\wedge^2\otimes \eta,\psi)\gamma(s,\pi,Sym^2\otimes\eta,\psi)$$ $$\gamma(s,(\rho\otimes\rho)\otimes\eta,\psi)=\gamma(s,\wedge^2\rho\otimes\eta,\psi)\gamma(s,Sym^2\rho\otimes\eta,\psi),$$ and by the fact that LLC preserves the $L$- and $\epsilon$-factors of pairs, we see that the proposition is true for the case when $n=2$ and any character $\chi$. Now $\rho$ is an irreducible n-dimensional representation of $W_F$, let $\pi=\pi(\rho)$ be its corresponding supercuspidal representation of $\textrm{GL}_n(F)$. Take $\omega_0=\omega_{\pi}$ in Proposition 3.2, then there exists an irreducible n-dimensional representation $\rho_0$ of $W_F$ and its corresponding supercuspidal representation $\pi_0=\pi(\rho_0)$ of $\textrm{GL}_n(F)$ such that $\omega_{\pi}=\omega_{\pi_0}$, $\det (\rho)=\det(\rho_0)$ and $\gamma(s, \textrm{Sym}^2(\rho_0\otimes\chi)\otimes\eta,\psi)=\gamma(s, \pi_0\otimes\chi, \textrm{Sym}^2\otimes\eta,\psi)$. Take $\chi$ sufficiently ramified such that Proposition 3.3 holds for the pair $(\rho,\rho_0) $, and Proposition 3.4 holds for the pair $(\pi,\pi_0)$. Then for such $\chi$ we have $$\gamma(s, \textrm{Sym}^2(\rho\otimes\chi)\otimes\eta,\psi)=\gamma(s, \textrm{Sym}^2(\rho_0\otimes\chi)\otimes\eta,\psi)$$$$=\gamma(s, \pi_0\otimes\chi,\textrm{Sym}^2\otimes\eta,\psi)=\gamma(s,\pi\otimes\chi, \textrm{Sym}^2\otimes\eta,\psi)$$ The degree of ramification now depends on $(\rho,\pi)$ and $(\rho_0,\pi_0)$, so one needs to fix such a base point $(\rho_0,\pi_0)$ for every character $\omega_0$. As in \[7\], this can be reduced to just fix the character $\omega_0$ since twisting by unramified characters can be absorbed into the complex parameter $s$ of the $\gamma$-factors. This completes the proof of Proposition 3.1. Next we extend our result to Weil-Deligne representations. Let $\rho$ be a continuous n-dimensional $\Phi$-semisimple complex representation of the Weil-Deligne group $W_F'$, and $\eta$ a fixed character of $F^{\times}$. Then for sufficiently ramified characters $\chi$ of $F^{\times}$ we have $$\gamma(s, \textrm{Sym}^2(\rho\otimes\chi)\otimes\eta,\psi)=\gamma(s, \pi(\rho)\otimes\chi, \textrm{Sym}^2\otimes\eta, \psi).$$ The corollary follows from the following facts: (1) the compatibility of the construction of $\Phi$-semisimiple representations of $W_F'$ from irreducible representations of $W_F$ and the Bernstein-Zelevinsky construction \[3\] of irreducible representations of $\textrm{GL}_n(F)$ from supercuspidals; (2) the local $\gamma$-factors attached to $\rho$ only depends on its semisimplification(as representtions of $W_F$)(page 201, \[4\]); (3) LLC is compatible with pairs of local L-factors and the twisted symmetric square L-factors on both the arithmetic and the analytic sides, and under highly ramified twists these become 1 \[12\]; (4), the additivity of the arithmetic local $\gamma$-factors \[7\] and the multiplicativity of the analytic local $\gamma$-factors, which was proved by an induction argument as in Proposition 3.2. (**General analytic stability for the twisted symmetric square $\gamma$-factors**) Let $\pi_1$ and $\pi_2$ be two irreducible admissible representations of $GL_n(F)$ with $\omega_{\pi_1}=\omega_{\pi_2}$, $\eta$ is a fixed character of $F^{\times}$. Then for any sufficiently ramified character $\chi$ of $F^{\times}$ we have $$\gamma(s, \pi_1\otimes\chi, Sym^2\otimes\eta, \psi)=\gamma(s, \pi_2\otimes\chi, Sym^2\otimes\eta, \psi)$$ Let $\rho_1$ and $\rho_2$ be two continuous n-dimensional $\Phi$-semisimple representations of the Weil-Deligne group $W_F'$ and $\pi_i=\pi(\rho_i)$ (i=1,2) be their corresponding irreducible admissible representations of $\textrm{GL}_n(F)$. By corollary 3.5 we have $\gamma(s, \textrm{Sym}^2(\rho_i\otimes\chi)\otimes\eta, \psi)=\gamma(s, \pi_i\otimes\chi,\textrm{Sym}^2\otimes\eta,\psi).$ Then we can see that the result would follow if we have the analogue of Proposition 3.3 for Weil-Deligne representations. On the other hand, we know that the arithmetic $\gamma$-factors depend only on the semisimplification, i.e., we have $\gamma(s, \rho,\psi)=\gamma(s,\rho^{ss},\psi)$. Since the semisimplification does not change the determinant $\det{\rho}$ and $\dim(\rho_1)=\dim(\rho_2)=n$, so again since the local arithmetic $\epsilon$-factors depend only on $\det(\rho) $ and $\dim(\rho)$ under suitably highly ramified twist by $\chi$, as we mentioned earlier. So we can take $\chi$ sufficiently ramified such that the arithmetic stability of $\gamma$-factors follows for Weil-Deligne representations. That is, $\gamma(s, \textrm{Sym}^2(\rho_1\otimes\chi)\otimes\eta,\psi)=\gamma(s, \textrm{Sym}^2(\rho_2\otimes\chi)\otimes\eta,\psi).$ Then the result follows immediately from Corollary 3.5. PROOF OF THE MAIN THEOREM ========================= In this section we will prove our main theorem(Theorem 1.1), by assuming the analytic stability of the twisted symmetric square $\gamma$-factors attached to supercuspidal representations(Proposition 3.4). Before we proceed, as in \[7\], we make a remark on the additive character $\psi$ of $F$. Take $a\in F^{\times}$ and fix a non-trivial additive character $\psi$ of $F$. Let $\psi^a$ denote the character given by $\psi^a(x)=\psi(ax).$ By the study of Henniart \[11\] and Deligne \[9\] respectively, it turns out that as a function of $a\in F^{\times}$, both the analytic $\gamma$-factors $\gamma(s, \pi, r, \psi^a)$ and the corresponding arithmetic $\gamma$-factors $\gamma(s, r\circ \rho, \psi^a)$ vary in the same way. Therefore it suffices to prove the result for a fixed $\psi$. We will first establish the equality for the $\gamma$-factors, and then use it to obtain the equality for L-factors. We begin with some lemmas: (**Equality for monomial representations**) Let $E/F$ be a finite Galois extension of degree n contained in a fixed algebraic closure $\overline{F}$ of $F$, and $\eta$ be a fixed character of $F^{\times}$. Denote $G=\operatorname{Gal}(E/F)$. Let $F\subset L\subset E$ be an intermediate extension and $\chi$ be a finite-order character of $H=\operatorname{Gal}(E/L)$. Let $\rho=\textrm{Ind}_H^G(\chi)$, then $$\gamma(s, \textrm{Sym}^2\rho\otimes\eta,\psi)=\gamma(s,\pi(\rho), \textrm{Sym}^2\otimes\eta,\psi)$$ This is the same globalization method as used in Lemma 3.2 in \[7\], one may simply replace the $\wedge^2$ there by $\textrm{Sym}^2\otimes\eta$, change the equalities in the proof accordingly and use Proposition 3.1 and 3.2. (**Equality for Galois representations**) Let $\rho$ be an irreducible continuous n-dimensional representation of $W_F$ with $\det(\rho)$ being a character of finite order, and $\eta$ be a fixed character of $F^{\times}$. Then $$\gamma(s, Sym^2\rho\otimes\eta,\psi)=\gamma(s, \pi(\rho), Sym^2\otimes\eta, \psi).$$ This is also a straightforward analogue of Lemma 3.3 in \[7\]. A very similar argument shows that the arithmetic and analytic twisted symmetric square local $\gamma$-factors satisfy the same formalism, then we use additivity and multiplicativity of the arithmetic and analytic twisted symmetric square $\gamma$-factors respectively, together with Lemma 4.1 then we are done. Now we have all the ingredients for the proof of Theorem 1.1. (**Proof of Theorem 1.1**) First we prove the equality of $\gamma$-factors. By Lemma 4.2, we have the equality of the local twisted symmetric square $\gamma$-factors for irreducible continuous representations of $W_F$ with finite order determinant. After tensoring with an unramified character, we can extend the result to any irreducible continuous n-dimensinal representation of $W_F$. Both LLC and the formalism of the twisted symmetric square $\gamma$-factors are compatible with twisting by characters. Since LLC also preserves the local $\gamma$-factors for direct sums of representations on the arithmetic side with isobaric sums of the corresponding representations on the analytic side, we can further extend the result in Lemma 4.2 to arbitrary continuous $n$-dimensional representations of $W_F$. Next, as in the proof of Corollary 3.5, we can extend the result to all continuous $\Phi$-semisimple n-dimensional representations of the Weil-Deligne group $W_F'$. This completes the proof of the equality of the twisted symmetric square $\gamma$-factors in Theorem 1.1. We are left with the equality of L-factors. We use a similar argument as Henniart’s proof in \[12\] to show that the equality of $\gamma$-factors imply the equality of their corresponding L-factors. Recall that $\pi$ is an irreducible representation of $\textrm{GL}_n(F)$. Suppose $\pi\leftrightarrow \rho$ under LLC, where $\rho=(\rho',V, N)$. In general if $r$ is any analytic representation of $\textrm{GL}_n(\mathbb{C})$ we have that $r\circ\rho=(r\circ\rho', r(V), \frac{d}{dx}\vert_{x=0}(r\circ\rho)(x))$ is also a Weil-Deligne representation, where $r(V)$ is the space given by $r$ and $V$, i.e., $r:{^LG}=GL_n(\mathbb{C})\longrightarrow GL(r(V)).$ Notice that the monodromy operater $N$ satisfies $\rho(x)v=\exp(xN)v$ for all $v\in V$ and $x\in \mathbb{G}_a$. Recall that $W_F'\simeq W_F\rtimes \mathbb{G}_a$. So $N=\frac{d}{dx}\vert_{x=0}\rho(x)$, therefore in general the monodromy operator $T$ for $r\circ\rho$ is given by $T=\frac{d}{dx}\vert_{x=0}(r\circ\rho)(x)$. Following Henniart’s terminology in \[12\], we say a Weil-Delinge representation $\rho$ is tempered if all its indecomposable constituents are of the form $\rho_i'\otimes Sp(m_i)$ where $\rho_i'$ is an irreducible unitary representation of $W_F$ and $Sp(m_i)$ is a special representation of dimension $m_i$, corresponding to a Steinberg representation of $\textrm{GL}_{m_i}(F)$. Or equivalently, if we define the Weil-Delinge group to be $W_F\rtimes \textrm{SL}_2(\mathbb{C})$, then the image of $W_F$ is bounded in $\textrm{GL}(V)$. Since we have the exact sequence $$0\rightarrow I_F\rightarrow W_F\rightarrow \mathbb{Z}\rightarrow 0$$ where $I_F$ is the inertial subgroup, which is compact, it is the same as saying that the image of the geometric Frobenius is a unitary operator on $V$. For this purpose here we use another definition of the Weil-Deligne group given by $W_F\rtimes \textrm{SL}_2(\mathbb{C})$. By Theorem 2.8 of \[20\], the triple $\rho=(\rho',V,N)$ is equivalent to a representation $\varphi: W_F\rtimes \textrm{SL}_2(\mathbb{C})\rightarrow \textrm{GL}_n(\mathbb{C})$ such that $\varphi$ is trivial on an open subgroup of $I_F$, $\varphi(\Phi)$ is semi-simple and $\varphi\vert_{\textrm{SL}_2(\mathbb{C})}$ is algebraic. By Lemma 2.9 of \[23\], there exists a unique $\mathfrak{sl}_2$-triple $(e,f,h)$ such that $e=N=\mathfrak{gl}_n^{\rho(I_F)}(\Phi)(q^{-1})$, $f=\mathfrak{gl}_n^{\rho(I_F)}(q)$, and $h=\mathfrak{gl}_n^{\rho(W_F)}=\mathfrak{gl}_n^{\rho(I_F)}(1)$, where $q=\vert\mathcal{O}_F/\mathfrak{m}_F\vert$ is the cardinality of the residue field and $V(q)$ denotes the q-eigenspace of the action of $\rho(\Phi)$ on V. Then the corresponding representation $\varphi: W_F\rtimes \textrm{SL}_2(\mathbb{C})\rightarrow \textrm{GL}_n(\mathbb{C})$ is given by $\varphi(w)=\exp(\frac{-v(w)}{2}\log q\cdot h)\rho(w).$ First we assume that $\pi$ is tempered and $\eta$ is unitary. Then it follows that the representation $\sigma_\eta$ of $M_H(F)$ is tempered. We show $\textrm{Sym}^2\rho\otimes\eta$ is also tempered. $\rho=(\rho',V,N)$ implies that $\textrm{Sym}^2\rho\otimes\eta=(\textrm{Sym}^2\rho'\otimes\eta, \textrm{Sym}^2(V), 1\otimes N+N\otimes 1)$, here we identify $\textrm{Sym}^2\rho$ as a subspace of $\rho\otimes\rho$ generated by $e_i\otimes e_j+e_j\otimes e_i$ where $\{e_i\}_{i=1}^n$ is a basis of $V$. Now if $\rho$ is given by $\varphi$ as above, then $\textrm{Sym}^2\rho\otimes\eta$ is given by $\tilde{\varphi}: W_F\rtimes \textrm{SL}_2(\mathbb{C})\rightarrow \textrm{GL}_n(\mathbb{C})$ by $\tilde{\varphi}(w)=\exp(\frac{-v(w)}{2}\log q\cdot H)\textrm{Sym}^2\rho\otimes\eta(w)=\exp(\frac{-v(w)}{2}\log q\cdot H)(\rho\otimes \rho)\vert_{\textrm{Sym}^2(V)}(w)\cdot \eta(w),$ where $H=1\otimes h+h\otimes 1.$ Notice that if $e=N, f, h$ form an $\mathfrak{sl}_2$-triple, then $E=1\otimes N+N\otimes 1, F=1\otimes f+ f\otimes 1$, and $H=1\otimes h+h\otimes 1$ also form an $\mathfrak{sl}_2$-triple. $\pi$ being tempered implies that $\rho$ is tempered, therefore $U=\varphi(\Phi)=\exp(\frac{1}{2}\log q\cdot h)\rho(\Phi)$ is unitary. Since $\eta$ is unitary, it suffices to show that $\exp(\frac{1}{2}\log q\cdot (1\otimes h+h\otimes 1))(\rho\otimes\rho)\vert_{\textrm{Sym}^2(V)}(\Phi)$ is unitary, thus it suffices to show that $\exp(\frac{1}{2}\log q\cdot (1\otimes h+h\otimes 1))(\rho\otimes\rho)(\Phi)$ is unitary. We have $\exp(\log \sqrt{q}(1\otimes h+h\otimes 1))(\rho\otimes\rho)(\Phi)=\exp(\log \sqrt{q}(1\otimes h))\cdot\exp(\log \sqrt{q}(h\otimes 1))((1 \otimes \rho)(\Phi)\cdot(\rho\otimes 1)(\Phi))=\exp(1\otimes \log\sqrt{q}\cdot h)(1\otimes \rho(\Phi))\cdot \exp(\log \sqrt{q}\cdot h\otimes 1)(\rho\otimes 1)(\Phi)=(1\otimes U)\cdot (U\otimes 1)$ is unitary since $U$ is unitary. Therefore $\textrm{Sym}^2\rho\otimes \eta$ is tempered. In this case we have that $L(s, \textrm{Sym}^2\rho\otimes \eta)$ has no poles for $Re(s)>0$, and for the same reason we have that $L(1-s, \textrm{Sym}^2{\rho}^{\vee}\otimes\eta^{-1})$ has no poles for $Re(s)<1$. By Langlands-Shahidi method we have $$\gamma(s, \pi, \textrm{Sym}^2\otimes \eta,\psi)=\epsilon(s, \textrm{Sym}^2\rho\otimes \eta, \psi)\frac{L(1-s, \textrm{Sym}^2\rho^{\vee}\otimes\eta^{-1})}{L(s,\textrm{Sym}^2\rho\otimes\eta)}$$ Moreover, $\gamma(s,\pi,\textrm{Sym}^2\otimes \eta,\psi)$ is a rational function of $q^{-s}$. To be precise, $\gamma(s,\pi,\textrm{Sym}^2\otimes\eta,\psi)=F(q^{-s})$ where $F(X)=c X^a\frac{P(X)}{Q(X)}$ with $P(X),Q(X)\in \mathbb{C}[X]$ such that $P(0)=Q(0)=1$, $c\in \mathbb{C}$ and $a\in \mathbb{Z}$. We also know that $\epsilon(s, \textrm{Sym}^2\rho\otimes\eta,\psi)$ is a monomial of $q^{-s}$. The local tempered L-factor is defined as $L(s,\pi, \textrm{Sym}^2\otimes \eta)=P(q^{-s})$. Since $L(s, \textrm{Sym}^2\rho\otimes\eta)$ and $L(1-s,\textrm{Sym}^2\rho^{\vee}\otimes \eta^{-1})$ have no poles in common, similar to Henniart’s proof in \[12\], we can conclude that $L(s,\pi, \textrm{Sym}^2\otimes\eta)=L(s, \textrm{Sym}^2\rho\otimes\eta).$ Now if $\sigma_{\eta}$ is quasi-tempered, then $\pi$ is quasi-tempered and $\eta$ is arbitrary. Let $\tau_0:M(F)\simeq \textrm{GL}_n(F)\times \textrm{GL}_1(F)\rightarrow \mathbb{C}^{\times}$ be an unramified character of $M(F)$ given by $\tau_0=\vert \det(\cdot)\vert^{s_1} \vert\cdot\vert ^{s_2}$, where $s_1, s_2\in \mathbb{C}$. The fundamental weight attached to $\alpha$ is given by $\hat{\alpha}=\langle \rho, \alpha \rangle^{-1}\rho$ where $\rho$ is half of the sum of positive roots in $N_H$. In our case $\alpha=\alpha_n=e_n$ and $\rho=\frac{1}{2}(\sum_{1\leq i<j\leq n}(e_i+e_j)+\sum_{i=1}^n e_i)=\frac{n}{2}\sum_{i=1}^n e_i,$ therefore we have $$\langle \rho, \alpha\rangle=\frac{2(\rho,\alpha)}{(\alpha,\alpha)}=\frac{2(\frac{n}{2}\sum_{i=1}^n e_i, e_n)}{(e_n,e_n)}=n$$ where $(\cdot,\cdot)$ is a Weyl group invariant non-degenerate bilinear form on $\mathfrak{a}^*=X^*(H)\otimes_{\mathbb{Z}}\mathbb{R}.$ So $\hat{\alpha}=\langle \rho, \alpha\rangle ^{-1}\rho=n^{-1}(\frac{n}{2}\sum_{i=1}^ne_i)=\frac{1}{2}\sum_{i=1}^ne_i.$ For $s\in \mathbb{C}$, define $\sigma_{\eta,s}=\sigma_\eta\otimes q^{\langle s\hat{\alpha}, H_M(\cdot)\rangle}\simeq(\sigma_s)_\eta$ where $\sigma_s$ is the lift of the representation $ \pi\otimes \vert\det(\cdot)\vert^{\frac{s}{2}}$ of $\textrm{GL}_n(F)$ to $M_H(F)$. So for $v\in V_{\pi}$, $\sigma_s(m(g,a))v=\vert \det(g)\vert^{\frac{s}{2}}\pi(g)v$ and $\sigma_{\eta,s}(m(g,a))v=\eta^{-1}(a)\vert \det(g)\vert^{\frac{s}{2}}\pi(g)v$. Let $\eta_s=\eta\cdot \vert\cdot \vert^s$, then $\sigma_\eta\otimes \tau_0\simeq \sigma_{\eta_{-s_2},2s_1}.$ Now if $\eta=\eta_0\vert\cdot \vert^{z_0}$ where $\eta_0$ is unitary and $z_0\in \mathbb{C}^*$, take $s_2=z_0$, and take $s_1$ such that $\pi\otimes \vert\det(\cdot)\vert^{s_1}$ is tempered, then by the previous case we have $$L(s, \sigma_\eta\otimes\tau_0, r)=L(s, (\sigma_{2s_1})_{\eta_0}, r)=L(s, \textrm{Sym}^2(\rho\otimes\vert\vert\cdot\vert\vert^{s_1})\otimes \eta_0)$$ $$=L(s+2s_1, \textrm{Sym}^2\rho\otimes\eta_0)=L(s+2s_1+s_2,\textrm{Sym}^2\rho\otimes \eta).$$ On the other hand, we apply section 2.7 of \[12\], which states how the local analytic $\gamma$-factor shifts under twists by unramified character of the maximal split quotient of $M_H$, to our case. The maximal split quotient $T_0$ of $M_H\simeq \textrm{GL}_n\times \textrm{GL}_1$ is isomorphic to $\textrm{GL}_1\times \textrm{GL}_1$, since the derived group $M_{H,{der}}$ of $M_H$ is isomorphic to $\textrm{SL}_n$. The adjoint action $r:{^L}M_H\longrightarrow \textrm{GL}({^L}\mathfrak{n}_H)$ is irreducible, so its restriction on the torus $\hat{T}_0$ is given by a character $\chi_r: \hat{T}_0\longrightarrow \mathbb{C}^{\times}.$ In our case, $r$ is given by the symmetric square action twisted by a character given by the $\textrm{GL}_1$ part of $^LM_H$. A direct calculation shows that $\chi_r: \hat{T}_0\longrightarrow GL({^L}{\mathfrak{n}_H})$ is given by $(xI_n, y)\mapsto x^2y$. Taking dual of this map we obtain a one-parameter subgroup $\hat{\chi}_r: F^{\times}\longrightarrow T_0\simeq \textrm{GL}_1\times \textrm{GL}_1$ given by $x\mapsto (x^2,x)$. Notice that $\tau_0\in X_{un}(M)$, and $M_{H,der}\subset \ker(H_{M_H})$, where $H_{M_H}: M_H(F)\longrightarrow \mathfrak{a}_{M_H}=\operatorname{Hom}(X(M_H)_F, \mathbb{Z})\otimes \mathbb{R}$ is the Harish-Chandra map. Therefore $\tau_0$ defines an unramified character on $T_0(F)$, say $\overline{\tau}_0: T_0(F)\longrightarrow \mathbb{C}^{\times}$ such that $\overline{\tau}_0\circ (\det\times id)=\tau_0$. Since $\tau_0=\vert\det(\cdot)\vert^{s_1}\vert\cdot\vert^{s_2}$, we see that $\overline{\tau}_0=\vert\cdot\vert^{s_1}\vert\cdot\vert^{s_2}$. Following \[12\], this defines an unramified character $\overline{\tau}_0\circ \hat{\chi}_r: F^{\times}\longrightarrow \mathbb{C}^{\times}$ given by $x\mapsto \vert x^2\vert^{s_1}\vert x\vert^{s_2}=\vert x\vert^{2s_1+s_2}$. Therefore by section 2.7 of \[12\] we obtain $\gamma(s+2s_1+s_2,\pi, \textrm{Sym}^2\otimes\eta,\psi )=\gamma(s, \sigma_\eta\otimes \tau_0, r,\psi),$ therefore also $L(s+2s_1+s_2,\pi, \textrm{Sym}^2\otimes\eta)=L(s, \sigma_{\eta}\otimes\tau_0,r)$, by the previous argument on the tempered case. Compare it with the arithmetic side we obtain $L(s+2s_1+s_2, \pi, \textrm{Sym}^2\otimes\eta)=L(s+2s_1+s_2, \textrm{Sym}^2\rho\otimes\eta).$ Then by the uniqueness of complex meromorphic functions we see that $L(s,\pi,\textrm{Sym}^2\otimes\eta)=L(s, \textrm{Sym}^2\rho\otimes\eta).$ This shows the case when $\sigma_{\eta}$ is quasi-tempered. In general, if $\rho$ is an n-dimensional $\Phi$-semisimple representation of $W_F'$, then $\rho=\oplus_{i=1}^r\rho_i$, where each $\rho_i$ is indecomposable and $\rho_i\simeq \rho_i'\otimes Sp(m_i)$, where each $\rho_i'$ is an irreducible $n_i'$-dimensional representation of $W_F$. Let $\pi'_i=\pi(\rho_i')\leftrightarrow \rho_i'$ under LLC, and let $\Delta_i$ be the segment $\{\pi_i',\pi_i'(1),\cdots,\pi_i'(m_i-1)\}$ where $\pi_i'(j)=\pi_i'\otimes\vert \det(\cdot)\vert^j$. Then the Bernstein-Zelevinsky’s classification \[3\] tells us that $\rho_i\leftrightarrow Q(\Delta_i)$, where $Q(\Delta_i)$ is the unique irreducible subquotient of $\textrm{Ind}_{\textrm{GL}_{n_i}(F)^m}^{\textrm{GL}_{n_im_i}(F)}\pi_i'\otimes \pi_i'(1)\otimes\cdots\otimes\pi_i'(m_i-1)$ and $\pi(\rho)$ is the unique irreducible subquotient of $\textrm{Ind}_{\prod \textrm{GL}_{n_im_i}(F)}^{\textrm{GL}_n(F)}Q(\Delta_1)\otimes Q(\Delta_2)\otimes\cdots\otimes Q(\Delta_r)$. To simplify the notation we use $Q(\Delta_1)\times\cdots\times Q(\Delta_r)$ to denote this induced representation. For each $1\leq i\leq r$ there exists a unique $\beta_i\in \mathbb{R}$ such that $Q(\Delta_i)(-\beta_i)$ is square integrable, thus tempered. We can order the $\Delta_i$’s such that $\alpha_1:=\beta_1=\beta_2=\cdots=\beta_{m_1}>\alpha_2:=\beta_{m_i+1}=\cdots=\beta_{m_2}>\cdots>\alpha_s:=\beta_{m_{s-1}+1}=\cdots=\beta_r.$ In this order $\Delta_i$ does not precede $\Delta_j$ for $i<j$ and all $\Delta_i$’s corresponding to the same $\alpha_j$ are not linked. For $1\leq j\leq s$, let $\pi_j=Q(\Delta_{m_{j-1}+1})(-\alpha_j)\times\cdots\times Q(\Delta_{m_j})(-\alpha_j)$ where $m_0=0$ and $m_s=r$. Then all the $\pi_j$’s are irreducible tempered representations, and $\pi=\pi(\rho)$ is the unique irreducible subquotient of $\pi_1(\alpha_1)\times\cdots\times\pi_s(\alpha_s)$. This gives the Langlands classification \[13\]. We denote the corresponding parabolic subgroup by $P$ and let $\sigma=\pi_1\times\cdots\times\pi_s$, $\nu=\vert\det(\cdot)\vert^{\alpha_1}\otimes\vert\det(\cdot)\vert^{\alpha_2}\otimes\cdots\otimes\vert\det(\cdot)\vert^{\alpha_s}$, and $\pi=\pi(\rho)=J(P,\sigma,\nu)$. On the other hand, by section 1.4\* of \[18\] we know that $J(P,\sigma,\nu)=\tilde{I}(P,\tilde{\sigma},-\nu)$ where $\tilde{}$ denotes the contragredient, and $I(P,\sigma,\nu)$ denotes the unique irreducible subrepresentation of the parabolic induction $\textrm{Ind}_P^G (\sigma\otimes \nu)$ \[5\]. By Langlands-Shahidi method we know the multiplicativity of the local analytic $\gamma$-factors attached to generic representations which appear as subrepresentations of parabolic inductions from irreducible generic representations. We also have the multiplicativity of their corresponding local analytic $L$-factors. Using $J(P,\sigma,\nu)=\tilde{I}(P,\tilde{\sigma},-\nu)$ and the local functional equation $\gamma(s, \pi, \textrm{Sym}^2\otimes\eta,\psi)\gamma(1-s, \tilde{\pi}, \textrm{Sym}^2\otimes\eta^{-1},\overline{\psi})=1$, we obtain the multiplicativity of $\gamma(s,\pi, \textrm{Sym}^2\otimes\eta,\psi)$ and $L(s,\pi, \textrm{Sym}^2\otimes\eta)$ with respect to their quasi-tempered inducing data. Since we already showed the equality of $L$-factors for quasi-tempered case, we finally obtain that $L(s, \pi(\rho), \textrm{Sym}^2\otimes\eta)=L(s, \textrm{Sym}^2\rho\otimes\eta).$ By the symmetry between $\wedge^2$ and $\textrm{Sym}^2$ we also obtain that $L(s, \pi(\rho), \wedge^2\otimes\eta)=L(s, \wedge^2\rho\otimes\eta).$ So far we have successfully reduced the problem to the supercuspidal stability(Proposition 3.4), which will be established in the rest part of this paper. We will start with some preparations in section 5, in which we will obtain a formula of the local coefficients in our case as the Mellin transform of some partial Bessel functions, and relate the partial Bessel functions with partial Bessel integrals. Then we will study the analysis of partial Bessel integrals in section 6 and obtain their asymptotic expansion formulas, generalizing the results in \[7\]. PREPARATIONS FOR SUPERCUSPIDAL STABILITY ======================================== We’ve already seen that the adjoint action $r: \leftidx{^L}M_H\longrightarrow \textrm{GL}(\leftidx{^L}{\mathfrak{n}_H})$ gives the twisted symmetric L- and $\gamma$-factors. Moreover, since $r$ is irreducible we have that the local coefficient $C_{\psi}(s,\pi)=\gamma(s, \pi, \textrm{Sym}^2\otimes \eta,\psi)$(Chapt. 5, \[14\]). So it reduces the proof of Proposition 3.4 to the stability of local coefficients. The local coefficients can be written as the Mellin transform of certain partial Bessel functions under some conditions (Theorem 6.2, \[19\]). In order to study the Mellin transform in our case, we need to understand the following things at first: the structure of $H=\textrm{GSpin}_{2n+1}$, the structure and measure of the orbit space that the partial Bessel function is integrating on, and certain Bruhat decomposition. THE STRUCTURE OF $\textrm{GSpin}_{2n+1}$ ---------------------------------------- Let $H=\textrm{GSpin}_{2n+1}$. We want to understand its structure and its relationship with $H_D=\textrm{Spin}_{2n+1}$ and $\textrm{SO}_{2n+1}$. We have an exact sequence $$1\longrightarrow \mathbb{Z}/2\mathbb{Z}\longrightarrow Spin_{2n+1}\xrightarrow{\varphi} SO_{2n+1}\longrightarrow 1$$ where $\varphi$ is the covering map. We fix the standard Borel subgroup $B=TU$ of $\textrm{SO}_{2n+1}$, and denote the corresponding Borel subgroup of $H$(resp. $H_D$) by $B_H=T_HU_H$(resp. $B_{H_D}=T_{H_D}U_{H_D}$). We see that $U\simeq U_{H_D}\simeq U_{H}$. As in the proof of Proposition 2.4 of \[1\], we start by fixing a basis $f_1,\cdots, f_n$ of the character lattice $X^*(T)$ of $\textrm{SO}_{2n+1}$. The root datum of $SO_{2n+1}$ can be given as follows: $$X^*(T)=\mathbb{Z}f_1\oplus\mathbb{Z}f_2\oplus\cdots\oplus\mathbb{Z}f_n$$ $$\Delta=\{\gamma_1=f_1-f_2, \gamma_2=f_2-f_3,\cdots, \gamma_{n-1}=f_{n-1}-f_n, \gamma_n=f_n\}$$ $$X_*(T)=\mathbb{Z}f_1^*\oplus\mathbb{Z}f_2^*\oplus\cdots\oplus\mathbb{Z}f_n^*$$ $$\Delta^{\vee}=\{\gamma_1^{\vee}=f_1^*-f_2^*,\gamma_2^{\vee}=f_2^*-f_3^*,\cdots, \gamma_{n-1}^{\vee}=f_{n-1}^*-f_n^*,\gamma_n^{\vee}=2f_n^*\}.$$ Then the weight lattice $\textrm{P}_{\textrm{SO}_{2n+1}}=\{\lambda\in X^*(T): \langle\lambda,\gamma^{\vee}\rangle\in \mathbb{Z}, \forall \gamma\in \Phi\}$. If $\langle\Sigma c_if_i,\gamma_i^{\vee}\rangle\in \mathbb{Z}$, for $1\leq i \leq n-1$, this implies that $c_i-c_{i+1}\in\mathbb{Z}$, and if $i=n$, this implies that $2c_n\in \mathbb{Z}$. Therefore $\textrm{P}_{\textrm{SO}_{2n+1}}=\{\Sigma c_if_i: c_i\in \frac{\mathbb{Z}}{2}, c_i-c_j\in \mathbb{Z}\},$ hence equal to the $\mathbb{Z}$-span of $f_1\cdots,f_n,\frac{f_1+f_2\cdots+f_n}{2}$. The group $\textrm{Spin}_{2n+1}$ is the simply connected double cover of $\textrm{SO}_{2n+1}$, hence its character lattice is equal to the root lattice of $\textrm{SO}_{2n+1}$, and its cocharacter lattice is the root lattice of type $C_n$, so we obtain the root datum of $H_D=\textrm{Spin}_{2n+1}$: $$X^*(T_{H_D})=\mathbb{Z}f_1\oplus\mathbb{Z}f_2\oplus\cdots\oplus\mathbb{Z}f_n+\mathbb{Z}\frac{f_1\cdots+f_n}{2}$$ $$\Delta_{H_D}=\{\beta_1=f_1-f_2,\beta_2=f_2-f_3,\cdots, \beta_{n-1}=f_{n-1}-f_n,\beta_n=f_n\}$$ $$X_*(T_{H_D})=\mathbb{Z}\beta_1^{\vee}\oplus\mathbb{Z}\beta_2^{\vee}\oplus\cdots\oplus\mathbb{Z}\beta_n^{\vee}$$ $$\Delta^{\vee}_{H_D}=\{\beta_1^{\vee}=f_1^*-f_2^*,\beta_2^{\vee}=f_2^*-f_3^*,\cdots,\beta_{n-1}^{\vee}=f_{n-1}^*-f_n^*,\beta_n^{\vee}=2f_n^*\}.$$ We can realize $$H=\textrm{GSpin}_{2n+1}=(\textrm{GL}_1\times \textrm{Spin}_{2n+1})/ \{(1,1), (-1,\beta^{\vee}_n(-1))\}.$$ We add another character $f_0$ so that the character lattice of $\textrm{GL}_1\times \textrm{Spin}_{2n+1}$ is spanned by $f_0,f_1,f_2,\cdots,f_n,\frac{f_1+\cdots f_n}{2}$. Taking the ones that are trivial on $(-1,\beta^{\vee}(-1))$, we see that the character lattice of $\textrm{GSpin}_{2n+1}$ is spanned by $e_0=f_0+\frac{f_1+\cdots f_n}{2}, e_1=f_1,e_2=f_2,\cdots,e_n=f_n.$ Taking the dual basis, we have that the cocharacter lattice of $\textrm{GSpin}_{2n+1}$ is spanned by $e_0^*=f_0^*$, $e_1^*=f_1^*+\frac{f_0^*}{2},e_2^*=f_2^*+\frac{f_0^*}{2},\cdots, e_n^*=f_n^*+\frac{f_0^*}{2}$. Therefore thw root datum of $H=\textrm{GSpin}_{2n+1}$ is given by: $$X^*(T_H)=\mathbb{Z}e_0\oplus\mathbb{Z}e_1\oplus\cdots\oplus\mathbb{Z}e_n$$ $$\Delta_H=\{\alpha_1=e_1-e_2,\alpha_2=e_2-e_3,\cdots,\alpha_{n-1}=e_{n-1}-e_n,\alpha_n=e_n\}$$ $$X_*(T_H)=\mathbb{Z}e_0^*\oplus\mathbb{Z}e_1^*\cdots\oplus\mathbb{Z}e_n^*$$ $$\Delta^{\vee}_H=\{\alpha_1^{\vee}=e_1^*-e_2^*,\alpha_2^{\vee}=e_2^*-e_3^*,\cdots,\alpha_{n-1}^{\vee}=e_{n-1}^*-e_n^*,\alpha^*_n=2e_n^*-e_0^*\}.$$ It is easy to see that the three groups share the same root system, and we can identify $\alpha_i=\beta_i=\gamma_i$ for all $1\leq i\leq n$. Take the Siegel Levi $M_H=M_{\theta}$ where $\theta=\Delta-\{\alpha_n\}$. We have $M_H\simeq \textrm{GL}_n\times \textrm{GL}_1$. Accordingly we will have that the Siegel Levi subgroup $M$ of $\textrm{SO}_{2n+1}$ is isomorphic to $\textrm{GL}_n$. Let $M_{H_D}$ be the corresponding Levi subgroup of $\textrm{Spin}_{2n+1}$. In the rest part of this section we will realize $M_{H_D}$ inside $M_H$. It is crucial for the Bruhat decomposition in section 5.3. The covering map $\varphi$ induces a surjective map on the two corresponding Levi subgroups, then we have the following commutative diagram: $$\begin{tikzcd} \textrm{GL}_n\times \textrm{GL}_1\simeq & M_H \arrow[r, two heads,"pr"] & M &\simeq \textrm{GL}_n\\ & M_{H_D}\arrow[u, hook, "j"]\arrow[ru, two heads, "\varphi"] \end{tikzcd}$$ where $j$ is the injection map and $pr$ is the projection of $M_H\simeq\textrm{GL}_n\times \textrm{GL}_1$ onto the $\textrm{GL}_n$-factor. Note that $j$ is induced from the surjective homomorphism of the character groups $X^*(T_H)\twoheadrightarrow X^*(T_{H_D})$ by mapping $e_i$ to $f_i$ for $1\leq i\leq n-1$ and $e_0\mapsto f_0+\frac{f_1+\cdots+f_n}{2}$. Since $\textrm{Spin}_{2n+1}$ is simply connected, any element in its maxmal torus can be uniquely written as $t=\prod_{i=1}^n\beta^{\vee}(x_i)$. Any element in $T_H$ is of the form $\prod_{i=0}^n e_i^*(t_i)$. Hence if $t=\prod_{i=1}^n\beta_i^{\vee}(x_i)\in T_H$, since $\beta_i^{\vee}=\alpha_i^{\vee}$ for all $1\leq i\leq n$, we have $$t=\prod_{i=1}^n\beta_i^{\vee}(x_i)=\prod_{i=1}^n\alpha_i^{\vee}(x_i)=\prod_{i=1}^{n-1}(e_i^*-e_{i+1}^*)(x_i)\cdot (2e_n^*-e_0^*)(x_n)$$$$=e_1^*(x_1)e_2^*(\frac{x_2}{x_1})\cdots e_{n-1}^*(\frac{x_{n-1}}{x_{n-2}})e_n^*(\frac{x_n^2}{x_{n-1}})e_o^*(x_n^{-1}).$$ Therefore the injection $j: T_{H_D}\hookrightarrow T_H\simeq T_n\times T_1$ is given by $\prod_{i=1}^n\beta_i^{\vee}(x_i)\mapsto \prod_{i=1}^n e^*_i(t_i)\mapsto e_1^*(x_1)e_2^*(\frac{x_2}{x_1})\cdots e_{n-1}^*(\frac{x_{n-1}}{x_{n-2}})e_n^*(\frac{x_n^2}{x_{n-1}})e_o^*(x_n^{-1})$ for all $x_i\in \mathbb{G}_m$. On the other hand, the covering map $\varphi$ induces a surjective map $\varphi:M_{H_D}\twoheadrightarrow M$. Since $\textrm{Spin}_{2n+1}$ and $\textrm{SO}_{2n+1}$ share the same roots, $\varphi$ is given by the surjective map $T_{H_D}\twoheadrightarrow T$, hence by the injection $X^*(T)\hookrightarrow X^*(T_{H_D})$, $f_i\mapsto f_i$, $1\leq i\leq n$, and in return by the surjective map $X_*(T_{H_D})\twoheadrightarrow X_*(T)$, $\beta_i^{\vee}\mapsto \gamma_i^{\vee}$, $1\leq i\leq n$. As a result, $T_{H_D}\twoheadrightarrow T$ can be explicitly written as $$\prod_{i=1}^n\beta_i^{\vee}(x_i)\mapsto \prod_{i=1}^n\gamma_i^{\vee}(x_i)=\prod_{i=1}^{n-1}(f_i^*-f_{i+1}^*)(x_i)\cdot (2f_n^*)(x_n)$$ $$=f_1^*(x_1)f_2^*(\frac{x_2}{x_1})\cdots f_{n-1}^*(\frac{x_{n-1}}{x_{n-2}})f_n^*(\frac{x_n^2}{x_{n-1}}).$$ The kernel of this map is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ with generator $\beta_n^{\vee}(-1)$. The above discussion shows that we have a commutative diagram on the corresponding tori: $$\begin{tikzcd} T_n\times T_1 \simeq&T_H \arrow[r, two heads,"pr"] & T_n \\ & T_{H_D} \arrow[u, hook, "j"]\arrow[ru, two heads, "\varphi"] \end{tikzcd}$$ where $T_n$ and $T_1$ are the maximal tori of $\textrm{GL}_n$ and $\textrm{GL}_1$ respectively. Taking the isomorphisms on the root subgroups and Weyl groups of these groups, and using the Bruhat decomposition, we get the commutative diagram of Levi subgroups we discussed earlier. Moreover, from this we can also realize $M_{H_D}\subset M_H\simeq \textrm{GL}_n\times \textrm{GL}_1$ by $$M_{H_D}=\{m(g,a)\in M_H, \det(g)a^2=1\}^\circ,$$ where $\circ$ means taking the connected component. THE SPACE $Z_{M_H}^0U_{M_H}(F)\backslash N_H(F)$, ITS ORBIT REPRESENTATIVES AND MEASURE --------------------------------------------------------------------------------------- The partial Bessel functions that we are going to define will be integrating over this space. We proceed by first work on the space $U_{M_H}(F)\backslash N_H(F)$, then define $Z_{M_H}^0$ and consider its action after that. Let $H=\textrm{GSpin}_{2n+1}$, as an algebraic group defined over $F$. We fix the Borel subgroups $B_H=T_HU_H$, $B=TU$ of $H$ and $\textrm{SO}_{2n+1}$ respectively as in section 5.1. Notice that the Siegel parabolic $P_H=M_HN_H$ of $\textrm{GSpin}_{2n+1}$ share the same unipotent radical $N_H$ with the corresponding parabolic subgroup $P=MN$ of $\textrm{SO}_{2n+1}$. Let $U_{M_H}=U_H\cap M_H$, and $U_M=U\cap N$. We need to study the $U_{M_H}$-action on the $N_H$ by conjugation, both of which lie in the derived group of $H$. We have $U_{M_H}\simeq U_{M}$, and $N_H\simeq M$, and the action of $U_{M_H}$ on $N_H$ in $H=\textrm{GSpin}_{2n+1}$ is compatible with the $U_M$-action on $N$ in $\textrm{SO}_{2n+1}$. Therefore $U_{M_H}\backslash N_H\simeq U\backslash N$. Hence it suffices to study the $U_M$-action on $N$. We realize $\textrm{SO}_{2n+1}$ as $$SO_{2n+1}=\{h\in \textrm{GL}_{2n+1}:\leftidx{^t}h\tilde{J}h=\tilde{J} \},$$ where $\tilde{J}= \begin{bmatrix} \ \ & \ \ & J' \\ \ \ & 1 & \ \ \\ \leftidx{^t}J' & \ \ & \ \ \\ \end{bmatrix}$ and $J'=\begin{bmatrix} \ \ & \ \ & \ \ & 1 \\ \ \ & \ \ & -1 \ \ \\ \ \ & \reflectbox{$\ddots$} & \ \ & \ \ \\ (-1)^{n-1} & \ \ & \ \ & \ \ \\ \end{bmatrix}.$ An easy calculation shows that the $M =\{m=m(g)=\begin{bmatrix} g & \ \ & \ \ \\ \ \ & 1 & \ \ \\ \ \ & \ \ & J'\leftidx{^t}g^{-1}J'^{-1} \\ \end{bmatrix}:g\in \textrm{GL}_n\}$. Consequently $U_{M}=\{ \begin{bmatrix} u & \ \ & \ \ \\ \ \ & 1 & \ \ \\ \ \ & \ \ & J'\leftidx{^t}u^{-1}J'^{-1}\\ \end{bmatrix}: u\in U_n\}$, where $U_n$ is the unipotent radical of the standard Borel subgroup of $\textrm{GL}_n$ consists of upper triangular unipotent matrices. And the unipotent radical of $P=MN$ is $$N=\{n=n(X,\alpha)=\begin{bmatrix} I & \alpha & X\\ \ \ & 1 & -\leftidx{^t}\alpha J' \\ \ \ & \ \ & I \\ \end{bmatrix}: X\leftidx{^t}J'+J' \leftidx{^t}X+\alpha\leftidx{^t}\alpha=0 \ \ (*)\}$$ A simple calculation shows that the conjugate action of $U_{M}(F)$ on $N(F)$ is equivalent to $$X\mapsto uXJ'\leftidx{^t}uJ'^{-1}, \alpha\mapsto u\alpha.\ \ \cdots\cdots (a)$$ Let $Z=X\leftidx{^t}J' + \frac{\alpha\leftidx{^t}\alpha}{2}$, then $(*)\Leftrightarrow Z+\leftidx{^t}Z=0.$ Now $X=(Z-\frac{\alpha\leftidx{^t}\alpha}{2})\leftidx{^t}J'^{-1}=(Z-\frac{\alpha^t\alpha}{2})J'$. So $n=n(Z,\alpha)\in N_H(F)$ is therefore parameterized by $Z\in Sk_n(F),$ the set of skew-symmetric matrices with $F$-coefficients, and $\alpha\in F^n$. The action (a) translates into $$Z\mapsto uZ\leftidx{^t}u, \alpha\mapsto u\alpha. \ \ \cdots\cdots (a'),$$ since if we denote $X'=uXJ'\leftidx{^t}uJ'^{-1}, \alpha'=u\alpha$, then the corresponding $$Z'=X'^tJ'+\frac{\alpha'\leftidx{^t}{\alpha'}}{2}=(uXJ'\leftidx{^t}uJ'^{-1})^tJ'+\frac{u\alpha\leftidx{^t}{\alpha'}\leftidx{^t}u}{2}=u(X^tJ'+\frac{\alpha^t\alpha}{2})^tu=uZ^tu.$$ Now it is equivalent to find the orbit representatives for the action of $U_n(F)$ on $Sk_{n+1}(F)$ because $Sk_n(F)\times F^n\longrightarrow Sk_{n+1}(F)$ defined by $(Z,\alpha)\mapsto \begin{bmatrix}Z & \alpha\\ -^t\alpha & 0 \\ \end{bmatrix}$ is a homeomorphism of p-adic manifolds. If we identify $U_n(F)$ with its image in $U_{n+1}(F)$ by the embedding $u \mapsto \begin{bmatrix} u & \ \ \\ \ \ & 1 \\ \end{bmatrix}$, we also have $\begin{bmatrix} u & \ \ \\ \ \ & 1 \\ \end{bmatrix} \begin{bmatrix} Z & \alpha \\ -\leftidx{^t}\alpha & 0 \end{bmatrix} \begin{bmatrix} \leftidx{^t}u & \ \ \\ \ \ & 1 \\ \end{bmatrix} =\begin{bmatrix} uZ\leftidx{^t}u & u\alpha\\ -\leftidx{^t}(u\alpha) & 0 \\ \end{bmatrix}.$ So it suffices to find orbit representatives of the action of $U_n(F)$ on $Sk_{n+1}(F)$ by $u.\tilde{Z}=\begin{bmatrix} u & \ \ \\ \ \ & 1 \\ \end{bmatrix}\tilde{Z} \begin{bmatrix} \leftidx{^t}u & \ \ \\ \ \ & 1 \\ \end{bmatrix}$ where $u\in U_n(F)$ and $\tilde{Z}\in Sk_{n+1}(F).$ For our concern it suffices to find such orbit representatives for an open dense subset of $N(F)$ under the p-adic topology. We will define this open dense subset inductively. Let’s begin with a few lemmas: If $\varphi: M\rightarrow N$ is a surjective submersion of manifolds, if we have an open dense subset $V\subset N $, then $U=\varphi^{-1}(V)$ is open dense in M. It suffices to show this locally. Thus without loss of generality, assume $M\simeq F^m$ and $N\simeq F^n$ with $m\ge n$, and $\varphi=pr: F^m\rightarrow F^n$ is the projection map. Then if $V$ is dense in $F^n$, we have $\varphi^{-1}(V)=pr^{-1}(V)\simeq V\times F^{m-n}$. So $\overline{\varphi^{-1}(V)}\simeq \overline{V\times F^{m-n}}\simeq \overline{V}\times F^{m-n}\simeq F^n\times F^{m-n}\simeq F^m\simeq M$. Since $\overline{\varphi^{-1}(V)}\subset M$, we have $\overline{\varphi^{-1}(V)}=M$. Let $\varphi_i: Sk_{i+1}(F)\longrightarrow Sk_i(F)$ be defined by $Z=\begin{bmatrix} Z' & \beta \\ -^t\beta & 0 \\ \end{bmatrix}\mapsto u_i Z' {^t}{u_i}$ where $u_i=\begin{bmatrix} I_{i-1} & \gamma \\ 0 & 1 \\ \end{bmatrix}$, $\beta=\begin{bmatrix} \beta' \\ b_i \\ \end{bmatrix}$ with $b_i\neq 0$, $I_{i-1}$ denotes the $(i-1)\times (i-1)$ identity matrix and $\gamma=-b_i^{-1}\beta'$. Then $\varphi_i$ is a surjective submersion of p-adic manifolds. Write $Z'=\begin{bmatrix} Z'' & \alpha' \\ -^t\alpha' & 0 \\ \end{bmatrix}$ with $Z''\in Sk_{i-1}(F)$. Also notice that $u_iZ'^tu_i=\begin{bmatrix} I_{i-1} & \gamma \\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} Z'' & \alpha' \\ -^t\alpha' & 0 \\ \end{bmatrix} \begin{bmatrix} I_{i-1} & 0 \\ ^t\gamma & 1 \\ \end{bmatrix}=\begin{bmatrix} Z''-\gamma^t\alpha'+\alpha'^t\gamma & \alpha'\\ -^t\alpha' & 0 \\ \end{bmatrix}$. The map $$Sk_{i-1}(F)\times F^{i-1}\times F^{i-1}\times F^*\longrightarrow Sk_{i-1}(F)\times F^{i-1}$$ $$(Z'',\alpha',\beta',b_i)\mapsto (Z''-\gamma^t\alpha'+\alpha'^t\gamma, \alpha')$$ is a submersion because the Jacobian of this map contains an $i\times i$ identity matrix, due to that the coefficient of $Z''$ is 1 on both hand sides. The surjectivity is clear by the definition of $\varphi_i$. Denote $V_i=\{Z\in Sk_i(F): z_{i-1,i}\neq 0\}$ and let $$V=\{Z\in Sk_{n+1}(F): \varphi_{n-i}\circ \varphi_{n-i+1}\circ\cdots \circ \varphi_n(Z)\in V_{n-i-1}, \forall 0\leq i \leq n-2 \}$$ where $\varphi_i: Sk_{i+1}(F)\longrightarrow Sk_i(F)$ as in Lemma 5.2, which is a surjective submersion. Then V is open dense in $Sk_{n+1}(F).$ By the previous two lemmas, each $V_i$ is open dense in $Sk_i(F)$. Since the composition of surjective submersions is still a surjective submersion, the topology of $Sk_i(F)\hookrightarrow Sk_{i+1}(F)$ is the induced topology. So the subset $V$, which is defined inductively, is a finite intersection of open dense subsets, therefore open dense. Based on the above discussion, we obtain Let $N(F)'=\{n=\begin{bmatrix} I & \alpha & (Z-\frac{\alpha^t\alpha}{2})J' \\ \ \ & 1 & -^t\alpha J' \\ \ \ & \ \ & I \\ \end{bmatrix}:\begin{bmatrix} Z & \alpha \\ -^t\alpha & 0 \\ \end{bmatrix}\in V \}.$ Then $N(F)'\subset N(F)$ is open dense. Moreover, for $\forall n(Z,\alpha)\in N(F)'$, $\exists u\in U_n(F)$, such that $u\cdot n(Z,\alpha)=n(uZ^tu,u\alpha)$ where $$\begin{bmatrix} uZ{^tu} & u\alpha \\ -^t(u\alpha) & 0 \\ \end{bmatrix}=\begin{bmatrix} 0 & a_1 & \ \ & \ \ & \ \ \\ -a_1 & 0 & \ \ & \ \ & \ \ \\ \ \ & \ \ & \ddots & \ \ & \ \ \\ \ \ & \ \ & \ \ & 0 & a_n \\ \ \ & \ \ & \ \ & -a_n & 0 \\ \end{bmatrix}$$ with $a_i\in F^*$. This gives a set of orbit representative for the adjoint action of $U_{M}(F)\simeq U_n(F)$ on $N(F)'.$ First, by the previous argument, $N(F)'$ is open dense in $N(F)$ under the p-adic topology. Now take $u_n$ as in Lemma 3.3 and write $\tilde{Z}=\begin{bmatrix} Z & \alpha \\ -^t\alpha & 0 \\ \end{bmatrix}$. Then we have $u_nZ^tu_n=\varphi_n(\tilde{Z})\in V_n$ and $u_n\alpha=[0,\cdots,0,a_n]^t$ with $a_n\neq 0$ by the construction of $N(F)'$. Now $u_nZ^tu_n\in V_n\subset Sk_n(F)$, by induction on $n$ we end up with some $u\in U_n(F)$ as stated in the lemma. Let $R$ denote this orbit representatives, as we saw above it is homeomorphic to $(F^*)^n$. So we have a continuous surjective map: $U_n(F)\times R\longrightarrow V$ given by $(u, (a_1,\cdots,a_n))\mapsto \begin{bmatrix} u & \ \ \\ \ \ & 1 \\ \end{bmatrix} \begin{bmatrix} 0 & a_1 & \ \ & \ \ & \ \ \\ -a_1 & 0 & \ \ & \ \ & \ \ \\ \ \ & \ \ & \ddots & \ \ & \ \ \\ \ \ & \ \ & \ \ & 0 & a_n \\ \ \ & \ \ & \ \ & -a_n & 0 \\ \end{bmatrix} \begin{bmatrix} ^tu & \ \ \\ \ \ & 1 \\ \end{bmatrix}$. The map is clearly continuous. It has a inverse. In fact, the inverse map is just given by the process of finding the orbit representatives as we showed above, which is apparently continuous since all maps showed up are again just matrix multiplications. Hence to show it is a homeomorphism, we only need to show that any two matrices of this form lie in different orbits. This follows easily by induction on the size of the matrix. Indeed, suppose $u=\begin{bmatrix} u' & \gamma \\ \ \ & 1 \\ \end{bmatrix}$ and let $\tilde{Z}=\begin{bmatrix} 0 & a_1 & \ \ & \ \ & \ \ \\ -a_1 & 0 & \ \ & \ \ & \ \ \\ \ \ & \ \ & \ddots & \ \ & \ \ \\ \ \ & \ \ & \ \ & 0 & a_n \\ \ \ & \ \ & \ \ & -a_n & 0 \\ \end{bmatrix}=\begin{bmatrix} \tilde{Z}_1 & \alpha \\ -^t\alpha & 0 \\ \end{bmatrix}$ with $\alpha=[0,\cdots,0,a_n]^t$ and $\tilde{Z}_1$ the principal $(n-1)\times(n-1)$ block of $\tilde{Z}$. Now suppose $\tilde{Z}'$ is another such matrix with entries $a_i'$ and $\begin{bmatrix} u & \ \ \\ \ \ & 1 \\ \end{bmatrix}\tilde{Z}\begin{bmatrix} ^tu & \ \ \\ \ \ & 1 \\ \end{bmatrix}=\tilde{Z}',$ and similarly we define $\tilde{Z}'_1$ and $\alpha'$. This implies that $u\alpha=\alpha'$, hence $u$ has to be the form $u=\begin{bmatrix} u' & 0 \\ 0 & 1 \\ \end{bmatrix}$. This gives that $\begin{bmatrix} u' & \ \ \\ \ \ & 1 \\ \end{bmatrix}\tilde{Z}_1\begin{bmatrix} ^tu' & \ \ \\ \ \ & 1 \\ \end{bmatrix}=\tilde{Z}'_1$ where $\tilde{Z}_1$ and $\tilde{Z}'_1$ are of the same form as $\tilde{Z}$ and $\tilde{Z}'$ respectively, but of strictly smaller size, so by induction hypothesis, we derive that $u'=I_{n-1}$, which also means that $u=I$. This forces $\tilde{Z}=\tilde{Z}'$, so $a_i=a'_i$ for $1\leq i \leq n$. Moreover, the action is simple, i.e., if $u\cdot Z=Z$, then $u=I$. To see this, just take $\tilde{Z}'=\tilde{Z}$ in the above argument, and a similar process gives $u=I$. Now we have a homeomorphism $U_{M}(F)\times R\simeq N(F)'\subset N(F)$ with $N(F)'\subset N(F)$ open dense. Recall that we have isomorphism of algebraic groups $U_{M_H}\simeq U_M$, $N_H\simeq N$, given by identifying the corresponding root subgroups. So we obtain homeomorphisms of p-adic manifolds: $U_{M_H}(F)\simeq U_M(F)$ and $N_H(F)\simeq N(F) $. Denote the homeomorphic image of $N(F)'$ in $N_H(F)$ by $N_H(F)'$, then it’s clear that $N_H(F)'\subset N_H(F)$ is also open dense. Moreover, the $U_{M_H}(F)$-action on $N_H(F)$ is compatible with the $U_M(F)$-action on $N(F)$. From now on we identify the p-adic manifolds: $U_{M_H}(F)\simeq U_M(F)$, $N_H(F)\simeq N(F)$, $N_H(F)'\simeq N(F)'$, and $U_{M_H}(F)\backslash N_H(F)\simeq U_M(F)\backslash N(F)$. We also identify $R$ as the orbit space representatives of $U_{M_H}(F)\backslash N_H(F)$. Now let’s discuss the invariant measure on the orbit space. Any measurable function $f$ on $N_H(F)$ can be viewed as a function on $U_{M_H}(F)\times R$. Let $du$ and $dn$ the Haar measure on $U_{M_H}(F)$ and $N_H(F)$ respectively. Let $da$ be the measure on $R$ such that the integration formula $\int_{U_{M_H(F)}}\int_Rf(u\cdot a) du da= \int_{N_H(F)}f(n)dn$ holds. We also need to construct an invariant measure on $R$. When the dimension $n=2$, $U_{M_H}(F)\simeq U_2(F)=\{\begin{bmatrix} 1 & x \\ \ \ & 1 \\ \end{bmatrix}:x\in F\}\simeq F$, $R\simeq \{\begin{bmatrix} 0 & a_1 & 0 \\ -a_1 & 0 & a_2\\ 0 & -a_2 & 0 \end{bmatrix}: a_1, a_2\in F^*\}\simeq (F^*)^2$, and $N_H(F)\simeq \{n(Z,\alpha):Z\in Sk_2(F),\alpha\in F^2\}\simeq F^3$. The action of $U_2(F)$ on $R$ is give by $$\begin{bmatrix} 1 & x & \ \ \\ 0 & 1 & \ \ \\ \ \ & \ \ & 1 \\ \end{bmatrix}\begin{bmatrix} 0 & a_1 & 0 \\ -a_1 & 0 & a_2 \\ 0 & -a_2 & 0 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 & \ \ \\ x & 1 & \ \ \\ \ \ & \ \ & 1 \\ \end{bmatrix}=\begin{bmatrix} 0 & a_1 & a_2x \\ -a_1 & 0 & a_2 \\ -a_2x & a_2 & 0 \\ \end{bmatrix}$$ So $$F\times (F^*)^2\simeq U_{M_H}(F)\times R\longrightarrow N_H(F)\simeq F^3$$ is given by $$(x, a_1, a_2)\mapsto (a_1, a_2x, a_2).$$ So we can write $f(u\cdot a)=f(a_1,a_2x,a_2)$. Let $da=da_1\vert a_2 \vert da_2$, then $$\int_{U_{M_H}(F)}\int_R f(u\cdot a)du da=\int_{(a_1,a_2)\in (F^*)^2}\int_{x\in F}f(a_1, a_2x, a_2)dx da_1\vert a_2 \vert da_2.$$ Let $x'=a_2x, a_1'=a_1,a_2'=a_2$, then $dx'=\vert a_2 \vert dx$. Then the above integral $$=\int_F\int_{(F^*)^2}f(a'_1,x',a_2')\frac{dx'}{\vert a_2'\vert}da_1' \vert a_2'\vert da_2'=\int_F\int_{(F^*)^2}f(a_1',x',a_2')dx'da_1'da_2'$$ $$=\int_{F^3}f(a_1',x',a_2')dx'da_1'da_2'=\int_{N_H(F)}f(n)dn.$$ It is straightforward to show by induction on the dimension $n$ that the invariant measure on the space of orbits $R$ is given by $da=\prod_{i=1}^n\vert a_i\vert^{i-1} da_i=\prod_{i=1}^n\vert a_i \vert^i d^{\times}a_i.$ Next, we define $Z^0_{M_H}$ and consider its action on $U_{M_H}(F)\backslash N_H(F)$. $H=\textrm{GSpin}_{2n+1}$. Let $Z_H$ and $Z_{M_H}$ denote the center of $H$ and $M_H$ respectively, then $Z_H=\{e^*_0(\lambda): \lambda\in GL_1\}$ and $Z_{M_H}=\{e^*_0(\lambda)e^*_1(\mu)\cdots e^*_n(\mu):\lambda,\mu\in GL_1\}$. There exists an injection: $\alpha^{\vee}:F^{\times}\hookrightarrow Z_H\backslash Z_{M_H} $ such that $\alpha(\alpha^{\vee}(t))=t$ for $\forall t\in F^*$. The structure of $Z_H$ and $Z_{M_H}$ follows from Proposition 2.3 of \[2\], for the second part of the lemma, take $\alpha^{\vee}:t\mapsto Z_H(e_1^*(t)\cdots e^*_n(t))$. Then $\alpha^{\vee}$ is an injection, since if $Z_H(e_1^*(t)\cdots e_n^*(t))=Z_H$, then $e^*_1(t)\cdots e_n^*(t)\in Z_H,$ therefore $e^*_1(t)\cdots e_n^*(t)=e^*_0(\lambda)$ for some $\lambda\in GL_1$, but the cocharacters are independent since they form a basis for the cocharacter lattice, it forces $e_1^*(t)=e^*_2(t)=\cdots=e^*_n(t)=e^*_0(\lambda)=1,$ this implies $t=1$. Moreover, since $\alpha=\alpha_n=e_n$, we have $\alpha(\alpha^{\vee}(t))=e_n(e^*_1(t)\cdots e_n^*(t))=e_n(e^*_n(t))=t$. Let $Z_{M_H}^0=\{ \alpha^{\vee}(t):t\in F^*\}$ be the image of the map $\alpha^{\vee}$ we just constructed. For $z=\alpha^{\vee}(t)=\prod_{i=1}^ne^*_i(t)$ and $n(Z,\alpha)\in N_H(F)$ as before it’s easy to see that $$\alpha^{\vee}(t) n(Z,\alpha)\alpha^{\vee}(t)^{-1}= n(t^2Z,t\alpha).$$ Therefore the $Z_{M_H}^0$-action on $N_H(F)$ induces an action $Z_{M_H}^0\times R\longrightarrow R$, given by $(t,(a_1,\cdots, a_n))\mapsto (t^2a_1,\cdots t^2 a_{n-1},t a_n).$ We also need to define a measure on the space of orbits $R'$ of $Z_{M_H}^0U_{M_H}\backslash N_H$ such that it is compatible with the measure on $R$ we constructed. We can take $a_n=1$ to identify $R'$ with $\{(a_1',\cdots,a_{n-1}',1):a_i'\in F^*\}$. By the measure on $R$ we can see that the measure on $R'$ is of this form $da'=\prod_{i=1}^{n-1}\vert a'_i \vert^{k_i}da'_i$ with $k_i\in \mathbb{Z}.$ Recall that $\rho$ is the half of the sum of positive roots in $N_H$, as we computed before $\rho=\frac{n}{2}\sum_{i=1}^n e_i.$ So for $z=\alpha^{\vee}(t)$, we have $q^{\langle 2\rho, H_{M_H}(z)\rangle}=\vert n\sum_{i=1}^n e_i(\prod_{i=1}^n e^*(t))\vert=\vert t\vert^{n^2}$. Then we should have $$\int_Rf(a)da=\int_{Z_{M_H}^0}\int_{R'}f(z\cdot a')q^{\langle 2\rho, H_{M_H}(z) \rangle}da'dz$$ $$=\int_{F^*\times R'}f(t^2a_1',\cdots,t^2a_{n-1}',t)\vert t\vert^{n^2-1}\prod_{i=1}^{n-1}\vert a'_i\vert^{k_i}da'_idt.$$ Let $a_i=t^2a'_i$ for $1\leq i\leq n-1$, and $a_n=t$. Then $da'_i=\vert t\vert^{-2}da_i$ and $da_n=dt$. So the above integral $$=\int_{F^*\times R'}f(a_1,\cdots,a_{n-1},a_n)\vert a_n \vert^{n^2-1}\prod_{i=1}^{n-1}\vert t^{-2}a_i\vert^{k_i}\vert a_n\vert^{-2(n-1)}da_ida_n.$$ On the other hand, we should also have $$\int_{(F^*)^n}f(a_1, \cdots, a_n)\prod_{i=1}^n\vert a_i\vert^{i-1}da_i=\int_{R}f(a)da.$$ By comparing this with the above discussion we can see that it forces each $k_i=i-1$. This means that $$da'=\prod_{i=1}^{n-1}\vert a_i'\vert^{i-1}da'_i$$ gives the desired measure on the space of orbits $R'$ of $Z_{M_H}^0U_{M_H}(F)\backslash N_H(F)$. A BRUHAT DECOMPOSITION ---------------------- Theorem 6.2 of \[19\] allows us to write the local coefficients as the Mellin transform of some partial Bessel functions, whose definitions rely on a Bruhat decomposition. We will study the Bruhat decomposition in this section. As before $H=\textrm{GSpin}_{2n+1}$. Let $w_H$ and $w_{\theta}$ be the long Weyl group element of $H$ and $M_{\theta}=M_H$ respectively. We denote the length of $w$ by $l(w)$. Then $l(w_H)=n^2$ and $l(w_\theta)=\frac{n(n-1)}{2}$, since in general $l(w)$ is the number of positive roots that are mapped to negatives ones by $w$. Their reduced decompositions can be given as follows: $$w_H=w_{\alpha_{n-1}}(w_{\alpha_{n-2}}w_{\alpha_{n-1}})\cdots (w_{\alpha_{2}}\cdots w_{\alpha_{n-1}})(w_{\alpha_1}\cdots w_{\alpha_{n-1}})$$ $$\cdot w_{\alpha_n}(w_{\alpha_{n-1}}w_{\alpha_{n}})\cdots(w_{\alpha_2}\cdots w_{\alpha_n})(w_{\alpha_1}\cdots w_{\alpha_n})$$ and $$w_\theta=w_{\alpha_{n-1}}(w_{\alpha_{n-2}}w_{\alpha_{n-1}})\cdots (w_{\alpha_2}\cdots w_{\alpha_{n-1}})(w_{\alpha_1}\cdots w_{\alpha_{n-1}})$$ In general there is a canonical way to pick the Weyl group representative $\dot{w}$ of $w\in W$ by a given splitting $\{u_{\alpha}: \mathbb{G}_m\rightarrow U_{\alpha}\}_{\alpha\in \Phi^+}$: fix a reduced decomposition $w=\prod_{\alpha}w_{\alpha}$ with each $w_\alpha$ a simple reflection, there is a unique $y_\alpha\in \mathbb{G}_m$ such that $w_{\alpha}(1)w_{-\alpha}(y_\alpha) w_\alpha(1)$ normalizes the maximal torus. For each $w_\alpha$ pick $\dot{w}_\alpha=u_{\alpha}(1)u_{-\alpha}(y_\alpha)u_{\alpha}(1)$ and let $\dot{w}=\prod_{\alpha}\dot{w}_\alpha$. This makes each $\dot{w}_\alpha$ the image of $\begin{bmatrix} \ \ & 1 \\ -1 & \ \ \\ \end{bmatrix}$ under the homomorphism $\textrm{SL}_2\rightarrow H$ attached to the $\mathfrak{sl}_2$-triple $\{X_\alpha, H_\alpha, H_{-\alpha}\}$. One can compute that we should pick $\dot{w}_{\alpha_i}=u_{\alpha_i}(1)u_{-\alpha_i}(-1)u_{\alpha_i}(1)$ for $1\leq i\leq n-1$ and $\dot{w}_{\alpha_n}=u_{\alpha_n}(1)u_{-\alpha_n}(-2)u_{\alpha_n}(1).$ Now we pick $\dot{w}_H$ and $\dot{w}_{\theta}$ as in the above process and let $\dot{w}_0=\dot{w}_H \dot{w}_\theta^{-1}$. Moreover, given $\psi :F\rightarrow \mathbb{C}^*$ a non-trivial additive character, recall that we can define a generic character of $U_H(F)$ by setting $\chi(u)=\psi(\sum_{\alpha\in \Delta}u_{\alpha})$. We can identify $u=m(u',1)\in U_{M_H}(F)\simeq U_n(F)$ with $m(u')\in U_{M}$, where $u'\in U_n$. Then a straightforward calculation shows that the generic character $\chi$ is compatible with the choice of the Weyl group representative $\dot{w}_0$, i.e., we have $\chi(\dot{w}_0u \dot{w}_0^{-1})=\chi(u)$. We will still use $\psi$ to denote the generic character since we want to distinguish it from $\chi$, which we will use to denote the highly ramified character. Let $\overline{N}_H=\dot{w}_H N_H \dot{w}_H^{-1}$. We need to find some open dense subset of $N_H(F)$ such that the Bruhat decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ holds for $n$ lying in this open dense subset, where $m\in M_H$, $n'\in N_H$ and $\bar{n}\in \overline{N}_H$. Observe that in this decomposition $m$ is uniquely determined by $n$. Since $n, n'$ and $\overline{n}$ are all in the derived group $H_D=\textrm{Spin}_{2n+1}$, so is $m$. Instead of doing this directly in $\textrm{Spin}_{2n+1}$(or in $\textrm{GSpin}_{2n+1}$), we first do it in $\textrm{SO}_{2n+1}$. We identify the Weyl group elements in $H=\textrm{GSpin}_{2n+1}$ and $\textrm{SO}_{2n+1}$. A direct computation in $\textrm{SO}_{2n+1}$ shows that we should pick $$\dot{w}_H=\begin{bmatrix} \ \ & \ \ & (-\frac{1}{2})J' \\ \ \ & (-1)^n & \ \ \\ (-2)\cdot \leftidx{^t}J' & \ \ & \ \ \\ \end{bmatrix},\dot{w}_{\theta}=\begin{bmatrix} J' & \ \ & \ \ \\ \ \ & 1 & \ \ \\ \ \ & \ \ & J' \\ \end{bmatrix}.$$ Hence $\dot{w}_0=\dot{w}_H\dot{w}_\theta^{-1}=\begin{bmatrix} \ \ & \ \ & (-\frac{1}{2})I \\ \ \ & (-1)^n & \ \ \\ (-1)^n2I & \ \ & \ \ \\ \end{bmatrix}$. Therefore $$\dot{w}_0^{-1}=\begin{bmatrix} \ \ & \ \ & (-1)^n\frac{1}{2}I \\ \ \ & (-1)^n & \ \ \\ -2I & \ \ & \ \ \\ \end{bmatrix}=\begin{bmatrix} (-\frac{1}{2})I & \ \ & \ \ \\ \ \ & 1 \ \ \\ \ \ & \ \ & -2I \\ \end{bmatrix}\cdot \begin{bmatrix} \ \ & \ \ & (-1)^{n-1}I \\ \ \ & (-1)^n & \ \ \\ I & \ \ & \ \ \\ \end{bmatrix}.$$ Let $\tilde{w}_0^{-1}=\begin{bmatrix} \ \ & \ \ & (-1)^{n-1}I \\ \ \ & (-1)^n & \ \ \\ I & \ \ & \ \ \\ \end{bmatrix}$, then the above formula shows that $$\dot{w}_0^{-1}=m(-\frac{1}{2}I)\tilde{w}_0^{-1}$$ To simplify our computation, let’s first compute the decomposition $\tilde{w}_0^{-1}n=m(g)n'\overline{n}$ in $\textrm{SO}_{2n+1}$. We have $$\tilde{w}_0^{-1}n=\begin{bmatrix} \ \ & \ \ & (-1)^{n-1}I \\ \ \ & (-1)^n & \ \ \\ I & \ \ & \ \ \\ \end{bmatrix} \begin{bmatrix} I & \alpha & X \\ \ \ & 1 & -\leftidx{^t}{\alpha}J' \\ \ \ & \ \ & I \\ \end{bmatrix}= \begin{bmatrix} \ \ & \ \ & (-1)^{n-1}I \\ \ \ & (-1)^n & (-1)^{n-1}\leftidx{^t}{\alpha}J' \\ I & \alpha & X \\ \end{bmatrix}$$ and if we assume $m(g)=\begin{bmatrix} g & \ \ & \ \ \\ \ \ & 1 & \ \ \\ \ \ & \ \ & J'\leftidx{^t}g^{-1}J'^{-1}\\ \end{bmatrix}$ with $g\in GL_n$, $n'=\begin{bmatrix} I & \beta & Y' \\ \ \ & 1 & -\leftidx{^t}{\beta}J' \\ \ \ & \ \ & I \\ \end{bmatrix}$ and $\bar{n}=\begin{bmatrix} I & \ \ & \ \ \\ (-1)^{n}2\leftidx{^t}{\gamma} & 1 & \ \ \\ 4\leftidx{^t}{J'}Z\leftidx{^t}{J'} & (-1)^{n-1}2\leftidx{^t}J'\gamma & I\\ \end{bmatrix}$. Let $\gamma'=-2\gamma$ and $Z'=4Z$, then $$m(g)n'\bar{n}=\begin{bmatrix} g & \ \ & \ \ \\ \ \ & 1 & \ \ \\ \ \ & \ \ & J'\leftidx{^t}g^{-1}J'^{-1}\\ \end{bmatrix} \begin{bmatrix} I-(-1)^n\beta\leftidx{^t}{\gamma'}+Y'\leftidx{^t}J' Z' \leftidx{^t}J' & \beta+(-1)^nY' \leftidx{^t}J'\gamma' & Y' \\ (-1)^{n-1}\leftidx{^t}{\gamma'}-\leftidx{^t}{\beta}Z'\leftidx{^t}J' & 1-(-1)^n\leftidx{^t}{\beta}\gamma' & -\leftidx{^t}{\beta}J' \\ \leftidx{^t}J' Z' \leftidx{^t}J' & (-1)^n\leftidx{^t}J'\gamma' & I \\ \end{bmatrix}$$ $$=\begin{bmatrix} g(I-(-1)^n\beta\leftidx{^t}{\gamma'}+Y'\leftidx{^t}J' Z' \leftidx{^t}J') & g(\beta+(-1)^nY' \leftidx{^t}J'\gamma') & g Y' \\ (-1)^{n-1}\leftidx{^t}{\gamma'}-\leftidx{^t}{\beta}Z'\leftidx{^t}J' & 1-(-1)^n\leftidx{^t}{\beta}\gamma' & -\leftidx{^t}{\beta}J' \\ (-1)^{n-1}J'{^tg^{-1}} Z' {^tJ'} & -J'{^tg^{-1}}\gamma' & J'{^tg^{-1}}J'^{-1} \\ \end{bmatrix}.$$ Assume that $\det(X)\neq 0$, then the equality $\tilde{w}_0^{-1}n=m(g)n'\bar{n}$ in our case is equivalent to the following conditions: \(1) $I-(-1)^n\beta{^t\gamma'}+Y'{^tJ'}Z'{^tJ'}=0$; (2) $\beta+(-1)^nY'{^tJ'}\gamma'=0;$ (3) $gY'=I;$ (4)$(-1)^{n-1}{^t\gamma'}-{^t\beta}Z'{^tJ'}=0;$ (5) $1-(-1)^n{^t\beta}\gamma'=(-1)^n;$ (6) $(-1)^{n-1}{^t\alpha}J'=-{^t\beta}J';$ (7) $(-1)^{n-1}J'{^tg^{-1}}Z'{^tJ'}=I;$ (8) $-J'{^tg^{-1}}\gamma'=\alpha;$ (9) $J'{^tg^{-1}}J'^{-1}=X.$ We also recall that by the definition of $N_H(F)$, we also have \(i) $X{^tJ'}+J'{^tX}+\alpha{^t\alpha}=0\Longleftrightarrow {^tJ'}X+{^tX}J'+{^tJ'}\alpha{^t\alpha}J'=0$ We need to simplify this first. Note that $(9)\Longleftrightarrow g=J'{^tX^{-1}}J'^{-1};$ $(6)\Longleftrightarrow \beta=(-1)^n\alpha$; $(7)\Longleftrightarrow Z'={^tJ'}X^{-1}J'^{-1};$ $(8)\Longleftrightarrow \gamma'=-{^tJ'}X^{-1}\alpha;$ $(3)\Longleftrightarrow Y'=g^{-1}=J'{^tX}J'^{-1}.$ Next, we have $(5)\Longleftrightarrow (-1)^n-{^t\beta}\gamma'=1\Longleftrightarrow {^t\beta}\gamma'=(-1)^n-1\Longleftrightarrow {^t\alpha}{^tJ'}X^{-1}\alpha=(-1)^n-1$ We call this formula (ii). Also we have $(4)\Longleftrightarrow (-1)^{n-1}\gamma'-J'{^tZ'}\beta=0\Longleftrightarrow (-1)^{n-1}(-{^tJ'}X^{-1}\alpha)-J'({^tJ'^{-1}}{^tX^{-1}}J')(-1)^n\alpha=0 \Longleftrightarrow {^tJ'}X^{-1}\alpha-J'J'{^tX^{-1}}J'\alpha=0\Longleftrightarrow ({^tJ'}X-(-1)^{n-1}{^tX}J')X^{-1}\alpha=0$. We call the last formula (4’). Also notice that $(2)\Longleftrightarrow (-1)^n\alpha+(-1)^n(J'{^tXJ'^{-1}}){^tJ'}(-{^tJ'}X^{-1}\alpha)=0\Longleftrightarrow \alpha+J'{^tX}(-1)^{n-1}(-{^tJ'}X^{-1}\alpha)=0\Longleftrightarrow \alpha-(-1)^{n-1}J'{^tX}{^tJ'}X^{-1}\alpha=0\Longleftrightarrow \alpha-J'^{-1}{^tX}{^tJ'}X^{-1}\alpha=0\Longleftrightarrow {^tX^{-1}}J'\alpha-{^tJ'}X^{-1}\alpha=0\Longleftrightarrow ({^tX^{-1}}J'-{^tJ'}X^{-1})\alpha=0\Longleftrightarrow ({^tJ'}X-(-1)^{n-1}{^tX}J')X^{-1}\alpha=0 \Longleftrightarrow (4')$. So $(2)\Longleftrightarrow (4')\Longleftrightarrow (4).$ Next we show that $(i)+(ii)\Longrightarrow (4').$ Notice that $(i)\Longleftrightarrow{^tJ'}X+{^tX}J'+{^tJ'}\alpha{^t\alpha}J'=0\Longleftrightarrow {^tJ'}X+{^tX}J'+(-1)^{n-1}J'\alpha{^t\alpha}J'=0\Longleftrightarrow {^tJ'}X+{^tX}J'+J'\alpha{^t\alpha}{^tJ'}=0,$ multiply this by $X^{-1}\alpha$ we obtain ${^tJ'}\alpha+{^tX}J'X^{-1}\alpha+J'\alpha((-1)^n-1)=0$. When $n$ is even, this is equal to ${^tJ'}\alpha+{^tX}J'X^{-1}\alpha=0,$ on the other hand in this case we have $(4')\Longleftrightarrow ({^tJ'}X+{^tXJ'})X^{-1}\alpha=0\Longleftrightarrow {^tJ'}\alpha+{^tX}J'X^{-1}\alpha=0;$ When $n$ is odd, this is saying that ${^tJ'}\alpha+{^tX}J'X^{-1}\alpha-2J'\alpha=0$, but since ${^tJ'}=(-1)^{n-1}J'=J'$ in this case, we have that this is the same as saying ${^tJ'}\alpha-{^tX}J'X^{-1}\alpha=0,$ while $(4')\Longleftrightarrow ({^tJ'}X-{^tX}J')X^{-1}\alpha=0\Longleftrightarrow {^tJ'}\alpha-{^tX}J'X^{-1}\alpha=0.$ Hence in both cases we have that $(i)+(ii)\Longrightarrow (4')$, and this is the same as saying that $(5)+(i)\Longleftrightarrow (i)+(ii)\Longrightarrow (2)\&(4).$ So we obtain that $(1)+(2)+\cdots+(9)+(i)\Longleftrightarrow (i)+(ii)+(1).$ We are left with (1). We have $(1)\Longleftrightarrow I-\alpha(-{^t\alpha}{^tX^{-1}}J')+({^tJ'}{^tX}J'^{-1})$ $\cdot{^tJ'}({^tJ'}X^{-1}J'^{-1}){^tJ'}=0 \Longleftrightarrow I+\alpha{^t\alpha}{^tX^{-1}}J'+J'{^tX}J'X^{-1}=0,$ we call the last formula $(iii)$. We show that if we pick $n\in N_H(F)'$, the open dense subset of $N_H(F)$ constructed in the last section, then both $(ii)$ and $(iii)$ are implied by $(i)$. If we let $Y=X{^tJ'}$ and $Z=X{^tJ'}+\frac{\alpha{^t\alpha}}{2}=Y+\frac{\alpha{^t\alpha}}{2}$ as in the previous section in which we find orbit representatives for $U_{M_H}(F)\backslash N_H(F)$, then there exists $u\in U_n(F)$ such that $uZ{^tu}=\begin{bmatrix} 0 & a_1 & \ \ & \ \ & \ \ \\ -a_1 & 0 & \ \ & \ \ & \ \ \\ \ \ & \ \ & \ddots & \ \ & \ \ \\ \ \ & \ \ & \ \ & 0 & a_{n-1} \\ \ \ & \ \ & \ \ & -a_{n-1} & 0 \\ \end{bmatrix}$, we denote this matrix by $Z(a_1,\cdots, a_{n-1})$, and we also have $u\alpha=[0,\cdots,0,a_n]^t$, hence $uY{^tu}=\begin{bmatrix} 0 & a_1 & \ \ & \ \ & \ \ \\ -a_1 & 0 & \ \ & \ \ & \ \ \\ \ \ & \ \ & \ddots & \ \ & \ \ \\ \ \ & \ \ & \ \ & 0 & a_{n-1} \\ \ \ & \ \ & \ \ & -a_{n-1} & -\frac{a_n^2}{2} \\\end{bmatrix}$, we denote this matrix by $Y(a_1,\cdots, a_n)$. Then we see that $(i)\Longleftrightarrow Y+{^tY}+\alpha{^t\alpha}=0\Longleftrightarrow u(Y+{^tY}+\alpha{^t\alpha}){^tu}=0\Longleftrightarrow uY{^tu} +{^t(uY{^tu})}+(u\alpha){^t(u\alpha)}=0;$ $(ii)\Longleftrightarrow {^t\alpha}Y^{-1}\alpha=-1-(-1)^{n-1}\Longleftrightarrow {^t(u\alpha)}(uY{^tu})^{-1}(u\alpha)=-1-(-1)^{n-1};$ $(iii)\Longleftrightarrow I+(-1)^{n-1}\alpha{^t\alpha}{^tY^{-1}}+{^tY}Y^{-1}=0\Longleftrightarrow u(I+(-1)^{n-1}\alpha{^t\alpha}{^tY^{-1}}+{^tY}Y^{-1})u^{-1}=0\Longleftrightarrow I+(-1)^{n-1}(u\alpha){^t(u\alpha)}{^t(uY{^tu})^{-1}}+{^t(uY{^tu})}(uY{^tu})^{-1}=0$. Therefore, without loss of generality, we can assume that $Y=Y(a_1,\cdots,a_n)$ and $\alpha=[0,\cdots,0,a_n]^t$ with all $a_i\neq 0$ in this proof. We work on the cases when the size of the matrix $n$ is even or odd separately. Case 1: When $n$ is even; Now we have that ${^tJ'}=J'^{-1}=(-1)^{n-1}J'=-J'.$ So $(ii)\Longleftrightarrow {^t\alpha}Y^{-1}\alpha=0,$ notice that $\alpha$ is a vector with only the last entry non-zero, so only the last entry in $Y^{-1}$ contributes. Let $Y_{i,j}^*$ denote the $(i,j)$-th entry of the adjoint matrix of $Y$. Then we see that ${^t\alpha}Y^{-1}\alpha=a_n^2(\det Y^{-1})Y_{n,n}^*$. But since $n$ is even and therfore the $(n,n)$-th minor of $Y$ is an $(n-1)\times (n-1)$ skew-symmetric matrix of odd size, thus $Y_{n,n}^*=0$, hence ${^t\alpha} Y^{-1}\alpha=0;$ And we also have that $(iii)\Longleftrightarrow I-\alpha{^t\alpha}{^tY^{-1}}+{^tY}Y^{-1}=0\Longleftrightarrow I-Y^{-1}\alpha{^t\alpha}+{^tY^{-1}}Y=0.$ But $(i)\Longleftrightarrow Y+{^tY}+\alpha{^t\alpha}=0 \Longleftrightarrow {^tY^{-1}}Y+I+{^tY^{-1}}\alpha{^t\alpha}=0$, so if we replace ${^tY^{-1}}Y$ by $-I-{^tY^{-1}}\alpha{^t\alpha}$ in the last formula for (iii) right above, then we have $(iii)\Longleftrightarrow (Y^{-1}+{^tY^{-1}})\alpha{^t\alpha}=0$. But now $\alpha{^t\alpha}$ is a matrix with only the last entry non-zero and equals $a_n^2$, so only the last column of $Y^{-1}+{^tY}^{-1}$ contribute. For the same reason we have that $Y_{n,n}^*={^tY_{n,n}^*}=0$. On the other hand, for the matrix $Y$, we see that $Y_{i,j}=-Y_{j,i}$ for all $(i,j)\neq (n,n)$, so we see that ${^tY_{i,n}^*}=(-1)^{n-1}Y_{i,n}^*=-Y_{i,n}^*$ for all $1\leq i\leq (n-1)$. This implies that $(Y^{-1}+{^tY^{-1}})\alpha{^t\alpha}=0.$ Case 2: When $n$ is odd. Now $(ii)\Longleftrightarrow {^t\alpha}Y^{-1}\alpha=-2$. We see that $Y_{n,n}^*=\det Y_{n-1}$ where $Y_{n-1}$ is the principal $(n-1)$-th minor of $Y$, therefore one can easily prove by induction that $\det Y_{n-1}=\prod_{k\ \ odd,k\neq n}a_k^2$ but on the other hand $\det Y=-\frac{1}{2}\prod_{k\ \ odd}a_k^2$, which can also be proved by induction on the size. Therefore we have ${^t\alpha}Y^{-1}\alpha=(\det Y)^{-1}Y_{n,n}^*a_n^2=\frac{\prod_{k\ \ odd,k\neq n}a_k^2}{-\frac{1}{2}\prod_{k\ \ odd}a_k^2}\cdot a_n^2=-2.$ We also have $(iii)\Longleftrightarrow I+\alpha{^t\alpha}{^tY^{-1}}+{^tY}Y^{-1}=0\Longleftrightarrow I+Y^{-1}\alpha{^t\alpha}+{^tY^{-1}}Y=0$. Again by $(i)$ we have ${^tY^{-1}}Y=-I-{^tY^{-1}}\alpha{^t\alpha}$, so $(iii)\Longleftrightarrow (Y-{^tY^{-1}})\alpha{^t\alpha}=0$. But in this case $Y^*_{n,n}={^tY^*_{n,n}}=\prod_{k\ \ odd,k\neq n}a_k^2$, and ${^tY_{i,n}^*}=(-1)^{n-1}Y_{i,n}^*=Y_{i,n}^*$, therefore it shows that $(Y-{^tY^{-1}})\alpha{^t\alpha}=0.$ From the above argument we see that in both cases if we pick $n=n(X,\alpha)\in N_H(F)'$, with $\det X\neq 0$ then $(i)\Longleftrightarrow (i)+(ii)+(iii)\Longleftrightarrow (i)+(1)+\cdots +(9).$ We have showed that for $n=n(X,\alpha)\in N_H(F)$, assume $\det(X)\neq 0$, then $\tilde{w}_0^{-1}n(X,\alpha)=m(J'\leftidx{^t}Y^{-1})n'\overline{n}$. Since $\dot{w}_0^{-1}=m(-\frac{1}{2}I)\tilde{w}_0^{-1}$, we see that $\dot{w}_0^{-1}n=m(-\frac{1}{2}I)m(J'\leftidx{^t}Y^{-1})n'\bar{n}=m(-\frac{1}{2}J'\leftidx{^t}Y^{-1})n'\overline{n}$ holds for $n\in N_H(F)'$, which already implies that $\det{X}\neq 0$ since $X=Y\leftidx{^t}J'^{-1}=YJ'$, and $\det(Y)=\det(Y(a_1,\cdots,a_n))\neq 0$. This gives the decomposition in $SO_{2n+1}$. The decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ in $\textrm{SO}_{2n+1}$ and $\textrm{Spin}_{2n+1}$ differ only by the $m$ part. Recall that at the end of section 5.1 we have $M_{H_D}=\{m(g,a)\in M_H\simeq\textrm{GL}_n\times \textrm{GL}_1: \det(g)a^2=1\}^\circ$, and the covering map $\varphi:M_{H_D}\rightarrow M\simeq \textrm{GL}_n$ is given by $m(g,a)\mapsto m(g)\mapsto g$. So for $n\in N_H(F)'$, we see that $\dot{w}_0^{-1}n=m(g,a(g) )n'\overline{n}$, where $g=(-\frac{1}{2})J'\leftidx{^t}Y^{-1}$, and $a(g)$ is uniquely determined by the relation $\det(g)\cdot a(g)^2=1$, since from the realization of $M_{H_D}$ in $M_H$ the $F$-points of $M_{H_D}$ is given by a pair $(g,a)\in \textrm{GL}_n(F)\times \textrm{GL}_1(F)$ such that $\det(g)=a^{-2}$ is a square in $F^\times$ and this $a$ is the unique square root of $\det(g)$ that lies in the identity component of the $F$-points of the variety $\{(g,a)\in \textrm{GL}_n\times \textrm{GL}_1: \det(g)a^2=1\}$. If $Y=Y(a_1\cdots, a_n)$, we can see that $\det(g)=\det((-\frac{1}{2})J' \leftidx{^t}Y(a_1,\cdots,a_n)^{-1})=\frac{(-\frac{1}{2})^n}{\prod_{k\ \ odd}a_k^2}$ if $n$ is even, and $(-\frac{1}{2})^{n}\cdot\frac{-2}{\prod_{k\ \ odd}a_k^2}=\frac{(-\frac{1}{2})^{n-1}}{\prod_{k\ \ odd}a_k^2}$ if $n$ is odd. Hence $a(g)=\frac{(\frac{1}{2})^{\frac{n}{2}}}{\prod_{k\ \ odd}a_k}$ if $n$ is even, and $\frac{(\frac{1}{2})^{\frac{n-1}{2}}}{\prod_{k\ \ odd}a_k}$ if $n$ is odd. So we obtain the desired Bruhat decomposition in $\textrm{Spin}_{2n+1}$ and therefore in $H=\textrm{GSpin}_{2n+1}$. LOCAL COEFFICIENTS AND PARTIAL BESSEL FUNCTIONS ----------------------------------------------- Now we are ready to apply Theoerem 6.2 of \[19\] to express the local coefficients as the Mellin transform of partial Bessel functions in our setting. Recall that we have an injection $\alpha^{\vee}:F^*\hookrightarrow Z_H\backslash Z_{M_H}$ and $\alpha(\alpha^{\vee}(t))=t$ for $t\in F^*$(Lemma 5.5). By the last section we also obtained that the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ holds for $n\in N_H(F)'\subset N_H(F)$. Moreover, by the work of R. Sundaravaradhan in \[22\], we have that except for a set of measure zero on $N_H(F)$, $U_{M_H,n}=U_{M_H,m}'$, where $U_{M_H,n}=\{u\in U_{M_H}: unu^{-1}=n\},$ and $U_{M_H,m}'=\{u\in U_{M_H}: mum^{-1}\in U_{M_H}\ \ \& \ \ \chi(mum^{-1})=\chi(u)\}$. The above two properties imply that the assumptions for Theorem 6.2 in \[19\] are satisfied. Let $\pi$ be a $\psi$-generic representation of $\textrm{GL}_n(F)$ and $\eta$ a character of $F^\times$, and $\lambda$ be a Whittaker functional attached to $\pi$. Since $U_{M_H}\simeq U_n$, $\psi$ can be viewed as a character of $U_{M_H}$. The representation $\sigma_\eta$ of $M_H(F)$ is also generic. Since $\psi(u)\lambda(v)=\lambda(\pi(u)v)=\lambda(\sigma_\eta(m(u,1)v))$, $\lambda$ can also be viewed as a Whittaker functional of $\sigma_\eta$. Let $ \mathfrak{a}^*_{H,\mathbb{C}}=\mathfrak{a}_H^*\otimes_{\mathbb{R}}\mathbb{C}$, where $\mathfrak{a}_H^*=X(M_H)_F\otimes_{\mathbb Z}\mathbb{R}$, and $\mathfrak{a}_H=\operatorname{Hom}(X(M_H)_F,\mathbb{R})$ is the real Lie algebra. The Harish-Chandra map $H_{M_H}:M_H\longrightarrow \mathfrak{a}_H$ is defined by $q^{\langle \chi, H_{M_H}(m)\rangle}=\vert \chi(m)\vert_F$ for all $\chi \in X(M_H)_F.$ Given $\mu\in \mathfrak{a}_{H,\mathbb{C}}^*$, let $I(\mu, \sigma_\eta)=\operatorname{Ind}_{M_HN_H}^H((\sigma_\eta\otimes q^{\langle \mu, H_{M_H}(\cdot)})\otimes 1_{N_H})$ be the induced representation, and denotes its space by $V(\mu,\sigma_\eta).$ As before let $\sigma_{\eta,s}$ denote the representation $\sigma_\eta \otimes q^{\langle s\hat{\alpha}, H_{M_H}(\cdot)\rangle}$, where $\hat{\alpha}=\langle \rho, \alpha\rangle ^{-1}\rho=\frac{(\alpha,\alpha)}{2(\rho,\alpha)}\rho= \frac{(e_n,e_n)}{2\cdot\frac{n}{2}(\sum_{i=1}^ne_i,e_n)}$ $\cdot(\frac{n}{2}\sum_{i=1}^ne_i)=\frac{1}{2}\sum_{i=1}^ne_i.$ For $s\in \mathbb{C}$, define $I(s,\sigma_\eta)=I(s\hat{\alpha},\sigma_\eta)$ and let $V(s,\sigma_\eta)$ be its space. The local standard intertwining operator $A(s,\sigma_\eta):I(s,\sigma_\eta)\longrightarrow I(-s,w_0(\sigma_\eta))$ is defined by $A(s.\sigma_\eta)f(h)=\int_{N_H}f(\dot{w}_0^{-1}nh)dn$ for $\forall h\in H$ and $f\in V(s,\sigma_\eta)$. We identify $\lambda$ as a Whittaker functional for $\sigma_\eta$, and denote $\lambda_{\psi}(s,\sigma_\eta)$ the Whittaker functional for $I(s,\sigma_\eta)$ given by $\lambda$, defined as $\lambda_{\psi}(s,\sigma_\eta)(f)=\int_{N_H}\langle f(\dot{w}_0^{-1}n),\lambda\rangle\cdot \psi^{-1}(n)dn.$ Then since $\psi$ is compatible with $\dot{w}_0$, $\lambda_{\psi}(-s,w_0(\sigma_\eta))\circ A(s,\sigma_\eta)$ defines another Whittaker functional for $I(s,\sigma_\eta)$. So by uniqueness of the local Whittaker functionals we obtain that the local coefficient $C_{\psi}(s,\sigma_\eta)$ is defined by $\lambda_{\psi}(s,\sigma_\eta)=C_{\psi}(s,\sigma_\eta)\cdot \lambda_{\psi}(-s, w_0(\sigma_\eta))\circ A(s,\sigma_\eta).$ As in \[19\] we will choose $\overline{N}_0\subset \overline{N}_H(F)$ to be open compact so that $\alpha^{\vee}(t)\overline{N}_0\alpha^{\vee}(t)^{-1}$ depends only on $\vert t\vert$ for all $t\in F^*$. Define $\varphi_{\kappa}(X)=1$ if $\vert X_{i,j}\vert\leq q^{(i+j-1)\kappa}$ , and $0$ otherwise. From the calculation of the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ in the last section we see that if $n=n(X,\alpha)$ with $\det(X)\neq 0$, then $\bar{n}=\begin{bmatrix} I & \ \ & \ \ \\ -^t(J'X^{-1}\alpha) & 1 & \ \ \\ X^{-1} & X^{-1}\alpha & I \\ \end{bmatrix}$, we denote $\begin{bmatrix} I & \ \ & \ \ \\ -^t(J'\tilde{X}\alpha) & 1 & \ \ \\ \tilde{X} & \tilde{X}\alpha & I \\ \end{bmatrix}$ by $\bar{n}(\tilde{X},\alpha)$. Let $$\overline{N}_{0,\kappa}=\{\bar{n}=\bar{n}(\tilde{X},\alpha): \varphi_{\kappa}(-\frac{1}{8}\varpi^{2(d+f)}\cdot\leftidx{^t}{\tilde{X}}{J'}^{-1})=1\},$$ where $d$ is the conductor of $\chi$ and $f$ is the conductor of $w_{\pi}^{-1}(w_0w_{\pi})$. And let $\varphi_{\overline{N}_0,\kappa}$ be the characteristic function of $\overline{N}_{0,\kappa}$. Let $n\in N_H(F)'$ with $w_0^{-1}n=mn'\bar{n}$, and let $z\in Z_M^0=\{\alpha^{\vee}(t): t\in F^*\}$. As in (6.21) of \[19\], the partial Bessel function on $M_H(F)\times Z^0_{M_H}$ is defined by $$j_{\sigma_{\eta,s},\kappa}(m,z)=\int_{U_{M_H}} W_{\sigma_{\eta,s},v}(mu^{-1})\varphi_{\overline{N}_{0,\kappa}}(zu^{-1}\bar{n}uz^{-1})\psi^{-1}(u)du$$ where $W_{\sigma_{\eta,s},v}\in W(\sigma_{\eta,s})$ is a Whittaker model attached to the $\sigma_{\eta,s}$, with $v$ a fixed vector in the represenattion space. For partial Bessel functions for quasi-split groups, we refer the reader to \[6\]. In our case $m=m(g,a(g))$ with $\det(g)a(g)^2=1$, and $u=m(u',1)$ for $u'\in U_n$. Hence $W_{\sigma_{\eta,s},v}(m(g,a(g)))=\lambda(\sigma_{\eta,s}(m(g,a(g)))v)=\eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}}\lambda(\pi(g)v)=\eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}}W_{\pi,v}(g).$ Moreover, let $z=\alpha^{\vee}(\varpi^{d+f}u_{\alpha_n}(\dot{w}_0\bar{n}\dot{w}_0^{-1}))$, and define for $g\in \textrm{GL}_n(F)$, $$j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)=j_{\sigma_{\eta,s},\kappa}(m,\alpha^{\vee}(\varpi^{d+f}u_{\alpha_n}(\dot{w}_0\bar{n}\dot{w}_0^{-1}))),$$ where $m=m(g,a(g)).$ This defines the partial Bessel function on $\textrm{GL}_n(F)$ in our case. Now apply Theorem 6.2 in \[19\], we obtain Let $\pi$ be an irreducible admissible $\psi$-generic representation of $\textrm{GL}_n(F)$, lifted as a $\psi$-generic representation $\sigma$ of $M_H(F)\simeq \textrm{GL}_n(F)\times \textrm{GL}_1(F)$ by pull-back through the projection on the $\textrm{GL}_n$-factor. $\eta:F^\times\rightarrow \mathbb{C}^{\times}$ is a fixed continuous character. Define the representation $\sigma_\eta$ as before. Suppose that $\omega_{\sigma_\eta}(w_0\omega_{\sigma_\eta}^{-1})$ is ramified as a character of $F^{\times}$. Then for all sufficiently large $\kappa$ we have $$C_{\psi}(s,\sigma_\eta)^{-1}=\gamma(2\langle \hat{\alpha}, \alpha^{\vee}\rangle)s, \omega_{\sigma_\eta}(w_0w_{\sigma_\eta}^{-1})\circ \alpha^{\vee},\psi)^{-1}$$ $$\cdot\int_{Z_{M_H}^0 U_{M_H}\backslash N_H} j_{\pi,\eta,\dot{w}_\theta, \kappa}(g)\omega_{\sigma_{\eta,s}}^{-1}(\alpha^{\vee}(u_n))(w_0\omega_{\sigma_{\eta,s}})(\alpha^{\vee}(u_n))q^{\langle s \hat{\alpha}+\rho,H_M(m)\rangle}d\dot{n}$$ where off a set of measure zero, the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ holds as in the previous section. Here $u_n=u_{\alpha_n}(\dot{w}_0\bar{n}\dot{w}^{-1}_0)\in U_{\alpha}$, $\gamma(2\langle \hat{\alpha}, \alpha^{\vee}\rangle s, \omega_{\sigma_\eta}(w_0\omega_{\sigma_\eta}^{-1})\circ \alpha^{\vee},\psi)$ is an abelian $\gamma$-factor depending only on $\omega_{\pi}$ and $\eta$. Let’s simplify this formula. First recall that in our case $\alpha=e_n$, $\rho$ is the half of the sum of roots in $N_H$. The roots in $N_H$ are $e_i+e_j(1\leq i<j\leq n)$ and $e_i(1\leq i\leq n)$, so $\rho=\frac{1}{2}(\sum_{1\leq i<j\leq n}(e_i+e_j)+\sum_{i=1}^n e_i)=\frac{n}{2}\sum_{i=1}^n e_i$. We have $$\langle \rho, \alpha\rangle=\frac{2(\rho,\alpha)}{(\alpha,\alpha)}=\frac{2(\frac{n}{2}\sum_{i=1}^n e_i, e_n)}{(e_n,e_n)}=n.$$ So $\hat{\alpha}=\langle \rho, \alpha\rangle ^{-1}\rho=n^{-1}(\frac{n}{2}\sum_{i=1}^ne_i)=\frac{1}{2}\sum_{i=1}^ne_i$. Since $\alpha^{\vee}=\sum_{i=1}^ne_i^*,$ so we have for $\forall t\in F^*$, $t^{\langle \hat{\alpha}, \alpha^{\vee}\rangle}=\hat{\alpha}(\alpha^{\vee}(t))=\frac{1}{2}\sum_{i=1}^ne_i(\prod_{i=1}^ne_i^*(t))=t^{n/2}.$ Therefore $\langle \hat{\alpha},\alpha^{\vee}\rangle=\frac{n}{2}.$ This implies that $q^{\langle s\hat{\alpha}, H_{M_H}(m)}=q^{\langle s\hat{\alpha}, H_{M_H}(m(g,a(g)))}=\vert \det(g)\vert^{s/2}.$ Then $\omega_{\sigma_{\eta,s}}(m(g,a(g)))$ $=\omega_{\sigma_{\eta}}(m(g,a(g)))\vert \det g\vert^{s/2}=\eta^{-1}(a(g))\vert \det(g)\vert^{\frac{s}{2}}\omega_\pi(g).$ Secondly, since we have $w_0=w_H\cdot w_{\theta}$, where $\theta=\Delta-\{\alpha_n\}=\Delta-\{\alpha\},$ and $w_H: e_i\mapsto -e_i$, $w_{\theta}: e_i\mapsto e_{n+1-i}$, we obtain $w_0^{-1}\cdot\prod_{i=1}^ne_i^*(t)\cdot w_0=\prod_{i=1}^n(-e^*_{n+1-i}(t))=\prod_{i=1}^n(-e^*_i(t)).$ This implies that $$\omega_{\sigma_\eta}(w_0\omega_{\sigma_\eta}^{-1})(\alpha^{\vee}(t))=\omega_{\sigma_\eta}(\prod_{i=1}^ne_i^*(t))\cdot \omega_{\sigma_\eta}^{-1}(w_0^{-1}\cdot \prod_{i=1}^ne_i^*(t)\cdot w_0)=$$$$\omega_{\sigma_\eta}(\prod_{i=1}^ne^*_i(t))\cdot \omega_{\sigma_\eta}^{-1}(\prod_{i=1}^n(-e^*_i(t)))=\omega_{\sigma_\eta}(\prod_{i=1}^ne^*_i(t))\cdot \omega_{\sigma_\eta}(\prod_{i=1}^ne^*_i(t))=\omega_{\sigma_\eta}^2(\alpha^{\vee}(t))=\omega_{\pi}^2(t),$$ since $\eta$ is trivial on the $\textrm{GL}_n$-component of $M_H$. So $\omega_{\pi}(w_0 \omega_{\pi}^{-1})\circ \alpha^{\vee}=\omega_{\pi}^2.$ Similarly $$\omega_{\sigma_{\eta,s}}^{-1}(w_0 \omega_{\sigma_{\eta,s}})(\alpha^{\vee}(t))=\omega_{\sigma_{\eta,s}}^{-1}(\prod_{i=1}^ne^*_i(t))\cdot \omega_{\sigma_{\eta_s}}(\prod_{i=1}^n(-e^*_i(t)))=\omega_{\sigma_{\eta_s}}^{-2}(\prod_{i=1}^ne^*_i(t))$$$$=\omega_{\pi}^{-2}(\alpha^{\vee}(t))\cdot \vert t^{n}\vert^{-(s/2)\cdot 2}\cdot =\omega_{\pi}^{-2}(t)\cdot \vert t\vert ^{-ns}.$$ So $\omega_{\sigma_{\eta,s}}^{-1}(w_0\omega_{\sigma_{\eta,s}})\circ \alpha^{\vee}=\omega_{\pi}^{-2}(\cdot) \vert \cdot \vert ^{-ns}.$ Finally $$q^{\langle s\hat{\alpha}+\rho,H_{M_H}(m)\rangle}=\vert(\frac{s}{2}\sum_{i=1}^ne_i+\frac{n}{2}\sum_{i=1}^ne_i)(m(g,a(g)))\vert=$$ $$\vert \frac{(s+n)}{2}\sum_{i=1}^ne_i(m(g,a(g))))\vert=\vert\det(g)\vert^{\frac{s+n}{2}}.$$ From the above discussion we obtain a simplified version of the local coefficient formula in our case, namely Let $\pi$ be an irreducible admissible $\psi$-generic representation of $\textrm{GL}_n(F)$, lifted as a $\psi$-generic representation $\sigma$ of $M_H(F)\simeq \textrm{GL}_n(F)\times \textrm{GL}_1(F)$ by pull-back through the projection on the $\textrm{GL}_n$-factor. $\eta:F^\times\rightarrow \mathbb{C}^{\times}$ is a fixed continuous character. Define the representation $\sigma_\eta$ as before. Suppose that $\omega_{\sigma_\eta}(w_0\omega_{\sigma_\eta}^{-1})$ is ramified as a character of $F^{\times}$. Then for all sufficiently large $\kappa$ we have $$C_{\psi}(s,\sigma_\eta)^{-1}=\gamma(ns, \omega_{\pi}^2,\psi)^{-1}$$$$\cdot\int_{Z^0_{M_H}U_{M_H}\backslash N_H}j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)\omega_{\pi}^{-2}(u_n)\vert u_n\vert^{-ns}\vert \det(g)\vert^{\frac{s+n}{2}}d\dot{n}.$$ where off a set of measure zero, the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ holds as in the previous section. Here $u_n=u_{\alpha_n}(\dot{w}_0\bar{n}\dot{w}^{-1}_0)\in U_{\alpha_n}=U_{\alpha}$. And $\gamma(ns, \omega_{\pi}^2,\psi)$ is an abelian $\gamma$-factor depending only on $\omega_{\pi}$. In the proof of stability, we also need an integral formula for the local coefficient $C_{\psi}(s,(\sigma_\eta \otimes \chi))^{-1}$ for a sufficiently ramified character $\chi$ of $F^{\times}$, viewed as a character of $M_H(F)$ by $\chi(m(g,a))=\chi(\det(g))$. Therefore it is important to be able to choose $\kappa$ or equivalently, $\overline{N}_0\subset \overline{N}_H(F)$ to be independent of $\chi$. To make this work, as in the proof of Theorem 6.2 in \[19\] and the corresponding discussion in \[7\], if we fix an irreducible generic representation $\pi'$ of $G$ such that $\omega_{\sigma_\eta'}$ is ramified, where $\sigma'$ is the lift of $\pi'$, $\sigma_\eta'$ is defined in the same way as $\sigma_\eta$. Then $\overline{N}_0$ is chosen to satisfy (1) $\exists f\in V(s,\sigma_\eta')$ such that $f$ is supported in $P_H\overline{N}_0$; (2) $\overline{N}_0$ is large enough such that $\alpha^{\vee}(t)\overline{N}_0\alpha^{\vee}(t)^{-1}$ depends only on $\vert t\vert $ for all $t\in F^{\times}.$ Note that here (2) does not depend on $\pi'$. For (1), as in the proof of Theorem 6.2 in \[19\], there exist $f\in V(s,\sigma_\eta') $ s.t. $f$ is compactly supported modulo $P_H$. Fix such an $f$ and choose $\overline{N}_0$ sufficiently large such that it contains the support of $f$, then $f$ is supported in $P_H\overline{N}_0$. Now let’s get back to our case. We fix a character $\chi_0$ of $F^{\times}$ such that $\omega_{\sigma_\eta}\chi_0^n=\eta^{-1}\omega_\pi\chi_0^n=\omega_{\sigma_\eta\otimes \chi_0}$ is ramified. Then we take $\kappa_0$ such that both conditions (1) and (2) above are satisfied for $\overline{N}_{0,\kappa_0}$ and $f_{\chi_0}\in V(s,\sigma_\eta\otimes\chi_0 )$. Also note that if $\kappa\ge \kappa_0$, we have $\overline{N}_{0,\kappa_0}\subset \overline{N}_{0,\kappa}$. Therefore (1) and (2) hold for $\sigma_\eta\otimes \chi_0$ and all $\kappa\ge \kappa_0.$ Let $\chi$ be any other character of $F^{\times}$ such that $\omega_{\sigma_\eta}\chi^n$ is ramified. Then as discussed above we can choose $f_{\chi}\in V(s,\sigma_\eta\otimes\chi )$ which is supported in $P_H\overline{N}_{0,\chi}$ for some open compact $\overline{N}_{0,\chi}\subset\overline{N}_H$. Now if $\overline{N}_{0,\chi}\subset\overline{N}_{0,\chi_0}$, then Proposition 5.7 holds for $\sigma_\eta\otimes \chi$ and all $\kappa\ge \kappa_0$. While if not, note that $\alpha^{\vee}(t)=\prod_{i=1}^ne_i^*(t)\in M_H,$ then $R(\alpha^{\vee}(t)^{-1})f$ will be supported in $P_H(\alpha^{\vee}(t)^{-1}\overline{N}_{0,\chi}\alpha^{\vee}(t))$. To see this, note that for $\bar{n}(\tilde{X},\alpha)\in \overline{N}_H(F)$, we have $\alpha^{\vee}(t)^{-1}\bar{n}(\tilde{X},\alpha)\alpha^{\vee}(t)=\bar{n}(t^2\tilde{X}, t\alpha).$ Therefore if we take $\vert t\vert $ sufficiently small, we will have $\alpha^{\vee}(t)^{-1}\overline{N}_{0,\chi}\alpha^{\vee}(t)\subset \overline{N}_{0,\kappa_0}$. So if we take such a $t$ and replace $f$ with $f'_{\chi}=R(\alpha^{\vee}(t)^{-1})f_{\chi}$, we see that $f'_{\chi}$ will be supported in $P_H\overline{N}_{0,\kappa_0}$ and Proposition 5.7 holds for $\sigma_\eta\otimes \chi$ and for all $\kappa\ge \kappa_0$. Now we obtain a stronger version of Proposition 5.7. Let $\pi$ be an irreducible admissible $\psi$-generic representation of $\textrm{GL}_n(F)$, lifted as a $\psi$-generic representation $\sigma$ of $M_H(F)\simeq \textrm{GL}_n(F)\times \textrm{GL}_1(F)$ by pull-back through the projection on the $\textrm{GL}_n$-factor. $\eta:F^\times\rightarrow \mathbb{C}^{\times}$ is a fixed continuous character. Define the representation $\sigma_\eta$ as before. Suppose that $\omega_{\sigma_\eta}(w_0\omega_{\sigma_\eta}^{-1})$ is ramified as a character of $F^{\times}$. Then there exist a $\kappa_0$ such that for all $\kappa\ge \kappa_0$ and all $\chi$ such that $\omega_{\sigma_\eta}\chi^n$ is ramified, we have $$C_{\psi}(s,\sigma_\eta\otimes \chi)^{-1}=\gamma(ns, (w_{\pi}\chi)^{2n},\psi)^{-1}\int_{Z^0_{M_H}U_{M_H}\backslash N_H}j_{\pi\otimes \chi,\eta,\dot{w}_\theta,\kappa}(g)(\omega_{\pi}\chi^n)^{-2}(u_n)$$$$\cdot \vert u_n\vert^{-ns}\vert \det(g)\vert^{\frac{s+n}{2}}d\dot{n}.$$ where off a set of measure zero, the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ holds as in the previous section. Here $u_n=u_{\alpha_n}(\dot{w}_0\bar{n}\dot{w}^{-1}_0)\in U_{\alpha_n}=U_{\alpha}$. And $\gamma(ns,(\omega_{\pi}\chi)^{2n},\psi)$ is an abelian $\gamma$-factor depending only on $\omega_{\pi}$ and $\chi$. Next, we use our orbit space representatives and measure to further simplify the integral in the local coefficient formula. Recall that we have the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$ holds for $n$ lying in the open dense subset $N_H(F)'$ of $N_H(F)$. Now for $n=n(X,\alpha)$, let $Y=X^tJ'=(Z-\frac{\alpha^t\alpha}{2})J'^tJ'=Z-\frac{\alpha^t\alpha}{2}$. Then by section 5.2 on orbit space and measure, if $n\in N_H(F)'$, then $Z$ can be taken as $Z(a_1,\cdots,a_{n-1})$ and $\alpha$ can be taken as $[0,\cdots,0,a_n]^t$, consequently $Y$ can be given as $Y(a_1,\cdots,a_n)$(see the notations on Page 22). Also recall that the calculation of the decomposition $w_0^{-1}n=mn'\bar{n}$ gives $m=m(g,a(g))$ where $g=(-\frac{1}{2})J'^tY^{-1}$ and $a(g)=\frac{(\frac{1}{2})^{\frac{n}{2}}}{\prod_{k\ \ odd}a_k}$ if $n$ is even and $a(g)=\frac{(\frac{1}{2})^{\frac{n-1}{2}}}{\prod_{k\ \ odd}a_k}$ if $n$ is odd. We have seen that in the decomposition $\dot{w}_0^{-1}n=mn'\bar{n}$, if $n=n(X,\alpha)$, then the corresponding $\bar{n}=\bar{n}(X^{-1},\alpha)=\begin{bmatrix} I & \ \ & \ \ \\ -^t(J'X^{-1}\alpha) & 1 & \ \ \\ X^{-1} & X^{-1}\alpha & I \\ \end{bmatrix}.$ So $$\dot{w}_0\bar{n}\dot{w}_0^{-1}=\begin{bmatrix} \ \ & \ \ & (-\frac{1}{2})I \\ \ \ & (-1)^n & \ \ \\ (-1)^{n}2I & \ \ & \ \ \\ \end{bmatrix} \begin{bmatrix} I & \ \ & \ \ \\ -^t(J'X^{-1}\alpha) & 1 & \ \ \\ X^{-1} & X^{-1}\alpha & I \\ \end{bmatrix}$$$$\cdot \begin{bmatrix} \ \ & \ \ & (-1)^{n}\frac{1}{2}I \\ \ \ & (-1)^n & \ \ \\ -2I & \ \ & \ \ \\ \end{bmatrix} =\begin{bmatrix} I & (-1)^{n-1}\frac{1}{2}X^{-1}\alpha & (-1)^{n-1}\frac{1}{4}X^{-1} \\ \ \ & 1 & -\frac{1}{2}^t(J'X^{-1}\alpha)' \\ \ \ & \ \ & I \\ \end{bmatrix}$$ So $u_n=u_{\alpha_n}(\dot{w}_0\bar{n}\dot{w}_0^{-1})$ is the last entry of $(-1)^{n-1}\frac{1}{2}X^{-1}\alpha$. Since only the last entry of $\alpha$ is non-zero, $u_n=(-1)^{n-1}\frac{1}{2}(\det X)^{-1}X_{n,n}^*a_n$, where $X^*_{n,n}$ is the $(n,n)$-th entry of the adjoint matrix of $X$. Since $X=Y{^tJ'^{-1}}=YJ'$, $Y$ is the matrix given as above, it is not hard to see that $X_{n,n}^*=(-1)^{n-1}\prod_{i=1}^{n-1}a_i.$ Therefore we have that $u_n=\frac{1}{2}(\det X)^{-1}\prod_{i=1}^n a_i.$ Also notice that $X=YJ'$ and $\det{J'}=1$, so $\det(X)=\det(Y)$. Hence $u_n=\frac{1}{2}(\det Y)^{-1}\prod_{i=1}^na_i.$ Next, we work on $zu^{-1}\bar{n}uz^{-1}.$ Let $z_0=\varpi^{d+f}u_n=\frac{1}{2}\varpi^{d+f}(\det Y)^{-1}\prod_{i=1}^na_i,$ let $t=(\det Y)^{-1}\prod_{i=1}^na_i\in F^{\times}$, then $z_0=\frac{1}{2}\varpi^{d+f}t$. Let $u=m(u_0,1)$ and $z=\alpha^{\vee}(z_0)=m(z_0I,1)=$ with $u_0\in U_n(F)\subset GL_n(F).$ Since $Y=X^tJ'$, so $X^{-1}=^tJ'Y^{-1}$, therefore $\bar{n}(X^{-1},\alpha)=\bar{n}(^tJ'Y^{-1},\alpha)=\begin{bmatrix} I & \ \ & \ \ \\ -{^t\alpha}{^tY^{-1}} & 1 & \ \ \\ {^tJ'}Y^{-1} & {^tJ'}Y^{-1}\alpha & I \\ \end{bmatrix}$. Then a direct calculation shows that $u^{-1}\bar{n}(^tJ'Y^{-1},\alpha)u=\bar{n}({^tJ}'{^tu}_0Y^{-1}u_0,u_0^{-1}\alpha)$. This implies that $zu^{-1}\bar{n}(^tJ'Y^{-1},\alpha)uz^{-1}=\bar{n}(z_0^{-2}\cdot\leftidx{^t}J'{^tu}_0Y^{-1}u_0,z_0u_0^{-1}\alpha)$. We have $z_0=\frac{1}{2}\varpi^{d+f}t$, with $t=(\det Y)^{-1}\prod_{i=1}^na_i\in F^{\times}.$ Let $Y'=t^2Y$ and $\alpha'=t\alpha$. Recall that $\overline{N}_{0,\kappa}=\{\bar{n}=\bar{n}(\tilde{X},\alpha): \varphi_{\kappa}(-\frac{1}{8}\varpi^{2(d+f)}\cdot\leftidx{^t}{\tilde{X}}{J'}^{-1})=1\}.$ Therefore $\varphi_{\overline{N}_{o,\kappa}}(zu^{-1}\bar{n}uz^{-1})=\varphi_{\kappa}(-\frac{1}{8}\varpi^{2(d+f)}\cdot(\frac{1}{2}\varpi^{d+f}t)^{-2}\cdot\leftidx{^t}(^tJ'\leftidx{^t}u_0Y^{-1}u_0)J'^{-1})$ $=\varphi_\kappa(-\frac{1}{2}t^{-2}(^tu_0\leftidx{^t}Y^{-1}u_0J')J'^{-1})=\varphi_{\kappa}(-\frac{1}{2}t^{-2}\cdot\leftidx{^t}u_0\leftidx{^t}Y^{-1}u_0)=\varphi_\kappa(-\frac{1}{2}\leftidx{^t}u_0\leftidx{^t}Y'^{-1}u_0)$. We pick the long Weyl group representative of $G=\textrm{GL}_n$ by $\dot{w}_G=J'$, then $$j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)=j_{\pi,\eta,\dot{w}_\theta,\kappa}(-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1})$$$$=\int_{U_{M_H}}W_{\sigma_{\eta,s},v}(m(-\frac{1}{2}\dot{w}_G{^tY^{-1}},a(g))u)\varphi_{\overline{N}_{0,\kappa}}(zu^{-1}\bar{n}uz^{-1})\psi^{-1}(u)du$$$$=\int_{U_n} \eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}} W_{{\pi},v}(-\frac{1}{2}\dot{w}_G{^tY^{-1}}u_0)\varphi_{\kappa}(-\frac{1}{2}\leftidx{^t}u_0{^tY'^{-1}}u_0)\psi^{-1}(u_0)du_0$$$$=\eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}}\int_{U_n} W_{\pi,v}(gu)\varphi_{\kappa}(\leftidx{^t}u \dot{w}^{-1}_G g' u)\psi^{-1}(u)du.$$ where $g'=-\frac{1}{2}\dot{w}_G\leftidx{^t}Y'^{-1}$(so $g=t^2g'$), $U_n$ is the upper triangular unipotent matrices of size $n$ in $\textrm{GL}_n$. We also used the fact that $W_{\pi,v}(g)=\lambda(\pi(g)v)$, therefore $W_{\sigma_{\eta,s},v}(m(g,a(g)))=\lambda(\sigma_{\eta,s}(m(g,a(g))))=\eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}}\lambda(\pi(g)v)=\eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}}W_{\pi,v}(g)$. Moreover, substitute $u_n=\frac{1}{2}(\det Y)^{-1}\prod_{i=1}^na_i$ into the local coefficient formula, and use the orbit space measure we constructed earlier. After some simplifications, we obtain Let $\pi$ be an irreducible admissible $\psi$-generic representation of $\textrm{GL}_n$, lifted as a $\psi$-generic representation $\sigma$ of $M_H(F)\simeq \textrm{GL}_n(F)\times \textrm{GL}_1(F)$ by pull-back through the projection on the $\textrm{GL}_n$-factor. $\eta:F^\times\rightarrow \mathbb{C}^{\times}$ is a fixed continuous character. Define the representation $\sigma_\eta$ as before. Suppose that $\omega_{\sigma_\eta}(w_0\omega_{\sigma_\eta}^{-1})$ is ramified as a character of $F^{\times}$. Then for all sufficiently large $\kappa$, we have $$C_{\psi}(s,\sigma_\eta)^{-1}=\gamma(ns, \omega_{\pi}^2,\psi)^{-1}\int_{F^{\times}\backslash R}j_{\pi,\eta,\dot{w}_\theta,\kappa}(-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1})$$$$\cdot\omega_{\pi}(4\det(Y)^2\prod_{i=1}^n a_i^{-2}) \vert\frac{1}{2}\vert^{\frac{n(n-s)}{2}}\vert \det(Y)\vert^{\frac{2ns-s-n}{2}} \prod_{i=1}^n\vert a_i \vert ^{i-1-ns}da_i$$ In addition, there exists a constant $\kappa_0$ such that for all $\kappa\ge \kappa_0$ and all $\chi$ such that $\eta^{-1}\omega_{\pi}\chi^n$ is ramified, we have $$C_{\psi}(s,\sigma_\eta\otimes\chi)^{-1}=\gamma(ns, (\omega_{\pi}\chi^n)^2,\psi)^{-1}\int_{F^{\times}\backslash R}j_{\pi,\eta,\dot{w}_\theta,\kappa}(-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1})$$$$\cdot (\omega_{\pi}\chi^n)(4\det(Y)^2\prod_{i=1}^n a_i^{-2})\vert \frac{1}{2}\vert^{\frac{n(n-s)}{2}} \vert \det(Y)\vert^{\frac{2ns-s-n}{2}} \prod_{i=1}^n\vert a_i \vert ^{i-1-ns}da_i.$$ Partial Bessel integrals ------------------------ For the proof of the stability of local coefficients, it is important to relate partial Bessel functions with partial Bessel integrals, which have nice asymptotic expansions under some conditions. Let $\textbf{G}$ be a split reductive group over $F$, and $G=\textbf{G}(F)$. Fix a Borel subgroup $\textbf{B}=\textbf{AU}$ and let $B$, $A$, $U$ denote the groups of their $F$-points respectively. Suppose $\Theta:G\longrightarrow G$ is an involution defined over $F$, i.e., $\Theta^2=1$ and $\Theta\neq 1$. Let $\pi$ be a $\psi$-generic supercuspidal representation of $G$ with its central character $\omega_\pi$. Let $f\in \mathcal{M}(\pi)$ be a matrix coefficient of $\pi$. Then $f\in C^\infty_c(G;\omega_\pi)$, the space of smooth functions on $G$ with compact support modulo the center $Z_G$ such that $f(zg)=\omega_\pi(z)f(g)$ for $z\in Z_G$ and $g\in G$. We associate $f$ with the Whittaker function $W^f(g)=\int_Uf(u'g)\psi^{-1}(u')du'$. The integral convergences since the coset $UZg$ is closed in $G$ and $f\in C^\infty_c(G;\omega_\pi)$. We can normalize it by choosing $f\in \mathcal{M}_\pi$ such that $W^f(e)=1$, where $e\in G$ is the identity element. We define the twisted centralizer of $g\in G$ by $$U_g=\{u\in U: \Theta(u^{-1})gu=g \}.$$ Suppose $G=Z_GG'$, write $g=zg'$ with $z\in Z_G$, $g\in G'$. Then we define the partial Bessel integral $$B^G_{\tilde{\varphi}}(g,f)=\int_{U_g\backslash U}W^f(gu)\tilde{\varphi}(\Theta(u^{-1})g'u)\psi^{-1}(u)du,$$ where $\tilde{\varphi}$ is some cut-off function. Note that the above definitions can also be applied to any Levi subgroup $\textbf{M}$ of $\textbf{G}$. If we apply the above settings to the case $\textbf{G}=\textrm{GL}_n$, $\Theta(g)=\dot{w}_G \leftidx{^t}g^{-1}\dot{w}_G^{-1}$, and $\tilde{\varphi}=L_{\dot{w}_G}\varphi$, where $L_{s}\varphi(g)=\varphi(s^{-1}g)$ is the left translation of $\varphi$, we obtain $$B^G_{\varphi}(g, f)=\int_{U_g\backslash U} W^{f}(gu)\varphi(\leftidx{^t}u\dot{w}^{-1}_G g' u)\psi^{-1}(u)du,$$ which is the definition of partial Bessel integrals in \[7\]. And in this case the twisted centralizer of $g$ is given by $$U_g=\{u\in U: \leftidx{^t}u\dot{w}_G^{-1}gu=\dot{w}_G^{-1}g\}$$ We will only use this definition for partial Bessel integrals and twisted centralizers in the rest part of the paper. On the other hand, it is not hard to see by induction on the size $n$ that if $g=-\frac{1}{2}\dot{w}_G \leftidx{^t}Y^{-1}$ for $Y=Y(a_1,\cdots,a_n)$ with $(a_1\cdots,a_n)\in (F^{\times})^n$ as in the last part of section 5.4, the twisted centralizer $U_g$ is trivial. Hence the partial Bessel integral $$B^G_{\varphi}(g, f)=\int_U W^{f}(gu)\varphi(\leftidx{^t}u\dot{w}^{-1}_G g' u)\psi^{-1}(u)du,$$ where $g=zg'$, $z\in Z$. Now choose $f\in \mathcal{M}(\pi)$ such that $W_{\pi,v}=W^f$, and $W^f(e)=1$. Take $\varphi=\varphi_{\kappa}$. From the calculations right before Proposition 5.9, we have $$j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)=\eta(a(g))^{-1}\vert \det(g)\vert^{\frac{s}{2}}\int_{U_n} W_{\pi,v}(gu)\varphi_{\kappa}(\leftidx{^t}u \dot{w}^{-1}_G g' u)\psi^{-1}(u)du.$$ Therefore we obtain Let $f\in \mathcal{M}(\pi)$ such that $W^f(e)=1$, and let $\varphi=\varphi_{\kappa}$, then $$j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)=\eta(a(g))^{-1}\vert\det(g)\vert^{\frac{s}{2}}\cdot B^G_{\varphi}(g,f),$$ for $g=-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1}$, where $Y=Y(a_1,\cdots,a_n)$ with all $a_i\in F^{\times}.$ Now we have successfully related our partial Bessel functions with partial Bessel integrals, whose asymptotic expansions will lead to the proof of stability. ANALYSIS OF PARTIAL BESSEL INTEGRALS ==================================== Let $\textbf{G}$ be a split connected reductive group over $F$. Fix a Borel subgroup $\textbf{B}=\textbf{AU}$, and let $\textbf{U}^-$ be the unipotent group generated by all the negative roots. We use $G$, $B$, $A$, $U$, $U^-$ to denote their groups of $F$-points respectively. Denote the Weyl group of $\textrm{G}$ by $W$. We begin by stating some basic facts and properties. - [**.**]{} Define the subset of $W$ that supports Bessel functions by $B(G)=\{w\in W: \alpha\in \Delta \ \ s.t. \ \ w\alpha>0 \Rightarrow w\alpha\in \Delta \}$, or equivalently, $B(G)=\{w\in W: w_G w=w_M \ \ \textrm{for} \ \ \textrm{some} \ \ \textrm{standard} \ \ \textrm{Levi} \ \ M\subset G\}$. We take the representatives of such $w\in B(G)$ so that $\dot{w}=\dot{w}_G\dot{w}_M^{-1}$. Then there is a one-to-one correspondence between elements in $B(G)$ and Levi subgroups standard parabolic subgroups of $G$. To be precise, to a $w\in B(G)$ we associate $\theta_w^+=\{\alpha\in \Delta: w\alpha>0\}\subset \Delta$ which determines a standard parabolic subgroup $P_w=M_wN_w$, such that $M_w=Z_G(\cap_{\alpha\in \theta_w^+}\ker \alpha)$. We also have that $\theta^+_w=\theta_{w_M}^-=\Delta_M\subset \Delta$, where $w_M$ is the long Weyl group element of $M$. - $\textbf{U}_w^+, \textbf{U}_w^-.$ For each $w\in W$ we define two unipotent subgroups $U^+_w$ and $U_w^-$ of $U$ to be $U_w^+=\{u\in U: wuw^{-1}\in U\}$ and $U_w^-=\{u\in U:wuw^{-1}\in U^-\}.$ In other words, $U_w^+$(resp. $U_w^-$) is generated by those roots that are made positive(resp. negative) by $w$. One can see that $U^+_w=U\cap w^{-1}Uw$, $U^-_w=U\cap w^{-1} U^-w,$ and $U=U^+_wU^-_w$. Moreover, if $w\in B(G)$, suppose $\dot{w}=\dot{w}_G\dot{w}_M^{-1}$, so $w$ associates the Levi $M=M_w$ of $G$. Let $U_M=U\cap M$, then $U_M$ is the standard maximal unipotent subgroups of $M$. If we denote $N_M$ to be the unipotent radical of the corresponding parabolic, i.e., $P_M=MN_M$. Then $U=U_MN_M$. Now for $w=w_M$, we can see that $U_{w_M}^+=N_M, U_{w_M}^-=U_M$ and for $w=w_G$, we have $U_{w_G}^+=\{e\}, U_{w_G}^-=U$. In general for $w=w_Gw_M$ we have $U_w^+=U_M, U_{w}^-=N_M.$ - **Bessel distance** For $w,w'\in B(G)$ with $w>w'$ we define the Bessel distance as follows: $d_B(w,w')=max\{m: \exists w_i\in B(G)\ \ s.t \ \ w=w_m>w_{m-1}>\cdots>w_0=w' \}$. And if we denote $\Delta_{M_w}$ to be the set of simple roots associated with the standard Levi $M_w$, we have $\Delta_{M_w}\subset \Delta_{M_w'}$, and $d_B(w,w')=\vert \Delta_{M_{w'}}-\Delta_{M_w}\vert$. - **Bruhat order** For $w\in W$ we denote the Bruhat cell by $C(w)=UwAU$, we define the Bruhat order on $W$ by $w\leq w'\Longleftrightarrow C(w)\subset \overline{C(w')}.$ - **The relevant torus $\textbf{A}_w$**. For $w\in B(G)$, define $A_w=\{a\in A: a \in \cap_{\alpha\in \theta_w^+}\ker \alpha\}\subset A$, which is also the center $Z_{M_w}$ of $M_w$. - **The relevant Bruhat cell** $\textbf{C}_{r}(w)$. We call $C_r(w)=UwA_wU$ the relevant part of the Bruhat cell $C(w)$. - **Transverse tori** Let $w,w'\in B(G)$ and let $M=M_w$ and $M'=M_{w'}$ be their associated Levi subgroups respectively. Suppose $w'\leq w$. Then $M\subset M'$ and $A_{w'}\supset A_w$. Let $A^{w'}_w=A_w\cap M^d_{w'}=Z_M\cap (M')^d$. Note that in particular $A^w_w=Z_M \cap M^d$ is finite since $M$ is reductive and in general we have that $M^d\cap R(M)=M^d\cap Z^0$ is finite, where $Z^0$ is the connected component of $Z$ and $R(M)$ is the radical of $M$. In the case of $G=GL_n$ the center is connected, and $A^w_w$ consists of certain roots of unity on the diagonal blocks of $M$. Similarly $A^{w'}_w\cap A_{w'}=A^{w'}_{w'}$ is finite and the subgroup $A^{w'}_w A_{w'}\subset A_w$ is open and of finite index. So this decomposition is essentially a “transfer principal” for relevant tori, from the larger one $A_w$ to the smaller one $A_{w'}$ which differs by the transverse torus $A^{w'}_w$, on which the germ functions live on, as we will see later. Here are some useful properties of $B(G)$: **1,** For $w,w'\in B(G)$. Then $w'\leq w\Longleftrightarrow M_w\subset M_{w'}\Longleftrightarrow A_w\supset A_{w'}.$ (Lemma 5.1 in \[7\]) **2,** For each $w\in B(G)$, say $\dot{w}=\dot{w}_G\dot{w}_M^{-1}$. Then for all $u\in U_w^+=U_M$, we have $\psi(\dot{w}u\dot{w}^{-1})=\psi(u)$, where $\psi$ is the generic character. (Proposition 5.1 in \[7\]) **3,** Let $\Omega_w=\bigsqcup_{w\leq w'} C(w'),$ we see that $\Omega_w$ is invariant under the two-sided action of $U\times U$ and as in Lemma 5.2 in \[7\], $\Omega_w$ is an open subset of $G$ and $C(w)$ is closed in $\Omega_w$. As stated in \[7\] we also have: For $w\in B(G)$ and suppose $w$ is associated with a standard Levi $M$ of $G$, then we have $\Omega_{w}\simeq U^-_{(w)^{-1}}\times \dot{w}M\times U_w^-.$ This decomposition is unique. Suppose $\pi$ is a generic representation of $M(F)$. Let $C^{\infty}_c(\Omega_w;w_{\pi})$ denote the space of smooth functions of compact support modulo the center $Z$, so $\forall g\in \Omega_w$ and $z\in Z$, $f(zg)=w_{\pi}(z)f(g)$. Since $\Omega_w$ is open in $G$, we have $C_c^{\infty}(\Omega_w;w_{\pi})\subset C^{\infty}_c(G;w_{\pi}).$ There is a surjective map: $C^{\infty}_c(M;w_{\pi}){\rightarrow\mathrel{\mkern-14mu}\rightarrow}C^{\infty}_c(\Omega_{w'};w_{\pi})$ given by $h=h_f\mapsto f$ where $h(m)=h_f(m)=\int_{U^-_{w'}}\int_{U^-_{(w')^{-1}}}f(x^-\dot{w}mu^-)\psi^{-1}(x^-u^-)dx^-du^-.$ See Lemma 5.9 \[7\]. Partial and full Bessel integrals --------------------------------- Let $w\in B(G)$ and $g=u_1\dot{w}au_2\in C_r(w)$, the relevant cell associated to $w$. Let $M=M_w$ be the Levi subgroup of $G$ such that $w=w_Gw_M.$ Then we have For $g=u_1\dot{w}au_2\in C_r(w)$ with $w=w_Gw_M\in B(G)$, then $$U_g\subset u_2^{-1}U^+_w u_2=u_2^{-1} U_M u_2$$ $u\in U_g\Longleftrightarrow ^tu\dot{w}_G^{-1}u_1\dot{w}_G\dot{w}_M^{-1}au_2u=\dot{w}_G^{-1}u_1\dot{w}_G\dot{w}_M^{-1}au_2$. Let $\overline{u_1}=\dot{w}_G^{-1}u_1\dot{w}_G\in U^-$, then this is equivalent to $ (\overline{u_1})^{-1}{^tu} \overline{u_1}\dot{w}_M^{-1}au_2uu_2^{-1}=\dot{w}_M^{-1}a$, which is the same as $ (\overline{u_1})^{-1}{^tu}\overline{u_1}=\dot{w}_Mau_2u^{-1}u_2^{-1}a^{-1}\dot{w}_M$. Notice that $(\overline{u_1})^{-1}{^tu}\overline{u_1}\in U^-$, and $au_2u^{-1}u_2^{-1}a^{-1}\in U$. This implies that $au_2u^{-1}u_2^{-1}a^{-1}\in U_{w_M}^-=U_M$. Therefore $ u_2u^{-1}u_2^{-1}\in a^{-1}U_Ma=U_M$ since $a\in A_w$. So $u^{-1}\in u^{-1}_2U_Mu_2$, thus $u\in u^{-1} U_M u_2=u_2^{-1}U_{w}^+u_2$. Next, we will show an equality that relates partial Bessel integrals with full Bessel integrals. First, decompose $U=u_2^{-1}Uu_2=(u^{-1}_2U^+_wu_2)(u_2^{-1}U_w^-u_2)$ and for $u\in U$, write $u=u'^+(u_2^{-1}u^-u_2)$ with $u'^+=u_2^{-1}u^+u_2$ where $u^+\in U^+_w$, and $u^-\in U^-_w.$ Since by lemma 6.3, $U_g\subset u_2^{-1}U_w^+u_2$, we have $$B^G_{\varphi}(g,f)=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(xgu'^+u_2^{-1}u^-u_2)$$$$\cdot\varphi(^t{(u_2^{-1}u^-u_2)}^t{u'^+}\dot{w}_G^{-1}g'u'^+u_2^{-1}u^-u_2) \psi^{-1}(x)\psi^{-1}(u'^+u_2^{-1}u^-u_2)dxdu^-du'^+$$ $$=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(xu_1\dot{w}a(u_2u'^+u_2^{-1})u^-u_2)$$$$\cdot\varphi(^t{(u_2^{-1}u^-u_2)}^t{u'^+}\dot{w}_G^{-1}u_1\dot{w}a'(u_2u'^+u_2^{-1}) u^-u_2) \psi^{-1}(x)\psi^{-1}(u'^+u_2^{-1}u^-u_2)dxdu^-du'^+$$ $$=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(xu_1\dot{w}au^+u^-u_2)\varphi(^t{u_2}^t{u^-}^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+u^-u_2)$$ $$\cdot \psi^{-1}(x)\psi^{-1}(u_2^{-1}u^+u^-u_2)dxdu^-du^+$$ Now since $a\in A_w$, we have $au^+=u^+a$. So the above integral $$=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(xu_1(\dot{w}u^+\dot{w}^{-1})\dot{w}au^-u_2)\varphi(^t{u_2}^t{u^-}^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+u^-u_2)$$ $$\cdot \psi^{-1}(x)\psi^{-1}(u_2^{-1}u^+u^-u_2)dxdu^-du^+$$ Let $x'=xu_1(\dot{w}u^+ \dot{w}^{-1})$ and $u'^-=u^-u_2$, then $dx'=dx$ and $du'^-=du^-$. After this change of variable we have the above integral $$=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(x'\dot{w}au'^-)\varphi(^t{u'^-}^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+u'^-)$$ $$\cdot \psi^{-1}(x'(u_1\dot{w}u^+\dot{w}^{-1})^{-1})\psi^{-1}(u_2^{-1}u^+u'^-)dxdu'^-du^+$$ $$=\psi(u_1)\psi(u_2)\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(x'\dot{w}au'^-)\varphi(^t{u'^-}^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+u'^-)$$ $$\cdot \psi^{-1}(x')\psi(\dot{w}u^+\dot{w}^{-1})\psi^{-1}(u^+)\psi^{-1}(u'^-)dxdu'^-du^+$$ By compatibility of $\psi$ and $\dot{w}$, we have $\psi(\dot{w}u^+\dot{w}^{-1})=\psi(u^+)$, so $$B^G_{\varphi}(g,f)=\psi(u_1)\psi(u_2)\int_{U_g\backslash u_2^{-1}U^+_wu_2}\int_{U^-_w}\int_U f(x'\dot{w}au'^-)$$$$\cdot\varphi(^t{u'^-}^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+u'^-) \psi^{-1}(x')\psi^{-1}(u'^-)dxdu'^-du^+$$ Now take $f\in C_c^{\infty}(\Omega_w;w_{\pi})$. Since $g$ is fixed, $a$ is fixed. Then there exists open compact subsets $U_1\subset U$ and $U_2\subset U_w^-$ such that the support of the function $(x,u^-)\mapsto f(x\dot{w}au^-)$ lies in $U_1\times U_2$. Take $N$ large enough such that $\varphi=\varphi_N$ is invariant under the left and right action of $U_2$ as in Lemma 4.2 in \[7\], i.e., $\varphi(^tugu)=\varphi(g)$ for all $u\in U_2$. Then we have $\varphi(^t{u'^-}^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+u'^-)=\varphi(^t{u^+}^t{u_2^{-1}}\dot{w}_G^{-1}u_1\dot{w}a'u^+)$. Define $$\tilde{\varphi}^G_M(g')=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\varphi(^t{u^+}\leftidx{^t}{u_2^{-1}}u_1\dot{w}a'u^+)du^+$$ Then $$\tilde{\varphi}^G_M(g')=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\varphi(^t{u^+}\leftidx{^t}{u_2^{-1}}\dot{w}_G^{-1}g'u_2^{-1}u^+)du^+$$ $$=\int_{U_g\backslash u_2^{-1}U^+_wu_2}\varphi(^t{u_2^-}\leftidx{^t}{u'^+}\dot{w}_G^{-1}g'u'^+u_2^{-1})du'^+$$ So we have $$B^G_{\varphi}(g,f)=\psi(u_1)\psi(u_2)\tilde{\varphi}^G_M(g')\int_{U_w^-}\int_Uf(x\dot{w}au^-)\psi^{-1}(x)\psi^{-1}(u^-)dxdu^-$$ $$=\psi(u_1)\psi(u_2)\tilde{\varphi}^G_M(g')B^G(\dot{w}a,f)=\tilde{\varphi}^G_M(g')B^G(g,f)$$ We just showed the following result: For $w\in B(G)$ and any $g=u_1\dot{w}au_2\in C_r(w)$, $g'=u_1\dot{w}a'u_2$ where $a=za'$, $z\in Z$ and $a'\in A'$, we have $$B^G_{\varphi}(g,f)=\tilde{\varphi}^G_M(g')B^G(g,f).$$ where $$B^G(g,f)=\int_{U\times U_{w}^-}f(xgu^-)\psi^{-1}(x)\psi^{-1}(u^-)dxdu^-$$ is the full Bessel integral and $\tilde{\varphi}^G_M(g')$ as defined above. Twisted centralizer and transfer principle ------------------------------------------ For $\textbf{G}=\textrm{GL}_n$, $G=\textbf{G}(F)$, and $f\in C^{\infty}_c(G;w_{\pi})$, we defined the partial Bessel integral as $$B^G_\varphi(g,f)=\int_{U_g\backslash U}W^f(gu)\varphi(^tu\dot{w}_G^{-1}g'u)\psi^{-1}(u)du$$ $$=\int_{U_g\backslash U}\int_U f(xgu)\varphi(^tu\dot{w}_G^{-1}g'u)\psi^{-1}(x)\psi^{-1}(u)dxdu,$$ where $\varphi$ is the characteristic function of some compact neighborhood of zero in $Mat_n(F)$. Now for any Levi subgroup $M$ of $G$, we define the twisted centralizer of $m\in M$ in $U_M=U\cap M$ to be $U_{M,m}=\{u\in U_M: \leftidx{^t}u\dot{w}_M^{-1}mu=\dot{w}_M^{-1}u\}$. Let $h\in C^{\infty}_c(M;w_{\pi})$, the space of smooth functions of compact support modulo $Z$ on $M$, satisfying $h(zm)=w_{\pi}(z)h(m),$ for $z\in Z=Z_G$. The partial Bessel integral on $M$ is then given by $$B^M_{\varphi}(m,h)=\int_{U_{M,m}\backslash U_M}\int_{U_M}h(xmu)\varphi(^tu\dot{w}_M^{-1}m'u)\psi^{-1}(xu)dxdu,$$ where $m'$ is obtained by $m$ from the decomposition $Z_M=ZA_M'$, i,e., if $m\in U_M \dot{w} A_M U^-_{M,w}$, then $m'\in U_M\dot{w} A_M' U^-_{M,w},$ $z\in Z$ and $m=zm'$. Now Let $L\subset M \subset G$ be standard Levi subgroups of G, as before let $w_G$, $w_M$ and $w_L$ be the long Weyl group elements of $G, M$ and $L$ respectively. And let $\dot{w}_G,\dot{w}_M,$ and $\dot{w}_L$ be their representatives chosen to be compatible with $\psi$ as before. Now denote $w^M_L=\dot{w}_M\cdot \dot{w}_L^{-1}$, similarly if $M$ is replaced by $G$. Take $g\in C_r(w^G_L),$ the relevant cell for $w^G_L$, and suppose $g=u_1 w^G_L a u_2$ is the Bruhat decomposition of $g$, where $a\in A_{w^G_L}=Z_L$. Decompose $u_1=u_1^-u_1^+\in U^-_{(w')^{-1}} U^+_{(w')^{-1}}=U$, also $u_2=u_2^+u_2^-\in U^+_{w'}U_{w'}^-=U_M N_M=U$, where $w'=w^G_M$. Therefore $g=u_1w'au_2=u_1^-u_1^+w'au_2^+u_2^-=u_1^-w'(w'^{-1})u_1^+w'au_2^+u_2^-$. Since $C_r(w^G_L)\subset \Omega_{w'}$, by Lemma 6.1, $g$ has a unique decomposition $g=u_1^-w'mu_2^-$, $u_1^-\in U_{(w')^{-1}}^-$, and $u_2^-\in U_{w'}^-$. On the other hand, since $w'(w'^{-1}u_1^+w')w'^{-1}=u_1^+\in U$, so by definition $(w'^{-1})u_1^+w'\in U_{w'}^+=U_M\subset M$. Therefore $(w'^{-1})u_1^+w'au_2^+\in M$. Now compare the two decompositions and by uniqueness of Lemma 6.1, we see that $m=(w'^{-1})u_1^+w'au_2^+$. Now we prove the following transfer principal for partial Bessel integrals: (**Transfer principle for partial Bessel integrals**)For any given $g\in C_r(w^G_L)$, suppose $g=u_1^- w' m u_2^-$, then $$B^G_{\varphi}(g, f)=\psi(u_1^-)\psi(u_2^-)Vol(U_{M,m}\cap n_0 U_{M,m}n_0^{-1}\backslash U_{M,m}) B^M_{\varphi}(m, h_f).$$ where $h_f\mapsto f$ through the surjective map: $C_c^{\infty}(\Omega_{w'};w_{\pi})\twoheadrightarrow C_c^{\infty}(M; w_{\pi}),$ and $n_0=\leftidx{^t}(\overline{u_1^-})(u_2^-)^{-1}\in N_M$. To prove this, we first need to deal with the twisted centralizers in the above two partial Bessel integrals. Suppose that we have a chain of standard Levi subgroups $L\subset M\subset G$ with associated Weyl group elements $w^G_L\in B(G)$ and $w^M_L\in B(M)$ respectively. Then for $g\in C_r(w^G_L)$ with $g=u_1w^G_La u_2=u_1^-w'mu_2^-\in C_r(w^G_L)\subset \Omega_{w'}\simeq U^-_{(w')^{-1}}\times \dot{w}'M\times U_{w'}^-,$ where $a\in A_{w^G_L}=Z_L$ and $w'=w^G_M,$ $u=u_1^-u_1^+\in U^-_{(w')^{-1}} U^+_{(w')^{-1}}=U$, also $u_2=u_2^+u_2^-\in U^+_{w'}U_{w'}^-=U_M N_M=U.$ Then the twisted centralizer of $g$ and $m$ satisfies $$U_g=(\leftidx{^t}(\overline{u_1^-})^{-1}U_{M,m} \leftidx{^t}{\overline{u_1^-}})\cap ((u_2^-)^{-1}U_{M,m}u_2^-)$$ where $\overline{u_1^-}=\dot{w}_G^{-1}u_1^- \dot{w}_G.$ We have $g=u_1 w^G_L a u_2 =u^-_1 u_1^+w' w^M_L a u_2^+u_2^-=u_1^-w'(w'^{-1}u_1^+ w' w^M_L a u_2^+)u_2^-=u_1^-w'mu_2^-$ where $m=w'^{-1}u_1^+w'w^M_Lau_2^+$. Notice that we have $w'^{-1}U^+_{(w')^{-1}}w'=U^+_{w'}=U_M.$ The above decomposition is unique by Lemma 6.1. Now we show that $\overline{u^-_1}=\dot{w}_G^{-1}u_1^-\dot{w}_G\in N_M^-$, or equivalently, $^t\overline{u_1^-}\in U^-_{(w')}=N_M$. To see this, since $u_1^-\in U^-_{(w')^{-1}}\subset U$, $\overline{u^-_1}=\dot{w}_G^{-1}u^-_1\dot{w}_G\in U^-$. On the other hand, we have that $w'^{-1}u_1^-w'=\dot{w}_M\dot{w}_G^{-1}u_1^-\dot{w}_G \dot{w}_M^{-1}=\dot{w}_M\overline{u^-_1}\dot{w}_M^{-1}\in U^-$ by the definition of $u_1^-$. Taking transpose and using the fact that $^t\dot{w}_M=\dot{w}_M^{-1}$ by the way we choose the Weyl group representatives, we see that this is the same as saying $\dot{w}_M\leftidx{^t}{\overline{u_1^-}}\dot{w}_M^{-1}\in U$, this shows that $^t\overline{u^-_1}\in U_{w_M}^-=N_M$. Next, we see that $$u\in U_g\Longleftrightarrow\dot{w}_G^tu\dot{w}_G^{-1}gu=g$$ $$\Longleftrightarrow\dot{w}_G\leftidx{^t}{u^-}\leftidx{^t}{u^+}\dot{w}_G^{-1}u_1^-w'mu_2^-u^+u^-=u_1^-w'mu_2^-$$ $$\Longleftrightarrow \dot{w}_G\leftidx{^t}{u^-}\leftidx{^t}{u^+}\dot{w}_G^{-1}u^-_1\dot{w}_G\dot{w}_M^{-1}mu_2^-u^+u^-=u_1^-w'mu_2^- \cdots\cdots (w'=\dot{w}_G\dot{w}_M^{-1})$$ $$\Longleftrightarrow \dot{w}_G\leftidx{^t}{u^-}\leftidx{^t}{u^+}\overline{u^-_1}\dot{w}_M^{-1}mu^-_2 u^+u^-=u_1^-w'mu_2^-\cdots\cdots(\overline{u_1^-}=\dot{w}_G^{-1}u^-_1\dot{w}_G\in N_M)$$ $$\Longleftrightarrow \dot{w}_G\leftidx{^t}{u^-}\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1}\leftidx{^t}{u^+}\dot{w}_M^{-1}mu_2^-u^+u^-=u_1^-w'mu_2^-$$ $$\Longleftrightarrow \dot{w}_G\leftidx{^t}{u^-}(\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1})\dot{w}_G^{-1}(\dot{w}_G\dot{w}_M^{-1})\dot{w}_M\leftidx{^t}{u^+}\dot{w}_M^{-1}mu^+(u^+)^{-1}u_2u^+u^-=u_1w'mu_2^-$$ $$\Longleftrightarrow (\dot{w}_G\leftidx{^t}{u^-}(\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1})\dot{w}_G^{-1})w'(\dot{w}_M\leftidx{^t}{u^+}\dot{w}_M^{-1}mu^+)((u^+)^{-1}u_2^-u^+u^-)=u_1^-w'mu_2^-$$ We call the last equality $(A)$. Now notice that $\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1}=\leftidx{^t}{(({u^+})^{-1}\leftidx{^t}{\overline{u_1^-}}u^+)}$, and $(({u^+})^{-1}\leftidx{^t}{\overline{u_1^-}}u^+)\in N_M$ since we showed that $\leftidx{^t}{\overline{u_1^-}}\in N_M$ and $u^+\in U_M$, $U_M$ normalizes $N_M$. So we have $\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1}\in N_M^-$. Next, we claim that $\dot{w}_G\leftidx{^t}{u^-}(\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1})\dot{w}_G^{-1}\in U_{(w')^{-1}}^-$. To see this, notice that this is equivalent to $ w'^{-1}\dot{w}_G\leftidx{^t}{u^-}(\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1})\dot{w}_G^{-1} w'\in U^-$, which is the same as saying $\dot{w}_M\leftidx{^t}{u^-}\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1}\dot{w}_M^{-1}\in U^-$, since $ w'^{-1}=\dot{w}_M\dot{w}_G^{-1}.$ Also note that $^t{u^-}\in N_M^-$ and $\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1}\in N_M^-,$ and it is not hard to see that $\dot{w}_MN^-_M\dot{w}_M^{-1}\subset U^-$, so the claim follows. Moreover, clearly we have $\dot{w}_M\leftidx{^t}{u^+}\dot{w}_M^{-1}mu^+\in M$ and $(u^+)^{-1}u_2^-u^+u^-\in N_M$. Summarize what we obtained so far, we have $\dot{w}_G\leftidx{^t}{u^-}(\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1})\dot{w}_G^{-1}\in U_{(w')^{-1}}^-$, $\dot{w}_M\leftidx{^t}{u^+}\dot{w}_M^{-1}mu^+\in M$ and $(u^+)^{-1}u_2u^+u^-\in U_{w'}^-$. In addition, by the uniqueness of the decomposition $\Omega_{w'}=U^-_{(w')^{-1}}\times w'M\times U_{w'}^-$ as in Lemma 6.1 and equality $(A)$, the following three equalities hold at the same time: $(a),\ \ \dot{w}_G\leftidx{^t}{u^-}(\leftidx{^t}{u^+}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1})\dot{w}_G^{-1}=u^-_1;$ $(b), \ \ \dot{w}_M\leftidx{^t}{u^+}\dot{w}_M^{-1}mu^+=m$; $(c), \ \ (u^+)^{-1}u_2u^+u^-=u^-_2$. Notice that $(a)\Longleftrightarrow \leftidx{^t}{u}\overline{u_1^-}(\leftidx{^t}{u^+})^{-1}=\overline{u_1^-}\Longleftrightarrow (u^+)^{-1}\leftidx{^t}{\overline{u_1^-}}u=\leftidx{^t}{\overline{u_1^-}}\Longleftrightarrow u^+=\leftidx{^t}{\overline{u_1^-}}u(\leftidx{^t}{\overline{u_1^-}})^{-1}\Longrightarrow \leftidx{^t}{\overline{u_1^-}}u(\leftidx{^t}{\overline{u_1^-}})^{-1}=u^+\in U_M$. On the other hand, from $(b)$ we see that $u^+\in U_{M,m}$, so $(a)$&$(b)$ implies that $\leftidx{^t}{\overline{u_1^-}}u(\leftidx{^t}{\overline{u_1^-}})^{-1}\in U_{M,m}$. Since we started with $u\in U_g$, we see that $U_g\subset (\leftidx{^t}{\overline{u_1^-}})^{-1} U_{M,m} \leftidx{^t}{\overline{u_1^-}}. $ Similarly, $(c)\Longleftrightarrow u_2^-u(u_2)^{-1}=u^+\Longrightarrow u_2^-u(u_2)^{-1}=u^+\in U_M$ and again by $(b)$ we have $u^+\in U_{M,m}$, therefore $u_2^-u(u_2)^{-1}\in U_{M,m}$. So $(b)\&(c)$ implies that $U_g\subset (u_2^-)^{-1}U_{M,m}u_2^-$. We conclude that $U_g\subset(\leftidx{^t}(\overline{u_1^-})^{-1}U_{M,m} \leftidx{^t}{\overline{u_1^-}})\cap ((u_2^-)^{-1}U_{M,m}u_2^-)$. Conversely, if $u=\leftidx{^t}(\overline{u_1^-})^{-1}u'\leftidx{^t}{\overline{u_1^-}}=(u_2)^{-1}u''u_2^-$ with $u',u''\in U_{M,m}$, we see that $u^+u^-=u=u'(u')^{-1}\leftidx{^t}(\overline{u_1^-})^{-1}u'\leftidx{^t}{\overline{u_1^-}}=u'((u')^{-1}\leftidx{^t}(\overline{u_1^-})^{-1}u')\leftidx{^t}{\overline{u_1^-}}$. Since $u'\in U_M,(u')^{-1}\leftidx{^t}(\overline{u_1^-})^{-1}u'\in N_M$, $U=U_M\times N_M$ and $U_M\cap N_M=\{1\}$, $u^+=u'$ and $u^-=(u')^{-1}\leftidx{^t}(\overline{u_1^-})^{-1}u'\leftidx{^t}{\overline{u_1^-}}.$ Replace $\leftidx{^t}{\overline{u_1^-}}$ by $u_2^-\in N_M$ in the above argument we also obtain $u^+=u''$. This implies $(b)$. Morover, from $u=\leftidx{^t}(\overline{u_1^-})^{-1}u'\leftidx{^t}{\overline{u_1^-}}=(u_2)^{-1}u''u_2^-$, we see that $\leftidx{^t}(\overline{u_1^-})u\leftidx{^t}{\overline{u_1^-}}^{-1}=u'=u^+\Longleftrightarrow (a)$ and $u_2^-u(u_2^-)^{-1}=u''=u^+\Longleftrightarrow (c).$ Since $u\in U_g$ is equivalent to $(a),(b),(c)$ to hold at the same time, hence it proves the reverse inclusion $U_g\supset (\leftidx{^t}(\overline{u_1^-})^{-1}U_{M,m} \leftidx{^t}{\overline{u_1^-}})\cap ((u_2^-)^{-1}U_{M,m}u_2^-)$. So we finally obtain that $U_g=(\leftidx{^t}(\overline{u_1^-})^{-1}U_{M,m} \leftidx{^t}{\overline{u_1^-}})\cap ((u_2^-)^{-1}U_{M,m}u_2^-)$. Now we can show the proposition based on the above lemma: (Proposition 6.5) For any given $g\in C_r(w^G_L)$, $$g=u_1w^G_L a u_2=u_1^- w' m u_2^-\in C_r(w^G_L)\subset \Omega_{w'}=U_{(w')^{-1}}^-\times w'M\times U_{w'}^-$$ By lemma 6.6, $U_g= (\leftidx{^t}(\overline{u_1^-})^{-1}U_{M,m} \leftidx{^t}{\overline{u_1^-}})\cap ((u_2^-)^{-1}U_{M,m}u_2^-)$. To simplify the notations, we denote $n=\leftidx{^t}({\overline{u_1^-}})^{-1}$ and $n_0=\leftidx{^t}{\overline{u_1^-}}(u_2^-)^{-1}$, then they both lie in $N_M$. Since $n\in U$, we have $U=nUn^{-1}=(nU_Mn^{-1})\times(nN_Mn^{-1})$. For $f\in C^{\infty}_c(\Omega_{w'};w_{\pi})$ we have $B_{\varphi}^G(g,f)=\int_{U_g\backslash U}\int_U f(xgu)\varphi(^tu\dot{w}_G^{-1}g'u)\psi^{-1}(xu)dxdu$. Make a change of variable $u\mapsto nun^{-1}$, and decompose $U$ as $U=nUn^{-1}=(nU_Mn^{-1})\times(nN_Mn^{-1})$. Then $U_g=(\leftidx{^t}(\overline{u_1^-})^{-1}U_{M,m} \leftidx{^t}{\overline{u_1^-}})\cap ((u_2^-)^{-1}U_{M,m}u_2^-)=n(U_{M,m}\cap n_0U_{M,m}n_0^{-1})n^{-1}.$ We can rewrite the integral as $$B^G_{\varphi}(g,f)=\int_{n(U_{M,m}\cap n_0U_{M,m}n_0^{-1})n^{-1}\backslash nU_Mn^{-1}}\int_{U_{w'}^-}\int_{U^+_{(w')^{-1}}}\int_{U^-_{(w')^{-1}}}$$ $$f(x^-x^+u_1^-w'mu_2^-nu^+u^-n^{-1})\varphi(^t{n^{-1}}^t{u^-}^t{u^+}^tn\dot{w}_G^{-1}u_1^-\dot{w}_G\dot{w}_M^{-1}m'u_2^-nu^+u^-n^{-1})$$ $$\cdot \psi^{-1}(x^-x^+)\psi^{-1}(nu^+u^-n^{-1})dx^-dx^+du^-du^+$$ $$=\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{N_M}\int_{U_{(w')^{-1}}^+}\int_{U_{(w')^{-1}}^-}f(x^-x^+u_1^-(x^+)^{-1}w'(w')^{-1} x^+$$ $$\cdot w'mu^+(u^+)^{-1}u_2^-nu^+u^-n^{-1})\varphi(^t{n^{-1}}^t{u^-}^t{u^+}(\overline{u_1^-})^{-1}\overline{u_1^-}\dot{w}_M^{-1}m'u_2^-nu^+u^-n^{-1})$$ $$\cdot \psi^{-1}(x^-x^+)\psi^{-1}(nu^+u^-n^{-1})dx^-dx^+du^-du^+$$ $$=\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{N_M}\int_{U_{(w')^{-1}}^+}\int_{U_{(w')^{-1}}^-}f(x^-x^+u_1^-(x^+)^{-1}w'(w')^{-1} x^+$$ $$\cdot w'mu^+(u^+)^{-1}u_2^-nu^+u^-n^{-1})\varphi(^t{n^{-1}}^t{u^-}^t{u^+}\dot{w}_M^{-1}m'u_2^-nu^+u^-n^{-1})$$ $$\cdot \psi^{-1}(x^-x^+)\psi^{-1}(nu^+u^-n^{-1})dx^-dx^+du^-du^+$$ Now let $x'=w'^{-1}x^+w'$, then $x'\in U_M$, and by compatibility we have $\psi(x')=\psi(x^+)$. Moreover, let $y^-=x^-x^+u_1^-(x^+)^{-1}$, then since $U_{(w')^{-1}}^+$ normalizes $U_{(w')^{-1}}^-$, we see that $x^+u_1^-(x^+)^{-1}\in U_{(w')^{-1}}^-$. As a result, we have $y^-\in U_{(w')^{-1}}^-$. Let $v^-=(u^+)^{-1}u_2^-nu^+u^-n^{-1}\in N_M$. And also let $u'=u^+$. Then since all variables live in unipotent subgroups therefore are all unimodualr, we see that $dy^-=dx^-$, $dv^-=du^-$, and $du'=du^+$. After making the above change of variable, the above integral $$=\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{N_M}\int_{U_{(w')^{-1}}^-}\int_{U_M}f(y^-w'x'mu'v^-)$$ $$\cdot \varphi(^t{v^-}^t{u'}^t{(u_2^-)^{-1}}^t{n^{-1}}\dot{w}_M^{-1}m'u'v^-)\psi(u_1^-)\psi(u_2^-)\psi^{-1}(y^-)\psi^{-1}(x')\psi^{-1}(v^-)\psi^{-1}(u')$$ $$dx'dy^-dv^-du'$$ Since here $f\in C^{\infty}_c(\Omega_{w'};w_{\pi})$, the decomposition $\Omega_{w'}=U^-_{(w')^{-1}}\times w' M \times U_{w'}^-$ implies that there exists open compact subsets $U_1\subset U_{(w')^{-1}}^-$, and $U_2 \subset U_{w'}^- $ such that $f(y^-w'x'mu'v^-)\neq 0\Longrightarrow y^-\in U_1, v^-\in U_2 $. Therefore we can take $N$ large enough, such that $\varphi=\varphi_N$ is invariant under large open compact subgroups of $U_{w'}^-$, as in Lemma 4.2 \[7\]. Consequently, $\varphi(^t{v^-}^t{u'}^t{(u_2^-)^{-1}}^t{n^{-1}}\dot{w}_M^{-1}m'u'v^-)=\varphi(^t{u'}^t{(u_2^-)^{-1}}^t{n^{-1}}\dot{w}_M^{-1}m'u').$ So now we have $$B^G_{\varphi}(g,f)=\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{N_M}\int_{U_{(w')^{-1}}^-}\int_{U_M}f(y^-w'x'mu'v^-)$$ $$\cdot\varphi(^t{u'}^t{(u_2^-)^{-1}}^t{n^{-1}}\dot{w}_M^{-1}m'u')\psi(u_1^-)\psi(u_2^-)\psi^{-1}(y^-)\chi^{-1}(x')\psi^{-1}(v^-)\psi^{-1}(u')$$ $$dx'dy^-dv^-du'$$ $$=\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{U_{w'}^-}\int_{U_{(w')^{-1}}^-}\int_{U_M}f(y^-w'x'mu'v^-)$$ $$\cdot\varphi(^t{u'}^t{n_0^{-1}}\dot{w}_M^{-1}m'u')\psi(u_1^-)\psi(u_2^-)\psi^{-1}(y^-)\psi^{-1}(x')\psi^{-1}(v^-)\psi^{-1}(u')$$ $$dx'dy^-dv^-du'$$ Now by Lemma 6.2, there exists an $h=h_f\in C^{\infty}_c(M;w_{\pi})$ such that $$h(m)=h_f(m)=\int_{U^-_{w'}}\int_{U^-_{(w')^{-1}}}f(x^-\dot{w}mu^-)\chi^{-1}(x^-u^-)dx^-du^-.$$ This implies that $$B^G_{\varphi}(g,f)=\chi(u_1^-)\chi(u_2^-)\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{U_M}h_f(x'mu')$$ $$\cdot\varphi(^t{u'}^t{n_0^{-1}}\dot{w}_M^{-1}m'u')\chi^{-1}(x')\chi^{-1}(u')dx'du'$$ Since $h_f\in C^{\infty}_c(M;w_{\pi})$, $x',u'\in U_M$, and $m$ is fixed (since $g$ is fixed, and $m$ is given by $g$ uniquely through the decomposition $g=u_1^-w'm u_2^-$). This implies that there exist open compact subset $U'\subset U_M$ such that $h_f(x'mux)\neq 0\Longrightarrow x',u'\in U'$, since $U_MmU_M$ is closed in M and $\textrm{Supp}(h_f)$ is compact in $M$ modulo $Z$, and $U_M\cap Z=\{1\}$. Now since $u'\in U'$ which is compact, so is its continuous image $\{^tu'{^t}{n_0^{-1}}\leftidx{^t}{(u')^{-1}}: u\in U'\}$. Again by Lemma 4.2 \[7\] we can take $N$ large enough so that $\varphi=\varphi_N$ is left invariant under some large open compact subgroup of $U_M$ which contains the compact subset $\{^tu'^t{n_0^{-1}}^t{(u')^{-1}}: u'\in U'\}.$ Therefore we have $$\varphi(^t{u'}^t{n_0^{-1}}\dot{w}_M^{-1}m'u')=\varphi(^t{u'}^t{n_0^{-1}}^t{(u')^{-1}}^t{u'}\dot{w}_M^{-1}m'u')=\varphi(^t{u'}\dot{w}_M^{-1}m'u')$$ From this we see that the above integral $$B^G_{\varphi}(g,f)=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}\int_{U_M}h_f(x'mu')\varphi(^t{u'}\dot{w}_M^{-1}m'u')$$ $$\cdot \psi^{-1}(x')\psi^{-1}(u')dx'du'$$ Break the integral over ${U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_M}$ into integral over $U_{M,m}\backslash U_M$ and $U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}$. Write $u'=\tilde{u}u''$ with $\tilde{u}\in U_{M,m}$, $u''\in U_M$ and denote the measure on $U_{M,m}\backslash U_M$ and $U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}$ by $du''$ and $d\tilde{u}$ respectively. We have $$B^G_{\varphi}(g,f)=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}\int_{U_M}h_f(x'm\tilde{u}u'')$$ $$\cdot \varphi(^t{u''}^t{\tilde{u}}\dot{w}_M^{-1}m'\tilde{u}u'')\psi^{-1}(x')\psi^{-1}(\tilde{u}u'')dx'd\tilde{u}du''$$ Now since $\tilde{u}\in U_{M,m}$, by definition $^t{\tilde{u}\dot{w}_M^{-1}m'\tilde{u}}=\dot{w}_M^{-1}m'$ (recall that $m=zm'$ with $z\in Z$). Therefore the above integral $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}\int_{U_M}h_f(x'm\tilde{u}u'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(x')\psi^{-1}(\tilde{u}u'')dx'd\tilde{u}du''$$ $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}(\int_{U_M}h_f(x'm\tilde{u}u'')\psi^{-1}(x')dx')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(\tilde{u}u'')d\tilde{u}du''$$ $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}W^{h_f}(m\tilde{u}u'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(\tilde{u}u'')d\tilde{u}du''$$ Using $^t{\tilde{u}\dot{w}_M^{-1}\tilde{u}}=\dot{w}_M^{-1}m$ again, we see that this $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}W^{h_f}(\dot{w}_M\leftidx{^t}{\tilde{u}}^{-1}\dot{w}_Mmu'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(\tilde{u}u'')d\tilde{u}du''$$ $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}\psi(\dot{w}_M\leftidx{^t}{\tilde{u}}^{-1}\dot{w}_M)W^{h_f}(mu'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(\tilde{u}u'')d\tilde{u}du''$$ By compatibility of $\psi$ and $\dot{w}_M$, we have $\psi(\dot{w}_M\leftidx{^t}{\tilde{u}}\dot{w}_M^{-1})=\psi(\tilde{u})$, so this $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}\psi(\tilde{u})W^{h_f}(mu'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(\tilde{u}u'')d\tilde{u}du''$$ $$=\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}\int_{U_{M,m}\cap n_0U_{M,m}n_0^{-1}\backslash U_{M,m}}W^{h_f}(mu'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(u'')d\tilde{u}du''$$ $$=Vol(U_{M,m}\cap n_0 U_{M,m}n_0^{-1}\backslash U_{M,m})\psi(u_1^-)\psi(u_2^-)\int_{U_{M,m}\backslash U_M}W^{h_f}(mu'')$$ $$\cdot \varphi(^t{u''}\dot{w}_M^{-1}m'u'')\psi^{-1}(u'')d\tilde{u}du''$$ $$=\psi(u_1^-)\psi(u_2^-)Vol(U_{M,m}\cap n_0 U_{M,m}n_0^{-1}\backslash U_{M,m}) B^M_{\varphi}(m, h_f)$$ $$=c(u_1^-,u_2^-)B^M_{\varphi}(m,h_f),$$ where $c(u_1^-,u_2^-)=\psi(u_1^-)\psi(u_2^-)Vol(U_{M,m}\cap n_0 U_{M,m}n_0^{-1}\backslash U_{M,m})$ is a constant depending smoothly on $u_1^-$ and $u_2^-$. Recall that $n_0=^t{\overline{u_1^-}}(u_2^-)^{-1}\in N_M$. The convergence of the above integrals are controlled by the convergence of $B^G_{\varphi}(g,f)$ we started with. So there is no convergence issue in the above process. Small cell Analysis ------------------- The philosophy to prove supercuspidal stability is to analyze the asymptotic behavior of the partial Bessel integrals through looking at the contribution of each Bruhat cell inductively. In this section we will analyze the small cell of both $G$ and its Levi subgroups. The following lemmas(lemma 6.7, 6.8, 6.9), which were proved in \[7\], show that the non-zero contributions are only from the relevant parts of those Bruhat cells that support Bessel functions. We will use them, together with the transfer principal(proposition 6.5) to obtain the asymptotic expansion for partial Bessel integrals. Let $w\in B(G)$ and $f\in C_c^{\infty}(\Omega_w;\omega_{\pi}).$ Suppose $B^G_{\varphi}(\dot{w}a,f)=0$ for all $a\in A_w$. Then there exists $f_0\in C^{\infty}(\Omega'_w;\omega_{\pi})$, where $\Omega'_w=\Omega_w-C_r(\dot{w})$, such that for sufficiently large $\varphi$ depending only on $f$, we have $B^G_{\varphi}(g,f)=B^G_{\varphi}(g,f_0)$ for all $g\in G$. See Lemma 5.12, \[7\]. Let $w\in B(G)$ and $f\in C^{\infty}_c(\Omega;\omega_{\pi})$, $\Omega_w^\circ=\Omega_w-C(w)$. Suppose $B^G(\dot{w}a,f)=0$ for all $a\in A_w$. Then there exists $f_0\in C^{\infty}_c(\Omega_w^\circ,\omega_{\pi})$ such that, for all sufficiently large $\varphi$ depending only on $f$, we have $B^G_{\varphi}(g,f)=B^G_{\varphi}(g,f_0)$ for all $g\in \Omega_w$. See Lemma 5.13, \[7\]. Let $w=w_Gw_M\in B(G)$. Let $\Omega_{w,0}$ and $\Omega_{w,1}$ be $U\times U $ and $A$-invariant open sets of $\Omega_w$ such that $\Omega_{w,0}\subset \Omega_{w,1}$ and $\Omega_{w,1}-\Omega_{w,0}$ is a union of Bruhat cells $C(w')$ such that $w'$ does not support a Bessel function, i.e, $w'\notin B(G)$. Then for any $f_1\in C^{\infty}_c(\Omega_{w,1};\omega_{\pi})$, there exists $f_0 \in C^{\infty}_c(\Omega;\omega_{\pi})$ such that, for all sufficiently large $\varphi$ depending only on $f_1$, we have $B^G_{\varphi}(g,f_0)=B^G_{\varphi}(g,f_1)$ for all $g\in G$. See Lemma 5.14, \[7\]. Now let’s work on the inductive process of the asymptotic expansion of partial Bessel integrals. We begin with the analysis of the small cell of $G$. Consider $e$ as a Weyl group element, then $M_e=G, A_e=Z_G=Z$, and $U_e^+=U.$ We also have $\Omega_e=\bigsqcup_{e\leq w}C(w)=G$. Take the representative of $e$ to be $\dot{e}=I.$ Take $f\in \mathcal{M}(\pi)\subset C^{\infty}_c(G;\omega_{\pi})$ with $W^f(e)=1.$ We also fix an auxiliary function $f_0\in C^{\infty}_c(G;\omega_{\pi})$ such that $W^{f_0}(e)=1$. Decompose $G=G^dA_e=G^dZ$, where $G^d$ is the derived group of $G$. Since $G^d\cap Z$ is finite, if we write $g=g_1c$ for $g\in G$ and $g_1\in G^d$, $c\in Z$, then there are only finitely many such decompositions and they differ by elements in the transverse torus $A^e_e$. In the case of $G=GL_n$, $A^e_e$ consists of diagonal matrices whose entries are n-th roots of unity, and notice that there is no such decomposition if $\det(g)$ is not an $n$-th power in $F^\times$. Now let $$f_1(g)=\sum_{g=g_1c}f_0(g_1)B^G(\dot{e}c, f)=\sum_{g=g_1c}f_0(g_1)\omega_{\pi}(c)$$ if $\det(g)$ is an $n$-th power in $F^\times$, and $f_1(g)=0$ otherwise. Then $f_1(g)\in C_c^{\infty}(G;\omega_{\pi})$, since the subgroup of all $g\in G$ such that $\det(g)$ is an $n$-th power in $F^\times$ is open in $G$. We have $B^G_{\varphi}(\dot{e}a,f_1)=B^G_\varphi(\dot{e}a, f)$ for all $a\in A_e=Z.$ See Lemma 5.15, \[7\]. Fix an auxiliary function $f_0\in C^{\infty}_c(G;\omega_{\pi})$ with $W^{f_0}(e)=1$. Then for each $f\in C_c^{\infty}(G;\omega_{\pi})$ with $W^f(e)=1$ and for each $w'\in B(G)$ with $d_B(e,w')=1$, there exists a function $f_{w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$ such that for any $w\in B(G)$ and any $g=u_1\dot{w}au_2 \in C_r(w)$ we have $$B^G_{\varphi}(g,f)=\sum_{w'\in B(G), d_B(w',e)=1}B^G_{\varphi}(g,f_{w'})+\sum_{a=bc}\omega_\pi(c)B^G_{\varphi}(u_1\dot{w}bu_2,f_0)$$ where $a=bc$ runs over the possible decompositions of $a\in A_w$ with $b\in A^e_w$ and $c\in A_e=Z.$ We construct $f_1$ from $f_0$ as above. By Lemma 6.10, $B^G_{\varphi}(\dot{e}a,f-f_1)=0$ for all $a\in A_e=Z$. We have $C_r(e)=A_eU=ZU\subset C(e)=AU$ and $\Omega_e^{\circ}=\Omega_e-C(e)=G-AU=\bigsqcup_{w\neq e}C(w)$. Then by Lemma $6.8$, there exists an $f_2'\in C^{\infty}_c(\Omega_e^{\circ};\omega_{\pi})$ such that $B^G_{\varphi}(g,f-f_1)=B^G_{\varphi}(g,f'_2)$ for all $g\in G$. Let $\Omega_1=\bigcup_{w\in B(G), w\ne e}\Omega_w=\bigcup_{w'\in B(G), d_B(w',e)=1}\Omega_{w'}=\bigsqcup_{w''\ge w'\in B(G), d_B(w',e)=1}C(w'')$ and $\Omega_0=\Omega_e^{\circ}=G-C(e)=\bigsqcup_{w\neq e}C(w).$ So $\Omega_0 - \Omega_1$ is a union of Bruhat cells $C(w)$ such that $w\notin B(G)$, since $d_B(w',e)=1$ in the definition of $\Omega_1$. By Lemma 6.9, there exists $f_2\in C^{\infty}_c(\Omega_1,\omega_{\pi})$ such that for sufficiently large $\varphi$ we have $B^G_{\varphi}(g,f_2)=B^G_{\varphi}(g,f'_2)=B^G_{\varphi}(g,f-f_1)$ for all $g\in G$. Then we use a partition of unity argument, to get $f_2=\sum_{w'\in B(G),d_B(w',e)=1}f_{w'}$ with $f_{w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi}).$ Thus for any $w\in B(G)$ and any $g\in C_r(w)$ we have $$B^G_{\varphi}(g,f)=B^G_{\varphi}(g,f_1)+\sum_{w'\in B(G),d_B(w',e)=1}B^G_{\varphi}(g,f_{w'})$$ Now we work with $B^G_{\varphi}(g, f_1)$ for $g\in C_r(w)$. We have $$B^G_{\varphi}(g,f_1)=\int_{U_g\backslash U}\int_Uf_1(xgu)\varphi(^tu\dot{w}_G^{-1}g'u)\psi^{-1}(x)\psi^{-1}(u)dxdu$$ $$=\int_{U_g\backslash u_2^{-1}U^+_w u_2}\int_{U_w^-}\int_Uf_1(xgu'^+u_2^{-1}u^-u_2)\varphi(^t{(u_2^{-1}u^-u_2)}^t{u'^+}\dot{w}_G^{-1}g'u'^+u_2^{-1}u^-u_2)$$ $$\cdot \psi^{-1}(x)\psi^{-1}(u'^+u_2^{-1}u^-u_2)dxdu^-du'^+$$ Since $f_1(g)=\sum_{g=g_1c}f_0(g_1)B^G(\dot{e}c, f)=\sum_{g=g_1c}f_0(g_1)\omega_{\pi}(c),$ we need to decompose $xgu'^+u_2^{-1}u^-u_2=g_1c$ with $g_1\in G^d$ and $c\in Z$. Write $g=u_1\dot{w}au_2$, then $g_1=xu_1\dot{w}ac^{-1}u_2u'^+u_2^{-1}u^-u_2\in G^d$. So $1=\det(g_1)=\det(ac^{-1}).$ This says that $b=ac^{-1}\in A_w^e=SL_n(F)\cap Z_L$, where $L=L_w$ is the Levi given by $w=w_Gw_L\in B(G)$. We decompose $A_w=ZA_{w'}$, then $a=za'$ and $a'=(bc)'=b'.$ Therefore we have $$f_1(xgu'^+u_2^{-1}u^-u_2)=\sum_{a=bc}f_0(xu_1\dot{w}bu_2u'^+u_2^{-1}u^-u_2)\omega_{\pi}(c).$$ So eventually we have $$B^G_{\varphi}(g,f_1)=\int_{U_g\backslash U}\int_Uf_1(xgu)\varphi(\leftidx{^t}u\dot{w}_G^{-1}g'u)\psi^{-1}(x)\psi^{-1}(u)dxdu$$ $$=\int_{U_g\backslash u_2^{-1}U^+_w u_2}\int_{U_w^-}\int_Uf_1(xgu'^+u_2^{-1}u^-u_2)\varphi(\leftidx{^t}{(u_2^{-1}u^-u_2)}\leftidx{^t}{u'^+}\dot{w}_G^{-1}g'u'^+u_2^{-1}u^-u_2)$$ $$=\omega_{\pi}(c)\sum_{a=bc}\int_{U_g\backslash u_2^{-1}U^+_w u_2}\int_{U_w^-}\int_Uf_0(xu_1\dot{w}bu_2u'^+u_2^{-1}u^-u_2)\varphi(\leftidx{^t}{(u_2^{-1}u^-u_2)}\leftidx{^t}{u'^+}\dot{w}_G^{-1}$$ $$\cdot u_1\dot{w}bu_2u'^+u_2^{-1}u^-u_2)\psi^{-1}(x)\psi^{-1}(u'^+u_2^{-1}u^-u_2)dxdu^-du'^+$$ $$=\sum_{a=bc}\omega_{\pi}(c)\int_{U_g\backslash U}\int_U f(xu_1\dot{w}bu_2u)\varphi(\leftidx{^t}u\dot{w}_G^{-1}u_1\dot{w}bu_2u)\psi^{-1}(x)\psi^{-1}(u)dxdu$$ $$=\sum_{a=bc}\omega_{\pi}(c)B^G_{\varphi}(u_1\dot{w}bu_2,f_0).$$ A very similar process works for Levi subgroups $M\subset G$. If $w'=w_Gw_M\in B(G)$, then $A^{w'}_{w'}=Z_M\cap M^d$, which is also finite. In the case $G=\textrm{GL}_n$, $M^d\simeq SL_{n_1}\times \cdots\times SL_{n_t}$ for some $t\ge 1$, and $A^{w'}_{w'}=A_w\cap (M_{w'})^d$ consists of $n_i$-th roots of unity in the $i$-th block of $M$. Let’s analyze the small cell of $M$. For $h\in C_c^{\infty}(M;\omega_{\pi})$, and $c\in Z_M=A_{w'}$, define the Bessel integral on $M$ by $B^M(c,h)=\int_{U_M}h(xc)\psi^{-1}(x)dx$. Take $h_0\in C_c^{\infty}(M;\omega_{\pi})$, such that $B^M(e, h_0)=\frac{1}{\kappa_M}$, where $\kappa_M=\vert Z\cap A_{w'}^{w'}\vert <\infty$, and $B^M(b, h_0)=0$ for $b\in A^{w'}_{w'}$ but $b\notin Z\cap A^{w'}_{w'}$. Decompose $M=M^d Z_M$, where $M^d\cap Z_M=A^{w'}_{w'}$ is finite. Define $h_1$ on $M$ by $h_1(m)=\sum_{m=m'c}h_0(m')B^M(c,h)$ with $m'\in M^d$ and $c\in Z_M=A_{w'}$. Similar to the case for $G$, if $m=\textrm{diag}\{m_1,m_2,\cdots, m_r\}$, $\det(m_i)$ is not an $n_i$-th power on each block, then $h_1(m)=0$. We have $B^M_{\varphi}(a, h_1)=B^M_{\varphi}(a,h)$ for all $a\in Z_M=A_{w'}$. See Proposition 5.4, \[7\]. Now suppose $g\in C_r(\dot{w}_G)$ with $g=u_1\dot{w}_G a u_2$, then for $w'=\dot{w}_G\dot{w}_M^{-1}$ we have $$C_r(\dot{w}_G)\subset \Omega_{w'}=U_{(w')^{-1}}^-\times w'M\times U_{w'}^-.$$ We further decompose $g$ as $=u_1^-u_1^+\dot{w}_Gau_2^+u_2^-$ with $u_1^-\in U_{(w')^{-1}}^-$, $u_1^+\in U_{(w')^{-1}}^+$, $u_2^+\in U_{w'}^+$, $u_2^-\in U_{w'}^-$, $u_1=u_1^-u_1^+$, $u_2=u_2^+u_2^-$. Then $$g=u_1^-w'(w')^{-1}u_1^+w'\dot{w}_Mau_2^+u_2^-=u_1^-w'mu_2^-$$ where $m=(w')^{-1}u_1^+w'\dot{w}_Mau_2^+\in C_r^M(\dot{w}_M)$, the relevant cell of $\dot{w}_M$ in $M$, and $a\in A_{w_G}=A$. Then $$B^M_{\varphi}(m,h_1)=\int_{U_{M,m}\backslash U_M\times U_M}h_1(xmu)\varphi(^tu\dot{w}_M^{-1}m'u)\psi^{-1}(xu)dxdu$$ where $m'=(w')^{-1}u_1^+w'\dot{w}_Ma'u_2^+$. Here $a=za'$ is the decomposition of $a\in A=ZA'$. It follows that $B^M_{\varphi}(m,h_1)=\omega_{\pi}(z)B^M_{\varphi}(m', h_1)$. Since $h_1(m)=\sum_{m=m_1c}h_0(m_1)B^M(m',h)$ with $m_1\in M^d$ and $c\in Z_M$, to compute the above integral, we need to decompose $xm'u=m_1c$. This gives $xw'^{-1}u_1^+w'\dot{w}_M^{-1}a'u_2^+uc^{-1}=m_1\in M^d$. Since $x,w',\dot{w}_M, u,u_1^+,u_2^+\in M^d$, it suffices to decompose $a'=bc$ for $b\in A\cap M^d$ and $c\in Z_M$. Now we can write $$h_1(m')=\sum_{a'=bc}h_0(xw'^{-1}u^+_1w'\dot{w}_Mbu_2^+u)B^M(c,h)$$ Decompose $b=z_bb'$ and $c=z_cc'$, with $z_b,z_c\in Z$, $b'\in A'$ and $c'\in Z_M'$. Then $a'=bc=z_bz_cb'c'\Longrightarrow a'=b'c'$, and $z_bz_c=1.$ As $h,h_0\in C^{\infty}_c(M;\omega_{\pi})$, we have $$h_0(xw'^{-1}u^+_1w'\dot{w}_Mbu^+_2u)B^M(c,h)=\omega_{\pi}(z_bz_c)h_0(xw'^{-1}u_1^+w'\dot{w}_Mb'u_2^+u)B^M(c',h)$$ $$=h_0(xw'^{-1}u_1^+w'\dot{w}_Mb'u_2^+u)B^M(c',h)$$ Thus $$B^M(m',h_1)=\int_{U_{M,m}\backslash U_M\times U_M}\sum_{a=bc}h_0(xw'^{-1}u_1^+w'\dot{w}_Mb'u_2^+u)B^M(c',h)$$ $$\cdot \varphi(^tu \dot{w}_M^{-1}w'^{-1}u_1^+w'\dot{w}_Mb'u^+_2uc')\psi^{-1}(xu)dxdu$$ $$=\sum_{a'=b'c'}B^M(c',h)\int_{U_{M,m}\backslash U_M\times U_m}h_0(xw'^{-1}u_1^+w'b'u^+_2u)$$ $$\cdot \varphi(^tu \dot{w}_M^{-1}w'^{-1}u_1^+w'\dot{w}_Mb'u^+_2uc')\psi^{-1}(xu)dxdu$$ Now since $a'=b'c'$, $c'\in Z_M'\subset Z_M$, let $m_{b'}=w'^{-1}u_1^+w'\dot{w}_Mb'u^+_2$, then $$m'=w'^{-1}u_1^+w'\dot{w}_Ma'u_2^+=w'^{-1}u_1^+w'\dot{w}_Mb'c'u_2^+=m_{b'}c'$$ Meanwhile we have $U_{M,m'}=\{u\in U_M: {^tu}\dot{w}_M^{-1}m'u=\dot{w}_M^{-1}m'\}=\{u\in U_M: \dot{w}_M{^tu}\dot{w}_M^{-1}m'u=m'\}=\{u\in U_M: \dot{w}_M{^tu}\dot{w}_M^{-1}m_{b'}c'u=m_{b'}c'\}=\{u\in U_M: \dot{w}_M{^tu}\dot{w}_M^{-1}m_{b'}uc'=m_{b'}c'\}=\{u\in U_M:\dot{w}_M{^tu}\dot{w}_M^{-1}m_{b'}u=m_{b'}\}=U_{M,m_{b'}}$ So we obtain $$B^M_{\varphi}(m',h_1)=\sum_{a'=b'c'}B^M(c',h)\int_{U_{M,m_{b'}}\backslash U_M\times U_M}h_0(xm_{b'}u)\varphi({^tu}\dot{w}_M^{-1}m_{b'}uc')\psi^{-1}(xu)dxdu$$ $$=\sum_{a'=b'c'}B^M(c',h)B^M_{\varphi^{c'}}(m_{b'},h_0)$$ where $\varphi^{c'}(m)=\varphi(mc')$ for $c'\in Z_M'$. Uniform smoothness ------------------ The key to prove supercuspidal stability is that the asymptotic expansions of partial Bessel integrals have two parts, one part depends only on the central character of $\pi$, the other is a uniform smooth function on certain torus. Therefore under highly ramified twist, the uniform smooth part becomes zero. We study the uniform smoothness in this section. A smooth function $B$ on a torus $T\subset A$ is **uniformly smooth** if there exists a fixed open compact subgroup $T_0\subset T$ such that $B(tt_0)=B(t)$ for $t_0\in T_0$ and all $t\in T$. For $m=m(a)=\tilde{u}_1(a)\dot{w}_Ma \tilde{u}_2(a)\in C^M_r(\dot{w}_M)$, $a\in A^{w'}_{w_G}A_{w'}\subset A_{w_G}=A$, $\tilde{u}_1(a)$ and $\tilde{u}_2(a)$ are rational functions(as morphisms of algebraic varieties) of $a$. Let $a=bc$ be a fixed decomposition with $b\in A^{w'}_{w_G}$ and $c\in A_{w'}$. Then all decompositions are of the form $a=(b\zeta^{-1})(\zeta c)$ with $\zeta\in A^{w'}_{w'}=A^{w'}_{w_G}\cap A_{w'}$, a finite set with appropriate roots of unity on the diagonal. Moreover, if $c=c'z$ with $c'\in A_{w'}'=Z_M'$ and $z\in Z$, then $$B^M_{\varphi}(m(a),h_1)=\omega_{\pi}(z)B^M_{\varphi}(\tilde{u}_1(bc'z)\dot{w}_Mbc'\tilde{u}_2(bc'z), h_1)$$ is uniformly smooth as a function of $c'\in Z_M'$. First fix one decomposition $a=bc$. To simplify the notation, we denote $\tilde{u}_i=\tilde{u}_i(a)$. Then we have $B^M_{\varphi}(m,h_1)=\sum_{a=bc}B^M(c,h)B^M_{\varphi^c}(m_{b'}, h_0)=\sum_{\zeta}B^M(\zeta c, h)B^M_{\varphi^{\zeta c}}(\tilde{u}_1\dot{w}_Mb\zeta^{-1}\tilde{u}_2,h_0)$. Since $\vert\zeta\vert=1$, so we have $\varphi^{\zeta c}=\varphi^c$. This implies that $B^M_{\varphi}(m,h_1)=\sum_{\zeta}B^M(\zeta c, h)B^M_{\varphi^c}(\tilde{u}_1\dot{w}_Mb\zeta^{-1}\tilde{u}_2,h_0).$ Now $$B^M(\zeta c, h)=\int_{U_M}h(x\zeta c)\psi^{-1}(x)dx=\omega_{\pi}(\zeta_1 z)\int_U h(x\zeta'c')\psi^{-1}(x)dx$$ where $\zeta=diag(\zeta_1 I_{n_1},\cdots, \zeta_t I_{n_t})$ and $\zeta'=diag(I_{n_1},\zeta_1^{-1}\zeta_2I_{n_2}\cdots, \zeta_1^{-1}\zeta_{t}I_{n_t}).$ Since $h\in C^{\infty}_c(M;\omega_{\pi})$, $x\zeta c\in A_MU_M=B_M$ and $C^M(e_M)=B_M$ is closed in $M$, there exists compact subsets $U_1\subset U$, $K''\subset A'$ s.t. $h(x\zeta'c')\neq 0 \Longrightarrow x\in U_1, \zeta'c'\in K'.$ Moreover, since $Z_M'\subset A'$ is closed and $\zeta'c'\in Z_M'$, there exists a further compact subset $K''\subset Z_M'$ s.t. $h(x\zeta'c')\neq 0 \Longrightarrow x\in U_1, \zeta'c'\in K''.$ Write $a=bc=bc'z$, we see that $$B^M_{\varphi}(m,h_1)=\omega_{\pi}(z)\sum_{\zeta}B^M(\zeta c',h)B_{\varphi^c}^M(\tilde{u}_1\dot{w}_Mb\zeta^{-1}\tilde{u}_2, h_0)$$ is zero unless $c'\in \bigcup_{\zeta'}(\zeta')^{-1}K''$, which is compact since it is a finite union of compact subsets. So $$B^M_{\varphi}(m,h_1)=B^M_{\varphi}(\tilde{u}_1\dot{w}_Ma\tilde{u}_2, h_1)=B^M_{\varphi}(\tilde{u}_1\dot{w}_Mbc'\tilde{u}_2,h_1)$$ $$=\omega_{\pi}(z)B^M_{\varphi}(\tilde{u}_1\dot{w}_Mbc'\tilde{u}_2, h_1)$$ has compact support on $c'\in Z_M'$, depending only on $h$ through the choice of $K''$ and $A_{M^d}\cap Z_M$. Thus independent of $a$ and $b$. Since $h$ is smooth and its support in $c'$ is compact, there exists uniform compact subset $C_1\subset Z_M'$ s.t. $h(x\zeta c'c_1)=h(x\zeta c')$ for all $c_1\in C_1$, $x\in U_1$, and $c'\in Z_M'.$ Shrink $C_1$ if necessary, we may assume that $C_1\subset Z_M'(\mathcal{O}_F)$. Notice that even if here $B^M_{\varphi^c}(\tilde{u}_1\dot{w}_Mb\zeta^{-1}\tilde{u}_2, h_0)$ also depends on $c'$, the uniform smoothness is already well controlled by the behavior of $B^M(\zeta c', h)$ by the above analysis. So we have proved that $$B^M_{\varphi}(\tilde{u}_1(a)\dot{w}_Mac_1\tilde{u}_2(a),h_1)=B^M_{\varphi}(\tilde{u}_1(bc'z)\dot{w}_Mbzc'c_1\tilde{u}_2(bc'z),h_1)$$$$=B^M_{\varphi}(\tilde{u}_1(bc'z)\dot{w}_Mbc'z\tilde{u}_2(bc'z),h_1)=\omega_{\pi}(z)B^M_{\varphi}(\tilde{u}_1(bc'z)\dot{w}_Mbc'\tilde{u}_2(bc'z),h_1)$$ for all $c_1\in C_1$, $a=bc.$ Finally note that since $A_{w_G}^{w'}A_{w'}\subset A_{w_G}=A$ is open of finite index, one can extend $B^M_{\varphi}(\tilde{u}_1(a)\dot{w}_Ma\tilde{u}_2(a), h_1)$ on all of $A$. Asymptotic expansions --------------------- We are ready to establish a more general version of the asymptotic expansion formula for partial Bessel integrals as in \[7\]. The formula that will be established works for all elements in the relevant Bruhat cells. The following proposition is the key to prove the main results in this section. Let $w'=\dot{w}_G\dot{w}_M^{-1}\in B(G)$, and $f_{w'}\in C_c^{\infty}(\Omega_{w'};\omega_{\pi})$. There exists $f_{1,w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$, such that (1), $\exists$ a family of functions $ \{f_{w''}\}_{w''\in B(G)}$ with $d_B(w'',w')=1$, $w''>w'$, such that $f_{w''}\in C_c^{\infty}(\Omega_{w''};\omega_{\pi})$, and for $\forall w\in B(G)$ and $g\in C_r^G(w)$, we have $$B^G_{\varphi}(g,f_{w'})=B^G_{\varphi}(g,f_{1,w'})+\sum_{w''\in B(G),w''>w', d_B(w'',w')=1} B^G_{\varphi}(g,f_{w''});$$ (2), Let $g=u_1(a)\dot{w}_G a u_2(a)\in C^G_r(\dot{w}_G)$, where $u_i(a)$’s are rational functions(as algebraic varieties) of $a$. Write $u_1(a)=u_1^-(a)u_1^+(a)\in U_{(w')^{-1}}^-U_{(w')^{-1}}^+=U$ and $u_2(a)=u_2^+(a)u_2^-(a)\in U_{w'}^+U_{w'}^-=U$, then $u_i^{\pm}(a)$’s are all rational functions of $a\in A$. Then $g=u^-_1(a)w' m(a) u^-_2(a)$ and $m(a)=w'^{-1}u_1^+(a)w'\dot{w}_Mau_2^+(a)=\tilde{u}_1(a)\dot{w}_Ma\tilde{u}_2(a)$ where $\tilde{u}_1=w'^{-1}u_1^+w'$, $\tilde{u}_2=u^+_2.$ And we have $$B^G_{\varphi}(g,f_{1,w'})=c(u_1^-(a),u_2^-(a)) B^M_{\varphi}(m(a), h_{1,w'})$$ $$=c(u_1^-(a),u_2^-(a)) B^M_{\varphi}(\tilde{u}_1(a)\dot{w}_M a \tilde{u}_2(a), h_{1,w'})$$ $$=w_{\pi}(z)c(u_1^-(a),u^-_2(a))B^M_{\varphi}(\tilde{u}_1(bc'z)\dot{w}_Mbc' \tilde{u}_2(bc'z), h_{1,w'})$$ $$=w_{\pi}(z)B^G_{\varphi}(u_1(bc'z)\dot{w}_G bc' u_2(bc'z),f_{1,w'})$$ is uniformly smooth as a function of $c'\in A'_{w'}=Z'_M$, where $h_{1,w'}\in C^{\infty}_c(M;\omega_{\pi})$ is mapped to $f_{1,w'}$ through the surjective map $C^\infty_c(M;\omega_\pi)\twoheadrightarrow C^\infty_c(\Omega_{w'};\omega_\pi)$, and $$c(u_1^-(a),u_2^-(a))=\psi(u_1^-(a))\psi(u_2^-(a))Vol(U_{M,m(a)}\cap n_0(a) U_{M,m(a)}n_0(a)^{-1}\backslash U_{M,m(a)})$$ is a smooth function of $a\in A$. Take $h=h_{f_{w'}}\in C^{\infty}_c(M,\omega_{\pi})$ which maps to $f_{w'}$ under the surjective map $C^{\infty}_c(M;\omega_{\pi})\twoheadrightarrow C^{\infty}_c(\Omega_{w'}, \omega_{\pi})$ in Lemma 6.2. Construct $h_1$ based on $h$ as Lemma 6.12 such that $B^M_{\varphi}(a,h_1)=B^M_{\varphi}(a,h)$ for all $z\in Z_M=A_{w'}$. We have $h_1\in C^{\infty}_c(M;\omega_{\pi})$. Let $f_1$ be the image of $h_1$ under the map $C^{\infty}_c(M;\omega_{\pi})\twoheadrightarrow C^{\infty}_c(\Omega_{w'}, \omega_{\pi})$. Then by the transfer principal of partial Bessel integrals (Proposition 6.5), we have for Levi subgroups $L$, $M$ of $G$ with $A\subset L \subset M\subset G$, and $g=u_1\dot{w}^G_Lau_2=u_1^-w'mau_2^-\in C_r(w^G_L)$, $$B^G_{\varphi}(g, f_1)=\psi(u_1^-)\psi(u_2^-)Vol(U_{M,m}\cap n_0 U_{M,m}n_0^{-1}\backslash U_{M,m}) B^M_{\varphi}(m, h_1)$$$$=c(u_1^-,u_2^-)B^M_{\varphi}(m,h_1)$$ where $c(u_1^-,u_2^-)=\psi(u_1^-)\psi(u_2^-)Vol(U_{M,m}\cap n_0 U_{M,m}n_0^{-1}\backslash U_{M,m})$ is a constant depending smoothly on $u_1^-$ and $u_2^-$. Apply this with the case when $L=M$, and $g=w'a$, $a\in A_{w^G_M}=A_{w'}=Z_M$ we have $$B^G_{\varphi}(w'a,f_1)=B^M_{\varphi}(a, h_1)=B^M_{\varphi}(a,h)=B^G(\dot{w}'a,f_{w'})$$ by Lemma 6.12. So $B^G_{\varphi}(\dot{w}'a,f_{w'}-f_1)=0$ for all $a\in A_{w'}=Z_M$ and $f_{w'}-f_1\in C^{\infty}_c(\Omega_{w'};\omega_{\pi}).$ Therefore by Lemma 6.7, Lemma 6.8, and Lemma 6.9, in addition with a partition of unity argument, we can find a family of functions $\{f_{w'}: w''\in B(G), w''>w', d_B(w'',w')=1, f_{w''}\in C^{\infty}_c(\Omega_{w''};\omega_{\pi})\}$ such that for any $w\in B(G)$ and any $g\in C_r(w)$, we have $$B^G_{\varphi}(g, f_{w'})=B^G_{\varphi}(g,f_1)+\sum_{w''\in B(G), w''>w', d_B(w'',w')=1}B^G_{\varphi}(g,f_{w''}).$$ Moreover for each $f_{w''}$ we have $w''=w^G_{M''}$, this will be used for induction later. On the other hand if we apply the transfer principal (Proposition 6.5) for partial Bessel integrals to the case $L=A$, then for $g=u_1\dot{w}_Gau_2=u_1^-w'mu_2^-\in C_r(w_G)=C(w_G)$, where $m=w'^{-1}u_1^+w'\dot{w}_Mau_2^+\in C^M_r(\dot{w}_M)=C^M(\dot{w}_M)$, we obtain that $$B^G_{\varphi}(g, f_1)=B^G_{\varphi}(u_1\dot{w}_Ga u_2, f_1)=c(u_1^-,u_2^-)B^M_{\varphi}(w'^{-1}u_1^+w'\dot{w}_Mau_2^+, h_1)$$ $$=c(u_1^-,u_2^-)B^M_{\varphi}(m,h_1)$$ If we decompose $A^{w'}_{w_G}A_{w'}$ as $a=bc$, and assume that $u_1=u_1(a)=u_1^-(a)u_1^+(a)$, $u_2=u_2(a)=u_2^+(a)u_2^-(a)$ are rational maps in $a$, then $g=g(a)=u_1(a)\dot{w}_Gau_2(a)$ is rational in $a$ as well. Then by proposition 6.14 we have $$B^G_{\varphi}(g,f_1)=B^G_{\varphi}(g(a), f_1)=B^G_{\varphi}(u_1(a)\dot{w}_Ga u_2(a), f_1)$$ $$=c(u_1^-(a), u_2^-(a))B^M_{\varphi}(w'^{-1}u_1^+(a)w'\dot{w}_Mau_2^+(a),h_1)$$$$=\omega_{\pi}(z)c(u_1^-(bc'z),u_2^-(bc'z))B^M_{\varphi}(w'^{-1}u_1^+(bc'z)w'\dot{w}_Mbc'u_2^+(bc'z), h_1)$$ is compactly supported in $c'\in A'_{w'}=Z'_M$, and therefore $B^G_{\varphi}(g(bc'z),f_1)$ is uniformly smooth as a function of $c'\in Z_M'$. Next we are going to perform an induction on the Bessel distance $d_B(w,e)$, to obtain the following main proposition for our final proof of supercuspidal stability: Fix an auxiliary function $f_0\in C^{\infty}_c(G;\omega_{\pi})$ with $W^{f_0}(e)=1$. Let $f\in M(\pi)$ with $W^f(e)=1$, and $m\in \mathbb{Z}$ with $1\leq m\leq d_B(w_G, e)+1$. Then \(1) there exists a function $f_{1,e}\in C^{\infty}_c(G;\omega_{\pi})$; \(2) for each $w'\in B(G)$ with $1\leq d_B(w',e)$ there exists $f_{1,w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$, and for each $w''\in B(G)$ with $d_B(w'',e)=m$ there exists a function $f_{w''}\in C^{\infty}_c(\Omega_{w''};\omega_{\pi})$ such that for sufficiently large $\varphi$ we have \(a) for any $w\in B(G)$ we have $$B^G_{\varphi}(g, f)=B^G_{\varphi}(g,f_{1,e})+\sum_{1\leq d_B(w',e)<m}B^G_{\varphi}(g, f_{1,w'})+\sum_{d_B(w'',e)=m}B^G_{\varphi}(g, f_{w''})$$ for $\forall g\in C_r(w)$; \(b) for each $w\in B(G)$,$B^G_{\varphi}(g, f_{1,e})$ depends only on the auxiliary function $f_0$ and $w_{\pi}$ for all $g\in C_r(w)$; \(c) for each $w'\in B(G)$ with $1\leq d_B(w',e)<m$, and $g=g(a)=u_1(a)w^G_M a u_2(a)\in C_r(w)$, parameterized by $a\in A$ and such that $u_i(a) $’s are both rational functions of $a\in A$, we have that $$B^G_{\varphi}(g(a), f_{1,w'})=w_{\pi}(z)B^G_{\varphi}(u_1(bc'z)\dot{w}_Gbc' u_2(bc'z), f_{1,w'})$$ is uniformly smooth as a function of $c'\in A'_{w'}=Z'_M$, where $B^G_{\varphi}(g(a),f_{1,w'})$ defined apriori on $a=bc=bc'z \in A^{w'}_{w_G}A_{w'}\subset A_{w_G}=A$ and finally extended on all $a\in A$. First we fix an auxiliary function $f_0\in C^{\infty}_c(G;\omega_{\pi})$ with $W^{f_0}(e)=1$. Take $f\in M(\pi)\subset C^{\infty}_c(G,\omega_{\pi})$ normalized such that $W^f(e)=1$. Then by Proposition 6.11, we have the following result: There exists $f_{1,e}\in C^{\infty}_c(G;\omega_{\pi})$ and, for each $w'\in B(G)$ with $d_B(w',e)=1$, there exists a function $f_{w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$ such that for sufficiently large $\varphi$, \(i) For any $w\in B(G)$, we have $$B^G_{\varphi}(g, f)=B^G_{\varphi}(g,f_{1,e})+\sum_{w'\in B(G), d_B(w',e)=1}B^G_{\varphi}(g, f_{w'})$$ for all $g\in C_r(w)$, the relevant cell attached to $w$; \(ii) For each $w\in B(G)$, the partial Bessel integral $B^G_{\varphi}(g, f_{1,e})$ in (i) depends only on the auxiliary function $f_0$ and the central character $\omega_{\pi}$ for all $g\in C_r(w)$. (This can be seen directly from the expansion formula for $B^G_{\varphi}(g,f_{1,e})$ as in the proof of Proposition 6.11.) By proposition 6.15, we also have that for each $f_{w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$, there exists $f_{1,w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$ such that for sufficiently large $\varphi$, \(i) There exists a family of functions $\{f_{w',w''}\}\in C^{\infty}_c(\Omega_{w''}; \omega_{\pi})$, parameterized by $w''\in B(G)$ with $w''>w'$ and $d_B(w'',w')=1$ such that for any $w\in B(G)$ and any $g\in C_r(w)$, we have $$B^G_{\varphi}(g,f_{w'})=B^G_{\varphi}(g,f_{1,w'})+\sum_{w''\in B(G),w''>w', d_B(w'',w')=1} B^G_{\varphi}(g,f_{w',w''});$$ \(ii) Let $g=u_1(a)\dot{w}_G a u_2(a)\in C^G_r(\dot{w}_G)=C^G(\dot{w}_G)$, where $u_i(a)$’s are rational functions of $a\in A$. Write $u_1(a)=u_1^-(a)u_1^+(a)\in U_{(w')^{-1}}^-U_{(w')^{-1}}^+=U$ and $u_2(a)=u_2^+(a)u_2^-(a)\in U_{w'}^+U_{w'}^-=U$, then $u_i^{\pm}(a)$’s are all rational functions of $a\in A$, then $g=u^-_1(a)w' m(a) u^-_2(a)$ and $m(a)=w'^{-1}u_1^+(a)w'\dot{w}_Mau_2^+(a)=\tilde{u}_1(a)\dot{w}_Ma\tilde{u}_2(a)$ where $\tilde{u}_1=w'^{-1}u_1^+w'$, $\tilde{u}_2=u^+_2.$ And we have $$B^G_{\varphi}(g,f_{1,w'})=w_{\pi}(z)B^G_{\varphi}(u_1(bc'z)\dot{w}_G bc' u_2(bc'z),f_{1,w'})$$ is uniformly smooth as a function of $c'\in A'_{w'}=Z'_M$. Combine the above two results we obtain that for any $w\in B(G)$, $$B^G_{\varphi}(g,f_{w'})=B^G_{\varphi}(g,f_{1,w'})+\sum_{ d_B(w',e)=1} B^G_{\varphi}(g,f_{1,w'})$$$$+\sum_{d_B(w'',w')=d_B(w',e)=1}B^G_{\varphi}(g, f_{w',w''})$$ $$=B^G_{\varphi}(g,f_{w'})=B^G_{\varphi}(g,f_{1,w'})+\sum_{ d_B(w',e)=1} B^G_{\varphi}(g,f_{1,w'})+\sum_{d_B(w'',e)=2}B^G_{\varphi}(g, f_{w',w''})$$ for any $g\in C_r(w)$. Let $f_{w''}=\sum_{d_B(w'',w')=1}f_{w',w''},$ then we see that $f_{w''}\in C^{\infty}_c(\Omega_{w''};\omega_{\pi})$. Hence for any $w''\in B(G)$ with $d_B(w'',e)=2$, there exist $f_{w''}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$ such that for sufficiently large $\varphi$ \(i) for any $w\in B(G)$ and $g\in C_r(w)$ we have $$B^G_{\varphi}(g,f_{w'})=B^G_{\varphi}(g,f_{1,w'})+\sum_{ d_B(w',e)=1} B^G_{\varphi}(g,f_{1,w'})+\sum_{d_B(w'',e)=2}B^G_{\varphi}(g, f_{w''});$$ \(ii) for each $w\in B(G)$, $B^G_{\varphi}(g, f_{1,e})$ depends only on the auxiliary function $f_0$ and the central character $\omega_{\pi}$ for all $g\in C_r(w)$; \(iii) for $g=u_1(a)\dot{w}_G a u_2(a)\in C^G_r(\dot{w}_G)=C^G(\dot{w}_G)$, parametrized by $a$, where $u_i(a)$’s are rational functions of $a$, we have $$B^G_{\varphi}(g,f_{1,w'})=w_{\pi}(z)B^G_{\varphi}(u_1(bc'z)\dot{w}_G bc' u_2(bc'z),f_{1,w'})$$ is uniformly smooth as a function of $c'\in A'_{w'}=Z'_M$. We proceed by induction on $m=d_B(w,e)$ with $w\in B(G)$, and use Proposition 6.15 on each step, we obtain the statements in the Proposition. Now if we apply Proposition 6.16 to the case when $m=d_B(w_G, e)+1$, we obtain a final result that we need for the proof of supercuspidal stability in our case: Fix an auxiliary function $f_0\in C^{\infty}_c(G;\omega_{\pi})$ with $W^{f_0}(e)=1$. Let $f\in M(\pi)$ with $W^f(e)=1$, Then \(1) there exists a function $f_{1,e}\in C^{\infty}_c(G;\omega_{\pi})$; \(2) for each $w'\in B(G)$ with $1\leq d_B(w',e)$ there exists $f_{1,w'}\in C^{\infty}_c(\Omega_{w'};\omega_{\pi})$ such that for sufficiently large $\varphi$ we have \(a) $$B^G_{\varphi}(g, f)=B^G_{\varphi}(g,f_{1,e})+\sum_{1\leq d_B(w',e)}B^G_{\varphi}(g, f_{1,w'})$$ for $g\in C_r(\dot{w}_G)=C(\dot{w}_G)$; \(b) $B^G_{\varphi}(g, f_{1,e})$ depends only on the auxiliary function $f_0$ and $w_{\pi}$ for all $g\in C(\dot{w}_G)$; \(c) for each $w'\in B(G)$ with $1\leq d_B(w',e)$, and $g=g(a)=u_1(a)\dot{w}_G a u_2(a)\in C(\dot{w}_G)$, parameterized by $a\in A$ and such that $u_i(a) $’s are both rational functions of $a\in A$, we have that $$B^G_{\varphi}(g(a), f_{1,w'})=w_{\pi}(z)B^G_{\varphi}(u_1(bc'z)\dot{w}_Gbc' u_2(bc'z), f_{1,w'})$$ is uniformly smooth as a function of $c'\in A'_{w'}=Z'_M$. SUPERCUSPIDAL STABILITY ======================= Now we have all the ingredients for the final proof of supercuspidal stability in our case. First recall that we have reduced Proposition 3.4 to the proof of the stability of local coefficient, since the adjoint action $r: \leftidx{^L}M_H\longrightarrow \textrm{GL}(\leftidx{^L}{\mathfrak{n}_H})$ is irreducible. And from Langlands-Shahidi method, $C_{\psi}(s,\pi)=\gamma(s, \pi, \textrm{Sym}^2\otimes \eta,\psi)$. We wrote the local coefficients as the Mellin transform of the partial Bessel functions $j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)$, where $g=-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1}$. By an apropriate choie of orbit space representatives of the space $U_{M_H}\backslash N_H$, we can pick $Y=Y(a_1,\cdots, a_n)$. Then by induction on $n$ we can show that such $g$ lies in the big cell. Let $g=u_1 \dot{w}_G a u_2$ be its Bruhat decomposition. Since $g\mapsto u_1$, $g\mapsto a$, $g\mapsto u_2$ are all morphisms of algebraic varieties, we see that here the entries of $a$, $u_1=u_1(a)$, and $u_2=u_2(a)$ are all rational functions of $(a_1,a_2,\cdots,a_n)\in (F^\times)^n.$ We have $g=u_1 \dot{w}_G a u_2=u_1(a)\dot{w}_Ga u_2(a)\in C_r(\dot{w}_G)=C(\dot{w}_G)\subset \Omega_{w'}$, write $$g=u_1 \dot{w}_G a u_2=u_1^-u_1^+\dot{w}_G a u_2^+u_2^-=u_1^-w'mu_2^-,$$ where $m=(w')^{-1}u_1^+w' \dot{w}_M a u_2^+\in C_r^M(\dot{w}_M)$ with $u_1^-\in U^-_{(w')^{-1}}$, $u_1^+\in U_{(w')^{-1}}^+$, $u_2^+\in U_{w'}^+$, $u_2^-\in U_{w'}^-$, $u_1=u_1^-u_1^+$, $u_2=u_2^+u_2^-.$ Since $u_1(a)$ and $u_2(a)$ are both rational functions of $a$, the projection maps $u_i(a)\mapsto u_i^{\pm}(a)$ are rational maps, so $u_i^{\pm}(a)$’s are all rational functions of $a$. So we can apply Proposition 6.14 to our case with $\tilde{u}_1(a)=(w')^{-1}u_1^+(a)w'$, $\tilde{u}_2(a)=u_2^+(a)$. Now we see that the conditions for Proposition 6.17 are all satisfied for our $g$. By Proposition 5.9, $$C_{\psi}(s,\sigma_\eta)^{-1}=\gamma(ns, \omega_{\pi}^2,\psi)^{-1}\int_{F^{\times}\backslash R}j_{\pi,\eta,\dot{w}_\theta,\kappa}(-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1})$$$$\cdot\omega_{\pi}(4\det(Y)^2\prod_{i=1}^n a_i^{-2}) \vert\frac{1}{2}\vert^{\frac{n(n-s)}{2}}\vert \det(Y)\vert^{\frac{2ns-s-n}{2}} \prod_{i=1}^n\vert a_i \vert ^{i-1-ns}da_i$$ In the Bruhat decomposition $g=-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1}=u_1(a)\dot{w}_G au_2(a)$ if we write $a=\textrm{diag}\{ d_1,\cdots, d_n\}$, then a direct calculation shows that $$d_1=\frac{\prod_{j\ \ even}a_j^2}{\prod_{k\ \ odd}a_k^2}, d_2=\frac{\prod_{k\neq 1,\ \ odd}a_k^2}{4\prod_{j\ \ even}a_j^2},d_3=\frac{\prod_{j\neq 2,\ \ even}a_j^2}{\prod_{k\neq 1, \ \ odd}a_k^2},$$ $$d_4=\frac{\prod_{k\neq 1, 3,\ \ odd}a_k^2}{4\prod_{j\neq 2,\ \ even}a_j^2},\cdots,d_n$$ and $d_n=\frac{1}{4a_n^2}$ if $n$ is even, $d_n=\frac{1}{a_n^2}$ if $n$ is odd. And no matter $n$ is even or odd we have $d_i\cdot d_{i+1}=\frac{1}{4a_i^2}$ for all $1\leq i \leq n-1$. Recall that the action of $F^{\times}$ on $R\simeq (F^{\times})^n$ is given by $t\cdot (a_1,\cdots,a_n)=(t^2 a_1, t^2 a_2,\cdots t^2 a_{n-1}, t a_n)$. From the above observation, it is clear that this action is equivalent to the action of $F^{\times}$ on $A=\{\textrm{diag}\{d_1,\cdots,d_n):d_i\in F^{\times}\}$ by $t\cdot \textrm{diag}(d_1, d_2,\cdots,d_n)=\textrm{diag}(\frac{d_1}{t^2},\frac{d_2}{t^2},\cdots,\frac{d_n}{t^2}).$ Thus the action of $F^{\times }$ on $R$ translates into the action of $Z$ on $A$. Meanwhile the change of variable $(a_1,\cdots,a_n)\mapsto (d_1,\cdots,d_n)$ translates the measure given by the $a_i$’s into a unique measure given by the $d_i$’s, with the determinant of the Jacobian matrix a rational function of the $d_i$’s. Recall that by the computation at the end of section 5.3, $\det(g)=\det(Y)^{-1}=\frac{(-\frac{1}{2})^n}{\prod_{k\ \ odd}a_k^2}$, if $n$ is even; $\det(g)=\det(Y^{-1})=\frac{(-\frac{1}{2})^{n-1}}{\prod_{k\ \ odd}a_k^2}$ if $n$ is odd. In both cases $\det(Y^{-1})\in (F^{\times})^2$. On the other hand, $\det(Y)^2=\frac{1}{(d_1\cdots d_n)^2}=\frac{1}{d_1(d_1d_2)(d_2d_3)\cdots(d_{n-1}d_n)d_n} $. The last expression is equal to $\frac{1}{d_1}(4a_1^2)(4a_2^2)\cdots(4a_{n-1}^2)(4a_n^2)$ if $n$ is even, and $\frac{1}{d_1}(4a_1^2)(4a_2^2)\cdots (4a_{n-1}^2)a_n^2$ if $n$ is odd. Therefore $\det(Y)^2\prod_{i=1}^n a_{i}^{-2}=\frac{4^n}{d_1}$ if $n$ is even and $\frac{4^{n-1}}{d_1}$ if $n$ is odd. Meanwhile, $\prod_{i=1}^n\vert a_i\vert^{i-1-ns}=\prod_{i=1}^n \vert a_i^2\vert ^{\frac{i-1-ns}{2}}=\prod_{i=1}^{n-1}(\vert\frac{1}{4d_i\cdot d_{i+1}}\vert^{\frac{i-1-ns}{2}})\cdot\vert\frac{1}{4d_n}\vert^\frac{n-1-ns}{2}=\vert\frac{1}{2}\vert^{\frac{n(n+1)}{2}-ns-1}\cdot\prod_{i=1}^{n-1}(\vert\frac{1}{d_i\cdot d_{i+1}}\vert^{\frac{i-1-ns}{2}})\cdot\vert\frac{1}{d_n}\vert^\frac{n-1-ns}{2}$ if $n$ is even, and $\prod_{i=1}^n\vert a_i\vert^{i-1-ns}=\prod_{i=1}^{n-1}(\vert\frac{1}{4d_i\cdot d_{i+1}}\vert^{\frac{i-1-ns}{2}})\cdot\vert\frac{1}{d_n}\vert^\frac{n-1-ns}{2}=\vert\frac{1}{2}\vert^{\frac{n(n-1)}{2}-ns-1}\cdot \prod_{i=1}^{n-1}(\vert\frac{1}{d_i\cdot d_{i+1}}\vert^{\frac{i-1-ns}{2}})\cdot\vert\frac{1}{d_n}\vert^\frac{n-1-ns}{2}$ if $n$ is odd. Let $\nu(n,s)=\frac{n(n-s)}{2}+\frac{n(n+1)}{2}-ns-1$ if $n$ is even and $\frac{n(n-s)}{2}+\frac{n(n-1)}{2}-ns-1$ if $n$ is odd. Let $A=A'Z$, which gives $d'_i=d_i/d_1$, ($1\leq i\leq n$), then since $d_1'=1$, $\omega_\pi(4\det(Y')^2\prod_{i=1}^na_i'^{-2})=\omega_\pi(4^{n+1})$ if $n$ is even and $\omega_\pi(4^n)$ if $n$ is odd, denote this number by $c_{\pi}$. From the above observations we see that there exists complex numbers $\tau(i,s)$, which are of the form $\tau(i,s)=p_i+sq_i$, $s\in \mathbb{C}$ with $p_i,q_i\in \mathbb{Q}$ depending only on $1\leq i \leq n$, such that $$C_{\psi}(s,\sigma_\eta)^{-1}=c_\pi\vert\frac{1}{2}\vert^{\nu(n,s)}\gamma(ns, w_{\pi}^2,\psi)^{-1}\int_{A'}j_{\pi,\eta,\dot{w}_\theta,\kappa}(g'(a))\prod_{i=2}^n\vert d_i'\vert^{\tau(i,s)}\prod_{i=2}^2 d^{\times}d'_i$$ where $g'=g(a')=u_1(a')\dot{w}_G a' u_2(a')$ with $a=a'z$, and $a'=\textrm{diag}\{d'_1,\cdots,d'_n\}$. Now let’s prove Proposition 3.4 (**Proof of Proposition 3.4**) If we are given two irreducible supercuspidal representations $\pi_1$ and $\pi_2$ of $\textrm{GL}_n(F)$ with the same central character $w_{\pi_1}=w_{\pi_2}$, lift them to representations of $M_H(F)$ and denote them by $\sigma_1$ and $\sigma_2$ respectively, then by Proposition 5.9 and the above argument, $$C_{\psi}(s,\sigma_{1,\eta}\otimes \chi)^{-1}-C_{\psi}(s,\sigma_{2,\eta}\otimes \chi)^{-1}=c_\pi\vert\frac{1}{2}\vert^{\nu(n,s)}\gamma(ns,(w_{\pi}\chi^n)^2,\psi)^{-1}D_{\chi}(s)$$ where $$D_{\chi}(s)=\int_{A'} (j_{\pi_1\otimes\chi, \eta,\dot{w}_\theta,\kappa}(g(a'))-j_{\pi_2\otimes\chi,\eta,\dot{w}_\theta,\kappa}(g(a')))\prod_{i=2}^n\vert d'_i\vert^{\tau(i,s)}\prod_{i=2}^nd^{\times}d'_i$$ Pick $f_i\in M(\pi_i)$ such that $W^{f_i}(e)=1$, for $i=1,2$, and such that for $g=-\frac{1}{2}\dot{w}_G\leftidx{^t}Y^{-1}=g(a)=u_1(a)\dot{w}_G a u_2(a).$ By Proposition 5.10, $$j_{\pi_i,\eta,\dot{w}_\theta,\kappa}(g(a),f_i)=\eta(a(g))^{-1}\vert\det(g)\vert^{\frac{s}{2}}B^G_{\varphi}(g(a), f_i).$$ For convenience let $J_{\pi_i,\eta,\dot{w}_\theta,,\kappa}(g,f_i)=\eta(a(g))\vert\det(g)\vert^{-\frac{s}{2}}\cdot j_{\pi_i,\eta,\dot{w}_\theta,,\kappa}(g,f_i)$. We may also assume that $\kappa $ is sufficiently large so that Proposition 6.17 holds for both $f_1$ and $f_2$ with the same auxiliary function $f_0$. Then apply Proposition 6.17 (2)(a), we have $$J_{\pi_1,\eta,\dot{w}_\theta,\kappa}(g(a'))-J_{\pi_2,\eta,\dot{w}_\theta,\kappa}(g(a'))=B^G_{\varphi}(g(a'),f_1)-B^G_{\varphi}(g(a'), f_2)$$ $$=B^G_{\varphi}(g(a'),f_{1,1,e})-B^G_{\varphi}(g(a'),f_{2,1,e})+\sum_{1\leq d_B(w',e)}(B^G_{\varphi}(g(a'),f_{1,1,w'})-B^G_{\varphi}(g(a'),f_{2,1,w'}))$$ Now since both $B^G_{\varphi}(g(a'),f_{1,1,e})$ and $B^G_{\varphi}(g(a'),f_{2,1,e})$ depend only on the auxiliary function $f_0$, the central character $\omega_{\pi}=\omega_{\pi_1}=\omega_{\pi_2}$, and $\eta$, we see that $$B^G_{\varphi}(g(a'),f_{1,1,e})-B^G_{\varphi}(g(a'),f_{2,1,e})=0.$$ So we are left with $$J_{\pi_1,\eta,\dot{w}_\theta,\kappa}(g(a'))-J_{\pi_2,\eta,\dot{w}_\theta,\kappa}(g(a'))=\sum_{1\leq d_B(w',e)}(B^G_{\varphi}(g(a'),f_{1,1,w'})-B^G_{\varphi}(g(a'),f_{2,1,w'}))$$ Meanwhile, notice that $j_{\pi\otimes\chi, \eta,\dot{w}_\theta,\kappa}(g)=\chi(\det(g))j_{\pi,\eta,\dot{w}_\theta,\kappa}(g)$. So we have $$j_{\pi_1\otimes\chi, \eta,\dot{w}_\theta,\kappa}(g(a'))-j_{\pi_2 \otimes\chi,\eta,\dot{w}_\theta,\kappa}(g(a'))$$$$=\chi(\det(a'))(j_{\pi_1,\eta,\dot{w}_\theta,\kappa}(g(a'))-j_{\pi_2,\eta,\dot{w}_\theta,\kappa}(g(a'))).$$ Moreover, since $\det(g')=\det(a')=\frac{d_1\cdots d_n}{d_1^n}$, and as we saw before both $d_1\cdots d_n$ and $d_1$ are in $(F^\times)^2$, so $\det(g')\in (F^\times)^2$. Recall that at the end of section 5.1, we have $M_{H_D}=\{(g,a)\in M_H:\det(g)a(g)^2=1\}^\circ$, there is a unique $a(g)\in F^\times$ such that $\det(g)a(g)^2=1$, denote it by $\det(g)^{-\frac{1}{2}}$. Then $\eta(a(g'))=\eta(\det(g')^{-\frac{1}{2}})=\eta(\det(a')^{-\frac{1}{2}})$. Now put everything together we obtain that $$D_{\chi}(s)=\int_{A'} (\sum_{1\leq d_B(w',e)}(B^G_{\varphi}(g(a'),f_{1,1,w'})-B^G_{\varphi}(g(a'),f_{2,1,w'})))\chi(\det(a'))$$$$\cdot \eta(\det(a')^{-\frac{1}{2}})^{-1}\vert\det(a')\vert^{\frac{s}{2}}\prod_{i=2}^n\vert d'_i\vert^{\tau(i,s)}\prod_{i=2}^nd^{\times}d'_i$$ $$= \sum_{1\leq d_B(w.e)}\int_{A^{w'}_{w_G}}(\int_{A'_{w'}}(B^G_{\varphi}(g(bc'),f_{1,1,w'})-B^G_{\varphi}(g(bc'),f_{2,1,w'}))\prod_{i=2}^n\vert c'_i\vert^{\tau(i,s)}\chi(\det(c'))$$$$\cdot\eta(\det(c')^{-\frac{1}{2}})^{-1}\vert \det(c')\vert^{\frac{s}{2}}dc')\chi(\det(b))\eta(\det(b)^{-\frac{1}{2}})^{-1}\vert\det(b)\vert^{\frac{s}{2}}\prod_{i=2}^n\vert b_i\vert^{\tau(i,s)}db.$$ where $a=diag(d_1,\cdots,d_n)=bc=bc'z$ gives the corresponding entries $b_i$ of $b$ and $c'_i$ of $c'$ for $1\leq i \leq n$, and the measure $db$ and $dc'$ on $A^{w'}_{w_G}$ and $A_{w'}$ respectively. Notice that in the inner integral the function $$(B^G_{\varphi}(g(bc'),f_{1,1,w'})-B^G_{\varphi}(g(bc'),f_{2,1,w'}))\prod_{i=2}^n\vert c'_i\vert^{\tau(i,s)}$$ is uniformly smooth as a function of $c'\in A_{w'}$ since both $B^G_{\varphi}(g(bc'),f_{1,1,w'})$ and $B^G_{\varphi}(g(bc'),f_{2,1,w'})$ are by Proposition 6.17. And $\prod_{i=2}^n\vert c'_i\vert^{\tau(i,s)}$, being a power of the p-adic absolute values of $c_i'$’s, is by definition also uniformly smooth. Therefore if we take $\chi$ to be sufficiently ramified, we see that the inner integral $$\int_{A'_{w'}}(B^G_{\varphi}(g(bc'),f_{1,1,w'})-B^G_{\varphi}(g(bc'),f_{2,1,w'}))\prod_{i=2}^n\vert c'_i\vert^{\tau(i,s)}\chi(\det(c'))$$$$\cdot\eta(\det(c')^{-\frac{1}{2}})^{-1}\vert\det(c')\vert^{\frac{s}{2}}dc'=0$$ So we obtain that $D_{\chi}(s)=0$, and therefore $$C_{\psi}(s, \sigma_{1,\eta}\otimes \chi)=C_{\psi}(s, \sigma_{2,\eta}\otimes \chi).$$ ACKNOWLEDGEMENTS ================ I would first like to sincerely thank my advisor, Dr. Freydoon Shahidi, for recommending this problem to me, for many of his deep insights and brilliant ideas in mathematics which greatly helped me to increase my research abilities through my Ph.D career, and for all the enlightening discussions with him that give me long-term visions toward my future research. I would like to acknowledge James Cogdell, Mahdi Asgari, Chung Pang Mok, and David Goldberg for several helpful discussions on various related topics. I would also like to thank my friend Daniel Shankman for many years of mutual learning, and for his willingness and patience to carefully check several technical parts of my proof. My research was supported by the NSF grant DMS-1500759. REFERENCES ========== \[1\] M. Asgari, “Local L-function for Split-spinor Groups”, Cand J. Math Vol.54(4), 2002, pp. 673-693. \[2\] M. Asgari, F. Shahidi, “Generic Transfer for General Spin Groups”, Duke Mathematical Journal, Vol. 132, No.1, 2006, pp. 137–190. \[3\] I.N. Bernstein, A.V. Zelevinsky, “Induced representations of reductive p-adic groups(I)”, Annales Scientifiques de l’École Normale Supérieure, Série 4, 10 (4): pp. 441–472, 1977. ISSN 0012-9593, MR 0579172. \[4\] C.J. Bushnell, G. Henniart, “The Local Langlands Conjecture for $GL(2)$”, Grundlehren der Math. Wiss. 335, Springer-Verlag, 2006. \[5\] W. Casselman, “Introduction to the theory of admissible representations of p-adic reductive groups”, preprint, 1995. \[6\] J.W. Cogdell, I.I. Piatetski-Shapiro, F. Shahidi, “Partial Bessel functions for Quasi-split Groups.” In “automorphic Representations, L-functions and Applications: Progress and Prospects.” Berlin: Walter de Gruyter, 2005, pp. 95-128. \[7\] J.W. Cogdell, F. Shahidi, T.-L. Tsai, “Local Langlands Correspondence for $GL_n$, and the Exterior and Symmetric Square $\epsilon$-factors”, Duke Mathematical Journal, Vol 166, No. 11, 2017, p. 2053-2132. \[8\] S. Gelbart, H. Jacquet, “A relation between automorphic representations of GL(2) and GL(3)”, Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), pp. 471–542. \[9\] P. Deligne, “Les constantes des equations fonctionnelles des fonctions L” in Modular Functions of One Variable, II(Antwerp,1972) Lecture Notes in Math. 349, Springer, Berlin, 1973, pp. 501-597. MR 0349635. \[10\] M. Harris, R. Taylor, “The Geometry and Cohomology of Some Simple Shimura Varieties”. Annals of Mathematics Studies, No. 151, Princeton University Press, 2001. \[11\] G. Henniart, “Correspondance de Langlands et fonctions L des carrés extérieur et symétrique”, Int. Math. res. Not. IMRN 2010, no. 4, pp. 633-673. MR 2595008. \[12\] G. Henniart, “Une preuve simple des conjectures de Langlands pour $\textrm{GL}(n)$ sur un corps p-adique,” Invent. Math. 139(2000), pp. 439-455. MR 1738446. \[13\] R.P. Langlands, “On the classification of irreducible representations of real algebraic groups”, in ’Representation Theory and Harmonic Analysis on Semisimple Lie Groups’ (Editors: P.J. Sally, Jr. and D.A. Vogan), Mathematical Surveys and Monographs, AMS, Vol.31, 1989, pp. 101-170. \[14\] G. Laumon, M. Rapoport, U. Stuhler, “D-elliptic sheaves and the Langlands correspondence”, Invent. Math. 113:2 (1993), pp. 217–338. MR 96e:11077. \[15\] P. Scholze, “The local Langlands correspondence for $GL_n$ over p-adic fields”. Invent. Math. 192.3 (2013), pp. 663–715. \[16\] F. Shahidi, “A proof of Langlands’s conjecture on Plancharel Measures, Complementary series for p-adic Groups”, Annals of Mathematics 132, No.2, 1990, pp. 273–330. MR 91m:11095. \[17\] F. Shahidi, “Eisenstein Series and Automorphic L-Functions”, American Mathematical Society, Colloquium Publications, volume 58. \[18\] F. Shahidi, “Local coefficients as Artin factors for real groups”, Duke Math. J. 52(1985), pp. 973-1007. MR 0816396. DOI 10. 1215/S0012-7094-85-0525204.(2064, 2070) \[19\] F. Shahidi, “Local Coefficients as Mellin Transforms of Bessel Functions Towards a General Stability”, IMRN 2002, No. 39, pp. 2075-2119. \[20\] F. Shahidi, “On non-vanishing of twisted symmetric and exterior square L-functions for GL(n)”, Olga TausskyTodd: in memoriam. Pacific J. Math. 1997, Special Issue, pp. 311–322. \[21\] A. J. Silberger, Ernst-Wilhelm Zink, “Langlands classification for L-parameters”, J. Algebra 511 (2018), pp. 299–357, \[22\] R. Sundaravaradhan, “Some Structural Results for the Stability of Root Numbers”, Int. Math. Res. Not. IMRN 2007, No. 2, pp. 1–22. \[23\] I. Vogt, “The local Langlands correspondence for $\textrm{GL}_n$ over a p-adic field”, Available at: http://web.stanford.edu/ vogti/LLC.pdf, pp. 1-18.

epsf.sty Introduction {#sec_intro} ============ The intimate connection between the novel superconductivity and magnetism found in the high-[*T$_{c}$*]{} cuprates is believed to be fundamental to the underlying superconducting mechanism. [@Kastner_98] Extensive neutron scattering measurements of La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO) have revealed the doping dependence of the low-energy magnetic spin fluctuations over a wide range of doping. [@Yamada_98] More recent studies have focused attention on the static or quasi-static magnetic ordering that coexists or phase-separates with the superconductivity. [@Tranquada_95] Incommensurate (IC) elastic peaks were first observed in La$_{1.48}$Nd$_{0.4}$Sr$_{0.12}$CuO$_{4}$ around the ($\pi$,$\pi$) position, which corresponds to (1/2,1/2,0) in tetragonal notation as shown in the right inset of Fig. 1(b). Similar peaks have also been observed in La$_{1.88}$Sr$_{0.12}$CuO$_4$ [@Suzuki_98; @Kimura_99] and La$_2$CuO$_{4.12}$, [@YoungLee_99] although the peak positions are slightly shifted towards a more rectangular arrangement compared to the square geometry found in La$_{1.48}$Nd$_{0.4}$Sr$_{0.12}$CuO$_{4}$. [@Tranquada_96] In these superconducting systems, since the spin modulation vector is parallel [@Tranquada_96] or almost parallel to the Cu-O bond [@YoungLee_99; @Kimura_00], we use the term parallel spin modulation or correlations. Recently, another type of IC magnetic order was discovered by Wakimoto [*et al*]{}. in the insulating spin-glass phase at [*x*]{} = 0.03, 0.04 and 0.05. [@Waki_99_1; @Waki_00_1] The IC peaks observed at these Sr concentrations are located at the diagonal positions depicted in the reciprocal lattice diagram shown in the left inset of Fig. 1(b). Another important discovery is that the spin modulation is observable only along the orthorhombic [*b*]{}-axis. Such a one-dimensional nature for the spin correlations is consistent with a stripe-like ordering of the holes in the CuO$_{2}$ planes. More recently, similar diagonal IC peaks were found in LSCO samples with [*x*]{} = 0.024, just above the critical concentration for three-dimensional N$\acute{e}$el order. [@Matsuda_00] Therefore, the diagonal spin density modulation is considered to be an intrinsic property of the entire insulating spin-glass region, and stands in stark contrast to the parallel spin modulation observed in the superconducting phase. These results strongly suggest that a drastic change takes place in the spin modulation vector, from diagonal to parallel, near the lower critical concentration for superconductivity $x_{c}$$\approx$0.055. Important questions to be resolved are how the change in the spin density modulation occurs and how it is related to the insulating-to-superconducting phase transition. In addition, the nature of the doping-induced superconducting phase transition itself is a key issue related to the position of quantum critical point in the high-[*T$_{c}$*]{} cuprate phase diagram. To shed more light on these questions, we have carried out a systematic series of elastic neutron scattering experiments on single crystals of LSCO with [*x*]{} = 0.053, 0.056, 0.06 and 0.07 that spans the insulating-superconducting phase boundary. Quantitative analyses are also presented here on data obtained from other samples =2.65in with [*x*]{} = 0.03 and 0.04 which were not shown in detail in Ref. 10. From all of these data we can confirm that the appearance of the parallel spin correlations coincides with that of the superconductivity. On the other hand, the diagonal spin correlations persist into the superconducting state near the phase boundary, where they also exhibit an anomalous broadening of the peak-width. The incommensurability $\delta$, for both the diagonal and parallel peaks, varies monotonously across the phase boundary. The preparation and characterization of our LSCO single crystals and experimental details are described in Section II of this paper. Data from the neutron scattering measurements taken on single crystals in both the insulating spin-glass and superconducting phases are introduced in Section III. Finally, in Section IV we discuss the nature of the doping-induced superconducting phase transition. IEXPERIMENTAL DETAILS ===================== Sample Preparation and Characterization --------------------------------------- A series of single crystals with [*x*]{} = 0.053, 0.056, and 0.07 were grown using a traveling-solvent floating-zone method. [@Tanaka_89] By utilizing large focusing mirrors in our new furnaces we are able to keep the temperature gradient around the molten zone stable and sharp for more than 150 hours . As a result, we can make the molten zone smaller, which helps to keep the growth conditions stable. Such stability is required to grow large crystals with narrow mosaic spreads and small concentration gradients. The shapes of the resulting crystals are columnar, with typical dimensions of 7-8 mm in diameter and  100 mm in length. Crystal rods near the final part of the growth were cut into $\sim$30 mm long pieces for neutron scattering measurements. All crystals were annealed under oxygen gas flow at 900 $^{\circ}$C for 50 hours, cooled to 500 $^{\circ}$C at a rate of 10 $^{\circ}$C/h, annealed at 500 $^{\circ}$C for 50 hours, and finally furnace-cooled to room temperature. The sample with [*x*]{} = 0.06 is the same crystal as that used for a previous neutron scattering study. [@Waki_99_2] The upper and lower parts of the crystals used for neutron scattering measurements were cut into $\sim$1 mm thick pieces in order to measure the superconducting shielding signal with a SQUID magnetometer. As shown in Fig. 1(a), superconducting transitions are observed in those samples with [*x*]{} = 0.056, 0.06, and 0.07, having onset temperatures [*T$_{c}$*]{} = 6.3 K, 11.6 K, and 17.0 K, respectively. These values are almost identical to those previously obtained on powder samples as shown in Fig. 1(b). [@Takagi_89] The difference in [*T$_{c}$*]{} between the upper and lower parts of each crystal was found to be less than 0.1 K, which indicates the absence of any significant gradients in the Sr concentration [*x*]{}. On the other hand, no evidence for superconductivity is found for samples with [*x*]{} = 0.053 down to 2 K. Based on these results we estimate the lower critical concentration for superconductivity to lie between [*x*]{} = 0.053 and 0.056. The orthorhombic distortion ([*b*]{}/[*a*]{}-1) measured at lowest temperature ($\sim$2 K) for all samples is shown in Fig. 2, along with data obtained on other crystals during our previous neutron diffraction measurements. We note that this distortion depends mainly on the Sr concentration, and is independent of the oxygen content. Therefore the linear fit to these data implies that we have achieved a systematic and well-defined Sr concentration in our samples. The slight scatter in the data is due primarily to error from experimental uncertainties caused by the use of different spectrometer configurations during different experiments. Similarly, both [*T$_{c}$*]{} and the lattice constants, which do depend on the oxygen content, can be used to gauge the oxygen stoichiometry. Fig. 1(b) shows a smooth and monotonic variation of [*T$_{c}$*]{} with [*x*]{}. Since the Sr content is known to vary systematically from our data on the =2.85in orthorhombic distortion, the absence of any significant scatter in these data likewise suggests the absence of any scatter in the oxygen contents of these crystals. As an additional cross check, we measured the [*c*]{}-axis lattice constant for crystals with [*x*]{} = 0.053, 0.06, and 0.07 using an identical x-ray powder diffraction setup. Again, the [*c*]{}-axis lattice constant, shown in the inset to Fig. 2, also changed monotonically with Sr doping. Based on these results, and the agreement between current and prior results concerning the Sr-concentration dependence of [*T$_{c}$*]{}, we believe that the oxygen content in these crystals is stoichiometric. The maximum deviation of the Sr-concentration from the average value in each crystal is estimated to be $\sim$0.004, which is comparable to that previously evaluated for LSCO single crystals for 0.06 $\leq$ [*x*]{} $\leq$ 0.12, since the same growth techniques are utilized. [@Yamada_98] The hole concentration is equal to the Sr concentration. Neutron Scattering Measurements ------------------------------- The primary elastic neutron-scattering measurements reported here were performed on the cold neutron triple-axis spectrometers HER, located at the Japan Atomic Energy Research Institute (JAERI) JRR-3M reactor, and SPINS, located at the National Institute of Standards and Technology (NIST) Center for Neutron Research. Most experiments were carried out using incident and scattered neutron energies of 4.25 meV at HER, and 3.5 meV at SPINS, selected via Bragg diffraction from the (0,0,2) reflection from highly-oriented pyrolytic graphite crystals. Elastic scattering measurements were also performed on the thermal neutron triple-axis spectrometer TOPAN, located at JAERI. In this case the initial neutron energy was fixed at 14.7 meV. Horizontal collimations used were 32$^{\prime}$-100$^{\prime}$-Be-S-80$^{\prime}$-80$^{\prime}$ at HER, 32$^{\prime}$-Be-80$^{\prime}$-S-BeO-80$^{\prime}$-150$^{\prime}$ at SPINS and 40$^{\prime}$-100$^{\prime}$-S-PG-60$^{\prime}$-80$^{\prime}$ at TOPAN. Here “S” denotes the sample position. Be, BeO and PG filters were used to eliminate contamination from higher-order wavelengths in the incident and scattered neutron beams. The resultant elastic energy resolution is about 0.2 meV and 2.0 meV FWHM (full-width at half-maximum) for cold and thermal neutron spectrometers, respectively. Crystals were mounted in the ([*h*]{},[*k*]{},0) zone, and sealed in an aluminum can with He gas for thermal exchange. The aluminum cans were then attached either to the cold plate of a $^{4}$He-closed cycle refrigerator, or to a top-loading liquid-He cryostat, which is able to control the temperature from 1.5 K to 300 K. All crystals examined in this study have a twinned-domain orthorhombic structure and contain two types of domains. The volume ratio of the two domains is approximately 2:1. For samples with [*x*]{} = 0.053, 0.056, and 0.06, domain-selective scans were done as described in Ref. 11. It is convenient to use a polar coordinate [*Mod*]{}-[*q*]{}, measured in reciprocal lattice units (r.l.u.) of the high-temperature tetragonal structure (1 r.l.u. $\sim$ 1.65 ${\AA}^{-1}$), to describe the distance between incommensurate peaks, irrespective of the propagation vector. As illustrated in Fig. 3, the polar coordinate [${Mod}$]{}-[${q}$]{} = [$Q$]{} - [$Q$]{}$_{center}$, where [$Q$]{}$_{center}$ is the vector from the =2.6in origin to the orthorhombic (1,0,0) position of the largest domain (represented by the solid square), and [$Q$]{} is the momentum transfer. We define $\delta$, often called the incommensurability, as half the distance between the pair of IC peaks. The orientation of the peaks is described by a polar angle $\alpha$ that is measured with respect to [$Q$]{}$_{center}$ and [${Mod}$]{}-[${q}$]{}. We note that this polar-geometry description of $\delta$ is also useful to describe the incommensurate peak positions observed in La$_{2}$CuO$_{4.12}$ [@YoungLee_99] and La$_{1.88}$Sr$_{0.12}$CuO$_{4}$. [@Kimura_99] In this paper we present most of our [*q*]{}-scans using [${Mod}$]{}-[${q}$]{} as the horizontal axis. MAGNETIC SCATTERING CROSS SECTION ================================= In this section we present neutron elastic magnetic scattering data for samples with [*x*]{} = 0.04, 0.053 and 0.06. Fig. 4 shows the elastic scattering profiles of the magnetic peaks measured along the diagonal scan trajectory (see the top panel inset) for the insulating [*x*]{} = 0.04 and 0.053 samples at 1.5 K, along with prior results obtained for [*x*]{} = 0.05. [@Waki_00_1] The elastic peaks observed in the insulating [*x*]{} = 0.04 and 0.053 samples are located at diagonal reciprocal lattice positions along the orthorhombic \[0,1,0\] direction, which is in complete agreement with the results obtained by Wakimoto [*et al*]{}. for [*x*]{} = 0.05. [@Waki_00_1] By contrast, scans on the superconducting [*x*]{} = 0.06 sample (near [*x$_{c}$*]{}) indicate the presence of elastic peaks at parallel reciprocal lattice positions along the tetragonal \[1,0,0\] direction, which is consistent with previous results on superconducting samples [@Waki_99_2] (Fig. 5(a)). To our surprise, however, we also observe [*diagonal*]{} elastic peaks at low temperatures for this same superconducting [*x*]{} = 0.06 sample as shown in Fig. 5(b). The data represented by the open circles in this panel were taken using the same scan at 40 K, and demonstrate that these diagonal peaks vanish at high temperatures, and are therefore genuine. To clarify this observation we performed a circle scan denoted in the inset of Fig. 5(c), thereby obtaining a more detailed two-dimensional peak profile of these unexpected diagonal peaks. We note, as will be shown later, that the incommensurabilities of the peaks at the diagonal and parallel positions are nearly the same. In other words, the diagonal and parallel peaks are nearly equidistant from the center of our polar coordinate system ([${Mod}$]{}-[${q}$]{} = 0), and thus all lie on a circle of radius $\delta$. Therefore, the circular scan is able to survey each of these peak profiles at once. As shown in Fig. 5(c), the circular scan reveals a single broad peak centered at $\alpha$ = 90$^{\circ}$ (see Fig. 3). From these scans, we can conclude that a pair of crescent-shaped peaks is present for [*x*]{} = 0.06, and is centered at the diagonal positions. A similar feature was observed in the [*x*]{} = 0.056 sample. We remark here that these measurements were performed by choosing magnetic peaks from the major domain, and are free from contamination by the corresponding peaks from the minor domain. The crescent-shaped peak revealed by three scans in Figs. 5(a)-(c) is quite different from the situation found for [*x*]{} = 0.05 [@Waki_99_1] or 0.12. [@Kimura_99] In these two cases, a pair of nearly isotropic diagonal peaks was observed for the [*x*]{} = 0.05 sample, whereas four isotropic peaks were observed at parallel positions for the [*x*]{} = 0.12 sample. In order to analyze the data for [*x*]{} = 0.06, we used a model in which distinct, isotropic peaks were assumed to coexist at both diagonal and parallel positions, as shown in the inset of Fig. 5(c), hereafter referred to as the coexistence model. The fitting parameters were refined so as to reproduce three profiles shown in Figs. 5(a)-(c) simultaneously. =2.95in The obtained parameters for the diagonal peaks incommensurability $\delta$, and peak-width $\kappa$, are 0.053$\pm$0.002 (r.l.u.) and 0.039$\pm$0.004 ($\AA$$^{-1}$), respectively. The corresponding values for the parallel peaks are $\delta$ = 0.049$\pm$0.003 (r.l.u.) and $\kappa$ = 0.03$\pm$0.006 ($\AA$$^{-1}$). The resulting intensity ratio between the sum of the four parallel peaks and the two diagonal peaks is approximately 1:2. Note that in the [*x*]{} = 0.07 sample obvious IC peaks are observed at the parallel positions although the elastic magnetic signal is relatively weaker compared with the data for [*x*]{} = 0.06 samples and parallel component dominates the diagonal one. Another new finding in this study is the observation of an anomalous dependence of the peak-width $\kappa$ on doping. In Fig. 6 we plot the doping dependence of $\kappa$, measured along the spin modulation vector, together with data from previous studies for both sides of the boundary. [@Kimura_99]$^{,}$ [@Waki_00_1]$^{,}$ [@Matsuda_00]$^{,}$ [@Matsushita] A remarkable enhancement of $\kappa$ for both the diagonal and parallel elastic peaks are clearly observed in the superconducting phase near [*x$_{c}$*]{}. We remark that this enhancement is not simply caused by the overlap of two peaks because our analysis fits the width of each peak separately. We also note that a similar enhancement was already shown in Ref. 9 in which, however, the peak-width $\kappa$ was evaluated without domain-selective scans for [*x*]{} $\leq$0.05 and using low-energy inelastic signals for [*x*]{} $\geq$0.06. On the other hand, we show here the $\kappa$ using only elastic signals taken by domain-selective scans. Figs. 7(a) and (b) show the $\delta$ and peak-angle $\alpha$ (defined in Fig. 3) versus Sr concentration [*x*]{} for both the parallel (open circles) and diagonal (solid circles) elastic peaks. In Fig. 7(b), the size of the circles represents the relative intensities of the peaks, and the vertical bars correspond to the peak-width FWHM measured along the circle of =2.75in =2.9in radius $\delta$. For samples on which we did not perform any circular scans, peak-width in angle unit is calculated from that perpendicular to the spin modulation vector. The value of $\delta$ for both peaks approximately follows the simple linear relation $\delta$ = [*x*]{}, except for [*x*]{} = 0.024 as is discussed in Ref. 11. However, as seen in Fig. 7(b), the intensity at the parallel positions appears beyond [*x$_{c}$*]{}, in accordance with the onset of the superconductivity. These results suggest that when the parallel component first appears at [*x$_{c}$*]{}, it does so with a finite incommensurability. Here we remark that the value of $\delta$ near the phase boundary exhibits no discontinuous change, but does show a slight downward deviation away from the relation $\delta$ = [*x*]{}. Discussion ========== In this study we experimentally clarified how the change in the spin modulation vector takes place upon crossing the phase boundary at [*x*]{} = [*x$_{c}$*]{}. In the superconducting phase well-defined peaks appear at parallel positions, although the diagonal component seen in the spin-glass phase persists. The intensity of the parallel component becomes dominant with the development of superconductivity upon further doping. The coincidence of the parallel component and the superconductivity with Sr doping indicates an intimate connection between the parallel spin modulation and the superconductivity (or the itinerant holes on the antiferromagnetic Cu-O square lattices). Another important result is that the incommensurabilities for both diagonal and parallel peaks monotonously connect at [*x*]{} = [*x$_{c}$*]{} (see Fig. 7(a)). Hence it is revealed that over a wide range of Sr concentration [*x*]{}, the value of $\delta$ for both the diagonal and parallel components follows the linear relation $\delta$ = [*x*]{}, even though the spin modulation vectors for the two components are entirely different. Note that if two types of hole stripes coexist corresponding to the two types of spin modulations, the average hole density for each stripe phase is nearly the same at [*x*]{} = [*x$_{c}$*]{}. We believe that having different spin modulation vectors present on either side of the phase boundary is a key to understanding the nature of the doping-induced superconducting phase transition. As already mentioned in the previous section, the coexistence of isotropic diagonal and parallel peaks in the superconducting phase near [*x$_{c}$*]{} reproduces the observed spectra in Fig. 5 quite well. In the [*x*]{} = 0.06 sample, both insulating and superconducting phases may coexist or phase-separate either microscopically or mesoscopically, and our neutron scattering data strongly suggest the former (latter) phase is accompanied with the diagonal (parallel) spin modulation. We now briefly remark an alternative model to describe the drastic change in the spin modulation vector at [*x*]{} = [*x$_{c}$*]{}. In this model, it is assumed that a pair of diagonal peak splits continuously into four crescent-shaped peaks centered between the diagonal and parallel positions. Although this model also reproduces three profiles in Fig. 5, the fitted parameters for the incommensurability and the peak-width along the radial direction are approximately the same as those obtained by the coexistence model. Therefore, results for the incommensurability and the peak-width shown in Fig. 6 and Fig. 7(a) do not depend on the details of the change in the spin modulation vector near [*x$_{c}$*]{}. Moreover, if inhomogeneous hole-distribution exists in these samples by either extrinsic or intrinsic reason, it is very difficult to distinguish two models or two situations. The continuous change in the spin modulation vector near [*x$_{c}$*]{}, if it occurred, is difficult to detect experimentally. On the other hand, in general case with such inhomogeneous hole-distribution the continuous change turns into the two phase-mixing at the phase boundary. In the present system, we, therefore, expect two phase separation rather than continuous change of the direction of spin modulation vector. The anomalous broadening of peak-width is also consistent with the existence of two phases. If both phases microscopically phase-separate, then one phase will impede the other from expanding the size of its ordered regions upon cooling, and hence result in broadened peaks. Phase separation is also suggested by the incomplete Meissner effect observed near the phase boundary, and the subsequent increase of the superconducting volume fraction upon further doping. Recall that the incommensurability $\delta$ of the parallel peak when it first appears is non-zero above [*x$_{c}$*]{}, and that the $\delta$ of the diagonal peak connects smoothly to that of the parallel one at [*x$_{c}$*]{}. Assuming that a proportional relationship exists between [*T$_{c}$*]{} and $\delta$ in the superconducting phase, [@Yamada_98] a non-zero value for $\delta$ at [*x$_{c}$*]{} would imply a finite value for [*T$_{c}$*]{} at [*x$_{c}$*]{}. Within our experimental uncertainties for [*T$_{c}$*]{}, this conclusion is not inconsistent with the experimental results presented in Fig. 1(b). Previously, in the hole-doped high-[*T$_{c}$*]{} cuprates, the transition temperature [*T$_{c}$*]{} has been considered to decrease continuously down to zero with decreasing hole content. However, the present neutron scattering results strongly suggest the discontinuous change in [*T$_{c}$*]{} at the phase boundary. Such a first-order transition of superconductivity at the boundary is supported by the result of recent resistivity measurement. [@Fujita_00] In conclusion, we confirmed the coincident appearance of the parallel spin modulation and the superconductivity upon hole-doping. The diagonal spin modulation persists into the superconducting state near the phase boundary. The incommensurabilities for the diagonal and parallel spin modulations monotonously connect at the boundary. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank R. J. Birgeneau, V. J. Emery, Y. Endoh, K. Hirota, K. Kimura, K. Machida, M. Matsuda, J. Tranquada, and Y. S. Lee for valuable discussions. We also would like to acknowledge Y. Ikeda for his assistance with the sample characterization. This work was supported in part by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research on Priority Areas (Novel Quantum Phenomena in Transition Metal Oxides), 12046239, 2000, for Scientific Research (A), 10304026, 2000, for Encouragement of Young Scientists, 13740216, 2001 and for Creative Scientific Research (13NP0201) “Collaboratory on Electron Correlations - Toward a New Research Network between Physics and Chemistry -”, by the Japan Science and Technology Corporation, the Core Research for Evolutional Science and Technology Project (CREST). Work at Brookhaven National Laboratory was carried out under Contract No. DE-AC02-98-CH10886, Division of Material Science, U. S. Department of Energy. We also acknowledge the support of the National Institute of Standards and Technology (NIST), U.S. Department of Commerce, in providing the neutron facilities used in this work. Work performed on SPINS spectrometer in NIST was also based upon activities supported by the National Science Foundation under Agreement No. DMR-9423101. The work at MIT was supported by the NSF under Grant No. DMR0071256 and by the MRSEC Program of the National Science Foundation under Grant No. DMR98-08941. Now at the Institute for Material Research, Tohoku University, Katahira, Sendai 980-0821. Also at the University of Maryland, College Park, MD 20742 Massachusetts Institute of Technology, Cambridge, MA 02139. Present address: Department of Physics, University of Toronto, Toronto, Ontario, Canada M5A 1A7. M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. Endoh, Rev. Mod. Phys. [**70**]{}, 897 (1998). K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R.J. Birgeneau, M. Greven, M.A. Kastner, and Y.J. Kim, Phys. Rev. B [**57**]{}, 6165 (1998). J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature (London) [**375**]{}, 561 (1995). T. Suzuki, T. Goto, K. Chiba, T. Shinoda, T. Fukase, H. Kimura, K. Yamada, M. Ohashi, and Y. Yamaguchi, Phys. Rev. B [**57**]{}, 3229 (1998). H. Kimura, K. Hirota, H. Matsushita, K. Yamada, Y. Endoh, S. H. Lee, C. F. Majkrzak, R. Erwin, G. Shirane, M. Greve, Y. S. Lee, M. A. Kastner, and R. J. Birgeneau, Phys. Rev. B [**59**]{}, 6517 (1999). Y. S. Lee, R. J. Birgeneau, M. A. Kastner, Y. Endoh, S. Wakimoto, K. Yamada, R. W. Erwin, S. H. Lee, and G. Shirane, Phys. Rev. B [**60**]{}, 3643 (1999). J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, Phys, Rev. B [**54**]{}, 7489 (1996). H. Kimura, H. Matsushita, K. Hirota, Y. Endoh, K. Yamada, G. Shirane, Y. S. Lee, M. A. Kastner, and R. J. Birgeneau, Phys. Rev. B [**61**]{}, 14366 (2000). S. Wakimoto, G. Shirane, Y. Endoh, K. Hirota, S. Ueki, K. Yamada, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, P. M. Gehring, and S. H. Lee, Phys. Rev. B [**60**]{}, R769 (1999). S. Wakimoto, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, R. Erwin, P. M. Gehring, S. H. Lee, M. Fujita, K. Yamada, Y. Endoh, K. Hirota, and G. Shirane, Phys. Rev. B [**61**]{}, 3699 (2000). M. Matsuda, M. Fujita, K. Yamada, R. J. Birgeneau, M. A. Kastner, Y. Endoh, S. Wakimoto, and G. Shirane, Phys. Rev. B [**62**]{}, 9148 (2000). I. Tanaka, K. Yamane, and H. Kojima, J. Crystal Growth [**96**]{}, 711 (1989).; S. Hosoya, C. H. Lee, S. Wakimoto, K. Yamada, and Y. Endoh, Physica C 235-240, [**547**]{} (1994).; C. H. Lee, N. Kaneko, S. Hosoya, K. Kurahashi, S. Wakimoto, K. Yamada, and Y. Endoh, Supercond. Sci. Technol. [**11**]{}, 891 (1998). S. Wakimoto, K. Yamada, S. Ueki, G. Shirane, Y. S. Lee, S. H. Lee, M. A. Kaster, K. Hirota, P. M. Gehring, Y. Endoh, and R. J. Birgeneau, J.Phys. Chem. Solid, [**60**]{}, 1079 (1999). H. Takagi, T. Ido, S. Ishibashi, M. Uota, and S. Uchida, Phys. Rev. B [**40**]{}, 2254 (1989).; H. Takagi, B. Batlogg. H. L. Kao, J. Kwo, R. J. Cava, J. J. Krajewski, and W. F. Peck, Jr, Phys. Rev. Lett. [**69**]{}, 2975 (1992). H. Matsushita, H. Kimura, M. Fujita, K. Yamada, K. Hirota, and Y. Endoh, J. Phys. Chem. Solid [**60**]{}, 1071 (1999). S.Wakimoto, S.Ueki, Y.Endoh, and K.Yamada, Phys. Rev. B [**62**]{}, 3547 (2000). T. Fujita, J. Hori, T. Goko, N. Kikugawa, and S. Iwata, Inst. Phys. Chem. Res. [**27**]{}, 75 (2000).

--- abstract: 'We present the asymptotic formula for the Wigner $9j$-symbol, valid when all quantum numbers are large, in the classically allowed region. As in the Ponzano-Regge formula for the $6j$-symbol, the action is expressed in terms of lengths of edges and dihedral angles of a geometrical figure, but the angles require care in definition. Rules are presented for converting spin networks into the associated geometrical figures. The amplitude is expressed as the determinant of a $2\times 2$ matrix of Poisson brackets. The $9j$-symbol possesses caustics associated with the fold and elliptic and hyperbolic umbilic catastrophes. The asymptotic formula obeys the exact symmetries of the $9j$-symbol.' address: 'Department of Physics, University of California, Berkeley, California 94720 USA' author: - 'Hal M. Haggard and Robert G. Littlejohn' title: 'Asymptotics of the Wigner $9j$-Symbol' --- Introduction ============ The asymptotic behavior of spin networks has played a significant role in simplicial approaches to quantum gravity. Indeed, the field began with the observation that the Ponzano-Regge action (1968) for the semiclassical $6j$-symbol is identical to the Einstein-Hilbert action of a tetrahedron in 3-dimensional gravity in the Regge formulation (Regge, 1961; see also Williams and Tuckey 1992 and Regge and Williams 2000). More recently, semiclassical expansions have been used to study the low-energy or classical limit of quantum gravity as well as to derive quantum corrections to the classical theory. Asymptotic studies in this area have included treatments of the $10j$-symbol (Barrett and Williams 1999, Baez 2002, Barrett and Steele 2003, Freidel and Louapre 2003), amplitudes in the Freidel-Krasnov model (Conrady and Freidel 2008), LQG fusion coefficients (Alesci 2008), and the EPRL amplitude (Barrett 2009). In addition, the venerable $6j$-symbol and Ponzano Regge (1968) formula continue to receive attention (Roberts 1999, Barrett and Steele 2003, Freidel and Louapre 2003, Gurau 2008, Charles 2008, Littlejohn and Yu 2009, Depuis and Livine 2009, Ragni 2010), not to mention the $q$-deformed $6j$-symbol (Nomura 1989; Taylor and Woodward 2004, 2005). In this article we present the generalization of the Ponzano-Regge formula to the Wigner $9j$-symbol, as well as some material relevant for the asymptotics of arbitrary spin networks. The Ponzano-Regge formula (Ponzano and Regge 1968) gives the asymptotic expression for the Wigner $6j$-symbol when all quantum numbers are large. The $9j$-symbol is the next most complicated spin network after the $6j$-symbol, with features that are found in all higher spin networks. In this article we present only the asymptotic formula itself for the $9j$-symbol and some salient facts surrounding it. We defer a derivation and deeper discussion of the formula to a subsequent publication. Our derivation has quite a few steps, and some of them at this point are supported by numerical evidence only. Thus, we do not now have a rigorous derivation of our result. We believe it is correct, however, on the basis of direct numerical comparisons with the exact $9j$-symbol, the fact that our formula obeys all the symmetries of the exact $9j$-symbol, and the plausibility and numerical support for the conjectures involved in the parts of the derivation currently lacking proofs. The proofs do not seem difficult, and we hope to fill in the gaps in our future work. Although most of the papers cited above have dealt with the asymptotics of specific spin networks, usually there are special values of the angular momenta that are used. For example, the $10j$-symbol involves balanced representations of $SO(4)$, which means that some pairs of $j$’s are equal, while the $9j$-symbols that appear in LQG fusion coefficients have two columns in which one quantum number is the sum of the other two. In addition, $j$’s are sometimes set equal because this is regarded as the most interesting regime from a physical standpoint. As a result, the spin networks that have been studied tend to fall on caustics where the asymptotic behavior is not generic. At such points, the value of the spin network (the wave function) is not oscillatory in a simple sense, instead it has the form of a diffraction catastrophe (Berry 1976). In addition, the wave function scales as a higher (less negative) power of the scaling parameter (effectively, $1/\hbar$). This type of behavior has been noted in several places in the quantum gravity literature, although as far as we can tell no one has noted that it is related to standard caustic and catastrophe types. In this article we give a rather complete picture of the $9j$-symbol for all possible parameters in the classically allowed region, including all phases and Maslov indices. We also indicate the subsets upon which the behavior is nongeneric and described by various types of caustics. We believe that this is the first time that such information has been available for any spin network more complicated than the $6j$-symbol. Another reason for interest in the $9j$-symbol is that it is the nontrivial part of the Clebsch-Gordan coefficient for $SO(4)$. Basic references on the Wigner $9j$-symbol include Edmonds (1960), Biedenharn and Louck (1981ab) and Varshalovich (1981). Recent work on the $9j$-symbol has included new asymptotic forms when some quantum numbers are large and others small (Anderson 2008, 2009). We also note the use of $SU(2)$ spin networks in quantum computing (Marzuoli and Rasetti 2005). In Sec. \[asymptoticformula\] we present the asymptotic formula for the $9j$-symbol and draw comparisons with the Ponzano-Regge formula to introduce its geometrical content. A detailed explanation of the notation follows in later sections. In Sec. \[tog\] we present general rules for converting spin networks into surfaces composed of oriented edges and oriented triangles, and illustrate them for the $9j$-symbol. In Sec. \[findingvectors\] we explain how the geometrical objects (pieces of oriented surfaces) corresponding to the $9j$-symbol can be constructed in 3-dimensional space. In Sec. \[visualizing\] we explain the configuration space of the $9j$-symbol and the classically allowed subset thereof. In Sec. \[theamplitude\] we define the amplitude of the asymptotic formula and discuss the manifolds (the caustics) upon which it diverges as well as the diffraction catastrophes that replace the simple asymptotic form in the neighborhood of the caustics. In Sec. \[thephase\] we explain the phase of the semiclassical approximation, a generalization of the Ponzano-Regge action that requires careful definitions of dihedral angles. In Sec. \[symmetries\] we show that the asymptotic formula correctly obeys the symmetries of the $9j$-symbol. Finally, in Sec. \[conclusions\] we present some comments and conclusions. The Asymptotic Formula {#asymptoticformula} ====================== The asymptotic expression for the $9j$-symbol is $$\left\{\begin{array}{ccc} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9 \end{array}\right\} = A_1 \cos S_1 + A_2 \sin S_2, \label{theresult}$$ where $A_{1,2}$ are positive amplitudes, $S_{1,2}$ are phases, and each term is roughly similar to the single term in the Ponzano-Regge formula for the $6j$-symbol. The right hand side is the leading term in an asymptotic expansion in powers of $1/k$ of the $9j$-symbol when all nine $j$’s are scaled by a positive factor $k$ that is allowed to go to infinity ($k$ plays the role of $1/\hbar$ in the asymptotic expansion). The $k$’s are suppressed in (\[theresult\]), but the expression on the right scales as $1/k^3$. Equation (\[theresult\]) applies only in the classically allowed region. We do not present the analog of (\[theresult\]) in the classically forbidden region. Equation (\[theresult\]) breaks down near caustics, where the $9j$-symbol scales with a higher (less negative) power of $k$ than $1/k^3$. In the neighborhood of caustics, the $9j$-symbol is approximated by diffraction catastrophes, including the fold and hyperbolic and elliptic umbilic. These are discussed more fully in Sec. \[theamplitude\]. To explain the meaning of (\[theresult\]) some analogies with the Ponzano-Regge formula for the $6j$-symbol are useful. In the classically allowed region, the Ponzano-Regge formula associates a given $6j$-symbol with a real tetrahedron whose six edge lengths are $J_i = j_i+1/2$, where the six $j$’s are those appearing in the $6j$-symbol. More precisely, there are two tetrahedra, related by spatial inversion, that is, time-reversal. Except for flat configurations, the two tetrahedra are not related by proper rotations in $SO(3)$. We recall that time-reversal, not parity, inverts the direction of angular momentum vectors. The two tetrahedra correspond to the two stationary phase points of the $6j$-symbol, which make contributions to the asymptotic expression that are complex conjugates of each other. The result is the real cosine term in the Ponzano-Regge formula. One can say that semiclassically the $6j$-symbol is a superposition of two amplitudes, corresponding to a tetrahedral geometry and its time-reversed image, that produce oscillations in the result. We shall use lower case $j$’s for quantum numbers, and capital $J$’s for the lengths of the corresponding classical vectors. These are always related by $J_i=j_i+1/2$. The $1/2$ is a Maslov index (Maslov and Fedoriuk 1981, Mischenko 1990, de Gosson 1997), and the manner in which it arises in this context is explained in Aquilanti (2007). In the case of the $9j$-symbol in the classically allowed region, there are four geometrical figures associated with a given set of nine $j$’s, consisting of two pairs related by time-reversal. The four geometrical figures correspond to the four real stationary phase points of the $9j$-symbol. Each pair of figures is associated with an “admissible” root (defined momentarily) of a certain quartic equation. There are two admissible roots in the classically allowed region, labeled 1 and 2, corresponding to the two terms in (\[theresult\]). Each trigonometric term in (\[theresult\]) consists of an exponential and its complex conjugate, corrresponding to a geometrical figure and its time-reversed image. One can say that semiclassically the $9j$-symbol is a superposition of four amplitudes corresponding to four geometries, consisting of two pairs of a geometry and its time-reversed image. We now explain these geometries and how they are specified by the nine $j$’s that appear in the symbol. Triangles, Orientations and Geometries {#tog} ====================================== The $9j$-symbol specifies the lengths $J_i=j_i+1/2$ of nine classical angular momentum vectors $\Jvec_i$ but not their directions. Therefore we inquire as to how the directions may be determined, and geometrical figures constructed out of the resulting vectors. Actually, it is convenient to double this set and speak of 18 classical vectors $\Jvec_i$, $\Jvec'_i$, $i=1,\ldots,9$. A doubling of this kind was introduced by Roberts (1999), who gave a highly symmetrical way of writing the $6j$-symbol as a scalar product in a certain Hilbert space. Although Roberts only worked with the $6j$-symbol, his method is easily generalized to an arbitary spin network. Ponzano and Regge (1968) also gave hints that doubling of angular momentum vectors are important in the asymptotic analysis of spin networks. We now describe rules that take an arbitary spin network (with at most trivalent vertices) and transcribe it into relations among a doubled set of classical angular momentum vectors, defining a set of oriented triangles and oriented edges of a geometrical figure. We exemplify these rules only in the case of the $9j$-symbol, but they are easily applied to any spin network. The reader may find it illuminating to apply our rules to the $6j$-symbol, starting with the usual spin network (the Mercedes graph). Figure \[network\] illustrates the spin network of the $9j$-symbol. See also Fig. 18.1 of Yutsis (1962). Each edge of the spin network, labeled by $j_i$, is associated with two classical angular momentum vectors $\Jvec_i$ and $\Jvec'_i$ that are required to satisfy $$|\Jvec_i|=|\Jvec'_i| = J_i=j_i+1/2 \label{lengtheqn}$$ and $$\Jvec_i+\Jvec'_i=0. \label{JJprimeeqn}$$ Vectors $\Jvec_i$ and $\Jvec'_i$ have the same length and point in opposite directions. Each vertex of the spin network, where three edges meet, corresponds to three vectors that add to zero. The three vectors are associated with the three edges. If the arrow on an edge ending at the vertex is pointing away from the vertex, then the angular momentum vector is unprimed; if it is pointing toward the vertex, then the vector is primed. This rule applied to Fig. \[network\] gives $$\eqalign{ \Jvec_1 + \Jvec_2 + \Jvec_3 &=0, \qquad \Jvec'_1 + \Jvec'_4 + \Jvec'_7=0, \\ \Jvec_4 + \Jvec_5 + \Jvec_6 &=0, \qquad \Jvec'_2 + \Jvec'_5 + \Jvec'_8 =0, \\ \Jvec_7 + \Jvec_8 + \Jvec_9 &=0, \qquad \Jvec'_3 + \Jvec'_6 + \Jvec'_9=0.} \label{triangleeqns}$$ These are a set of classical triangle relations, one for each vertex of the spin network. In the case of the $9j$-symbol, they are obviously related to the rows and columns of the symbol. Although the vector addition in (\[triangleeqns\]) is commutative, we agree to write the vectors in each equation in counterclockwise order (around the vertex of the spin network) for a vertex with $+$ orientation, and in clockwise order for a vertex with $-$ orientation, modulo cyclic permutations. Thus the ordering of the vectors is the same as the ordering of the columns of the $3j$-symbol implied by the vertex of the network. This ordering is used to define a set of oriented triangles. We take the three vectors of any one of the equations (\[triangleeqns\]) and place the base of one vector at the tip of the preceding one, to create the three edges of a triangle. In this process we parallel translate the vectors (in $\Reals^3$) but do not rotate them. The triangle is given an orientation (a definition of a normal) by taking the cross product of any two successive vectors defining the edges. For example, the normal to the 123-triangle is $\Jvec_1\times\Jvec_2$, and that of the $1'4'7'$-triangle is $\Jvec'_1\times\Jvec'_4$, which, in view of (\[JJprimeeqn\]), is the same as $\Jvec_1\times\Jvec_4$. Next, we take the triangles and displace them so that the edge $\Jvec_i$ of one triangle is adjacent to the edge $\Jvec'_i$ of another triangle. In this process, the triangles are displaced but not rotated. If we do this with the six triangles defined by (\[triangleeqns\]) in the case of the $9j$-symbol, we find that six pairs of edges can be made adjacent, as illustrated by the central six triangles of Fig. \[triangles\]. In this “central region” six pairs of vectors $\Jvec_i$ and $\Jvec'_i$ are adjacent for $i=1,2,5,6,7,9$. There is some arbitrariness in choosing which six pairs of edges will be made adjacent. If we wish that the remaining edges $i=3,4,8$ also be paired, we can duplicate three of the triangles and attach them to the periphery of the central region, as illustrated in Fig. \[triangles\]. This amounts to a kind of “analytic continuation” of the central region. Figure \[triangles\] is highly schematic. In general, the triangles are not equilateral, the surface that is formed by attaching them together is not planar, and the triangles may fold under one another. The central region in Fig. \[triangles\] is a piece of an oriented surface, that is, all the normal vectors (by our convention) are pointing on the same side. In the case of the $6j$-symbol, our rules produce a closed surface (the usual tetrahedron), with normals all pointing either outward or inward (time-reversal converts one into the other). In the case of the $9j$-symbol, the surface is not closed. There is some suggestion that this surface represents a triangulation of $\Reals P^2$ but for this article we shall view it as living in $\Reals^3$. Finally, we orient each edge by choosing the direction of the vector $\Jvec_i$ (not $\Jvec'_i$). We will be interested in finding solutions $\{\Jvec_i, \Jvec'_i, i=1,\ldots,9\}$ of (\[lengtheqn\]), (\[JJprimeeqn\]) and (\[triangleeqns\]), modulo overall proper rotations (in $SO(3)$). That is, although we do not rotate vectors or faces when forming our surface with oriented faces and egdes, we are allowed to rotate the whole surface once completed. We notice that if $\{\Jvec_i, \Jvec'_i, i=1,\ldots,9\}$ is a solution of these equations, then the time-reversed set $\{-\Jvec_i, -\Jvec'_i, i=1,\ldots,9\}$ is also a solution. If we apply our rules for converting vectors into a surface, we will find in general that the time-reversed set produces a different surface (not equivalent under $SO(3)$). We apply time-reversal only to the vectors, not the rules; for example, the ordering of the time-reversed vectors is the same as the original vectors. The central six triangles of the time-reversed surface are illustrated in Fig. \[trev\]. To visualize the surfaces in Figs. \[triangles\] and \[trev\], we may imagine that the central region of Fig. \[triangles\] bulges out of the paper, like the northern hemisphere of a sphere (whether it does or not depends on the parameters, but this is one possibility). Then the time-reversed surface in Fig. \[trev\] bulges into the paper, since spatial inversion is equivalent, modulo $SO(3)$, to reflection in a plane. Then the central region of Fig. \[triangles\] can be glued to the time-reversed surface in Fig. \[trev\], bringing edge $\Jvec'_3$ adjacent to edge $-\Jvec_3$, etc, and producing a surface homeomorphic to $S^2$. This is the hexagonal bipyramid constructed by Ponzano and Regge (1968). The conventional normals are pointing outward in the northern hemisphere, and inward on the southern. As noted by Ponzano and Regge, this bipyramid is bisected by three planes passing through a common line, namely the “axis” of the sphere, which cut the bipyramid into three pairs of congruent tetrahedra. These correspond to the three $6j$-symbols in the representation of the $9j$-symbol as a sum over products of $6j$-symbols (see Edmonds (1960) Eq. 6.4.3), in which the variable of summation is the common edge of the tetrahedra (the axis of the sphere). Finding the Vectors {#findingvectors} =================== To find a solution of (\[lengtheqn\]), (\[JJprimeeqn\]) and (\[triangleeqns\]) we notice that all 18 vectors are determined if only four of them, $\{\Jvec_1,\Jvec_2,\Jvec_4,\Jvec_5\}$ are given. We let $G$ be the $4\times 4$ Gram matrix constructed out of these vectors, that is, the $4\times 4$, real symmetric matrix of dot products of these vectors among themselves. Of the ten independent dot products, eight can be determined from the given lengths $J_i$, $i=1,\ldots,9$. That is, the diagonal elements are $J_i^2$, $i=1,2,4,5$, while $$\eqalign{ \Jvec_1\cdot\Jvec_2 &= (J_3^2-J_1^2-J_2^2)/2,\qquad \Jvec_1\cdot\Jvec_4 = (J_7^2-J_1^2-J_4^2)/2,\\ \Jvec_2\cdot\Jvec_5 &= (J_8^2-J_2^2-J_5^2)/2,\qquad \Jvec_4\cdot\Jvec_5 = (J_6^2-J_4^2-J_5^2)/2.} \label{dotprods}$$ The two dot products that cannot be determined from the given lengths are $u=\Jvec_1\cdot\Jvec_5$ and $v=\Jvec_2\cdot\Jvec_4$, which we regard as unknowns. These satisfy a linear equation obtained by squaring $\Jvec_9 = -\Jvec_3-\Jvec_6$, $$J_9^2=J_3^2+J_6^2 + 2(u+v +\Jvec_1\cdot\Jvec_4 +\Jvec_2\cdot\Jvec_5). \label{xyeqn}$$ Another equation connecting $u$ and $v$ is $\det G=0$, which holds since the four vectors lie in $\Reals^3$ and the 4-simplex defined by them is flat. This is a quartic equation in $u$ and $v$, which by using (\[xyeqn\]) to eliminate $v$ can be converted into a quartic equation in $u$ alone. We write this quartic as $Q(u)=0$. We find the roots $u$ of this quartic, solve for $v$ by using (\[xyeqn\]), whereupon all components of the Gram matrix become known (there is one Gram matrix for each root). Ponzano and Regge (1968) discussed this procedure in somewhat different language, and apparently believed that all four roots would contribute to the asymptotics of the $9j$-symbol. In fact, they do, if one wishes to work in the classically forbidden region and/or take into account tunnelling and exponentially small corrections in the neighborhood of internal caustic points (more about these below). But in the classically allowed region the asymptotics of the $9j$-symbol are dominated by the contributions from “admissible” roots, namely, those roots that produce Gram matrices that can be realized as dot products of real vectors $\Jvec_i$. Only these correspond to real geometrical figures of the type we have described. If a root $u$ of $Q(u)=0$ is complex, then it produces a complex Gram matrix that cannot be realized with real vectors, and so $u$ is inadmissible. But a real Gram matrix can be realized as the dot products of real vectors if and only if it is positive semidefinite, so even if $u$ is real it will still be inadmissible if $G$ has negative eigenvalues. We define the classically allowed region of the $9j$-symbol as the region in which $Q(u)$ has at least one admissible root. In fact, in the classically allowed region $Q(u)$ has four real roots of which two are generically admissible. We order the four real roots of $Q(u)$ in the classically allowed region in ascending order and label them by $k=0,1,2,3$. It turns out that the two admissible roots are the middle two, $k=1,2$, corresponding to the two terms of (\[theresult\]) with the same subscripts, $k=1,2$. For a given admissible root, that is, a positive semidefinite Gram matrix, we wish to find the vectors $\Jvec_i$, $i=1,2,4,5$. We arrange the four unknown vectors as the columns of a $3\times4$ matrix $F$, so that $G=F^TF$. To find $F$ given $G$, we diagonalize $G$, $G=VKV^T$, where $V\in O(4)$ and $K$ is diagonal with nonnegative diagonal entries (the eigenvalues of $G$). At least one of these eigenvalues must be 0; we place it last, and write $K=D^TD$ where $D$ is a real, $3\times 4$ diagonal matrix. Then $F=UDV^T$, where $U$ is an arbitrary element of $O(3)$. This generates all possible sets of vectors whose dot products are realized in $G$; it amounts to using the singular value decomposition of $F$. If $U=R\in SO(3)$ then we generate a set of surfaces related by overall rotations; if $U=-R$ we generate the time-reversed set. In this way a single Gram matrix, corresponding to a single admissible root of the quartic, produces a geometry and its time-reversed image. Altogether, the two admissible roots imply the four geometries in (\[theresult\]). This method of finding $F$ is discussed in the context of the $6j$-symbol by Littlejohn and Yu (2009), where it is also applied in the classically forbidden region. There we find complex angular momentum vectors that satisfy the required algebraic relations. This carries over to the $9j$-symbol in the classically forbidden region. In the literature on the $6j$-symbol it is common to state that a Euclidean group applies in the classically allowed region and a Lorentz group in the classically forbidden region; but for the $9j$-symbol the groups are actually $SO(3,\Reals)$ and $SO(3,\Complexes)$. The Classically Allowed Region and Configuration Space {#visualizing} ====================================================== The classically allowed region is a subset of full dimensionality of the 9-dimensional parameter space of the $9j$-symbol, itself a convex subset of $\Reals^9$ defined by the triangle inequalities. To visualize this and other subsets of the parameter space it helps to fix seven of the $j$’s to obtain a 2-dimensional slice. Figure \[configspace\] illustrates such a slice for the case $$\left\{ \begin{array}{ccc} 129/2 & 137/2 & j_3 \\ 113/2 & 121/2 & j_6 \\ 64 & 108 & 90 \end{array}, \right\} \label{slice}$$ in which only $j_3$ and $j_6$ are allowed to vary. The choice of $j_3$ and $j_6$ for this purpose is not arbitrary, since these two $j$’s are quantum numbers for a pair of commuting operators on a space of 5-valent $SU(2)$ intertwiners. They are like $x$ and $y$ for a wave function $\psi(x,y)$. In this analogy, we think of $(j_3,j_6)$-space as a “configuration space” for the $9j$-symbol and the $9j$-symbol itself as a “wave function” $\psi(j_3,j_6)$. We will mostly use the variables $J_3=j_3+1/2$, $J_6 = j_6+1/2$ to describe this space. When thinking in classical terms, $J_3$ and $J_6$ are continuous variables (not quantized). Figure \[configspace\] illustrates a convex region of the $J_3$-$J_6$ plane, bounded by straight lines and defined by the classical triangle inequalities, $$\eqalign{ \max(|J_1-J_2|,|J_6-J_9|) &\le J_3 \le \min(J_1+J_2,J_6+J_9)\\ \max(|J_4-J_5|,|J_3-J_9|) &\le J_6 \le \min(J_4+J_5,J_3+J_9).} \label{triangleinequals}$$ Properly speaking, configuration space is this convex region, not the whole plane. The unshaded area inside the convex region is the classically allowed region, surrounded by the shaded classically forbidden region. The caustic curve separates the classically allowed from the classically forbidden regions; it has kinks (discontinuities in slope) at points $B$, and is tangent to the boundary of the convex region at several points. Other features of this figure are explained below. Given a point $(J_3,J_6)$ of the classically allowed region, the procedure described in Secs. \[tog\] and \[findingvectors\] produces a quartic polynomial $Q(u)$ whose two middle roots $k=1,2$ are admissible. These can be thought of as specifying a two-branched “root surface” that sits over the classically allowed region. The two middle roots coalesce as we approach the caustic curve, and become (inadmissible) complex conjugates as we move beyond. Thus, the two root surfaces can be thought of as being glued together on the caustic curve. Corresponding to each root there are two geometries modulo $SO(3)$, related by time-reversal, so there is a two-fold “geometry surface” sitting above each root surface, or four geometry surfaces sitting above the classically allowed region. These four geometry surfaces are actually branches of the projection of an invariant 2-torus onto configuration space, and correspond to the four exponential terms in (\[theresult\]). This 2-torus sits in the phase space of the $9j$-symbol, a 4-dimensional, compact symplectic manifold. This symplectic manifold is only one of several phase spaces that describe the classical mechanics of the $9j$-symbol, but all the others have higher dimensionality so we call this one the “phase space of minimum dimensionality.” It is one of the symplectic manifolds discovered by Kapovich and Millson (1996). Its analog in the case of the $6j$-symbol is a spherical phase space, which has been studied by Charles (2008) and by Littlejohn and Yu (2009). The phase space of minimum dimensionality is related to other phase spaces for the $9j$-symbol by a combination of symplectic reduction (Marsden and Ratiu 1999) and the elimination of constraints. We have found it useful to employ all these spaces in our work on the $9j$-symbol. The Amplitude and Caustics {#theamplitude} ========================== The amplitudes of semiclassical approximations are notorious for the computational difficulties they cause. For example, several authors have resorted to computer algebra and/or numerical experimentation to check the amplitude determinant in the Ponzano-Regge formula. Actually, this amplitude (due originally to Wigner (1959)) is given by a single Poisson bracket between intermediate angular momenta (Aquilanti 2007, and, in more detail, Littlejohn and Yu 2009), which can be evaluated in a single line of algebra. More generally, semiclassical amplitudes are easily found in terms of matrices of Poisson brackets. In the case of the $9j$-symbol we define $$V_{ijk} = \Jvec_i \cdot (\Jvec_j \times \Jvec_k), \label{Vijkdef}$$ which is six times the signed volume of the tetrahedron specified by edges $i$, $j$, $k$ (it is the volume of the corresponding parallelepiped). Then the amplitudes $A_1$, $A_2$ in (\[theresult\]) are given by $$A= \frac{1}{4\pi \sqrt{|\det D|}}, \label{Adef}$$ where $$D= \left(\begin{array}{cc} V_{124} & V_{215} \\ V_{451} & V_{542} \end{array}\right). \label{Ddef}$$ The subscripts 1,2 are omitted on $A$ in (\[Adef\]) because the same formula applies for both terms in (\[theresult\]), but $A_1 \ne A_2$ in general because the formula is evaluated on two different geometries (associated with the two admissible roots). The quantitity $\det D$ is even under time-reversal, so the same amplitude applies to both a geometry and its time-reversed image. The volumes in matrix $D$ are Poisson brackets of intermediate angular momenta in a recoupling scheme for the $9j$-symbol, which are most easily evaluated in the phase space of minimum dimensionality. We omit details; suffice it to say for now that the derivation of the matrix (\[Ddef\]) in terms of Poisson brackets and thence the amplitude is extremely easy. We define the caustic set as the subset of the $9j$-parameter space where $\det D=0$. Its intersection with the 2-dimensional slice seen in Fig. \[configspace\] consists of the union of the caustic curve (the curve separating the classically allowed from the classically forbidden region) with the two points marked $I$. In addition, the caustic set includes the continuation of the caustic curves from points $B$ into the classically forbidden region. The points $I$ are “internal” caustics, that is, internal to the classically allowed region. While the caustic curve has codimension 1, the internal caustics have codimension 2. The quantity $\det D$ is nonzero away from the caustics. It turns out that the sign of $\det D$ distinguishes the two root surfaces, with $\det D >0$ on root surface 1 and $\det D <0$ on root surface 2. The caustics of the $6j$-symbol occur at the flat configurations (flat tetrahedra), as appreciated by Ponzano and Regge (1968) and Schulten and Gordon (1975a,b). The caustics of the $9j$-symbol, however, are not in general flat, that is, $\det D=0$ does not imply that the configuration is flat. The flat configurations of the $9j$-symbol, however, do lie on the caustic set. In a given $J_3$-$J_6$ slice, there are precisely four flat configurations. In the example of Fig. \[configspace\], these are marked $B$ and $I$. The points $B$ are flat configurations lying on the boundary of the classically allowed region (the caustic curve), while points $I$ are internal flat configurations. As we vary the seven $j$’s that are fixed in Fig. \[configspace\], the number of flat configurations on the boundary varies from 2 to 4; those not on the boundary are internal. In the usual manner of semiclassical approximations, (\[theresult\]) breaks down in a neighborhood of the caustic set (it diverges exactly at the caustic), and must be replaced by a diffraction function associated with a catastrophe (Berry 1976). In the case of the $6j$-symbol, the only catastrophe that occurs is the fold, yielding an Airy function as the semiclassical approximation, as noted by Ponzano and Regge (1968) and Schulten and Gordon (1975). This is the normal situation for systems of one degree of freedom. The $9j$-symbol, however, possesses two degrees of freedom, and other types of catastrophes occur. The fold catastrophe applies at most points along the caustic curve, where the $9j$-symbol is approximated by an Airy function; but at flat configurations there is an umbilic catastrophe, hyperbolic for those ($B$) falling on the boundary (caustic) curve and elliptic for the internal caustics ($I$). See Trinkhaus and Dreper (1977) for illustrations of the associated diffraction functions. The umbilic catastrophes are generic in systems of three degrees of freedom but occur in the $9j$-symbol (with only two) because of time-reversal symmetry. However, only sections of the full three-dimensional umbilic wave forms appear (Berry 1976). The cusp catastrophe, which can be expected in generic systems of two degrees of freedom, does not occur in the classically allowed region of the $9j$-symbol. Caustics are associated with the coalescence of branches of the projection of a Lagrangian manifold in phase space onto configuration space. In the case of the $9j$-symbol, the Lagrangian manifold is the invariant 2-torus mentioned in Sec. \[visualizing\]. Along the boundary of the classically allowed region, the two admissible roots coalesce, which means that the four geometries merge into two. At most points on the boundary curve, the two remaining geometries are not equal, but are related by time-reversal. At such points we have a fold catastrophe, and the $9j$-symbol is approximated by an Airy function (modulated by a cosine term). At points $B$, however, the two geometries related by time-reversal merge into a single flat configuration, producing the hyperbolic umbilic catastrophe. At internal caustic points $I$ the geometry and its time reversed image for one of the two admissible roots coalesce to produce a flat configuration. The two geometries of the other root surface, however, do not coalesce. Thus at internal caustics $I$ there are three geometries. Only the flat configuration associated with one of the roots produces the elliptic umbilic catastrophe; thus, only one of the two terms in (\[theresult\]) is replaced by the elliptic umbilic diffraction function, while the other remains as shown in (\[theresult\]). The $9j$ symbol is a linear combination of these two terms, but the elliptic umbilic diffraction function dominates when the scaling factor $k$ is large. The caustics have a certain size, that is, a distance around the caustic set over which diffraction functions must be used instead of (\[theresult\]). This distance $\Delta j$ scales as $k^{1/3}$ for all three catastrophe types (fold and elliptic and hyperbolic umbilic) discussed here. In the neighborhood of fold catastrophes the wave function scales as $k^{-17/6}$, that is, $k^{1/6}$ higher than the $k^{-3}$ of the two terms in (\[theresult\]). In the neighborhood of umbilic catastrophes the scaling is $k^{-8/3}$, that is, with another factor of $k^{1/6}$. For large values of $k$ the $9j$-symbol is largest near the points $I$, $B$. Linear combinations with different scaling behaviors have been observed by Barrett and Steele (2003) and by Freidel and Louapre (2003) in their studies of the $10j$-symbol. It seems that the $9j$-symbol is the simplest spin network in which this phenomenon occurs. The Phase {#thephase} ========= The phases $S_1$ and $S_2$ in (\[theresult\]) each have the form $$S = \sum_{i=1}^9 J_i \theta_i, \label{Sdef}$$ where $\theta_i$ is the angle between normals of adjacent faces of the geometrical figure. This of course is similar to the Ponzano-Regge formula, but the $6j$-tetrahedron is convex and all dihedral angles can be taken in the interval $[0,\pi]$. The dihedral angles for the $9j$-symbol, on the other hand, must be allowed to lie in a full $2\pi$ interval, as explained momentarily. The subscripts 1,2 are omitted on $S$ in (\[Sdef\]) because the same formula applies to both terms in (\[theresult\]). The formula must be evaluated, however, on two different geometries, so $S_1$ and $S_2$ are not equal. In addition, the angles $\theta_i$ lie in different intervals for the two geometries. Each edge $i$ of the geometrial figure is adjacent to two faces, for example, edge 4 in Fig. \[triangles\] is adjacent to faces $1'4'7'$ and $456$. One face adjacent to edge $i$ contains vector $\Jvec_i$, and the other $\Jvec'_i$. Let the two normals of these two faces, according to the conventions given above, be $\hat{\bf n}$ and $\hat{\bf n}'$. Then we define $\theta_i$ as the angle such that $$R(\hat{\bjmath},\theta_i)\hat{\bf n} = \hat{\bf n'}, \label{thetadef}$$ where $\hat{\bjmath}$ is the unit vector along $\Jvec$, specifying the axis of a rotation $R$ by angle $\theta_i$ using the right-hand rule. In the Ponzano-Regge formula one can compute the dihedral angle from its cosine, but for the $9j$ one must also use the sine of the angle. That is, (\[thetadef\]) is equivalent to $$\hat{\bf n}' = \cos\theta_i \,\hat{\bf n} + \sin\theta_i \,\hat{\bjmath} \times \hat{\bf n}. \label{altthetadef}$$ This determines $\theta_i$ to within an additive integer multiple of $2\pi$. We add the further requirement that for the geometries associated with the first root (the cosine term in (\[theresult\])), $-\pi\le \theta_i < +\pi$, while for the second root (the sine term in (\[theresult\])), $0\le\theta_i < 2\pi$. These ranges for the angle $\theta_i$ are chosen because they give a continuous branch for the angle over the two root surfaces. It turns out that $\theta_i$ never crosses $\pm \pi$ on the surface for root 1, and it never crosses 0 or $2\pi$ on the surface for root 2. The rules given in Secs. \[tog\] and \[findingvectors\] for converting vectors into surfaces with oriented edges and triangles are an essential part of the definition of the dihedral angles $\theta_i$. It is of interest to see how the angles change when a set of vectors or the associated geometry is subjected to some symmetry. Under time-reversal, the orientation of all triangles reverses, that is, the normal vectors stay the same but the vectors defining the edges are inverted. This means that the angles $\theta_i$ go into $-\theta_i$ on root surface 1, while they go into $2\pi-\theta_i$ on root surface 2 (both changes guarantee that the angles remain within their respective ranges). Thus $S$ goes into $-S$ on root surface 1 and $$S \to -S + 2\pi\nu+ 9\pi \label{S2map}$$ on root surface 2, where $\nu$ is the integer, $$\nu=\sum_{i=1}^9 j_i. \label{nudef}$$ These guarantee that $\cos S_1$ and $\sin S_2$ are invariant under time-reversal. Since the same applies to the amplitudes $A_1$ and $A_2$, one can choose either a geometry or its time-reversed image, for each root, when evaluating (\[theresult\]). This completes the definition and geometrical interpretation of all the notation used in (\[theresult\]). Symmetries of the $9j$-symbol {#symmetries} ============================= The formula (\[theresult\]) transforms correctly under the symmetries of the $9j$-symbol (Varshalovich 1981, Sec. 10.4), which state that the $9j$-symbol suffers a phase change of $(-1)^\nu$ under odd permutations of rows or columns or under transposition. Consider, for example, the swapping of the first two columns, and let $P$ be the permutation of indices, so that $P1=2$, $P2=1$, $P3=3$, etc. This maps an old set of nine $j$’s into a new set, and old quartic $Q(u)=0$ into a new one, etc. We find that the $u$ root of the old quartic becomes the $v$ root of the new one, which amounts to saying that the root 1 surface of the old geometry is mapped into the root 2 surface of the new one, and vice versa. Also, the orientations of the three unprimed triangles reverse, but not those of the primed ones, causing all nine dihedral angles to be incremented or decremented by $\pi$ (depending on the range). If we let $\theta_i$ be the original angles and ${\tilde\theta}_i$ the new ones, then when $\theta_i$ is on root surface 1 we find ${\tilde\theta}_{Pi} = \theta_i +\pi$, which means that the new angle is in the right range since it is on root surface 2. Similarly, when $\theta_i$ is on root surface 2 then ${\tilde\theta}_{Pi} = \theta_i-\pi$, which is in the right range since ${\tilde\theta}_{Pi}$ is on root surface 1. As a result, when the original geometry is on root surface 1, we have $$\sum_{i=1}^9 J_i{\tilde\theta}_i = \sum_{i=1}^9 J_i\theta_i +\nu\pi + \frac{9\pi}{2},$$ so that $\sin {\tilde S}_2 = (-1)^\nu \cos S_1$, while if the original geometry is on root surface 2, we have $$\sum_{i=1}^9 J_i{\tilde\theta}_i = \sum_{i=1}^9 J_i\theta_i -\nu\pi -\frac{9\pi}{2},$$ so that $\cos{\tilde S}_1 = (-1)^\nu \sin S_2$. The sine and cosine terms in (\[theresult\]) swap under column swap, and the result acquires an overall phase of $(-1)^\nu$, as required. The specified ranges on the dihedral angles on the two root surfaces are necessary for this to work out. Comments and Conclusions {#conclusions} ======================== It is easy to derive the expression (\[Sdef\]) by the method of Roberts (1999), which involves rotating faces by an angle of $\pi$ about their normals, and edges by an angle of $\pi$ about a normal to them. The phase (\[Sdef\]) (times 2) is then an action integral along one Lagrangian manifold and back along another (the analogs of the $A$- and $B$-manifolds of Aquilanti (2007)). Similar expressions apply to any spin network of any complexity. But the contours chosen for the integration are not unique, in that one can add any multiples of quantized loops on the two manifolds. These modify both the actions and the Maslov indices, and amount to changing the choice of branch for the angles $\theta_i$, that is, adding an integer multiple of $2\pi$ to these angles. This does not leave the trigonometric functions in (\[theresult\]) invariant because the angles are multiplied by the $J_i$, which may be half-integers. The result is that the phase of the approximation to the $9j$-symbol depends on the contours. A more serious worry is that the contours, that is, the branches for the $\theta_i$, may change as we move around in the parameter space of the $9j$-symbol. This would amount to crossing a branch cut for the angles $\theta_i$ (and there are different branch cuts for different angles). In addition, as we move around in parameter space we can make any two adjacent faces rotate relative to one another around their common edge as many times as we want. Although the phases in question are “only” powers of $-1$, straightening out this issue was by far the hardest part of this work. In the end we realized that the ranges $[-\pi,+\pi)$ on root surface 1 and $[0,2\pi)$ on root surface 2 guarantee that there are no branch cuts and hence no discontinuities. The ranges specified for the angles $\theta_i$ give us in effect a global, smooth definition of contours for carrying out action integrals. We present several numerical comparisons of (\[theresult\]) with the exact $9j$-symbol. In Fig. \[comparison\] the approximation (\[theresult\]) (smooth curve) may be compared to the exact $9j$-symbol (sticks) as a function of $j_3$ for fixed values of the other $j$’s. The range chosen lies inside the classically allowed region, far from a caustic. Fig. \[comparefold\] shows the comparison in a range that crosses a fold catastrophe, and Fig. \[compareumbilic\] shows the comparison in an interval that passes near a hyperbolic umbilic catastrophe (the upper point $I$ in Fig. \[configspace\]). The approximation (\[theresult\]) is too large near the point $I$. Varshalovich (1981) present an asymptotic approximation for the $9j$-symbol without citation (their Eq. (10.7.1)). We believe their formula must be an asymptotic expression for the $9j$-symbol in a different sense than we have defined it, or else it is incorrect. The two terms in (\[theresult\]) have different trigonometric functions (sine and cosine) because there is a relative Maslov index of 2 between the two root surfaces. The relative Maslov index between a geometry and its time-reversed image is 0, a somewhat surprising result because in mechanical systems and in the $6j$-symbol the Maslov index between a branch or geometry and its time-reversed image is 1. When an interior caustic occurs on a root surface, the two geometries that sit above it form a double cover, in the manner of the Riemann sheet for the square root function. The internal caustic point $I$ is a branch point for the cover. Geometries transform continuously into their time-reversed images as we go around the point $I$, without crossing a caustic. Several studies of the asymptotics of spin networks have started with an integral representation of the network, to which the stationary phase approximation is applied. Roberts (1999) represented the $6j$-symbol as a scalar product in a certain Hilbert space, which was put into the coherent state representation, whereupon the integral was evaluated by the stationary phase approximation. Coherent states have played a prominent role in many recent semiclassical studies. Our approach has been to work as much as possible in a representation-independent manner. For example, the stationary phase points are seen as intersections of Lagrangian manifolds. Some of the basics of this approach were presented in Aquilanti (2007). We have not specifically used the coherent state or any other representation. Some aspects of this calculation carry through in an obvious way to higher spin networks, while for others nontrivial generalizations seem to be required. But we believe that an understanding of the $9j$ results are necessary for a full understanding of the asymptotics of higher spin networks. We will report in more detail on the derivation of (\[theresult\]) in a later publication. The authors would like to thank Enzo Aquilanti, Mauro Carfora, Annalisa Marzuoli and Carlo Rovelli for encouragement, many useful pieces of information, many stimulating conversations, and much warm hospitality during the progress of this work. We would also like to thank Cynthia Vinzant for discussion of positive semidefinite completion, which helped greatly in proving that there are generically two admissible roots in the classically allowed region. This work was supported by a grant from the France-Berkeley Fund. References {#references .unnumbered} ========== Alesci E, Bianchi E, Magliaro E and Perini C 2008 preprint gr-qc 0809.3718 Anderson R W, Aquilanti V and Ferreira C da S 2008 [*J. Chem. Phys.*]{} [**129**]{} 161101 Anderson R W, Aquilanti V and Marzuoli A 2009 [*J. Phys. Chem. A*]{} [**113**]{} 15106 Aquilanti V, Haggard H M, Littlejohn R G and Yu L 2007 [*J Phys A*]{} [**40**]{} 5637 Baez J C Christensen J D and Egan G 2002 [*Class. Quantum Grav.*]{} [**19**]{} 6489 Barrett J W, Dowdall R J, Fairbairn W J, Gomes H and Hellmann F 2009 preprint gr-qc 0902.1170 Barrett J W and Steele C M 2003 [*Class. Quantum Grav.*]{} [**20**]{} 1341 Barrett J W and Williams R M 1999 [*Adv. Theor. Math. Phys.*]{} [**3**]{} 209 Berry M V 1976 [*Adv. Phys.*]{} [**25**]{} 1 Biedenharn L C and Louck J D 1981a [*Angular Momentum in Quantum Physics*]{} (Reading, Massachusetts: Addison-Wesley) 1981b [*The Racah-Wigner Algebra in Quantum Theory*]{} (Reading, Massachusetts: Addison-Wesley) Charles L 2008 preprint math 0806.1585 Conrady F and Freidel L 2008 [*Phys. Rev. D*]{} [**78**]{} 104023 de Gosson M 1997 [*Maslov Classes, Metaplectic Representation and Lagrangian Quantization*]{} (Berlin: Akademie Verlag) Dupuis M and Livine E R 2009 [*Phys. Rev. D*]{} [**80**]{} 024035 Edmonds A R 1960 [*Angular Momentum in Quantum Mechanics*]{} (Princeton: Princeton University Press) Freidel L and Louapre D 2003 [*Class. Quantum Grav.*]{} [**20**]{} 1267 Gilmore R 1981 [*Catastrophe Theory for Scientists and Engineers*]{} (New York: Dover) Gurau R 2008 [*Annales Henri Poincaré*]{} [**9**]{} 1413 Kapovich M and Millson J J 1996 [*J. Differential Geometry*]{} [**44**]{} 479 Littlejohn R G and Yu L 2009 [*J. Phys. Chem. A*]{} [**113**]{} 14904 Marsden J E and Ratiu T 1999 [*Introduction to Mechanics and Symmetry*]{} (New York: Springer-Verlag) Marzuoli A and Rasetti M 2005 [*Ann. Phys.*]{} [**318**]{} 345 Maslov V P and Fedoriuk M V 1981 [*Semi-Classical Approximation in Quantum Mechanics*]{} (Dordrecht, Holland: D. Reidel) Mishchenko A S, Shatalov V E and Sternin B Yu 1990 [*Lagrangian manifolds and the Maslov operator*]{} (Berlin: Springer Verlag) Nomura M 1989 [*J. Math. Phys.*]{} [**30**]{} 2397 Ponzano G and Regge T 1968 in [*Spectroscopy and Group Theoretical Methods in Physics*]{} ed F Bloch  (Amsterdam: North-Holland) p 1 Ragni M, Bitencourt A C P, Ferreira C da S, Aquilanti V, Anderson R W and Littlejohn R G 2010 [*Int. J. Quantum Chem.*]{} [**110**]{} 731 Regge T 1961 [*Nuovo Cimento*]{} [**19**]{} 558 Regge T and Williams R M 2000 [*J. Math. Phys.*]{} [**41**]{} 3964 Roberts J 1999 [*Geometry and Topology*]{} [**3**]{} 21 Schulten K and Gordon R G 1975a [*J. Math. Phys.*]{} [**16**]{} 1961 1975b [*J. Math. Phys.*]{} [**16**]{} 1971 Taylor Y U and Woodward C T 2004 preprint math 0406.228 2005 [*Sel. Math. New ser.*]{} [**11**]{} 539 Trinkhaus H and Drepper F 1977 [*J. Phys. A*]{} [**10**]{} L11 Varshalovich D A, Moskalev A N and Khersonskii V K 1981 [*Quantum Theory of Angular Momentum*]{} (Singapore: World Scientific) Wigner E P 1959 [*Group Theory*]{} (Academic Press, New York) Williams R M and Tuckey P A 1992 [*Class. Quantum Grav.*]{} [**9**]{} 1409 Yutsis A P, Levinson I B and Vanagas V V 1962 [*The Theory of Angular Momentum*]{} (Jerusalem: Israel Program for Scientific Translations)

--- abstract: 'A continuous quadratic form (“quadratic form”, in short) on a Banach space $X$ is: (a) [*delta-semidefinite*]{} (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $T\colon X\to X^*$ factors through a Hilbert space; (b) [*delta-convex*]{} (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional $L_p(\mu)$ space ($1\le p \le \infty$) is: (a) delta-semidefinite if[f]{} $p \ge 2$; (b) delta-convex if[f]{} $p>1$. Some other related results concerning delta-convexity are proved and some open problems are stated.' address: - | Department of Mathematics\ University of Missouri-Columbia\ Columbia, MO 65211, U.S.A. - | Department of Mechanics and Mathematics\ Moscow State University\ Moscow, 119992, Russia - | Dipartimento di Matematica\ Università degli Studi di Milano\ Via C. Saldini, 50\ 20133 Milano, Italy author: - Nigel Kalton - 'Sergei V. Konyagin' - Libor Veselý date: 'May 15, 2006' title: | Delta-semidefinite and delta-convex\ quadratic forms in Banach spaces --- [^1] Introduction {#introduction .unnumbered} ============ Let $X$ be a real Banach space. Recall that a function $q\colon X\to {\mathbb{R}}$ is a [*continuous quadratic form*]{} (more precise would be “continuous purely quadratic form”) if there exists a continuous bilinear form $b\colon X\times X\to{\mathbb{R}}$ such that $q(x)=b(x,x)$ for each $x\in X$. In the present paper, we are interested mainly in the following two isomorphic properties of $X$. 1. [*Each continuous quadratic form on $X$ is [*delta-semidefinite*]{}, i.e., it can be represented as a difference of two nonnegative continuous quadratic forms.*]{} 2. [*Each continuous quadratic form on $X$ is [*delta-convex*]{}, i.e., it can be represented as a difference of two continuous convex functions.*]{} Since nonnegative quadratic forms are convex, (D) always implies (dc). The reverse implication is not true, as we shall see in Section \[S:three\]. In Section \[S:one\], we characterize delta-semidefinite continuous quadratic forms on $X$ as precisely those whose corresponding symmetric linear operator $T\colon X\to X^*$ is factorizable through a Hilbert space. This leads, via known results on factorizability, to sufficient conditions for a Banach space $X$ to satisfy (D). The characterization also implies that the property (D) passes to quotients, and the spaces $\ell_p$, $1\le p<2$, do not satisfy (D). In Section \[S:two\], we use $X$-valued Walsh-Paley martingales to prove that a continuous quadratic form on $X$ is delta-convex if and only if the corresponding symmetric linear operator is a UMD-operator. It follows that $\ell_1$ not only fails (D) but it also fails (dc). In Section \[S:three\], we discuss relationships between the properties (D), (dc) and the following property. It is easy to see that also (Cdc) implies (dc). We show that (dc) and (Cdc) pass to quotients. For each of the properties (D), (dc), (Cdc), we characterize those $p$’s in $[1,\infty]$ for which an infinite-dimensional $L_p(\mu)$ space satisfies the property (Theorem \[T:L\_p\]). It follows that (dc) implies neither (D) nor (Cdc). (The latter should be compared with a result from [@Du-Ve-Za] which says that all Banach space-valued quadratic mappings on $X$ are delta-convex if and only if all Banach space-valued $C^{1,1}$ mappings on $X$ are delta-convex.) We also solve a problem from [@Ve-Za] by proving existence of a function $f$ whose compositions with all “delta-convex curves” (in the sense of [@Ve-Za]) are delta-convex while $f$ is not locally delta-convex. Some of these counterexamples use a result by M. Zelený [@Zel]. Finally, we show that the property (dc) is not stable with respect to direct sums, and we state some open problems. Delta-semidefinite quadratic forms {#S:one} ================================== In what follows, the term “operator” means “bounded linear operator”. Recall that an operator $T\colon X\to X^*$ is called [*symmetric*]{} if $\langle Tx,y\rangle = \langle Ty,x\rangle$ for all $x,y\in X$ (equivalently: $T^*=T$ on $X$). It is easy to see that the formula $$\label{E:qT} q(x)=\langle Tx,x\rangle$$ defines a one-to-one correspondence between the continuous quadratic forms $q$ on $X$ and the symmetric operators $T\colon X\to X^*$.\ (Indeed, if $q$ is generated by a continuous bilinear form $b$, it is generated also by the symmetric bilinear form $\frac{b(x,y)+b(y,x)}{2}$. Moreover, there is a unique symmetric bilinear form $b$ that generates $q$; this follows from the formula $$\label{E:polar} 2b(x,y)=b(x+y,x+y)-b(x,x)-b(y,y)$$ valid for symmetric $b$. The rest follows from the well-known one-to-one correspondence, via the formula $b(x,y)=\langle Tx,y\rangle$, between the continuous bilinear forms $b$ on $X\times X$ and the operators $T\colon X\to X^*$.)\ If (\[E:qT\]) holds for each $x\in X$, we say that [*$T$ generates $q$*]{}. The formula (\[E:polar\]) also implies the following \[F:p’\] Each continuous quadratic form $q$ on $X$ is everywhere Fréchet differentiable. Moreover, its Fréchet derivative at $x$ is given by $q'(x)=2Tx$ where $T$ is the symmetric operator that generates $q$. The following theorem characterizes delta-semidefinite continuous quadratic forms. (Recall that a continuous quadratic form is called [*delta-semidefinite*]{} if it is the difference of two nonnegative continuous quadratic forms.) An operator $T\colon X\to Y$ is said to be [*factorizable through $Z$*]{} if there exist operators $A\colon X\to Z$ and $B\colon Z\to Y$ such that $T=BA$. \[T:kalton\] Let $q$ be a continuous quadratic form on a Banach space $X$, and $T\colon X\to X^*$ be the symmetric operator that generates $q$. Then the following assertions are equivalent: 1. $q$ is delta-semidefinite; 2. there exists a continuous quadratic form $p$ on $X$, such that $|q|\le p$; 3. $T$ is factorizable through a Hilbert space.  \ $(iii)\Rightarrow(i)$. If $T=BA$ where $A\colon X\to H$ and $B\colon H\to X^*$ are operators, and $H$ is a Hilbert space, then we have $$q(x)=\langle BAx,x\rangle=\langle Ax, B^*x\rangle_H= {\textstyle\frac{1}{4}}\|Ax+B^*x\|^2_H- {\textstyle\frac{1}{4}}\|Ax-B^*x\|^2_H$$ which shows that $q$ is difference of two nonnegative quadratic forms. $(i)\Rightarrow(ii)$. If $q=p_1-p_2$ where $p_i$ ($i=1,2$) are nonnegative continuous quadratic forms, then $|q|\le p_1+p_2=:p$. $(ii)\Rightarrow(iii)$. Let (ii) hold, and let $S\colon X\to X^*$ be the symmetric operator such that $p(x)=\langle Sx,x\rangle$. The function $$[\cdot,\cdot]\colon X/\mathrm{Ker}(S) \times X/\mathrm{Ker}(S) \to{\mathbb{R}}, \quad [\xi,\eta]:=\langle Sx,y\rangle\ \text{where $x\in\xi$, $y\in\eta$,}$$ is well-defined, bilinear, symmetric, and $[\xi,\xi]\ge0$ for each $\xi\in X/\mathrm{Ker}(S)$. Moreover, if $p(x)=0$ for some $x\in X$, then $x$ is a minimizer for $p$, and hence $0=p'(x)=2Sx$ by Fact \[F:p’\]. In other words, $[\xi,\xi]=0$ implies $\xi=0$. Consequently, $[\cdot,\cdot]$ is an inner product on $X/\mathrm{Ker}(S)$. Let $H$ be the completion of the inner product space $\bigl(X/\mathrm{Ker}(S),[\cdot,\cdot]\bigr)$. Consider the operator $J=iQ\colon X\to H$ where $Q\colon X\to X/\mathrm{Ker}(S)$ is the quotient map and $i\colon X/\mathrm{Ker}(S)\hookrightarrow H$ is the inclusion map. ($J$ is continuous since $\|Qx\|^2_H=\langle Sx,x\rangle\le\|S\|\cdot\|x\|^2$ for all $x\in X$.) If $x\in\mathrm{Ker}(S)$, then $p(x)=q(x)=0$. Since $p+q$ is a nonnegative quadratic form generated by the symmetric operator $T+S$, the same argument as above shows that $Tx+Sx=0$. This proves that $\mathrm{Ker}(S)\subset\mathrm{Ker}(T)$. Consequently, the operator $$T_0\colon X/\mathrm{Ker}(S)\to X^*,\quad T_0\xi:=Tx\ \text{where $x\in\xi$},$$ is well-defined. We claim that $T_0$ is continuous also in the norm generated by the inner product $[\cdot,\cdot]$. To prove this, consider $\xi\in X/\mathrm{Ker}(S)$ and $y\in X$ such that $\|\xi\|_H \le1$ and $\|y\| \le1$. Fix $x\in\xi$, and denote $\eta=Qy$. Then $\|\eta\|^2_H=\langle Sy,y\rangle\le\|S\|$, and $ \bigl|\langle T_0\xi,y\rangle\bigr| \le \bigl|\langle Tx,y\rangle\bigr| = \frac{1}{2}\bigl|q(x+y)-q(x)-q(y)\bigr|\le \frac{1}{2}\bigl[p(x+y)+p(x)+p(y)\bigr]= \frac{1}{2}\bigl[\|\xi+\eta\|^2_H +\|\xi\|^2_H+ \|\eta\|^2_H\bigr]\le \frac{1}{2}\bigl[(1+\|S\|^{1/2})^2+1+\|S\|\bigr]. $ Thus $T_0$ has a unique extension to an operator from $H$ into $X^*$; let us denote it by $T_0$ again. Then $T=T_0J$ is the desired factorization through $H$. \[C:quotient\] 1. A Banach space $X$ has the property (D) (see Introduction) if and only if each symmetric operator $T\colon X\to X^*$ is factorizable through a Hilbert space. 2. The property (D) passes to quotients, and hence also to complemented subspaces. \(a) follows immediately from Theorem \[T:kalton\]. Let us show (b). Let $X$ satisfy (D), and let $L$ be a closed subspace of $X$. Let $T\colon X/L\to (X/L)^*=L^\perp$ be a symmetric operator. Consider the operator $S=iTQ\colon X\to X^*$ where $Q\colon X\to X/L$ is the quotient map, and $i\colon L^\perp\to X^*$ is the inclusion isometry. Since $Q^*=i$, we have $\langle Sx,y\rangle=\langle T(Qx),(Qy)\rangle$ ($x,y\in X$), which shows that $S$ is symmetric. By (a), $S$ is factorizable through a Hilbert space. Now, Proposition 7.3 in [@DJT] implies that $T$ factors through a Hilbert space, too. Operators that are factorizable through a Hilbert space were intensively studied (the main reference is [@Pisier], see also [@DJT]), and there exist many sufficient conditions for factorizability of all operators between two given spaces. Thus, by Theorem \[T:kalton\], we obtain various sufficient conditions for validity of the property (D) (defined in Introduction); we collect them in the following theorem. For the classical notion of modulus of smoothness, see e.g. [@Lin-Tz]. For the notion of type and cotype, see e.g. [@Lin-Tz] or [@DJT]. We shall need the following notion of second order differentiability, studied in [@Bor-Noll]. 1. Let $f$ be a continuous convex function on a Banach space $X$. We say that $f$ is [*second order differentiable*]{}at a point $x_0\in X$ if there exist $x_0^*\in X^*$ and a continuous quadratic form $q$ on $X$ such that, for each $v\in X$, $$f(x_0+tv)=f(x_0)+x^*_0(v)t+q(v)t^2+o(t^2)\qquad \hbox{as\ } t\to0.$$ 2. A [*second order differentiable norm*]{} is a norm which is second order differentiable at each nonzero point. It follows from results in [@Bor-Noll] that a norm $\|\cdot\|$ on $X$ is second order differentiable if[f]{} it is Fréchet (equivalently: Gâteaux) smooth and its derivative $\|\cdot\|'\colon X\setminus\{0\}\to X^*$ is ${\rm weak}^*$-Gâteaux differentiable. \[T:D\] Let $X$ be a Banach space. Each continuous quadratic form on $X$ is delta-semidefinite (and hence delta-convex), provided [*at least one*]{} of the following conditions is satisfied. 1. $X$ has type 2. 2. $X^*$ has cotype 2, and $X$ has the approximation property. 3. $X^*$ has cotype 2, and $X$ does not contain $\ell_1(n)$’s uniformly. 4. $X^*$ has cotype 2, and $X$ is a Banach lattice. 5. $X=C(K)$ for some compact space $K$. 6. $X=L_p(\mu)$ for $2\le p\le\infty$ and some positive measure $\mu$. 7. $X=c_0(I)$ for some set $I$. 8. $X$ admits a uniformly smooth renorming with modulus of smoothness of power type 2 (i.e., $\rho_X(\tau)\le a\tau^2$ for some $a>0$). 9. $X$ has the Radon–Nikodým property and admits an equivalent second order differentiable norm. \(a) follows from Corollary 3.6 and Proposition 3.2 in [@Pisier].\ (b): see Theorem 4.1 in [@Pisier].\ (c) follows from (a) by [@Pi1 Corollary 2.5].\ (d) follows from Theorems 8.17 and 8.11 in [@Pisier].\ (e) follows e.g. from (d) since $C(K)^*$ has cotype 2 (see [@Pisier p.34]).\ (f): the case $p<\infty$ follows from (a) (see [@Lin-Tz p.73]); the case $p=\infty$ follows from (e) (see [@Lin-Tz Theorem 1.b.6]).\ (g) follows from (e) by Corollary \[C:quotient\](b), since $c_0(\Gamma)$ is a closed hyperplane in $c(\Gamma)=C(K)$ where $K$ is the one-point compactification of the discrete set $\Gamma$.\ (h) follows from (a) (see Theorem 1.e.16 in [@Lin-Tz]).\ (i) follows from (h) by the following reasoning. If the norm of $X$ is second order differentiable, then this norm is Lipschitz-smooth at each point of $S_X$ by [@Bor-Noll]; this implies (by [@FWZ Lemma 2.4]) that the gradient of the norm is pointwise Lipschitz at each point of $S_X$. By [@DGZ Corollary III.2], if $X$ has also the RNP, then it satisfies (h). Results in [@SSTT Section$\,$5] imply that, for each $1<p<2$, there exists an operator $U\colon\ell_p\to\ell_{p^*}$ (where $\frac{1}{p}+\frac{1}{p^*}=1$) such that $U$ is not factorizable through a Hilbert space. Since $U$, constructed in [@SSTT], is also symmetric, we obtain one more corollary to Theorem \[T:kalton\]. \[C:badp\] The space $\ell_p$ fails the property (D) whenever $1\le p<2$.\ [(The case of $p=1$ follows from Corollary \[C:quotient\](b) and from the well-known fact that each separable Banach space is isometric to a quotient of $\ell_1$.) ]{} Delta-convex quadratic forms {#S:two} ============================ Let $X$ be a Banach space. Recall that a continuous function $\phi\colon X\to{\mathbb{R}}$ is [*delta-convex*]{} if it is the difference of two continuous convex functions on $X$. It is easy to see that $\phi$ is delta-convex if and only if there exists a (necessarily convex) continuous function $\psi$ on $X$ such that both $\pm\phi +\psi$ are convex. Every such function $\psi$ is called a [*control function*]{} for $\phi$. Denoting $$\Delta^2\phi(x,y):=\phi(x+y)+\phi(x-y)-2\phi(x)\,,\qquad x,y\in X,$$ it is easy to see that $\psi$ is a control function for $\phi$ if and only if $|\Delta^2\phi(x,y)| \le \Delta^2\psi(x,y)$ for all $x,y\in X$. Since every nonnegative quadratic form is convex, each delta-semidefinite quadratic form is delta-convex. As we shall see in Section 3, the converse is not true in general. In this section, we use $X$-valued Walsh-Paley martingales to study delta-convexity of quadratic forms. We recall all needed definitions and properties to make our exposition self-contained. Let $n\ge1$ be an integer, $\Gamma=\{-1,1\}$, $f\colon\Gamma^n\to X$. Then the [*expectation*]{} of $f$ is defined as ${\mathbb{E}}f=2^{-n}\sum_{\eta\in\Gamma^n} f(\eta)=\int_{\Gamma^n} f\,d{\mathbb{P}}$, where ${\mathbb{P}}={\mathbb{P}}_n$ is the uniformly distributed probability measure on $\Gamma^n$. For $0\le k\le n$, consider the $\sigma$-algebra $\Sigma_k=\{A\times \Gamma^{n-k}: A\subset\Gamma^k\}.$ Obviously, a function $f\colon\Gamma^n\to X$ is $\Sigma_k$-measurable if and only if $f$ depends only on the first $k$ coordinates (in particular, all $\Sigma_0$-measurable functions are constant). For this reason, we sometimes view $\Sigma_k$-measurable functions on $\Gamma^n$ as functions on $\Gamma^k$. For $f\colon\Gamma^n\to X$ and $0\le k\le n$, the [*conditional expectation of $F$ w.r.t. $\Sigma_k$*]{} is the $\Sigma_k$-measurable function ${\mathbb{E}}(f|\Sigma_k)\colon \Gamma^n\to X$ which has the same integral (w.r.t. ${\mathbb{P}}$) as $f$ over each element of $\Sigma_k$. It is easy to see that it is given by $$\textstyle {\mathbb{E}}(f| \Sigma_k)(\omega)=\int_{\Gamma^{n-k}}f(\omega,\cdot)\,d{\mathbb{P}}_{n-k} \,,\qquad \omega\in\Gamma^k.$$ Note that ${\mathbb{E}}(f|\Sigma_0)\equiv{\mathbb{E}}f$, ${\mathbb{E}}(f|\Sigma_n)=f$, and ${\mathbb{E}}({\mathbb{E}}(f|\Sigma_k))={\mathbb{E}}f$. In this paper, we consider only Walsh-Paley martingales of finite length. Let $X$ be a Banach space. An [*$X$-valued Walsh-Paley martingale*]{} is any finite sequence $(f_0,\ldots, f_n)$ of $X$-valued functions on $\Gamma^n$ such that $f_k={\mathbb{E}}(f_n|\Sigma_k)$ for each $0\le k\le n\,;$ or equivalently, each $f_k$ is $\Sigma_k$-measurable, and $$\textstyle f_k(\omega)=\frac{1}{2}f_{k+1}(\omega,-1)+\frac{1}{2}f_{k+1}(\omega,1) \quad\text{whenever $0\le k<n$ and $\omega\in\Gamma^k$.}$$ Given a Walsh-Paley martingale $(f_0,\ldots,f_n)$, the corresponding [*martingale differences*]{} are the functions $df_k=f_k-f_{k-1}$   $(1\le k\le n).$ \[O:mart\] Let $(f_0,\ldots,f_n)$ be an $X$-valued Walsh-Paley martingale. The above definition easily implies the following properties. 1. ${\mathbb{E}}(df_k | \Sigma_j)=0$ whenever $0\le j< k\le n$. 2. $(f_0,\ldots,f_n,f_n,\ldots,f_n)$ is a Walsh-Paley martingale no matter how many times $f_n$ is repeated. 3. $(Tf_0,\ldots,Tf_n)$ is a $Y$-valued Walsh-Paley martingale whenever $T\colon X\to Y$ is linear. 4. If $0\le m< n$ and $\overline{\omega}\in\Gamma^m$, then the finite sequence $(g_0,\ldots,g_{n-m})$, where $g_k:=f_{m+k}(\overline{\omega},\cdot)\colon\Gamma^{n-m}\to X$, is a Walsh-Paley martingale. 5. $df_k(\omega)=\frac{1}{2}\omega_k \bigl[f_k(\omega_1,\ldots,\omega_{k-1},1)-f_k(\omega_1,\ldots,\omega_{k-1},-1)\bigr]$ whenever $1\le k\le n$ and $\omega\in\Gamma^k$. 6. $f_{k-1}(\omega)\pm df_k(\omega)=f_k(\omega_1,\ldots,\omega_{k-1},\pm\omega_k)$ whenever $1\le k\le n$ and $\omega\in\Gamma^k$. A finite sequence $(\epsilon_1,\ldots,\epsilon_n)$ of functions on $\Gamma^n$ is said to be [*predictable*]{} if $\epsilon_k$ is $\Sigma_{k-1}$-measurable for each $1\le k\le n$. \[L:mart\] Let $(f_0,\ldots,f_n)$ be an $X$-valued Walsh-Paley martingale. 1. If $\phi\colon X\to{\mathbb{R}}$ is a function, then ${\mathbb{E}}\sum_{k=1}^n \Delta^2\phi(f_{k-1},df_k)=2{\mathbb{E}}\phi(f_n)- 2\phi(f_0)$. 2. If $1\le k\le n$ and $w\colon\Gamma^n\to{\mathbb{R}}$ is $\Sigma_{k-1}$-measurable, then ${\mathbb{E}}(w\,df_k)=0$. 3. If $(g_0,\ldots,g_n)$ is an $X^*$-valued Walsh-Paley martingale and $(\epsilon_1,\ldots,\epsilon_n)$ is a predictable sequence of real-valued functions, then ${\mathbb{E}}\langle df_k,\epsilon_j dg_j\rangle=0$ whenever $k\ne j$ and $k,j\in\{1,\ldots,n\}$. 4. $({\mathbb{E}}\max\limits_{0\le k\le n}\|f_k\|^p)^{1/p}\le\frac{p}{p-1}({\mathbb{E}}\|f_n\|^p)^{1/p}$ for each real $p>1$. \(a) Recall that $f_0$ is constant. By Remark \[O:mart\](f), the left-hand side equals $$\begin{gathered} \textstyle \sum_{k=1}^n \int_{\Gamma^k} \bigl[ \phi(f_k(\omega))+\phi(f_k(\omega_1,\ldots,\omega_{k-1},-\omega_k))-2\phi(\omega) \bigr]\,d{\mathbb{P}}_k(\omega) \\ \textstyle =2\sum_{k=1}^n \bigl( {\mathbb{E}}\phi(f_k) - {\mathbb{E}}\phi(f_{k-1})\bigr)= 2{\mathbb{E}}\phi(f_n) - 2{\mathbb{E}}\phi(f_0).\end{gathered}$$ (b) follows easily from Remark \[O:mart\](a).\ (c) Let, e.g., $k<j$. Obviously, $\langle df_k,\epsilon_jdg_j\rangle$ is $\Sigma_{j}$-measurable. For each fixed $\omega\in\Gamma^{j-1}$, we have $${\mathbb{E}}(\langle df_k,\epsilon_jdg_j\rangle|\Sigma_{j-1})(\omega)= {\mathbb{E}}\langle df_k(\omega),\epsilon_j(\omega) dg_j(\omega,\cdot)\rangle =\langle df_k(\omega), \epsilon_j(\omega) {\mathbb{E}}dg_j(\omega,\cdot)\rangle=0$$ by Remark \[O:mart\](d),(a). Thus ${\mathbb{E}}\langle df_k,\epsilon_j dg_j\rangle= {\mathbb{E}}\bigl({\mathbb{E}}(\langle df_k,\epsilon_j dg_j\rangle|\Sigma_k)\bigr)=0$. The case $k>j$ is similar.\ (d) It is easy to verify that $h(\eta):=\tilde{h}(\eta) v(\eta)$ works, where $\tilde{h}\colon\Gamma^n\to[0,\infty)$ satisfies ${\mathbb{E}}\tilde{h}^2=1$ and $({\mathbb{E}}\|g\|^2)^{1/2}={\mathbb{E}}(\tilde{h}\cdot\|g\|)$, and $v(\eta)\in S_X$ is such that $\|g(\eta)\|\le 2\langle v(\eta),g(\eta)\rangle$.\ (e) For real-valued martingales, this is the well-known Doob’s $L_p$-inequality (see e.g. [@williams Theorem 14.11]). In the general case, consider the Walsh-Paley martingale $(g_0,\ldots,g_n)$ given by $g_k={\mathbb{E}}(\|f_n\|\,|\Sigma_k)$. Then $\|f_k\|=\| {\mathbb{E}}(f_n|\Sigma_k)\|\le g_k$ and $\|f_n\|=g_n$. Hence, using the scalar case, we get $({\mathbb{E}}\max\limits_{0\le k\le n}\|f_k\|^p)^{1/p}\le ({\mathbb{E}}\max\limits_{0\le k\le n}g_k^p)^{1/p}\le \frac{p}{p-1} ({\mathbb{E}}g_n^p)^{1/p}= \frac{p}{p-1}({\mathbb{E}}\|f_n\|^p)^{1/p}$. \[1\] Let $\phi$ be a continuous real function on a Banach space $X$. In order that $\varphi$ is delta-convex it is necessary and sufficient that there is a continuous function $\rho\colon X\to [0,\infty)$ such that if $(f_0,\ldots,f_n)$ is an $X$-valued Walsh-Paley martingale then $$\label{E:1} \mathbb E \sum_{k=1}^n |\Delta^2\varphi(f_{k-1},df_k)|\le \mathbb E\rho(f_n).$$ Moreover, in this case, $\rho$ can always be taken convex. Let $\phi$ be delta-convex with a control function $\psi$. By adding a suitable affine function, we can (and do) suppose that $\psi\ge0$. Now Lemma \[L:mart\](a) implies $$\begin{aligned} \textstyle \mathbb E \sum\limits_{k=1}^n |\Delta^2\varphi(f_k,df_k)|& \textstyle \le \mathbb E \sum\limits_{k=1}^n \Delta^2\psi(f_k,df_k)\\ &= 2\mathbb E\psi(f_n)-2\psi(f_0)\\ &\le 2\mathbb E\psi(f_n).\end{aligned}$$ Thus holds with $\rho=2\psi$ (which is convex). Conversely, if holds, we may define $$\label{control} \textstyle \psi(x)= \frac12\,\inf\Big\{ \mathbb E\rho(f_n)- \mathbb E \sum\limits_{k=1}^n |\Delta^2\phi(f_{k-1},df_k)|\Big\}$$ where the infimum is taken over all $X$-valued Walsh-Paley martingales with $f_0= x.$ Suppose $x,u\in X$, $y=x+u,z=x-u$ and $\epsilon>0.$ Using Remark \[O:mart\](b) and the definition of $\psi$, pick $X$-valued Walsh-Paley martingales $(f_0,\ldots,f_n)$ and $(g_0,\ldots,g_n)$ such that $f_0= y$, $g_0= z$ and $$\begin{aligned} \psi(y)&\textstyle >\frac12\,\mathbb E\rho(f_n)-\frac12\,\mathbb E \sum\limits_{k=1}^n |\Delta^2\phi(f_{k-1},df_k)|-\epsilon\\ \psi(z)&\textstyle >\frac12\,\mathbb E\rho(g_n)-\frac12\,\mathbb E \sum\limits_{k=1}^n |\Delta^2\phi(g_{k-1},dg_k)|-\epsilon.\end{aligned}$$ Form a new Walsh-Paley martingale $(h_0,\ldots,h_{n+1})$ by setting $h_0=x$ and, for $1\le k\le n+1$, $$h_k(\eta_1,\ldots,\eta_{n+1})=\begin{cases} f_{k-1}(\eta_2,\ldots,\eta_{n+1}) \qquad \text{if } \eta_1=1\\ g_{k-1}(\eta_2,\ldots,\eta_{n+1})\qquad \text{if } \eta_1=-1.\end{cases}$$ Note that, for example, $$\textstyle {\mathbb{E}}\rho(h_{n+1})= \int\limits_{\{\eta_1=1\}} \rho(h_{n+1})\,d{\mathbb{P}}_{n+1}+\int\limits_{\{\eta_1=-1\}} \rho(h_{n+1})\,d{\mathbb{P}}_{n+1} =\frac12{\mathbb{E}}\rho(f_n) +\frac12{\mathbb{E}}\rho(g_n).$$ Thus $$\begin{aligned} 2\psi(x) &\textstyle \le {\mathbb{E}}\rho(h_{n+1})- {\mathbb{E}}\sum\limits_{k=1}^{n+1} |\Delta^2\phi(h_{k-1},dh_{k})|\\ &\textstyle =\textstyle{\frac12}\,{\mathbb{E}}\rho (f_n) + {\frac12}\,{\mathbb{E}}\rho (g_n) \\ &\qquad\quad\textstyle - |\Delta^2\phi(x,u)|- {\frac12}\,{\mathbb{E}}\sum\limits_{k=1}^n |\Delta^2\phi(f_{k-1},df_k)|- {\frac12}\,{\mathbb{E}}\sum\limits_{k=1}^n |\Delta^2\phi(g_{k-1},dg_k)| \\ &\le \psi(x) + \psi(y) + 2\epsilon - |\Delta^2\phi(x,u)|.\end{aligned}$$ Since $\epsilon>0$ was arbitrary, it follows that $|\Delta^2\phi(x,u)|\le \Delta^2\psi(x,u)$ whenever $x,u\in X$. Thus $\psi$ is a midconvex (or Jensen convex) function which is locally bounded since $0\le\psi\le\rho/2$. Consequently (see [@Ro-Va p.215]), $\psi$ is a continuous convex function. Thus $\psi$ is a control function for $\phi$. \[obs\] For each $g\colon\Gamma^n\to[0,\infty)$ and $p>0$, we have $$\sum_{j=1}^\infty 2^{jp}\,{\mathbb{P}}(g>2^{j}) \le \frac{2^p}{2^p-1}\,{\mathbb{E}}g^p.$$ Indeed, $ \int g^p\,d{\mathbb{P}}\ge \sum\limits_{j=1}^\infty \int\limits_{\{2^{j-1}<g\le 2^j\}}g^p\,d{\mathbb{P}}\ge \sum\limits_{j=1}^\infty 2^{(j-1)p}\,\bigl[{\mathbb{P}}(g>2^{j-1})-{\mathbb{P}}(g>2^j)\bigr] = \sum\limits_{j=0}^\infty 2^{jp}\, {\mathbb{P}}(g>2^j) - \sum\limits_{j=1}^\infty 2^{(j-1)p} \,{\mathbb{P}}(g>2^j) \ge (1-2^{-p}) \sum\limits_{j=1}^\infty 2^{jp}\, {\mathbb{P}}(g>2^j). $ Let $p>0$. Recall that a function $\phi\colon X\to {\mathbb{R}}$ is called [*positively $p$-homogeneous*]{} if $\phi(tx)=t^p\phi(x)$ whenever $t\ge0$, $x\in X$. \[2\] Suppose $p>1$. A continuous positively $p$-homogeneous function $\phi\colon X\to{\mathbb{R}}$ is delta-convex if and only if there is a constant $C$ such that for all $X$-valued Walsh-Paley martingales $(f_0,\ldots,f_n)$ we have $$\label{E:2} \mathbb E\sum_{k=1}^n|\Delta^2\varphi(f_{k-1},df_k)|\le C\mathbb E\|f_n\|^p.$$ Assume $\phi$ is delta-convex. Let $\rho$ be the corresponding continuous function from Lemma \[1\]. Choose $r>0$ so that $C_0:=\sup\{\rho(x):\|x\|\le r\}<\infty.$ Then $$\textstyle {\mathbb{E}}\sum\limits_{k=1}^n|\Delta^2\varphi(f_{k-1},df_k)|\le {\mathbb{E}}\rho(f_n)\le C_0 \qquad\text{whenever $\|f_n\|_\infty\le r,$}$$ where $\|g\|_\infty=\max_{\eta\in \Gamma^n}|g(\eta)|$ as usual. Hence, for an arbitrary Walsh-Paley martingale $(f_0,\ldots,f_n)$, $p$-homogeneity implies that $$\label{ellinfty} \textstyle {\mathbb{E}}\sum\limits_{k=1}^n|\Delta^2\varphi(f_{k-1},df_k)| \le C_1\|f_n\|_\infty^p$$ where $C_1=C_0/r^p$. Now, fix any $X$-valued Walsh-Paley martingale $(f_0,\ldots,f_n)$ with ${\mathbb{E}}\|f_n\|^p=1$. (By $p$-homogeneity, it suffices to prove for such martingales.) Let $\eta\in\Gamma^n$. We define $m_0(\eta)=0$ and, for any integer $r\ge 1$, $$M_r(\eta)=\bigl\{0\le k<n: \max\{\|f_k(\eta)\pm df_{k+1}(\eta)\|\}>2^r\bigr\}$$ and $$m_r(\eta)=\begin{cases} \min M_r(\eta) &\text{if $M_r(\eta)\ne\emptyset$,}\\ n &\text{if $M_r(\eta)=\emptyset$.} \end{cases}$$ For each $m\in\{0,\ldots,n\}$, the set $\{m_r=m\}$ belongs to $\Sigma_m$ by Remark \[O:mart\](f). (Thus the functions $m_r$ are so-called “stopping times”.) Hence it can be written in the form $$\{m_r=m\}=A_{r,m}\times\Gamma^{n-m}, \qquad\text{where $A_{r,m}\subset\Gamma^m$.}$$ We have $$\begin{aligned} {\mathbb{E}}\sum_{m_{r-1}<k\le m_r} & |\Delta^2\phi(f_{k-1},df_k)| \\ &= \sum_{m=0}^{n-1}\, \int\limits_{\{m_{r-1}=m\}} \sum_{m<k\le m_r(\eta)} |\Delta^2\phi(f_{k-1}(\eta),df_k(\eta))|\,d{\mathbb{P}}_n(\eta) \\ &= \sum_{m=0}^{n-1}\, \int\limits_{A_{r-1,m}}\left( \,\,\int\limits_{\Gamma^{n-m}} |\Delta^2\phi(f_{k-1}(\omega,\xi),df_k(\omega,\xi))|\,d{\mathbb{P}}_{n-m}(\xi) \right)\,d{\mathbb{P}}_m(\omega)\,.\end{aligned}$$ The expression in parentheses can be seen as $$\label{exp} \textstyle {\mathbb{E}}\sum\limits_{m<k\le n} |\Delta^2\phi(g_{k-1}(\omega,\cdot),dg_k(\omega,\cdot))|\,,$$ where $$g_k(\omega,\xi)=\begin{cases} f_k(\omega,\xi) &\text{if $m<k\le m_r(\omega,\xi)$,} \\ f_{m_r(\omega,\xi)}(\omega,\xi) &\text{if $m_r(\omega,\xi)<k\le n$.} \end{cases}$$ Since $(g_k(\omega,\cdot))_{k=m}^n$ is a Walsh-Paley martingale by Remark \[O:mart\](d), and the definition of $m_r$ implies $\|f_{m_r(\eta)}(\eta)\|=\|f_{m_r(\eta)-1}(\eta)+df_{m_r(\eta)}(\eta)\|\le 2^r$, we can majorize the expression (using ) by $$C_1\|g_n(\omega,\cdot)\|_\infty^p= C_1\|f_{m_r(\omega,\cdot)}(\omega,\cdot)\|_\infty^p\le C_1 2^{rp}.$$ Thus $$\begin{aligned} \textstyle {\mathbb{E}}\sum\limits_{k=1}^n|\Delta^2\varphi(f_{k-1},df_k)| &\textstyle = \sum\limits_{r=1}^\infty \,{\mathbb{E}}\sum\limits_{m_{r-1}<k\le m_r} |\Delta^2\phi(f_{k-1},df_k)| \\ &\textstyle \le C_1\sum\limits_{r=1}^\infty 2^{rp} \sum\limits_{m=0}^{n-1} {\mathbb{P}}_m(A_{r-1,m}) = C_1 \sum\limits_{r=1}^\infty 2^{rp} \,{\mathbb{P}}(m_{r-1}<n).\end{aligned}$$ Now, for $r>1$, Remark \[O:mart\](f) implies that $${\mathbb{P}}(m_{r-1}<n) \le {\mathbb{P}}\bigl(\,\max\limits_{1\le k<n}\max\{\|f_k\pm df_{k+1}\|\}>2^{r-1}\bigr) \le 2 {\mathbb{P}}\bigl(\,\max\limits_{0\le k\le n}\|f_k\|>2^{r-1}\bigr).$$ This gives (via Observation \[obs\] and Lemma \[L:mart\](d)) $$\begin{aligned} \textstyle {\mathbb{E}}\sum\limits_{k=1}^n|\Delta^2\varphi(f_{k-1},df_k)| &\textstyle \le C_1 2^p + C_1 2\sum\limits_{r=2}^\infty 2^{rp}\, {\mathbb{P}}(\max\limits_{0\le k\le n}\|f_k\|>2^{r-1}) \\ &\textstyle = C_1 2^p + C_1 2^{p+1}\sum\limits_{j=1}^\infty 2^{jp}\, {\mathbb{P}}(\max\limits_{0\le k\le n}\|f_k\|>2^j) \\ &\textstyle\le C_1 2^p + C_1\frac{2^{2p+1}}{2^p-1}\, {\mathbb{E}}\max\limits_{0\le k\le n}\|f_k\|^p \\ &\le C_2 \bigl(1+{\mathbb{E}}\|f_n\|^p\bigr)=2C_2\,,\end{aligned}$$ where $C_2$ is a suitable constant depending only on $p$. Thus holds. The converse follows trivially from Lemma \[1\] by putting $\rho(x)=C\|x\|^p$. \[p-homog\] Suppose $p\ge1$. Then every positively $p$-homogeneous delta-convex function $\phi\colon X\to {\mathbb{R}}$ has a control function which is positively $p$-homogeneous. The case $p=1$ was proved in [@Ve-Za Lemma 1.21]. Assume $p>1$. By Lemma \[2\], holds with $\rho(x)=C\|x\|^p$. By the proof of Lemma \[1\], the formula defines a positively $p$-homogeneous control function for $\phi$. Let us remark that natural analogues of Lemma \[1\], Lemma \[2\] and Corollary \[p-homog\] hold also for mappings $\Phi\colon X\to Y$ (instead of functions $\phi\colon X\to{\mathbb{R}}$), where “delta-convex function” is replaced by “delta-convex mapping” (as defined in [@Ve-Za]) and, in the terms involving $\Phi$, the absolute value is replaced by the norm of $Y$. This follows from [@Ve-Za Proposition 1.13]. Let $X$ and $Y$ be Banach spaces. We say that a linear operator $T\colon X\to Y$ is a [*UMD-operator*]{} if there exists a constant $C>0$ such that $$\label{UMD} \textstyle {\mathbb{E}}\|\sum_{k=1}^n \epsilon_k\, Tdf_k\|^2 \le C\, {\mathbb{E}}\|f_n\|^2$$ whenever $(f_0,\ldots,f_n)$ is an $X$-valued Walsh-Paley martingale and $\epsilon_1,\ldots,\epsilon_n$ are numbers in $\{-1,1\}$. We say that $X$ is a [*UMD-space*]{} if the identity $I\colon X\to X$ is a UMD-operator. \[R:UMD\] 1. It is easy to see that a composition of two bounded linear operators is a UMD-operator whenever at least one of them is. In particular, if at least one of $X,Y$ is a UMD-space, then each bounded linear operator $T\colon X\to Y$ is a UMD-operator. 2. Suppose $p>1$ is a real number. Then $X$ is a UMD-space if and only if is a constant $c_p>0$ such that $$\textstyle {\mathbb{E}}\|\sum_{k=1}^n \epsilon_k \,df_k\|^p \le c_p\, {\mathbb{E}}\|f_n\|^p$$ whenever $\epsilon_k=\pm1$ ($1\le k\le n$) and $(f_0,\ldots,f_n)$ is a Walsh-Paley martingale. Moreover, in this case the above inequality holds also for general (i.e. not necessarily Walsh-Paley) martingales. (See p.67 and Lemma 7.1 in [@B-LNM].) 3. Every UMD-space is superreflexive. (See e.g. [@Pi-LNM p.222] or [@aldous Proposition 2].) For us the following result, which was proved in [@wenzel], will be important. Let us remark that a similar result for general martingales in UMD-spaces was proved by Burkholder [@B] and, as remarked in [@BD p.502], his proof can be easily modified to prove the same for UMD-operators defined using general martingales. \[UMD-MT\] Let $T\colon X\to Y$ be a UMD-operator between Banach spaces $X,Y$. Then there exists a constant $C>0$ such that holds whenever $(f_0,\ldots,f_n)$ is an $X$-valued Walsh-Paley martingale and $(\epsilon_1,\ldots,\epsilon_n)$ is a predictable sequence of $\{\pm1\}$-valued functions (i.e., each $\epsilon_k$ is $\Sigma_{k-1}$-measurable). (See [@wenzel].) \[3\] Let $q$ be a continuous quadratic form on $X$ and $T\colon X\to X^*$ the symmetric operator that generates $q$. Then $q$ is delta-convex if and only if $T$ is a UMD-operator. Let $T$ be a UMD-operator, let $(f_0,\ldots,f_n)$ be an $X$-valued Walsh-Paley martingale. Then, for $\eta\in\Gamma^n$ and $1\le k\le n$, we have $$|\Delta^2 q(f_{k-1}(\eta),df_k(\eta))| = 2|q(df_k(\eta))| = 2\epsilon_k(\eta) q(df_k(\eta)) =2\epsilon_k(\eta) \langle df_k(\eta),Tdf_k(\eta)\rangle,$$ where $\epsilon_k(\eta)=1$ if $q(df_k(\eta))\ge0$, $\epsilon_k(\eta)=-1$ if $q(df_k(\eta))<0$. Observe that $\epsilon_k$ is $\Sigma_{k-1}$-measurable by Remark \[O:mart\](e) since $q$ is an even function; in other words, the sequence $(\epsilon_1,\ldots,\epsilon_n)$ is predictable. Using Lemma \[L:mart\], we can write $$\begin{aligned} \textstyle {\mathbb{E}}\sum\limits_{k=1}^n |\Delta^2 q(f_{k-1},df_k)| &\textstyle= 2 {\mathbb{E}}\sum\limits_{k=1}^n \langle df_k,\epsilon_k Tdf_k\rangle = 2{\mathbb{E}}\langle\sum\limits_{j=1}^n df_j\,,\sum\limits_{k=1}^n \epsilon_k Tdf_k\rangle\\ &\textstyle= 2{\mathbb{E}}\langle f_n-f_0,\sum\limits_{k=1}^n \epsilon_k Tdf_k\rangle = 2{\mathbb{E}}\langle f_n,\sum\limits_{k=1}^n \epsilon_k Tdf_k\rangle\\ &\textstyle\le 2 ({\mathbb{E}}\|f_n\|^2)^{1/2} \bigl({\mathbb{E}}\|\sum\limits_{k=1}^n \epsilon_k Tdf_k\|^2\bigr)^{1/2} \le 2\sqrt{C}\,\, {\mathbb{E}}\|f_n\|^2\,,\end{aligned}$$ where $C$ is the constant from Fact \[UMD-MT\]. For the converse, suppose that $q$ is delta-convex. Consider any $X$-valued Walsh-Paley martingale $(f_0,\ldots,f_n)$ with ${\mathbb{E}}\|f_n\|^2=1$, and numbers $\epsilon_k\in\{-1,1\}$ ($1\le k\le n$). It is easy to see that there exists $h_n\colon\Gamma^n\to X$ such that ${\mathbb{E}}\|h_n\|^2=1$ and $$\textstyle \bigl(\|\sum\limits_{k=1}^n \epsilon_k Tdf_k\|^2\bigr)^{1/2}\le 2 {\mathbb{E}}\langle h_n, \sum\limits_{k=1}^n \epsilon_k Tdf_k\rangle\,;$$ indeed, denoting $g=\sum_{k=1}^n \epsilon_k Tdf_k$, one can put $h(\eta):=t(\eta) v(\eta)$, where $t\colon\Gamma^n\to[0,\infty)$ satisfies ${\mathbb{E}}t^2=1$ and $({\mathbb{E}}\|g\|^2)^{1/2}={\mathbb{E}}(t\cdot\|g\|)$, and $v(\eta)\in S_X$ is such that $\|g(\eta)\|\le 2\langle v(\eta),g(\eta)\rangle$. Let $(h_0,\ldots,h_n)$ be the Walsh-Paley martingale given by $h_n$ (i.e., $h_k={\mathbb{E}}(h_n|\Sigma_k)$, $1\le k\le n$). Then Lemma \[2\] and the identities $$\langle x,Ty\rangle = (1/4)\bigl(q(x+y)-q(x-y)\bigr)\ \ \ \text{and}\ \ \ q(y)=(1/2)\Delta^2q(x,y)$$ imply $$\begin{aligned} \textstyle \bigl(\|\sum\limits_{k=1}^n \epsilon_k Tdf_k\|^2\bigr)^{1/2}& \textstyle\le 2 {\mathbb{E}}\langle h_n, \sum\limits_{k=1}^n \epsilon_k Tdf_k\rangle\,= \qquad\text{(as above)}\\ &\textstyle= 2{\mathbb{E}}\sum\limits_{k=1}^n \langle dh_k, \epsilon_k Tdf_k\rangle\le 2{\mathbb{E}}\sum\limits_{k=1}^n \bigl| \langle dh_k, Tdf_k\rangle \bigr|\\ &\textstyle\le \frac12 {\mathbb{E}}\sum\limits_{k=1}^n \bigl|q(d(f_k+h_k))\bigr| + \frac12 {\mathbb{E}}\sum\limits_{k=1}^n \bigl|q(d(f_k-h_k))\bigr| \\ &\textstyle\le \frac14 C\, {\mathbb{E}}\|f_n+h_n\|^2 + \frac14 C\, {\mathbb{E}}\|f_n-h_n\|^2 \\ &\textstyle\le \frac12 C {\mathbb{E}}\bigl(\|f_n\|+\|h_n\|\bigr)^2 \le C\bigl({\mathbb{E}}\|f_n\|^2 +{\mathbb{E}}\|h_n\|^2\bigr)\le 2C.\end{aligned}$$ Thus $T$ is a UMD-operator. The following theorem is an immediate consequence of Theorem \[3\] and Remark \[R:UMD\](a). \[4\] Let $X$ be a Banach space. Then every continuous quadratic form on $X$ is delta-convex if and only if every symmetric operator $T:X\to X^*$ is a UMD-operator. In particular, if $X$ is a UMD-space, then every continuous quadratic form on $X$ is delta-convex. \[T:counterexample\] There exists a continuous quadratic form on $\ell_1$ which is not delta-convex. Let $J\colon \ell_1\to\ell_\infty$ be an isometric embedding (recall that every separable Banach space isometrically embeds into $\ell_{\infty}$). Consider the continuous quadratic form on $\ell_1=\ell_1\oplus_1\ell_1$, given by $$q(x,y)=\langle y, Jx \rangle + \langle x, Jy \rangle\,, \qquad x,y\in\ell_1\,,$$ which is generated by the symmetric operator $T(x,y)=(Jy,Jx)$. If $q$ were delta-convex, $T$ would be a UMD-operator. But then $J$ would be a UMD-operator; consequently, $\ell_1$ would be a UMD-space. But this is false by Remark \[R:UMD\](c). Let us conclude with a simple but useful proposition. \[P:d.c.\] Let $p>0$. Let $\phi\colon X\to{\mathbb{R}}$ be a $p$-homogeneous function on a Banach space $X$. Then $\phi$ is delta-convex if and only if $\phi$ is delta-convex on a convex neighborhood of the origin. Let $\phi$ be delta-convex on a convex neighborhood $U$ of the origin, and let $\psi\colon U\to{\mathbb{R}}$ be a corresponding control function. There exists $\delta>0$ such that $\psi$ is bounded on $\delta B_X$. A simple homogeneity argument shows that $\phi$ is delta-convex on each $r B_X$ ($r>0$) with a bounded control function of the form $\rho(x)=c_1\psi(c_2 x)$. Then $\phi$ is delta-convex on $X$ by [@Kop-Maly Theorem 16]. Further results and open problems {#S:three} ================================= We shall consider the following three properties of a Banach space $X$, defined already in Introduction. 1. [*Each continuous quadratic form on $X$ is delta-semidefinite.*]{} 2. [*Each continuous quadratic form on $X$ is delta-convex.*]{} 3. [*Each $C^{1,1}$ function $f\colon X\to{\mathbb{R}}$ is delta-convex.*]{} Recall that a function (or mapping) $f$ is $C^{1,1}$ if the Fréchet derivative $f'(x)$ exits for each $x$ and the mapping $f'$ is Lipschitz. We have seen that (D) passes to quotients (Corollary \[C:quotient\]). Let us observe the same result for properties (dc) and (Cdc). \[dcquotient\] If $X$ is a Banach space with property (dc) (respectively, (Cdc)), then for any closed subspace $E$ of $X$, the quotient $X/E$ has property (dc) (respectively, (Cdc)). Let $Q\colon X\to X/E$ be the quotient map and let $f\colon X/E\to\mathbb R$ be a continuous function such that $f\circ Q$ is delta-convex. We show that $f$ is delta-convex (which proves both assertions). Let $\psi\colon X\to\mathbb R$ be a continuous function such that $\psi\pm f$ is convex; we can assume $\psi\ge 0$. Define $\hat\psi\colon X/E\to\mathbb R$ by $\hat\psi(y)=\inf\{\psi(x): Qx=y\}.$ Then it is easy to prove that $\hat\psi\pm f$ is convex. Moreover, the (convex) function $\hat\psi$ is continuous since it is easily seen to be bounded on a neighborhood of the origin. Since each continuous quadratic form is $C^{1,1}$ (by Fact \[F:p’\]), we always have the implications $$({\rm D})\ \Longrightarrow\ ({\rm dc})\ \Longleftarrow\ ({\rm Cdc})\,.$$ As we shall see in the next theorem, no two of the above three properties are equivalent. Let us start with the following corollary of [@Du-Ve-Za Theorem 11]. A norm on $X$ is said to have modulus of convexity of power type 2 if, for some $c>0$, $\delta_{X}(\epsilon)\ge c\cdot\epsilon^2$ whenever $\epsilon\in(0,2]$ (where $\delta_X$ is the usual modulus of convexity; see e.g. [@Lin-Tz]). \[F:powertype\] Let $X$ be a Banach space that admits a uniformly convex renorming with modulus of convexity of power type 2. Then $X$ satisfies (Cdc) and hence also (dc). By an $L_p(\mu)$ space we mean an infinite-dimensional space $L_p(\Omega,\Sigma,\mu)$ where $(\Omega,\Sigma,\mu)$ is a positive measure space. This class includes the spaces $L_p(0,1)$ and $\ell_p$. For such spaces, we have the following theorem which summarizes results of [@Du-Ve-Za], [@Zel] and of the present paper. \[T:L\_p\] Let $X$ be an infinite-dimensional $L_p(\mu)$ space with $1\le p \le \infty$. 1. $X$ satisfies [(D)]{} if and only if $p\ge2$. 2. $X$ satisfies [(Cdc)]{} if and only if $1<p\le2$. 3. $X$ satisfies [(dc)]{} if and only if $p>1$. [*(a)*]{} If $p\ge2$ then $L_p(\mu)$ satisfies (D) by Theorem \[T:D\](f). If $p<2$ then $L_p(\mu)$ fails (D) since it contains a complemented copy of $\ell_p$ which fails (D) by Corollary \[C:badp\]. [*(b)*]{} If $1<p\le2$, then the standard norm on $X=L_p(\mu)$ has modulus of convexity of power type 2 (see [@DGZbook Corollary V.1.2]). By Fact \[F:powertype\], each such space satisfies (Cdc). $L_1(\mu)$ fails (Cdc) since it fails (dc) (see (c) below). Now, let $2<p<\infty$. For such $p$, M. Zelený [@Zel] proved that $\ell_p$ fails (Cdc); thus $L_p(\mu)$ fails (Cdc), too. Finally, to see that also $L_\infty(\mu)$ fails (Cdc), it suffices to show that $L_\infty(0,1)$ fails (Cdc); indeed, the spaces $L_\infty(0,1)$ and $\ell_\infty$ are isomorphic by [@pelcz], and $L_\infty(\mu)$ contains a complemeted copy of $\ell_\infty$. By [@Lin-Tz Corollary 2.f.5], $(\ell_4)^*=\ell_{4/3}$ isometrically embeds in $L_1(0,1)$; consequently, $\ell_4$ is isomorphic to a quotient of $L_\infty(0,1)$. Hence $L_\infty(0,1)$ fails (Cdc) by Lemma \[dcquotient\], since we already know that $\ell_4$ fails (Cdc). [*(c)*]{} $L_1(\mu)$ fails (dc) since it contains a complemented copy of $\ell_1$ which fails (dc) by Theorem \[T:counterexample\]. For $p>1$, the space $L_p(\mu)$ satisfies (dc) since, by (a) and (b), it satisfies (D) (if $p\ge2$) or (Cdc) (if $p\le2$). Alternately one may observe that $L_p(\mu)$ is a UMD-space if $1<p<\infty$ using Remark \[R:UMD\](b) and then apply Theorem \[4\]. By Theorem \[T:L\_p\], ${\rm(dc)}\not\Rightarrow{\rm(D)}$ and also ${\rm(dc)}\not\Rightarrow{\rm(Cdc)}$. It is interesting to compare the second non-implication with the following result from [@Du-Ve-Za] about vector-valued mappings: [*every Banach space-valued continuous quadratic mapping on $X$ is delta-convex if and only if every Banach space-valued $C^{1,1}$ mapping on $X$ is delta-convex.*]{} Delta-convex functions via delta-convex curves {#delta-convex-functions-via-delta-convex-curves .unnumbered} ---------------------------------------------- There is another corollary to the above results. It is connected with Problem 6 in [@Ve-Za]. In that paper, delta-convex mappings between Banach spaces (a generalization of delta-convex functions) were defined and widely studied. We do not state the definition here; it can be found also in [@Du-Ve-Za] together with a survey of principal results. We confine ourselves to stating an equivalent definition (see [@Ve-Za Theorem 2.3]) of a delta-convex mapping of one real variable. Let $I\subset{\mathbb{R}}$ be an open interval, $X$ be a Banach space. A mapping $\phi\colon I\to X$ is [*delta-convex*]{} on $I$ if the right derivative $\phi'_+(t)$ exists at each $t\in I$ and the mapping $\phi'_+$ has bounded variation on each compact subinterval of $I$. For real-valued functions, Problem 6 in [@Ve-Za] asks: [*suppose that $X$ is a Banach space and $f\colon X\to{\mathbb{R}}$ is a function such that $f\circ\phi$ is a delta-convex function on $(0,1)$ whenever $\phi\colon(0,1)\to X$ is a delta-convex mapping; is then $f$ locally delta-convex?*]{} The following example answers in negative this problem. (Let us remark that the vector-valued case was solved in negative already in [@Du-Ve-Za].) \[E:composition\] Let $X$ be an infinite-dimensional $L_p(\mu)$ space where either $p=1$ or $2<p<\infty$. Then there exists a continuous function $f\colon X\to{\mathbb{R}}$ such that $f$ is delta-convex on no neighborhood of $0$, and $f\circ\phi$ is delta-convex on $(0,1)$ for each delta-convex mapping $\phi\colon(0,1)\to X$. The case $p=1$. By Theorem \[T:L\_p\](b), there exists a continuous quadratic form $q$ on $X$ such that $q$ is not delta-convex. By Proposition \[P:d.c.\], $q$ is delta-convex on no neighborhood of $0$. By Proposition 14 in [@Du-Ve-Za], $q\circ\phi$ is delta-convex for each delta-convex “curve” $\phi$. Thus we can put $f=q$. The case $2<p<\infty$ follows in a similar way using [@Zel] instead of Theorem \[T:L\_p\]. Indeed, by [@Zel], there exists a $C^{1,1}$ function $g\colon\ell_p\to{\mathbb{R}}$ that is not delta-convex. A careful look at the proof in [@Zel] shows that the function constructed therein is d.c. on no neighborhood of $0$. Consider $\ell_p$ as a complemented subspace of $X=L_p(\mu)$ and extend $g$ to a $C^{1,1}$ function on the whole $X$ by $f=g\circ P$ where $P$ is a bounded linear projection of $X$ onto $\ell_p$. Then $f$ has the desired property by [@Du-Ve-Za Proposition 14] again. It is natural to ask the following Does there exist a function $f$ as in Example \[E:composition\] for each at least two-dimensional Banach space $X$? Or, at least, for each infinite-dimensional Banach space $X$? Stability with respect to direct sums {#stability-with-respect-to-direct-sums .unnumbered} ------------------------------------- Consider two Banach spaces $X_1$ and $X_2$. Since $(X_1\oplus X_2)^*=X_1^*\oplus X_2^*$, each bounded linear operator operator $T\colon X_1\oplus X_2 \to (X_1\oplus X_2)^*$ can be represented as an operator-valued matrix $T=\textstyle\left( \begin{array}{cc} T_{11} & T_{12} \\ T_{21} & T_{22} \end{array}\right)$ where $T_{ij}\colon X_j\to X_i^*$. It is an easy exercise to verify that: 1. $T$ is factorizable through a Hilbert space if and only if each $T_{ij}$ is; 2. $T$ is a UMD-operator if and only if each $T_{ij}$ is; 3. $T$ is symmetric if and only if $T_{ij}^*=T_{ji}$ on $X_i$ whenever $i,j\in\{1,2\}$ (equivalently, $T_{11}$ and $T_{22}$ are symmetric and $T^*_{12}=T_{21}$ on $X_1$). Hence we have the following consequence of Corollary \[C:quotient\] and Theorem \[3\]. (Given Banach spaces $X$ and $Y$, we denote by $\mathcal{L}(X,Y)$ the set of all bounded linear operators from $X$ into $Y$.) \[C:sum\] Let $X_1$ and $X_2$ be Banach spaces. Then: 1. $X_1\oplus X_2$ has the property (D) if and only if $X_1$ and $X_2$ have (D) and every element of $\mathcal{L}(X_1,X_2^*)$ is factorizable through a Hilbert space; 2. $X_1\oplus X_2$ has the property (dc) if and only if $X_1$ and $X_2$ have (dc) and every element of $\mathcal{L}(X_1,X_2^*)$ is a UMD-operator. In particular, if $X$ is isomorphic to $X^2$ then 1. $X$ has the property (D) if and only if every element of $\mathcal{L}(X,X^*)$ is factorizable through a Hilbert space; 2. $X$ has the property (dc) if and only if every element of $\mathcal{L}(X,X^*)$ is a UMD-operator. The following Corollary is immediate using Remark \[R:UMD\](a) for part (b). Let $X$ be a Banach space. 1. If $X$ satisfies (D) and $H$ is a Hilbert space, then $X\oplus H$ satisfies (D). 2. If $X$ satisfies (dc) and $U$ is a UMD-space, then $X\oplus U$ satisfies (dc). \[adjoint\] The adjoint $T^*$ is a UMD-operator if and only if $T$ is a UMD-operator. In particular, $X$ is a UMD-space if and only if $X^*$ is.\ To see this, note that $T\colon X\to Y$ is a UMD-operator if and only if the operators $$\textstyle T_{n,\epsilon}:=\sum_{k=1}^n \epsilon_k T(E_{k,X}-E_{k-1,X}) \colon L_2(\Gamma^n,X)\to L_2(\Gamma^n,Y),\quad\; n\in{\mathbb{N}},\;\epsilon\in\{-1,1\}^n,$$ are equi-bounded, where $E_{k,X}:={\mathbb{E}}(\cdot|\Sigma_k)\colon L_2(\Gamma^n, X)\to L_2(\Gamma^n, X)$. But then also the corresponding adjoints are equi-bounded. Moreover, it is easy to see that $$\textstyle T_{n,\epsilon}^*= \sum_{k=1}^n \epsilon_k (E_{k,X^*}-E_{k-1,X^*}) T^*= \sum_{k=1}^n \epsilon_k T^*(E_{k,Y^*}-E_{k-1,Y^*})$$ which means that $T^*$ is a UMD-operator. The reverse implication follows easily from this one. \[XxX\*\] Let $X$ be a Banach space. Let $Q$ be the continuous quadratic form on $X\oplus X^*$, given by $Q(x,x^*)=x^*(x)$. Then the following three assertions are equivalent: 1. $X\oplus X^*$ satisfies (D); 2. $Q$ is delta-semidefinite; 3. $X$ is isomorphic to a Hilbert space. And also the following three assertions are equivalent: 1. $X\oplus X^*$ satisfies (dc); 2. $Q$ is delta-convex; 3. $X$ is a UMD-space. The implications (iii)$\Rightarrow$(i)$\Rightarrow$(ii) are obvious. Assume (ii). Since $Q$ is generated by the symmetric operator $T\colon X\oplus X^*\to X^*\oplus X^{**}$, $T(x,x^*)=\frac12(x^*,x)$, it follows from Theorem \[T:kalton\] that the identity $I\colon X\to X$ factors through a Hilbert space. It is easy to see that this implies (iii). (Indeed, if $I=BA$ is a factorization through a Hilbert space $H$, then $AB$ is a bounded linear projection onto a closed subspace $H_0=A(X)$ of $H$. Then $A$ is a linear isomorphism between $X$ and $H_0$.) The implication (i’)$\Rightarrow$(ii’) is obvious. If (ii’) holds, then (as above, via Theorem \[3\]) the identity $I\colon X\to X$ is a UMD-operator, which gives (iii’). Finally, if (iii’) holds, then $X\oplus X^*$ is a UMD-space (indeed, it suffices to apply (II) before Corollary \[C:sum\] to the identity operator of $X\oplus X^*$, taking into account Observation \[adjoint\]). Hence (i’) holds by Theorem \[4\]. As far as we know, the following question is open. \[DDD\] Is the property (D) stable with respect to making direct sums of two spaces? Equivalently, if Banach spaces $X_1$ and $X_2$ have property (D), does it imply that each $S\in\mathcal{L}(X_1,X_2^*)$ is factorizable through a Hilbert space? We conjecture that the answer is negative, but we do not know any counterexample. However, the following observation shows that a possible counterexample cannot be found by using only spaces provided by Theorem \[T:D\]. Let each of given two Banach spaces $X_1$ and $X_2$ satisfy at least one of the conditions (a)–(i) in Theorem \[T:D\]. Then $X_1\oplus X_2$ has (D). By Corollary \[C:sum\], it suffices to show that every operator $S\in\mathcal{L}(X_1,X_2^*)$ factors through a Hilbert space. By the proof of Theorem \[T:D\], each of the spaces $X_i$ (i=1,2) has at least one of the following three properties: 1. $X_i$ has type 2; 2. $X_i^*$ has cotype 2, and $X_i$ has the approximation property; 3. $X_i^*$ has cotype 2, and $X_i$ is a Banach lattice. First observe that this implies that $X_2^*$ has cotype 2 (see e.g.[@Pisier Proposition 3.2]). Now, if $X_1$ satisfies ($\alpha$), apply [@Pisier Corollary 3.6]; if $X_1$ satisfies ($\beta$), apply [@Pisier Theorem 4.1]; if $X_1$ satisfies ($\gamma$), apply [@Pisier Theorems 8.17 and 8.11]. Note that Proposition \[XxX\*\] implies that Problem \[DDD\] will have a negative answer once the following problem is solved in negative. Let $X$ and $X^*$ satisfy (D). Does it imply that $X$ is isomorphic to a Hilbert space? For the property (dc) we have the following theorem. The property (dc) is not stable under making direct sums. By [@bourgain], there exists a Banach lattice $X$ such that $X$ is not a UMD-space and $X$ satisfies an upper-3 estimate and a lower-4 estimate (see [@Lin-Tz] for definitions). By [@Lin-Tz Theorem 1.f.7], $X$ is 2-convex and 5-concave. Then $X$ admits a uniformly smooth renorming with modulus of smoothness of power type 2 by [@Lin-Tz Theorem 1.f.1], which implies that $X$ has (D) (see Theorem \[T:D\](h)), and hence (cd). By duality (see [@Lin-Tz p.63]), $X^*$ admits a uniformly convex renorming with modulus of convexity of power type 2; consequently, $X^*$ has (dc) by Fact \[F:powertype\]. By Proposition \[XxX\*\], $X\oplus X^*$ fails (dc) since $X$ is not a UMD-space. [WW]{} D.J. Aldous, [*Unconditional bases and martingales in $L_p(F)$*]{}, Math. Proc. Camb. Phil. Soc. [**85**]{} (1979), 117–123. J.M. Borwein and D. Noll, [*Second order differentiability of convex functions in Banach spaces*]{}, Trans. Amer. Math. Soc. [**342**]{} (1994), 43–81. J. Bourgain, [*Some remarks on Banach spaces in which martingale difference sequences are unconditional*]{}, Ark. Mat. [**21**]{} (1983), 163–168. J. Bourgain and W.J. Davis, [*Martingale transforms and complex uniform convexity*]{}, Trans. Amer. Math. Soc. [**294**]{} (1986), 501–515. D.L. Burkholder, [*A geometrical characterization of Banach spaces in which martingale difference sequences are inconditional*]{}, Ann. Probab. [**9**]{} (1981), 997–1011. D.L. Burkholder, [*Martingales and Fourier analysis in Banach spaces*]{}, Probability and Analysis (Varenna, 1985), Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, 61–108. R. Deville, G. Godefroy and V. Zizler, [*Smooth bump functions and geometry of Banach spaces*]{}, Mathematica [**40**]{} (1993), 305–321. R. Deville, G. Godefroy and V. Zizler, [*Smoothness and Renormings in Banach Spaces*]{}, Pitman Monographs and Surveys in Pure and Applied Math., vol. 64, Longman Scientific & Technical, Harlow (in USA: John Willey & Sons, Inc., New York), 1993. J. Diestel, H. Jarchow and A. Tonge, [*Absolutely Summing Operators*]{}, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, 1995. J. Duda, L. Veselý and L. Zajíček, [*On d.c. functions and mappings*]{}, Atti Sem. Mat. Fis. Univ. Modena [**51**]{} (2003), 111–138. M. Fabián, J.H.M. Whitfield and V. Zizler, [*Norms with locally Lipschitzian derivatives*]{}, Israel J. Math. [**44**]{} (1983), 262–276. E. Kopecká and J. Malý, [*Remarks on delta-convex functions*]{}, Comment. Math. Univ. Carolin. [**31**]{} (1990), 501–510. J. Lindenstrauss and L. Tzafriri, [*Classical Banach Spaces II: Function Spaces*]{}, Springer, Berlin and New York, 1979. A. Pe[ł]{}czyński, [*On the isomorphism of the spaces $m$ and $M$*]{}, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. [**6**]{} (1958), 695–696. G. Pisier, [*Holomorphic semigroups and the geometry of Banach spaces*]{}, Ann. of Math. (2) [**115**]{} (1982), 375–392. G. Pisier, [*Factorization of Linear Operators and Geometry of Banach Spaces*]{}, CBMS Regional Conference Series in Mathematics, vol. 60, Amer. Math. Soc., Providence R.I., 1985. G. Pisier, [*Probabilistic methods in the geometry of Banach spaces*]{}, Probability and Analysis (Varenna, 1985), Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, 167–241. A.W. Roberts and D.A. Varberg, [*Convex Functions*]{}, Pure and Applied Mathematics, vol. 57, Academic Press, Neww York-London, 1973. B. Sari, T. Schlumprecht, N. Tomczak-Jaegerman and V.G. Troitsky, [*On norm closed ideals in $L(\ell_p\oplus\ell_q)$*]{}, Studia Math. [**179**]{} (2007), 239–262. L. Veselý and L. Zajíček, [*Delta-convex mappings between Banach spaces and applications*]{}, Dissertationes Math. (Rozprawy Mat.) [**289**]{} (1989), 52 pp. J. Wenzel, [*Ideal norms associated with the UMD-property*]{}, Arch. Math. [**69**]{} (1997), 327–332. D. Williams, [*Probability with Martingales*]{}, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. M. Zelený, [*An example of a $C^{1,1}$ function, which is not a d.c. function*]{}, Comment. Math. Univ. Carolin. [**43**]{} (2002), 149–154. [^1]: The first author was supported by NSF grant DMS-0555670. The second author was supported by the Russian Foundation for Basic Research, Grant 05-01-00066, and by Grant NSh-5813.2006.1. The third author was supported in part by the Ministero dell’Università e della Ricerca of Italy.

--- abstract: 'We demonstrate an all-optical magnetometer capable of measuring the magnitude and direction of a magnetic field using nonlinear magneto-optical rotation in a cesium vapor. Vector capability is added by effective modulation of the field along orthogonal axes and subsequent demodulation of the magnetic-resonance frequency. This modulation is provided by the AC Stark shift induced by circularly polarized laser beams. The sensor exhibits a demonstrated rms noise floor of 50 fT/[$\sqrt{\textrm{Hz}}$ ]{}in measurement of the field magnitude and 0.5 mrad/[$\sqrt{\textrm{Hz}}$ ]{}in the field direction; elimination of technical noise would improve these sensitivities to 12 fT/[$\sqrt{\textrm{Hz}}$ ]{}and 5 $\mu$rad/[$\sqrt{\textrm{Hz}}$ ]{}, respectively. Applications for a precise all-optical vector magnetometer would include magnetically sensitive fundamental physics experiments, such as the search for a permanent electric dipole moment of the neutron.' address: - 'Department of Physics, University of California, Berkeley, CA 94720-7300' - 'Physik-Department, Technische Universität München, 85748 Garching, Germany' - 'Department of Physics, University of California, Berkeley, CA 94720-7300' - 'Southwest Sciences Ohio Operations, Cincinnati, OH 45244' - 'Department of Physics, University of California, Berkeley, CA 94720-7300' - 'Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720' - 'Helmholtz Institute, Johannes Gutenberg University, 55099 Mainz, Germany' author: - 'B. Patton' - 'E. Zhivun' - 'D. C. Hovde' - 'D. Budker' bibliography: - 'VectorMagBib.bib' title: 'All-Optical Vector Atomic Magnetometer' --- Spin-precession magnetometers [@OpticalMagnetometryBook; @Budker_NP07] have found widespread application in disciplines ranging from geophysics [@Dang_APL10] to medicine [@Bison_APL09; @Johnson_PMB13] and fundamental physics [@Vasilakis_PRL09; @Altarev_PRL09]. Alkali-vapor magnetometers in particular have experienced great advances in recent years, with sensitivities at or below the fT/[$\sqrt{\textrm{Hz}}$ ]{}level demonstrated in the laboratory [@Dang_APL10; @Ledbetter_PRA08; @Griffith_OE10; @Smullin_PRA09]. Because these devices measure the Larmor precession frequency of atomic spins, they are intrinsically sensitive to the magnitude of an applied field rather than its projection along a particular direction. This can be advantageous in that precision of the scalar field measurement is not limited by physical alignment of the sensors, as it can be in the case of triaxial fluxgates or superconducting quantum interference devices (SQUIDs). Nevertheless, in many situations it is desirable to have full knowledge of a field’s vector components. There are several ways to derive vector field information from a scalar magnetometer. In bias-field nulling, calibrated magnetic fields are imposed upon the magnetometer in order to achieve a zero-field magnetic resonance condition [@Slocum_IEEE63; @Seltzer_APL04; @Dong_IEEE13]. With finite-field sensors using radiofrequency coils to drive the resonance (e.g., $M_x$ magnetometers [@Bloom_AO62]), one may add secondary continuous light beams and measure their modulation to extract vector information [@Fairweather_JPE72; @Vershovskii_TPL11]. It is also possible to detect magnetically sensitive resonances in electromagnetically induced transparency (EIT) schemes; the amplitudes of different EIT resonances can yield information about the relative angle between the laser polarization and the field [@Lee_PRA98; @Cox_PRA11]. Synchronously pumped magnetometers employing atomic alignment can also yield partial vector information when the magnetic field is not wholly perpendicular to the linear polarization of the pump beam [@Pustelny_PRA06]. Perhaps the simplest way to adapt a scalar magnetometer for vector measurements is to operate it in the finite-field regime (e.g., through synchronous optical pumping [@Bell_PRL61; @Gawlik_APL06; @Higbie_RSI06]) and apply time-varying fields to it. By applying orthogonal fields modulated at different frequencies, it is possible to demodulate the magnetic-resonance signal and determine which applied fields add linearly with the ambient field and which add in quadrature with it [@Rasson_GT91; @Gravrand_EPS01; @Vershovskii_TP06]. Although this is effective, there are some situations where this approach is infeasible or undesirable. One example would be the case of remote magnetometry [@Patton_APL12; @Higbie_PNAS11], where it would be impractical to apply fields to a distant atomic sample. A different limitation appears in certain precision physics applications, such as the search for a neutron electric dipole moment (nEDM) [@Baker_PRL06; @Altarev_PRL09; @Knowles_NIMPRA09; @Altarev_NC12]. In such experiments alkali-vapor magnetometers can reduce systematic error by providing crucial magnetic-field information, but only if these sensors do not themselves produce field contamination. All-optical alkali-vapor magnetometers are particularly well suited for nEDM tests as they can be designed to produce no significant static or radiofrequency fields [@Patton_EDMmag]. Here we demonstrate an all-optical vector magnetic sensor based upon nonlinear magneto-optical rotation in a cesium vapor. The effective magnetic field seen by the atoms is modulated by AC Stark shifts (“light shifts”) induced by orthogonally propagating laser beams. Since the light shift of a circularly polarized beam is analogous [^1] to an effective magnetic field oriented along its propagation direction [@Mathur_PR68; @Cohen-Tannoudji_PRA72; @Moriyasu_PRL09], a comparison of the Larmor frequency shifts induced by these beams yields a measurement of the field angle. If technical noise were eliminated, this magnetometer would have 12 fT/[$\sqrt{\textrm{Hz}}$ ]{}precision in measurement of the field magnitude and 5 $\mu$rad/[$\sqrt{\textrm{Hz}}$ ]{}in the field direction. ![\[VectorMagSetup\] Experimental schematic. An amplitude-modulated, circularly polarized pump beam (not shown) propagates in the $\hat{x}$ direction. The local oscillator (LO) controls the pump AOM and serves as a reference to the lock-in amplifier (LIA), whose analog output is recorded by a data acquisition card (DAQ) and read into a computer (PC). A linearly polarized probe beam passes through the cell and is split by the polarizing beamsplitter (PBS) of a balanced polarimeter; the output of this polarimeter is demodulated by the lock-in. Two circularly polarized light-shift beams $LS_y$ and $LS_z$ are independently modulated and sent through the cell along $\hat{y}$ and $\hat{z}$. Coils allow the magnetic field $\textbf{B}_0$ to be tilted in the $\hat{y}$–$\hat{z}$ plane.](Setup_v07.pdf) The experimental setup is shown in Fig. \[VectorMagSetup\]. The heart of the sensor is a cylindrical antirelaxation-coated [@Seltzer_JCP10] Cs vapor cell, approximately 5 cm diameter and 5 cm in length, with a longitudinal spin relaxation time of 0.7 seconds. This cell is enclosed within four layers of $\mu$-metal magnetic shielding; measurements were performed at ambient temperature. Coils wound on a frame within the innermost shield allow magnetic fields and gradients to be applied to the cell. The field component oriented along $\hat{z}$ is produced by a current generated by a custom supply which can provide up to 150 mA (Magnicon GmbH). This supply is housed in a temperature-stabilized enclosure and exhibits a relative drift of $\sim$10$^{-7}$ over 100 seconds. A second current supply (Krohn Hite 523) is connected to the coil in the $\hat{y}$ direction, allowing the net field $\textbf{B}_0$ to be tilted in the $\hat{y}$–$\hat{z}$ plane. The pump beam which drives the magnetic resonance is generated by a distributed feedback (DFB) diode laser that is locked with a dichroic atomic vapor laser lock (DAVLL) [@Yashchuk_RSI00] to the Cs $D1$ transition at 894 nm. The $\hat{x}$-directed pump is circularly polarized and amplitude modulated with an acousto-optic modulator (AOM) at the $^{133}$Cs Larmor frequency $\omega_{L}$ to achieve synchronous optical pumping; the modulation waveform is a square wave with a duty cycle of 5%. A separate linearly polarized probe beam, generated by a DFB locked with a DAVLL to the Cs $D2$ transition, traverses the cell in the $\hat{y}$ direction. The probe experiences optical rotation [@Budker_RMP02] in the polarized Cs sample, modulating its polarization at $\omega_{L}$. This is detected by a balanced polarimeter with a differential transimpedance amplifier; its output is fed into a digital lock-in amplifier (Stanford Research Systems SR830) whose reference frequency is provided by the local oscillator which drives the pump AOM. The phase of the lock-in amplifier is chosen such that the () output displays an absorptive (dispersive) Lorentzian as the driving frequency is scanned across the resonance. Directly on resonance, the output is maximum and the output is nulled; small shifts in the magnetic-resonance frequency $\omega_L$ cause a linear change in the output about zero. With a time-averaged pump power of 2.5 $\mu$W and a probe power of 10 $\mu$W, the peak optical rotation signal is 5 mrad and the magnetic-resonance linewidth is 2.9 Hz. The dominant contributions to this linewidth are alkali–alkali spin-exchange broadening and slight power broadening due to the pump and probe beams. The beam powers and optical detunings were chosen to optimize the scalar sensitivity of the magnetometer. In addition to the pump and probe, a third DFB laser tuned near the Cs $D2$ transition can be used to apply light-shift beams $LS_y$ and $LS_z$ in the $\hat{y}$ and $\hat{z}$ directions. The optical frequency of the light-shift laser is actively controlled using a wavelength meter (Ångstrom/HighFinesse WS-7) and computer control of the laser current. An optimal detuning of $\sim$5 GHz blue-shifted from the center of the $F\!\!=\!4 \rightarrow F'\!\!=\!5$ $D2$ transition was chosen to allow a large effective magnetic field ($\sim$1 nT/mW) with minimal (${\raisebox{0.05em}{\parbox[][][t]{0.7em}{\scalebox{0.8}[0.7]{$\lesssim$}}}}$0.5 Hz/mW) broadening of the magnetic-resonance line. This beam is split into two paths and sent through independent AOMs, then coupled into two polarizing [^2] fiber patch cables (Fibercore HB830Z). After the fibers, the light-shift beams are sent through quarter-wave plates to generate circularly polarized beams which pass through the cell along the $\hat{y}$ and $\hat{z}$ axes. Optical pickoffs (not shown in Fig. \[VectorMagSetup\]) and photodiodes directly before the shields allow the power of each light-shift beam to be measured. In an evacuated antirelaxation-coated cell, the alkali atoms rapidly sample the internal volume of the cell and experience a light shift equivalent to the volume-averaged intensity of the laser beam within the cell. Thus two beams of the same power will possess slightly different light-shift coefficients (measured in nT/mW) when propagating in different directions due to asymmetry of the cell dimensions. Nevertheless, their ratio will remain independent of the optical detuning of the light-shift laser. ![\[tanplot\] Above: Depiction of the $LS_y$ and $LS_z$ beam powers versus time and the resulting (simulated) change in the lock-in output about zero. Demodulation of the latter yields the contributions of $LS_y$ and $LS_z$ to the shift in the magnetic-resonance frequency. Below: The ratio of the Larmor frequency shift induced by the $LS_y$ and $LS_z$ beams, plotted as a function of field angle $\theta$ from the $\hat{z}$ axis. The curve shows a fit to Eq. (\[taneq\]). Each data point resulted from 20 seconds of averaging; uncertainties in the data points are uniformly below $10^{-2}$.](TanPlot_v06.pdf) To demonstrate the effective magnetic fields produced by $LS_y$ and $LS_z$, we recorded the data shown in Fig. \[tanplot\]. For this measurement, the primary $\hat{z}$ field was held constant at 946.5 nT and an additional $\hat{y}$ field was varied between -1180.5 nT and +1177 nT. Thus the field’s magnitude $B_0$ changed with its angle $\theta$ in the $\hat{y}$–$\hat{z}$ plane, requiring the local oscillator and the lock-in reference phase to be reset for each measurement. At each field, the respective light shifts produced by the $LS_y$ and $LS_z$ beams were measured by modulating the two beam intensities at different frequencies (12 and 20 Hz) and demodulating the lock-in $\textsf{Y}$ output in software. The average intensity of each light-shift beam was 0.5 mW. Although it improves precision to measure the system response to both beams simultaneously, it is important to alternate the fast and slow modulation in each channel, as shown in Fig. \[tanplot\]. This is because the atomic system acts as a low-pass filter for fast field perturbations, since it is in effect a driven oscillator with a damping rate on the order of the magnetic-resonance linewidth. Assume that the magnetometer is operated in the finite-field regime, such that the magnetic resonance frequency is much higher than the resonance linewidth. The modulated $LS_y$ beam produces an effective magnetic field of magnitude $B_y = P_y \alpha_y [$ + $\sin(\omega_y t)]$, where $P_y$ is the beam power, $\alpha_y$ is its effective light-shift coefficient, and $\omega_y$ the amplitude-modulation frequency. Similarly, $LS_z$ produces $B_z = P_z \alpha_z [$ + $\sin(\omega_z t)]$. To maintain the synchronous pumping condition, the fields $B_y$ and $B_z$ are assumed to be comparable to the resonance linewidth (in field units). Adding these fields to the vector components of $\textbf{B}_0$, the total field magnitude becomes: $$\begin{aligned} B_{\textrm{tot}} & = & B_0 \sqrt{1+2\frac{B_y \sin \theta + B_z \cos \theta}{B_0} + \frac{B_y^2 + B_z^2}{B_0^2}} \nonumber \\ & \approx & B_0 + B_y \sin \theta + B_z \cos \theta + \zeta, \label{LSfield01}\end{aligned}$$ where the approximation is valid for $B_y, B_z \ll B_0$ and the small quadratic correction $\zeta$ is given by: $$\zeta = \frac{\left(B_y \cos \theta - B_z \sin \theta \right)^2}{2B_0}. \label{zeta}$$ Since the lock-in output is proportional to the change in effective Larmor frequency induced by the light-shift fields, demodulation of the signal at frequencies $\omega_y$ and $\omega_z$ will extract the terms in Eq. (\[LSfield01\]) proportional to $B_y$ and $B_z$. Thus the ratio of the measured light shifts is: $$\frac{(\Delta B_{\textrm{tot}})_{LSy}}{(\Delta B_{\textrm{tot}})_{LSz}} \approx \frac{P_y \alpha_y}{P_z \alpha_z} \tan \theta. \label{taneq}$$ Here we have ignored the contribution from the terms in $\zeta$ and other terms of higher power in $(B_{y,z}/B_0)$, which cause modulation of $B_{\textrm{tot}}$ at harmonics other than $\omega_y$ and $\omega_z$ or scale by powers of $\left| B_{y,z}/B_0 \right|$ (here ${\raisebox{0.05em}{\parbox[][][t]{0.7em}{\scalebox{0.8}[0.7]{$\lesssim$}}}}~10^{-3}$). The data shown in Fig. \[tanplot\] were fit to Eq. (\[taneq\]). The best-fit ratio $(P_y \alpha_y / P_z \alpha_z)$ was measured to be (${0.94 \pm 0.01}$) rather than unity, possibly due to slight asymmetry in the cell dimensions or systematic uncertainty of the beam powers within the cell. With no added light-shift beams, the synchronously pumped scalar sensor has sensitivity of 48 fT/[$\sqrt{\textrm{Hz}}$ ]{}for integration times of 1 second, as calculated from the power spectral density (PSD) of the measured magnetic field, shown in Fig. \[PSDplot\]. To confirm this sensitivity in the time domain, we stepped the local oscillator frequency by $\pm$0.875 mHz around $\omega_L$ and observed shifts in the lock-in output with a signal-to-noise ratio of 7.2. Given the lock-in’s equivalent noise bandwidth (ENBW) of 1.25 Hz, this corresponds to a sensitivity of 62 fT/[$\sqrt{\textrm{Hz}}$ ]{}. To assess the uncertainty in the field angle, we recorded data with the $\hat{z}$ field held constant and the $\hat{y}$ field toggled between two small values. The lock-in signal was demodulated at $\omega_y$ and $\omega_z$, and the ratio of these two responses converted to a measured magnetic-field angle according to the best-fit curve shown in Fig. \[tanplot\]. The resulting plot of $\theta$ vs. time is shown in Fig. \[StepPlot\]. The modulation of the field angle is clearly visible, and the rms noise in the ratio corresponds to 0.47 mrad/[$\sqrt{\textrm{Hz}}$ ]{}precision in the measured angle of the magnetic field. (This takes into account the measured ENBW of the software demodulation procedure.) ![\[StepPlot\] Measured field angle $\theta$ as a function of time while the applied $\hat{y}$ field is being switched. The average rms noise for a constant field angle translates to a precision of 0.47 mrad/[$\sqrt{\textrm{Hz}}$ ]{}in measurement of the field direction. The steps in the plotted ratio are slightly low-pass filtered due to the time constants of the lock-in amplifier and the secondary demodulation at $\omega_y$ and $\omega_z$.](StepPlot_v04.pdf) In the present setup, the precision of the measurement of $\theta$ is limited by apparent magnetic noise induced by fluctuations in the light-shift beam powers. With $LS_z$ set to 1 mW without modulation and the field along $\hat{z}$, the smallest observable magnetic-field step with 1 Hz ENBW was 1.3 pT – a factor of 21 worse than the same data recorded without the light-shift beams. Power fluctuations in the $LS_y$ and $LS_z$ beams were recorded and converted into effective magnetic-field fluctuations according to the observed light-shift coefficients $\alpha_{y,z}$. As shown in Fig. \[PSDplot\], the predicted magnetic noise floor matches that observed in the magnetic-field PSD. Better control of intensity noise within the light-shift beams should allow dramatically improved scalar measurements and correspondingly better sensitivity to the field angle. The scalar sensitivity of the magnetometer would be 12 fT/[$\sqrt{\textrm{Hz}}$ ]{}if the polarimeter and amplifiers operate at the photon shot-noise limit. By eliminating these sources of technical noise, it should be possible to reach a sensitivity of in the measurement of the magnetic-field direction. ![\[PSDplot\] Power-spectral-density plot of the scalar field measurement with the $LS_z$ beams turned off (blue) and turned on at a constant power of 1 mW (red). For these data, $\theta = 0$ and the light-shift beam power was not actively controlled. The black trace is the predicted noise floor the scalar field measurement taken from a (separate) recording of the light-shift beam power, from which a PSD was derived and the effective magnetic field calculated using the observed light-shift coefficients $\alpha_{y}$ and $\alpha_{z}$.](PSD_v02.pdf) Expanding the vector measurement to three dimensions will simply require adding another light-shift beam in the $\hat{x}$ direction. The bandwidth of the vector measurement is presently limited by the narrow magnetic-resonance line, but this can be expanded by power-broadening the resonance with the probe beam or heating the cell to increase the Cs density and spin-exchange-broadened linewidth. Either technique would allow more rapid measurement of the vector field components with little if any loss in sensitivity. As discussed in the Supplemental Material, the uncertainty in the measured angle $\theta$ has no intrinsic dependence on the magnitude of the ambient field $B_0$. Consequently, this technique should be applicable for vector magnetometry in geophysical fields with comparable precision, provided that a similar scalar sensitivity can be achieved. In summary, we have demonstrated a method for measuring the magnitude and direction of a magnetic field through all-optical interrogation of an atomic sample. This technique offers advantages over other methods (such as EIT vector magnetometry) because it relies on measuring changes in the magnetic-resonance frequency, rather than resonance amplitudes which can be affected by many experimental factors. Further optimization of the apparatus will allow for a compact, magnetically inert vector magnetometer well-suited for precision physics experiments or geophysical field measurement. We thank Douglas Beck, Michael Sturm, David Wurm, and Peter Fierlinger for their helpful input, as well as Mikhail Balabas for preparation of the antirelaxation-coated cesium cell. We also thank Arne Wickenbrock for contributions to the measurement. This work was funded in part by a University of California UC Discovery Proof of Concept grant (award 197073) and NASA SBIR contract NNX13CG20P, and is supported by National Science Foundation Grant No. PHY-1068875. B. Patton is supported by DFG Priority Program SPP1491, ‘Precision measurements with cold and ultracold neutrons’. [^1]: See the Supplemental Information for a discussion of the difference between the shifts induced by a light-shift beam and a magnetic field. [^2]: Unlike conventional polarization-maintaining fiber, the HB830Z only transmits light which is linearly polarized along one of the axes of the anisotropic fiber core; the other polarization experiences large attenuation.

--- abstract: 'We report on observations of four early-type galaxies performed with the Rutgers Fabry-Perot in order to search for Planetary Nebulae (PNe) in these systems. The aim is to use the PNe as kinematic tracers of the galaxy potential. We describe our data reduction and analysis procedure and show that the proper calibration of our detection statistic is crucial in getting down to our limiting magnitude of $m_{5007} = 26.1$. In the case of the two Leo galaxies we find moderately sized samples: 54 PNe in NGC 3379 and 50 PNe in NGC 3384; NGC 4636 (2 PNe) and NGC 1549 (6 PNe) are included for completeness. We present our samples in tabular form, as well as the spectrum for each PN. We constructed simple non-parametric spherical mass models for NGC 3379 using a Monte Carlo Markov Chain method to explore the space of likely mass models. We find a remarkably constant mass-to-light ratio within five half-light radii with an overall $B$ band mass-to-light ratio $\sim 5$. A simple mass-to-light estimate for NGC 3384 yields $\Upsilon_B \sim 11$, but is likely an overestimate.' author: - 'Arend P. N. Sluis and T.B. Williams' title: 'Uncovering Planetary Nebulae in Early Type Galaxies using the Rutgers Fabry-Perot' --- Introduction ============ The existence of Dark Matter (DM) has been postulated to resolve the conflict between dynamical estimates of galaxy and galaxy cluster masses, and estimates of their luminous or baryon mass. DM, as a source of gravitational potential, is an essential ingredient in the current $\Lambda$CDM concordance cosmology of the Universe as derived from observations of the Cosmic Microwave Background [@sperg+03] and high redshift Supernova searches [@ries+98; @perl+99]. Little consensus exists, however, on its properties at smaller galactic scales with observations [e.g. @debl+03] and numerical simulations [e.g., @diem+04] currently being at odds. Indeed, the lack of a consistent understanding of DM has led some to question the necessity of the DM postulate [@sk01]. An important diagnostic in the study of DM is its distribution throughout a galaxy. The strongest constraints come from observations of spiral galaxies, where neutral hydrogen gas can be used to detect the influence of DM beyond the stellar disk [@sv01]. The case for DM in the more massive elliptical galaxies is less concrete. Unlike the simpler geometry of disk galaxies, the intrinsic three-dimensional shape of an elliptical cannot be uniquely determined from its projected shape. Similarly, the orbital structure of ellipticals is dominated by random motion, whereas the stars and gas in spiral galaxies move on nearly circular orbits. These fundamental limitations are exacerbated by practical ones. Because the galaxy surface brightness drops steeply as a function of radius, measurements of the stellar kinematics in ellipticals do not extend far into the galaxy halo, and require substantial observational effort. Furthermore, their gas content is low, and surveys of tracer populations such as globular clusters and planetary nebulae have until recently generated only modestly sized samples, the one exception being Cen A [@hea95]. The picture is only discouraging in contrast to the observations of disk galaxies, however, and—taken in their own right—the recent data on elliptical galaxies have greatly improved the understanding of these systems. Measurements of the stellar kinematics in ellipticals now include higher order moments of the line-of-sight velocity distribution (LOSVD) as a matter of course [e.g., @caroea95]. Knowledge of the actual LOSVD, as opposed to only the first and second moments, can lift the degeneracy between the mass distribution and the velocity structure [@ger93; @mer93]. Along with more sophisticated modeling techniques, the observations, sometimes going out as far as two effective (half-light) radii $(R_e)$, have allowed for a better constrained dynamical picture of a growing number of ellipticals [for references, see @kronea00]. Most intriguingly, the circular velocity curves in the @kronea00 sample are quite flat, indicative of the presence of a DM halo. Despite this, no clear trends have emerged about the DM distribution in ellipticals, with some galaxies showing evidence for DM within $2R_e$ (mass-to-light ratio in the $B$ band $\Upsilon_B \sim 20 \textrm{-}30$), and others showing little or no evidence [@kronea00]. Gas is scarce in early-type galaxies, but not rare. Polar ring galaxies, a rare breed of S0s with a ring of neutral hydrogen perpendicular to the galaxy’s symmetry plane, have been used to constrain the mass and shape of DM halos, but much depends on the assumption of self-gravity of the ring [@spar02]. The most consistent evidence for a DM halo around ellipticals comes from studies of their hot gas, which radiates in the X-ray around 1 keV. Using a sample of giant ($L>L_*$), ellipticals @lw99 conclude that DM does not dominate within an effective radius, but that the DM fraction within $6R_e$ is in the range of 39-85%. Because early type galaxies with $L<L_*$ have faint X-ray spectra which tend to be dominated by the contribution from the stellar component, e.g. X-ray binaries, this finding cannot be extended to fainter ellipticals. Imaging by X-ray satellites *Chandra* and *XMM-Newton*, which shows bubbles and other structures, has complicated the picture presented by @lw99, which assumes—as is typical—sphericity and (isothermal) hydrostatic equilibrium. For a recent review of the topic of hot gas in ellipticals, see @mb03. Gravitational lensing of light by elliptical galaxies is an additional probe of DM halos, especially at large radii. @wilsea01 have used the distortion (shear) of faint background galaxies by a foreground galaxy to measure the mass-to-light ratio of early-type galaxies. Adding the shear signal from many hundreds of early type lens galaxies in their fields they derive $\Upsilon_B \sim 121\pm28$ within $\sim 100{\ensuremath{ \mathrm{\ kpc} }}$. Strong lensing, where the lens galaxy produces multiple—typically two or four—images of a background source, yields more specific information, but the sample of lens systems is limited $(<50)$. @keet01 deduces that DM cannot account for more than 33% of the mass within one $R_e$, or 40% within $2R_e$, consistent with the findings of @lw99. Finally, globular clusters (GCs), planetary nebulae (PNe), and satellite galaxies provide discrete tracers of a galaxy’s kinematics and, with suitable assumptions, of the dynamics [e.g., @rk01]. GCs are not as numerous as PNe, and the additional difficulty of absorption line spectroscopy on faint objects has limited the use of GCs as kinematic tracers [e.g., @bridea03; @cotea03]. The analysis is further complicated by the fact that GCs do not generally trace the light distribution of their host galaxy. In this paper we focus on the Planetary Nebulae (PNe) that can be used to investigate the dynamics of early-type galaxies. This approach stretches back to @nf86 and their study of M32. PNe are part of the brief phase in the life of an intermediate mass (0.8–8 $M_\odot$) star when it evolves from a red giant to a white dwarf; this makes PNe a good tracer of old stellar populations (but see @pengea05 for a discussion on “young” PNe). During this transition the star ejects most of its mass, leaving a hot stellar remnant which rapidly ($\sim10^4$ years) cools to a white dwarf. The PN proper is an expanding shell of dense gas being photo-ionized by the central stellar remnant. PNe are bright objects, emitting on the order of a few hundred $L_\odot$ in a few emission lines, most notably \[\] $\lambda5007$ [up to 15% of the total luminosity; @djv92]. This not only enables us to detect them at extragalactic distances (out to 10–15 Mpc on 4-m class telescopes), but also to measure their velocity along the line of sight. We have collected PNe samples for a set of early-type galaxies with the Rutgers Fabry-Pérot (RFP), which can be thought of as a narrow band ($\sim 2~\textrm{\AA}$) filter with a tunable central wavelength. Scanning around the (redshifted) \[\] $\lambda5007$ emission line, the RFP generates a three-dimensional data cube where the galaxy background has been strongly reduced and monochromatic point sources, such as PNe, stand out against the background. Detection and measurement of the radial velocity are all achieved in one observing run. Other techniques that look for extragalactic PNe include the Planetary Nebula Spectrograph [@dea02], an instrument using slitless spectroscopy, and on/off band photometry in conjunction with multi-object spectroscopy [@pengea04; @menea01]. The paper is outlined as follows: §\[s:observations\] summarizes the observing strategy and conditions; §\[s:reduction\] and §\[s:calibration\] discuss the data reduction and calibration; §\[s:extraction\] to §\[s:compcont\] show the process of our PN candidate selection, including accounting for completeness and interlopers; and we present the samples in full in §\[s:results\]. Considering the recent interest in NGC 3379 [@romea03] we present our mass models for this galaxy in §\[s:massmodels\]. A summary of our findings is given in §\[s:wrapup\]. Observations {#s:observations} ============ We observed the four galaxies in our sample with the RFP over the course of two runs at the CTIO 4 m Blanco telescope (f/8 Cassegrain focus). Details of the setup are given in Table \[t:setup\]. The RFP has a circular field of view with a diameter of $2\farcm8$, and a spectral response function that is well approximated by a Voigt profile with a $\mathrm{FWHM} \simeq 2$ Å, equivalent to $120\ \mathrm{km}\ \mathrm{s}^{-1}$ at 5000 Å. There is a wavelength gradient between the center and the edge of an image of 4.6 Å. We typically scanned a wavelength span of 25 Å around the appropriately redshifted 5007 Å \[\] emission line in steps of approximately 1 Å. At this wavelength the free spectral range of the RFP is 48 Å and we used one of two blocking filters with a FWHM of 44 Å to ensure that only one order was transmitted. We used Tek $1024\times1024$ pixel CCDs, with $0\farcs35$ pixels (binned to $0\farcs70$ on the 1994 run), but only read out the portion of the CCD illuminated by the RFP. At the start of each run we observed a set of emission lines around 5000 Å using a calibration lamp. The resulting “ring” images allowed us to establish the relationship between the gap setting $z$ and the transmitted wavelength $\lambda_c$ at the center of the image; to determine the free spectral range of the etalon; and to measure the shape of the spectral response function. To monitor changes in the wavelength calibration we took further images of the lamp throughout each night. We discuss the wavelength calibration in more detail in §\[s:wlcal\]. For each run we obtained one series of dome flats at settings spaced $\sim1\ \textrm{\AA}$ apart. Twilight flats are not advisable, because of the spectral structure of the sky at the resolution of the RFP. Additionally we observed spectrophotometric standard stars on each run: six 90 s exposures of LTT 3218 in 1994 and six 300 s exposures of LTT 2415 in 1995 [@sb83]. The Leo galaxies, NGC 3379 and NGC 3384, were observed with multiple pointings, typically one pointing per night. NGC 1549 and NGC 4636, on the other hand, had only one pointing each, but were observed throughout the course of a run to get the full wavelength coverage. NGC 1549 was observed at the start of each night, NGC 4636 towards the end. The exposure time for each image was 900 s. To avoid systematic effects in the photometry consecutive exposures were not sequential in wavelength and were dithered by a few pixels. The appropriate blocking filter was determined by the central wavelength in an exposure being smaller or larger than the switching wavelength $\lambda_s$ (see Table \[t:setup\]). Exposures were repeated if the seeing or the photometric conditions were particularly poor. The observations are summarized in Table \[t:fields\]. The seeing was poor during both runs: $\sim1.5\arcsec$ in 1995 and in the range 1.7–2.2 in 1994. The latter is partly due to difficulties in focusing the telescope and problems with the auto-guider. Conditions were mostly photometric in 1994, with the exception of the last half of the first night, which only affected the observations of NGC 4636. For the 1995 run we were less fortunate. The first two nights are fully photometric, but not the two following nights. As a result only half of our fields was observed completely under photometric conditions. As we will show below, this affected the quality of our PN photometry, but not of our radial velocity measurements. Data Reduction {#s:reduction} ============== We used IRAF[^1] for most of our data reduction. Each image was overscan corrected, trimmed and bias subtracted. Since the dark current contribution was negligible for the length of our exposures, no dark frames were taken. Each image was flatfielded using the flatfield image with a central wavelength closest to that of the image. To aid the cosmic ray removal process and the photometry (see §\[s:flcal\]) we aligned the images for each pointing (a “stack”) using the few stars that were visible in each field (see Table \[t:fields\]). In all cases a simple shift over a few pixels proved to be adequate. Since the PSF was well resolved, we shifted the images by integer pixels to avoid interpolation effects. Cosmic rays were tagged, not removed, by combining all the images in a stack using the IRAF task `imcombine` with the `crreject` rejection algorithm, creating a mask for each image which covered the comic ray events. To be conservative we added a rim of masked pixels around each event. The padded masks covered on average 8% of the field of view. The final step in preparing the images in a stack was the removal of the background light. In our case the background is mainly light from the galaxy and its ghost, with the sky contributing an almost negligible amount. Ghosts are the result of the reflections between the CCD and the etalon, which create a faint in-focus replica of the original object, mirror reflected about the optical axis. We removed the background to reduce gradients which would bias the photometry, allowing us to detect PNe closer to the galaxy. We wrote a Fortran program which estimated the background with a ring filter [@sec95], where we replaced the median with a more robust estimator of the mean [@hmt83]. A ring filter removes objects at scales smaller than the diameter of the ring and we chose our diameter to be about twice the seeing radius. The shifted, masked and background subtracted stacks formed the basis for our further analysis. Calibration {#s:calibration} =========== The calibration of the RFP stacks consists of three parts: (i) image registration, i.e. assigning celestial coordinates to each pixel position; (ii) spectral calibration; (iii) flux normalization, i.e. accounting for variations in the observing conditions. Image Registration {#s:imgreg} ------------------ In each field we had only a few stars to register the images (see Table \[t:fields\]). These astrometric reference stars generally did not have published positions, because of their proximity to the galaxy being observed, and we had to determine their coordinates from observations with a larger field of view. For NGC 3379 and NGC 3384 we had deep broad band ($V$-like) images with $0\farcs4$ pixels; for NGC 1549 and NGC 4636 we used images from the Digitized Sky Survey[^2] (DSS) with $1\farcs7$ pixels. Using the USNO-A2.0 catalog [@mea98] we were able to identify 80 or more stars in each of these images and consequently measured the celestial coordinates of the RFP reference stars. The systematic uncertainty in a single RFP coordinate, as estimated from the variance in the reference star positions, is $0\farcs6$ for NGC 1549 and NGC 4636, compared to about $0\farcs1$ for the other two galaxies. In the case of the Leo I galaxies (NGC 3379 and NGC 3384) we noticed a systematic $\sim 1''$ offset between the @cjf89 coordinates and our own when we compared our PNe sample with their lists (see §\[s:samp3379\]). We found that the offset can be attributed completely to the difference in astrometric reference star coordinates. Since our project does not require absolute astrometry we did not investigate the matter further. Spectral Calibration {#s:wlcal} -------------------- The transmitted wavelength $\lambda$ at a position ${\ensuremath{ \mathbf{x} }}=(x,y)$ on an RFP image taken at a gap setting $z$ is given by $$\lambda(x,y,z) = (a+bz) (1+ |{\ensuremath{ \mathbf{x} }}-{\ensuremath{ \mathbf{x} }}_c|^2 / f^2)^{-1/2}, \label{e:wlgrad}$$ where $a$ and $b$ are parameters relating $z$ to the central wavelength of an RFP image, ${\ensuremath{ \mathbf{x} }}_c$ is the position of the optical axis of the RFP and $f$ is the focal length of the camera lens of the RFP. To calibrate this relation we observe several emission lines from a calibration lamp at a range of $z$ settings at the beginning of a run. During a run we take additional calibration images (“night rings”), eight to ten per night, to ensure the stability of our calibration. Over the course of our runs $b$ and $f$ were stable, consistent with our experience from other runs. The value for ${\ensuremath{ \mathbf{x} }}_c$, however, fluctuates as a result of flexure in the spectrograph; likewise, the wavelength zero-point $a$ shifts due to drift in the control electronics. The overall drift in $a$ was a little larger than 1 Å over a run; the overall shift in ${\ensuremath{ \mathbf{x} }}_c$ never larger than 6 pixels during a night. We interpolated the values for $a$ and ${\ensuremath{ \mathbf{x} }}_c$ for each image from the values obtained from the night rings. The uncertainty in $\lambda$ due to the calibration is 0.05 Å ($3\ \textrm{km}\ \textrm{s}^{-1}$ at 5007 Å), based on the scatter in our calibration fits with the dominant source of uncertainty being $f$. The initial calibration run also allows us to calibrate the spectral response function (SRF) of the RFP, i.e. the line shape of a monochromatic source. Although for a perfect etalon the SRF is an Airy function, experience shows that the SRF is better fitted by a Voigt function, the convolution of a Gaussian with a Lorentzian [@vhr47]. The shape is determined by two parameters: the Gaussian width $\Delta\lambda_G$ and the Lorentzian width $\Delta\lambda_L$ (see Table \[t:setup\]). Note that the shape parameters are significantly different between the two runs, but that the full width at half maximum (FWHM) of the profile is practically the same at $\sim2$ Å. We intentionally did not want to resolve the intrinsic structure of the PN line profile (see Section \[s:modspectra\]), since it would reduce the depth of our survey and would complicate our study unnecessarily. Flux Normalization {#s:flcal} ------------------ In each field of view we use a reference object to provide us with a fiducial flux for every image in a stack, to calibrate the extinction due to light cirrus. Considering the narrow wavelength range we are observing it is reasonable to assume a flat spectrum for each reference object. We can then calculate normalization factors for each image in a straightforward manner. Only the spectrum of the reference star in the West field of NGC 3379 showed an indication of absorption lines. For NGC 3379 we therefore decided to use the galaxy flux within a $10\arcsec$ aperture as a reference source. The galaxy spectrum has a slight curvature at the blue edge, which we modeled with a smoothing spline. The uncertainty in each normalization factor is 3-5%. We were unable to get an independent determination of the atmospheric extinction during our runs, because we observed our spectrophotometric standard stars only once a night and all at approximately the same airmass. Instead, we adopted a value for the extinction of 0.19 magnitude per airmass as given for CTIO [@hea92]; on photometric nights the flux normalization factors were found to be consistent with assumed extinction. Finally, using the spectrophotometry for the standard stars, we converted our instrumental fluxes to physical units. Comparison with previously published PN magnitudes shows no systematic effects (see below). Extracting Spectra {#s:extraction} ================== We extract spectra for every independent point in the field of view of a stack and for each spectrum decide whether or not an emission line is present. A more targeted approach, where we would look for emission line point sources in each image of a stack using, for instance, DAOPHOT [@pbs87], requires extensive fine tuning due to the low signal-to-noise ratio of the objects and still produces numerous spurious detections. (@tmw95 used this approach to look for PNe around NGC 3384.) Our method is not computationally expensive and has no bias due to a search algorithm. The fluxes for each spectrum were extracted using the `apphot` aperture photometry package in IRAF. We performed a set of Monte Carlo simulations of our instrumental setup (see Table \[t:setup\]) to compare the results of PSF and aperture photometry. PSF photometry, assuming perfect knowledge of the PSF, produced an appreciably larger scatter in the derived fluxes than aperture photometry did. Because we have only one star in each field bright enough to determine a PSF, there was the additional concern of the effects of a PSF mismatch. Both considerations lead us to use aperture photometry. Each field of view was sampled on a triangular grid, the most isotropic choice. The spacing between sample points was chosen to Nyquist sample the image, i.e. sample points are separated by half the smallest seeing FWHM in a stack, to guarantee no information was lost. Following @nay98 we chose our aperture radius to be slightly larger than two thirds of the seeing FWHM in order optimize the signal to noise ratio within the aperture. In this case the photometry apertures overlap slightly more than 50% in area. If the aperture contained masked pixels, e.g. a cosmic ray, the data point was removed from the spectrum. In cases where we measured a negative flux, we estimated the uncertainty from the uncertainties in the positive fluxes near the continuum level. No aperture corrections are needed, since we photometer a PN candidate and a reference star in the same way. The final data product, then, is a set of $M$ spectra where each spectrum is a list of the form $\{\lambda_i, f_i, \sigma_{f_i}, e_i\}_{i=1}^N$ with $N$ the number of images in a stack, $\lambda_i$ the wavelength, $f_i$ the flux, $\sigma_{f_i}$ the uncertainty in the flux and $e_i$ a possible photometric error flag. In the following section we discuss how we determine which spectra indicate the presence of an emission line. Candidate Selection {#s:selection} =================== To establish the presence of a PN candidate in a spectrum we fit a flat continuum model and an emission line model to each spectrum, measuring the difference in the goodness-of-fit using a statistic $S$. After calibrating the distribution of $S$ with Monte Carlo simulations we choose a significance level $\alpha$ and select all spectra with a value $S>S(\alpha)$. Finally, we visually inspect each stack at the candidate positions to verify the selection and avoid contamination from CRs and image artifacts. The details of the procedure are described below. Modeling the spectra {#s:modspectra} -------------------- We expect most extracted spectra to have a flat, practically zero, flux distribution, since we removed the background in each image. In addition a flat model spectrum should encompass the spectra from foreground stars in our field. Hence, our null hypothesis is a simple constant flux model. Modeling spectra with a linear function $f_i = a\lambda_i+b$ yielded no significant improvements over the constant model, because of the flux uncertainties. The alternative hypothesis is that a spectrum contains an emission line. Such a model needs to take into account the intrinsic structure of the \[\] line, since PNe, as observed in our own Galaxy, often have doubly peaked emission lines [@pot84]. The intrinsic width of each of these peaks is small ($\sim10\ \textrm{km}\ \textrm{s}^{-1}$), and the nebular expansion velocity $v_{\ensuremath{ \mathrm{exp} }}$, defined as half the separation between the two peaks, has a distribution with a mode around $10\ \textrm{km}\ \textrm{s}^{-1}$, but a mean of $25\ {\ensuremath{ \mathrm{km} }}\ {\ensuremath{ \mathrm{s} }}^{-1}$ due to the large tail at higher velocities [@phi02]. Based on the latter we calculate that the broadening of our *observed* line profile (see below) due to line structure is $\lesssim9\%$, negligible considering the wavelength sampling of our spectra (see Table \[t:setup\]). We will assume from now on that PNe can be treated monochromatic point sources. Consequently, the line profile is determined by the effective filter transmission $T$ and the RFP spectral response $R_s$. We separate the two because they are measured separately from each other. The measurement of the latter is described in section \[s:wlcal\]; for the former we use the filter curves provided by CTIO. We assumed that temperature effects on $T$ are unimportant. The f/7.5 beam used for our observations is slow enough to have a negligible effect on the transmission properties of our filters [@cjf89]. The effective filter transmission $T$ changes the line shape from a simple Voigt profile in two ways. The first modification is a discontinuous jump in the emission line profile, which is the result of changing from the “blue” filter $T_b$ to “red” filter $T_r$ whenever the central wavelength of an image is larger than the switching wavelength $\lambda_s$. The total flux of a PN observed at $\lambda_{\ensuremath{ \mathrm{pn} }}$ will differ between the two filters, since in general $T_b(\lambda_{\ensuremath{ \mathrm{pn} }}) \neq T_r(\lambda_{\ensuremath{ \mathrm{pn} }})$. Hence, a spectrum will show a jump at $\lambda_{\ensuremath{ \mathrm{s} }}$. The jump is appreciable when $\lambda_{\ensuremath{ \mathrm{pn} }}$ close to $\lambda_{\ensuremath{ \mathrm{s} }}$, but is otherwise negligible. The second modification stems from the fact that the flatfield images are based on exposures of continuum sources. A flatfield image taken on the red edge of a filter includes light from the neighboring order of the RFP on the blue edge. As a result we overestimate the sensitivity of the detector when applying the flatfield to an emission line object, which has flux in one order only. Ignoring this effect would lead to a wavelength dependent bias in the measured total flux of a PN. We can adjust for this effect by calculating a correction factor $\Gamma$, the integral over the filter transmittance and the SRF. Figure \[f:profiles\] illustrates how the two effects modify the line profile. To account for the above two effects we characterize the spectrum of a PN candidate extracted from a stack of $N$ images in the following way: $$\label{e:emmodel} M(f_{\ensuremath{ \mathrm{pn} }}, \lambda_{\ensuremath{ \mathrm{pn} }}, c_{\ensuremath{ \mathrm{pn} }}|{\ensuremath{ \mathbf{x} }}, z_i) = f_{\ensuremath{ \mathrm{pn} }} { R_s(\lambda_{\ensuremath{ \mathrm{pn} }}, {\ensuremath{ \mathbf{x} }},z_i) T(\lambda_{\ensuremath{ \mathrm{pn} }},z_i) \over \Gamma({\ensuremath{ \mathbf{x} }},z_i) } + c_{\ensuremath{ \mathrm{pn} }}, i=1,\cdots,N,$$ where $M$ is the flatfielded and normalized flux measured at position ${\ensuremath{ \mathbf{x} }}$ and RFP setting $z_i$, $f_{\ensuremath{ \mathrm{pn} }}$ is the total flux received from the PN candidate, $R_s$ is the SRF, $T(\lambda_{\ensuremath{ \mathrm{pn} }})$ is the appropriate filter transmission, and $c_{\ensuremath{ \mathrm{pn} }}$ is the continuum level. Since the spectra we extract from the RFP stacks are critically sampled, we cannot constrain the shape of the line profile in addition to fitting a peak wavelength, total intensity and background flux. Hence, we fixed $\Delta\lambda_G$ and $\Delta\lambda_L$ to the values determined from the calibration run in our further data analysis. The factor $\Gamma({\ensuremath{ \mathbf{x} }},z_i)$ corrects for the luminosity bias; it depends implicitly on $R_s$ and $T$. Note that $c_{\ensuremath{ \mathrm{pn} }}$ can potentially give us a handle on the contamination of our sample (see section \[s:cont\]). We fit the two models to each spectrum by minimizing a goodness-of-fit statistic $s^2(f_{\ensuremath{ \mathrm{pn} }},\lambda_{\ensuremath{ \mathrm{pn} }}, c_{\ensuremath{ \mathrm{pn} }})$ defined by $$s^2 = \sum_{i=1}^N {[f({\ensuremath{ \mathbf{x} }}, z_i) - M(f_{\ensuremath{ \mathrm{pn} }}, \lambda_{\ensuremath{ \mathrm{pn} }}, c_{\ensuremath{ \mathrm{pn} }}|{\ensuremath{ \mathbf{x} }}, z_i)]^2 \over \sigma_{f_i}^2 }.$$ We set $f_{\ensuremath{ \mathrm{pn} }} \equiv 0$ for the continuum model and $f_{\ensuremath{ \mathrm{pn} }} \geqslant 0$ for the emission line model. The minimum $s^2$ value was found using the E04UNF routine from the NAG numerical library, which is designed to solve nonlinear least-squares programming problems in the presence of constraints on the parameters. Since the uncertainties in our data points are not gaussian, the uncertainties in our best fit parameter values do not necessarily correspond to a $1\sigma$ error. We used simulated observations (see Section \[s:finsel\]) to verify the plausibility of uncertainties found in the fitting procedure. Choosing the correct model -------------------------- The continuum model $M_c$ and the emission line model $M_l$ are nested models, i.e. the parameters in $M_c$ form a subset of the parameters in $M_l$ (the additional parameters of $M_l$ are fixed to some default value). The likelihood-ratio (LR) test or the $F$-test are conventionally used to decide between the null hypothesis $M_c$ and the alternative hypothesis $M_l$ [@eea71; @bea97]. Each of these tests uses a test statistic $S$ which is some simple function of $\hat s^2_c$ and $\hat s^2_l$, the best fit values of $s^2$ for $M_c$ and $M_l$, respectively. Under certain regularity conditions these test statistics have analytically known reference distributions (e.g. @pea02), usually in the limit of large $N$, which allow a level of significance to be determined for the observed value of $S$. In our case two of the regularity conditions are not met. First, the tests assume that the data are independent, identically distributed random variables. As a result of the flatfielding the data points in our spectra are not identically distributed. Second, the default values for the additional parameters in $M_l$ cannot be on the boundary of parameter space. The null hypothesis assumes that $f_{\ensuremath{ \mathrm{pn} }} \equiv 0$, clearly on the boundary of our parameter space. This is not an academic point: @mea96 show that when this condition is violated the actual reference distribution is markedly different from the nominal reference distribution (see their Figure 3). The problem of including boundary values in a parameter space has no standard analytical solution. @pea02 present a more comprehensive introduction. We decided to use the LR test statistic $S = \hat s^2_c - \hat s^2_l$, but to calibrate its reference distribution by way of Monte Carlo simulations instead of using the nominal reference distribution. (The choice of the LR test over the $F$-test is motivated in the next section.) The reference distribution $p(S|M_c) {\ensuremath{ \mathrm{d} }} S$ specifies the probability that the test statistic has an observed value $S$ under the condition that the null hypothesis is true. To calibrate $p(S|M_c)$ we simulated stacks of images that contain no emission line objects, preserve the noise characteristics of the original observations, and are reduced in the same way as the original observations. Every field of view was calibrated separately. As an example, we show the result of these simulations for the East field of NGC 3379 in Figure \[f:falsepos\_hist\]. The nominal reference distribution of $S$ in our case is a $\chi_2^2$ distribution [@eea71]. In this particular case the actual reference distribution can be approximated by a $\chi^2_\nu$ distribution, as shown by the best-fitting $\chi_{4.6}^2$ curve, but this is not generally true. A proper calibration of the test statistic is clearly essential in order to avoid a plethora of false positives. For each field of view we choose the value of $S_c(\alpha)$ that corresponds to a significance level $\alpha = 0.01$ and select all spectra with $S>S_c(\alpha)$. Our experience shows that at this significance level we can observe a PN candidate in at least two frames. Final selection {#s:finsel} --------------- Since the photometric apertures overlap, PNe candidates in each field of view tend to cluster in contiguous groups of 3-6 spectra sharing a similar central wavelength $\lambda_{\ensuremath{ \mathrm{pn} }}$. Instead of developing an algorithm to make a final selection of candidates from these clusters, we performed the final selection by hand. First we remove any obviously false identifications due to CRs and image artifacts, e.g. the ghost image of a bright star. Next we choose the brightest spectrum in a cluster as the candidate spectrum and visually inspect its position in every image of the stack. Once we are convinced we have a bona fide PNe candidate, we determine the exact position of the candidate and redo the photometry of the candidate to provide the final parameters for the object. We have two estimates for the uncertainties in $f_{\ensuremath{ \mathrm{pn} }}$, $\lambda_{\ensuremath{ \mathrm{pn} }}$, and $c_{\ensuremath{ \mathrm{pn} }}$: the variances from the formal best-fit covariance matrix [@eea71], and the empirical estimates from the artificial PNe simulations (see section \[s:compcont\]). For PN candidates with $S>S_c(\alpha)$ we found that the best-fit parameters are only slightly correlated and that the two uncertainty estimates are consistent with each other. Hence, we used the best-fit variances as our measure of the uncertainties in the magnitude and line-of-sight velocity. We note that $\lambda_{\ensuremath{ \mathrm{pn} }}$ is in our experience significantly better constrained than $f_{\ensuremath{ \mathrm{pn} }}$; the continuum level $c_{\ensuremath{ \mathrm{pn} }}$ is always consistent with zero, as expected. The false-negative simulations also allow us to estimate the random uncertainty in a PN position at $0\farcs3$, which does not include the systematic uncertainty discussed in section \[s:imgreg\]. In the same way that one can determine an aperture radius that optimizes the signal-to-noise ratio $Q$ within some photometric aperture [@nay98], we determine a wavelength window centered on the peak wavelength which optimizes $Q$ in the spectral domain. Numerical experiments show that a window width of 4/3 the spectral FWHM is optimal. To calculate the signal-to-noise ratio we simply added the signal from all frames within 1.4 Å of $\lambda_{\ensuremath{ \mathrm{pn} }}$ and divided this by the sum in quadrature of the uncertainties (all quantities were converted to photons). Taking into account the seeing, the background flux, the sampling and the photometric conditions of our observations, we find that our empirical values are in agreement with the theoretical expression of @dea02 [eq. 6]. Completeness and Contamination {#s:compcont} ============================== To characterize our PN samples fully we need to address two issues: completeness and contamination. The former describes the probability that our experiment, i.e. the whole of observations, data reduction and analysis, would not detect a PN with a flux $f_{\ensuremath{ \mathrm{pn} }}$. The latter specifies the extent to which candidates in our samples are not actually PNe. In the context of our experiment the completeness is closely related to the (statistical) power of the test statistic and we show that the LRT is preferred over the $F$-test because it gives us a more complete sample. It is more difficult to quantify the contamination, mainly because the distribution of possible contaminating sources is still poorly understood. Based on our estimates, discussed in more detail below, contamination is not a serious concern in our samples. Completeness and power ---------------------- For statisticians the usefulness or *power* of a test lies in its ability to reject the null hypothesis ($f_{\ensuremath{ \mathrm{pn} }}=0$) when in fact the alternative hypothesis ($f_{\ensuremath{ \mathrm{pn} }}>0)$ is true.[^3] The power function $\beta(f_{\ensuremath{ \mathrm{pn} }})$ is quantified by $$\beta(f_{\ensuremath{ \mathrm{pn} }}) = p(S<S_c(\alpha)|f_{\ensuremath{ \mathrm{pn} }}),$$ i.e. the probability that, given a level of significance $\alpha$, we reject the true (alternative) hypothesis [@eea71]. The actual form of $\beta(f_{\ensuremath{ \mathrm{pn} }})$ is contingent upon our choice of $\alpha$, but is generally monotonically increasing. When multiple tests are available, statisticians look for the test with a minimum $\beta(f_{\ensuremath{ \mathrm{pn} }})$, the one least likely to lead to an invalid conclusion. Astronomers, on the other hand, think in terms of completeness: we want to maximize our ability to identify objects of a given type. Since our experiments are limited by observing conditions (exposure time, seeing, etc.) we rarely obtain a complete sample, instead quantifying the level of completeness in terms of a limiting magnitude. Typically, the depth of a survey is defined by the flux level where the probability of detecting an object is 50%: $\beta(f_{\ensuremath{ \mathrm{lim} }}) = 0.5$ [@har90]. Hence, the more powerful test is the test where $f_{\ensuremath{ \mathrm{lim} }}$ is the smallest and so allows the survey to go deepest. We considered two statistics commonly used in model comparisons: the LR statistic $$S_{\ensuremath{ \mathrm{LR} }} \equiv \hat s^2_c - \hat s^2_l,$$ and the $F$-statistic $$S_F \equiv { s^2_c - s^2_l \over s^2_l} \left/ {N-\Delta P \over P_l} \right.,$$ where $N$ is the number of data points in a spectrum, $P_l=3$ the number of parameters in $M_l$, and $\Delta P=2$ the difference in the number of parameters between $M_c$ and $M_l$. A third statistic that is sometimes used, the Goodness-Of-Fit statistic $S_{\ensuremath{ \mathrm{GOF} }} \equiv \hat s^2_c$, was not considered, since it does not take into account an alternative hypothesis and consequently is not a powerful test. Although $S_{\ensuremath{ \mathrm{LR} }}$ and $S_F$ yield about the same significance for a spectral feature [@bea97], the LR test is generally more powerful than the $F$-test, the latter being the more appropriate test when the uncertainties in the data are unknown [@fea99]. In order to ascertain the limiting magnitude of our observations we simulated stacks with artificial PNe and analyzed them as we would real observations. By comparing the analyses using $S_{\ensuremath{ \mathrm{LR} }}$ and $S_F$ we found that the former is the more powerful of the two tests, as expected. An illustration of this point is given in Figure \[f:3379maglim\]: in the case of our NGC 3379 observations $S_{\ensuremath{ \mathrm{LR} }}$ allows us to go $\sim0.3$ mag deeper than $S_F$. The limiting magnitudes for each field of view are shown in Table \[t:fields\]. Sources of contamination {#s:cont} ------------------------ Each PN detection is based on the assumption that the detected emission line is the \[\] 5007 line. A simple way to check if a candidate is actually a PN, short of taking a complete spectrum, is to measure the flux in the \[\] $\lambda4959$ emission line, since the line ratio $I(5007)/I(4959)=3$ is fixed [@fea00]. Our wavelength range, however, does not include the redshifted $\lambda4959$ line, and we have to rely on statistics to appraise the possibility of an interloper in our PN samples. The search for high-redshift emission line galaxies in order to determine their star-formation history [@hcm98; @sea00], as well as searches targeted to quantify the contamination of intracluster PN surveys [@kea00; @cea02] have improved our understanding of possible contaminants. The most likely candidates that—to our knowledge—could contaminate our samples are $\textrm{Ly}\alpha$ sources at $z = 3.13$, and \[\] $\lambda3727$ at $z = 0.35$ [@kea00]. In the absence of any other information, the most likely interloper is a $\textrm{Ly}\alpha$ source, since these tend to have significantly stronger lines than the \[\] sources [@cea02]. In particular, at the depth of our observations ($m_{5007} < 26.6$ or $f_{5007} > 0.7\times10^{-16} {\ensuremath{ \mathrm{\ erg\ cm^{-2}\ s^{-1}} }}$) the probability of the latter contaminating our samples is negligible. For the former @cea02 determine a surface density of $\sim 3500\ {\ensuremath{ \mathrm{deg^{-2}} }}$ per unit redshift for $\textrm{Ly}\alpha$ with $f_{5007} > 0.5\times10^{-16} {\ensuremath{ \mathrm{\ erg\ cm^{-2}\ s^{-1}} }}$, consistent with the determination of @cr03 for the Leo group of galaxies. Taking into account the redshift range ($\Delta z \sim 0.005$) and the effective area (5.8–16.2 ${\ensuremath{ \mathrm{arcmin} }}^{-2}$) of our surveys, we expect well fewer than one contaminating source in any of our samples. The estimate does not take into account any clustering of the background sources anticipated because of large scale structure, which could result in significant fluctuations in the surface density [@oea03]. In many cases, however, the ${\ensuremath{ \mathrm{Ly} }}\alpha$ line will be broad enough to be resolved by the RFP and have an asymmetric profile [e.g., @cr03]. Such sources are unlikely to be selected in our detection procedure and would be recognizable from their spectra. Additionally, these ${\ensuremath{ \mathrm{Ly} }}\alpha$ sources often have a faint continuum which can be used to identify an interloper, which partly motivated our inclusion of a continuum term in equation \[e:emmodel\]. All our candidate spectra show continuum levels fully consistent with a zero background. On the basis of these considerations we believe that the possibility of contamination by ${\ensuremath{ \mathrm{Ly} }}\alpha$ sources in our samples can be ignored. Results {#s:results} ======= Our galaxy sample consists of three close to round elliptical galaxies and one lenticular galaxy. An overview of the properties of our galaxy sample is given in Table \[t:galsamp\]. We present the spectra for each PN in our samples, as well as tables with positions, magnitudes, and line-of-sight velocities. A simple overview of the line-of-sight velocities for all four galaxies is shown in Figure \[f:rvfields\]. NGC 4636 -------- The E1 galaxy NGC 4636 lies on the Southern edge of the Virgo cluster and has been well studied because of its high X-ray luminosity. @rea01 note that the galaxy does not follow the Fundamental Plane relation for core galaxies (its surface brightness is too low for its absolute luminosity) and has an unusually diffuse core. This is reflected in the large effective radius of the system. All the NGC 4636 images were taken at the end of each night during the 1994 run. Of the 34 available image 28 were centered on the galaxies, but 6 were offset by half a field radius. Additionally, the seeing of these observations was large ($2\farcs2$) and the first night we had non-photometric conditions. Because of the lower surface brightness we were able to go deep enough to discover two PNe in this system, whose velocities are consistent with the systemic velocity [@mce95]. The spectra are shown in Figure \[f:n1549spectra\].1 and the properties of the PNe in Table \[t:1549-pne\]. NGC 1549 -------- The E1 galaxy NGC 1549 is in close interaction with its neighbor NGC 1553 as is evidenced by the strong isophote twisting [@fih89] and the faint shells surrounding it [@mc83]. The galaxy is the most distant in our sample and has not been targeted for a PN survey before. Photometric conditions varied throughout the five nights of observation, but this was mitigated by the relatively good seeing and the small airmass of the observations, which allowed us to go a little deeper than for the Leo galaxies. The galaxy was observed with one pointing, giving an effective survey area of $5.8{\ensuremath{ \mathrm{\ arcmin^{-2}} }}$. We discovered 6 PNe around this galaxy. The spectra are shown in Figure \[f:n1549spectra\].1 and the properties of the PNe in Table \[t:1549-pne\]. Their average velocity of $1279{\ensuremath{ \mathrm{\ km\ s^{-1}} }}$ is consistent with the $1220{\ensuremath{ \mathrm{\ km\ s^{-1}} }}$ systemic velocity of the galaxy and their velocity dispersion $\sigma = 217{\ensuremath{ \mathrm{\ km\ s^{-1}} }}$ matches the value found with absorption line spectroscopy [$220{\ensuremath{ \mathrm{\ km\ s^{-1}} }}$, @lea94]. NGC 3379 {#s:samp3379} -------- The E1 galaxy NGC 3379 (M105) forms with NGC 3384 (below) the central pair of the Leo (M96) group [@fs90]. Following the work by @dvc79 NGC 3379 is often considered the “standard” elliptical, although @cea91 have argued, purely on the basis of photometry, for a reclassification from E1 to S0. Because of its proximity NGC 3379 is an ideal candidate to look for extragalactic PNe and we observed it with two pointings, giving an effective survey area of $11.9{\ensuremath{ \mathrm{\ arcmin^{-2}} }}$, which excludes the bright central ($<10\arcsec$) part of the galaxy. The frames for each pointing were taken over two nights; each pointing had one night of non-photometric quality. Because one of our reference stars showed evidence of an absorption line in its RFP spectrum, we used the galaxy flux within $0.5R_e$ as the photometric reference for our flux normalization. We present our sample of PNe in Table \[t:3379-pne\]; the corresponding spectra are shown as Figures \[f:n1549spectra\].2–\[f:n1549spectra\].4 in the online version of the Journal. In NGC 3379 we find 54 PNe out to $\sim5R_e$. The sample is sparse and its distribution on the sky is consistent with the surface brightness of NGC 3379, taking into account the position dependent limiting magnitude. Using the on/off band technique @cjf89 obtained a sample of 93 PNe in NGC 3379. In a spectroscopic follow up @cjd93 measured the line-of-sight velocities of a subset of 29 PNe. In principle we can observe 57 of the @cjf89 candidates, and we recovered 30 of these. Similarly, we can measure 15 velocities from @cjd93 and we recover 7 of these. The overlap with the previous work gives us an independent check on our calibration: comparisons of the PN magnitudes and line-of-sight velocities are shown in Figures \[f:n3379magcomp\] and \[f:n3379velcomp\]. The magnitudes are consistent within their uncertainties, although there is clearly crowding near the limiting magnitudes of either survey. Considering that 50% of our data was taken under non-photometric conditions, the agreement is remarkable. The additional tight agreement between the two velocity measurements further convinces us that we calibrated our data properly and adequately. In our sample of PNe we have one candidate with an unusual velocity: E27 at $291{\ensuremath{ \mathrm{\ km\ s^{-1}} }}$. It is unlikely to be associated with NGC 3384, because of the latter’s velocity field, which would favor PNe with velocities over $720{\ensuremath{ \mathrm{\ km\ s^{-1}} }}$. Hence, it could either be an interloper or an intragroup PN, but we have no way of ascertaining either conjecture. In the mass models we present for NGC 3379, we exclude this object in our data analysis. The overall pattern of radial velocities (see Fig. \[f:rvfields\]) shows the signature of minor axis rotation. Although there are not enough data points for an informative independent estimate of the velocity field based on the PNe alone, simple smoothing spline estimates of the rotation and velocity dispersion (using code kindly provided by D. Merritt) are consistent with the long slit results of @ss99, yielding $V/\sigma\sim0.25$. Hence we can apply the Tracer Mass Estimator [TME; @evanea03] to give a first approximation of the total mass of NGC 3379. Assuming a number density $n\propto r^{-2.3}$ and a logarithmic potential, the TME yields $(2.1\pm0.1) \times 10^{11}\mathrm{\ M_\sun}$ within 8 kpc, where the uncertainties were derived assuming anisotropies $\beta=\pm0.3$. The implied $B$ band mass-to-light ratio is $\Upsilon_B=9.9\pm0.5$. NGC 3384 -------- The SB0 galaxy NGC 3384 is the closest companion of NGC 3379 and is one of the brightest members of the Leo (M96) group [@fs90]. There is some evidence for interaction from a faint tidal arm [@mal84] and the large HI ring that surrounds the galaxy pair [@schn89]. Three components contribute to the light distribution of NGC 3384: a small bulge with a complex structure which includes the bar within $\sim20\arcsec$, a lens that extends out to $160\arcsec$ and an outer exponential disk [@bea96]. The galaxy was also observed by @cjf89, who found a sample of 102 PNe (100 if we identify two pairs of PNe separated by less than $0.5\arcsec$). There was, however, no spectroscopic follow up for their NGC 3384 sample. NGC 3384 was observed with four pointings, giving an effective survey area of $16.2{\ensuremath{ \mathrm{\ arcmin^{-2}} }}$, which excludes the bright inner part of the galaxy, ghosts, and the brightest reference stars. Because the RFP produces ghost images of bright objects the survey area has a complicated topology; spectra will have as few as 19 data points in some regions and as many as 65 data points in others. We purposely avoided the region between NGC 3384 and NGC 3379. The galaxies have systemic velocities that are close enough to make it impossible to disentangle the correct host of each PN candidate in the intergalactic area, even more considering the possible interaction between the two. The NGC 3384 data have been discussed previously by @tmw95, but was based on a different data reduction and candidate selection process. We present our sample of PNe in Table \[t:3384-pne\]; the corresponding spectra are shown as Figures \[f:n1549spectra\].5–\[f:n1549spectra\].7 in the online version of the Journal. With our field of view we can, in principle, observe 82 PNe from the @cjf89 sample. We recover 37 PNe and discovered 13 new PNe. The comparison of the magnitudes of the PNe we have in common is shown in figure \[f:n3384magcomp\]. As before, the agreement is quite good. An interesting candidate is W05, the outlier at the bright end of the plot. Visual inspection shows possible structure around this candidate. @cjf89 interpreted this as two close candidates (their numbers 6 and 60), which seems unlikely since the two candidates are close both in velocity space and on the sky. Hence, W05 could be an interloper, but we include it for completeness. We can apply the TME to the NGC 3384 data, but we need to make some additional assumptions, since Figure \[f:rvfields\] shows the clear disk-like kinematics of the PN sample. We will assume that the S0 has a flat rotation curve at $120\mathrm{\ km\ s^{-1}}$ and that all PNe lie in the plane of the disk with a power law density distribution. The disk has a presumed inclination $i=62^\circ$; hence we can split the line-of-sight velocities of the PNe into a component due to the rotation and a random component. We apply the TME to the random component, adding $\langle v_\mathrm{rot} \rangle r/G$ for the rotational motion to arrive at the total mass: $(5.1\pm0.3) \times 10^{10}\mathrm{\ M_\sun}$ within 9 kpc. The implied $B$ band $\Upsilon_B$ within this radius is $10.9\pm0.7$. Mass modeling {#s:massmodels} ============= Physical Framework ------------------ Determining the potential of a spherical system, let alone an axisymmetric or triaxial system, using a discrete kinematic tracer population such as PNe would take on the order of $10^3$ radial velocities [@ms93]. With our samples ($N \lesssim 50$) clearly falling short of this requirement, we shift the objective of our modeling from finding the most likely mass model to assessing the constraints the PNe velocities can place on the distribution of mass at $r \gtrsim R_e$. Only the two Leo galaxies, NGC 3379 and NGC 3384, have PN sample sizes that are large enough to give potentially interesting results. In the following we consider spherical mass models for the E1 galaxy NGC 3379; attempts to build axisymmetric mass models for the S0 galaxy NGC 3384 were not successful and are not presented here [@sl04]. Our model space $M$ is predicated on the Jeans equation for a spherical non-rotating system, assuming a distribution function of the form $f(E,L^2)$: $$GM_<(r) = -r \sigma_r^2(r) \left( \frac{\mathrm{d} \ln \nu}{\mathrm{d} \ln r} + \frac{\mathrm{d} \ln \sigma_r^2}{\mathrm{d}\ln r} + 2\beta(r) \right), \label{e:sje}$$ where $M_<(r)$ is the mass interior to radius $r$, $\nu(r)$ the luminous density of the tracer population, $\sigma_r(r)$ the radial velocity dispersion of said population, and $\beta(r) \equiv 1 - \sigma_t^2(r) / \sigma_r^2(r)$ its velocity anisotropy. The gravitational potential of NGC 3379, an E1 galaxy, will be rounder than its mass distribution [@bt87] and deviate from sphericity by only a few per cent. Because the PNe data sets are highly incomplete within $1\ R_e$ we augment our kinematic data with the results of long slit spectroscopy to constrain the models in the inner regions, assuming both data sets represent the same tracer population. In the case of NGC 3379 we will use the results of @ss99. The kinematic data show an amount of rotation ($V/\sigma\sim0.25$) that is slight enough to reasonably model the galaxy as a non-rotating system. We derive the luminous density $\nu(r)$ from the surface brightness $\mu(R)$ by applying an Abel inversion in the fashion of @gebea96. For the surface brightness $\mu_B(r)$ we combined the groundbased data from @pelea90 with the *HST* data (F555W filter) from @gebea00 shifted to $B$ band by assuming a uniform color [@gouea94]. The combined surface brightness profile only extends out to $154''$, and we performed a linear extrapolation in $(\log r,\mu_B)$ for larger radii. Exploring the model space ------------------------- In looking for ways to analyze the range of plausible mass models for NGC 3379, we insisted that our method be non-parametric and use the data directly. A non-parametric method, though generally computationally intensive, will give a more conservative estimate of the range of possible mass models and will come closer to the most likely model than a parametric method. Indeed, the form of a parametric function can artificially constrain our search for plausible mass models. Furthermore, we want to be frugal with our data and avoid unnecessary binning of the PNe velocities. In other words, instead of relying on the unstable process of estimating and differentiating $\sigma_r^2(r)$ to infer $M_<(r)$ from Equation \[e:sje\], we posit $M_<(r)$ and measure its success in matching the observed data. The method developed by @mb01 [MB], which is briefly reiterated below, meets both these requirements. The MB analysis is Bayesian in outlook and quantifies the plausibility of a mass model $M$ given the data $D$: $$p(M|D) \propto p(D|M)p(M), \label{e:bt}$$ where the “prior” probability $p(M)$ encodes our prejudices about the most appropriate model $M$ absent any observations, the likelihood (“chi-squared”) $p(D|M)$ quantifies the probability of observing the data $D$ given a model $M$, and the “posterior” probability $p(M|D)$ quantifies the distribution of plausible models $M$ given our prejudices and the data. Credible regions—the Bayesian analog of confidence regions—are naturally constructed from the posterior and quantify the constraints the data place on the mass distribution. We specify our models as follows. Given a *local* mass-to-light profile $\Upsilon(r)$ to convert the luminous density $\nu(r)$ to a mass density $\rho(r)$, and an anisotropy profile $\beta(r)$ to determine $\sigma_t^2(r)$ from $\sigma_r^2(r)$, we can assay the likelihood $p(D|M)$ of the kinematic data by integrating equation \[e:sje\] to find the velocity dispersion along the line of sight. (In the case of the long slit data the projected velocity dispersion is convolved with a Gaussian to mimic the effect of seeing.) The shape of $\Upsilon(r)$ is specified in terms of its values $\Upsilon_i$ at a discrete set of sample points $r_i, i=1,\ldots,N_\Upsilon$, with intermediate values obtained through interpolation. On the other hand, we will assume that the anisotropy is constant with either $\beta=0$ or $\beta=0.3$, partly for ease of computation, partly due to the work by @gerea01 [fig. 4] who find a remarkably constant anisotropy ($\beta=0.3$) in their dynamical models of giant ellipticals. Having specified the projection of our model into observable space, we calculate the posterior probability $p(M|D)$ for a given trial model $M = \{\Upsilon(r),\beta\}$ from the likelihood and the prior probability. We define the likelihood as follows $$\begin{aligned} \ln p(D | \Upsilon(r), \beta) & = & -\frac{1}{2} \sum_{i} \left ( \frac{\sigma_i - \sigma_p(R_i)}{\Delta\sigma_i} \right)^2 \nonumber \\ & & - \sum_{j}\left ( \frac{v_j^2}{2\sigma_p^2(R_j)} - \ln \frac{\Delta v_j}{\sigma_p(R_j)} \right), \label{e:likelihood}\end{aligned}$$ where $\sigma_i$ and $\Delta\sigma_i$ are the observed velocity dispersions from the long slit spectra and their uncertainties, and $v_j$ and $\Delta v_j$ are the radial velocities of the PNe and their uncertainties. The first term quantifies the goodness-of-fit for the long slit data and the second term quantifies the likelihood of a radial velocity $v_j$ given the velocity dispersion along the line of sight. The prior probability of a trial model is in essence a smoothness constraint: $$\ln p(\Upsilon(r),\beta) = -\lambda \int \left[ \frac{{\ensuremath{ \mathrm{d} }}{}^2(\ln\Upsilon)} {{\ensuremath{ \mathrm{d} }}(\ln r)^2} \right]^2 {\ensuremath{ \mathrm{d} }}(\ln r), \label{e:prior}$$ where $\lambda$ specifies the amount of smoothness we prefer in our trial $\Upsilon(r)$. The prior gives a high probability to mass-to-light profiles that are close to a power-law, i.e. $\Upsilon(r) \sim r^\alpha$. In our experience the results of our calculation are not very sensitive to the value of $\lambda$: we settled on $\lambda=0.1$. Nor is the value critical, since it simply encoded our prejudice as to what constitutes $\Upsilon(r)$ with too much variation. The next step is to investigate the posterior $P(M|D)$. At this point we deviate from the MB method. Where they continue and include an ansatz for the distribution function of the system to ensure that the choices for $\Upsilon(r)$ correspond to a physical, i.e. non-negative, distribution function, we decided to forego this step in order to speed up the calculations. Our results might not always be physically plausible, but they are conservative, in the sense that the physical results are at worst a subset of our results. The high dimensionality of our model space ($N_\Upsilon=12$) precludes sampling the posterior $p(M|D)$ using a simple grid. Instead we explore the model space using a Monte Carlo Markov Chain (MCMC) method. A MCMC is no more than a random walk which, given enough steps, yields a set of trial models whose density distribution in the model space is proportional to the posterior distribution. The main difficulty in using MCMCs is to ensure that the random walk has wandered enough through model space for the density of trial models to have converged to the posterior distribution. To improve the convergence of our MCMCs we use the slice sampler [@neal03] to generate the random walk instead of the more conventional Metropolis algorithm [@metrea53]. Lacking a consensus among statisticians about the correct way to establish the convergence of an MCMC we use graphical checks of the random walks as well as the Gelman & Rubin $R$ statistic [see, e.g., @verdea03]. For a more detailed description of our implementation we refer the reader to @sl04. The results ----------- We performed two sets of MCMC calculations: one for isotropic $(\beta=0)$ mass models and one for radially anisotropic $(\beta=0.3)$ mass models. In both cases we sampled $\Upsilon(r)$ at $N_\Upsilon = 12$ logarithmically spaced radii. We performed some exploratory calculations with a range of values for $N_\Upsilon$. Our chosen value reflects the balance between an adequate resolution of $\Upsilon(r)$ and a reasonable convergence speed of the MCMCs. The MCMC for the isotropic models required a total of $1.8\times10^5$ iterations to converge; the anisotropic models a total of $4.6\times10^5$. In both cases we discarded a total of $2\times 10^4$ iterations as a “burn in” period for the chains. Once we have our set of trial models it is straightforward to establish the, e.g., 95% credible region for a given $\Upsilon_i$ by determining the range in $\Upsilon_i$ that encompasses 95% of all the trial models. Implicit in this determination is the marginalization of the posterior probability over all the other $\Upsilon_j, j\neq i$. Likewise we calculate the corresponding confidence regions for, say, $\sigma_p(R)$ and gauge how well our models can match the data. The results of our calculations are summarized in Figure \[f:n3379\_mcmc\] with credible regions of 99%, 90%, and 50%. To provide some context we obtained the best-fit models for three parametric mass models: a Hernquist density profile $\propto r^{-4}$ [@her90], an NFW profile $\propto r^{-3}$ [@nfw97], and a pseudo-isothermal profile $\propto r^{-2}$ [@bt87], where in all cases $r \gg r_s$ with $r_s$ the appropriate scale length; all three models have a second parameter $\rho_s$ that scales the density. The best-fit models maximize the likelihood of the data—defined in Equation \[e:likelihood\]—by varying $M_<(r|r_s,\rho_s)$, but keeping $\nu(r)$ fixed, i.e. we do not assume a constant mass-to-light ratio for these models. They appear as the (red) curves in Figure \[f:n3379\_mcmc\]. The bottom panels of Figure \[f:n3379\_mcmc\] show that the non-parametric description of $\Upsilon(r)$ has more leeway than the parametric models in matching the actual data, especially around 1 kpc where the two sides of the galaxy have significantly different velocity dispersions and at the outer edges, where the PNe show a steep drop in velocity dispersion. We emphasize that our method does not bin the PNe velocities; the figure shows binned PN data for clarity. The core radii of the parametric models are quite small, an indicator of the failure of these mass models: the anisotropic Hernquist model has the largest scalelength with 0.6 kpc, the pseudo-isothermal model the smallest with 1.15 pc. A pseudo-isothermal profile, unsurprisingly, is not able to reproduce the data, because of its nearly constant velocity dispersion; the Hernquist and NFW profiles do better, but still show a clear bias, justifying our choice for a non-parametric approach. The difference between the two sets of models, at least in terms of matching the data, is a widening of the credibility contours in the case of the anisotropic models. (The results from MB for NGC 3379 in their Figure 3 show a similar widening.) Whether or not this is a significant difference can be answered by calculating the odds ratio (see Equation 14 of MB) of the isotropic versus the anisotropic hypothesis: $$O = \frac{p(\beta_1|D)}{p(\beta_2|D)} = \frac{p(\beta_1)}{p(\beta_2)} \frac{\int p(D|\Upsilon,\beta_1) p(\Upsilon) {\ensuremath{ \mathrm{d} }} \Upsilon} {\int p(D|\Upsilon,\beta_2) p(\Upsilon) {\ensuremath{ \mathrm{d} }} \Upsilon}.$$ Typically, an odds ratio $O\sim10$ is considered to be conclusive in favoring one hypothesis over another. Since we do not have any *a priori* preference for the value of $\beta$, the odds ratio reduces to the ratio of the average values of the posterior distributions. In the case of the parametric models $O \sim 2$, implying either hypothesis is equally likely. From the MCMCs we find $O = 4.6$ in favor of isotropy, which is a hint, but hardly conclusive. As MB noted, this is not a surprising result: barring information about the higher order velocity moments $\Upsilon(r)$ has enough freedom to match to data, given some value (or even profile) for $\beta$. The little information of these higher order moments encoded in the PN velocities is clearly not enough to be a useful discriminant. The circular velocity in both sets of models is flat between 1 and 4 kpc at roughly $250\ \rm{km\ s^{-1}}$, but drops of rapidly outside the latter radius, matching the results of @kronea00 [Figure 18]. The agreement with the analysis of MB is less convincing at large radii, a consequence of their use of the smaller @cjd93 PN sample. The largest difference between the two sets is within 0.6 kpc, where the anisotropic models need significantly less mass than the isotropic models to produce the same velocity dispersion profile. This is simply a reflection of degeneracy between mass and velocity anisotropy, which can only be broken by the inclusion of higher order moments velocity data [see @mer93; @ger93; @vdmf93]. The top panels of Figure \[f:n3379\_mcmc\] show a constant $\Upsilon(r)$ between 0.1 and 2 kpc, before showing a small hump followed by a steep decrease at larger radii. The range in $\Upsilon(r)$ is consistent with the range found by @gerea01 for NGC 3379 from stellar population synthesis. The bump is related to the diverging velocity dispersion data points outside 1 kpc and creates a small plateau in the model velocity dispersion. The flaring of the $\Upsilon(r)$ credibility contours shows that the constraints on the mass-to-light profile from the PNe are weaker than those from the long slit spectra, but are nonetheless informative within 10 kpc. Compare the parametric models with their associated mass-to-light profiles: they agree fairly well with $\Upsilon(r)$ within 1 kpc (long slit spectra), but diverge strongly in the region dominated by the PN data. Despite this fact, the parametric models only allow us to conclude that the mass density should fall off steeper than $\propto r^{-2}$, but cannot discriminate between a Hernquist or an NFW profile. The non-parametric estimate $\Upsilon(r)$, however, not only matches the data better, but also gives us hints of a breakdown of our assumptions (a plateau instead of a flaring of confidence contours). Most remarkable, however, remains the relative flatness of $\Upsilon(r)$. The implied total mass within 8 kpc is $7.5\times 10^{11}\mathrm{\ M_\sun}$, a factor of a few smaller than the TME estimate in section \[s:samp3379\]. The total mass-to-light ratio implied is $\Upsilon_B\sim7$, consistent with the stellar population models of @gerea01. Discussion ---------- The conclusion that our data show little evidence for DM in the inner 8 kpc of NGC 3379 is hardly surprising in the light of the work done by @cjd93, @kronea00, and—more recently—@romea03. In particular, a cursory inspection of the PN data in Figure 3 of @cjd93 shows a striking resemblance to the similar data in our Figure \[f:n3379\_mcmc\]: a steep drop off in velocity dispersion between $1R_e$ and $3R_e$. Using a parametric model that guaranteed a positive distribution function they found no significant signature for the presence of DM within $3.5R_e$ with $M = 1 \times 10^{11}\mathrm{\ M_\sun}$ and $\Upsilon_B\sim7$ consistent with our own findings. The investigations by @kronea00, based on absorption line spectra out to $\sim2R_e$, arrive at the same but more general conclusion: ellipticals likely have nearly maximal mass-to-light ratios. Specifically, they find $\Upsilon_B=4.5$. On the basis of orbit superposition modeling @romea03 determined a mass-to-light ratio for NGC 3379 in the $B$ band of $7.1\pm0.6$. Their sample included about $\sim100$ PNe and also showed the radial decline in velocity dispersion. Indeed, they found the same decline in three more galaxies and concluded that elliptical galaxies contain little DM when compared to other galaxies. Considering the range in modeling procedures employed to study the dynamics of NGC 3379 and the consistency of the results, it is fair to say that it shows no evidence for dark matter within the inner few effective radii. At the same time, analysis of the HI ring around NGC 3379 and NGC 3384 implies an enclosed mass of $\sim 6\times10^{11}\mathrm{\ M_\sun}$ [@schnea89]. The discrepancy between the total dynamical mass-to-light ratio $\Upsilon_B=27$ within the 110 kpc ring [@schn89] and within a 10 kpc radius of NGC 3379 suggests that most of the dark matter is at large radii. Our models might not capture the complexity of NGC 3379 acceptably. The galaxy is not perfectly spherical and might even be an S0 seen face on [@cea91], but as @romea03 have observed, either possibility is unlikely to have a large effect on our final answer. The fact, noted by @ss99, that the long slit data show a bump in the $h_3$ moment at $\sim 15''$ is interesting in this regard as an indication of a more complex description for NGC 3379 than incorporated in our model. The twists in the photometric surface brightness and the kinematic velocity field (both roughly 5 degrees, but in opposite directions) already hint at such a departure from our assumptions. [@ss99] suggest that NGC 3379 might be triaxial. The system might also not have adequately relaxed. Indeed, the asymmetry of the velocity dispersion profile beyond $20''$ seems to corroborate the latter possibility, despite the lack of photometric evidence. A more worrisome development, from a recent paper @sgm05, is the discovery of two populations of PNe in NGC 4697, one younger and inherently brighter than the other, creating a bias in the measurements of the PN kinematics. Given its somewhat odd velocity field, it is entirely possible that NGC 3379 contains a bimodal population of PNe. A sample size of 50 PNe, however, is clearly unable to address these questions usefully. A larger sample of ellipticals combined with deeper observations are needed to investigate the relevant statistics. Conclusions {#s:wrapup} =========== We have reported on a search for PNe around four early-type galaxies using the RFP. We obtained reasonably sized samples in the case of the two Leo galaxies, NGC 3379 and NGC 3384, with adequate photometry and well determined radial velocities. The main limiting factor in our observations was the seeing; our data are read noise limited and better seeing would have improved the limiting magnitude of our survey. In our analysis of the RFP data cubes two realizations were essential in optimizing the size of the extracted PN samples. The power of our detections comes from using the data cube as a whole, as opposed to looking for point sources in each monochromatic RFP image. Secondly, applying the proper statistic is essential, along with a proper characterization of the reference distribution. A simple estimator yields mass-to-light ratios $\Upsilon_B \sim 10$ for the Leo galaxies, although a more sophisticated analysis for NGC 3379 gives an estimate that is a factor two lower, making it consistent with stellar population mass-to-light ratios. Although our models are relatively simple, they do produce conservative estimates of the mass-to-light ratios, and hence we do not find evidence for a dominant DM component inside a few $R_e$, confirming the recent work by @romea03. In order to address questions about the existence of multimodal PN populations or the properties of PN kinematics as a function of host galaxy luminosity, we clearly need a larger and deeper set of surveyed galaxies. The RFP has been decommissioned, but successor instruments will be coming online in the very near future. The Prime Focus Imaging Spectrograph on the Southern African Large Telescope (11 m) will have Fabry-Perot image spectroscopy as one of its modes and will be ideally suited to extend current surveys of extragalactic PNe. It will be both competitive, considering depth and field of view, and complementary, being on the Southern hemisphere, to the Planetary Nebulae Spectrograph [@dea02]. Benoit Tremblay collaborated in some of the observations of this study. The staff at CTIO provided their usual excellent support for these observations. AS would like to acknowledge helpful conversations and email exchanges with E. Barnes, D. Chakrabarty, R. Méndez, R. Ciardullo, and J.Magorrian. This work was supported in part by the National Science Foundation through grants AST9731052 and AST0098650. [*Facilities:*]{} [References that is]{} Band, D., Ford, L., Matteson, J., Briggs, M., Paciesas, W., Pendleton, G., & Preece, R. 1997, , 485, 747 Binney, J., & Tremaine, S. 1987, Galactic Dynamics (Princeton, NJ: Princeton University Press) Bridges, T., et al.  2003, Extragalactic Globular Clusters and their Host Galaxies, 25th meeting of the IAU, Joint Discussion 6, 17 July 2003, Sydney, Australia, 6, Busarello, G., Capaccioli, M., D’Onofrio, M., Longo, G., Richter, G. & Zaggia, S. 1996, , 314, 32 Capaccioli, M., Vietri, M., Held, E. V., & Lorenz, H. 1991, , 371, 535 Carollo, C. M., de Zeeuw, P. T., van der Marel, R. P., Danziger, I. J., & Qian, E. E., 1995, , 441, L25 Castro-Rodr[í]{}guez, N., Aguerri, J. A. L., Arnaboldi, M., Gerhard, O., Freeman, K. C., Napolitano, N. R., & Capaccioli, M. 2003, , 405, 803 Ciardullo, R., Feldmeier, J. J., Krelove, K., Jacoby, G. H., & Gronwal, C. 2002, , 566, 784 Ciardullo, R., Jacoby, G. H., & Dejonghe, H. B. 1993, , 414, 454 Ciardullo, R., Jacoby, G. H. & Ford, H. C. 1989, , 344, 715 C[\^ o]{}t[' e]{}, P., McLaughlin, D. E., Cohen, J. G., & Blakeslee, J. P. 2003, , 591, 850 de Blok, W. J. G., Bosma, A., & McGaugh, S. 2003, , 340, 657 de Vaucouleurs, G., & Capaccioli, M. 1979, , 40, 699 de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Buta, R. J., Paturel, G., Fouque, P. 1991, Third Reference Catalog of Bright Galaxies, (New York: Springer-Verlag) Diemand, J., Moore, B., & Stadel, J. 2004, , 353, 624 Dopita, M. A., Jacoby, G. H., & Vassiliadis, E. 1992, , 389, 27 Douglas, N. G., et al. 2002, , 114, 1234 Eadie, W. T., Drijard, D., James, F. E., Roos, M., & Sadoulet, B. 1971, Statistical Methods in Experimental Physics (New York: North-Holland) , N. W., [Wilkinson]{}, M. I., [Perrett]{}, K. M., & [Bridges]{}, T. J., 2003, , 583, 752 Ferguson, H. C., & Sandage, A. 1990, , 100, 1 Ferrarese, L., et al. 2000, , 529, 745 Franx, M., Illingworth, G., & heckman, T. 1989, , 98, 538 Freeman, P. E., Graziani, C., Lamb, D. Q., Loredo, T. J., Fenimore, E. E., Murakami, T., & Yoshida, A. 1999, , 524, 753 Freeman, K. C., et al. 2000, in ASP Conf. Ser. 197, Dynamics of Galaxies: from the Early Universe to the Present, 15th IAP meeting held in Paris, France, July 9-13, 1999, ed. F. Combes, G. A. Mamon, and V. Charmandaris (San Francisco: ASP), 389 Gebhardt, K., et al. 1996, , 112, 105 Gebhardt, K., et al. 2000, , 119, 1157 Gerhard, O. E. 1993, , 265, 213 Gerhard, O., Kronawitter, A., Saglia, R. P., & Bender, R. 2001, , 121, 1936 Goudfrooij, P., Hansen, L., Jørgensen, H. E., Nørgaard-Nielsen, H. U., de Jong, T., & van den Hoek, L. B. 1994, , 104, 179 Hamuy, M., Walker, A. R., Suntzeff, N. B., Gigoux, P., Heathcote, S. R., & Phillips, M. M. 1992, , 104, 533 Harris, W. E. 1990, , 102, 949 Hernquist, L. 1990, , 356, 359 Hoaglin, D. C., Mosteller, F., & Tukey, J. W. 1983, Wiley Series in Probability and Mathematical Statistics, New York: Wiley, 1983, edited by Hoaglin, David C.; Mosteller, Frederick; Tukey, John W. Hu, E. M., Cowie, L. L., & McMahon, R. G. 1998, , 502, L99 Hui, X., Ford, H. C., Freeman, K. C., & Dopita, M. 1995, , 449, 592 , C. R., 2001, , 561, 46 Kronawitter, A., Saglia, R. P., Gerhard, O., & Bender, R. 2000, , 144, 53 Kudritzki, R.-P., et al. 2000, , 536, 19 Loewenstein, M., & White, R. E., III 1999, , 518, 50 Longo, G., Zaggia, S. R., Busarello, G., & Richter, G. 1994, , 105, 433 Magorrian, J., & Ballantyne, D. 2001, , 322, 702 Malin, D. F. 1984, in IAU Coll. 58, Astronomy with Schmidt Type Telescopes, ed. M. Capaccioli (Dordrecht: Reidel), 73 Malin, D. F., & Carter, D. 1983, , 274, 534 , W. G. & [Brighenti]{}, F., 2003, , 41, 191 Mattox, J., et al. 1996, , 461, 396 McElroy, D. B., 1995, , 100, 105 Méndez, R. H., Riffeser, A., Kudritzki, R.-P., Matthias, M., Freeman, K. C., Arnaboldi, M., Capaccioli, M., & Gerhard, O. E. 2001, , 563, 135 Merritt, D. 1993, , 413, 79 Merritt, D., & Saha, P. 1993, , 409, 75 , N., [Rosenbluth]{}, A., [Rosenbluth]{}, M., [Teller]{}, A., & [Teller]{}, E., 1953, J. Chem. Phys., 21, 1087 Monet, D., Bird A., Canzian, B., Dahn, C., Guetter, H., Harris, H., Henden, A., Levine, S., Luginbuhl, C., Monet, A. K. B., Rhodes, A., Riepe, B., Sell, S., Stone, R., Vrba, F., Walker, R. 1998, The USNO-A2.0 Catalogue, (Washington DC: U.S. Naval Observatory) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 Naylor, T. 1998, , 296, 339 , R. M., 2003, Annals of Statistics, 31, 705 Nolthenius, R., & Ford, H. 1986, , 305, 600 Ouchi, M., et al. 2003, , 582, 60 Peletier, R. F., Davies, R. L., Illingworth, G. D., Davis, L. E., & Cawson, M. 1990, , 100, 1091 Peng, E. W., Ford, H. C., & Freeman, K. C. 2004, , 602, 685 Peng, E. W., Ford, H. C., & Freeman, K. C. 2005, preprint (astro-ph/0412449) Perlmutter, S., et al. 1999, , 517, 565 Phillips, J. P. 2002, , 393, 1027 Pottasch, S. R. 1984, Planetary Nebulae, (Dordrecht: Reidel) Protassov, R., van Dyk, D. A., Connors, A., Kashyap, V. L., & Siemiginowska, A. 2002, , 571, 545 Ravindranath, S., Ho, L. C., Peng, C. Y., Filippenko, A. V., & Sargent, W. L. 2001, , 122, 653 Riess, A. G., et al.  1998, , 116, 1009 Romanowsky, A. J., Douglas, N. G., Arnaboldi, M., Kuijken, K., Merrifield, M. R., Napolitano, N. R., Capaccioli, M., & Freeman, K. C. 2003, Sci., 301, 1696 , A. J. & [Kochanek]{}, C. S., 2001, , 553, 722 Sambhus, N., Gerhard, O., & Mendez, R. H. 2005, preprint (astro-ph/0510499) , S. E., 1989, , 343, 94 , S. E. et al., 1989, , 97, 666 Secker, J. 1995, , 107, 496 , J. A. & [Kosowsky]{}, A., 2001, in: [ASP Conf. Ser. 240: Gas and Galaxy Evolution,]{} 311 Sluis, A. P. N., PhD. Thesis, Rutgers, NJ , Y. & [Rubin]{}, V., 2001, , 39, 137 , L. S., 2002, in: [Proceedings of the Yale Cosmology Workshop “The Shapes of Galaxies and Their Dark Matter Halos”, New Haven, Connecticut, USA, 28-30 May 2001,]{} (Singapore: World Scientific), 178 Spergel, D. N., et al.  2003, , 148, 175 Statler, T. S., & Smecker-Hane, T. 1999, , 117, 839 Stern, D., Bunker, A., Spinrad, H., & Dey, A. 2000, , 537, 73 Stetson, P. B. 1987, , 99, 191 Stone, R. P. S. & Baldwin, J.A. 1983, , 204, 347 Tonry, J.L., Dressler, A., Blakeslee, J. P., Ajhar, E. A., Fletcher, A. B., Luppino, G. A., Metzger, M. R., & Moore, C. B. 2001, , 546, 681 Tremblay, B. T., Merritt, D., & Williams T. B. 1995, , 443, L5 van de Hulst, H. C. & Reesinck, J. M. 1947, , 106, 121 van der Marel, R. P., & Franx, M. 1993, , 407, 525 , L. et al., 2003, , 148, 195 , G., [Kaiser]{}, N., [Luppino]{}, G. A., & [Cowie]{}, L. L., 2001, , 555, 572 [lll]{} CCD & Tek 1k (CTIO \#1, VEB) & Tek 1k (CTIO \#2, Arcon)\ Gain & 1.73 $e^-\ {\ensuremath{ \mathrm{ADU^{-1}} }}$ & 1.25 $e^-\ {\ensuremath{ \mathrm{ADU^{-1}} }}$\ Read noise & 3.22 $e^-$ & 4.33 $e^-$\ Image scale & 070 ${\ensuremath{ \mathrm{pix^{-1}} }}$ & 035 ${\ensuremath{ \mathrm{pix^{-1}} }}$\ Voigt FWHM & 2.0 Å& 2.1 Å\ —Gaussian width $\Delta\lambda_G$ & 0.41 Å& 0.22 Å\ —Lorentzian width $\Delta\lambda_L$ & 0.89 Å& 1.0 Å\ Free spectral range $\Delta\lambda_\mathrm{FSR}$ & 48 Å& 48 Å\ Filters (at CTIO) & 5007-44, 5037-44 & 5007-44, 5037-44\ Switching wavelength $\lambda_{\ensuremath{ \mathrm{s} }}$ & 5019 Å& 5022 Å\ [lllcccccll]{} NGC 1549 & 04 15 45&$-55$ 35 27& 6& 28 (5)& 5016–5043& 1.0& 3& 1.5\*& 26.3\ NGC 3379 E& 10 47 54& +12 35 27 & 69& 29 (2)& 5011–5039& 1.0& 3& 1.6\*& 26.1\ NGC 3379 W& 10 47 45& +12 34 25 & 77& 33 (2)& 5011–5040& 1.0& 3& 1.5\*& 26.1\ NGC 3384 C& 10 48 17& +12 37 42 & 4& 27 (2)& 5012–5035& 0.6& 5& 1.8 & 26.2\ NGC 3384 N& 10 48 17& +12 38 14 & 31& 19 (1)& 5012–5035& 1.2& 5& 1.7 & 26.1\ NGC 3384 W& 10 48 12& +12 37 47 & 75& 19 (1)& 5012–5035& 1.2& 4& 2.0 & 26.2\ NGC 3384 E& 10 48 26& +12 39 34 & 171& 19 (1)& 5010–5028& 1.0& 6& 1.6 & 25.9\ NGC 4636 & 12 42 50& +02 41 15 & 4& 34 (4)& 5013–5034& 0.6& 3& 2.2\*& 26.7\ [lcccc]{} Right ascension (2000.0; hms) & 04 15 44.00 & 10 47 49.60 & 10 48 16.90 & 12 42 49.87\ Declination (2000.0; $\degr\;\arcmin\;\arcsec$) &-55 35 30.0& 12 34 53.9& 12 37 45.5& 02 41 16.0\ Type & E/S$0_1$ & E1 & SB(s)0- & E/S$0_1$\ Radial velocity $(\mathrm{km\ s^{-1}})$ & 1220 & 911 & 704 & 938\ Distance (Mpc) & 19.7 & 11.1 & 11.4 & 15.0\ Total $B$ magnitude & 10.72 & 10.24 & 10.85 & 10.34\ $M_{B_T}$ & -20.75 & -19.99 & -19.43 & -20.54\ $R_e$ ($\arcsec$) & 91 & 35 & 50 & 177\ [lccrrrrrrrr]{} N1-1& 4:15:35.96& -55:35:18.3& 5026.2& 1147& 14& 1.04& 26.21& 0.23& 23.1& 5.0\ N1-2& 4:15:42.65& -55:34:43.5& 5032.9& 1547& 17& 0.84& 26.45& 0.23& 22.8& 5.7\ N1-3& 4:15:44.86& -55:36:10.7& 5033.8& 1603& 18& 0.98& 26.28& 0.25& 19.6& 5.6\ N1-4& 4:15:48.31& -55:36:01.9& 5024.7& 1060& 14& 1.20& 26.07& 0.22& 26.7& 5.8\ N1-5& 4:15:51.14& -55:34:34.8& 5025.1& 1080& 15& 0.93& 26.34& 0.20& 31.9& 6.4\ N1-6& 4:15:51.75& -55:35:23.0& 5027.6& 1234& 12& 1.23& 26.03& 0.19& 34.1& 7.5\ N4-1& 12:42:51.53& 2:40:49.9& 5019.3& 739& 19& 0.77& 26.54& 0.26& 17.4& 4.8\ N4-2& 12:42:52.09& 2:40:24.9& 5025.1& 1086& 21& 0.75& 26.58& 0.23& 23.3& 7.1\ [lccrrrrrrrrrrr]{} E01& 10:47:48.73& 12:35:24.8& 5024.8& 1083& 13& 1.68& 25.69& 0.21& 27.5& 6.0& ..& ... & ...\ E02& 10:47:49.17& 12:35:42.4& 5017.6& 653& 17& 1.61& 25.74& 0.23& 26.3& 5.5& ..& ... & ...\ E03& 10:47:49.47& 12:35:35.3& 5023.6& 1009& 17& 1.52& 25.80& 0.19& 32.6& 7.0& 9& 25.66& ...\ E04& 10:47:49.66& 12:36:02.5& 5021.0& 853& 18& 1.11& 26.14& 0.23& 23.4& 5.8& ..& ... & ...\ E05& 10:47:49.94& 12:35:40.2& 5018.3& 694& 13& 1.97& 25.52& 0.22& 26.3& 6.1& 20& 25.84& ...\ E06& 10:47:50.30& 12:35:13.0& 5025.0& 1098& 12& 2.03& 25.49& 0.18& 37.1& 6.7& ..& ... & ...\ E07& 10:47:51.06& 12:35:01.5& 5023.0& 974& 17& 1.78& 25.64& 0.22& 25.4& 6.9& 12& 25.76& ...\ E08& 10:47:51.15& 12:35:15.4& 5018.8& 723& 19& 1.85& 25.59& 0.26& 18.8& 4.8& 36& 26.10& ...\ E09& 10:47:51.26& 12:34:51.2& 5025.8& 1141& 17& 1.73& 25.66& 0.21& 26.4& 6.3& ..& ... & ...\ E10& 10:47:51.40& 12:35:31.5& 5023.8& 1022& 13& 2.26& 25.37& 0.15& 54.7& 9.0& 2& 25.33& 1061\ E11& 10:47:51.59& 12:35:43.3& 5025.4& 1116& 13& 1.73& 25.66& 0.15& 53.4& 8.2& 6& 25.53& ...\ E12& 10:47:51.68& 12:34:47.3& 5023.6& 1013& 10& 2.44& 25.29& 0.17& 39.3& 6.9& 23& 25.92& ...\ E13& 10:47:52.03& 12:35:11.0& 5018.0& 675& 20& 1.76& 25.64& 0.22& 25.3& 4.7& ..& ... & ...\ E14& 10:47:52.11& 12:34:43.5& 5022.4& 942& 13& 2.19& 25.41& 0.15& 50.3& 9.1& 1& 25.28& 936\ E15& 10:47:52.28& 12:35:40.1& 5023.3& 991& 13& 1.81& 25.62& 0.17& 45.1& 8.0& 14& 25.77& ...\ E16& 10:47:52.33& 12:36:05.7& 5020.8& 844& 11& 1.75& 25.65& 0.18& 35.6& 6.7& 16& 25.78& 832\ E17& 10:47:53.34& 12:36:17.2& 5021.4& 881& 14& 1.25& 26.02& 0.22& 23.5& 6.0& ..& ... & ...\ E18& 10:47:54.83& 12:36:24.8& 5020.8& 843& 13& 1.47& 25.84& 0.20& 31.0& 5.6& 30& 26.00& ...\ E19& 10:47:55.12& 12:36:17.1& 5021.3& 873& 13& 1.43& 25.87& 0.20& 28.8& 6.7& 60& 26.37& ...\ E20& 10:47:55.43& 12:34:51.5& 5023.5& 1007& 17& 1.28& 25.99& 0.22& 24.2& 6.0& 38& 26.14& 1021\ E21& 10:47:55.50& 12:34:46.0& 5020.9& 850& 13& 1.57& 25.77& 0.21& 28.3& 6.6& ..& ... & ...\ E22& 10:47:55.76& 12:35:40.5& 5020.2& 806& 19& 1.43& 25.87& 0.27& 17.0& 4.9& ..& ... & ...\ E23& 10:47:56.07& 12:35:42.0& 5015.2& 510& 17& 1.46& 25.85& 0.22& 23.8& 5.4& ..& ... & ...\ E24& 10:47:56.86& 12:36:14.5& 5019.2& 748& 21& 1.16& 26.10& 0.27& 17.0& 4.3& 28& 25.98& ...\ E25& 10:47:56.90& 12:34:20.7& 5015.4& 521& 17& 1.30& 25.97& 0.27& 15.8& 4.6& ..& ... & ...\ E26& 10:47:56.92& 12:35:31.2& 5021.9& 910& 14& 1.36& 25.93& 0.20& 28.5& 6.2& 39& 26.15& ...\ E27& 10:47:58.10& 12:34:42.2& 5011.6& 291& 23& 1.71& 25.68& 0.30& 22.1& 5.4& ..& ... & ...\ E28& 10:47:59.21& 12:35:10.6& 5018.7& 719& 13& 1.79& 25.63& 0.20& 29.8& 6.8& 10& 25.68& ...\ W01& 10:47:41.41& 12:35:12.0& 5020.0& 797& 17& 1.09& 26.17& 0.22& 23.9& 3.6& ..& ... & ...\ W02& 10:47:41.67& 12:34:05.5& 5018.7& 720& 17& 1.15& 26.11& 0.23& 22.7& 5.5& ..& ... & ...\ W03& 10:47:41.71& 12:34:32.2& 5023.4& 1004& 15& 1.32& 25.96& 0.20& 28.4& 6.9& 48& 26.26& ...\ W04& 10:47:42.55& 12:35:31.7& 5015.3& 517& 17& 1.12& 26.14& 0.25& 18.6& 4.3& ..& ... & ...\ W05& 10:47:42.56& 12:35:08.9& 5023.8& 1025& 13& 1.39& 25.91& 0.15& 52.0& 8.9& 41& 26.18& ...\ W06& 10:47:42.92& 12:35:09.6& 5022.0& 919& 15& 1.25& 26.02& 0.21& 28.1& 6.8& 62& 26.40& ...\ W07& 10:47:44.73& 12:35:07.6& 5024.0& 1039& 12& 1.17& 26.09& 0.19& 34.7& 7.4& ..& ... & ...\ W08& 10:47:45.21& 12:34:59.7& 5023.1& 982& 12& 1.48& 25.83& 0.17& 39.5& 7.2& 29& 25.99& 977\ W09& 10:47:45.26& 12:35:17.3& 5018.7& 720& 11& 1.50& 25.82& 0.16& 48.2& 8.6& 19& 25.83& 716\ W10& 10:47:45.39& 12:33:37.9& 5018.2& 691& 18& 1.24& 26.03& 0.23& 25.0& 4.7& 50& 26.28& 726\ W11& 10:47:45.67& 12:34:13.4& 5017.4& 640& 18& 1.36& 25.93& 0.23& 25.2& 4.9& 63& 26.40& ...\ W12& 10:47:45.83& 12:35:06.7& 5025.7& 1138& 15& 1.06& 26.19& 0.21& 27.6& 7.3& ..& ... & ...\ W13& 10:47:46.23& 12:33:37.5& 5026.9& 1212& 16& 1.02& 26.24& 0.24& 20.4& 5.5& 40& 26.15& ...\ W14& 10:47:46.24& 12:34:10.5& 5024.7& 1078& 15& 1.08& 26.18& 0.22& 25.0& 5.7& 59& 26.37& ...\ W15& 10:47:46.72& 12:33:18.7& 5015.6& 531& 20& 1.16& 26.10& 0.26& 17.9& 4.3& ..& ... & ...\ W16& 10:47:46.94& 12:34:06.0& 5024.2& 1050& 16& 1.01& 26.25& 0.22& 24.0& 4.4& ..& ... & ...\ W17& 10:47:47.25& 12:35:01.3& 5028.6& 1310& 11& 1.53& 25.80& 0.18& 36.2& 6.9& 11& 25.75& ...\ W18& 10:47:47.29& 12:34:47.1& 5026.2& 1168& 12& 1.48& 25.83& 0.18& 37.6& 7.4& ..& ... & ...\ W19& 10:47:48.10& 12:35:00.1& 5020.0& 797& 13& 1.54& 25.79& 0.23& 22.2& 5.9& ..& ... & ...\ W20& 10:47:48.34& 12:34:38.0& 5024.3& 1053& 13& 1.91& 25.56& 0.15& 54.0& 9.8& 7& 25.63& 1060\ W21& 10:47:48.58& 12:33:32.9& 5019.3& 757& 18& 1.37& 25.92& 0.21& 27.0& 5.5& 49& 26.28& ...\ W22& 10:47:48.65& 12:35:03.5& 5022.3& 932& 10& 2.40& 25.31& 0.14& 61.5& 9.4& ..& ... & ...\ W23& 10:47:48.85& 12:34:41.6& 5020.2& 807& 17& 1.58& 25.76& 0.25& 21.4& 4.5& ..& ... & ...\ W24& 10:47:48.94& 12:33:40.3& 5023.4& 1001& 14& 1.22& 26.04& 0.19& 33.4& 6.3& 27& 25.96& 985\ W25& 10:47:48.97& 12:34:27.6& 5019.4& 758& 15& 1.67& 25.70& 0.18& 39.0& 7.3& 42& 26.20& ...\ W26& 10:47:49.97& 12:34:38.8& 5021.0& 858& 16& 1.95& 25.54& 0.21& 30.6& 5.9& ..& ... & ...\ [lccrrrrrrrrrrr]{} E01& 10:48:21.63& 12:38:43.9& 5015.2& 497& 14& 1.32& 25.96& 0.21& 27.5 (19)& 5.8& 49& 26.38\ E02& 10:48:24.42& 12:38:43.6& 5016.5& 576& 11& 1.55& 25.78& 0.18& 38.0 (19)& 6.8& 1& 25.59\ E03& 10:48:25.18& 12:39:36.2& 5016.3& 564& 13& 1.07& 26.19& 0.23& 23.4 (19)& 6.4& 11& 25.93\ W01& 10:48:09.00& 12:37:44.6& 5020.8& 812& 13& 1.10& 26.16& 0.24& 21.4 (19)& 6.7& 21& 26.05\ W02& 10:48:09.63& 12:37:36.7& 5021.2& 838& 8& 1.79& 25.63& 0.16& 48.9 (18)& 9.8& 10& 25.90\ W03& 10:48:10.43& 12:36:54.5& 5021.3& 844& 12& 1.31& 25.97& 0.21& 26.8 (19)& 8.2& 56& 26.43\ W04& 10:48:11.56& 12:37:32.9& 5020.6& 800& 10& 1.35& 25.93& 0.14& 61.5 (45)& 12.4& 7& 25.81\ W05& 10:48:11.85& 12:37:21.2& 5021.5& 857& 5& 2.58& 25.23& 0.08& 172.6 (44)& 17.1& 6& 25.73\ W06& 10:48:11.97& 12:36:27.4& 5022.0& 886& 8& 1.90& 25.56& 0.16& 48.6 (18)& 9.5& 8& 25.85\ W07& 10:48:12.29& 12:36:53.1& 5021.2& 838& 17& 0.90& 26.37& 0.20& 28.8 (35)& 8.4& 68& 26.58\ W08& 10:48:12.80& 12:36:53.8& 5022.2& 896& 8& 1.92& 25.55& 0.11& 107.0 (42)& 16.6& 2& 25.63\ W09& 10:48:13.25& 12:38:02.3& 5022.6& 920& 12& 0.80& 26.51& 0.17& 39.4 (61)& 9.6& 46& 26.36\ W10& 10:48:13.28& 12:37:52.1& 5020.6& 800& 9& 0.95& 26.32& 0.16& 45.8 (62)& 9.6& 43& 26.34\ W11& 10:48:13.41& 12:38:06.9& 5018.9& 698& 13& 0.92& 26.36& 0.18& 37.2 (61)& 8.7& 12& 25.94\ W12& 10:48:13.49& 12:37:10.5& 5021.0& 826& 11& 1.29& 25.99& 0.15& 59.9 (51)& 10.9& 28& 26.11\ W13& 10:48:13.51& 12:37:02.8& 5022.6& 923& 14& 0.86& 26.42& 0.23& 24.3 (44)& 7.2& 14& 25.96\ W14& 10:48:13.64& 12:36:46.1& 5021.9& 883& 7& 1.88& 25.57& 0.10& 112.8 (42)& 15.0& 9& 25.86\ W15& 10:48:14.17& 12:36:39.0& 5019.8& 754& 17& 0.86& 26.43& 0.21& 26.4 (45)& 6.9& ..& ...\ W16& 10:48:14.25& 12:37:04.6& 5020.6& 800& 11& 1.01& 26.25& 0.17& 39.5 (63)& 8.9& 59& 26.45\ W17& 10:48:14.37& 12:36:47.7& 5021.3& 842& 11& 1.01& 26.25& 0.18& 34.6 (44)& 8.2& 37& 26.30\ W18& 10:48:14.38& 12:37:23.9& 5021.0& 826& 11& 1.07& 26.19& 0.16& 45.3 (64)& 10.5& ..& ...\ W19& 10:48:14.60& 12:36:59.9& 5021.2& 839& 11& 1.23& 26.03& 0.17& 39.4 (47)& 8.3& ..& ...\ W20& 10:48:14.87& 12:37:25.8& 5021.4& 847& 9& 1.65& 25.71& 0.11& 93.2 (62)& 13.5& 3& 25.66\ W21& 10:48:15.12& 12:38:08.3& 5019.5& 735& 11& 0.99& 26.28& 0.19& 33.0 (62)& 8.9& 94& 26.96\ W22& 10:48:16.00& 12:37:03.3& 5020.3& 781& 14& 0.78& 26.54& 0.22& 22.9 (64)& 7.1& ..& ...\ W23& 10:48:16.03& 12:38:02.9& 5020.9& 821& 11& 1.22& 26.04& 0.18& 36.9 (61)& 8.4& 24& 26.07\ W24& 10:48:16.12& 12:37:17.0& 5023.2& 956& 9& 1.16& 26.10& 0.16& 46.3 (63)& 10.9& 34& 26.20\ W25& 10:48:16.17& 12:37:09.2& 5021.0& 824& 14& 0.94& 26.32& 0.21& 27.5 (61)& 9.1& ..& ...\ W26& 10:48:16.55& 12:38:09.9& 5019.8& 752& 17& 0.93& 26.34& 0.20& 28.4 (59)& 8.1& 33& 26.18\ W27& 10:48:16.81& 12:37:20.1& 5023.1& 951& 11& 1.42& 25.88& 0.16& 45.9 (59)& 10.4& 18& 25.99\ W28& 10:48:17.03& 12:38:27.3& 5019.6& 741& 11& 1.38& 25.91& 0.16& 44.8 (43)& 9.9& 4& 25.67\ W29& 10:48:17.52& 12:37:10.5& 5021.4& 847& 11& 1.20& 26.06& 0.16& 45.7 (44)& 9.6& 57& 26.44\ W30& 10:48:17.58& 12:38:07.8& 5018.8& 696& 15& 1.17& 26.09& 0.23& 23.3 (42)& 7.0& 5& 25.71\ W31& 10:48:17.91& 12:38:09.3& 5018.6& 679& 20& 0.90& 26.37& 0.26& 17.5 (46)& 6.3& ..& ...\ W32& 10:48:17.98& 12:38:22.2& 5017.2& 599& 14& 1.04& 26.22& 0.21& 28.8 (45)& 7.9& ..& ...\ W33& 10:48:18.07& 12:38:44.0& 5018.4& 671& 11& 1.76& 25.65& 0.19& 31.4 (25)& 7.3& ..& ...\ W34& 10:48:18.12& 12:36:36.5& 5019.4& 729& 14& 1.36& 25.93& 0.21& 27.2 (25)& 7.2& 22& 26.06\ W35& 10:48:18.58& 12:38:18.8& 5018.5& 676& 10& 1.51& 25.81& 0.17& 42.0 (43)& 9.1& 16& 25.98\ W36& 10:48:18.83& 12:37:15.4& 5019.3& 721& 11& 1.09& 26.17& 0.21& 28.2 (41)& 8.2& ..& ...\ W37& 10:48:19.18& 12:37:30.5& 5018.4& 672& 18& 0.85& 26.43& 0.25& 19.8 (41)& 6.9& 50& 26.38\ W38& 10:48:19.40& 12:38:16.4& 5017.3& 606& 16& 1.03& 26.23& 0.24& 22.4 (44)& 6.4& ..& ...\ W39& 10:48:19.64& 12:37:26.0& 5019.7& 747& 16& 1.10& 26.15& 0.23& 26.0 (42)& 6.6& ..& ...\ W40& 10:48:19.73& 12:36:44.5& 5020.7& 809& 11& 1.33& 25.95& 0.18& 38.6 (27)& 7.9& 44& 26.35\ W41& 10:48:19.73& 12:38:16.1& 5016.9& 583& 10& 1.26& 26.01& 0.18& 38.5 (45)& 8.8& 40& 26.31\ W42& 10:48:20.41& 12:38:03.5& 5018.2& 659& 16& 0.91& 26.36& 0.26& 18.8 (44)& 6.1& ..& ...\ W43& 10:48:20.50& 12:38:53.7& 5018.0& 643& 13& 1.52& 25.81& 0.23& 21.8 (19)& 6.5& 20& 26.04\ W44& 10:48:20.75& 12:38:21.7& 5016.8& 572& 11& 1.29& 25.99& 0.18& 36.0 (45)& 8.6& 19& 26.01\ W45& 10:48:20.88& 12:37:29.0& 5020.0& 766& 8& 1.42& 25.88& 0.14& 62.3 (45)& 10.8& 83& 26.73\ W46& 10:48:20.92& 12:38:05.7& 5018.0& 648& 19& 1.00& 26.26& 0.22& 24.2 (44)& 6.6& 53& 26.41\ W47& 10:48:20.93& 12:38:08.1& 5017.4& 607& 12& 1.03& 26.23& 0.23& 23.3 (43)& 6.8& ..& ...\ [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. [^2]: The Digitized Sky Survey was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. [^3]: For purposes of detection, $\lambda_{\ensuremath{ \mathrm{pn} }}$ and $c_{\ensuremath{ \mathrm{pn} }}$ are parameters of little interest (“nuisance parameters”) and we marginalize the distribution of $S$ over these two parameters.

--- abstract: 'Transition disk objects are pre-main-sequence stars with little or no near-IR excess and significant far-IR excess, implying inner opacity holes in their disks. Here we present a multifrequency study of transition disk candidates located in Lupus I, III, IV, V, VI, Corona Australis, and Scorpius. Complementing the information provided by [*[Spitzer]{}*]{} with Adaptive Optics (AO) imaging (NaCo, VLT), submillimeter photometry (APEX), and echelle spectroscopy (Magellan, Du Pont Telescopes), we estimate the multiplicity, disk mass, and accretion rate for each object in our sample in order to identify the mechanism potentially responsible for its inner hole. We find that our transition disks show a rich diversity in their SED morphology, have disk masses ranging from ${\raisebox{-0.4ex}{${\stackrel{<}{\scriptstyle \sim}}$}}$ 1 to 10 [M$_{\mathrm{JUP}}$]{} and accretion rates ranging from ${\raisebox{-0.4ex}{${\stackrel{<}{\scriptstyle \sim}}$}}$ $10^{-11}$ to $10^{-7.7}$ M$_{\odot}$ yr$^{-1}$. Of the 17 bona fide transition disks in our sample, 3, 9, 3, and 2 objects are consistent with giant planet formation, grain growth, photoevaporation, and debris disks, respectively. Two disks could be circumbinary, which offers tidal truncation as an alternative origin of the inner hole. We find the same heterogeneity of the transition disk population in Lupus III, IV, and Corona Australis as in our previous analysis of transition disks in Ophiuchus while all transition disk candidates selected in Lupus V, VI turned out to be contaminating background AGB stars. All transition disks classified as photoevaporating disks have small disk masses, which indicates that photoevaporation must be less efficient than predicted by most recent models. The three systems that are excellent candidates for harboring giant planets potentially represent invaluable laboratories to study planet formation with the Atacama Large Millimeter/Submillimeter Array..' author: - 'Gisela A. Romero, Matthias R. Schreiber, Lucas A. Cieza, Alberto Rebassa-Mansergas, Bruno Merín, Analía V. Smith Castelli, Lori E. Allen, and Nidia Morrell' title: | The Nature of Transition Circumstellar Disks II.\ Southern Molecular Clouds --- Introduction {#intro} ============ Low-mass pre-main-sequence (PMS) stars are generally separated in two different classes, accreting classical TTauri stars (CTTSs) with broad H$\alpha$ emission lines, blue continuum and near infrared excess and non-accreting weak-line TTauri Stars (WTTSs) with narrow symmetric H$\alpha$ emission lines [e.g. @bertout84-1]. While CTTSs typically show large excess emission from the near-infrared to the millimeter, WTTSs often have no infrared (IR) excess at all. Only a relatively small fraction of TTauri stars are observed in an intermediate transition state with little or no near-IR excess and significant far-IR excess. This clearly indicates that once the inner disk starts to dissipate, the entire disk disappears very rapidly [@wolk+walter96-1; @andrews+williams05-1; @ciezaetal07-1]. The missing near-IR excess combined with the clear presence of an outer disk is the defining characteristic of transition disks. However, a precise and generally accepted definition of what constitutes a transition disk object does not yet exist. The most conservative definition of transition disks, often labeled [*classical transition disks*]{}, consists of objects with no detectable near-IR excess, steeply rising slopes in the mid-IR, and large far-IR excesses [e.g. @muzzerolleetal06-1; @Sicilia-Aguilaretal06-1; @muzzerolleetal10-1]. Being less restrictive, objects with small, but still detectable, near-IR excesses [e.g. @brownetal07-1; @merinetal10-1] can be included, until considering objects with decrement relative to the Taurus median Spectral Energy Distribution (SED) at any or all wavelengths [e.g. @najitaetal07-1; @ciezaetal10-1]. Throughout this paper we follow the latter and broader definition. However, one has to be aware that this broad definition still is mostly sensitive to inner opacity holes but may overlook pre-transitional disks with a gap separating an optically thick inner disk from an optically thick outer disk. Such systems have been identified from Spitzer IRS spectra [@espaillatetal07-1], but can be missed by photometric selection alone. The [*[Spitzer]{}*]{} Space Telescope generated a huge database containing IR observations of PMS stars in star-forming regions. Most importantly, [*[Spitzer]{}*]{} products such as the catalogs of the [*[Cores to Disks (c2d)]{}*]{}[^1] and [*[Gould Belt Spitzer (GB)]{}*]{} Legacy projects [@spezzietal11-1; @petersonetal11-1] provide SEDs from 3.6 to 24 $\mu$m for large numbers of PMS stars. One of the most interesting results concerning transition disk studies with [*[Spitzer]{}*]{} has been the great diversity of SED morphologies [see @williams+cieza11-1 for a review]. The widespread of IR SED morphologies found in transition disk objects cannot be adapted to the classical taxonomy to describe young stellar objects (YSOs) such as the Class I, II, III definitions from @lada87-1. [@ciezaetal07-1] quantified the richness of SED morphologies in terms of two parameters based on the SED shapes considering the longest wavelength at which the observed flux is dominated by the stellar photosphere, [$\lambda_{\mathrm{turn−off}}$]{}, and the slope of the infrared excess, [$\alpha_{\mathrm{excess}}$]{}, computed from [$\lambda_{\mathrm{turn−off}}$]{} to 24 $\mu$m. Studying the diverse population of transition disks is key for understanding circumstellar disk evolution as much of the diversity of their SED morphologies is likely to arise from different physical processes dominating the disk’s evolution. Evolutionary processes that may play an important role include viscous accretion [@hartmannetal98-1], photoevaporation [@alexanderetal06-1], the magneto-rotational instability ([*[MRI]{}*]{}) [@chiang+murray-clay07-1], grain growth and dust settling [@dominik+dullemond08-1], planet formation [@lissauer93-1; @boss00-1] and dynamical interactions between the disk and stellar or substellar companions [@artymowicz+lubow94-1]. As discussed by @najitaetal07-1 [@cieza08-1; @alexander08-1], one can distinguish between some of these processes if certain observational constraints, in addition to the SEDs, are available. To this end, we are performing an extensive ground-based observing program to obtain estimates for the disk masses (from submillimeter photometry), accretion rates (from the velocity profiles of the H$\alpha$ line), and multiplicity information (from AO observations) of [*[Spitzer]{}*]{}-selected disks in several nearby star-forming regions. Our recently completed study of Ophiuchus objects [@ciezaetal10-1 hereafter Paper I] confirms that transition disks are indeed a very heterogeneous group of objects with a wide range of SED morphologies, disk masses ($<$ 0.5 to 40 [M$_{\mathrm{JUP}}$]{}) and accretion rates . Since the properties of the transition disks in our sample point towards different processes driving the evolution of each disk, we have been able to identify strong candidates for the following disk categories: (giant) planet-forming disks, circumbinary disks, grain-growth dominated disks, photoevaporating disks, and debris disks. We here follow the same approach as in Paper I in performing multiwavelength observations to derive estimates on disk masses, accretion rates, and multiplicity. We present submillimeter wavelength photometry (from APEX), high-resolution optical spectroscopy (from the Clay, and Du Pont telescopes), and Adaptive Optics near-IR imaging (from the VLT) for [*[Spitzer]{}*]{}-selected transition circumstellar disks located in the following star forming regions: a) Lupus: I, III, IV, V, VI, b) Corona Australis (CrA), and c) Scorpius (Scp). Transition disks in Southern star-forming regions {#selection} ================================================= The Lupus clouds constitute one of the main southern nearby low-mass star-forming regions containing the following sub-clouds at slightly different distances: Lupus I, IV, V, VI at 150 $\pm$ 20 pc and Lupus III at 200 $\pm$ 20 pc [@comeron08-1]. The clouds are situated in the Lupus-Scorpius-Centaurus OB association spanning over 20 deg in the sky. Their population is dominated by mid M-type PMS stars, but some very late M stars or substellar objects have been found as well thanks to [*[Spitzer]{}*]{} capabilities [see @comeron08-1 for a review]. In general, the ages of the Lupus clouds are estimated to be $\approx\,1.5-4$ Myr, [@hughesetal94-1; @comeronetal03-1]. However, a comprehensive analysis using [*[Spitzer]{}*]{} IRAC and MIPS observations in combination with near-IR (2MASS) data has been performed for Lupus I, III, and IV by the [*[c2d]{}*]{} Legacy Project [@merinetal08-1] and Lupus V, VI by the [*[Gould Belt]{}*]{} Legacy Project [@spezzietal11-1] and revealed a significant difference between the sub-clouds. While Lupus I, III, and IV are dominated by Class II YSOs, Lupus V, VI mostly contain Class III objects. This has been interpreted as a consequence of Lupus V, VI being a few Myrs older than Lupus I, III, and IV by @spezzietal11-1. In any case, the Lupus star-forming regions represent an excellent test-bed for theories of circumstellar disk evolution as their stellar members should span all evolutionary stages. The Scorpius clouds [@nozawaetal91-1; @vilasboasetal00-1] lie on the edge of the Lupus-Scorpius-Centaurus OB association, just north of the well studied Ophiuchus molecular cloud, but it is highly fragmentary and presents much lower levels of star-formation. In fact, the [*[Gould Belt]{}*]{} project only finds 10 YSOs candidates in the 2.1 sq deg. mapped by IRAC and MIPS (Hatchell et al., in preparation). The age of Scp is estimated to be $\sim\,5$ Myr [@preibischetal02-1]. The CrA star-forming region, also mapped by the [*[Gould Belt]{}*]{} project [@petersonetal11-1], contains an embedded association known as the Coronet, a relatively isolated cluster containing HAeBe stars and TTauri stars [@chenetal97-1]. It is situated at a distance of 150 $\pm$ 20 pc out of the Galactic plane, at the edge of the [*[Gould Belt]{}*]{} [see @Sicilia-Aguilaretal08-1 and references therein]. With an age of $\approx$ 1 Myr, the Coronet is younger than the Lupus clouds and has been claimed to host an intriguingly high fraction of classical transition disks of $\approx~50\%$ [@Sicilia-Aguilaretal08-1]. However, @ercolanoetal09-1 convincingly shows that the dust emission in TTauri stars of spectral type M is very small short ward of $6\mu$m which might mimic an inner hole, the defining feature of typical transition disk systems. Based on this finding, @ercolanoetal09-1 estimate a much smaller fraction of transition disks in Coronet, of $\sim15\%$. Target selection ---------------- We have systematically searched the catalogs of the [*[c2d]{}*]{} and [*[Gould Belt]{}*]{} Legacy Projects[^2] applying the broad transition disk definition described in detail in Paper I to the Lupus I, III, IV, V, VI, Scp, and CrA clouds. In brief, we select systems that fulfill the following criteria. 1. Have Spitzer colors \[3.6\]-\[4.5\] $<$ 0.25, which excludes “fulls disks", i.e., optically thick disks extending inward to the dust sublimation radius except in cases with significant dust settling in inner disks around M stars [@ercolanoetal09-1]. 2. Have Spitzer colors \[3.6\]-\[24\] $>$ 1.5, to ensure that all targets have very significant excesses ($>$ 5-10 $\sigma$), unambiguously indicating the presence of circumstellar material. 3. Have S/N $\ge$ 7 in 2MASS, IRAC, MIPS (24 $\mu$m) bands to only include targets with reliable photometry. 4. Have K$_{s}$ $<$ 11 mag, driven by the sensitivity of our near-IR Adaptive Optics observations and to avoid extragalactic contamination. 5. Are brighter than R = 18 mag according to the USNO-B1 [@monetetal03-1], driven by the sensitivity of our optical spectroscopy observations. Compared to the c2d sample discussed in @merinetal08-1 our sample might be slightly biased against very low mass stars and deeply embedded objects because of this brightness limit. These selection criteria result in a primary target list of 60 objects that we did follow-up using different observational facilities to characterize our transition disk candidate sample. Observations {#obs} ============ We performed multiwavelength (optical, infrared, and submillimeter) observations of our targets with the aim to identify which physical process is primarily responsible for their transition disk nature. High resolution optical spectra can be used to estimate spectral types and accretion rates from the velocity dispersion of the H$\alpha$. Near-IR images allow to identify multiple star systems down to projected separations of 0.06-007$''$, corresponding to $8-14$ AU at distances of $130-200$ pc. From single dish submillimeter observations we inferred disk masses. In the following Section, we describe in detail the observations performed and the data reduction. Optical Spectroscopy -------------------- We obtained high resolution (R $>$ 20,000) spectra for our entire sample using 2 different telescopes: Magellan/Clay and Du Pont located at Las Campanas Observatory in Chile. ### Clay–Mike Observations We observed 49 of our 60 targets with the Magellan Inamori Kyocera Echelle (MIKE) spectrograph on the 6.5-m Clay telescope. The observations were performed on 2009 April 27–28 and 2010 June 11–13. Since the CCD of MIKE’s red arm has a pixel scale of 0.13$''$/pixel, we binned the detector by a factor of 3 in the dispersion direction and a factor of 2 in the spatial direction, thus reducing the readout time and readout noise. We used an 1$''$ slit width. The resulting spectra covered 4900–5000Å at a resolving power of 22,000. This corresponds to a resolution of $\sim$0.3 Å at the location of the H$\alpha$ line, and to a velocity dispersion of $\sim$14 km s$^{-1}$. For each object, we obtained a set of 3 or 4 spectra, with exposure times ranging from 3 to 10 minutes each, depending on the brightness of the targets. The data analysis was carried out with IRAF[^3] software. After bias subtraction and flat-field corrections with Milky Flats, the spectra were reduced using the standard IRAF package IMRED:ECHELLE. ### Du Pont–Echelle Observations The remaining 11 targets were observed with the Echelle Spectrograph on the 2.5-m Irénée du Pont telescope. The observations were performed in 2009 May 14–16, and we used an 1$''$ slit width. The CCD’s scale is 0.26$''$/pixel, and we consequently applied a 2$\times$2 binning. The wavelength coverage of the obtained spectra ranged between 4000 and 9000 Å at a resolving power of 32,000 in the red arm. This corresponds to a resolution of $\sim$0.2Å and a velocity dispersion of $\sim$9.4 km s$^{-1}$ in the vicinity of H$\alpha$. For each object we obtained a set of 3 to 4 spectra with exposure times ranging from 10 to 15 minutes each, depending on the brightness of the target. The data analysis was carried out with IRAF. After bias subtraction and flat-field corrections with Milky Flats, the spectra were reduced using the standard package IMRED:ECHELLE. Adaptive Optics Imaging {#s:NaCo} ----------------------- High spatial resolution near-IR observations of our 60 targets were obtained with NaCo (the Nasmyth Adaptive Optics Systems (NAOS) and the Near-IR Imager and Spectrograph (CONICA) camera at the 8.2-m telescope Yepun), which is part of the European Southern Observatory’s (ESO) Very Large Telescope (VLT) in Cerro Paranal, Chile. The data were acquired in service mode during the ESO’s observing period 083 (2009 April 1 – September 30). To take advantage of the near-IR brightness of our targets, we used the infrared wavefront sensor and the N90C10 dichroic to direct 90$\%$ of the near-IR light to the adaptive optics systems and 10$\%$ of the light to the science camera. We used the S13 camera (13.3 mas/pixel and 14$\times$14$''$ field of view) and the Double RdRstRd readout mode. The observations were performed through the K$_s$ and J-band filters at 5 dithered positions per filter. The total exposure times ranged from 1 to 50 s for the K$_s$-band observations and from 2 to 200s for the J-band observations, depending on the brightness of the target. The data were reduced using the Jitter software, which is part of ESO’s data reduction package Eclipse[^4]. Submillimeter Wavelength Photometry ----------------------------------- As discussed in the following section, our spectroscopic observations showed that our initial sample of 60 transition disk candidates was highly contaminated by asymptotic giant branch (AGB) stars. The 17 bona fide PMS stars were observed with the Atacama Pathfinder Experiment (APEX) [^5], the 12-m radio telescope located in Llano de Chajnantor in Chile. The observations were performed during period 083 (083.F-0162A-2009, 9.2 hrs) and period 085 (E-085.C-0571D-2010, 30.9 hrs). We used the APEX-LABOCA camera [@siringoetal09-1] at 870 $\mu$m (345 GHz) in service mode aiming for detections of the dust continuum emission. The nominal LABOCA beam is full width at half-maximum $18.6\pm1.0$” and the pointing uncertainty is $4$”. To obtain the lowest possible flux limit, the most sensitive part of the array was centered on each source. The observations were reduced using the Bolometer array data Analysis package BoA$\footnote{http://www.apex-telescope.org/bolometer/laboca/boa/}$. For both observing runs, Skydips were performed hourly and combined with radiometer readings to obtain accurate opacity estimates. The absolute flux calibration follows the method outlined by [@siringoetal09-1] and is expected to be accurate to within 10$\%$. The absolute flux scale pointing calibrators were determined through observations of either IRAS16342-38 or G34.3 while planets were used to focus the telescope. The telescope pointing was checked regularly with scans on nearby bright sources and was found to be stable within 3” (rms). The period 083 observations were performed using compact mapping mode with raster spiral patterns. The weather conditions were excellent with precipitable water vapor levels below $0.5$ mm. Eight sources (objects \# 1, 2, 5, 9, 12, 15, 16, 17) were observed. On-source integrations of 64 minutes were performed to achieve an rms $\sim$ 7 mJy/beam. The brightest object of the whole sample (\# 2) was the only source detected at submillimeter wavelengths in period 083. During the longer period 085 observing run, the beam switching mode using the wobbling secondary and mapping mode were employed. During this period, the remaining nine sources were observed and object \#12 was re-observed with higher sensitivity. The weather conditions were favorable with precipitable water vapour levels below 1.2 mm. The wobbler observations of each target consist of a set of two loops of 10 scans per target, reaching a total on-source observing time of 48 minutes. An average rms $\sim$ 4 mJy/beam was obtained. In the case of a signal detection on-source position, we took a few maps in order to check for emission contamination from the off-position. In all cases, the contamination was discarded and we confirmed the detection of six sources (\# 3, 7, 8, 10, 11, and 12). Results ======= AGB Contamination {#contamination} ----------------- AGB stars are surrounded by shells of dust and thus have small, but detectable, IR excesses. The [*[Spitzer]{}*]{}-selected YSO samples from [*[c2d]{}*]{} and [*[Gould Belt]{}*]{} catalogs are therefore contaminated by AGB stars. Using high resolution optical spectra, we discovered that 43 objects of our candidates are AGB stars, while the remaining 17 targets are spectroscopically confirmed TTauri stars. We separated contaminating AGB stars from genuine transition disk TTauri stars in the same way as in Paper I, i.e. based on the presence/absence of emission lines associated with chromospheric activity and/or accretion and the presence of the Li 6707 Å absorption line indicating stellar youth. The coordinates, [*[Spitzer]{}*]{} names, the USNO-B1 R-band magnitude, and the near to mid-IR fluxes of the AGB stars contaminating our sample of transition disks are compiled in Table \[t:giants\]. As shown in Table \[t:cont\] the fractional contamination due to AGB stars of our color selected transition disk candidates differs significantly between the different clouds. The number of transition disk candidates is far too small in the case of Lupus I and Scp to draw any conclusions. Our Lupus III, IV, and CrA samples are contaminated by a fraction of AGB stars that is more or less consistent with the contamination in Ophiuchus (see PaperI, section 4.1.2). The small number of transition disks in CrA seems to be in contradiction with the larger sample identified by @Sicilia-Aguilaretal08-1. However, our selection criteria contain relatively strong brightness constraints (in particular $K<11$) due to the design of our follow-up program which excludes most of the systems listed by them. In addition, as mentioned in the introduction, a large fraction of the transition disk candidates of @Sicilia-Aguilaretal08-1 might be classical M-dwarf TTauri stars with intrinsically little near IR excess due to the small color contrast between the disk and the stellar photosphere [@ercolanoetal09-1]. Apparently, LupusV and VI are dramatically more contaminated than Lupus III, IV, i.e. all the color-selected transition disk candidates are in fact AGB stars. This high percentage of contamination is perhaps related to the position in the Galaxy (see Table \[t:cont\]). The Lupus complex occupies 334 $<$ l $<$ 352, +5 $<$ b $<$ + 25, i.e. observing LupusV and VI we are looking towards the galactic center closer to the plane. In contrast, CrA and Ophiuchus (Paper I) are located at higher Galactic latitudes. In any case, the absence of any spectroscopically confirmed transition disk in LupusV, VI puts doubts on the finding of @spezzietal11-1 [see their section 5.1] that the high fraction of ClassIII Lada systems can [*[not]{}*]{} be explained by contamination. So far all ClassIII YSO candidates from these clouds that have been followed up spectroscopically are clearly contaminating background giants. Our sample of transition disks in Lupus V, VI shares 30 ClassIII objects and one ClassII object with the sample investigated by @spezzietal11-1. All these 31 objects turned out to be AGB stars which means that at least $\sim50\%$ and potentially much more of the ClassIII objects from @spezzietal11-1 are not YSOs but giant stars. This result also questions the conclusion of @spezzietal11-1 that Lupus V, VI are significantly older than Lupus I, III. Color selection of AGB star candidates -------------------------------------- The generally large fraction of giant stars in our sample of transition disk candidates allows to investigate possible refinements of our color selection algorithm. Figure \[f:ccont\] shows the color-color diagram of the 60 selected southern transition disk targets of this paper. All transition disk candidates in our sample with $[3.6]-[24] < 1.8$ turned out to be giant stars. This agrees quite well with the results obtained for the Ophiuchus sample where 4/6 transition disk candidates with \[3.6\]-\[24\] $<$ 1.8 had to be classified as giant stars (see PaperI, Fig.1). Consequently, one may derive an estimate of the contamination of YSO catalogs due to background giant stars by applying this simple color cut. Figure\[f:c2d\] and \[f:gb\] show the transition disk candidates and AGB candidates for both the [*[c2d]{}*]{} and [*[GB]{}*]{} catalogs including all star forming regions. Note, that we here apply color selection criteria only, i.e. the requirements of $R < 18$ mag and $K < 11$ mag that have been used to define the transition disk candidate sample for our multiwavelength follow-up program are not incorporated. Instead, here we are interested in estimating the fraction of YSO candidates in a given cloud that are likely to be AGB stars based on their very low $24\mu$m excess. The resulting rough estimates of giant star contamination are given in Table \[giants-all\] separated by catalog and cloud. According to these estimates, the AGB contamination is expected to vary significantly ranging from $\sim1-85$ per cent. This shows that AGB contamination can have an important impact on studies that are based on the pure numbers of YSOs as provided by the [*[c2d]{}*]{} and [*[GB]{}*]{} catalogs. For example, star formation rates as determined e.g. in @heidermanetal10-1 might become significantly smaller if AGB contamination is taken into account. Apparently, applying the new more restrictive color selection could also significantly increase the success-rate of identifying YSOs directly from color selection criteria and future follow-up studies may take this into account. Characterizing southern transition disks ---------------------------------------- The [*[Spitzer]{}*]{} and alternative names, 2MASS and [*[Spitzer]{}*]{} fluxes, and the USNO-B1 R-band magnitudes and the relevant information derived from our follow-up observations for the remaining 17 bona fide transition disk candidates are listed in Tables \[t:mags\] and \[t:obs\]. In what follows, we use the data discussed in Section \[obs\] to characterize our sample of transition disks. ### Spectral types {#s:steprop} In oreder to determine the spectral types of the transition disks in our sample we use the equations by @cruz+reid02-1 that empirically relate the spectral type with the strength of the TiO5 molecular band. The uncertainty of this method is estimated to be $\sim$0.5 subclasses. For most of our transition disk objects, estimates of the spectral types have been provided previously [@hughesetal93-1; @hughesetal94-1; @krautteretal97-1; @walteretal97-1; @Sicilia-Aguilaretal08-1]. The spectral types obtained by us and those given in the literature are listed in Table \[t:obs\] and we find good agreement. All but one system (\# 2) have been classified as M-dwarfs. For target \# 2, we adopt the spectral type K0 given by @hughesetal93-1. ### Multiplicity Binarity can play an important role in the context of transition disks as the presence of a close stellar companion may cause the inner hole, i.e. some of the transition disks in our sample might actually be nothing else but circumbinary disks. Some systems in our sample have been previously identified as wide binaries. @merinetal08-1 carried out an optical survey of the Lupus I, III, IV regions using bands Rc, Ic, ZwI of the Wide-Field Imager (WFI) attached to the ESO 2.2-m telescope at La Silla Observatory. Visual inspection of the images revealed that objects \# 1, \# 3, \# 4, and \# 9 are wide binary systems with companions at projected separations of 420, 1000, 600, and 560 AU considering the distance of the Lupus clouds. We have newly identified six multiple systems by visual inspection of the NaCo images, objects \# 6, \# 7, \# 11, \# 13, \# 16, and \# 17 (see Figure \[f:mult\]). The projected separations are 0.7$''$, 0.4$''$, 1.15$''$, 1.8$''$, 0.5$''$, 0.5$''$ corresponding to 140, 80, 230, 234, 75, 75 AU at the corresponding distances. Object \# 13 is a triple system, i.e. a binary with an additional faint companion at 3$''$ (390 AU). For each binary system, we searched for additional tight companions by comparing each other’s point-spread functions (PSFs). The PSF pairs were virtually identical in all cases, except for target \# 11. The south-west component of this target has a perfectly round PSF, while the south-east component, $\sim$1.5$''$ away, is clearly elongated (see Figure \[f:mult\]). Since variations in the PSF shape are not expected within such small angular distances and this behavior is seen in both the J- and K$_{s}$-band images, we conclude that target \# 11 is likely to be a triple system. We have also searched in the literature for additional companions in our sample that our VLT observations could have missed. In addition to multiple systems discussed above, we find that \# 14 has been reported by @kohleretal08-1 as a binary system with a projected separation of 0.13$''$ (corresponding to 20 AU) and flux ratio of 0.7 using speckle interferometry at the New Technology Telescope in 2001. We see no evidence for a bright companion in our NaCo images (see Figure \[f:mult\]). However, since our AO images were taken 8 years later than the speckle data, it is possible that the projected separation had changed enough in the intervening years for the binary to become unresolved. Hence, our sample consists of eleven multiple systems, i.e. nine binaries (objects \# 1, 3, 4, 6, 7, 9, 14, 16, 17), and two triples (objects \# 11 and 13). Only in the cases of the close B/C pair in object \# 11 and \# 14 the binary separation is small enough to suspect that the companions might have tidally disrupted the circumbinary disk thereby causing the inner hole inferred from the SED. However, in neither case the circumbinary nature can be confirmed because it is unknown whether the IR excess in object \# 11 originates in the wide A component or the close B/C pair and the multiplicity of object \# 14 is not confirmed by our observations. We therefore only consider these two objects to be circumbinary disk [*[candidates]{}*]{}. Table \[t:obs\] lists the projected angular separations of the systems. ### Disk Masses @andrews+williams05-1 [@andrews+williams07-1] modeled the IR and submillimeter SEDs of circumstellar disks and found a linear relation between the submillimeter flux and the disk mass that has been calibrated by @ciezaetal08-1 who obtained $$\begin{aligned} \label{eq:mass} M_{\rm{DISK}} \sim8.0 \times 10^{-2}~[\frac{F_{\nu} (0.86~\rm{mm})} {\rm{mJy}} \times (\frac{d_{1}}{140~\rm{pc}})^2]~M_{\rm{JUP}},\end{aligned}$$ where $d_{1}$ is the distance to the target. As described in Paper I, disk masses obtained with the above relation are within a factor of 2 of model derived values, which is certainly good enough for the purposes of our survey. However, one should keep in mind that larger systematic errors can not be ruled out [@andrews+williams07-1] as long as strong observational constraints on the grain size distributions and the gas-to-dust ratios are lacking. Adopting distances of 150 pc to Lupus IV, 200 pc to Lupus III [@comeron08-1], 130 pc to Scp (Hatchell et al. in preparation), and 150 pc to CrA [@Sicilia-Aguilaretal08-1] we use Equation (\[eq:mass\]) to estimate disk masses for the 17 systems in our sample (see Table \[t:derived\]). 50$\%$ (i.e. 7/17) of the targets have been detected at 8510 $\mu$m (Table \[t:obs\]). The corresponding disks masses range from $1-9$ [M$_{\mathrm{JUP}}$]{}. Adopting a flux value of 3$\sigma$ for targets with non-detected emission, we derive upper limits for the remaining targets of $\sim\,1-4$ [M$_{\mathrm{JUP}}$]{}. Most of our targets have disk masses $<~1-2$ [M$_{\mathrm{JUP}}$]{}, but 5 targets have disk masses typical for CTTSs ($\sim$3 – 10 [M$_{\mathrm{JUP}}$]{}). The most massive disks are detected around , with 9 and 6 [M$_{\mathrm{JUP}}$]{}, respectively. ### Accretion rates The accretion rate is the second crucial parameter necessary to distinguish between the different mechanisms that may form inner opacity holes in circumstellar transitions disks. Most PMS stars show H$\alpha$ emission, either generated from chromospheric activity or magnetospheric accretion [@nattaetal04-1]. While non-accreting objects show rather narrow ($<200$ km s$^{-1}$) and symmetric line profiles of chromospheric origin, the large-velocity magnetospheric accretion columns produce broad ($>200$ km s$^{-1}$) and asymmetric line profiles. As in PaperI we estimate the accretion rates of our transition disk systems according to the empirical relation obtained by [@nattaetal04-1], i.e. $$\label{eq:acc} log(M_{\mathrm{acc}} (M_{\odot}~\rm{yr}^{-1})) = -12.89 (\pm 0.3) + 9.7 (\pm 0.7) \times 10^{-3} \Delta V (\rm{km~s^{-1}})$$ which is supposed to be relatively well calibrated for velocity widths covering , which corresponds to accretion rates of $10^{-11}~\rm{M}_{\odot}~\mathrm{yr}^{-1}<\rm{M}_{\rm{acc}}<10^{-7}~ \rm{M}_{\odot}~\mathrm{yr}^{-1}$. However, the empirical dividing line between accreting and non-accreting objects has been placed slightly shifted by different authors at $\Delta\,V$ between 200 km s$^{-1}$ [@jayawardhanaetal03-1] and 270 km s$^{-1}$ [@white+basri03-1]. For systems with $\Delta\,V\sim\,200-300$ km s$^{-1}$ we therefore separate accreting and non-accreting objects based on the (a)symmetry of the H$\alpha$ emission line profile and take into account the spectral type because accreting lower mass stars tend to have narrower H$\alpha$ emission lines. To measure the H$\alpha$ velocity width $\Delta\,V$ we considered for each system the spectral range that corresponds to H$\alpha\pm 2500$ Km/s. The continuum plus emission profile were fitted with a Gaussian plus parabolic profile. The parabolic fit was then used to normalized the spectrum. A single Gaussian profile was sufficient here, being the emission single or double-peaked, as at this stage we were only interested in obtaining a good parabolic fit for the normalization. Once the continuum had been normalized we measured $\Delta\,V$ at 10 per cent of the peak intensity. The obtained velocity dispersion of the H$\alpha$ emission lines and the corresponding accretion rate estimates are given in Table \[t:obs\] and Table \[t:derived\], respectively. The obtained accretion rates should be considered order-of-magnitude estimates due to the large uncertainties associated with Equation (\[eq:acc\]) and the intrinsic variability of accretion in TTauri stars. Our sample shows a large diversity of H$\alpha$ signatures. Five targets are classified as non-accreting objects that clearly show symmetric and narrow H$\alpha$ emission line profiles ($<$ 200 km s$^{-1}$, see Figure \[f:acc1\]) as expected from chromospheric activity. For all these non-accreting objects, we estimate an upper limit of the accretion rate, i.e. . We classify the remaining 12 transition disk objects as accreting. The majority of them (10) show clearly broad and asymmetric emission-line profiles (see Figure \[f:acc2\]). However, targets \# 8 and \# 12 represent borderline cases with a rather small velocity dispersion for accreting systems ($\Delta\,V\sim200$ km s$^{-1}$) and not clearly asymmetric line profiles. Such borderline systems require a more detailed discussion. Both stars are of late spectral type (M4-M5) and very low-mass stars tend to have narrower H$\alpha$ lines than higher mass objects because of their lower accretion rates [@nattaetal04-1] and their lower gravitational potentials [@muzzerolleetal03-1]. Object \# 12 additionally shows $8\,\mu$m excess emission indicating the presence of an inner disk. Given all the available data, we classify targets \# 8 and \# 12 as accreting objects, but warn the reader that their accreting nature is less certain than that of the rest of the objects classified as CTTSs. As accretion in TTauri stars can well be episodic, mutli-epoch spectroscopy would be useful to unambiguously identify the accreting nature of these two systems. The mass accretion rates estimated for the 12 disks classified as accreting systems range from Discussion ========== The origin of the inner opacity hole ------------------------------------ With the collected information presented in the previous sections we have at hand the following information of the [*[Spitzer]{}*]{}-selected transition disks in our sample: - Detailed SEDs that we quantify with the two-parameter scheme introduced by [@ciezaetal07-1], which is based on the longest wavelength at which the observed flux is dominated by the stellar photosphere, [$\lambda_{\mathrm{turn−off}}$]{}[^6], and the slope of the IR excess, [$\alpha_{\mathrm{excess}}$]{}, computed as $d\log(F_{\lambda})/d\log(\lambda)$ between [$\lambda_{\mathrm{turn−off}}$]{} and $24$$\mu$m. - Multiplicity information from the literature and AO IR imaging. - Disk mass estimates based on measured submillimeter flux. - Accretion rate estimates derived from H$\alpha$ line profiles. This information allows to separate the sample according to the physical processes that are the most likely cause of the inner opacity hole: grain growth, planet formation, photoevaporation, or close binary interactions. In what follows we briefly review each process that might be responsible for the formation of the inner opacity holes, describe our criteria for classifying transition disks, and discuss the corresponding sub-samples of transition disks obtained. Accreting transition disks -------------------------- The presence of accretion in classical transition disk objects raises the question how the inner disk can be cleared of small grains while gas remains in the dust hole. The two mechanisms that can explain the coexistence of accretion and inner opacity holes are grain growth and dynamical interactions with (sub)stellar companions. ### Grain growth Due to both, the higher densities and the faster relative velocities of particles in the inner parts of the disk, this disk region offers much better conditions for dust agglomeration than the outer parts of the disk. Therefore, significant grain growth should start in the inner disk regions. As soon as the grains grow to sizes exceeding the considered wavelength ($r>>\lambda$), the opacity decreases until an inner opacity hole forms. Early models by @dullemond+dominik05-1 taking into account only dust coagulation predict much too short timescales of the order of $10^5$yrs to clear the entire disk of small grains, which is inconsistent with observed SEDs of most classical TTauri stars. A more reliable picture combines coagulation and collisional fragmentation or erosion of large dust aggregates [@dominik+dullemond08-1]. As a gradual transition between the inner and outer disk is predicted by grain growth and dust settling models [@dullemond+dominik08-1; @weidenschilling08-1], grain growth dominated disks should have $\alpha_{\mathrm{excess}}\le\,0$ (i.e., falling mid-IR SEDs) while $\lambda_{\mathrm{turnoff}}$, associated with the size of the hole, can differ over a rather wide range of values. Although grain growth does not directly affect the gas, it may increase accretion because the inner opacity hole can lead to efficient ionization and trigger the [*[MRI]{}*]{} instability [@chiang+murray-clay07-1]. Among the 17 transition disks in our sample, nine objects are accreting and are associated with [$\alpha_{\mathrm{excess}}$]{} $\leq\,0$. The corresponding SEDs are shown in Figure \[f:grain-seds\]. The grain growth candidate systems in our sample could be confounded with accreting classical TTauri M-stars as predicted by @ercolanoetal09-1. However, most of our grain growth dominated disks have SEDs close to the stellar photosphere up to $\sim8-10\,\mu$m and we therefore do not expect significant contamination by non transition disks. The only exceptions being objects \# 5 and 10 with a small value of [$\lambda_{\mathrm{turn−off}}$]{}$=2.2$ and to some extend \# 4 and \# 11 ([$\lambda_{\mathrm{turn−off}}$]{}$=4.5$). We recommend the reader to keep in mind the uncertain classification of these four systems. Compared with the Oph sample (Paper I) the accretion rates obtained for grain growth dominated disks are slightly smaller (i. e. $10^{-9.3}-10^{-11}$[M$_{\odot}$]{}yr$^{-1}$). This might be related to the slightly lower stellar masses or to advanced viscous evolution as discussed in @dalessioetal06-1. The grain growth dominated disks are located in the Lupus III, IV (8) and the CrA (1) star-forming regions. ### Dynamical interactions with (sub)stellar companions The truncation of the disk as the result of dynamical interactions with companions was first proposed by [@lin+papaloizou79-1]. More recently, it has been shown that most PMS stars are in multiple systems with a lognormal semi-major axis distribution centered at $\sim\,30$AU [e.g., @ratzkaetal05-1]. A significant fraction of the binaries in the star-forming regions considered here should therefore be tight binaries with separations of $1-20$AU. Disks in such close binary system will be tidally truncated at $\sim\,2~\times$ the binary separation and a circumbinary disk with an inner hole is formed [@artymowicz+lubow94-1]. The corresponding SED is that of a transition disk. However, the circumbinary nature does not exclude additional evolutionary processes to be at work and we therefore provide an additional classification based on the disk properties only (see Table \[t:derived\]). Identification of companions that may cause the formation of a circumbinary disk is possible either due to high-resolution imaging or by measuring radial velocity variations. As described in Sect. \[s:NaCo\], we identified 2 circumbinary disk candidates among our 17 transition disk systems. One of them, object \# 11 was discovered by inspecting the NaCo images obtained with the VLT, while the close binary nature of object \# 14 has been discovered by @kohleretal08-1 using speckle interferometry at the NTT. Object \# 11 shows signs of strong accretion and has a SED with $\alpha_{\mathrm{excess}}\sim-1$ in agreement with grain growth. It is currently not clear under which conditions gap-crossing streams can exist and allow accretion onto the central star to proceed, but signs of accretion in circumbinary systems [@carretal01-1; @espaillatetal07-1] indicate that accretion is likely to continue. On the other hand, object \# 14 is a non-accreting system such as the known binary CoKu Tau$/$4 [@ireland+kraus08-1]. According to its [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{} ratio we classify this system as a circumbinary/photoevaporation disk candidate. As a final note of caution, we would like to stress that both objects discussed above (\# 11 and \# 14) are circumbinary disk [*[candidates]{}*]{}. As all but one of our targets are M-type stars, most companions potentially responsible for their transition disk SEDs are expected to lie at closer separations than those probed by the AO images. Therefore, our sample of circumbinary disk candidates is incomplete and heavily biased towards large separations. Methods more sensitive to closer companions such as aperture masking and/or radial velocity observations are required to draw firm conclusions on circumbinary disk fractions. ### Giant Planet formation The most exciting way to produce a transition disk SED is by giant planet formation. According to early models as well as recent numerical simulations, the formation of giant planets involves the formation of gaps and holes in the circumstellar disk [@lin+papaloizou79-1; @artymowicz+lubow94-1]. As in the case of (sub)stellar companions it is uncertain if and to what extent accretion proceeds in the presence of a forming giant planet. Therefore, the most important sign of ongoing planet formation remains a sharp inner hole, usually corresponding to [$\alpha_{\mathrm{excess}}$]{}$>0$ (i.e., a rising mid-IR SED). However, although very useful, the definition of [$\alpha_{\mathrm{excess}}$]{} is incomplete, as the SED may also steeply rise at wavelengths longer than $24\,\mu$m. A spectacular example illustrating this is given by object \# 3. While  [$\alpha_{\mathrm{excess}}$]{}$\sim \,-2.2$, the SED steeply rises between $24\,\mu$m and $70\,\mu$m. Furthermore, object \#3 shows clear signs of accretion ($M_{\mathrm{acc}}=~10^{-10.1}$ [M$_{\odot}$]{}/yr) and of harboring a relatively massive disk ([M$_{\mathrm{DISK}}$]{}$ ~\sim 6 $ [M$_{\mathrm{JUP}}$]{}). Since this is a very atypical object, we verified that the large 70 $\mu$m flux is not contaminated by extended emission from the molecular cloud. We examined the 24 and 70$\mu$m mosaics and verified that the detections are consistent with point sources at the source location (see Figure \[f:tran6\]). A more typical transition disk system that might represent a currently planet forming disk is target \#15 with a clearly positive value of [$\alpha_{\mathrm{excess}}$]{} and a high accretion rate. A borderline case between grain growth and planet forming disks is object \# 2. A high accretion rate combined with [$\alpha_{\mathrm{excess}}$]{}$\sim0$ could be consistent with both scenarios. Keeping in mind the ambiguity, we classify object \# 2 as a planet forming disk candidate because it could potentially be an extremely interesting object. The SED of object \# 2 might be explained by a discontinuity in the grain size distribution rather than an inner opacity hole. While the inner part of the disk still contains small grains, outer regions of the disk might be dominated by slightly larger dust particles. Such a scenario is in excellent agreement with the predictions of numerical simulations performed by [@riceetal06-1]. They show that the planet–disk interaction at the outer edge of the gap cleared by a planet can act as a filter passed by small particles only which produces a discontinuity in the dust particle size. To firmly establish its nature object\# 2 deserves further follow-up observations (e.g., high resolution imaging with ALMA). The SEDs of the three candidates for ongoing giant planet formation in our sample are shown in Figure \[f:planet-seds\]. The hosting forming giant planets candidates are located in the Lupus III, IV (2) and the CrA (1) star-forming regions. Non-accreting objects --------------------- The second main class of transition disks are those that do not show signs of accretion. In such disks the inner opacity hole, i.e. the lack of small dust particles in the inner disk regions, is likely to coincide with a gas hole, i.e. the inner disk is completely drained. ### Photoevaporating disk {#sed_mor} The most important process for clearing the inner disk in transitions disks that do not accrete is photoevaporation [e.g. @alexanderetal06-1]. According to this model, extreme-ultraviolet (EUV) photons, originating in the stellar chromosphere, ionize and heat the circumstellar hydrogen which is then partly lost in a wind. This process is supposed to work in all circumstellar disks but becomes important only when the accretion rate drops to values similar to the photoevaporation rate. Then, the inner disk drains on the viscous timescale supported by the generation of the [*[MRI]{}*]{} [@chiang+murray-clay07-1]. Once an inner hole has formed, the inner disk rim is efficiently radiated and the entire disk should therefore quickly disappear. Photoevaporating disks should have negligible accretion [@williams+cieza11-1]. To separate photoevaporating disks from debris disks, we require the disk luminosity to be higher than that seen in the brightest bona-fide debris disks, i.e. [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{} $\geq\,10^{-3}$ [@brydenetal06-1; @wyatt08-1]. We thus obtained $70~\mu$m upper limits from the noise of the $70~\mu$m [*[Spitzer]{}*]{} mosaics at the source location and calculated [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{} for our sample by integrating the stellar fluxes and disk fluxes over frequency (see Section 5.1.3 in Paper I for details of the $70~\mu$m data analysis and the [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{} calculation). We classify three transition disks as photoevaporating disk candidates with negligible accretion (M$_{\mathrm{acc}}$ $<$ 10$^{-11}$ yr$^{-1}$) but [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{} $\geq\,10^{-3}$ (Table \[t:derived\]). According to our submillimeter measurements, all these three systems have small disk masses ($< 1-3.4$ [M$_{\mathrm{JUP}}$]{}, Table \[t:derived\]). In fact, for all photoevaporation candidates we could only derive upper limits on the disk mass. The SEDs of the three systems classified as photoevaporating disk objects are shown in Figure \[f:photo-seds\]. The photoevaporated disks are located in the Lupus III, IV (2) and CrA (1) star-forming regions. ### Debris disk Photoevaporation can be considered as a transition stage between primordial and debris disks. Debris disks contain a small amount of dust and are gas-poor. We find two debris disk candidates, i.e. non-accreting systems with [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{}$<\,10^{-3}$, among our 17 transition disks (see Figure \[f:debris-seds\]). The debris disks are located in the CrA (1) and Scp (1) star-forming regions. Implications for disk evolution ------------------------------- ### Heterogeneity of transition disks In the previous sections, we presented detailed follow-up observations of 60 [*[Spitzer]{}*]{}-selected transition disk candidates located in the southern star-forming regions Lupus I, III, IV, V, VI, CrA, and Scp. Optical spectroscopy revealed that only 17 systems of these candidates are genuine transition disk TTauri stars. Deriving estimates for the accretion rates, disk masses, and multiplicity of these 17 systems we classified them as dominated by grain growth (9), giant planet formation (3), photoevaporation (3), or being in the final debris disk stage (2). Two of these transition objects, one grain growth (\# 11) and one photoevaporating (\# 14), are circumbinary disk candidates, which offers the possibility of tidal truncation as mechanism responsible for an inner hole in the common/shared disk. Combining these results with those presented in PaperI, we now have at hand well-defined and well characterized samples of transition disks from several different star-forming regions. Figure \[f:alp-lam\] summarizes the properties of these samples based on [$\alpha_{\mathrm{excess}}$]{} and [$\lambda_{\mathrm{turn−off}}$]{}. The main aim of these series of papers is to progress with our understanding of circumstellar disk evolution and to compare transition disk samples of different clouds is key in this respect. Table \[t:fractions\] shows the fractions of different types of transition disks[^7] for Ophiuchus (age $\approx$ $0.3-2.1$ Myr, @wilkingetal05-1 [and references therein]), CrA (age $\approx$ 1 Myr, @Sicilia-Aguilaretal08-1), and Lupus I, III, IV (age $1.5-4$Myr, @merinetal08-1). All YSO candidates followed up spectroscopically located in Lupus V, VI turned out to be AGB stars. These clouds have been recently estimated to be a few Myrs older [@spezzietal11-1] based on the dominance of Class III systems. As we have shown in Section \[contamination\], at least $\sim50$% of the claimed class III systems located in Lupus V, VI are very likely to be AGB stars. This reduces the fraction of classIII objects to values similar to those obtained for Lupus III. Based on this, Lupus V, VI, and III could well be of a very similar age. The main result that can be obtained from Table \[t:fractions\] clearly is that young clouds (${\raisebox{-0.4ex}{${\stackrel{<}{\scriptstyle \sim}}$}}\ 1-4$ Myr) contain a mixture of grain growth, photoevaporating, debris, and tidally disrupted transition disks. It is clear that all states of disk evolution are already present at this age range, which implies that different disks evolve at different rates and/or through different evolutionary paths. An important difficulty in constraining disk evolution is that stellar ages obtained from isochrones are very uncertain for individual systems. An analysis of the stellar age distributions of each disk category is therefore postponed to paper III (Cieza et al. 2012, ApJ submitted), where we discuss a larger sample of well characterized transition disk objects including the systems presented here. ### Evidence for low photoevaporation rates The general picture of photoevaporation is the following. In very young disks, the accretion rate largely exceeds the evaporation rate and the disk evolves virtually unaffected by photoevaporation. As the accretion rate is decreasing with time, the disk necessarily reaches the time when the accretion rate equals the photoevaporation rate and the outer disk is no longer able to resupply the inner disk with material. At this point, the inner disk drains on the viscous timescale (${\raisebox{-0.4ex}{${\stackrel{<}{\scriptstyle \sim}}$}}$ $10^5$ yr) and an inner hole of a few AU in radius is formed in the disk. The inner disk edge is now directly exposed to the EUV radiation and the disk rapidly photoevaporates from the inside out. Early models of EUV photoevaporation predict evaporation rates of $10^{-10}-10^{-9}$[M$_{\odot}$]{}/yr [@hollenbachetal94-1]. More recent simulations taking into account X-ray [@owenetal10-1] and/or far-ultraviolet (FUV) irradiation [@gortietal09-1] in addition to the EUV photons, largely exceed these early predictions, reaching photoevaporation rates of the order of $10^{-8}$[M$_{\odot}$]{}/yr [see also @armitage11-1]. As in steady state accretion disks the mass transfer through the disk is roughly proportional to the mass accretion rate onto the star, a crucial prediction of the photoevaporation model is that high photoevaporation rates imply high disk masses at the time the inner disk is drained. In particular, models with efficient X-ray photoevaporation predict a significant population of relatively massive ($\sim$7 [M$_{\mathrm{JUP}}$]{}) non-accreting transition disks [@owenetal11-1]. Figure \[f:photo-mass\] shows the upper limits (derived from submillimeter non detections) on the disk masses of the photoevaporating transition disks in all the clouds we considered so far. Even taking into account uncertainties in our classification of photoevaporation candidates, it is evident that large numbers of non-accreting but massive disks do not exist. This indicates that photoevaporation is less efficient than predicted by the models described above. However, one has to take into account that the sample of transition disks considered here contains low-mass stars only while model calculations have been performed exclusively for more massive stars $\sim 1$[M$_{\odot}$]{}. Therefore, either a more homogeneous sample of photoevaporating disk systems covering a larger range of host star masses (earlier spectral types) or simulations of photoevaporation for disks around low-mass stars are required to provide a final answer on this issue. Current limitations and future perspectives ------------------------------------------- Of course, our classification of transition disk objects is based on rather rough empirical relations and requires to carefully consider possible caveats. An obvious uncertainty concerns our multiplicity survey. The method of direct detection of companions is obviously more sensitive to binaries with large separations and low inclinations. Our NaCo observations are sensitive to projected separations of $\sim10-15$AU given the distance to our targets and – depending on the intrinsic distribution of orbital separations – we may therefore miss a significant fraction of close binaries. To overcome this observational bias we are currently performing radial velocity measurements of our targets using VLT/UVES. The method of detecting radial velocity variations is more sensitive to small separations and high inclinations and therefore complements the imaging results presented here. We will present the results in a forthcoming paper. However, the fact that only 6 of the 43 transition disks studied herein and in Paper I are circumbinary disk candidates strongly suggests that binaries at the peak of their separation distribution ($\sim$ 30 AU) do not result in transition disk objects as such stellar binaries would be easily detectable by our AO observations. Instead, they are likely to destroy the disk rather quickly [@ciezaetal09-1]. Another uncertainty in our classification procedure is the rather ad-hoc separation between photoevaporating and debris disk systems by using a limit in [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{}. However, there is a physical and not only phenomenological difference between these two types of transition disks. Photoevaporating disks are dissipating primordial disks and should have gas rich outer disks while the debris disks should be gas poor. Molecular line observations with the Atacama Large Millimeter/Submillimeter Array (ALMA) of non-accreting disks will be able to distinguish between the two types of objects. A huge problem related to the process of photoevaporation is that the mass loss rates predicted by different models differ by up-to two orders of magnitude [see e.g. @williams+cieza11-1]. The disk mass at the time photoevaporation opens a hole in the disk is directly connected to the photoevaporation rates. Measuring the disk masses of photoevaporating disks could therefore significantly constrain theoretical models of photoevaporation. However, all of the photoevaporting disk candidates remain undetected and we can only put upper limits to their masses. Fortunately, ALMA will be much more sensitive than all presently available telescopes and will soon be able to measure the masses of many bona fide photoevaporating disks. ALMA should also be able to measure, through high resolution continuum observations at multiple wavelengths, the radial dependence in the grain size distribution expected in the grain-growth dominated disks. Finally, the recent identification using the aperture masking technique of what seems to be forming planets within the inner cavities of the transition disks around T Cha [@huelamoetal11-1] and LkCa 15 [@kraus+ireland12-1] strongly encourages to obtain similar observations for the three planet-forming disk candidates identified herein, objects \# 2, \# 3, and \# 15. Any system with a planet still embedded in a primordial disk would represent an invaluable laboratory to study planet formation with current and future instrumentation. Summary ======= We have carried out a multifrequency study of [*[Spitzer]{}*]{}-selected YSO transition disk candidates located in the Lupus complex (53), CrA (5), and Scp (2). We obtained submillimeter observations (APEX), optical high resolution echelle spectroscopy (Clay/Mike, Du Pont/echelle), and NIR images (from AO imaging VLT/NaCo). After deriving spectral types of each target, 43 AGB stars were removed (Lupus complex (41), CrA (1), and Scp (1)), leaving a sample of 17 genuine transition disk systems. We find that the vast majority of AGB stars have \[3.6\]-\[24\] $<$ 1.8, underscoring the need for a spectroscopic confirmation of YSO candidates with small 24$\mu$m excesses. The data obtained for the 17 transition systems allows to estimate multiplicity, stellar accretion rates, and disk masses thereby allowing to identify the physical mechanism that is most likely to be responsible for the formation of the inner opacity hole. The observational results of this study can be summarized as follows: 1. The derived spectral classification indicates that all but one (object \# 2, K0) central star are M-type stars, in agreement with previous results [@comeron08-1]. 2. 12/17 targets are accreting objects (i.e. asymmetric H$\alpha$ profile having a velocity width  200 km s$^{-1}$ at 10$\%$ of peak intensity). 3. $\sim$ 50$\%$ of the sample are multiple systems and among them, two triple systems. Two binary systems have small projected separations and are therefore candidates to host a circumbinary disk. 4. 7/17 targets have flux detection in the submillimeter. For the remaining systems, we derive and upper limit of the disk mass (corresponding to a flux of 3 $\times$ rms). The estimated disk masses for the detected objects cover the range 2 [M$_{\mathrm{JUP}}$]{}–10 [M$_{\mathrm{JUP}}$]{}. Combining the derived estimates of disk masses, accretion rates and multiplicity with the SED morphology and fractional disk luminosity ([L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{}) allows to classify the disks as strong candidates for the following categories: - 9/17 grain growth-dominated disks (accreting objects with negative SED slopes in the mid-IR, [$\alpha_{\mathrm{excess}}$]{}$<$ 0). - 3/17 photoevaporating disks (non-accreting objects with disk mass $<$ 3 [M$_{\mathrm{JUP}}$]{}, but ). - 2/17 debris disks (non-accreting objects with disk mass $<$ 2.1 [M$_{\mathrm{JUP}}$]{} and ). - 2/17 circumbinary disks (a binary tight enough to accommodate both components within the inner hole). - 3/17 giant planet forming disk (accreting systems with SEDs indicating sharp inner holes). Inspecting in more detail the different sub-clouds analyzed in this study we find the same heterogeneity of the transition disk population in Lupus III, IV, CrA as in our previous analysis of transition disks in Ophiuchus [@ciezaetal10-1 PaperI]. We therefore conclude that photoevaporation, giant planet formation, and grain growth produce inner holes on similar timescales. Not a single transition disks has been found in Lupus I, V, VI. All 33 candidates that have been spectroscopically followed up turned out to be AGB stars which questions the recent interpretation of @spezzietal11-1 that Lupus I, V, VI might be relatively old star forming regions dominated by ClassIII objects. In addition, our detailed observational analysis of transition disks provides clear constraints on theoretical models of disk photoevaporation by the central star. According to the large evaporation rates predicted by recent models [i.e. see @armitage11-1], large numbers of massive photoevaporating transition disks systems should exist. In contrast to this prediction, all photoevaporating disk candidates identified in this work and PaperI contain very little mass, indicating much smaller evaporation rates at least for the low-mass stars considered here. Similarly, the low incidence of circumbinary transition disk candidates ($\sim$ 10$\%$) supports the idea that most disks are destroyed rather quickly by companions at $\sim 10-40$ AU separations. Finally, we emphasize that the 43 transition disk systems discussed in this work and in PaperI represent the currently largest and most homogeneous sample of well-characterized transition disks. Further investigating these systems with new observing capabilities such as ALMA therefore holds the potential to significantly improvement our understanding of the physical processes driving circumstellar disk evolution. GAR was supported by ALMA/Conicyt (grant 31070021) and ESO/comité mixto. MRS acknowledges support from Millennium Science Initiative, Chilean ministry of Economy: Nucleus P10-022-F and Fondecyt (grant 1100782). LAC acknowledges support provided by NASA through the [*[Sagan]{}*]{} Fellowship Program. ARM thanks for financial support from Fondecyt in the form of grant number 3110049, ESO/comite mixto and Gemini/Conicyt (32080023). ASC was supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina, Agencia Nacional de Promoción Científica y Tecnológica and Universidad Nacional de La Plata (Argentina). We finally thank Dr. Giorgio Siringo and Dr. C. De Breuck for assistance with performing the APEX observations and the corresponding data reduction. We are also grateful for the support of the staff at Las Campanas Observatory. A special thank is given to Dr. M. Orellana and Evelyn Puebla for their help during the first Las Campanas observing run. APEX is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. This work makes use of data obtained with the [*[Spitzer]{}*]{} Space Telescope, which is operated by JPL/Caltech, under a contract with NASA. [*[Facilities]{}*]{}: [*[Spitzer]{}*]{} (IRAC, MIPS), VLT:Yepun, Magellan: Clay, Du Pont (Echelle) [70]{} natexlab\#1[\#1]{} , R. 2008, New Astron. Rev., 52, 60 , R. D., [Clarke]{}, C. J., & [Pringle]{}, J. E. 2006, MNRAS, 369, 229 , S. M., & [Williams]{}, J. P. 2005, ApJ, 631, 1134 —. 2007, ApJ, 671, 1800 , P. J. 2011, ARA&A, 49, 195 , P., & [Lubow]{}, S. H. 1994, ApJ, 421, 651 , C. 1984, Reports on Progress in Physics, 47, 111 , A. P. 2000, ApJ, 536, L101 , J. M., [et al.]{} 2007, ApJ, 664, L107 , G., [et al.]{} 2006, ApJ, 636, 1098 , J. S., [Mathieu]{}, R. D., & [Najita]{}, J. R. 2001, ApJ, 551, 454 , H., [Grenfell]{}, T. G., [Myers]{}, P. C., & [Hughes]{}, J. D. 1997, , 478, 295 , E., & [Murray-Clay]{}, R. 2007, Nature Physics, 3, 604 , L., [et al.]{} 2007, ApJ, 667, 308 , L. A. 2008, in ASP Conf. Ser., Vol. 393, New Horizons in Astronomy, ed. [A. Frebel, J. R. Maund, J. Shen, & M. H. Siegel]{}, 35 , L. A., [Swift]{}, J. J., [Mathews]{}, G. S., & [Williams]{}, J. P. 2008, ApJ, 686, L115 , L. A., [et al.]{} 2009, ApJ, 696, L84 —. 2010, ApJ, 712, 925 , F. 2008, The Lupus Clouds, Handbook of Star Forming Regions (London: The Southern Sky ASP Monograph Publications,Vol. 5. Edited by Bo Reipurth, p.295), 295–300 , F., [Fern[á]{}ndez]{}, M., [Baraffe]{}, I., [Neuh[ä]{}user]{}, R., & [Kaas]{}, A. A. 2003, , 406, 1001 , K. L., & [Reid]{}, I. N. 2002, , 123, 2828 , P., [Calvet]{}, N., [Hartmann]{}, L., [Franco-Hern[á]{}ndez]{}, R., & [Serv[í]{}n]{}, H. 2006, , 638, 314 , C., & [Dullemond]{}, C. P. 2008, A&A, 491, 663 , C. P., & [Dominik]{}, C. 2005, A&A, 434, 971 , B., [Clarke]{}, C. J., & [Robitaille]{}, T. P. 2009, , 394, L141 , C., [et al.]{} 2007, ApJ, 664, L111 , E., [et al.]{} 2006, ApJS, 165, 568 , U., & [Hollenbach]{}, D. 2009, , 690, 1539 , L., [Calvet]{}, N., [Gullbring]{}, E., & [D’Alessio]{}, P. 1998, ApJ, 495, 385 , P. H., [Allard]{}, F., & [Baron]{}, E. 1999, , 512, 377 , A., [Evans]{}, II, N. J., [Allen]{}, L. E., [Huard]{}, T., & [Heyer]{}, M. 2010, , 723, 1019 , D., [Johnstone]{}, D., [Lizano]{}, S., & [Shu]{}, F. 1994, , 428, 654 , N., [Lacour]{}, S., [Tuthill]{}, P., [Ireland]{}, M., [Kraus]{}, A., & [Chauvin]{}, G. 2011, , 528, L7 , J., [Hartigan]{}, P., & [Clampitt]{}, L. 1993, , 105, 571 , J., [Hartigan]{}, P., [Krautter]{}, J., & [Kelemen]{}, J. 1994, , 108, 1071 , M. J., & [Kraus]{}, A. L. 2008, ApJ, 678, L59 , R., [Mohanty]{}, S., & [Basri]{}, G. 2003, ApJ, 592, 282 , R., [Neuh[ä]{}user]{}, R., [Kr[ä]{}mer]{}, S., [Leinert]{}, C., [Ott]{}, T., & [Eckart]{}, A. 2008, , 488, 997 Kraus, A. L., & Ireland, M. J. 2012, , 745, 5 , J., [Wichmann]{}, R., [Schmitt]{}, J. H. M. M., [Alcala]{}, J. M., [Neuhauser]{}, R., & [Terranegra]{}, L. 1997, , 123, 329 , C. J. 1987, in IAU Symposium, Vol. 115, Star Forming Regions, ed. [M. Peimbert & J. Jugaku]{}, 1–17 , D. N. C., & [Papaloizou]{}, J. 1979, , 186, 799 , J. J. 1993, ARA&A, 31, 129 , B., [et al.]{} 2008, , 177, 551 —. 2010, , 718, 1200 , D. G., [et al.]{} 2003, AJ, 125, 984 , J., [Allen]{}, L. E., [Megeath]{}, S. T., [Hern[á]{}ndez]{}, J., & [Gutermuth]{}, R. A. 2010, ApJ, 708, 1107 , J., [Hillenbrand]{}, L., [Calvet]{}, N., [Brice[ñ]{}o]{}, C., & [Hartmann]{}, L. 2003, ApJ, 592, 266 , J., [et al.]{} 2006, ApJ, 643, 1003 , J. R., [Strom]{}, S. E., & [Muzerolle]{}, J. 2007, MNRAS, 378, 369 , A., [Testi]{}, L., [Muzerolle]{}, J., [Randich]{}, S., [Comer[ó]{}n]{}, F., & [Persi]{}, P. 2004, A&A, 424, 603 , S., [Mizuno]{}, A., [Teshima]{}, Y., [Ogawa]{}, H., & [Fukui]{}, Y. 1991, , 77, 647 , J. E., [Ercolano]{}, B., & [Clarke]{}, C. J. 2011, , 412, 13 , J. E., [Ercolano]{}, B., [Clarke]{}, C. J., & [Alexander]{}, R. D. 2010, , 401, 1415 , D. E., [et al.]{} 2011, , 194, 43 , T., [Brown]{}, A. G. A., [Bridges]{}, T., [Guenther]{}, E., & [Zinnecker]{}, H. 2002, , 124, 404 , T., [K[ö]{}hler]{}, R., & [Leinert]{}, C. 2005, A&A, 437, 611 , W. K. M., [Armitage]{}, P. J., [Wood]{}, K., & [Lodato]{}, G. 2006, MNRAS, 373, 1619 , A., [Henning]{}, T., [Juh[á]{}sz]{}, A., [Bouwman]{}, J., [Garmire]{}, G., & [Garmire]{}, A. 2008, , 687, 1145 , A., [et al.]{} 2006, ApJ, 638, 897 , G., [et al.]{} 2009, , 497, 945 , L., [et al.]{} 2011, ApJ, 730, 65 , J. W. S., [Myers]{}, P. C., & [Fuller]{}, G. A. 2000, , 532, 1038 , F. M., [Vrba]{}, F. J., [Wolk]{}, S. J., [Mathieu]{}, R. D., & [Neuhauser]{}, R. 1997, , 114, 1544 , S. J. 2008, Physica Scripta Volume T, 130, 014021 , R. J., & [Basri]{}, G. 2003, ApJ, 582, 1109 , B. A., [Meyer]{}, M. R., [Robinson]{}, J. G., & [Greene]{}, T. P. 2005, AJ, 130, 1733 , J. P., & [Cieza]{}, L. A. 2011, ARA&A , S. J., & [Walter]{}, F. M. 1996, AJ, 111, 2066 , M. C. 2008, ARA&A, 46, 339 ![ Spitzer color-color diagram for the southern sample. Contaminating AGB stars greatly dominate the sample with . []{data-label="f:ccont"}](Figure1.pdf){width="5.0in"} ![\[f:c2d\] Spitzer [*[c2d]{}*]{} systems classified into YSO candidates, AGB candidates, and transition disk candidates according to simple color cuts based on the results of our spectroscopic follow-up program (see text for more details). ](Figure2.pdf){width="5.0in"} ![\[f:gb\] Same as Fig.\[f:c2d\] but for the [*[GB]{}*]{} catalog. ](Figure3.pdf){width="5.0in"} ![The K$_{s}$-band images of the six multiple systems that have been detected with our VLT-AO observations (objects \# 6, 7, 11, 13, 16, and 17) and of object \# 14, which has been identified as a close binary with a 0.13$''$ separation from speckle observations in 2001 [@kohleretal08-1]. The putative companion in object \# 14 remains unresolved by our 2009 observations. Targets \# 11 and 13 are triple systems. In the former case, the tighter components are not fully resolved, but their presence can be inferred from the highly elongated image (lower right panel). []{data-label="f:mult"}](Figure4.pdf){width="5in"} ![\[f:acc1\]Normalized average H$\alpha$ profiles for non-accreting objects with clear H$\alpha$ emission. The horizontal dashed line indicates the 10% peak intensity, where $\Delta$V is measured. The velocity widths are $<$ 200 km s$^{-1}$ and the line profiles are symmetric.](Figure5.pdf){width="5in"} ![\[f:acc2\] Normalized H$\alpha$ profiles of the 12 accreting objects. These systems are considered accreting because either the velocity width is $>$ 200 km s$^{-1}$ or the line profile is asymmetric. Note that objects \#8 and 12 represent borderline systems as the $\Delta$V$\sim200$km/s and the line is not clearly asymmetric (see text for more details).](Figure6.pdf){width="5in"} ![SEDs of the 9 grain growth candidates. Disk masses range from $<$ 1.1 to 5.1 [M$_{\mathrm{JUP}}$]{}, and accretion rates from $10^{-11}$ to $10^{-9.3}$ . The filled circles are detections, while the arrows represent 3$\sigma$ limits. The open squares correspond to the observed optical and near-IR fluxes before being corrected for extinction as in Paper I. For each object, the average of the two R-band magnitudes (from the USNO-B1 catalog) listed in Table \[t:mags\] has been used. The classification of object \# 5 and 10 as grain growth dominated is slightly uncertain as classical TTauris stars of late M-types can have similar SEDs [@ercolanoetal09-1]. The solid line represents the stellar photosphere normalized to the extinction-corrected J band. The dotted lines correspond to the median mid-IR SED of CTTSs calculated by [@furlanetal06-1]. The dashed lines are the quartiles. []{data-label="f:grain-seds"}](Figure7.pdf){width="5in"} ![ The 24 and 70 $\mu$m images of object \# 3. We find no evidence for extended emission from the molecular cloud. In both mosaics, the detections are consistent with a point source at the location of the target (marked by the crosshairs). See the electronic edition of the Journal for a color version of this figure. []{data-label="f:tran6"}](Figure8.pdf){width="5in"} ![SEDs of the three planet forming candidates. While object \# 15 can be considered a classical planet forming candidate system with [$\alpha_{\mathrm{excess}}$]{} $>0$ and , the other two systems are somewhat peculiar: target \# 3 shows a very steep rise in flux observed at , which indicates a very large inner hole and being relatively close to a full disk but with signs for a small and sharp inner hole. Disk masses are 9.1, 5.6 and  for objects \# 2, \# 3, and \# 15; respectively. The solid line as well as the dashes and dotted lines are the same as in Fig.\[f:grain-seds\]. []{data-label="f:planet-seds"}](Figure9.pdf){width="3.0in"} ![SEDs of the 3 transition disks classified as photoevaporating disk candidates. The symbols, solid line as well as the dashes and dotted lines are the same as in Figure \[f:grain-seds\]. None of these systems have been detected at submillimeter wavelength and consequently only upper limits for the disk masses could be derived. We conclude that photoevaporation seems to be less efficient than has recently been suggested. []{data-label="f:photo-seds"}](Figure10.pdf){width="8.0cm"} ![SEDs of the 2 debris disk candidates identified in our sample. We distinguish between debris disks and photoevaporation applying an ad-hoc limit on [L$_{\mathrm{DISK}}$]{}/[L$_{*}$]{} ${\stackrel{<}{\scriptstyle \sim}}$ 10$^{-3}$. The detection of the gas component in photoevaporating disks (e.g. with ALMA) may lead to a more physically motivated separation between the two sub-samples. The solid line as well as the dashes and dotted lines are the same as in Figure \[f:grain-seds\]. \[f:debris-seds\] ](Figure11.pdf){width="8cm"} ![ [[$\alpha_{\mathrm{excess}}$]{} vs [$\lambda_{\mathrm{turn−off}}$]{} for transition disks identified in PaperI and the present work. The locations of the Ophiuchus transition disk sample from this work (top panel) and PaperI (bottom panel) cover very similar ranges in the [$\alpha_{\mathrm{excess}}$]{}  versus [$\lambda_{\mathrm{turn−off}}$]{} plane. Different symbols indicate different formation processes of the inner opacity hole. In general, the Lupus, Cra, and Scp data confirm our previous findings: planet forming disks have large values of [$\alpha_{\mathrm{excess}}$]{}  and [$\lambda_{\mathrm{turn−off}}$]{}, grain growth dominated disks should have small [$\alpha_{\mathrm{excess}}$]{} , but cover the entire range of [$\lambda_{\mathrm{turn−off}}$]{}; and debris disks have extremely low values of [$\alpha_{\mathrm{excess}}$]{}  and the IR excess starts at long [$\lambda_{\mathrm{turn−off}}$]{}. However, two systems clearly show that [$\alpha_{\mathrm{excess}}$]{} and [$\lambda_{\mathrm{turn−off}}$]{} alone cannot fully characterize transition disks. We identified one planet forming disk candidate with [$\alpha_{\mathrm{excess}}$]{}$<0$ but indications of a sharp hole at longer wavelength (object \# 3) and one planet forming candidate with [$\lambda_{\mathrm{turn−off}}$]{} = 4.5 $\mu$m (object \# 2) have been found. ]{}[]{data-label="f:alp-lam"}](Figure12.pdf){width="8cm"} ![The masses of the photoevaporating disk candidates in our sample. Clearly, massive photoevaporating disks predicted by high evaporation rates are absent.[]{data-label="f:photo-mass"}](Figure13.pdf){width="5in"} [rrcrrrrrrrrrrrcc]{} 1 & 234.51292 & -33.23269 & SSTc2d\_J153803.1-331358 & 13.35 &3.91e+02 & 6.91e+02& 6.83e+02 & 3.47e+02& 2.20e+02 & 1.82e+02 & 1.26e+02 & 3.56e+01 & Lup I & 1\ 2 & 235.64750 & -34.37292 & SSTc2d\_J154235.4-342223 & 14.17 & 5.16e+02 & 8.86e+02& 8.43e+02 & 4.00e+02& 2.66e+02 & 2.03e+02 & 1.40e+02 & 4.96e+01 & Lup I &\ 3 & 239.93868 & -41.91590 & SSTc2d\_J155945.3-415457 & 13.18 & 3.64e+02 & 7.92e+02 & 8.86e+02 & 5.78e+02 & 3.35e+02 & 3.23e+02 & 3.80e+02 & 2.68e+02 &Lup IV &1\ 4 & 240.37369 & -42.13432 & SSTc2d\_J160129.7-420804 & 15.74 & 7.74e+01 & 1.57e+02 & 1.60e+02 & 9.11e+01 & 5.86e+01 & 4.89e+01 & 3.80e+01 & 1.79e+01 &Lup IV & 1\ 5 & 240.62463 & -41.85307 & SSTc2d\_J160229.9-415111 & 17.91 & 3.95e+01 & 8.07e+01 &8.73e+01 & 4.99e+01 & 3.01e+01 & 2.55e+01 & 1.82e+01 & 5.15e+00 & Lup IV &1\ 6 & 242.19953 & -38.83361 & SSTc2d\_J160847.9-385001 &13.71 & 1.12e+03 & 1.93e+03 & 1.89e+03 & 9.86e+02 & 4.50e+02 &4.37e+02 & 3.14e+02 & 1.41e+02& Lup III &\ 7 &242.39212 & -39.22835 &SSTc2d\_J160934.1-391342 & 15.11 & 5.36e+02 & 1.20e+03 & 1.43e+03 & 5.33e+02& 3.99e+02& 4.66e+02& 2.83e+02 & 5.35e+01& Lup III & 1\ 8 & 242.50045 & -38.90031 & SSTc2d\_J161000.1-385401 &17.47 & 2.25e+02 & 4.13e+02 &5.55e+02 & 3.66e+02 & 1.76e+02 &2.68e+02 & 2.06e+02 & 9.72e+01 &Lup III & 1\ 9 & 242.85827 & -39.18979 & SSTc2d\_J161126.0-391123 & 16.29 & 6.96e+02 & 1.43e+03& 1.71e+03& 1.01e+03 & 5.99e+02 & 5.56e+02 & 3.62e+02 & 1.05e+02& Lup III &1\ 10 & 243.21550 & -38.70443 & SSTc2d\_J161251.7-384216 &14.06 & 5.97e+02 & 1.04e+03 & 1.07e+03 & 6.37e+02 & 3.49e+02 &3.06e+02 & 2.02e+02 & 7.43e+01 &Lup III & 1\ 11 & 244.92350 & -37.78246 &SSTGBS\_J16194163-3746568 & 16.36 & 6.48e+01 & 1.27e+02 & 1.33e+02 & 8.56e+01& 5.64e+01& 4.77e+01& 3.26e+01 & 1.02e+01& Lup V &2\ 12 & 245.00856 & -41.62400 & SSTGBS\_J16200205-4137264 &15.37 & 6.17e+02 & 1.24e+03 & 1.35e+03 & 7.27e+02 & 4.28e+02 &3.60e+02 & 2.31e+02 & 9.00e+01 &Lup VI & 2\ 13 & 245.03960 & -41.43343 & SSTGBS\_J16200950-4126003 &15.78 & 7.69e+02 & 1.77e+03 & 2.05e+03 & 1.40e+03 & 8.77e+02 &7.88e+02 & 5.20e+02 & 1.61e+02 &Lup VI & 2\ 14 & 245.13167 & -37.51133 & SSTGBS\_J16203160-3730407 &16.96 & 1.40e+03 & 2.78e+03 & 3.38e+03 & 2.07e+03 & 1.24e+03 &1.27e+03 & 9.10e+02 & 2.11e+02 &Lup V & 2\ 15 & 245.22704 & -36.91195 & SSTGBS\_J16205449-3654430 & 16.03 & 1.89e+02 & 3.86e+02 & 4.12e+02 & 2.24e+02 & 1.16e+02& 9.97e+01& 6.43e+01 & 2.51e+01& Lup V& 2\ 16 & 245.35836 & -37.51710 & SSTGBS\_J16212600-3731015 &16.69 & 1.62e+02 & 4.21e+02 & 6.41e+02 & 7.57e+02 & 5.29e+02 & 5.18e+02 & 3.87e+02 & 1.61e+02& Lup V & 2\ 17 & 245.37467 & -37.03926 & SSTGBS\_J16212991-3702213 & 14.84& 1.68e+02 & 3.15e+02 & 3.31e+02 & 1.89e+02 & 9.94e+01 & 8.37e+01 & 5.79e+01 & 5.69e+01 &Lup V &2\ 18 & 245.41990 & -41.37274 & SSTGBS\_J16214077-4122218 &15.98 & 1.26e+02 & 2.51e+02& 2.64e+02 & 1.58e+02 & 8.63e+01 & 7.36e+01 & 4.86e+01 & 1.92e+01 &Lup VI & 2\ 19 & 245.45722 & -41.11716 & SSTGBS\_J16214973-4107017 &15.35 & 1.20e+0 & 2.30e+02& 2.65e+02 & 1.92e+02 & 1.34e+02 & 1.08e+02 & 7.48e+01 & 2.40e+01 &Lup VI &2\ 20 & 245.48435 & -37.22131 & SSTGBS\_J16215624-3713167 &17.18 & 1.95e+0 & 3.89e+02 & 4.34e+02 & 2.43e+02 & 1.31e+02 & 1.15e+02 & 7.38e+01 & 2.84e+01 &Lup V &2\ 21 & 245.50680 & -37.27693 & SSTGBS\_J16220163-3716369&16.20 & 5.48e+0 & 1.06e+02 & 1.09e+02 & 6.50e+01 & 4.23e+01 & 3.53e+01 & 2.48e+01 & 8.24e+00 &Lup V &2\ 22 & 245.57457 & -36.97127 & SSTGBS\_J16221789-3658165&16.80 & 2.68e+02 & 5.24e+02 &5.83e+02 & 3.42e+02 & 1.95e+02 & 1.68e+02 & 1.08e+02 & 3.44e+01 &Lup V &2\ 23 & 245.63924 & -41.05499 & SSTGBS\_J16223341-4103179 &16.34 & 5.72e+02 & 1.32e+03 &1.50e+03 & 8.22e+02 & 4.95e+02 & 4.26e+02 & 2.87e+02 & 1.21e+02 &Lup VI&2\ 24 & 245.67774 & -37.35645 & SSTGBS\_J16224265-3721232&16.08 & 1.97e+02 & 4.34e+02 &4.97e+02 & 3.61e+02 & 2.74e+02 & 2.47e+02 & 2.61e+02 & 1.98e+02 &Lup V &2\ 25 & 245.76534 & -37.49477 & SSTGBS\_J16230368-3729411 &16.02 & 6.05e+01 & 1.18e+02 &1.20e+02 & 6.64e+01 & 4.17e+01 & 3.55e+01 & 3.12e+01 & 1.86e+01 &Lup V &2\ 26 & 245.79592 & -41.28620 & SSTGBS\_J16231101-4117103&16.48 & 1.14e+02 & 2.64e+02 & 3.00e+02 & 1.67e+02 & 1.15e+02 & 9.21e+01 & 6.16e+01 & 1.79e+01 &Lup VI & 2\ 27 & 245.81152 & -40.28753 & SSTGBS\_J16231476-4017151 &16.96 & 1.69e+02 & 3.56e+02 &4.07e+02 & 2.20e+02 & 1.36e+02 & 1.15e+02 & 8.15e+01 & 2.86e+01 &Lup VI &2\ 28 & 245.81661 & -41.06693 & SSTGBS\_J16231598-4104009 &13.11 & 1.97e+03 & 4.16e+03 & 4.86e+03 & 1.80e+03 & 1.31e+03 & 1.12e+03 & 7.36e+02 & 2.41e+02 &Lup VI &2\ 29 & 245.85994 & -37.86896 & SSTGBS\_J16232638-3752082 &17.31 & 1.20e+02 & 2.35e+02 & 2.99e+02 & 3.24e+02 & 2.47e+02 & 2.33e+02 & 1.74e+02 & 4.96e+01 &Lup V &2\ 30 & 245.88526 & -37.87672 & SSTGBS\_J16233246-3752361 &15.74 & 4.72e+02 & 9.78e+02 & 1.06e+03 & 5.68e+02 & 3.23e+02 & 2.85e+02 & 1.92e+02 & 7.09e+01& Lup V &2\ 31 & 245.91108 & -37.52745 & SSTGBS\_J16233865-3731388 &15.80 & 1.27e+02 & 2.39e+02 & 2.53e+02 & 1.43e+02 & 7.59e+01 & 7.05e+01 & 5.16e+01 & 3.14e+01 &Lup V & 2\ 32 & 245.95431 & -40.43823 & SSTGBS\_J16234903-4026176 &17.12 & 9.57e+01 & 2.08e+02 & 2.33e+02 & 1.64e+02 & 1.13e+02 & 9.72e+01 & 7.79e+01 & 2.52e+01 &Lup VI &2\ 33 & 245.99615 & -37.89800 & SSTGBS\_J16235907-3753528 &14.28 & 2.23e+03 & 4.60e+03 & 5.14e+03 & 2.51e+03 & 1.61e+03 & 1.41e+03 & 9.50e+02 & 3.19e+02 &Lup V & 2\ 34 & 246.11110 & -37.96131 & SSTGBS\_J16242666-3757407 &17.52 & 3.13e+02 & 6.67e+02 & 7.63e+02 & 4.10e+02 & 2.14e+02 & 2.02e+02 & 1.29e+02 & 5.62e+01 &Lup V & 2\ 35 & 246.23294 & -40.19118 & SSTGBS\_J16245590-4011282 &15.15 & 1.06e+03 & 2.09e+03 & 2.65e+03 & 1.43e+03 & 9.49e+02 & 9.54e+02 & 6.74e+02 & 1.78e+02 & Lup VI& 2\ 36 & 246.27875 & -38.05595 & SSTGBS\_J16250690-3803214 &14.99 & 1.34e+03 & 2.71e+03 & 2.98e+03 & 1.64e+03 & 9.41e+02 & 8.20e+02 & 5.39e+02 & 1.78e+02 &Lup V & 2\ 37 & 246.46860 & -40.31346 & SSTGBS\_J16255246-4018484 &17.39 & 1.23e+02 & 2.63e+02 & 2.93e+02 & 1.23e+02 & 7.96e+01 & 7.38e+01 & 4.68e+01 & 1.28e+01 &Lup VI &2\ 38 & 246.49321 & -40.16614 & SSTc2d\_J16255837-4009581 &16.88 & 5.17e+02 & 1.20e+03 & 1.44e+03 & 7.77e+02 & 4.23e+02 & 4.06e+02 & 2.63e+02 & 1.05e+02 &Lup VI& 2\ 39 & 246.55582 & -39.83173 & SSTc2d\_J16261339-3949542 &14.83 & 1.68e+02 & 3.01e+02 & 3.55e+02 & 2.15e+02 & 1.60e+02 & 1.34e+02 & 1.01e+02 & 3.70e+01 &Lup VI& 2\ 40 & 246.60631 & -39.74646 & SSTGBS\_J16262551-3944472 &15.44 & 4.76e+01 & 9.31e+01 & 9.09e+01 & 4.70e+01 & 3.20e+01 & 2.52e+01 & 2.94e+01 & 1.53e+01 &Lup VI &2\ 41 & 246.96060 & -39.80278 & SSTGBS\_J16275054-3948100 &16.20 & 3.15e+02 & 6.61e+02 & 7.35e+02 & 4.71e+02 & 2.81e+02 & 2.39e+02 & 1.59e+02 & 5.74e+01 &Lup VI &2\ 42 & 252.27333 & -15.62027 & SSTc2d\_J16490560-1537129 &17.92 &1.40e+02 & 6.63e+02 & 1.12e+03 & 1.00e+03 & 7.78e+02 & 7.42e+02 & 5.17e+02 & 1.46e+02 & Scp & 4\ 43 & 285.78812 & -36.95611 & SSTGBS\_J19030915-3657220 &17.45 & 6.88e+01 & 1.97e+02 & 2.43e+02 & 1.63e+02 &1.21e+02 &1.01e+02 & 6.75e+01 & 2.04e+01 &Cra & 3 \[t:giants\] [cccccccc]{} Lupus I & 339, 16 & 2 & 0 & 2 & 100& 1.5–4& 150 $\pm$ 20\ Lupus III & 340, 9 & 15 & 10 & 5 & 33 &1.5–4 & 200 $\pm$ 20\ Lupus IV & 336, 8 & 5 & 2 & 3 & 60 & 1.5–4&150 $\pm$ 20\ Lupus V & 342, 9 & 16 & 0 & 16 & 100 & 10 & 150 $\pm$ 20\ Lupus VI & 342, 6 & 15 & 0 & 15 & 100 & 10 &150 $\pm$ 20\ Cra & 0,-19 & 5 & 4 & 1 & 20 & 1 &150 $\pm$ 20\ Scp & 250,18 & 2 & 1 & 1 & 50 & 5 &130 $\pm$ 20\ Oph (Paper I) & 353,18 & 34 & 26 & 8 & 24 & 2 &150 $\pm$ 20\ [c|c|c|c|c]{} & YSOc & AGB & TD & AGB\ & & candidates & candidates & candidates in TD region\ & & whole sample & & in TD region\ &\# & % &\# & %\ \ & 29 & 10.4 & 7 & 28.6\ [**LUP I**]{} & 20 & 15 & 8 & 25\ [**LUP III**]{} & 79 & 18.9 & 18 &11.1\ [**LUP IV**]{} & 12 & 25 & 5 & 20\ [**OPH**]{} & 297 & 7.7 & 52& 15.38\ [**PER**]{} & 387 & 2.6 & 56 & 10.7\ [**SER**]{} & 262 & 6.5 & 60 & 10\ \ & 174 & 1.7 & 28 & 7.1\ [**CrA**]{} & 45 & 4.4 & 7 & 14.2\ [**IC5146**]{} & 163 & 2.4 & 24 & 4.1\ [**LUP V**]{} & 44 & 47.7 & 22 & 36.3\ [**LUP VI**]{} & 46 & 67.3 & 21 & 57.1\ [**SERP-AQUILA**]{} & 1442 & 28.6 &641 & 32.6\ [**CHAM I**]{} & 93 & 1 &17 & 5.9\ [**CHAM III**]{} & 4 & 75 &1 & 100\ [**MUSCA**]{} & 13 &84.6 & 5 & 80.0\ [**CEPH**]{} & 119 &2.5 & 19 & 10.5\ [**SCO**]{} & 9 &11 & 4 & 25\ [rrcrrrrrrrrrrrc]{} 1 & SSTc2d\_J160026.1-415356 & ..... & 15.62& 15.54 & 2.97e+01 &3.70e+01 &3.25e+01 &2.15e+01 & 1.68e+01 & 1.42e+01 & 1.63e+01 & 2.40e+01 &$<$ 50 & Lup IV\ 2 & SSTc2d\_J160044.5-415531 & V\*MYLup & 11.22& 11.06 & 2.63e+02 &3.44e+02 &3.05e+02 &1.77e+02 & 1.41e+02 & 1.40e+02 & 2.13e+02 & 5.90e+02 &1.05e+03& Lup IV\ 3 & SSTc2d\_J160711.6-390348 & SZ91 & 13.61& 13.89 & 6.03e+01 &9.13e+01 &7.67e+01 &3.86e+01 & 2.47e+01 & 1.72e+01 & 1.09e+01 & 9.72e+00 &5.02e+02& Lup III\ 4 & SSTc2d\_J160752.3-385806 & SZ95 & 13.66& 14.02 & 6.28e+01 &7.89e+01 &6.61e+01 &4.21e+01 & 3.18e+01 & 2.73e+01 & 2.96e+01 & 3.00e+01 &$<$ 50 & Lup III\ 5 & SSTc2d\_J160812.6-390834 & SZ96 & 12.98& 13.66 & 1.42e+02 &1.87e+02 &1.74e+02 &1.68e+02 & 1.13e+02 & 1.38e+02 & 1.73e+02 & 2.41e+02 &1.54e+02& Lup III\ 6 & SSTc2d\_J160828.4-390532 & SZ101 & 13.52& 13.53& 1.10e+02& 1.32e+02& 1.17e+02 & 7.98e+01& 5.55e+01& 4.16e+01& 3.29e+01& 2.41e+01& $<$ 50& Lup III\ 7 & SSTc2d\_J160831.5-384729 & Lup338 & 12.70& 13.03 & 2.15e+02 & 2.75e+02 & 2.37e+02 & 1.49e+02& 9.25e+01& 6.98e+01& 5.16e+01& 2.85e+01& $<$ 50& Lup III\ 8 & SSTc2d\_J160841.8-390137 & SZ107 & 15.29& 15.47 & 5.05e+01 & 5.76e+01 & 5.01e+01 & 2.65e+01& 2.02e+01& 1.41e+01& 9.24e+00& 1.07e+01& $<$ 50& Lup III\ 9 & SSTc2d\_J160855.5-390234 & SZ112 & 14.57& 14.68 & 6.32e+01 & 7.87e+01 & 6.90e+01 & 4.87e+01& 3.80e+01& 3.04e+01& 2.48e+01& 1.24e+02& 1.20e+02&Lup III\ 10 & SSTc2d\_J160901.4-392512 & .... & 14.83& 14.89 & 3.62e+01 & 5.45e+01 & 5.09e+01 & 4.47e+01& 3.40e+01& 3.06e+01& 2.58e+01& 4.22e+01 & 1.14e+02& Lup III\ 11 & SSTc2d\_J160954.0-392328 & Lup359 & 12.96& 13.30 & 1.59e+02 & 2.13e+02 & 1.94e+02 & 1.35e+02& 1.01e+02& 8.44e+01& 7.93e+01& 9.65e+01& $<$ 50& Lup III\ 12 & SSTc2d\_J161029.6-392215 & ... & 15.69& 15.79 & 2.66e+01 & 3.18e+01 & 2.88e+01 & 1.91e+01& 1.42e+01& 1.15e+01& 1.09e+01& 3.37e+01& 1.10e+02& Lup III\ 13 & SSTc2d\_J162209.6-195301 & ... &14.48 &14.27 & 1.44e+02 &2.08e+02 &1.84e+02 &1.09e+02 & 7.15e+01 & 5.05e+01 & 3.27e+01 & 1.59e+01 & $<$ 50& Scp\ 14 & SSTGBS\_J190029.1-365604 & CrAPMS8 &13.80 &13.78 & 7.84e+01 &1.06e+02 &9.89e+01 &5.08e+01 & 3.60e+01 & 2.62e+01 & 1.83e+01 & 3.59e+01 & $<$ 100 & Cra\ 15 & SSTGBS\_J190058.1-364505 & CrA-9 &13.49 &13.57 & 1.12e+02 &1.61e+02 &1.40e+02 &5.81e+01 & 4.38e+01 & 3.18e+01 & 2.48e+01 & 1.78e+02 & $<$ 100 & Cra\ 16 & SSTGBS\_J190129.0-370148 & G-94 &15.53 &14.95 & 3.53e+01 &4.25e+01 &3.62e+01 &1.95e+01 & 1.38e+01 & 9.67e+00 & 6.52e+00 & 2.92e+00 & $<$ 50& Cra\ 17 & SSTGBS\_J190311.8-370902 & CrA-35 &17.20 &16.82 & 2.46e+01 &3.24e+01 &3.09e+01 &2.12e+01 & 1.58e+01 & 1.21e+01 & 1.06e+01 & 1.18e+01 & $<$ 50& Cra\ \[t:mags\] [rcccrcrrrcc]{} 1 & 240.10887 & -41.89877 & Clay & M5.25,M1$^f$ & 1,2 & 0.47 & 162 & $<$ 21 & 7 &2.8\ 2 & 240.18554 & -41.92534 & DuPont & K0 & 7 & 0.44 & 532 & 100 & 5 &\ 3 & 241.79833 & -39.06326 & DuPont & M1.5,M0.5 & 1,6 & 0.41 & 283 & 34.5 & 2.9 &5\ 4 & 241.96800 & -38.96840 & DuPont & M3.25,M1.5 & 1,6 & 0.46 & 321 & $<$ 9.9 & 3.3 &3\ 5 & 242.05258 & -39.14264 & DuPont & M2,M1.5 & 1,6 & 0.5 & 233 & $<$ 21 & 7 &\ 6 & 242.11837 & -39.09229& DuPont& M5,M4 & 1,6 & 0.28 & 343 & $<$ 10.8 & 3.6 & 0.7\ 7 & 242.13146 & -38.79148 & DuPont & M2.25,M2 & 1,4 & 0.25 & 382 & 6.7 & 2.2 & 0.4\ 8 & 242.17413 & -39.02695 & DuPont & M5.75,M5.5 & 1,6 & & 200 & 9.7 & 2.5 &\ 9 & 242.23133 & -39.04276 & DuPont & M6,M6 & 1,6 & & 189 & $<$ 21 & 7 &2.8\ 10 & 242.25583 & -39.41997 & Clay & M4,M4 & 1,2 & 0.45 & 369 & 31.4& 3.4 &\ 11 & 242.47496 & -39.39109 &DuPont & M2.75,M1.5 & 1,4 & 0.40 & 336 & 16.7& 3.3 & 1.15\ 12 & 242.62321 & -39.37076 & Clay & M4.5,M4 & 1,2& 0.52 & 180 & 23.2 & 4.7 &\ 13 & 245.54000 & -19.88357 & Clay & M3.7 &1 & 0.55 & 132 & $<$ 10.8 & 3.6 &1.8,3\ 14 & 285.12113 & -36.93437 & DuPont & M4,M3 &1,5 &0.30 & 93 & $<$ 10.5 & 3.5 &0.132\ 15 & 285.24187 & -36.75139 & Clay & M0.75 & 1 & 0.48 & 440 & $<$ 21 & 7 &\ 16 & 285.37088 & -37.03011 & DuPont & M3.75,M3.5 &1,3 & & 83 & $<$ 21 & 7 &0.5\ 17 & 285.79929 & -37.15055 & Clay & M5.0 &1 & 0.51 & 205 & $<$ 21 & 7 &0.5 . \[t:obs\] [lrrrrrcrc]{} 1 & $<$ -11 & $<$ 1.9 & 420 & 4.50 & -0.80 &-2.1 & 0.06 & photo. disk, Lup IV\ 2 & -7.7 & 9.1 & & 4.50 & -0.17 &-2.4 & 2.37 & giant planet, Lup IV\ 3 & -10.1 & 5.6 & 1000 & 8.00 & -2.18 & -2.6 & 0.39 & giant planet, Lup III\ 4 & -9.7 & $<$ 1.6 & 600 & 4.50 & -1.05 & -2.2& 0.30 & grain growth$^b$, Lup III\ 5 & -10.6 & $<$ 3.4 & & 2.20 & -0.93 & -1.4 & 0.86 & grain growth$^b$, Lup III\ 6 & -9.5 & $<$ 1.8 & 140 & 5.80 & -1.42 & -2.6 & 0.43 & grain growth, Lup III\ 7 & -9.1 & 1.1 & 80 & 8.00 & -1.55 & -2.9 & 1.27 & grain growth, Lup III\ 8 & -11 & 1.6 & 76 & 5.8& -0.86 & -2.8 & 0.17 & grain growth$^b$, Lup III\ 9 & $<$ -11 & $<$ 3.4 & 560 & 4.50 & -0.31 & -1.9 & 0.22& photo. disk, Lup III\ 10 & -9.3 & 5.1 & & 2.20 & -1.19 & -1.5 & 0.27 & grain growth$^b$, Lup III\ 11 & -9.6 & 2.7 & 230$^a$ & 4.50 & -1.05 & -2.3 & 0.96 &circumbinary/gr-grow$^b$, Lup III\ 12 & -11 & 3.8 & & 5.80 & -0.28 & -2.1 & 0.11 & grain growth$^b$, Lup III\ 13 & $<$ -11 & $<$ 0.8 & 234,390$^a$ & 8.00 & -1.67 & -3.7 & 0.48 & circumbinary/debris, Scp\ 14 & $<$ -11 & $<$ 1 & 20 & 5.80 & -0.42 & -2.7 & 0.14 & circumbinary/ photo. disk, CrA\ 15 & -8.6 & $<$ 2 & & 8.00 & 0.76 & -2.4 & 0.46 & giant planet, CrA\ 16 & $<$ -11 & $<$ 2 & 75 & 8.00 & -1.74 & -3.2 & 0.07 & debris disk, CrA\ 17 & -11 & $<$ 2 & 75 & 5.80 & -1.06 & -2.3 & 0.06 & grain growth, CrA\ \[t:derived\] [**Disk candidates**]{} [**Lupus III, IV**]{} [**Cra**]{} [**Oph$^\dag$**]{} ------------------------------- ----------------------- -------------- -------------------- [**debris**]{} – 1 (25%) 4 (15%) [**photoevaporated**]{} 2 (17%) 1 (25%) 5 (19%) [**grain growth**]{} 8 (66%) 1 (25%) 13 (50%) [**hosting giant planets**]{} 2 (17%) 1 (25%) 4 (15%) [**circumbinary**]{} 1 (8%) 1 (25%) 4 (15%) [**Total**]{} 12 4 26 [**Age \[Myr\]**]{} $\sim$ 1.5–4$^a$ $\sim$ 1$^b$ $\sim$ 2$^c$ $^\dag$ From Paper I\ References: $^a$ @comeron08-1; $^b$ @wilkingetal05-1; $^c$ @Sicilia-Aguilaretal08-1 [^1]: http://irsa.ipac.caltech.edu/data/SPITZER/C2D/doc/c2d$\_$del$\_$document.pdf [^2]: the former is available at http://irsa.ipac.caltech.edu/data/SPITZER/C2D/ [^3]: Image Reduction and Analysis Facility, distributed by NOAO, operated by AURA, Inc., under agreement with NSF [^4]: http://www.eso.org/projects/aot/eclipse/ [^5]: This publication is based on data acquired with APEX which is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. [^6]: To calculate [$\lambda_{\mathrm{turn−off}}$]{}, we compare the extinction-corrected SED with NextGen Models [@hauschildtetal99-1] normalized to the J-band and choose [$\lambda_{\mathrm{turn−off}}$]{}as the longest wavelength at which the stellar photosphere contributes over 50% of the total flux. The uncertainty of [$\lambda_{\mathrm{turn−off}}$]{}is roughly one SED point. [^7]: Circumbinary disks are included twice in the table as binarity does not exclude a second process to cause the inner opacity hole in the disk.

--- abstract: 'We investigate various properties associated with the tilt of isolated magnetic bipoles in magnetograms taken at the solar surface. We show that bipoles can be divided into two groups which have tilts of opposite signs, and reveal similar properties with respect to bipole area, flux and bipolar moment. Detailed comparison of these physical quantities shows that the dividing point between the two types of bipoles corresponds to a bipole area of about 300 millionths of the solar hemisphere (MHS). The time-latitude distribution of small bipoles differs substantially from that for large bipoles. Such behaviour in terms of dynamo theory may indicate that small and large bipoles trace different components of the solar magnetic field. The other possible viewpoint is that the difference in tilt data for small and large bipoles is connected with spectral helicity separation, which results in opposite tilts for small and large bipoles. We note that the data available do not provide convincing reasons to prefer either interpretation.' author: - 'E. $^{1}$, A. $^{2}$, D. $^{3}$' title: 'The Properties of the Tilts of Bipolar Solar Regions\' --- Introduction ============ The solar magnetic cycle is believed to be associated with dynamo action which occurs somewhere inside the solar convective zone. In turn, the solar dynamo is based on two processes. The differential rotation produces toroidal magnetic field ${\bf B}_T$ from poloidal magnetic field ${\bf B}_P$. Details of this process looks quite clear following the modern development of helioseismology (see [*e.g.*]{} the review by ). On the other hand, another process has to regenerate poloidal magnetic field ${\bf B}_P$ from toroidal. There are several solar dynamo models which suggest various physical mechanisms underlying this regeneration. In particular, the regeneration can involve sunspot formation and diffusion at the solar surface (Babcock-Leighton mechanism), or can be associated with cyclonic motions in a more or less deep layer of the convective zone only (the Parker mechanism). However a combined action of both mechanisms looks possible as well (see [*e.g.*]{} the review by ). The relative importance of both mechanisms for the solar cycle remains a topic of intensive debate. Of course, an observational clarification of the details of the regeneration process is a useful contribution to the above discussion and the tilt angle of solar bipolar regions provides direct observational information for this regeneration. Indeed, the tilt data show how the direction between the two poles of a magnetic bipole is inclined with respect to the solar equator. If this inclination angle differs systematically from zero, this would mean that poloidal field is produced from toroidal by a physical process, which is exactly the effect under discussion. It is still not the whole story, and various physical mechanisms acting alone or jointly can provide the non-vanishing tilt. Of course, the intention of deducing the tilt from the observational data is to clarify the physics underlying the link between toroidal and poloidal fields. However it is preferable not to assume any choice in advance, so we use below the wording ’$\alpha$-effect’ for brevity to refer to this effect. Tilt studies originated as early as in 1919 () and resulted in Joy’s law, simultaneously with formulation of the well-known Hale polarity law. According to Joy’s law, the average tilt is a non-vanishing quantity antisymmetric with respect to the solar equator and growing linearly with $\sin \theta$ ($\theta$ is solar co-latitude). This result is fully in accordance with expectations from solar dynamo theory, and it encourages the use of the tilt data as a valuable observational source to constrain the governing parameters of the solar dynamo. The reality is however more complicated. The point is that the tilt is quite small (several degrees only) and rather noisy. Moreover, the very concept of bipoles and their identification from magnetograms requires an algorithmic formalism in order to make comparable the results of independent analyses. This is probably why experts in solar dynamo theory did not pay attention to the tilt data for quite some time. The methods available for isolation of bipoles in magnetograms and the database of the tilts grew gradually until they became convincing, at least for some of the dynamo community. To us, the breakthrough was by . This paper confirms Joy’s law in a convincing way and does not recognize any cyclic variations of the slope of the relation between tilt and $\sin \theta$ (see also ). Further investigation of the tilt data was undertaken by who broadly confirmed the conclusions of for those bipoles which can mainly be identified with bipolar sunspot groups. A time-latitude (butterfly) diagram for the tilt averaged over appropriate time-latitude bins obtained in demonstrates that the tilt is indeed almost independent of the cycle phase: however some rather minor variations were isolated. The point however is that the analysis of included substantially more small bipoles than that of and the behaviour of the small bipoles is almost opposite to that of the large bipoles, which correspond to sunspots. Recall that in we referred to small bipoles those with areas below 300 millionths of the solar hemisphere (MSH[^1]), which are mostly ephemeral regions (note that we measure in MSH an area of domains with magnetic field exceeding the threshold level $B_{\rm min}=10~G$ isolated in solar magnetograms). In particular, the tilt angle of small bipoles is antisymmetric with respect to the solar equator, whereas the tilt of small bipoles in, say, the northern hemisphere is of the opposite sign to that of the large bipoles. stressed again that the analysis of does not identify any difference between the tilts of small and large bipoles. However this analysis is not focused on the small bipoles and the situation deserves further investigation and clarification. Indeed, used a substantially different approach to bipole identification and parameters of their algorithm were optimized for large bipoles only, thus the sample of small-scale bipoles was rather incomplete. Moreover taking into account the higher smoothing level applied by the authors to magnetograms, and the different type of structures, which were recognized as bipoles, we conclude, that the sample of small bipoles, used in , cannot be compared directly with bipoles, referred as “small” in , and they are of particular interest. The aim of this paper is to extend the analysis of the different behaviour of small and large bipoles to various quantities associated with bipoles. It generalizes the approach of , who, following the idea of the earlier research, concentrated on the relation between tilt and latitude. The sample of the positions of bipoles extracted has sufficient size to proceed further and clarify a possible contribution of other factors to the tilt distribution. We note that the different behaviour of small and large bipoles isolated at least from the sample of bipoles produced by the algorithm applied does not seem to represent fundamental problems for dynamo theory. In particular, an assumption that the bipoles trace the toroidal magnetic field looks straightforward for large bipoles at least, because they are sunspots which are considered as a tracer for the large-scale magnetic field generated by solar dynamo somewhere in the convective shell. It might be supposed that the small bipoles represent, for example, poloidal magnetic field and this solves the controversy. Of course, this is an option only and other explanations, including even a demonstration that the algorithm used becomes somehow inapplicable for small bipoles, have to be considered. In our opinion, such considerations have to be based on an examination of the scaling between various physical quantities associated with bipoles. These could include in a plausible way the size of bipole, [*e.g.*]{} its flux, instead of its area. This is a motivation of the research discussed here. Speaking broadly, we arrive at the conclusion that the distinction between small and large bipoles can be recognized in various physical quantities and confirms to some extent that small and large bipoles trace different magnetic field components. The Data ======== We used a sample of bipoles identified by the algorithm of . The method was applied to the magnetograms from *Kitt Peak Vacuum telescope* (KPVT) for the period 1975–2003, from the *Solar and Heliospheric Observatory Michelson Doppler Imager* (SOHO/MDI) (; soi.stanford.edu/magnetic/Lev1.8/) for the period 1996–2011 and from the *Helioseismic and Magnetic Imager* (HMI) () for the period 2010–2013. We used the same parameters for recognition of bipoles as and both data samples are identical. In particular we selected domains with magnetic field exceeding the threshold level $B_{\rm min}=10~G$ and area exceeding 50 MSH. The parameters applied involve a large amount of small ephemeral regions, for which it is difficult to prove that each isolated region corresponds to a physical entity and we can operate with their statistical properties only. The tests presented in this paper and in do not show any evidence, that they are caused by a bias in the computer algorithm (see in detail ). We stress however that an independent verification of the result by another algorithm looks highly desirable. Such an additional verification is obviously out of the scope of this paper. For a more correct determination of bipole positions we exploited only the central part of the solar magnetogram within 0.7 of the solar disk radius, because projection effects near the solar limb may distort the result substantially. Some bipoles may have inverse polarity and violate the Hale polarity law. This happens only in about 5% of cases for bipolar sunspot groups ([*e.g.*]{} ) but this quantity increases substantially in going to smaller areas. Note that the prevalent orientation is not prescribed in advance. The bipoles are distributed in two-year time bins and $5^\circ$ latitudinal bins and in each bin the prevalent orientation is defined as follows. For each of the two groups of bipoles with opposite leading polarity we compute the Gaussian approximation to the distribution of their tilt angles. The group with the largest amplitude defines thus the prevalent orientation in the bin. Normally this is just the group with the larger number of bipoles. The obtained sample is a base for further investigations according to additional criteria. For our analysis we use all bipoles in both hemispheres. We combine them together in such way that final angular distribution has a single peak, *i.e.* we subtract $180^{\circ}$ from the angles in the second and third quadrants, and reverse the distributions in the southern hemisphere. The combined sample contains tilt angles in the interval between $-90^\circ$ to $+90^\circ$. For both hemispheres the positive sign indicates that the domain of leading polarity is closer to the Equator than the trailing domain. The negative sign means on the contrary that the domain of trailing polarity is situated closer to the Equator. We use the median as a robust statistic to estimate a mean tilt, and the t-Student criterion for $95\%$ confidence intervals. The database used gives for the bipoles the following parameters: the time $t$ of observation, latitude $\theta$ of the centre of the bipole, the area $S$, flux $F$, distance $d$ between the poles and the tilt $\mu$. Results ======= We are interested mainly in correlations between the tilt $\mu$ and the other parameters describing a bipole. The dependence of the orientation of bipoles on the solar cycle is the well-known Hale’s polarity law, while the dependence of $\mu$ on the latitude is given by Joy’s law. We recall that the verification of Joy’s law, based on an algorithmic procedure to recognise bipolar regions, confirms the law (; ; ). Quite surprisingly found that tilt substantially depends on the area of bipoles and that the prevalent tilt for small bipoles has the opposite sign to that for large bipoles. This trend can be easily noticed in Figure \[data\], where we show the density of tilt angle distribution against bipole area. Indeed, for large bipoles (we assume the dividing point between large and small bipoles is the same as in , *i.e.* 300 MSH) we observe a pronounced peak in the domain of positive tilts. With smaller areas the peak becomes blurred (the distribution becomes rather non-gaussian), but the domain of increased bipole density turns smoothly down to the domain of negative tilts. The linear least-square fit confirms the visual trend and intersects the line of zero tilt exactly near 300 MSH. Difference of mean tilt signs for these two groups of bipoles is confirmed by simple statistical test based on Student’s $t$-test. It gives $t_{eq}=22.9$ under hypothesis that both samples have similar mean values and $t_{op}=0.98$ under hypothesis that mean values have similar absolute values but opposite signs. However the noisy distribution for small bipoles restricts the abilities of the $t$-test in some ways. ![Distribution of bipoles in the combined area and tilt angle domains, intensity of the colour indicates number of bipoles relative to the total number of bipoles with the same areas (MDI data for the period 1998–2007, bipoles were selected from latitudinal zone $|\theta|\geqslant10^{\circ}$). The red line corresponds to the linear least-square fit.[]{data-label="data"}](data.eps){width="0.7\linewidth"} Now, we analyse the a correlation between the parameters mentioned above and tilt on the basis of the observational data available. In particular, it is interesting to compare the correlations for large and small bipoles in order to gain a better understanding of the physical nature of the difference in behaviour between these bipoles which was found in . Cyclic Modulations of the Tilt ------------------------------ We start from a straightforward (and possibly not the most instructive) correlation property of the tilt angles, *i.e.* a correlation of the tilt averaged over 2-year bins with the phase of the cycle. The correlation calculated separately for large and small bipoles is presented in Figure \[t\_time\]. There are two messages from the plot. First of all, the KPVT data, MDI data and HMI data look to be more or less in agreement, at least at the epoch when the data overlap. This appears to confirm the self-consistency of the bipole database used. In contrast, the large and small bipoles demonstrate opposite behaviour for each set of observational data. ![Tilt (in degrees) averaged over 2 years time bins. Solid line is for large bipoles with areas $S>300$ MSH, dashed line is for small bipoles with areas $50<S<300$ MSH. Black colour shows KPVT data, blue is for MDI data, green is for HMI data.[]{data-label="t_time"}](kpvt-mdi300.eps){width="0.7\linewidth"} The other message from the plot is a pronounced cyclic modulation visible for both types of bipoles. A remarkable feature is that in the course of the cycle the absolute value of tilts of large bipoles decreases, while for small bipoles we observe an increase. Strictly speaking we have to distinguish cycles for both types of bipoles and as it will be shown later there are some reasons to suppose it. An interpretation of the correlation for the large bipoles looks quite straightforward and is consistent with the suggestion of that the slope in Joy’s law is independent of the phase of the cycle. Indeed, Joy’s law tells us that $\mu \propto \sin \theta$ where $\theta$ is the colatitude of the bipoles, which decays on average with the phase. This results in a decay of $<\mu>$. Obviously, this interpretation does not explain the behaviour of the small bipoles. In order to clarify the situation we present in Figure \[tilt-lat\] the behaviour of the tilt for small bipoles averaged in various latitudinal zones *versus* time, and the distribution of the small bipoles compared with that of the large. We see from this figure that the small bipoles demonstrate some kind of cyclic behaviour which is, however, quite different from that of the large. The time-latitude distribution of small bipoles demonstrates an equatorward propagating pattern as well as poleward. The cycle described by small bipoles looks shifted from that of the large. The tilt angles of the small bipoles are, as expected, determined mainly by bipoles located in the middle latitudes. In general, it looks plausible that small bipoles represent a different component of the solar magnetic field to that traced by the large ones. ![ Upper figure: time-latitude diagram for tilt according to KPVT data for bipoles with areas $50<S<300$ MSH. Blue shows negative tilt, red is for positive. The yellow points show the sunspot distribution. Lower figure: tilt evolution for bipoles with areas $50<S<300$ MSH in different latitudinal zones: black represents bipoles with $|\theta|<10^\circ$, green with $10^\circ\leqslant|\theta|<20^\circ$, red with $20^\circ\leqslant|\theta|<30^\circ$, and blue with $30^\circ\leqslant|\theta|<40^\circ$. []{data-label="tilt-lat"}](tilt-lat2.eps){width="0.8\linewidth"} Violations of Hale’s Law ------------------------ Now we examine how Hale’s polarity law works for large and small bipoles. Of course, there is a small fraction of bipoles which violate Hale’s law. It is natural to compare this fraction with the fraction of sunspot groups which violate the law (according to this fraction is about $5-7\%$). The fraction of bipoles which [*follow*]{} the Hale polarity law *versus* the bipole area is presented in Figure \[hale\]. The plot is organized as follows. We divide the bipole sample in bins according to their area. Dots in the plot correspond to centres of a bin. Then the fraction of bipoles in a bin which follow Hale’s law is shown by the vertical coordinate in the plot. In our analysis we used the MDI data at the maximum stage of the Cycle 23 (period 1998–2007). At the end of this cycle the overlap of bipoles with opposed orientations occurs because of the extended solar cycle at high latitudes [@T10]. For the largest bipoles the fraction which follow Hale’s law exceeds 90%. Such bipoles correspond to bipolar sunspot groups and the result, as expected, agrees with the estimate of . The fraction of bipoles which follow Hale’s law drops with the bipole area. This seems natural because it is more difficult to isolate small bipoles than the large ones, and the noise level in determination of the bipole orientation is larger for the small bipoles. The point however is that the plot in Figure \[hale\] shows specific slopes for large, $S^{0.05}$, and small, $S^{0.2}$, bipoles. This confirms that small and large bipoles represent physical entities of different natures. The slopes match near $S=500$ MHS. This confirms the validity of the area threshold chosen to separate small and large bipoles. The data for bipoles in these groups are shown in blue and red in Figure \[hale\]. The fraction of bipoles which follow the Hale’s law becomes as small as 50% near $S=50$ MHS and, then, it becomes fruitless to consider smaller bipoles. ![The fraction of bipoles oriented according to Hale’s polarity law (MDI data). Blue marks the segment with the slope of the line 0.2, red marks the segment with the slope 0.05. []{data-label="hale"}](shale.eps){width="0.6\linewidth"} Thus the estimate $5-7\%$ for the number of reversed bipoles is valid mainly for bipoles with areas greater than 1000 MSH. Flux and Tilt ------------- We now investigate in more detail the link between the size of a bipole and the tilt. There are two natural measures for the size of a bipole, its area $S$ and its magnetic flux $F$ (here and below $F$ is measured in $10^{20}$Mx). Fortunately, both these quantities are closely interrelated (Figure \[sf\]), $F(S)\sim S^{1.25}$ (blue line). This means that it is sufficient to study only the dependence on $F$. In the same figure we show that the large bipoles give the main contribution to the total magnetic flux. More precisely we plot with a black line a function $\Phi(S)$ which gives the contribution to the total flux of bipoles with areas greater than $S$. The line is fitted well by $\Phi(S)=\exp [-S/(2\times 10^3)]$. Figure \[sf\] shows that about $90\%$ of total flux comes from bipoles with areas $S>300$ MSH. However we appreciate that such an estimate does not, for instance, take into account that the lifetime of small bipoles (ephemeral regions) is shorter than that of the large bipoles (sunspots). Thus the contribution of the smaller bipoles to the total flux may be underestimated and a more detailed analysis is required. In particular, a measure of the flux regeneration rate would seem to be more suitable here. ![Black line shows the contribution of bipoles to the total flux according to MDI data (graphics of $\Phi(S)$). Blue dots indicate the mean flux for bipoles with different areas, the fitted line has a slope of 1.25.[]{data-label="sf"}](area-flux3.eps){width="0.6\linewidth"} ![Tilt against flux $F$ \[$10^{20}$ Mx\] for MDI data.[]{data-label="fl"}](f-tilt.eps){width="0.6\linewidth"} Figure \[fl\] shows that a negative tilt value is predominant for $F<10$. With $F>30$ it becomes positive. Comparing the plot with the previous Figure \[sf\] we conclude that the dividing point $F=20$ corresponds to an area $S=300$ MSH. The tendency of increasing tilt seems to remain for larger values of $F$. The investigation of the correlation between flux and tilt is interesting in the first place for estimating the contribution of bipolar moments to the formation of the poloidal component of magnetic field [@S13]. We recall that the bipolar moment is $B_m=F\cdot d$, where $F$ is the flux of a bipole and $d$ is the distance between the unipolar regions in the bipole. Furthermore, $d$ is another natural measure of the bipole size. The distance $d$ is defined as the distance in heliographic degrees between the geometrical centres of monopoles in a given bipole. This definition includes indirectly a contribution from the area of the bipole (a part of $d$ comes from the radii of the two opposite polarities of the bipole) and is strongly affected by the shape of the domains (complex configurations can even give zero $d$). Indeed, a simplistic presentation of bipoles as two near circular domains leads to $d$ increasing as $\sqrt{S}$, where $S$ is a measure of area of a bipole. ![Mean distance $d$ against size of the domains given by $\sqrt{S}$.[]{data-label="sd"}](ssd.eps){width="0.6\linewidth"} In fact Figure \[sd\] shows the slope of $d$ as function of $\sqrt{S}$ to be significantly less than one. This means that domain sizes increase faster than the distance between them. Again, behaviour of the plot is different for large and small bipoles (however the slope is almost the same), and the dividing point is close to $S=300$ MHS (note that the plot shows $\sqrt S$ rather than $S$). ![Bipolar moment against area of bipoles, MDI data. The fitted line has a slope of 1.5.[]{data-label="sbm"}](sbm.eps){width="0.6\linewidth"} For investigation of the bipolar moment, $Bm$, we consider first its dependence on the area of the bipoles. Figure \[sbm\] shows that $Bm$ is proportional to $S^{1.5}$. The mean tilt versus $Bm$ is shown in Figure \[tbm\]. We see that the tilt of small bipoles behaves again in the opposite way to that of the large bipoles and the dividing point ($Bm=90$) lies between $200-300$ MSH (Figure \[sbm\]). ![Tilt against bipolar moment $Bm$ \[$10^{20}$ Mx$^\circ$\] for MDI data.[]{data-label="tbm"}](bm3.eps){width="0.6\linewidth"} Figure \[tbm\] shows a moderate growth of the tilt with increasing bipolar moment for large bipoles. Note that did not find any significant variations of tilt angle with flux or bipolar moment. Now we can calculate the averaged effect of the tilt as follows. We consider the bipoles which follow Hale’s polarity law, multiply the tilt by the polarity $p=\pm1$ of the leading component of the bipole, and sum $F\sin(p\mu)$ over all bipoles (for the period 1998–2007 using MDI data). The quantity obtained is 10% of the total magnetic flux and shows which part of the toroidal magnetic field is converted to poloidal one. In other words, it is an estimate of the ratio $\alpha/v$, where $v$ is the r.m.s. velocity of the convection and $\alpha$ is the magnitude of the alpha-effect. The estimate is robust in sense that it remains stable if the large bipoles only are taken into account (small bipoles give only a minor contribution to the estimate), or if we consider bipolar moments instead of magnetic fluxes. The estimate is remarkably close to the order-of-magnitude estimate in and corresponds to a traditional expectation from dynamo theory. Discussion and Conclusions ========================== Summarizing, we conclude that we can recognize specific properties of small and large bipoles, which are represented mainly by ephemeral regions and sunspot groups respectively. The dividing point between the groups is located near a bipole area $S = 300$ MSH. However the separation of the groups can be based on other relevant quantities, *i.e.* bipolar moment, magnetic flux, distance between the bipole domains. All ways of separating the groups give similar results. The main difference between the two groups in the context of our research is the opposite sign of the bipole tilt. However specific properties of two groups of bipoles are visible in other respects as well. We note several other remarkable features in the results obtained. The time-latitude diagram for small bipoles (Figure \[tilt-lat\]) seems to agree with the concept of the extended solar cycle [@Wetal88]. In particular the wings of the butterfly diagrams for the tilt of small bipoles start at high latitudes 1 – 2 years earlier than the corresponding sunspot cycle, and then propagate towards the solar equator. For the large bipoles ($S > 300$ MSH), the tilt becomes larger for larger bipoles. The tendency is visible if the flux is considered as a measure of the size of bipoles (Figure \[fl\]) as well as if the bipole moment is considered as the measure (Figure \[tbm\]). Separation of bipoles into small and large with opposed properties seems to support the idea of that the regimes of dynamo action in the presence of sunspots (*i.e.* large bipoles) and in their absence are substantially different. Probably, this illuminates a difference between solar dynamo action during the Maunder minimum and in contemporary solar cycles. Of course, a simple explanation is that the fields responsible for the large and small bipoles just originate from different depths where the fields have different properties. Pragmatically, the contribution of large bipoles to the total $\alpha$-effect (that parametrizes solar dynamo action) dominates and it seems possible to ignore small bipoles when quantifing it. A deeper understanding of the processes underlying solar dynamo action deserves however a physical interpretation of the opposed properties of the large and small bipoles with respect to the tilt. First of all, we have to note that our research is inevitably based on a complicated algorithmic procedure leading to the isolation of bipoles from magnetograms. Our analysis does not show any trace of a bug in the algorithm used which could produce a difference between small and large bipoles. It is difficult however to exclude such an option completely based on the results of the application of just one algorithm for isolation of the bipoles. We thus stress the desirability of comparing our results with those from other algorithms which can isolate small bipoles from large. Future research in this direction is needed. suggested that the opposite sign of tilt for small and large bipoles can be understood as indicating that large bipoles represent toroidal magnetic field while the small ones are connected with poloidal magnetic field. Figure \[tilt-lat\] seems to support this interpretation. Indeed, the interpretation of large bipoles, *i.e.* sunspot groups as tracers of toroidal magnetic field, is standard and the problem is to identify what is traced by the small bipoles. Figure \[tilt-lat\] shows that the tilt patterns in the time-latitude diagram for small bipoles are pronouncedly dissimilar to the sunspot wings in the diagram. They are however quite similar to corresponding patterns of the large-scale surface magnetic field [@Oetal06] which presumably trace the solar poloidal magnetic field. The fact that we see specific slopes for small and large bipoles in other plots presented above seems to agree with this interpretation. However one could expect even more dramatic events at transitions in the plots between small and large bipoles, such as jumps. The point is that dynamo modelling predicts that toroidal magnetic field should be substantially stronger than the poloidal, which would be expected to result in jumps. Another possible interpretation might be based on the idea that the $\alpha$-effect is associated with hydrodynamic and magnetic helicities, which are inviscid invariants of motion (; ). Then accumulating helicity in one range of scales (or spatial region) would have to be compensated by growth of helicity of the opposite sign in the other range. A moderate growth of the tilt with bipolar moment for large bipoles in Figure \[tbm\] seems to be consistent with the idea of the helicity separation in Fourier space [@S96]: the point is that the Coriolis force is larger for larger bipoles. A continuous behaviour of the plots in figures which do not involve tilt directly becomes natural with this interpretation: all bipoles now represent toroidal field. In contrast, the form of the time-latitude diagrams in Figure \[tilt-lat\] now requires an explanation. We stress that the arguments in favour of either of the above interpretations do not seem convincing enough for us to prefer one to the other. The paper is supported by RFBR under grants 12-02-00170, 13-02-91158, 12-02-00884, 12-02-31128, 13-02-01183, 12-02-00614. We are grateful to D. Moss for a critical reading of the paper. Choudhuri, A.R., Karak, B.B.: 2012, [*Phys. Rev. Lett.*]{} **109**, 171103 Hale, G.E., Ellerman, F., Nicholson, S.B., Joy, A.H.: 1919, [*Astrophys. J.*]{} **49**, 153 Kosovichev, A.G.: 2008, [*Adv. Space Res.*]{} **41**, 830 Krivodubskii, V.N.: 1998, [*Astron. Rep.*]{} **42**, 122 Li, J., Ulrich, R.K.: 2012, [*Astrophys. J.*]{} **758**, 115 Obridko, V.N., Sokoloff, D.D., Kuzanyan, K.M., Shelting, B.D., Zakharov, V.G.: 2006, [*Mon. Not. Roy. Astron. Soc.*]{} **365**, 827 Pipin, V.V., in Kosovichev, A.G., de Gouveia Dal Pino, E.M., Yan, Y. (eds.), [*Solar and Astrophysical Dynamos and Magnetic Activity*]{}, *Proc. IAU Symp.* **294**, 375 Scherrer, P.H., Bogart, R.S., Bush, R.I., Hoeksema, J.T., Kosovichev, A.G., Schou, J., *et al.* and the MDI Engineering Team: 1995, [*Solar Phys.*]{} **162**, 129 Schou, J., Scherrer P.H., Bush, R.I., Wachter, R., Couvidat, S., Rabello-Soares M.C., Bogart R.S., Hoeksema J.T., Liu, Y., Duvall, T.L., Akin, D.J., Allard, B.A., Miles, J.W., Rairden, R., Shine, R.A., Tarbell, T.D., Title, A.M., Wolfson, C.J., Elmore, D.F., Norton, A.A., Tomczyk, S., 2012: [*Solar Phys.*]{} **275**, 229 Seehafer, N.: 1996, [*Phys. Rev. E*]{} **53**, 1283 Sokoloff, D.D., Khlystova, A.I.: 2010, [*Astron. Nachr.*]{} **331**, 82 Stenflo, J.O.: 2013, [*Astron. Astrophys. Rev.*]{} **21**, 66 Stenflo, J.O., Kosovichev, A.G.: 2012, [*Astrophys. J.*]{} **745**, 129 Tlatov, A.G., Illarionov, E.A., Sokoloff, D.D, Pipin, V.V.: 2013, [*Mon. Not. Roy. Astron. Soc.*]{} **432**, 2975 Tlatov, A.G., Vasil’eva, V.V., Pevtsov, A.A.: 2010, [*Astrophys. J.*]{} **717**, 357 Vitinsky, Yu.I., Kopecky M., Kuklin G.V.: 1986, *Statistics of the Sunspot Activity*, Nauka, Moscow Wilson, P.R., Altrocki, R.C., Harvey, K.L., Martin, S.F., Snodgrass, H.B.: 1988, [*Nature*]{} **333**, 748 [^1]: 1 MSH = $3.044\times10^6$ km$^2$. A round spot with area $S$ (in MSH) has a diameter of $d$ = $(1969\sqrt{S})$ km = $(0.1621 \sqrt{S})^{\circ}$ (see ).

--- abstract: 'We present a $4''/m''$-respecting crisscross AFM model in 2D and 3D, both belonging to the $Z_2$ classification and exhibiting interesting magnetic high-order topological insulating (HOTI) phases. The topologically nontrivial phase in the 2D model is characterized by the fractional charge localized around the corners and the quantized charge quadrupole moment. Moreover, our 2D model also exhibits the quantized magnetic quadrupole moment, which is a unique feature compared with previous studies. The 3D system stacked from layers of the 2D model possesses the HOTI phase holding chiral 1D metallic states on the hinge, which corresponds to the Wannier center flow between the valence and conduction bands. The novel transport properties such as the half-quantum spin-flop pumping phenomena on the side surfaces of the HOTI phase is also discussed.' author: - Jinyu Zou - Zhuoran He - Gang Xu bibliography: - 'HOTI.bib' title: 'Higher-order topological insulators in a crisscross antiferromagnetic model' --- *Introduction*.–Topological insulators (TIs) are distinctive quantum states characterized by the $Z_2$ invariant protected by time-reversal symmetry (TRS), which are stable under the continuous deformations of the band structure without closing the band gap [@Hasan2010; @Qi2011]. A striking feature of the TI phase is the $(D-1)$-dimensional bulk-boundary correspondence, where the $D$-dimensional bulk is insulating but supports $(D-1)$-dimensional robust gapless boundary states [@Kane2005; @Bernevig2006; @Fu2007; @ZhangHaijun2009]. In 2011, Fu *et al.* generalized the classification of topological materials to topological crystalline insulators (TCIs), where the gapless boundary modes are immune to local perturbations without breaking the crystalline symmetries [@Fu2011; @Hsieh2012; @Chiu2016; @LiuChaoXing2015; @Neupert2018; @Song2018]. Recently, the $(D-1)$-dimensional bulk-boundary correspondence principle was generalized to a higher-order correspondence, and the topological phase is called higher-order topological insulator (HOTI) [@Benalcazar2017; @BenalcazarPRB2017; @Song2017; @Langbehn2017; @Schindler2018; @SchindlerNP2018; @WangArxiv2018]. In a $D$-dimensional $n$th-order TI ($n\leq D$), the $D$-dimensional bulk and the $D-1, \cdots, D-n+1$ dimensional boundaries are all gapped, but there are $(D-n)$-dimensional gapless boundary states protected by the crystalline symmetries. Especially, in the 2D second-order and 3D third-order TIs, the protected 0D corner states correspond to the quantized charge quadrupole and octupole moments [@Benalcazar2017]. Thus, these topologically nontrivial states turn out to be very rich in nature. To determine their topological numbers, a method has been established based on the band representations of the crystallographic space group at high-symmetry points in the Brillouin zone (BZ), which is a promising road to searching and constructing topological materials [@Bradlyn2017; @Po2017; @Zhang2019; @Tang2019]. According to the robust hinge states flowing between the valence and conduction bands, the 3D second-order TIs can be divided into helical HOTIs and chiral HOTIs [@Schindler2018]. The helical HOTIs preserve the TRS and support bidirectionally propagating gapless modes on the hinges. SnTe was the first predicted helical HOTI by first-principle calculations in Ref[@Schindler2018], and the crystal bismuth was predicted and experimentally confirmed to possess the helical HOTI phase in Ref[@SchindlerNP2018]. The chiral HOTIs break the TRS and support unidirectionally propagating hinge states [@Miert2018; @Kooi2018; @Ezawa2018]. The Sm-doped Bi$_2$Se$_3$ [@Yue2019] and EuIn$_2$As$_2$ [@Xu2019] materials was proposed by first-principle calculations to exhibit the chiral HOTI phase, but the magnetic structure in Sm-doped Bi$_2$Se$_3$ is still under debate. Hence, the search for chiral HOTIs remains an important open question. ![The configurations of the crisscross AFM model in 2D and 3D. (a) Schematic illustration of the 2D crisscross AFM model. The four sites in each unit cell are labeled by their corresponding numbers, and the pinned spin directions are represented by red arrows. The intra- and inter-cell hopping amplitudes are $t_1$ and $t_3$, respectively. (b) Schematic illustration of the 3D crisscross AFM model.[]{data-label="models"}](models.eps "fig:"){width="8.5cm"}\ From a theoretical point of view, chiral HOTIs can be constructed from the perspective of magnetic groups [@Okuma2019], in which the TRS is broken but the combination of TRS with some crystalline symmetry is preserved. In this paper, we construct a crisscross antiferromagnetic (AFM) square lattice model satisfying $4'/m'$ magnetic point group (MPG), where the $m'=M_z T$ symmetry confines the spin polarizations in the $xy$-plane and then the $4'=C_{4z}T$ symmetry pins the spins in the directions as shown in Fig. \[models\]. Our 2D model can realize the nontrivial corner states with $1/4$ quantum magnetic quadrupole moment (MQM) and $1/2$ quantum charge quadrupole moment (CQM). By stacking the 2D lattice in $z$-direction, the system exhibits a novel 3D second-order TI phase, whose topological invariant can be determined from the band representations of $4'/m'$ at $C_{4z}T$-invariant points. The symmetry protected chiral states can exist robustly on the hinges of the 3D HOTI phase with insulating side surfaces, which can lead to topological magnetoelectric response and half-quantum spin-flop pumping behaviors. *2D Model*.–Here we introduce a crisscross AFM model on a 2D square lattice. As marked by the blue dotted square in Fig. \[models\]a, there are four sites in one unit cell. The spins on sites $1$ and $3$ point in the $\hat{y}$ and $-\hat{y}$ directions and the spins on sites $2$ and $4$ point in the $\hat{x}$ and $-\hat{x}$ directions. Obviously, such a 2D lattice model satisfies the MPG $4'/m'$ generated by $C_{4z}T$ and $PT$, where $T$ is the TRS and $P$ is the inversion symmetry. We set $c_{\alpha},c_{\alpha}^\dagger$ as the annihilation and creation operators on site $\alpha$ ($\alpha = 1,2,3,4$) in the spin directions given in Fig. \[models\]a, *i.e* $\left|\uparrow_y\right> = (1,i)^T/\sqrt{2}$, $\left|\uparrow_x\right> = (1,1)^T/\sqrt{2}$, $\left|\downarrow_y\right> = (1,-i)^T/\sqrt{2}$ and $\left|\downarrow_x\right> = (1,-1)^T/\sqrt{2}$. Then $PT$ and $C_{4z}T$ act on these basis states as: $$\label{Generators} \begin{split} &PT c^\dagger_{\alpha} (PT)^{-1} = (i)^{\alpha+2} c^\dagger_{\alpha +2}, \\ &C_{4z}T c^\dagger_{\alpha} (C_{4z}T)^{-1} = e^{-i\frac{\pi}{4}}(i)^{\alpha+2} c^\dagger_{\alpha+1}. \end{split}$$ Yielding to the constrain of $4'/m'$ symmetries, the intracell nearest-neighbor hopping from site $2$ to site $1$ can be defined as $t_1 = \sqrt{2}\,_{\!}\lambda_{1\,}e^{-i\pi/4}$, with $\lambda_1$ being real, then the $C_{4z}T$ symmetry immediately requires the hopping amplitude from site $3$ to site $2$ to be $-it_1^*$. Such constraint also applies to the intercell nearest-neighbor hopping $t_3$, which can be written as $t_3 = \sqrt{2}\,_{\!}\lambda_{3\,}e^{-i\pi/4}$ with $\lambda_3$ being real. We note that the intracell next-nearest-neighbor hopping $t_2$ is zero due to the $PT$ symmetry. By setting the lattice constants $|\mathbf{a}_1|=|\mathbf{a}_2|$ to be unity, the Hamiltonian under the constraints of the MPG $4'/m'$ in the momentum space with the basis $(c_{k1},c_{k2},c_{k3},c_{k4})^T$ are given by $$\label{HamiltonianAndEnergy2D} \begin{split} H(k) &= (\lambda_1 + \lambda_3\cos{k_x})(\sigma_x+\sigma_y)\tau_z \\ &+ (\lambda_1 + \lambda_3\cos{k_y})(\sigma_x+\sigma_y)\tau_y \\ &- \lambda_3[\sin{k_x}(\sigma_x -\sigma_y)\tau_0 + \sin{k_y}(\sigma_x +\sigma_y)\tau_x], \end{split}$$ where each term is a direct product of Pauli matrices $\sigma_{x,y,z}$ and $\tau_{x,y,z}$, and $\tau_0$ is the $2\times 2$ identity matrix. Then the everywhere doubly degenerate dispersion relation is given by $E(k)= \pm 2\sqrt{\lambda_1^2 + \lambda_3^2 + \lambda_1\lambda_3(\cos{k_x} + \cos{k_y})}$, which separates the phase diagram of the system into four regions with two topologically distinct insulating phases: $|\lambda_1|>|\lambda_3|$ and $|\lambda_1|<|\lambda_3|$, and get a gapless phase transition state at $\lambda_1 = \pm \lambda_3$. We plot the band dispersion $E(k)$ along high-symmetry line for $\lambda_1 = 2.3\lambda_3$, $\lambda_1=\lambda_3$ and $\lambda_3 = 2.5\lambda_1$ in Figs. \[2D\]a–\[2D\]c, respectively, and assume the Fermi level at zero. The two insulating phases shown in Fig.\[2D\]a and Fig.\[2D\]c can be distinguished by the position of the Wannier centers (WCs) of the occupied bands. In the trivial phase with $|\lambda_1| > |\lambda_3|$, the atomic orbitals on four sites hybridize to form $4$ Wannier orbitals and the WCs are located at the center of the unit cell. In the nontrivial phase with $|\lambda_1| < |\lambda_3|$, the WCs move to the corners of the unit cell, which means when cutting the infinite bulk into a square sample, zero-energy states will be left at the corners. Such a topological phase with insulating 2D bulk and 1D edge but 0D zero-energy modes is called 2D second-order TI [@Benalcazar2017]. ![The band structure of the 2D crisscross AFM model and the corner states. (2a-2c) The band structure along high-symmetry lines with the parameters $\lambda_1 = 2.3\,_{\!}\lambda_3$ (trivial), $\lambda_1 = \lambda_3$ (gapless) and $\lambda_3 = 2.5\,_{\!}\lambda_1$ (nontrivial), respectively. Here $\xi=\pm 1$ labels the representation of the symmetry $S_4=\xi\,_{\!}e^{-i\frac{\pi}{4}\gamma_z}$ with the third Pauli matrix $\gamma_z$ at high-symmetry point $\Gamma$ and $M$. The band gap closes when $\lambda_1 = \lambda_3$ and band inversion happens when $|\lambda_1| < |\lambda_3|$, leading to a nontrivial electronic structure. (2d) The energy levels for $20 \times 20$ unit cells of the nontrivial phase $\lambda_3 = 2.5\,_{\!}\lambda_1$. Four zero-energy modes emerge in the gap. (2e) The exponential distribution of the $e/2$ fractional charges carried by each corner state in $20 \times 20$ unit cells square.[]{data-label="2D"}](2D.eps "fig:"){width="8.5cm"}\ As discussed by Benalcazar et al [@Benalcazar2017; @Liu2019], the 2D second-order TI phase transition can also be understood from the change of the charge quadrupole moment (CQM), which is defined as $$\label{quadrupole} q_{xy} = \sum_{n=\mathrm{occupied}} P_x^n P_y^n /e,$$ where $P_i^n = \frac{e}{2\pi}\int d^2k \langle u_n(k)|\partial_i|u_n(k)\rangle$ denotes the charge polarization of the $n$th band along the $i=x,y$ direction. With the unitary rotoinversion symmetry $S_4=PC_{4z}=(PT)(C_{4z}T)$, the polarization can be expressed by the representation of $S_4$ at the high-symmetry points $\Gamma$ and $M$ [@Schindler2018] as follows: $$\label{polarization} P_{x/y}^n = \frac{e}{2}\,_{\!}(\frac{\eta_M^n}{\eta_\Gamma^n}\ \mathrm{modulo}\ 2),$$ where $\eta_{M/\Gamma}^n =\pm e^{\pm i\pi/4}$ denotes the $n^{th}$ band’s eigenvalues of $S_4$ at $M/\Gamma$, and one get $P_x=P_y$ due to the $S_4$ symmetry. In Figs. \[2D\]a and \[2D\]c, we have illustrated all bands’ representation matrices of $S_4$ as $\xi\,_{\!}e^{-i\pi/4\gamma_z}$ with $\xi = \pm 1$. Explicitly, for the HOTI phase in Fig.\[2D\]c, two occupied states’ $S_4$ representation at $M$ and $\Gamma$ points are $\eta_{M} =\{ e^{i\pi/4}, e^{ -i\pi/4}\}$ and $\eta_{\Gamma} =\{ -e^{ i\pi/4}, -e^{ -i\pi/4}\}$ respectively. Thus one can obtain a nonzero CQM $q_{xy} = e/2$ in the HOTI phase $|\lambda_1| < |\lambda_3|$. The nonzero CQM implies the fractional corner charges [@Benalcazar2017], which corresponding to the localized corner sates. In Fig. \[2D\]d, we plot the energy levels of a square sample of $20\times 20$ unit cells in the nontrivial phase with $\lambda_3 = 2.5\,_{\!}\lambda_1$, where four zero-energy corner states related by $C_{4z}T$ all appear in the gap. At half filling ($2e$ per unit cell, $2L^2$ total electrons), the four corner states will share two electrons and each corner state carries a fractional charge $e/2$, which is exponentially distributed around the corner, as shown in Fig.\[2D\]e. Besides the CQM and localized fractional corner charges, our model also exhibits the magnetic quadrupole moment (MQM), which is a unique property compared with previously studied 2D HOTI models constructed from the $M_x$ and $M_y$ symmetries [@Benalcazar2017] or the $S_4$ rotoinversion group [@Miert2018; @Ezawa2018]. In a magnetic lattice model, we can define the MQM in a unit cell as $\varrho_{ij} = \frac{1}{2}\,_{\!}(r_im_j+r_jm_i), i,j=x,y$, where $\mathbf{r} \ \mathrm{modulo}\ \mathbf{a}_1, \mathbf{a}_2$ is the position of the orbital carrying the magnetic moment $\mathbf{m}$. In our model, the magnetic moments are confined in the $xy$-plane by the $M_zT$ symmetry, and the $2$-fold rotation $C_{2z} = (C_{4z}T)^2$ requires that both the $x$ and $y$ coordinates of the WCs must be either $0$ or $1/2$ in units of the lattice constant. Thus, in our HOTI phase, when the WCs are not at $0$, the quantized MQM tensor is given by $$\label{MQM} \varrho_{ij} = \frac{1}{4}\,_{\!}g\,_{\!}\mu_B\left( \begin{array}{ccc} 1 & -1 & 0 \\ -1 & -1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right)\!,$$ where $g$ is the electron Lande factor and $\mu_B$ is the Bohr magneton. *3D Model*.–As discussed above, the 2D systems for $|\lambda_1| > |\lambda_3|$ and $|\lambda_1| < |\lambda_3|$ correspond to distinct insulating phase, in which WCs locate at the center and corner of the unit cell, respectively. Hence it is nature to build a 3D tight-binding model where the $k_z=0$ and $k_z=\pi$ plane belong to different 2D phase. Such 3D model turns out to be a 3D second-order topological insulator with chiral hinge states, as shown in Fig. \[3D\]d. For this purpose, we construct a 3D lattice structure as illustrated in Fig. \[models\]b, where the interlayer hopping parameters are restricted by $4'/m'$ satisfying $t_4=\sqrt{2}\,_{\!}\lambda_{4\,}e^{-i\pi/4}$, $t_5=\sqrt{2}\,_{\!}\lambda_{5\,}e^{-i\pi/4}$, and $t_z=-i\lambda_z$, with $\lambda_4$, $\lambda_5$, and $\lambda_z$ being all real. Since the real part of $t_z$ only gives an overall shift of the energy bands, we take $t_z$ to be purely imaginary. Setting the interlayer distance as unity, the Hamiltonian in 3D momentum space takes the form as $$\label{HamiltonianAndEnergy3D} \begin{split} H_\mathrm{3D}(k) &= (\lambda'_1 + \lambda'_3\cos{k_x})(\sigma_x+\sigma_y)\tau_z \\ &+ (\lambda'_1 + \lambda'_3\cos{k_y})(\sigma_x+\sigma_y)\tau_y \\ &- \lambda'_3[\sin{k_x}(\sigma_x -\sigma_y) \tau_0 + \sin{k_y}(\sigma_x +\sigma_y)\tau_x]\\ &+ 2\lambda_z\sin{k_z}\,_{\!}\sigma_z\tau_0, \\ \end{split}$$ where the effective parameters $\lambda'_1 (k_z)= \lambda_1 + 2\lambda_4\cos{k_z}$ and $\lambda'_3 (k_z) = \lambda_3 + 2\lambda_5\cos{k_z}$ depend on $k_z$. Obviously, the $4'/m'$ MPG is preserved in the $k_z = 0$ and $\pi$ planes, in which the 3D Hamiltonian can hence be reduced to 2D model described by the formula (\[HamiltonianAndEnergy2D\]) with parameters $\lambda'_1 = \lambda_1 \pm 2\lambda_4$ and $\lambda'_3 = \lambda_3 \pm 2\lambda_5$. Depending on whether the $k_z=0$ and $\pi$ planes holding the same or different 2D topological phases, our 3D model can exhibit either the normal insulator (NI) phase or the second-order TI phase, which can be distinguished by a topological number $v$ defined through the $S_4$ symmetry eigenvalues as $$\label{Z2} (-1)^v = \frac{\xi_R\xi_M}{\xi_Z\xi_\Gamma}.$$ Here, $\{\xi_\beta e^{i\pi/4}, \xi_\beta e^{-i\pi/4}\}$ ($\xi_\beta = \pm1$ and $\beta = R,M,Z,\Gamma$) are the $S_4$ eigenvalues of the occupied bands at the high-symmetry points. The NI phase has $v=0$ and the HOTI phase has $v=1$. For the $v=1$ phase, symmetry protected chiral hinge states should exist robustly respecting to all the $C_{4z}T$ symmetry preserving perturbations. For example, one can add opposite 2D Chern insulators on the $x$-terminated and $y$-terminated surfaces of the HOTI, respectively. Such perturbation will change the number of edge states on each hinge by an even number. In that sense, the chiral HOTI phase here is classified by a $Z_2$ topological invariant, which is different from EuIn$_2$As$_2$ that belongs to $Z_4$ classification with $v=2$ standing for HOTI and $v=1,3$ standing for TI phase [@Xu2019]. To drive the system to the HOTI phase, the parameters should satisfy $$\begin{split} [(\lambda'_1(0))^2-(\lambda'_3(0))^2][(\lambda'_1(\pi))^2-(\lambda'_3(\pi))^2] < 0. \end{split}$$ Thus the phase diagram is divided by the hyperplanes in the parameter space $(\lambda_1, \lambda_3, \lambda_4, \lambda_5)$. In Fig. \[3D\]a, we plot a section of the phase diagram as an example, by assuming $\lambda_1 = 1$ and $\lambda_3=0.8$. Evaluate the band dispersion $E(k) = \pm 2\sqrt{\lambda'^2_1 + \lambda'^2_3 + \lambda'_1\lambda'_3(\cos{k_x} + \cos{k_y}) + \lambda_z^2\sin^2k_z}$, the energy gap closes at one of the $S_4$ high-symmetry points ($\Gamma$, $M$, $R$, or $Z$), when the topological phase transition occurs. ![The band structure of the 3D crisscross AFM model, the Dirac surface states, and the chiral hinge states. (3a) Phase diagram in $\lambda_4/\lambda_1, \lambda_5/\lambda_1$ space with $\lambda_3 = 0.8\lambda_1$. (3b) The band structure along high-symmetry line for a HOTI with $\lambda_3 = 0.8\lambda_1, \lambda_4 = 0.41\lambda_1, \lambda_5 = -0.3\lambda_1, \lambda_z = 0.55\lambda_1$, where $\Gamma = (0,0,0), Z = (0,0,\pi), R = (\pi,\pi,\pi), M = (\pi,\pi,0)$. The inset shows the band structure of the HOTI with open boundary in $z$ direction and periodical boundary in $x,y$ direction. Two Dirac cone come from up and down surface degenerate at Fermi level because of the combined $PT$ symmetry. (3c) The band structure of the HOTI with open boundary in $x,y$ direction and periodical boundary in $z$ direction. Four edge states crossing the Fermi level correspond to the chiral hinge states. (3d) Schematic illustration of the Dirac surface states and chiral hinge states. The black arrows indicate the flow direction of the hinge states. $\vec{E}_1$ and $\vec{E}_2$ indicate the external electric field on $(001)$ and $(010)$ direction.[]{data-label="3D"}](3D.eps "fig:"){width="9cm"}\ The band dispersion $E(k)$ along high-symmetry line for the HOTI phase is plotted in Fig. \[3D\]b. As shown in Figs. \[3D\]c–\[3D\]d, when the open boundary is set in $x$ and $y$ direction, the 2D surfaces are all gapped but the unidirectionally metallic states can survive on the surfaces intersecting hinges, which are protected by the $C_{4z}T$ or $S_4$ symmetry [@Schindler2018]. However, if the open boundary is set in $z$ direction, as shown in the inset of Fig. \[3D\]b, the $(001)$ surface band structure is not gapped but exhibit a Dirac cone, which is immune to the perturbations that preserve the $C_{4z}T$ symmetry. To destroy the Dirac cone on $(001)$ surface, one can apply a magnetic field along the $z$ direction to break the $C_{4z}T$ symmetry but remain the $S_4$ symmetry, which will result in the connected hinge states in the hexahedron sample [@Ezawa2018]. *Discussion*.– Finally, we would like to discuss some unique transport properties on the surface of the 3D HOTI phase. The chiral second-order TI phase can be viewed as the result of introducing TRS-breaking but $4'/m'$-preserving interactions to a first-order TI. Hence, the Dirac cones on the side surfaces $(010)$, $(0\bar{1}0)$, $(100)$, and $(\bar{1}00)$ are all gapped by mass terms, resulting in the massive Dirac fermion behavior. Due to the $C_{4z}T$ symmetry, the mass terms on neighbouring side surfaces have opposite signs. As a result, the 1D metallic states are unavoidable on the domain walls, *i.e* the surfaces intersecting hinges, and are robust to any $C_{4z}T$-preserving local perturbations. It is known that massive Dirac fermions can provide a half-quantum Hall conductance given by $\frac{e^2}{2h}\mathrm{sgn}(m)$ [@Bernevig2013], where $m$ is the Dirac mass. Therefore, our chiral second-order TIs can also be viewed as axion insulators [@Wieder2018; @Essin2009; @Mong2010; @Yue2019] with the side surface holding half-quantum Hall conductances, which can lead to novel responses to external electric fields. If the electric field is applied along the $y$-direction (see $\vec{E}_2$ in Fig. \[3D\]d), the $(100)$ and $(\bar{1}00)$ surfaces will obtain opposite half-quantum Hall currents, which are connected by the surface states on $(001)$ and $(00\bar{1})$ surfaces. Such transport phenomena is a natural result of the topological magnetoelectric effect in the axion insulator with the effective action $$\label{TME} S_\theta = \frac{\theta e^2}{4\pi^2}\int dt d^3\mathbf{r}\mathbf{E}\cdot \mathbf{B}$$ where the axion angle $\theta=\pi$ [@Essin2009]. More interestingly, if the electric field is applied in the $z$-direction (see $\vec{E}_1$ in Fig. \[3D\]d), the charge will be pumped from two diagonal hinges to the other two hinges. Considering that the spin polarization directions of four hinge states are pinned in the $xy$-plane by $M_zT$ symmetry and must satisfy the $C_{4z}T$ symmetry, the spin direction will be deflected to the perpendicular direction through the pumping procedure on a side surface. Therefore, with the electric field applying in the $z$-direction, one can observe the half-quantum spin-flop pumping phenomena on the side surfaces. The authors thank C.-X.Liu, X.Liu, Z.-X.Liu and Z.-D.Song for helpful discussions. The authors acknowledge the support by the Ministry of Science and Technology of China (2018YFA0307000), and the National Natural Science Foundation of China (11874022).

--- abstract: | A broad swathe of astrophysical phenomena, ranging from tubular planetary nebulae through Herbig-Haro objects, radio-galaxy and quasar emissions to gamma-ray bursts and perhaps high-energy cosmic rays, may be driven by magnetically-dominated jets emanating from accretion disks. We give a self-contained account of the analytic theory of non-relativistic magnetically dominated jets wound up by a swirling disk and making a magnetic cavity in a background medium of any prescribed pressure, $p(z)$. We solve the time-dependent problem for any specified distribution of magnetic flux $P(R,0)$ emerging from the disk at $z=0$, with any specified disk angular velocity $\Omega_d(R)$. The physics required to do this involves only the freezing of the lines of force to the conducting medium and the principle of minimum energy. In a constant pressure environment the magnetically dominated cavity is highly collimated and advances along the axis at a constant speed closely related to the maximum circular velocity of the accretion disk. Even within the cavity the field is strongly concentrated toward the axis. The twist in the jet’s field $<B_{\phi}>/<|B_z|>$\ is close to $\sqrt2$ and the width of the jet decreases upwards. By contrast when the background pressure falls off with height with powers approaching $z^{-4}$ the head of the jet accelerates strongly and the twist of the jet is much smaller. The width increases to give an almost conical magnetic cavity with apex at the source. Such a regime may be responsible for some of the longest strongly collimated jets. When the background pressure falls off faster than $z^{-4}$ there are no quasi-static configurations of well twisted fields and the pressure confinement is replaced by a dynamic effective pressure or a relativistic expansion. In the regimes with rapid acceleration the outgoing and incoming fields linking the twist back to the source are almost anti-parallel so there is a possibility that magnetic reconnections may break up the jet into a series of magnetic ‘smoke-rings’ travelling out along the axis. author: - | D. Lynden-Bell$^{1,2}$\ $^{1}$Institute of Astronomy, The Observatories, Cambridge, CB3 0HA\ $^{2}$Clare College date: 'This version dated 24 January 2006, revised 3 February 2006, revised following reviewer’s comments 20 March 2006' title: Magnetic Jets from Swirling Disks --- \[firstpage\] jets, MHD, Cosmic Rays Introduction ============ Orders of magnitude – the voltages generated -------------------------------------------- We consider accretion disks of bodies in formation when the differential rotation drags around magnetic field lines. Although the moving magnetic fields inevitably generate electric fields, the resulting EMFs are perpendicular to the magnetic fields in the perfect conduction approximation. Such EMFs do not accelerate particles to high energies. However the world is not a perfect place; in regions where perfect MHD predicts very high current densities there may be too few charge carriers to carry those currents. In those regions perfect conductivity is not a good approximation and the fields are modified to allow an electric field component along the magnetic field so that the larger current can be generated by the few charge carriers available. With this background idea it becomes interesting to get rough estimates of EMFs that are likely to be around whatever their direction. It turns out that these EMFs scale with $(v/c)^3$ where $v$ is the maximum velocity in the accretion disk and is independent of the size or mass of the system. Relativistic systems in formation, however small, can generate EMFs whose voltages at least match those of the highest energy cosmic rays. However the timescales over which these systems persist and the total energies available over those time scales do of course depend on the mass of the system. Very crude estimates of the EMFs can be made as follows. Let us suppose that there is a central object, a star or a black hole, of mass $M$ surrounded by an accretion disk of mass $\zeta M$. Further let us suppose that a small fraction $\eta$ of the binding energy of the accretion disk is converted into magnetic field energy. Letting R be the inner radius of the disk we put $(4\pi/3)R^3(B^2/(8\pi))=G\zeta\eta M^2/R$ so $B=(6\zeta\eta/G)^{1/2}v^2/R$., where $v^2=GM/R$. Fields of a few hundred Gauss are found in T Tauri stars, so putting $R=10^{11}$cm $v=300$km/s we find $\zeta \eta=10^{-11}$. We shall use this dimensionless number to make estimates elsewhere. Now the EMF generated around a circuit that goes up the axis and back to the disk around a field line and finally along the disk back to the centre is about $(v/c)BR$ in esu and 300 times that in volts so EMF= 300 $(6\zeta \eta/G)^{1/2}(v/c)^3 c^2=(v/c)^3 10^{22}$ volts. Kronberg et al (2004) considered such accelerative processes in radio galaxies. Again only a fraction of this voltage will be available to accelerate particles; in the exact flux freezing case it is ALL perpendicular to the fields, nevertheless it is an estimate of what might be available where the perfect conductivity approximation breaks down. Although the argument is crude the answer is interesting in that it suggests that cosmic rays generated in microquasars may reach the same individual particle energies as those generated in quasars and radio galaxies which may match the highest cosmic ray energies. Jets ---- Curtis photographed the jet in M 87 from the Lick Observatory in 1918. While it was soon found to be blue, the emission process was not understood despite Schott’s detailed calculations of synchrotron radiation in his 1912 book. Finally after Shklovskii introduced this emission mechanism into astrophysics Baade (1956) showed the jet to be highly polarized which clinched it. Although Ryle’s Cambridge group (1968) found many double radio lobes around large galaxies the next obvious jet came with the identification of the first quasar 3C273 in which the dominant radio source is not at the nucleus but at the other end of the optical jet (Hazard, Mackey & Shimmins 1963; Schmidt 1963). There were difficulties in understanding the powering of the radio lobes of galaxies. If all the energy were present as the lobes expanded outwards there would be more bright small ones. Finally this led Rees (1971) to suggest that the lobes must be continuously powered by as-yet-unseen jets feeding energy into the visible lobes. As radio astronomy moved to higher frequencies with greater resolution these jets duly appeared in both radio galaxies and quasars. All the above jets have dynamically significant magnetic fields seen via their synchrotron radiation; a particularly fine study of one, Herculis A, is found in Gizani & Leahy (2003). Magnetic fields are less obvious in the Herbig-Haro objects first found in star-forming regions in 1951. However it took the development of good infra-red detectors before the heavy obscuration was penetrated in the 1980s to reveal the jets feeding the emission. These jets have velocities of one or two hundred km/s, far less than the $0.1c - c$ speeds of the extragalactic jets. The jets around young stars are seen to be perpendicular to the accretion discs that generate them. Since the giant black-hole accretion disk theory of quasars Salpeter (1964), Lynden-Bell (1969), Bardeen (1970), Lynden-Bell & Rees (1971) Shakura & Sunyaev (1973, 1976) many have come to believe that the radio galaxies and quasars likewise have jets perpendicular to their inner disks. However it was not until the wonderful work on megamasers (Miyoshi et al. 1995) that such inner disks around black holes were definitively confirmed. In 1969 when I predicted that they would inhabit the nuclei of most major galaxies including our own, M31, M32, M81, M87, etc., the idea was considered outlandish, but now most astronomers take it for granted. Fine work by Kormendy (1995) and others on external galaxies and the beautiful results of Genzel (2003) and Ghez (2004) on our own has totally transformed the situation. Meanwhile many jets have been found in objects associated with dying stars, SS433 and the micro-quasars Mirabel & Rodriguez (1999) being prominent examples within the Galaxy. Less energetic but more beautiful examples may be the tubular planetary nebulae that are associated with accretion discs of central mass-exchanging binary stars. These were brought to my attention by Mark Morris and recent evidence indicates that magnetism is important here too (Vlemmings et al 2006). Much more spectacularly the $\gamma$-ray bursts are now thought to come from accretion-disks around black holes within some supernova explosions. Poynting flows of electromagnetic energy appear to be one of the best ways of extracting the energy from beneath the baryons that would otherwise absorb the $\gamma$-rays, Uzdensky & MacFadyen (2006). Jets and collimated outflows have also been invoked for giant stars. Remarkable examples are R Aquarii (Michalitsanos et al. 1988) and the Egg nebula (Cohen et al. 2004) and the collimated outflow in IRG 10011 (Vinković 2004). The systematic features of these diverse objects are that an accretion disk is present and the jet emerges along the rotation axis of the inner disk. Magnetic fields are important in the radio objects and may be important in all. The jet velocities are strongly correlated with the escape velocities and therefore with the circular velocities in the disk close to the central object. In very collapsed objects these velocities are relativistic but in star-forming regions the jet velocities are less than $10^{-3}c$. The obvious similarity of the jet structures and collimation despite such velocity differences strongly suggest that relativity is not a determining factor in the making of these jets. The thesis that I put forward in 1996 (paper II) and repeat here, is that all these jets are magnetically driven, the common feature being that the Poynting flux dominates the energy transport in the jet. However I do not exclude the possibility of material being entrained as the jet makes its way through the material that surrounds it. Nevertheless we consider that magnetism is the driver and the prime reason for collimation is the magnetic twisting combined with a weak external pressure which is dynamically enhanced by inertia. It is the electromagnetic field that generates the relativistic particles in radio jets so, even if their total energies evolve to become comparable, the magnetic energy comes first and drives the whole phenomenon. While this paper is concerned with jets from systems with accretion disks, the jets near pulsars are probably closely related. There the magnetic field is rotated by the neutron star but the weaker field at large radii may be heavily loaded by the inertia close to the light cylinder. Field lines may be twisted from below far faster than they can rotate across the light cylinder. The resulting twisting up of the field within the cylinder may result in jet phenomena with similarities to those described here. The problem of jet collimation was emphasised by Wheeler (1971) at the Vatican Conference on the Nuclei of Galaxies in 1970. He drew attention to the computations of Leblanc & Wilson (1970) who found a remarkable jet generated on the axis of a rotating star in collapse. This may be hailed as the first gamma-ray burst calculation and the magnetic cavities discussed here are in essence analytic calculations based on the same mechanism. We have concentrated on the simplest case of force-free magnetic fields within the cavity. In earlier work others produced good collimation by twisting up a magnetic field that was imposed from infinity. An early paper on this was by Lovelace (1976) and the rather successful simulations of Shibata & Uchida (1985, 1986) are based on this theme. I consider that the imposition of a straight field from infinity ducks the question of why collimation exists. The straight field imposes it. Much work has concentrated on the hard problem of winds carrying a significant mass flux from a rotating star. This subject is well covered in Mestel’s book (1999) Li et al (2001), Lovelace et al (2002) and Sakarai (1987) found a weak asymptotic collimation and Heyvaerts & Norman (2002) have recently concluded a thorough study of the asymptotics of wind collimation. Bogovalov & Tsinganos (2003) have tackled the difficulties of collimating mass loaded flows from central objects and give models with the near-axis part of the flow well collimated despite the centrifugal force. Blandford & Payne (1982) discuss winds launched by centrifugal force. Lovelace & Romanova (2003) Li et al (2006) made numerical calculations based upon the differential winding we proposed in Papers I & II. The stability problems of twisted jets have been tackled in both the linear and non-linear regimes by Appl, Lery & Baty (2000) and Lery, Baty & Appl (2000) while Thompson, Lyutikov and Kulkarni (2002) have applied to magnetars the self-similar fields found in Paper I. In section 5 we find that some magnetic cavities float upwards like bubbles thus fulfilling the ideas of Gull & Northover (1976). The laboratory experiments of Lebedev et al. (2005) give some support for the type of models given here. A fine review of extragalactic jets was given by Begelman et al. (1984). See also the more recent work of Pudritz et al 2006, Ouyed et al 2003. Outline of this paper --------------------- This is the fourth paper of this series and puts a new emphasis on regions in which the ambient pressure decreases with height like $z^{-4}$. It also gives detailed analytical solutions of the $dynamical$ problem for the first time. Papers II and III concentrated on why there are collimated jets at all. Here we concentrate on the dynamic magnetic configurations generated. Paper I (Lynden-Bell & Boily 1994) showed that when an inner disk was rotated by $208$ degrees relative to an outer disk field lines that had connected them splayed out to infinity in the absence of any confining external pressure. At greater angles there was no torque as the inner and outer disks were magnetically disconnected. Paper II (Lynden-Bell 1996) demonstrated that inclusion of a weak uniform external pressure led to a strong collimation after many turns with the magnetic field creating towers with jet-like cores whose height grew with each turn. Again the inner disk was rotated rigidly relative to the outer disk. Paper III (Lynden-Bell 2003) was a refined version of a conference paper (Lynden-Bell 2001). In these the differential rotation of the accretion disk and external pressure variation with height were included and the shape of the magnetic cavity was calculated as a function of time. However the fields were not calculated in detail and the treatment used a static external pressure which was assumed to fall less rapidly than $z^{-4}$. In this Paper IV we show that these quasi-static models are actually dynamically correct provided that the motions generated in the field lines by the twisting of the accretion disk never become relativistic, but we then explore the consequences of the magnetic cavity expanding into a region where the pressure variation approaches $z^{-4}$. This results in a dramatic acceleration of the top of the magnetic cavity along a cone whose angle gradually narrows at greater distances. The sudden expansion out to infinity when the twist exceeded a critical angle, found in paper I when there was no confining medium, is still present in modified form when beyond some height the external ambient pressure falls as $z^{-4}$ or faster; once the field has penetrated to that region there is no longer a quasi-static configuration for the system to go to, so the jet accelerates to reach either dynamic ram-pressure balance whenever the background density falls less fast than $z^{-6}$ or failing that relativistic speeds. Dynamics from statics --------------------- In the standard MHD approximation the displacement current is neglected so curl${\bf B}=4\pi {\bf j}$. If the magnetic field dominates over any other pressure or inertial forces, then, neglecting those, the magnetic force density has nothing to oppose it, so ${\bf j\times B=0}$ and we deduce that the currents flow along the lines of force. This implies that ${\bf j=}\tilde \alpha{\bf B}$ and as both div${\bf B}$ and div${\bf j}$ are zero $\tilde \alpha$ is constant along the lines of force. We shall be considering problems with the normal component of magnetic field specified on an accretion disk at $z=0$ and the field does not penetrate the other boundary where an external pressure $p(z)$ balances $B^2/8\pi $. The past motion of the accretion disk has produced a twist in the field-lines which emerge from the disk and return thereto further out. Those with past experience will know that the above conditions supplemented by expressions for $B_n$ on the disk, the twists $\Phi$ on each line and $p(z)$, serve to define the problem, so the magnetic field is then determined. The other Maxwell equations curl${\bf E}=-\partial{\bf B}/\partial ct$, and that giving div${\bf E}$, are not needed in the determination of the magnetic field. Notice that all the equations used to find the magnetic field do not involve the time. Thus if we specify the twist angles together with the normal field component on the disk and the boundary pressure $p(z)$ at any time, the whole field configuration at that time is determined. Now let us suppose that we know how to solve this static problem but that $B_n(R),\Phi(R)$ and $p(z)$ are continuously specified as functions of time. Then the solution for the magnetic field in this dynamic problem is found by merely taking the sequence of static problems parameterised by $t$. Such a procedure will give us ${\bf B(r},t)$ but in the dynamic problem the motions of the lines of magnetic force generate electric fields that must be found also. However these can be found almost as an afterthought because we can determine how the lines of force move and their motion is the $c{\bf E\times B/B^2}$ drift. Assuming perfect conductivity there is no component of ${\bf E}$ along ${\bf B}$ so ${\bf E=B\times u}/c$. Our knowledge of ${\bf u}$ on the accretion disk tells us how the lines move everywhere, which in turn tells us the electric field. We conclude that the crux of the problem lies not in difficult and dangerous dynamics but in the staid simplicity and safety of statics. That said we need whole sequences of static solutions that allow us to turn up the twists and parameterise the external pressures. Even the static problem is no walkover and we would have found it impossible to get general solutions were it not for the energy principle that the magnetic field adopts the configuration of minimum energy subject to the flux, twist and pressure constraints imposed at the boundary. Here we have already demonstrated why an evolution of the magnetic field structure through quasi-static models gives the solution to the dynamic problem. Section 2 details the specification of the relevant static problem and the methods of solving it. In Section 3 we solve it developing further the approximate method of paper III. This gives us the mean fields within magnetic cavities whose shapes we calculate for any prescribed external pressure distribution. Emphasis is placed on solutions that access regions with $p$ falling like $z^{-4}$ and the very fast expansions then generated. We then generalise our results to allow for a dynamic ram-pressure and discover the shapes of inertially confined jets. In Section 4 we calculate the detailed magnetic fields within the cavity. Section 5 finds the electric fields generated as the magnetic cavity grows and categorises the types of solution. Section 6 gives exact solutions of special cases with the dynamical electric field also calculated. The Magnetic Problems to be Solved ================================== A magnetic flux $P(R_i,0)$ rises out of an accretion disk on $z=0$ at radii up to $R_i$ . The lines of magnetic force on a tube encircling the flux $P$ eventually return to the disk at an outer radius $R_o(P)$ after a total twist around the axis of $\Phi(P)$. The magnetic field above the disc is force free with current flowing along the field lines and there is negligible gas pressure within the magnetic-field-dominated cavity. However the magnetic cavity is bounded by a surface at which an external pressure $p(z)$ is specified. Later we shall consider the case of a dynamical pressure $p(z,t)$. Our problem is to find the magnetic field and the shape of the cavity containing it when the functions $P(R,0),\Phi(P)$ and $p(z)$ are specified. Axial symmetry is assumed. We think of $\Phi$ as due to the disk’s past differential rotation and sometimes write $\Phi(P)=[\Omega_d(R_i)-\Omega_d(R_o)]t=\Omega(P)t$ where the suffix $d$ refers to the rotation of the disk itself. The equation div${\bf B}=0$ implies that the magnetic field components in cylindrical polar coordinates $(R,\phi,z)$ may be written in terms of the flux function $P(R,z)$ which gives the flux through a ring of radius $R$ at height $z$, and the gradient of the azimuthal coordinate $\phi$, in the form, $$\label{1} {\bf B}=(2\pi)^{-1}\nabla P\times\nabla\phi +B_{\phi}\hat\phi.$$ The force-free condition $4\pi{\bf j\times B}=$curl${\bf B\times B}=0$ then tells us that $B_\phi$ takes the form, $$\label{2} B_\phi=(2\pi R)^{-1}\beta(P).$$ The function $\beta$ is constant along each field line (so it is a function of P) and must be determined from the solution so that the total twist on that field line is $\Phi(P)$. Finally the azimuthal component of the force free condition yields the equation $$\label{3} \nabla^2 P-\nabla \ln R^2.\nabla P =R{\partial\over \partial R}({1\over R}{\partial P\over \partial R})+{\partial^2P\over\partial z^2}=-\beta ^\prime(P)\beta(P)=-8\pi^2Rj_\phi.$$ This is to be solved within an unknown surface S given by $R=R_m(z)$ in which the field lies and on which ${\bf B^2}=8\pi p(z)$. The difficulties of this problem are: - the equation is non-linear - the function $\beta$ in it is not known and can only be determined from $\Phi(P)$ via the solution. - the bounding surface S is unknown. Luckily there is a different way of tackling these magnetostatic problems. The energy of the magnetic field and of the external gas-pressure must be a minimum subject to the flux and twist conditions on the accretion disk. The pressure energy stored in making a cavity whose area at height $z$ is $A(z)$ against an external pressure $p(z)$ is $$\label{4} W_p=\int p(z)A(z)dz.$$ The energy principle that applies even outside axial symmetry is that $W$ must be a minimum, where $$\label{5} 8\pi W=\int\left [\int\int (B_R^2+B_\phi^2+B_z^2)Rd\phi dR+8\pi pA\right]dz,$$ and ${\bf B}$ satisfies the flux and twist conditions on $z=0$. Two exact theorems follow; they were proved in paper III by expanding a horizontal slice through the configuration first vertically and then horizontally. Earlier more specialised versions appeared in papers I and II. These theorems are true even without axial symmetry. Defining averages $<.. >$ over a horizontal plane at height $z$, $$\label{6} <B_R^2>+<B_\phi^2>=<B_z^2>+8\pi\left( p(z)+[A(z)]^{-1}\int_z A(z')[dp/dz']dz'\right);$$ the final integral from $z$ up is negative whenever pressure falls at height. Minimum energy for horizontal displacements of the slice gives, $$\label{7} <B_z^2>=8\pi p(z)-(4\pi/A)dW_0(z)/dz,$$ where $$\label{8} 4\pi W_0=\int\int B_RB_zR^2d\phi dR,$$ and the integration is over the plane at height $z$. We shall show presently that after much twisting the magnetic configuration becomes very tall as compared to its width. As no extra radial flux is created the radial field lines become widely spread out over the height of the resultant tower and the gradients with height likewise become small. When we neglect terms involving $B_R$ equation (\[7\]) becomes $$\label{9} <B_z^2>=8\pi p(z),$$ and with a similar neglect in the use of equation (\[6\]) we find $$\label{10} <B_\phi^2>/(2-s) = <B_z^2> = 8\pi p(z),$$ where we have defined a dimensionless s (positive when pressure falls at greater height) by, $$\label{11} s(z)=-\int_z A(z')(dp/dz')dz'/[A(z)p(z)].$$ Notice that $s=0$ for the constant pressure case and that $(1-s)/s$ is the constant $d\ln(A)/d\ln(p)$ when $A$ the cross sectional area varies as a power of $p$. Two possible approaches to solving this problem are: I. FORWARDS METHOD We use the energy variational principle choosing such simple trial functions that the boundary conditions can be applied and the resulting variational equations can be solved. We can then solve the problem for all choices of the prescribed functions $P(R,0),\Phi(P)$ and $p(z)$ but the accuracy of the solution is limited by the imposed form of the trial function. II\. BACKWARDS METHOD Here we solve the exact equation (\[3\]) but make special choices of $\beta(P)$ and of the surface S so that we can solve the problem. Once we have the solution we discover the functions $P(R,0),\Phi(P),$ and $p(z)$ for which we have the solution. While this method is limited to a very few special cases, at least for them the solutions are exact. This enables us to check the accuracy and the validity of the more general solutions given by the Forwards method. Solution by Variational Principle ================================= This method was developed in paper III and with an extremely crude but instructive trial function it was used in paper II. There we showed that if each field line turned N times around the axis and the magnetic cavity was taken as a cylinder of height $Z$ and radius $R$ then, if the total uprising poloidal flux was $F$, very crude estimates of the field components are: $B_z=2F/(\pi R^2);B_R=F/(\sqrt2\pi RZ);B_\phi =NF/(RZ)$. Squaring these estimates, adding the external pressure $p$ and multiplying by the volume $\pi R^2Z$ we find $8\pi^2W=F^2[(1/2 +N^2\pi^2)Z^{-1}+(4R^{-2}+8\pi^3pR^2F^{-2})Z]$ where $p$ is assumed independent of $z$. Minimising over $R$ gives $R=(2\pi^3p/F^2)^{-1/4}$ so R is determined by the external pressure and the flux. With this result the final round bracket in $W$ reduces to $8R^{-2}$ and minimising $W$ over all $Z$ we find $4Z/R=\sqrt{1+2N^2\pi^2}\rightarrow \sqrt2\pi N$ which shows a remarkable collimation which improves with every turn! The method of paper III involved a much improved trial function which allows each field line, $P$, to attain whatever maximum height, $Z(P)$ it likes, allows for the different total twists $\Phi(P)$ of the different field lines and properly accounts for the variation of external pressure with height. Because of the great heights of the magnetic towers generated after many turns, most of the field energy is high above the accretion disk and the detailed distribution of the flux in $P(R,0)$ no longer plays a part in the distant field. At each height, z, mean fields are defined by $$\label{12} \overline B_\phi(z)=R_m^{-1}\int_0^{R_m}B_\phi(R,z)dR,$$ where $R_m(z)$ is the radius of the magnetic cavity at height $z$, and $A(z)=\pi R_m^2$. Also $$\label{13} \overline{ |B_z|}=A^{-1}\int_0^{R_m}|B_z|2\pi RdR=A^{-1}\int|\partial P/\partial R| dR=2P_m/A.$$ The last expression arises because $P$ is zero at both $R=0$ and $R=R_m$ and achieves its maximum $P_m(z)$ at an intermediate point. We also define $$\label{14} I^2=<B_z^2>/ \overline{| B_z|}^2,$$ and $$\label{15} J^2=<B_\phi^2>/\overline B_\phi^2;$$ although $I$ and $J$ are in principle functions of height, for tall towers we expect them to settle down to some typical values which we determine later. In what follows we neglect any variation of J with height; variation of I does not change the result. We now explain the basis of the trial function used. If a poloidal flux $dP$ is twisted once around the axis, it generates a toroidal flux $dP$. If its twist is $\Phi(P)$ the toroidal flux generated is $(2\pi)^{-1}\Phi(P)dP$. First consider distributing this toroidal flux uniformly over the height $Z(P)$ to which this field line reaches as was done in paper III. Unlike the use of $Z$ in the crude calculation in its new meaning it depends on the $P$ of the field line. In a small height interval $dz$ the element of flux $dP$ contributes a toroidal flux $[2\pi Z(P)]^{-1}\Phi(P)dPdz$ whenever the height is less than $Z(P)$. The total toroidal flux through $dz$ is contributed by all lines of force that reach above that height; that is by those with $P<P_m(z)$, where $P_m(z)$ is the maximum value that $P$ achieves at height $z$. Hence there is a toroidal flux through the area $R_m dz$ of $$R_m\overline B_\phi dz=(2\pi)^{-1}\int_0^{P_m}(\Phi/Z)dP dz;$$ $\Phi(P)$ is specified in terms of the accretion disk’s twist. $Z(P)$ is to be varied. However while this makes a possible trial function it is not generally true that the flux is so distributed. Indeed in the limiting case of the exact self-similar solutions discussed in the next paper the twist is strongly concentrated toward the top of each field line. The extreme alternative is found by putting all the twist at the top of each field line. Then the toroidal flux in any height increment $dz$ depends on the twist of those field lines whose tops lie in $dz$ so $R_m\overline B_\phi dz=(2\pi)^{-1}\Phi(P_m)(-dP_m/dz)dz$. However the truth must lie between the extremes of uniform twist and all twist at the top of each line. We therefore take the geometric mean of these two expressions for the toroidal flux distribution and obtain: $$\label{16} R_m\overline{B_\phi}=(2\pi)^{-1}[\Phi(P_m)(-dP_m/dz)\int_0^{P_m} (\Phi/Z) dP]^{1/2}.$$ Now $Z(P)$ is in the variational principle and $P_m(z)$ has the property that $P_m[Z(P)]=P$, so $P_m$ is the functional inverse of $Z(P)$; so whenever $Z(P)$ is varied $P_m(z)$ automatically varies in concert. We now omit the $B_R^2$ term in the energy principle as it is much smaller than the others once the towers grow tall, cf the crude estimate above equation (\[12\]). Writing $W$ in terms of the expressions discussed above $$\label{17} 8\pi W=\int \left[(4\pi)^{-1} J^2 \Phi(P_m) (-dP_m/dz)\int_0^{P_m(z)}\Phi/Z.dP +4I^2P_m^2/A+8\pi p(z)A\right]dz$$ Varying $A(z)$ gives $$\label{18} A=\pi R_m^2=IP_m/\sqrt{2\pi p(z)}$$ which using (\[13\]) and (\[14\]) is equivalent to $<B_z^2>=8\pi p$, as found in equation (\[9\]). Putting this expression back into equation(17) we find that the last two terms there combine to make $P_md\Pi/dz$ where $\Pi=\int_0^z4I\sqrt{2\pi p(z')}dz'$. We now integrate both terms in $W$ by parts and then rename the dummy variables in the integrals that result from the first term; these operations give $$8\pi \label{19} W=(4\pi)^{-1}J^2\int_0^F[\Phi(P_m)/Z(P_m)]\int_{P_m}^F\Phi (P)dPdP_m+\int_0^F\Pi dP_m,$$ where $\Pi$ is merely $\Pi(z)$ re-expressed as a function of $P_m$ $$\label{20} \Pi(P_m)=\int_0^{Z(P_m)}4I\sqrt{8\pi p(z)}dz$$ Most remarkably our adopted geometric mean between the uniform twist and top- twist cases has resulted in a $W$ of precisely the form obtained in the uniform twist case of paper III except that the first term is now exactly half of its former value. Since this clearly reduces $W$ the new trial function is clearly better than the old in that it brings us closer to the true minimum. Minimising $W$ over all choices of $Z(P_m)$ we find from (\[19\]) and (\[20\]) $$\label{21} Z^2\sqrt{8\pi p(Z)}=[J^2/(16\pi I)]P_m\overline \Phi(P_m)^2$$ where $$\label{22} \overline\Phi^2=\Phi(P_m)P_m^{-1}\int_{P_m}^F\Phi(P)dP$$ Equation (\[21\]) gives the function $Z(P_m)$ and its inverse $P_m(z)$ in terms of the given $p(z)$ and $\Phi(P)$. In line with section two’s introduction we write $\Phi=\Omega t$ and $\overline\Phi=\overline{ \Omega} t$. From equation (21) we notice that as a result of this $t$ dependence $\overline\Phi$ and $Z$ grow with $t$ at each value of $P_m$. This growth is at constant velocity if the external pressure is independent of height. However if that pressure falls off with height the velocity $dZ/dt$ accelerates. e.g. if $p\propto (a+Z)^{-n}$ with $n=2$ then $Z$ has nearly constant velocity until it reaches ’$a$’ but at greater heights it behaves as $t^2$ with constant acceleration. The shape of the magnetic cavity at each moment is given by plotting $Z$ against $R_m(Z)$. We find the relationship by substituting $P_m$ from equation (18) into equation (21) to give the equation for $Z$ as a function of $R_m$. Viz $Z=[J/(4\sqrt{2}I)]R_m\overline\Omega [\pi I^{-1}R_m^2\sqrt{2\pi p(Z)}].t, $ where $\overline\Omega $ is evaluated at the value of $P_m$ indicated in its square bracket. Dividing equation (21) by equation (18) we find the collimation at any given $P_m$ is just $$\label{23} Z/R_m=[J/(4\sqrt{2}I)].\overline\Omega(P_m)t,$$ which still grows linearly with time even when the external pressure decreases with height. However $p(z)$ should not decrease too fast else equation (\[21\]) will not have a sensible solution. To understand this we see from equation (\[22\]) that $P_m\overline\Phi^2$ decreases as $P_m$ increases since the twist $\Phi(P_m)$ is greatest for the field lines rising nearest the centre of the disk. Also $P_m$ decreases at greater heights since less magnetic flux reaches there. Hence $P_m\overline\Phi^2$ increases with height $Z$. ![Figure 1 shows the time evolution of the height of the jet in different pressure environments. On the left for the lowest graph the jet penetrates a constant pressure environment, n=0, at constant speed; the higher graphs are for n=2,3,4,6, and n=2 has almost constant velocity up to Z=a but feels the pressure decrease and thereafter proceeds at almost constant acceleration. At higher n the acceleration increases and there are no continuing solutions with magnetic pressure in balance beyond n=4. In fact the n=6 curve turns back unphysically to earlier times because the external pressure is too weak to withstand the magnetic field at larger heights. In reality the magnetic field springs outward at such a speed that dynamic ram-pressure comes into play. On the right we see the equivalent figure for ram-pressure with density profiles m=0,2,3,4,6,8 see equation (\[26\]) ](./fheadgrowth1.ps "fig:") ![Figure 1 shows the time evolution of the height of the jet in different pressure environments. On the left for the lowest graph the jet penetrates a constant pressure environment, n=0, at constant speed; the higher graphs are for n=2,3,4,6, and n=2 has almost constant velocity up to Z=a but feels the pressure decrease and thereafter proceeds at almost constant acceleration. At higher n the acceleration increases and there are no continuing solutions with magnetic pressure in balance beyond n=4. In fact the n=6 curve turns back unphysically to earlier times because the external pressure is too weak to withstand the magnetic field at larger heights. In reality the magnetic field springs outward at such a speed that dynamic ram-pressure comes into play. On the right we see the equivalent figure for ram-pressure with density profiles m=0,2,3,4,6,8 see equation (\[26\]) ](./fheadgrowth2.ps "fig:") \[fig1\] This must be true of the left hand side of equation (\[21\]) too and indeed it is obviously so when $p$ is constant so there is then a sensible solution. However should $p(Z)$ decrease faster than $Z^{-4}$ the left hand side of (\[21\]) would decrease with $Z$ so there would be no such solution. If the cavity accesses regions in which $p$ decreases a little less fast than $z^{-4}$ then the field rapidly expands outwards and this gives a most interesting jet model. For example if $p\propto(a+z)^{-(4-\delta)}$ then $Z$ would grow with a very high power of $t$, $t^{2/\delta}$ at each $P_m$ so high expansion speeds would be achieved quite soon; however ram pressure will increase the effective pressure at the jet’s head. This is explored in a later section. When relativistic speeds are achieved our approximation fails but analytical progress with relativistic jets can now be achieved following the lead of Prendergast (2005). For static pressures with $p=p_0/[1+(z/a)]^n$ we illustrate the motion of the jet-head in Figure 1. For $Z<a$ the velocity is almost constant but as the jet encounters the decrease in pressure it accelerates. For $n=2$ the acceleration becomes uniform, but for larger $n$ it increases and for $n=4$ infinite speeds would occur in finite time if our equations still held. For $n>4$ the graph turns back so the solutions indicate an infinite speed and then turn back giving no solutions for later times. These results are of course modified when ram-pressure is included as described later. CAVITY SHAPES FOR SPECIFIC MODELS --------------------------------- We have the solution for any specified distributions of $p(z)$ and for any twist $\Phi(P)=\Omega(P)t$. However before we can draw any cavity shapes we must also specify $\Omega(P)$ or equivalently $\overline{\Omega}(P)$. These two are connected through the definition $P[\overline\Omega(P)]^2=\Omega(P)\int_P^F\Omega(P')dP'=-(1/2)d/dP[\int_P^F\Omega(P')dP']^2$. This relationship is easily inverted. Evidently $\int_P^F\Omega(P')dP'=[\int_P^F2P\overline{\Omega}^2dP]^{1/2}$ so finally $\Omega(P)=P\overline{\Omega}^2[\int_P^F2P\overline{\Omega}^2dP]^{-1/2}.$ We expect the behaviour of $\Omega(P)$ near $P=F$ should be proportional to $(F-P)^2$ since on the disk $B_z=0$ there. From the definition above this implies that $\overline{\Omega}^2$ should be proportional to $(F-P)^3$ near there. It is also true that if $\Omega$ tends to $\Omega_0$ as $P$ tends to zero then $\overline{\Omega}$ will be proportional to $P^{-1/2}$ near there.     \[fig2\] The simplest expression with these properties is the Simple Model $ \overline{\Omega}=(2)^{-1/2}\Omega_0(P/F)^{-1/2}(1-P/F)^{3/2}$. This steadily falls from its central value and corresponds to $\Omega(P)=\Omega_0(1-P/F)$. A model with a more physical motivation is a uniformly magnetised rapidly rotating star or black hole giving an initial flux distribution on the disk $P(R,0)=FR^2/a^2$ for $R<a$ and $P=Fa/R$ for $R>a$. This Dipole Model has a sudden reversal of field at $R=a$. We combine this with a rotation $\Omega_d=\Omega_*---R<a;\Omega_d=\Omega_*(R/a)^{-3/2}---R>a$. Setting $Y=P/F, \Omega=\Omega_*[1-Y^{3/2}]$ and on integration $\overline\Omega(P)=\Omega_*Y^{-1/2}\sqrt{(1-Y^{3/2})[3/5-Y+(2/5)Y^{5/2}]}$. The shapes of some of the dynamic magnetic cavities generated by combining these $\Omega(P)$ distributions with simple ambient external pressure distributions $p(z)$ are illustrated in Figure 2. Figure 2 shows the time evolution of the magnetic cavity for the $n=3$ pressure distribution for the Dipole and the Simple models. They do not differ much. Very different cavities and velocities arise when the pressure distribution in the ambient medium is changed. In particular larger $n$ albeit less than 4 gives a fatter jet at a given length and a longer jet at a given time as illustrated in figure 2, however the collimation as determined by the length to width ratio is governed by equation (\[23\]) and so remains about the same at a given time. With the shapes of the cavities known it is now possible to calculate their areas at each height for each external pressure and thus discover how $s(z)$ varies with height. This is interesting as, from (\[10\]), $<B_\phi^2>/<B_z^2>=2-s$. For $z>>a$ the distribution of $2-s$ with $(z/Z_h)^2$ is given for various values of $n$ using the Simple Model. These graphs illustrate that the twist is less for the higher values of $n$ but for them it is more concentrated toward the top of the jets. The integrations for equation (\[11\]) were performed by expressing the area as a function of pressure via equations (\[18\]), (\[21\]) and (\[23\]). The final function for $z>>a$ which is plotted is $2-s=(1/2)[(4-n)/(n-1)]x^b[(1-x^c)/(1-x^b)]$ where $x^2=(z/Z_h) ;b=(4-n)/3;c=4(n-1)/3 $  \[fig3\] FAST JETS – RAM PRESSURE ------------------------ Our most suggestive finding thus far is that there are NO quasi-static solutions of large total twist $ \Phi$ when the external pressure falls like $z^{-4}$ or faster. This result is easily understood; A purely radial field in a bottom-truncated cone $r > a$ of solid angle $\omega$ will fall like $r^{-2}$. If the total poloidal flux both outwards and inwards is $F$ the magnetic field would be $2F/[\omega r^2]$ and it would deliver a pressure $F^2/[2\pi \omega^2 r^4]$ on the walls. The total field energy would be $F^2/[2\pi\omega a]$ and an equal amount of work would be needed to inflate the magnetic cavity against the external pressure so the total energy would be $F^2/[\pi \omega a]$. If instead we had a pure potential field with the same flux its field would fall as $r^{-(l+2)}$ so its energy would be about $F^2/[2\pi(2l+1)\omega a]$ where $l$ is the order of the Legendre polynomial that fits into the solid angle; so the energy of the purely radial field to infinity and back is only about $2l+1$ times the energy of the potential field. In practice $l=(4\pi /\omega)^{1/2}.$ In paper V we find that a total twist of the upgoing flux relative to the downcoming flux, of $\pi/sin[\theta_m/2]$ is sufficient for the flux to extend to infinity within a cone of semi-angle $\theta_m$. When the pressure falls with a power a little less negative than minus four we already demonstrated that continued twisting leads to the top of the tower or jet head advancing with a very high power of the time. However that was on the basis of a static pressure which will be enhanced by the dynamic ram-pressure once speeds comparable with the sound speed are achieved. Applying Bernoulli’s equation in the frame with the jet-head at rest we have $ v^2/2+[\gamma/(\gamma-1)]p/\rho = $constant, so at the stagnation point $p_s = p[1+(\gamma-1)/2\gamma .\rho v^2/p]^{\gamma/(\gamma-1)}$. If $p >>\rho v^2$ we get $p_s=p+\rho v^2/2$. Although that is much used we are more interested in the general case. When the flow becomes supersonic a stand-off shock develops so the above formula needs to be modified. Just behind the bow shock the pressure and density are given in terms of the upstream Mach number $M$ by $p_2= p(\gamma+1)^{-1}[2\gamma M^2-(\gamma-1)]; \rho_2=\rho (\gamma+1)M^2/[(\gamma-1)M^2+2]$; also $ \rho_2v_2=\rho v.$ Applying Bernoulli’s theorem after the shock results in a formula which for Mach numbers of two or more is well approximated by $p_s=\Gamma(\gamma)\rho v^2$. Here $\Gamma=[(\gamma+1)/2]^{(\gamma+1)/(\gamma-1)}\gamma^{-\gamma/(\gamma-1)}$ which is $1,0.93,0.88$ for $\gamma=1,4/3,5/3,$ respectively. The full formula for $p_s$ has a further factor on the right $[1-(\gamma-1)/(2\gamma M^2)]^{-1/(\gamma-1)}$ which clearly tends to one at large $M^2$. For $\gamma=4/3$ it ranges between $1.49 --> 1.05$ as $M$ ranges from one to three. For $\gamma=5/3$ the range is from $1.41 --> 1.03$. A useful global formula that somewhat approximates the trans-sonic pressure but is good in both the static and strongly supersonic limits is $$\label{24} p_s=p+\rho v^2.$$ This is the stagnation pressure that will be felt at the head of the jet. Away from the head the pressure is reduced both because the magnetic cavity expands less rapidly in the direction of its normal there and because a considerable fraction of the velocity is now parallel to the surface of the cavity. We return to this in the paragraph on Inertially Confined Jets. MOTION OF THE JET-HEAD ---------------------- From equations (\[21\]) & (\[22\]) the position of the jet head at time $T$ is $Z_h$ where $$\label{25} Z_h^2\sqrt{8\pi p_s}=J^2[16\pi I]^{-1}\Omega(0)\int_0^F\Omega(P)dP.t^2$$ where, because we are dealing with the head, we have replaced the pressure with the stagnation pressure and set $P_m=0$. With the stagnation pressure given by (\[24\]) but $\dot{Z}_h$ replacing $v$, equation (\[25\]) is now an equation of motion for the jet-head. As only $Z_h$ occurs in this subsection we shall usually drop the suffix $h$ which will be understood. When formulae are to be used in other sections we shall resurrect the suffix so that they can be lifted unchanged. Squaring (\[21\]) $p(Z)+\rho(Z)\dot{Z}^2=L^2(t/Z)^4$ where $p(z)$ and $\rho(z)$ are the undisturbed pressure and density at height $z$ and $L$ is the constant $(1/2)(8\pi)^{-3/2}J^2I^{-1}\Omega(0)\int_0^F\Omega(P)dP$. We are interested in this equation when pressure falls with height. If the initial velocity given by neglecting the $\dot{Z}^2$ term is subsonic then initially we shall have results very similar to the quasi-static case illustrated on the left of figure 1 but all those solutions accelerate as the pressure decreases so the velocity will become sonic and then supersonic so that the $\rho(Z)\dot{Z}^2$ term will dominate over the falling pressure. Now neglecting the pressure and taking the square root we have $Z^2[\rho(Z)]^{1/2}\dot{Z}=Lt^2.$ Setting $$\label{26} \rho(Z)=\rho_0[1+(Z/a)^3]^{-m/3},$$ so that the density behaves as $z^{-m}$ at large heights, we integrate to find $$\label{27} [1+(Z/a)^3]^{1-m/6}-1=(1-m/6)L_1t^3.$$ where $L_1=L\rho_0^{-1/2}a^{-3}$. For all $m$ the solutions for small heights are $Z=a (L_1)^{1/3}t$ and for large heights $$\label{28} Z_h=a[(1-m/6)L_1]^{2/(6-m)}t^{6/(6-m)}.......m<6.$$ Once again such solutions accelerate but less rapidly; however if the density ever falls as fast as height to the minus six even the ram-pressure is too weak and the solutions rise formally to infinite speed before failing altogether when $t=[6/(m-6)]^{1/3}F^{1/6}L_1^{-1/3}$. The trouble arises because the rapid fall in density gives too small a ram-pressure to resist the acceleration of the jet. In practice either the excess inertia due to relativistic motion or the fact that the density does not fall below intergalactic values will avoid this behaviour. In the former case we need to develop the relativistic MHD jet theory. In the latter we may use our $m<6$ model modified by replacing the density by the intergalactic one whenever our formula yields a smaller value, i.e. at $Z_h>a[(\rho_0/\rho_i)^{3/m}-1]^{1/3}$. As it reaches this region the velocity reaches $\dot{Z}_h=[L( \rho_i)^{-1/2}]^{1/3}$ and thereafter it remains at that value. So $Z_h(t)$ is given by formula (\[28\]) displayed in Figure 1 (right) until it reaches that region but then maintains its constant speed. INERTIALLY CONFINED SUPERSONIC JETS ----------------------------------- Landau and Lifshitz in their Fluid Mechanics book paragraph 115 give an elegant theory of supersonic flow past a pointed body, and the pressure on the body may be found by using the stress tensor $\rho (\partial\phi/\partial R)^2$ of the velocity potential given in their formula 115.3. However there are two drawbacks. Firstly it is not likely that our jet will constitute a pointed body rather than a blunt one, even if it could be treated as a body at all, and secondly the resulting formulae involve integrals over the shape of the body which we can only discover AFTER the pressure is known. We shall circumvent such difficulties while maintaining momentum balance by treating the medium into which the jet penetrates as ionised dust each particle of which collides with the magnetic cavity. Taking axes that move with the top of the field lines labelled by $P$ i.e. with velocity $\dot Z(P)$ we find a ram-pressure on the cavity wall at height $Z$ of $ 2\rho\dot Z^2 cos^2\theta$ where $cos^2\theta=\frac{1}{[1+(\partial Z/\partial R)^2_t]}$. The factor two arises from the assumption of specular reflection from the cavity wall and at normal incidence the formula gives a factor two more than the stagnation pressure at the jet head. This suggests that a dead-cat-bounce off the cavity wall may be a better approximation than a specular reflection so we shall omit the factor two in what follows. However even the resulting formula is hard to apply in practice so we now proceed to simplify it further. Most of our jets are not far from parabolic at the front, in which case $Z_h-Z=\kappa(t)R_m^2$. Then $(\partial Z/\partial R_m)^2=4(Z_h-Z)^2/R_m^2$. Hence we shall adopt for the dynamic pressure at $Z(P)$ $$\label{29} p_d=\rho \dot Z(P)^2/[1+4(Z_h-Z)^2/R_m^2]$$ To find the shape of a dynamically confined jet we need to solve equation (\[21\]) with the dynamic pressure $p_d$ replacing $p$ and use equations (\[23\]) and (\[26\]) for $R_m$ and $\rho(z)$. For $Z>>a$ equation (\[21\]) becomes $$\label{30}Z^{2-m/2}\dot Z \sqrt{8\pi \rho_0a^m/[1+4(Z_h-Z)^2/R_m^2]}=[J^2/(16\pi I)] P_m\bar\Omega^2t^2.$$ Except near the jet-head $Z_h-Z>>R_m/2$ so we may neglect the one and ,using (23) for $R_m$ the above equation simplifies to $$\label{31} Z^{3-m/2}\dot Z=K_1\bar\Omega^3t^3(Z_h-Z).$$ For an initial orientation we take $Z_h>>Z$. We may then integrate directly using our former result that $Z_h\propto t^{1/(1-m/6)}$ and obtain $Z\propto t^{(5-2m/3)/[(1-m/6)(4-m/2)]}$, whence it follows that $Z$ grows with a higher power of $t$ than $Z_h$ (at least for $Z$ small) indeed $X=Z/Z_h\propto t^{1/(4-m/2)}$. When $Z$ is sizeable the above overestimates $\dot Z$ so it overestimates $dlnX/dlnt$ which must in reality lie between zero and $(4-m/2)^{-1}$ which is itself less than $(1/4)(1-m/6)^{-1}$. We get a better estimate of the behaviour of $X$ by writing equation (\[31\]) in the form $$\label{32} X^{4-m/2}[(1-m/6)(1-X)]^{-1}[1+(1-m/6)dlnX/dlnt]=K_2t,$$ where at given $P,K_2$ is constant. Now the second square bracket above only varies between one and 5/4. If at lowest order we neglect its variation, we may solve for $t(X)$. We may then evaluate $$\label{33} dlnX/dlnt=(1-X)/[4(1-m/8)(1-X)+X],$$ (\[32\]) then determines $t$ as a function of $X$. Evidently the $t^{1/(4-m/2)}$ behaviour of $X$ persists approximately until $X$ is near unity; however $Z_h-Z\propto t^{1/(1-m/6)}(1-X)$ and this actually grows because with $X$ near unity the second square bracket in equation (\[32\]) is one and so $Z_h-Z\propto [K_2(1-m/6)]^{-1/(1-m/6)}X^{(4-m/2)/(1-m/6)}(1-X)^{-m/(6-m)}$, so as $X$ grows $Z_h-Z$, the distance from the head actually increases. $Z$ never catches up with $Z_h$ despite the fact that $X$ tends to one. All the above rests on the premise that $Z_h-Z>>R_m/2$ so we now investigate the behaviour when $Z$ is close to $Z_h$ so that $X$ is close to one but not so close that $(1-X)Z/R_m$ has to be small, Writing equation (\[30\]) in terms of $X$, but omitting the $dlnX/dlnt$ term as this vanishes with $X$ near unity, and dividing it by the same equation for $Z_h$ we find $1+4(Z_h-Z)^2/R_m^2=X^{6-m}/[K_4(P_m)]^2$ where $K_4$ is $ P_m\bar\Omega^2(P_m)$ divided by its non-zero value when $P_m$ tends to zero. At constant $P_m$, $(Z_h-Z)/R_m$ clearly increases as $X$ increases but it tends to the limiting value $(1/2)\sqrt{K_4^{-2}-1}$ as $X$ tends to one. As $K_4$ depends only on $P_m$ the shape of the jet-head is determined by the $\bar\Omega(P)$ function though its size and position depend on $t$ also. Thus the whole head of the jet and all parts with $X$ near one grow self similarly. The shape of the these parts is even independent of the details of the density fall-off embodied in $m$ but the size of these parts of the jet is proportional to $t^{m/(6-m)}$ so the rate of self-similar growth depends on $m$. As explained above the whole length of the jet grows with a different power of the time so it is just the part with $X$ close to one that grows self-similarly. To find the shape of the whole cavity we notice that elimination of Omega-bar between equations (\[23\]) and (\[30\]) or (\[21\]) coupled with use of (\[33\]) leads to an expression for $P_m$ in terms of $Z,Z_h,R_m,t$ $$\label{34} P_m= \frac{\pi}{2I\left(1-\frac{m}{6}\right)}(R^2_mZ/t) \left[1+\frac{(1-m/6)(1-Z/Z_h)}{[4(1-m/8)(1-Z/Z_h)+Z/Z_h]}\right] \sqrt{{8\pi \rho_0 a^m(Z^3+a^3)^{-m/3}\over[1+4(Z_h-Z)^2/R_m^2]}}$$ This value of $P_m$ is substituted into the Omega-bar of equation (23) to yield the equation relating $Z$ and $R_m$ at each time, $Z_h(t)$ being already known. Thus we get the shapes of the inertially confined jet cavities. Figure 4 displays one of these at two different times. ![An inertially confined jet cavity at two different times, the inner edge of the black gives the cavity in an m=3 density distribution for the Simple model magnetic flux. The times to these heights are given by equation (\[27\])[]{data-label="horseshoes"}](./fhorseshoes.ps) \[fig4\] THE FIELD IN THE MAGNETIC CAVITY ================================ Our solution of the variational principle has improved on paper III and given us dynamical solutions for the cavity’s shape and the mean field at each height. We now seek the detailed field structure within the cavity. At each height $z$ we define $\lambda$ to be $[R/R_m(z)]^2$ so $P$ may then be written $$\label{35} P(R,z)=P_m(z)f(\lambda).$$ Since by its definition $P_m(z)$ is the maximum value that $P$ takes at height $z$, it follows that $f$ achieves its maximum of unity at each height. Furthermore, as $P$ is zero both on axis and at the surface of the magnetic cavity, it follows that $f(0)=0=f(1)$. Thus $f$, which is positive, is highly circumscribed rising from zero to one and falling again to zero at one. Although $f$ may in principle depend on height (especially near the disk or at the top), nevertheless so circumscribed a function is unlikely to have a strong height dependence. We shall make a second approximation that an average profile will do well enough over at least a local region of the tower’s height. Thus we adopt the form given in equation (\[35\]) with $f$ a function of lambda but not of height. Now in the tall tower limit both $R_m(z)$ and $P_m(z)$ are only weakly dependent on $z$, so squares of their first derivatives and their second derivatives may be neglected. Equation (\[3\]) then takes the form $$\label{36} 4P_mR_m^{-2}\lambda. \partial^2f/\partial\lambda^2=-\beta.\partial\beta/ \partial P= -(\partial\beta^2/\partial\lambda)/(2P_mf^{'}).$$ Multiplying by $2P_mf'$ and integrating $d\lambda$ we find $\beta^2=4P_m^2R_m^{-2}(\int_0^{\lambda}f^{'2}d\lambda-\lambda f^{'2})$, so $\beta (P)$ is of the form $\beta[P_mf(\lambda)]=(2P_m/R_m)G(\lambda)$. Evidently $\beta$ is a product of a function of $z$ and a function of $\lambda$ but it is also a function of such a product. It is readily seen that a power law form for $\beta$ achieves this and we readily prove that the power law is the only possibility that allows it. Hence we may write $$\label{37} \beta=C_1P^\nu .$$ Thus $\beta'\beta\propto \nu P^{2\nu-1}$. Inserting this into equation (\[36\]) and calling the value of $\lambda$ where $f$ achieves its maximum of one, $\lambda_1$, we find on separating the variables $\lambda$ and $z$ both $$\label{38} \lambda d^2f/d\lambda^2=-C^2 \nu f^{2 \nu-1},$$ and $$\label{39} R_m=C_2P_m^{1-\nu},$$ where $$\label{40} C^2 = C_1^2C_2^2/4.$$ Now $P_m(z)$ decreases because not all flux reaches up to great heights, so equation (\[39\]) tells us that the radius of the magnetic cavity decreases there when $\nu<1$ and increases when $\nu>1$. Of course this only holds once the tower is tall so that its lateral confinement is due to the ambient pressure rather than the flux profile on the disk itself. The constant $C$ is determined by the requirement that $f=1$ at its maximum because $f(\lambda)$ already has to obey the boundary conditions at 0 and 1. Sometimes we find it convenient to use $\alpha=2\nu-1$ in place of $\nu$. We already know how to calculate the shape of the magnetic cavity. Knowing $A(z)$ and $p(z)$ we can calculate $s=-(Ap)^{-1}\int_zA(z')(dp/dz')dz'$ at every height $z$. We now show that associated with each value of $s$ there is a profile $f(\lambda)$. Equation (\[38\]) governs the possible profiles. Multiplying it by $-f/\lambda$ and integrating by parts we find $\int_0^1(f')^2d\lambda =C^2\nu\int_0^1f^{2\nu}\lambda^{-1}d\lambda$. However $P_mf'=A(2\pi R)^{-1}\partial P/\partial R=AB_z$ so the first integral is related to $<B_z^2>$ and the right hand one is similarly related to $<B_\phi^2>$. Multiplying both sides by $P_m^2/A^2$ we find $$\label{41} <B_z^2>={\nu}<B_\phi^2>$$ so comparing this with equation (\[10\]) we find $s=(2\nu -1)/\nu$, so $$\label{42} \nu=1/(2-s).$$ Thus for each height we have an $s$ and hence a $\nu$ for that height and the associated profile is given by solving equation (\[38\]) with $f=0$ at $\lambda=0$ and $\lambda=1$ and the value of $C$ is determined by the condition that $f$ is one at its maximum. Notice that by this means we have determined the weak variation of the profile with height (see Figure 3). PROFILE FOR CONSTANT EXTERNAL PRESSURE -------------------------------------- From equations (\[18\]) and (\[39\]) constant p corresponds to $\nu=1/2$, $\alpha=0$, a case of great simplicity. Integrating equation (38) with the boundary conditions that $f=0$ at zero and one, we find $f'=-C^2(1/2) $ln$(\lambda/\lambda_1)$ and $f=-C^2(1/2)\lambda$ln$\lambda$ with $\lambda_1=1/e$. The requirement that $f=1$ at maximum then gives $C^2=2e$ so $$\label{43} f=-e\lambda ln\lambda.$$ With $f$ known we may now evaluate the dimensionless integrals $I$ and$J$, $I^2=<B_z^2>/\overline{|B_z|}^2=(1/4)\int_0^1(df/d\lambda)^2d\lambda=e^2/4$, hence $$\label{44} I=e/2=1.359.$$ Likewise $$\label{45} J=2\sqrt{\int_0^1(f/\lambda) d\lambda\over\int_0^1(f^{1/2}/\lambda)d\lambda}=\sqrt{2/\pi}=0.798.$$ Thus once we specify $\Omega(P)$ and $p$, our solution for the shape of the magnetic cavity $R_m(z)$ is known via equations (\[21\]), (\[22\]), and (\[23\]), and within that cavity the structure of the field is given via the poloidal and toroidal flux functions $P$ and $\beta$. $$\label{46} P=P_m(z)e\lambda \ln(1/\lambda)$$ $$\label{47} \beta(P)=4(2\pi^3p)^{1/4}P^{1/2}$$ The magnetic fields are given by $B_R=-(2\pi R)^{-1}\partial P/\partial z, B_\phi = (2\pi R)^{-1}\beta, B_z=(2\pi R)^{-1}\partial P/\partial R$. These are surprisingly interesting, $$\label{48} B_z={1\over\pi R_m^2}{\partial P\over \partial \lambda}={eP_m(z)\over \pi R_m^2}[\ln(1/\lambda)-1],$$ which is positive for $\lambda<1/e$, zero at $\lambda=1/e$, and negative for $\lambda>1/e$ reaching the value $-eP_m/(\pi R_m^2)$ at the boundary $\lambda=1$. By equation (\[18\]) the magnetic pressure precisely balances the external pressure there. Unexpectedly we find that $B_z$ is infinite on the axis $$\label{49} B_z\propto \ln(R_m/R),$$ however the flux near the axis is small because $P$ behaves like $R^2\ln(R_m/R) $ there. Likewise the contribution to the energy from the magnetic energy-density near the axis behaves as $R^2[\ln(R_m/R)]^2$so the infinity in the magnetic field appears to be harmless. A greater surprise comes from the behaviour of $B_\phi$ near the axis. $$\label{50} B_\phi={2eP_m\over\pi R_m^2}\sqrt{\ln(R_m/R)},$$ whereas we expected to find $B_\phi$ to be zero on axis, it is actually infinite! However the ratio $$\label{51} B_\phi/B_z\rightarrow [\ln(R_m^2/R^2)]^{-1/2}\rightarrow 0$$ thus although the field lines near the axis do wind around it helically, the helix gets more and more elongated as the axis is approached so the axis itself is a line of force. Faced with this example it is evident that the normal boundary condition that on axis $B_\phi$ should be zero should be replaced by the condition that $B_\phi/B_z$ should be zero on axis. Since the field is force-free the currents flow along the field lines and $4\pi{\bf j}=\beta'(P){\bf B}=(C_1/2)P^{-1/2}{\bf B}$. As $P$ is zero on axis ${\bf j}$ is even more singular on axis than the magnetic field. Nevertheless the total current parallel to the axis and crossing any small area is finite and tends to zero as the area shrinks onto the axis. The exact Dunce’s Cap model of section 6 gives the same field structure close to the axis. In the next section we find that the infinite fields on axis are replaced by large finite ones when the pressure decreases at greater heights. Current sheets are of course a common feature of idealised MHD and we have one of necessity on the boundary of the magnetic cavity. The infinite field strengths and current densities on axis encountered in this solution suggest that very large current densities occur close to the axis in reality. The lack of sufficient charge carriers there will lead to a breakdown of the perfect conductivity approximation with large EMFs appearing up the axis resulting in particle acceleration along the axis. What observers ‘see’ as a jet may be just this very high current-density region where the particles are accelerated, i.e. only the central column of the whole magnetic cavity which may be considerably wider, cf the observations of Herculis A, Gizani & Leahy (2003). PROFILES WHEN PRESSURE DECREASES WITH HEIGHT -------------------------------------------- Even when the pressure varies we know how to calculate the shape of the magnetic cavity, so all we need is the profile of the field across the cavity. So we need to solve equation (\[38\]). For $\nu=1/2$ equation (\[43\]) gives the solution, but we need it for more general $\nu$. While this is easily computed an analytical approximation is more useful. For $\nu<1$ a good approximation is given by noting that for $\alpha=2\nu-1 $ small $-\ln \lambda \approx(1-\lambda^\alpha)/\alpha$; using this in our $\alpha=0$ solution for $f$ suggests the form $f\propto \lambda(1-\lambda^\alpha)/\alpha$ but a better approximation is given by $f=g(\lambda)/g(\lambda_1)$ where $$\label{52} g(\lambda)=\alpha^{-1}\lambda(1-\lambda^\alpha)/(1+ a_2\lambda^\alpha),$$ and $\lambda_1$ is given by $g'(\lambda_1)=0$. Notice that the denominator in $g$ is constant when $\alpha=0$ so it makes no difference to that solution. This form for $f$ automatically satisfies both the boundary conditions and the one-at-maximum condition. The equation that gives $\lambda_1$ is a quadratic in $\lambda_1^{\alpha}$ $$\label{53} 1-[\alpha+1-a_2(1-\alpha)]\lambda_1^\alpha- a_2\lambda_1^{2\alpha}=0.$$ We use it to find $a_2$ in terms of $\lambda_1$ which we determine below $$\label{54} a_2=[1-\lambda_1^{\alpha}(1+\alpha)]/[\lambda_1^{\alpha} [1-(1-\alpha)\lambda_1^{\alpha}]].$$ The logarithmic infinities in the fields on axis found in equations (\[48\])-(\[50\]) occur because of the log term in equation (46). The form of equation (\[52\]) shows that large finite fields replace those infinities when $\alpha>0$ and so $B_\phi\rightarrow 0$ on axis. For $\alpha=1$ there is another exact solution in terms of the Bessel function $J_1$, $$\label{55} f=k_1\sqrt\lambda J_1(k_1\sqrt\lambda)/[k_0J_1(k_0)];$$ here $k_0=2.405$ is the first zero of the Bessel function $J_0$ and $k_1=3.832$ is the first zero of $J_1$. The value of $\lambda_1$ is therefore $[2.405/3.832]^2=0.3939$. This is not too far from the value 1/e=0.3679 obtained for our $\alpha=0$ solution. This suggests that linear interpolation i.e. $\lambda_1= e^{-1}(1+0.07073\alpha)$ and $a_2$ determined via equation (\[53\]) will give a good fit and indeed numerical computations show the fit is excellent all the way from 0 to 1,the Bessel case. In the latter the expression $k_0J_1(k_0)=1.249$. The integrals $I$ and $J$ can be evaluated for the Bessel solutions and we give their values for the $\alpha=0$ case in brackets for comparison. $I=1.179(1.359);J=1.098(0.799)$. As expected $I$ and $J$ do change with $\alpha$ but remain within 15% of their means. Solutions to equation (\[38\]) for large $\alpha =2\nu-1$ were used in paper I to solve a different problem but that method works well and can be extended to work for all $\alpha$ above unity. Approximate solutions for large $\nu$ of the form $f=1-\nu^{-1}\ln(\cosh\Lambda)$ where $\Lambda=2\Lambda_1(\lambda-\lambda_1) -\Delta \ln(\cosh[\nu C(\lambda-\lambda_1)])$ with $\Delta=(6\Lambda_1)^{-1}$; $\Lambda_1=ch^{-1}e^\nu\simeq\nu$; $C\simeq 2\lambda_1^{1/2}$; $\lambda_1 \simeq \frac{1}{2}$ are deduced in the appendix as well as the generalised form $$\label{56} f=1-H^{-1}\ln(\cosh\Lambda);....;\Lambda= C_*(\lambda-\lambda_1) -\Delta \ln(\cosh\Lambda),$$ with $H=\nu-1/42-5/26.\nu^{-1};~h=\sqrt{H\nu};~\Delta =1/2.\nu^{-1}-1/9.\nu^{-2}+1/12. \nu^{-3};$ $\lambda_1=1/2-7/31.\nu^{-1}+4/41.\nu^{-2}$ and $C_*=hC\lambda_1^{-1/2}=2\nu+\ln4-4/15.\nu^{-1}$. Notice that with $\Delta $ small equation (\[56\]) with $H=\nu$ $C_*=\nu$, $C\lambda^{-1/2}_1$ reduces to our former solution. Inclusion of the $\Delta$ term allows for the asymmetry around the maximum caused by the variation of the initial $\lambda$ in equation (\[38\]). These freedoms allow us to fit exactly, not only the curvature at the maximum and the boundary conditions, but also the gradients at which the solution reaches zero at the ends of the range. Some details are given in the Appendix. The solutions for $f$ are plotted in Figure 5.  \[fig5\] By taking $\nu$ to be given by equation (\[42\]) in terms of $s(z)$ we now have a solution of the form $P=P_mf(\lambda,\nu)$ with $\lambda=(R/R_m)^2$ and $P_m, R_m$ and $\nu$ all depending weakly on $z$, see equations (\[18\]), (\[21\]) and (\[42\]). With $P$ so determined the function $\beta(P)$ is found from $\Phi(P)$ by integrating along field lines. These are given by ${dR\over B_R}={Rd\phi\over \\B_\phi}={dz\over B_z}$. Hence $$\label{57} {dR\over-\partial P/\partial z}={Rd\phi\over\beta(P)}={dz\over\partial P/\partial R}$$ The equality of the first and last terms merely tells us that $P$ is constant along field lines. The equality of the second and last terms gives us on integration with $P$ held fixed $$\label{58} \beta(P)={2P\Phi(P)\over\int (d\ln f/d\ln\lambda)^{-1}dz},$$ where the integration is along the curve $P$ constant from one foot point to the other. While the above procedure is the one to use when $P(R,0),\Phi(P)$ and $p(z)$ are specified there are special cases that are simpler. If we ask that $f$ be strictly independent of $z$ rather than that being an approximation, then equation (\[39\]) must hold exactly. Combining that with equation (\[18\]) we would have $p=I^2(2\pi)^{-1}(\pi C_2A^{-\alpha})^{-2/(1-\alpha)}\propto P_m^{2\alpha}$; from (11) this formula leads directly to $s=2\alpha/(\alpha+1)$ and equation (\[37\]) gives us the simple relationship $\beta(P)=C_1P^{(\alpha+1)/2}$ in place of equation (\[58\]), nevertheless these equations with s constant imply that $p$ vanishes, or becomes infinite, when the area $A$ vanishes, which will not be true. The exactly separable case is too restrictive near the top when the pressure varies. Electric Fields, Pushers, Floaters and Squirmers ================================================ We now determine the electric fields that occur when the accretion disk is in differential rotation. The velocity ${\bf u}$ of a magnetic field line is $c{\bf E\times B/B^2}$ and there is no ${\bf E}$ along ${\bf B}$ because of the perfect conductivity, hence ${\bf E=-u\times B}/c$ . To use this formula we first calculate the velocity of our field lines. On the accretion disk this velocity is that of the disk itself so the field line which initially intersected the disk at azimuth $\psi$ now has its outer intersection at $R_o(P)$ at azimuth $\phi_d=\psi+\Omega_d(R_o)t$. We now look for the angular velocity of the point at which this field line intersects a $z=$const plane. At each instant the field line obeys equation (\[57\]) so on integrating the second equality there $$\label{59} \phi=\psi+\Omega_d(R_o)t+\Omega(P)tq(z.P)$$ where we have written $\Omega(P)t$ for $\Phi(P)$ and $q=\int_0^z(d\ln f/d\ln\lambda)^{-1}dz/\int(d\ln f/d\ln\lambda )^{-1}dz$. Both integrations $dz$ are to be performed with $\lambda $ varying so as to keep $P$ constant. The final integral is to be performed from foot-point to foot-point. The physical meaning of q is the fraction of the total twist on the field line $P$ that occurs by height z. Since the field line reaches a maximum height $Z(P)$ and then returns, $q$ will be double valued as a function of $z$. To avoid this it is better to convert those $z-$integrations into integrations over $\lambda $, in which $q$ is single valued, rather than $z$. Such a conversion yields $q=\int_\lambda^{\lambda_o}Z'(P/f)(f\lambda)^{-1}d\lambda/\int_{\lambda_i}^{\lambda_o} Z'(P/f)(f\lambda)^{-1}d\lambda$, where we have written $P/f$ for $P_m$ to demonstrate that the $\lambda$ dependence arises through $f$ when $P$ is held fixed. The angular rotation rate of the intersection of this field line with a $z=$const plane is $$\label{60} \dot \phi=\Omega_d+\Omega(P)q+\Omega(P)t\dot q$$ $\dot q$ arises from two causes i) any explicit dependence of $Z'$ on $t$ that does not cancel between numerator and denominator in $q$, ii) the change in the lower end point $\lambda$ of the numerator’s integration. At constant $P$ and $z$ we have $\partial\ln P_m/\partial t=-(d\ln f/d\lambda)\dot\lambda$, which gives us the rate of change of the lower limit of the numerator. When $P(z)$ is either a power or constant the explicit dependence of $Z'$ on $t$ cancels between numerator and denominator so there is then no contribution from i). Of course had we left $q$ as a $z$-integration there would have been no contribution from the end point but then the contribution from the $t$-dependence of the $\lambda$ in the integrand must be re-expressed as a function of $P$ and $z$ via $f(\lambda)=P/P_m(z,t)$, is unwieldy. Now $R\dot \phi$ at constant $z$ is not the $\phi$-component of the velocity of the line of force but the velocity of the point of intersection of the line with the $z$=const plane. When the line of force is only slightly inclined to the plane the difference is obvious since the velocity of the line is perpendicular to the line. A little thought shows that in the intersection velocity the component of the velocity of the line in the $z$-plane parallel to the projection of the field into the plane is exaggerated by the factor $B^2/B_z^2$ relative to that component of the field line’s velocity. Thus writing hats for unit vectors and $\hat {\bf b}$ for the unit vector $(B_R\hat{\bf R}+B_\phi \hat {\bmath \phi})/\sqrt{B_R^2+B_\phi^2}$ in the $z$-plane considered and $ {\bf u}_\perp$ for the component of the field line’s velocity in that plane, the velocity of the intersection is ${\bf u}_\perp+[(B^2/B_z^2)-1]({\bf u_\perp.\hat b)\hat b}$. The component of this intersection’s velocity along ${\bf \hat \phi}$ is $$\label{61} R\dot\phi = u_\phi(1+B_\phi^2/B_z^2)+u_R(B_RB_\phi/B_z^2).$$ Now by Faraday’s law, the rate of change of the magnetic flux through a circle about the axis in the plane considered gives the EMF so $$\label{62} 2\pi RE_\phi= -\partial P/\partial ct=-\dot P/c=-(u_zB_R-u_RB_z)/c$$ This is a second equation for ${\bf u}$ and the third is just ${\bf u.B}=0$. Eliminating first $u_z$ and then $u_R$ we find $u_\phi=[(B_R^2+B_z^2)/B^2]R\dot\phi+[B_RB_\phi/(B^2B_z)]\dot P$. The other components are readily found from equations (\[61\]) & (\[62\]). The remaining components of ${\bf E}$ follow from $c{\bf E=B\times u}$. Like the electric fields on the accretion disk these fields are not far from cylindrically radial and directed away from the surface of greatest flux, $P$, at each height. We now turn our attention to categorising the different types of solutions. A Helium balloon needs a tether in tension if it is not to rise further. A water tank needs a support in compression if it is not to fall. If we cut a magnetic tower by a horizontal plane there is net tension across that plane due to the magnetic stresses if $\int(B_z^2-B_R^2-B_\phi^2)dA/(8\pi) > 0$ and net compression if that quantity is negative. We call these floaters or pushers respectively if they satisfy those criteria low down the tower. Floaters are are held down by magnetic tension; pushers are supported from the bottom by a net magnetic pressure. In the tall tower approximation the above criterion simplifies to $(<B_z^2>-<B_\phi^2>) >0$ for a floater. Comparing this criterion with equations (\[10\]) and (\[42\]) we see floaters have $s>1,\nu>1,\alpha>1$ and cavities whose radii increase at greater heights while pushers have $0<s\le 1$, $ 0.5<\nu\le 1$ and $0<\alpha\le 1$. Their cavities’ radii decrease with height. In a constant pressure medium all the solutions are pushers. If the pressure in a long narrow column supporting a weight is greater than the ambient pressure in the medium surrounding the column then any lateral bend bowing the column will be exaggerated as in the Euler strut problem. All our magnetic towers are stable against that bowing as their net support is atmospheric or less $(<B_R^2>+<B_\phi^2>-<B_z^2>)/(8\pi)-p=A^{-1}\int A(dp/dz)dz\le 0$. This criterion corresponds to $\alpha\ge 0,s\ge 0$. Thus squirmers do not occur in the magnetostatics of our systems which have pressure that decreases with height. Exact Solutions, Backwards Method ================================= Here we postulate the forms of $\beta(P)$ and of the bounding surface S for which we can solve the differential equation (\[3\]). After we have solved it we discover what problem we have solved by finding $p(z)$ on the bounding surface and the twist angles $\Phi(P)$ and the boundary flux $P(R,0)$. The differential operator on the left of equation (\[3\]) separates in oblate or prolate spheroidal and rotational parabolic coordinates as well as in cylindrical and spherical polars. It is therefore important to take the surface S to be one of those coordinate surfaces. Even after choosing one of those the problem is still non-linear; but there are interesting exceptions. We have already seen that the case $\beta'\beta =$const. corresponds to a constant external pressure and it gives a linear but inhomogeneous equation. A second simple case is $\beta(P)\propto P\propto \beta'\beta $. This linear case is well known and much explored but is somewhat less interesting than the $\beta \propto P^{1/2}$ case. The combination $\beta'\beta \propto P+P_0$ though soluble has the idiosyncrasies of both the former cases without obvious advantages. A different approach with more interesting solutions is to accept the non-linear behaviour of equation (\[3\]) and look for self-similar solutions. Most of the above methods can be generalised using perturbation theory. For example if the bounding surface is a ‘cylinder’ of slowly varying radius $R_m(z)$ one can solve the problem in terms of the scaled radius $R/R_m$ as in section 4. SELF-SIMILAR SOLUTIONS ---------------------- Writing equation (\[3\]) in spherical polar coordinates $(r,\theta,\phi)$ and setting $\mu=\cos\theta $ we find $$\label{63} r^2\partial^2P/\partial r^2+(1-\mu^2)\partial^2P/\partial \mu^2=-r^2\beta'\beta(P).$$ Self-similar solutions only occur when $P\propto r^{-l},\,\, \beta(P)=C_1 P^\nu$. If we now set $P=r^{-l}M(\mu )$ then the left hand side is $r^{-l}$ times a function of $\mu$ so $\beta '\beta $ which is a function of such a product, must equal such a product. Hence $\beta'\beta =(1+1/l)C_1^2 P^{1+2/l}$. Thus with $\nu=1+1/l$ the $r^{-l}$ cancels out and writing $f=M/M_1$, where $M_1$ is the maximum value of $M$ $$\label{64} l(l+1)f+(1-\mu^2)d^2f/d\mu^2=-(1+1/l) C_3^2 f^{1+2/l}.$$ This is a version of the equation of Lynden-Bell and Boily (paper 1) for whom the case with $l$ small was of especial interest. To make contact with the work of section 4 we remark that for narrow jets it is only necessary to consider $\mu$ close to one. Supposing the walls at $\mu=\mu_m$, we write $\lambda=(1-\mu)/(1-\mu_m)$, where both $(1-\mu_m)$ and $(1-\mu)$ are small, the equation takes the form $$\label{65} \lambda d^2f/d\lambda^2+l(l+1)(1-\mu_m)f/2=-C^2(1+1/l)f^{1+2/l}.$$ We have approximated $1-\mu^2$ as $2(1-\mu)$, which is good to 1 per cent or better for opening angles of less than 23 degrees. When the $(1-\mu_m)l(l+1)f/2$ is neglected this equation will be recognised as equation (\[38\]) of section 3. The particular solutions of equation (\[64\]) to which we now turn are so simple that we obtain them without making this narrow cone approximation. THE DUNCE’S CAP MODEL --------------------- Unexpectedly we shall use spherical polar coordinates centred at the top of the tower-like magnetic cavity at $(0,0,Z)$. This use of $Z$ is similar to the very crude description as a cylindrical tower in the introduction but now the tower will be conical pointing down at the accretion disk. This $Z$ corresponds to $Z_h$ in sections 3 and 4. We measure $\theta$ from the downward axis pointing toward the centre of the accretion disk (see Figure 6). We solve the force free equations within the downward opening cone cos$\theta\ge\mu_m$ which forms a dunce’s cap configuration over the accretion disk. We take $l=-2$ which corresponds to our $\beta\propto P^{1/2}$ behaviour appropriate for a constant pressure. Equation (\[64\]) becomes $2f+ (1-\mu^2)d^2f/d\mu^2=-C_4$. Writing $f=(1-\mu^2)m$ our equation becomes $d/d\mu[(1-\mu^2)^2dm/d\mu]=-C_4$, so, using the boundary condition that $(1-\mu^2)m'$ is non-singular at $\mu=1$, we find, $m=C_4\int(1+\mu)^{-2}(1-\mu)^{-1}d\mu$. On integration by parts we get $$\label{66} P=r^2(1-\mu^2)m=C_4r^2(1-\mu^2)\left[{1\over 2}(1+\mu_m)-{1\over 2}(1+\mu)+{1\over 4}\ln\left({1+\mu\over1+\mu_m}.{1-\mu_m\over 1-\mu}\right) \right].$$ ![The poloidal lines of force in the Dunce’s cap model at two different times. The top advances at constant speed while the bottom only revolves in differential rotation[]{data-label="duncecap"}](./fduncecap1.ps "fig:") ![The poloidal lines of force in the Dunce’s cap model at two different times. The top advances at constant speed while the bottom only revolves in differential rotation[]{data-label="duncecap"}](./fduncecap2.ps "fig:") \[fig6\] Evidently $P=0$ on $\mu=\mu_m$ and on the axis. For tall narrow cones $\theta$ is small,$\mu,\mu_m$ are near one and the flux function becomes $P={1\over4}C_4r^2\theta^2[\ln(\theta_m^2/\theta^2)].$ At $r=Z$ the maximum value of $P$ is $F$ so $C_4=4eF/R_d^2$ where $R_d=\theta_mZ.$ Also $\beta(P)=\sqrt{2|C_4|}P^{1/2}= C_4r\theta[\ln(\theta_m/\theta)]^{1/2}$. Also, see Figure 7, $$\label{67} 2\pi B_r=(r^2\theta)^{-1}\partial P/\partial \theta=C_4[\ln(\theta_m/\theta)-1];\,2\pi B_\theta= C_4\theta\ln(\theta_m/\theta);\,2\pi B_\phi= C_4[\ln(\theta_m/\theta)]^{1/2}$$ ![$B_r,B_\theta$ and $B_\phi$ are all independent of $r$ inside the Dunce’s Cap so we plot them as functions of $\theta/\theta_{max}$. While $B_\theta$ is zero at both boundaries $ B_\phi$ is zero only at the outer one. $B_r$ passes through zero at an internal point. See text for the surprising behaviour near $\theta=0$[]{data-label="Bvstheta"}](./fBvstheta.ps) \[fig7\] The constant $C_4$ is determined by the external pressure $C_4 = 8\pi[\pi p]^{1/2}$. Near the axis these fields have precisely the same behaviour as those we found in equations (\[49\]) and (\[50\]). However the fields are in one sense even more interesting; they are all independent of $r$, the distance from the origin. By construction $B_\theta,B_\phi $ vanish at the edge of the cone but $B_r$ is constant along its generators all the way up to the origin. The origin itself is then a point at which the infinite field coming up the axis turns back and splays out down these generators at constant field strength. This is the way the constant external pressure is opposed. With the origin such a remarkable point, some may suspect that the forces do not balance there, that the whole configuration might be held up by some sky-hook pulling at the apex. However $P$ varies as $r^2$ and the Maxwell stresses integrated over horizontal cuts through the cone give a net force that vanishes like $r^2$ as the origin is approached. Thus no sky hook is necessary to balance forces at the origin. The radial fields reverse at an intermediate value $\theta_1=\theta_m/\sqrt e$ (for narrow cones) and some may suspect reconnection there but the $B_\phi$ field component maximises at that value of theta so there is plenty of magnetic pressure to prevent reconnection. With $\mu_m$ close to one the radii on the accretion disk are $R=\theta Z$ so the flux coming up within a circle of radius $R$ is $P(R,0)=Fe(R^2/R_d^2)[\ln(R_d^2/R^2)]$. The twist function $\Phi(P)$ is given by integration along field lines which is simple for this self-similar field. $rd\theta/B_\theta=r\sin\theta d\phi/B_\phi$ yields on writing $\eta=\theta_m/\theta$ $$\label{68} \Phi(P)=\Omega(P)t=(Z/R_d)\int_{\eta_i}^{\eta_o}(\ln\eta)^{-1/2}d\eta\simeq (Z/R_d)[(\eta_i-\eta_o)(\ln\eta_i)^{-1/2}],$$ where the integral is evaluated between foot points at $R_{i,o}(P)$ of the field line and $\eta_{o,i}(P)=R_d/R_{o,i}(P)$. Therefore $\eta_{o,i}$ are the roots for $\eta$ of $$\label{69} P/F=2e\eta^{-2}\ln \eta$$ When the inner foot is much further in than the outer foot then $\Omega_i>>\Omega_o$ so then $\Omega(P)$ is close to $\Omega_i(P)$. This allows us to estimate the angular velocity of the accretion disk required. In the central parts it behaves as $R^{-1}[\ln(R_d/R)]^{-1/2}$, so the circular velocity of the disk is almost constant but drops to zero at the very centre like $[\ln(R_d/R)]^{-1/2}$. From Equation (\[69\]) both roots $\eta_{o,i}$ are independent of time. Hence from equation (\[68\]) $Z\propto t$ as expected from section 4 so we may write $Z=Vt$. Writing our solution for P in terms of cylindrical polar coordinates now centred on the centre of the disk, $P=eF(R^2/R_d^2)\ln(R_m^2/R^2)$, where $R_m(z)=R_d(1-z/Z)$ for $z<\,Z$ and it is just the dependence of $Z$ on time that generates the evolution of the system. $\beta(P)=\sqrt{8eF}P^{1/2}/R_d$ with $\Phi(P)$ given by equation (\[69\]). The magnetic fields are given by equation (\[67\]) so we now calculate the electric fields following the method of section 5. The simplicity of the expression for $P$ means that $q$ is simpler and in place of equation (\[60\]) we have $\phi=\psi+\Omega_d(R_o)t+(Vt/R_d)\int_{\eta_o}^{R_m/R}(\ln\eta)^{-1/2}d\eta $. Now $R_m= (1-{z \over Vt})R_d$ so $\dot R_m=zR_d/(Vt^2)4$. At fixed height a point only remains on the same field line if $\dot R=-\dot P/(\partial P/\partial R)=-\dot P/(2\pi RB_z)$ Using these and differentiating $\phi$ at constant $z$ and $P$ $$\label{70} \dot \phi=\Omega_d+VR_d^{-1}\int_{\eta_o}^{R_m/R}(\ln \eta)^{-1/2}d \eta+[\ln(R_m/R)]^{-1/2}\left({z\over Rt}+{\dot PVt\over 2\pi R_dR^3B_z}\right).$$ As before Equations (\[61\]) and (\[62\]) give the velocities of the lines of force as, $$\begin{aligned} \label{71} u_\phi&=&[(B_R^2+B_z^2)/B^2]R\dot \phi +[B_RB_\phi/(B^2B_z)]\dot P\nonumber \\ u_R &=& [B_z^2/(B_RB_\phi)][R\dot \phi-u_\phi(1+B_\phi^2/B_z^2)]\\ u_z &=& (B_z/B_R)u_R+\dot P/B_R \nonumber\end{aligned}$$ Also $cE_\phi=-\dot P/(2\pi R)$ while the other components follow from $c{\bf E=B\times u}$. Notice that the meridional field velocity ${\bf u}_M=(u_R,u_z)$ is not perpendicular to the vector meridional component of the magnetic field. It is the full vector velocity that is perpendicular to the complete magnetic field. We have explored the Dunce’s Cap model in all this detail both to demonstrate its interesting field structure and to show how we get the fields in an explicit example. Figure 5 gives the poloidal structure of the magnetic fields which wind around these surfaces of constant flux. Conclusions =========== It is very remarkable that these simple analytic methods allow the solution of these highly non-linear problems with variable domain, not just for a few special cases but for all twist functions Omega and all pressure distributions. Furthermore we get time-dependent solutions for all time. The main secret lies in the variational principle coupled with a good form of trial function. Once the time-dependence becomes relativistic we lose this tool so the problems will become harder. However when the relativistic motion is only important for the rotation it is likely that another variational principle may exist. An important problem is to seek it out. In the foregoing we have explored the simplest case in which the magnetic cavity is empty and the Poynting flux carries both the energy and the momentum of the jet. I believe that the results justify the assertion that this simplest case has remarkable similarities to the observed world and this suggests that the extra complication of material winds is not essential to explaining the main phenomena. However especially for Pulsars Relativistic rotation is an essential ingredient of the problem not covered here. Acknowledgments =============== We are grateful to the Institute of Advanced Study for providing the environment at Princeton where much of this work was done thanks to the support of the Monell Foundation. It was started earlier on a visit to the Carnegie Observatories in Pasadena where most of paper III was completed. Discussions with N.O.Weiss and P.Goldreich proved most helpful; C.Pichon and G.Preston helped with computations and F.J. Dyson was a source of encouragement. [99]{} Appl, S., Lery, T. & Baty, H., 2000, A&A, 355, 818 Baade, W., 1956, ApJ, 439, L43 Bardeen, J., 1970, Nature, 226, 64 Begelman, M.C., Blandford, R.D. & Rees, M.J., 1984, Revs Mod Phys, 56, 255 Blandford, R.D. & Payne, D.G., 1982, MNRAS, 1999, 883 Bogovalov & Tsinganos, K., 2002, MNRAS, 325, 249 Bogovalov & Tsinganos, K., 2002, MNRAS, 337, 553 Cohen, M., Van Winckel, H., Bond, H.E. & Gull, T.R., 2004, A.J., 127, 2362 Curtis H.D., 1918, Pub Lick Observatory, 13, 31 Genzel, R., Schödel, R., Olt, T., Eckart, A., Alexander, T., Lacombe, F., Rouan, D. & Aschenback, B., 2003, Nature, 425, 934 Ghez, A.M. et al., 2004, ApJ, 601, L159 Gizani, N.A.B. & Leahy, J.P., 2003, MNRAS, 342, 399 Gull, S.R. & Northover, F.J.E., 1976, Nature, 263, 572 Hazard, C., Mackey, M.B. & Shimmins, T.A.R. , 1963, Nature, 197, 1037 Heyvaerts, J. & Norman, C.A., 2003, ApJ, 596, 1240, 1256, 1270 Kormendy, J. & Richstone, D., 1995, Astron Rev of A & A, 33, 581 Kronberg, P.P., Colgate, S.A., Li. H & Dufton, Q.W., 2004, ApJ, 604, L77 Lebedev, S.V., Ciardi, A., Ampleford, D.J., Bland, S.N., Bott, S.C., Chittenden, J.P., Hall, G.N., Rapley, J., Jennings, C.A., Frank, A., Blackman, E.G. & Lery, T., 2005, MNRAS, 361, 97L Leblanc, J.M. & Wilson, J.R., 1970, ApJ, 161, 541 Lery, T., Baty, H. & Appl, S., 2000, A&A, 355, 1202 Li, H., Lapenta, G., Finn, J.M., Li, S. & Colgate, S.A., 2006, preprint Lovelace, R.V.E, 1976, Nature, 262, 649 Lovelace, R.F.E., Li, H., Koldoba, A.V., Ustyugova, G.V. & Romanova, M.M., 2002, ApJ, 572, 445 Lovelace, R.V.E. & Romanova, M.M., 2003, ApJ, 596, L159 Lynden-Bell, D., 1969, Nature, 223, 690 Lynden-Bell, D. & Boily, C., 1994, MNRAS, 267, 146 (Paper I) Lynden-Bell, D., 1996, MNRAS, 279, 389 (Paper II) Lynden-Bell, D. & Rees, M.J., 1971, MNRAS, 152, 461 Lynden-Bell, D., 2001, in Knapen J.H. [*et al.*]{} eds.ASP Conf.Vol.249, 212 Lynden-Bell, D., 2003, MNRAS, 341, 1360 (Paper III) Mestel, L., 1999, Cosmical Magnetism, Oxford Michalitsanos, A.G., Oliverson, R.J., Kafatos, M., Gull, H.E. & Miller, R.J., 1988, A.J., 95, 1478 Mirabel, T.F. & Rodriguez, L.F., 1999, ARA&A, 37, 409 Miyoshi, M., Moran, J., Hernstein, J., Greenhill, L., Nakai, N., Diamond, P. & Inoue, M., 1995, Nature, 373, 127 Ouyed, R., Clarke, D.A. & Pudritz, R.E., 2003, ApJ, 582, 292 Prendergast, K., 2005, MNRAS, 359, 725 Pudritz, R.E., Rogers, C.S. & Ouyed, R., 2006, MNRAS, 365, 1131 Rees, M.J., 1971, Nature, 229, 312 Ryle, M, 1968, IAU Highlights of Astronomy, Ed. L.Perek, D.Reidel Sakarai, R., 1987, PASJ, 39, 821 Salpeter, E.E., 1964, ApJ, 140, 796 Schott, G.A., 1912, Electromagnetic Radiation & Reactions arising from it, Cambridge University Press, Cambridge Schmidt, M., 1963, Nature, 197, 1040 Shakura, N.I. & Sunyaev, R.A., 1973, Astron & Astrophys, 24, 37 Shakura, N.I. & Sunyaev, R.A., 1976, MNRAS, 173, 613 Shibata, K. & Uchida, Y., 1985, Pub Astron Soc Japan, 37, 31 Shibata, K. & Uchida, Y., 1986, Pub Astron Soc Japan, 38, 637 Stalin, C.S., Gupta, A.C., Gopal Krishna, Gopal Krishna, Wiita, P.J. & Sagar, R., 2005, MNRAS, 356, 2004 Thompson, C., Lyutikov, M. & Kulkarni, S.R., 2002, ApJ, 574, 332 Uzdensky, D.A. & MacFadyen, A.I., 2006, preprint Vinković, D., Blöcker, T., Hofmann, K.H., Elitzur, M. & Weigelt, G., 2004, MNRAS, 352, 852 Vlemmings, W.H.T., Diamond, P.J. & Imai, H., 2006 Wheeler, J.A., 1971, in Nuclei of Galaxies, Ed. D.J.K. O’Connell, Pontifical Acad Sci A useful mathematical technique =============================== We wish to solve a highly non-linear differential equation such as $$\label{A1} (1-\mu^2) \frac{d^2f}{d\mu^2}=-l(l+1)f-C^2\nu f^{2\nu-1}~;~\nu=1+\frac{1}{l}$$ under the boundary conditions that $f$ is zero at given end points and that $C$ is so chosen that the maximum of $f$ is one. There should be only only maximum of $f$ between the end points. We rewrite the equation in the form $$\label{A2} (1-\mu^2)\frac{d}{d\mu}\left(\frac{df}{d\mu}\right)^2 =\frac{d}{d\mu}S^2$$ where $$\label{A3} S^2=l(l+1)(1-f^2)+C^2(1-f^{2\nu})~.$$ We shall be particularly interested in the strongly non-linear case with $\nu$ large ($l$ small) so we start with $\nu$ large but later extend the technique down to $\nu\approx 1$. As we shall come across several variants of essentially the same problem we describe the technique in terms of the more general problem in which \[A2\] is replaced by $$\label{A4} G(\mu) \frac{d}{d\mu}\left(\frac{df}{d\mu}\right)^2 = \frac{d}{d\mu} [S(f)]^2$$ with $S^2(f)$ zero when $f$ reaches its maximum of 1. We shall assume that $S(f)$ depends on parameters such at $\nu$ or $C$ in (\[A1\]) and that we are primarily interested in the solution when $\nu$ is large in which case $dS^2/df$ is strongly peaked close to $f=1$ and away from there it is small. Under these conditions (which hold for (\[A1\])) most of the deceleration $-d^2f/d\mu^2$ of $f$ occurs in the region in which $f$ is close to one. Let that maximum be at $\mu=\mu_1$. Then at $\mu_1$ $$\label{A5} (d^2f/d\mu^2)_1 =\frac{1}{2G_1}\left[\frac{dS^2}{df}\right]_1= -g~{\rm say}$$ where $G_1=G(\mu_1)$. Near $\mu=\mu_1$ $$\label{A6} f= 1-\frac{1}{2}g(\mu-\mu_1)^2$$ and $S^2=(dS^2/df)_1(f-1)$ so $$\label{A7} \mu-\mu_1 =G_1^{-1/2} g^{-1}S~$$ $$\label{A8} G(\mu)\simeq G_1+G_1^\prime(\mu-\mu_1) \simeq G_1(1+a_1 S)^3$$ where $$\label{A9} a_1=\frac{1}{3}G_1^\prime G_1^{-3/2}g^{-1}$$ and $G_1^\prime =(dG/d\mu)$ at $\mu_1$. Writing (\[A8\]) as a cubic leads to simpler mathematics later. Near $\mu=\mu_1$ where most of the ‘deceleration’ of $f$ takes place we see from (\[A4\]) that $$\label{A10} \frac{d}{d\mu}\left(\frac{df}{d\mu}\right)^2 = \frac{2S dS/d\mu}{G_1(1+a_1 S)^3}$$ We have derived this equation near $\mu=\mu_1$ but away from there $f^{2\nu}$ is small and as $dS^2/d\mu$ is small and $1+a_1S$ is not near zero, the equation can be taken to hold everywhere. We now integrate (\[A10\]) and remembering that $df/d\mu$ is zero at $\mu_1$ we find $$\label{A11} (df/d\mu)^2=\frac{S^2}{G_1(1+a_1S)^2}$$ It was to get this simple result that we chose the cubic form in (\[A8\]). Taking the square root and multiplying up by $1/S+a_1$ we find $$\label{A12} \int^{\mu_1}_\mu \frac{df}{S} +a_1(1-f) = G_1^{-1/2}(\mu-\mu_1)$$ To proceed further we must integrate $df/S(f)$. As the major item in (\[A1\]) is $f^{2\nu -1}$ we shall assume that near the maximum of $f$, $S^2$ can be approximated as $C^2_1(1-f^{2\overline{\nu}})$. We do this by taking $$\label{A13} \overline{\nu} =\frac{1}{2}[d\ln[S^2(0)-S^2]/d\ln f]_{f=1}$$ and $$\label{A14} C_1^2=gG_1/\overline{\nu}$$ We now use the method of paper I to give us $\int (1-f^{2\overline{\nu}})^{-1/2}df$ for large $\overline{\nu}$. Set $f=1-\frac{1}{\overline{\nu}}\ln q;~df=-\frac{1}{\overline{\nu}}\frac{dq}{q}$. Then $f^{2\overline{\nu}}=(1-\frac{1}{2\overline{\nu}}\ln q^2)^{2\overline{\nu}}\rightarrow \exp -(\ln q^2)=1/q^2$ so\ $\int (1-f^{2\overline{\nu}})^{-1/2}df =\frac{1}{\overline{\nu}} \int(q^2-1)^{-1/2} dq =\frac{1}{\overline{\nu}} ch^{-1}q$ so setting $q=ch \Lambda$ we find $$\label{A15} f=1-\frac{1}{\overline{\nu}}\ln(ch\Lambda )$$ and from (\[A12\]( $\Lambda$ is given by $$\label{A16} \Lambda +a_1C_1 \ln (ch\Lambda) =\overline{\nu}C_1G_1^{-1/2}(\mu-\mu_1)$$ and we remember that $a_1 =\frac{1}{3}G^\prime_1G_1^{-3/2}g^{-1}=\frac{1}{3} \frac{G^\prime_1}{G_1^{1/2}\overline{\nu}C_1^2}$. To complete the solution we must impose the boundary conditions and so find the value of $C$ needed to make $f$ one at its maximum. However the boundaries vary from one application to another so it is time to consider the differential equations and their ranges one by one. The application in this paper is to the solution of equation (\[38\]) $\lambda d^2f/d\lambda^2=-C^2\nu f^{2\nu-1}$. To use the general theory above for this equation we write $\lambda$ for $\mu$ and $G(\lambda)=\lambda$, $G_1=\lambda_1$ and $S^2=C^2(1-f^{2\nu})$ the boundaries are at $\lambda=0$ and $\lambda=1$ and $\overline{\nu}=\nu$, $C=C_1$. At those boundaries we need $ch\Lambda =e^\nu$ so that $f=0$. If $\Lambda_1$ is the positive value of $\Lambda$ satisfying this then the negative solution is $-\Lambda_1$ so at $\lambda=1$ and $\lambda=0$ respectively $\Lambda_1+a_1C\nu=\nu C\lambda_1^{-1/2}(1-\lambda_1)$ $-\Lambda_1+a_1 C\nu = -\nu C \lambda_1^{1/2}$ where $a_1=\frac{1}{3\lambda_1^{1/2}\nu C^2}$. Subtracting $2\Lambda_1=\nu C \lambda_1^{-1/2}$ adding $a_1C\nu=\nu C\lambda_1^{-1/2}(\frac{1}{2} -\lambda_1)=2\Lambda_1 (\frac{1}{2}-\lambda_1)$ so using the value of $a_1$ above and eliminating $C$ $$\lambda_1\left(\frac{1}{2}-\lambda_1\right)=\nu/(12\Lambda^2_1)$$ Now $\Lambda_1 \simeq \nu$ or more accurately $\nu\left(1+\frac{\ln 2-\frac{1}{4}e^{-2\nu}}{\nu}\right)$ so the quantity on the right is $0\left(\frac{1}{12\nu}\right)$ which is small so $\lambda_1$ is nearly a half. Accurately $$\label{A17} \lambda_1=\frac{1}{2}\left[\frac{1+\sqrt{1- \frac{4\nu}{3\Lambda_1^2}}}{2}\right]$$ with $\lambda_1$ known $$\label{A18} C=2\lambda_1^{1/2}\Lambda_1/\nu$$ This completes the solution when $\nu$ is large $$\label{A19} f\simeq1-\frac{1}{\nu}\ln\left\{ch\left\{2\Lambda_1(\lambda-\lambda_1)- \frac{1}{6\Lambda_1} \{\ln ch[2\Lambda_1(\lambda-\lambda_1)]\}\right\}\right\}$$ Solutions when $\nu$ is no longer large --------------------------------------- When $\nu$ is lowered the solution will still have a single hump and will still fall to zero at the end points but it will no longer have precisely the form given in (\[A15\]),(\[A16\]) because there will be more deceleration of $f$ away from $\mu=\mu_1$ where our approximations for $G(\mu)$ and $S(f)$ are no longer valid. To allow for this we generalise the form of (\[A16\]) and (\[A17\]) to $$\label{A20} f=1-\frac{1}{H} \ln ch\Lambda$$ $$\label{A21} \Lambda + \Delta \ln ch \Lambda = h CG_1^{-1/2}(\mu-\mu_1)$$ where $H$ and $h$ can now be different and only for $\nu$ large will they become equal to $\nu$. Furthermore $\Delta$ is no longer restricted to the form it took previously in terms of $a_1,~\nu$ and $C$. We show in this section that with the four parameters $H,~\Delta,~h~C$ and $\mu_1$ we can ensure that the solution has the right position of the maximum, the right deceleration there and that the gradients of the solution at the end points are appropriate for the differential equation. Since the desired solution has just one hump and is zero at the end points it is no surprise that we can fit it well not only for $\nu$ large but for $\nu > 1$. In practice we find errors from computed solutions are less than 1% rising above 2% for $\nu=1$. Since $C$ is itself to be determined from the fitting of boundary conditions in the original equation we have actually five constants $H,~\Delta,~h,~C$ and $\mu_1$ to be determined so we impose the following five conditions on our proposed form of solution A $$\label{A22} 1/~\&~2/~f=0~{\rm at~the~end~points~so~there}~ch\Lambda_1=e^H$$ $$\label{A23} 3/~~~-\frac{d^2f}{d\mu^2}=-\frac{1}{2G_1}\left(\frac{dS^2}{df}\right)_{f=1} ~~~{\rm this~gives}~\frac{h^2C^2}{H}=\frac{1}{2}~\left.\frac{dS^2}{df}\right|_1$$ As we wish our proposed solution to solve (\[A4\]) as nearly as possible we now integrate (\[A4\]) by parts first from $\mu_1$ to 1 and then from the lower boundary $\mu_m$ to $\mu_1$. These give us $$\label{A24} G(1)\left(\frac{df}{d\mu}\right)^2_{\mu=1} - \int^1_{\mu_1}\left(\frac{df}{d\mu}\right)^2 G^\prime(\mu)d\mu=[S(0)]^2$$ and $$\label{A25} -G(\mu_m)\left(\frac{df}{d\mu}\right)^2_{\mu_m} + \int^{\mu_1}_{\mu_m} \left(\frac{df}{d\mu}\right)^2 G^\prime(\mu)d\mu=[S(0)]^2$$ We now use the forms (\[A20\]) and (\[A21\]) and substitute $$\label{A26} \left(\frac{df}{d\mu}\right)^2 =\frac{1}{H} th^2\Lambda \frac{d\Lambda}{d\mu} \frac{hC}{G_1^{1/2}[1+\Delta th\Lambda]}$$ into the integrals in (\[A24\]),(\[A25\]) but use for the end value in (\[A25\]) $$\label{A27} \left(\frac{df}{d\mu}\right)^2_{\mu_m} =\frac{1}{H^2} (1-e^{-2H}) \frac{h^2C^2}{G_1[1-\Delta\sqrt{1-e^{-2H}}]}$$ For $(f^\prime)^2_{\mu=1}$ we merely reverse the sign of $\Delta$ in (\[A27\]). We then perform the integrals, so (\[A22\])–(\[A25\]) provide us with the five equations to determine our five parameters. We wrote (\[A26\]) in that form because in the application to narrow jets $G$ is linear so that the integrals are readily performed using $\Lambda$ as the variable of integration. For that case the integral involved is $$\int \frac{th^2\Lambda}{1+\Delta th\Lambda}d\Lambda =\int \frac{1-sech^2\Lambda}{1+\Delta th\Lambda}d\Lambda =-\frac{1}{\Delta} (1+\Delta th\Lambda) +\int \frac{(e^{2\Lambda}+1)d\Lambda}{(1+\Delta)e^{2\Lambda}+1-\Delta}$$ $$=\frac{-1}{\Delta} \ln (1+\Delta th \Lambda) +\frac{\Lambda}{1-\Delta^2} - \frac{\Delta}{1-\Delta^2}\ln \left(\frac{1+\Delta}{1-\Delta} +e^{-2\Lambda}\right)$$ Applying these methods to the solution of equation (\[38\]), with $\lambda$ written for $\mu$ and $G(\lambda)=\lambda,~S^2=C^2(1-f^{2\nu})$ and boundaries at $\lambda=0$ and $\lambda=1$ the five equations are with $\Lambda_1=ch^{-1}e^H$ $$\label{A29} \Lambda_1 +\Delta \ln ch \Lambda_1 =hC\lambda_1^{-1/2}(1-\lambda_1)$$ $$\label{A30} -\Lambda_1 +\Delta \ln ch\Lambda_1 =-hC\lambda_1^{+1/2}$$ $$\label{A31} h^2/H=\nu$$ and from (\[A24\]) and (\[A25\]) $$\label{A32} C^2= \frac{1}{H^2} (1-e^{-2H}) \frac{h^2C^2}{\Lambda_1(1+\Delta\sqrt{1-e^{-2H}})} - \frac{hC\lambda_1^{-1/2}}{H^2} \left\{-\frac{1}{\Delta} \ln \left[1+\Delta\sqrt{1-e^{-2H}}\right]+\frac{\Lambda_1}{1+\Delta} -\frac{\Delta}{1-\Delta^2} \ln \left(\frac{1+\sqrt{1-e^{-2H}}}{1+\Delta\sqrt{1-e^{-2H}}}\right)\right\}$$ $$\label{A33} C^2= \frac{hC\lambda_1^{-1/2}}{H^2} \left\{+ \frac{1}{\Delta}\ln (1-\Delta\sqrt{1-e^{-2H}}) + \frac{\Lambda_1}{1\Delta} - \frac{\Delta}{1-\Delta^2} \ln \left( \frac{1-\sqrt{1-e^{-2H}}}{1-\Delta\sqrt{1-e^{-2H}}} \right) \right\}$$ In solving these one uses the sum and differences of (\[29\]) and (\[30\]) and thinks of $H$ rather than $\nu$ as the independent variable. Figure 5 demonstrates the accuracy of the results compared with computed solutions. Finally we rewrite the approximate results as power series in $\nu^{-1}$ $H=\nu-1/42-5/26.\nu^{-1}$\ $h=\sqrt{H\nu}$\ $\Delta=1/2.\nu^{-1}-1/9.\nu^{-2}+1/12. \nu^{-3}$\ $\lambda_1=1/2-7/31.\nu^{-1}+4/41.\nu^{-2}$\ $C_*=hC\lambda_1^{-1/2}=2\nu+\ln4-4/15.\nu^{-1}$ \[lastpage\]

--- abstract: 'We address the issue of adapting optical images-based edge detection techniques for use in Polarimetric Synthetic Aperture Radar (PolSAR) imagery. We modify the gravitational edge detection technique (inspired by the Law of Universal Gravity) proposed by Lopez-Molina et al, using the non-standard neighbourhood configuration proposed by Fu et al, to reduce the speckle noise in polarimetric SAR imagery. We compare the modified and unmodified versions of the gravitational edge detection technique with the well-established one proposed by Canny, as well as with a recent multiscale fuzzy-based technique proposed by Lopez-Molina et al. We also address the issues of aggregation of gray level images before and after edge detection and of filtering. All techniques addressed here are applied to a mosaic built using class distributions obtained from a real scene, as well as to the true PolSAR image; the mosaic results are assessed using Baddeley’s Delta Metric. Our experiments show that modifying the gravitational edge detection technique with a non-standard neighbourhood configuration produces better results than the original technique, as well as the other techniques used for comparison. The experiments show that adapting edge detection methods from Computational Intelligence for use in PolSAR imagery is a new field worthy of exploration.' address: - | Laboratório Associado de Computação e Matemática Aplicada, Instituto Nacional de Pesquisas Espaciais (LAC/INPE), Av. dos Astronautas, 1758, 12227–010,\ São José dos Campos, SP – Brazil - | Laboratório de Computação Científica e Análise Numérica (LaCCAN/UFAL), Universidade Federal de Alagoas, Av. Lourival Melo Mota, s/n, 57072-970,\ Maceió, AL – Brazil - 'Departamento de Automática y Computación, Universidad Pública de Navarra, Campus Arrosadía, 31006, Pamplona, Spain' author: - 'Gilberto P. Silva Junior' - 'Alejandro C. Frery' - Sandra Sandri - Humberto Bustince - Edurne Barrenechea - 'Cédric Marco-Detchart' bibliography: - 'posFlins14Pedro.bib' title: | Optical images-based edge detection in\ Synthetic Aperture Radar images --- Edge detection ,SAR images,Computational Intelligence,Gravitational method Introduction {#sec:intro} ============ Edge detection seeks to identify sharp differences automatically in the information associated with adjacent pixels in an image [@Gonzalez2006]. Edge detection for optical images is nowadays quite an established field. It is traditionally carried out using gradient-based techniques, such as the well-known Canny algorithm [@CANNY1986]. Techniques based on Computational Intelligence have also been proposed in the recent literature. Sun et al [@Sun2007] proposed the gravitational edge detection method, inspired by Newton’s Universal Law of Gravity. Lopez-Molina et al [@LopezMolina2010] proposed a fuzzy extension for this technique, allowing the use of T-norms, a large class of fuzzy operators; they also proposed small modifications in the basic formalism (see Section \[sec:relwork\]). Dănková et al [@Dankova2011] proposed the use of a fuzzy-based function, the F-transform; the original universe of functions is transformed into a universe of their “skeleton models” (vectors of F-transform components), making further computations easier to perform. Barrenechea et al [@barrechea11] proposed the use of interval-valued fuzzy relations for edge detection, using a T-norm and a T-conorm to produce a fuzzy edge image, that is then binarized. This approach was extended by Chang and Chang [@chang-chang13]. First of all, two new images are created—one rather dark and the other rather bright—by applying two different parameters on the linear combinations of the images obtained using $\min$ and $\max$ operators, respectively. Then, the fuzzy edge image is created by the difference between these two new images. Another recent approach from Computational Intelligence is the multiscale edge detection method proposed by Lopez-Molina et al [@lopez-molina-13-multiscale], using Sobel operators for edge extraction and the concept of Gaussian scale-space. SAR sensors are not as adversely affected by atmospheric conditions and the presence of clouds as optical sensors. Moreover, unlike the optical counterparts, SAR sensors can be used at any time of day or night. For these reasons, remote sensing applications using SAR imagery have been growing over the years [@Mott2006]. SAR images, however, contain a great amount of noise, known as *speckle*, that degrades the visual quality of the images. Caused by inherent characteristics of radar technology, this multiplicative non-Gaussian noise is proportional to the intensity of the received signal. Contrary to what happens with optical images, there are still few algorithms specifically dedicated to SAR images [@Fu2012]. One interesting means to create edge detection algorithms for SAR images is to modify those created for optical images. However, the use of these methods on SAR images is not straightforward, due to speckle. One can either adapt optical image techniques to meet SAR data properties, or first preprocess the images using filters and then apply the original optical techniques. The main purpose of our study is to investigate the application of the gravitational edge detection, Here we modify the original 3$\times$3 window: the value in each cell in the window is no longer the original one, but the aggregation of a set of neighbouring pixels, according to the larger $9\times9$ neighbourhood configuration proposed by Fu et al [@Fu2012]. We propose a typology of experiments to study the behaviour of the modified edge detection method, considering polarization, image aggregation, and image binarization. We focus on the use of the following processes: DAB (edge Detection on non-binary images, Aggregation of the resulting non-binary images, Binarization) and ADB (Aggregation of non-binary images, edge Detection on the resulting non-binary image, Binarization). We also investigate the use of noise-reduction filters in preprocessing the images, by making use of the the well-known Enhanced Lee filter [@Lopes90] and a filter recently proposed by Torres et al [@TorresPolarimetricFilterPatternRecognition]. Barreto et al [@Barrreto2013] describe a classification experiment, based on a full polarimetric image from an agricultural area in the Amazon region in Brazil. In that study, the authors estimated the parameters for probability distributions associated to each of the classes of interest, such as water and different types of vegetation and their phenology. They assessed their results in an image formed by a mosaic of the classes, with pixel values generated using the parameters found for each class. We apply all techniques addressed in this study on twenty simulated mosaics, using the parameters estimated in [@Barrreto2013], considering amplitude images derived from different polarizations. We assess the quality of the results, according to Baddeley’s Delta Metric (BDM) [@Baddeley1992]. We also apply the methods on the real images, but assessment is only visual. We compare our results with those produced by the use of Canny’s algorithm [@CANNY1986] and the recently proposed multiscale method by Lopez-Molina et al [@lopez-molina-13-multiscale]. The present study is an extended version of [@SilvaFLINS14], in which some of the main ideas of this paper were first delineated. However, the present study and [@SilvaFLINS14] differ in the scope of the proposed approach as well as in the reliability of the results. Indeed, in [@SilvaFLINS14], only one simulated image was used in the experiments and only Canny’s technique was compared to its results. Moreover, in the previous paper we only addressed the edge detection of the image resulting from the aggregation of the three simulated polarization images. In our first paper only ADB was addressed; edge detection on the individual polarization images as well as DAB strategy were not considered. The results from our current study show that adapting edge detection methods from Computational Intelligence to use in radar imagery is a new field worthy of exploration. In particular, our experiments show that modifying the gravitational method with Fu’s $9 \times 9$ neighbourhood produces better results than the unmodified method. They also show the importance of filtering when adapting edge detection techniques from optical to radar images. Basic concepts on SAR images ============================ Optical and SAR sensors measure the amount of energy reflected by a target in various bands of the electromagnetic spectrum. The bands employed in most imaging radars use frequencies in the to range, with wavelengths ranging from to . In this study, we used only the L-band with wavelengths of \[, \] and frequencies of \[, \]. SAR systems generate the image of a target area by moving along a usually linear trajectory, and transmitting pulses in lateral looks towards the ground, in either horizontal (H) or vertical (V) polarizations [@Richards2009], respectively denoted as H and V (see Figure \[fig:pol\]). In the past, the reception of the transmitted energy was made ​​solely on the same polarization of the transmission, generating images in the HH and VV polarizations. Currently, with the advent of polarized and fully polarimetric radars (PolSAR - *Polarimetric Synthetic Aperture Radar*), information about intensity and phase of the cross signals are also obtained, generating images relating to HV and VH polarizations. Usually, applications only consider the HH, VV, and HV polarizations. ![Horizontal and vertical signal polarizations transmitted by an antenna. Source: [@Meneses2012][]{data-label="fig:pol"}](./fig/experiments_polH "fig:") $\;\;$ ![Horizontal and vertical signal polarizations transmitted by an antenna. Source: [@Meneses2012][]{data-label="fig:pol"}](./fig/experiments_polV "fig:") The imaging can be obtained by gathering all the intensity and phase information data from the electromagnetic signal after it has been backscattered by the target in a given polarization [@LeePottier2009]. Each polarization in a given a scene generates a complex image, which can be thought of as two images, containing the real and imaginary values for the pixels, respectively. We denote the complex images from HH, VV, and HV polarizations as $S_{HH}$, $S_ {HV}$, and $S_ {VV}$. Multiplying the vector $[S_ {HH}\; S_ {HV}\; S_ {VV}]$ by its transposed conjugated vector $[S_ {HH}^*\; S_ {HV}^*\; S_ {VV}^*]^t$, we obtain a $3 \times 3$ covariance matrix. The main diagonal contains intensity values; taking their square root, we obtain amplitude values. We denote the intensity images by $I_{HH}$, $I_{HV}$, and $I_{VV}$ and their corresponding amplitude counterparts by $A_{HH}$, $A_{HV}$, and $A_{VV}$. In this paper, we only considered the amplitude images, such as those depicted in Figure \[fig:int-amp\]. ![Amplitude images for polarizations HH, VV, and HV from the same scene[]{data-label="fig:int-amp"}](./fig/experiments_amps_intens_amplitude_HH.png "fig:"){width=".3\linewidth"} ![Amplitude images for polarizations HH, VV, and HV from the same scene[]{data-label="fig:int-amp"}](./fig/experiments_amps_intens_amplitude_HV.png "fig:"){width=".3\linewidth"} ![Amplitude images for polarizations HH, VV, and HV from the same scene[]{data-label="fig:int-amp"}](./fig/experiments_amps_intens_amplitude_VV.png "fig:"){width=".3\linewidth"} $A_{HH} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; A_{HV} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; A_{VV}$ Speckle noise is multiplicative, non-Gaussian, and is proportional to the intensity of the received signal. Speckle degrades the visual quality of the displayed image by sudden variations in image intensity with a salt and pepper pattern, as can be seen in Figure \[fig:int-amp\]. It can be reduced with multiple looks in the generation of the complex images, causing degradation in spatial resolution. Another way to reduce noise is to employ filters, as will be discussed in the next section. In SAR image classification, one often uses samples from the classes in order to estimate the parameters of the distribution believed to underlie each class. Synthetic images can then be created using Monte Carlo simulation by taking the realization of the random variable associated to the class of each classified pixel. This artifice is useful to choose the most apt classifier for a given application: instead of relying solely on the original image, one takes the classifier that obtains the best average accuracy on the set of synthetic images. This methodology can also be used in other tasks, such as edge detection. Related Work {#sec:relwork} ============ One of the most successful edge detection algorithms for optical images was proposed by Canny [@CANNY1986], based on the following guidelines: i) the algorithm should mark as many real edges in the image as possible; ii) the marked edges should be as close as possible to the edge in the real image; iii) a given edge in the image should only be marked once; and iv) image noise should not create false edges. It makes use of numerical optimization to derive optimal operators for ridge and roof edges. The usual implementation of this method uses a $3 \times 3$ neighbourhood. A more recent multi-scale edge detection method was proposed by Lopez-Molina et al [@lopez-molina-13-multiscale], using Sobel operators for edge extraction and the concept of Gaussian scale-space. More specifically, the Sobel edge detection method is applied on increasingly smoother versions of the image. Then, the edges which appear on different scales are combined by performing coarse-to-fine edge tracking. The gravitational edge detection approach was first proposed by Sun et al [@Sun2007] and applied to optical images. It is based on Newton’s Universal Law of Gravity, described by Equation (\[eq:newton\]): $$f_{1,2} = G \times \frac{m_1 \times m_2}{\|\vec{r}\|^2} \times \frac{\vec{r}}{\|\vec{r}\|}, \label{eq:newton}$$ where $m_1$ and $m_2$ are the masses of two bodies; $\vec{r}$ is the vector connecting them; $\vec{f}_{1,2}$ is the gravitational force between them; $\|.\|$ denotes the magnitude of a vector; and $G$ is the gravitational constant. In the analogy proposed by Sun et al [@Sun2007]; the bodies are the gray level values of pixels in a grid; $G$ is a function of the values of the pixels in a given window; the distance between any two adjacent pixels is equal to 1; and, when computing the resulting force of the pixel in the center of a window; the pixels outside that window are considered negligible. Lopez-Molina et al [@LopezMolina2010] extended this technique, proposing the use of a Triangular Norm [@DP88] in place of the product between the two masses[^1], by first normalizing the gray level values to $[0,1]$. The authors treat edges as fuzzy sets for which membership degrees are extracted from the resulting gravitational force on each pixel. They take $G$ as a normalization constant, calculated so as to guarantee that the resulting forces lie in \[0,1\]. Also, in the normalization of gray level values into \[0,1\], a small value $\delta q$ is added beforehand to both the numerator and denominator so as avoid pixels with value 0, which would have too strong an effect on neighbouring pixels. The authors used $3 \times 3$ and $5 \times 5$ windows as well as several prototypical triangular norms. The so-called Lee (or sigma) filter introduced in 1983 [@LeeFilter1983], is still in use today due to its simplicity, its effectiveness in speckle reduction, and its computational efficiency. It is based on the fact that, under Gaussian distribution, approximately $95.5\%$ of the probability is concentrated within two standard deviations from the mean. The filter estimates the mean and the standard deviation of samples around each pixel, and only those values within this interval are used to compute the local mean. Lopes et al [@Lopes90] proposed an adaptive version for this filter, here referred to as “Enhanced Lee”. Torres et al [@TorresPolarimetricFilterPatternRecognition] recently proposed a nonlocal means approach for PolSAR image speckle reduction based on stochastic distances; the method can be tailored to any distribution, both univariate of multi-variate. It consists of comparing the distributions which describe the central observation for each pixel, and each of the observations which comprise a search region. The comparison is made through a goodness-of-fit test, and the $p$-value of the test statistic is used to define the convolution matrix which will define the filter: the higher the $p$-value the larger the confidence and, thus, the importance, each observation will have in the convolution. In Torres et al’s proposal, the tests are derived from $h$-$\phi$ divergences between multi-look scaled complex Wishart distributions for fully PolSAR data [@Frery2014]. Their results are competitive with classical and advanced polarimetric filters, with respect to usual quantitative measures of quality. Fu et al [@Fu2012] proposed a statistical edge detector suitable for SAR images which uses the squared successive difference of averages to estimate the edge strength from the sliding window. An interesting feature of this paper is the proposal of a specific type of $9 \times 9$ neighbourhood, shown in Figure \[fig:vizinhaca\_Fu\]. In a previous paper [@SilvaFLINS14], we proposed a modification of the gravitational approach using Fu et al’s neighbourhood: given a central pixel in a $3 \times 3$ window in an image, the values considered for the surrounding pixels in the window are no longer the ones in the original image, but the mean values in this new configuration. ![Standard $3 \times 3$ and Fu’s neighbourhood [@Fu2012][]{data-label="fig:vizinhaca_Fu"}](./fig/experiments_janelas_3x3_e_Fu.png) Materials and Methods {#sec:metod} ===================== We compare the edge detection methods proposed by Canny [@CANNY1986] and by Lopez-Molina et al [@lopez-molina-13-multiscale] to the modified gravitational approach using the product T-norm, followed by thresholding. The effect of preprocessing the images through filtering is also studied, using the filter described by Torres et al [@TorresPolarimetricFilterPatternRecognition] and the Enhanced Lee filter [@Lopes90]. We study the behaviour of Lopez-Molina’s method with the usual $3 \times 3$ window as well as a modified version of this approach, proposed in a previous paper [@SilvaFLINS14], involving the neighbourhood proposed by Fu et al [@Fu2012]. The input for Canny’s and Lopez-Molina’s edge detector are images in, respectively, $\{0, \dots, 255\}$ and $[0,1]$. Image values are, thus, mapped into these sets prior to edge detection. For the Lopez-Molina methods (the original and modified versions), we normalize further to $[0,1]$, using $\delta q=1$ and making $q'= ({q + 1})({255+1})^{-1}$, where $q$ and $q'$ are the old and new value of a given pixel, respectively. Working image ------------- We apply the methods on data derived from a fully polarimetric image, presented by Barreto et al [@Barrreto2013], from an agricultural area in the Amazon region in Brazil (see Figure \[aba:fig\_image\]). The authors describe a classification experiment using classes of interest from that area, such as water and different types of crops and natural vegetation, at different stages of growth. Samples from the classes from band L are used to estimate the parameters of the complex Wishart distribution associated to each class. The results are assessed using a mosaic with the classes that was created using the derived Wishart distributions. Figure \[aba:fig\_samples\] illustrates the approach. For our study we apply the edge detection methods on twenty independently simulated mosaics amplitude images, using the parameters estimated in [@Barrreto2013] to assess the quality of the methods. a\) ![Images derived from a scene in Bebedouro in Brazil (not registered): a) Landsat RGB composition and b) SAR L-band RGB composition (source: [@Barrreto2013])[]{data-label="aba:fig_image"}](fig/landsat.png "fig:") b) ![Images derived from a scene in Bebedouro in Brazil (not registered): a) Landsat RGB composition and b) SAR L-band RGB composition (source: [@Barrreto2013])[]{data-label="aba:fig_image"}](fig/im_banda_L.png "fig:")\ ![Images derived from a scene in Bebedouro in Brazil: a) training samples used to generate Wishart distributions and b) synthetic mosaic images generated using the Wishart distributions estimated in [@Barrreto2013] from image samples (source: [@Barrreto2013])[]{data-label="aba:fig_samples"}](fig/treinamento-bebedouro2.png "fig:") $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$\ a)![Images derived from a scene in Bebedouro in Brazil: a) training samples used to generate Wishart distributions and b) synthetic mosaic images generated using the Wishart distributions estimated in [@Barrreto2013] from image samples (source: [@Barrreto2013])[]{data-label="aba:fig_samples"}](fig/treinamento-bebedouro1.png "fig:") b)![Images derived from a scene in Bebedouro in Brazil: a) training samples used to generate Wishart distributions and b) synthetic mosaic images generated using the Wishart distributions estimated in [@Barrreto2013] from image samples (source: [@Barrreto2013])[]{data-label="aba:fig_samples"}](fig/treinamento-bebedouro-mosaico.png "fig:") Quality assessment ------------------ The quality of the results is assessed by the Baddeley’s Delta Metric (BDM) [@Baddeley1992], by comparison with what would be the perfect result, discarding those pixels close to the outer frame. Let $\bf{x}$ and $\bf{y}$ be two binary images, seen as mappings from $\Lambda$ to $[0,1]$, where $\Lambda$ is a set of sites arranged in a grid (positions). Let $\rho$ be a metric on $\Lambda$, such as the Euclidean distance, and $d(i, A)$ be the distance between a site $i$ and a set $A \subseteq \Lambda$, defined as $$d(i, A)=\min_{j \in A} \rho(i,j).$$ Let $b(\bf{x})= \{i \in \Lambda \mid x_i=1\}$ denote the set of foreground sites in $\bf{x}$. BDM between $\bf{x}$ and $\bf{y}$, denoted as $\Delta_{p,w.}(.,.)$, is then defined as $$\Delta_{p,w}({\bf{x}}, {\bf{y}})= \left(\frac{1}{\mid \Lambda \mid} \sum_{i \in \Lambda} \mid w(d(i,b({\bf x})) - w(d(i,b({\bf y}))\mid^p\right)^{\frac{1}{p}}, 1 \leq p \leq \infty \label{eq:BDM}$$ where $w$ is a strictly increasing concave function satisfying $w(0)=0$. Here we use $w(t)=t$ and $p=2$, as in [@LopezMolina2010]. Throughout the text, we display BDM results in $[0,100]$ instead of $[0,1]$, for the sake of readability. Proposed methodologies ====================== Edge detectors use a window around a center pixel to verify whether that pixel belongs to an edge or not. When adapting optical image edge detectors to radar imagery, we have to find the means to deal with speckle. The main contribution of this study is to modify the original $3\times3$ window used by the edge detection method proposed in [@LopezMolina2010] for use in radar imagery such that the value in each cell in the window is no longer the original one but the aggregation of set of neighbouring pixels, according to a larger $9\times9$ non-standard neighbourhood proposed by Fu et al [@Fu2012]. We here investigate this particular combination of method and neighbourhood, but the same procedure can be applied using other edge detection methods and/or non-standard filters. Frequently, a single band is used in edge-detection, resulting in a gray level-image that is then binarized at some point (the usual implementation of some methods, like Canny’s, already involve a binarization step). In radar imagery, very often one deals with more than one band at the same time (e.g. intensity images coming from different polarizations, or complex images in the fully polarimetric case), aiming at using the richness of information to compensate for the speckle noise. Therefore, the question of when to aggregate results has to be addressed. One may, for instance, first aggregate the bands and then apply the edge detector on the aggregated image, or else apply the edge detector on the individual bands and then aggregate the edge images. These two methods usually yield different results. Here, we propose a typology for experiments using radar imagery, considering different orderings of three steps: edge detection on gray level images, binarization of gray level images, and aggregation of results. In the aggregation step, the input may be either gray level or binary images, depending on whether the binarization is made immediately after edge detection or not. Three strategies can then be envisaged to perform edge detection experiments with radar images: - DAB (edge Detection on non-binary images, Aggregation of the resulting non-binary images, Binarization) and - ADB (Aggregation of non-binary images, edge Detection on the resulting non-binary image, Binarization). - DBA (edge Detection on non-binary images, Binarization, Aggregation of the resulting binary images). Options ABD, BAD and BDA are not considered, since that would mean applying edge detectors on the binary images. In this work, we focus on the DAB and ADB strategies. For both of them, we use the arithmetic mean to aggregate gray level images. Strategy DBA, involving the aggregation of binary images, is left for future study. When no aggregation is considered, the strategies are reduced to only edge detection and binarization. For example, for the HH, HV and VV polarizations, we obtain strategies DB-HH, DB-HV and DB-HH. Note that some methods already incorporate the binarization step in the edge detector. That is for instance the case of all methods discussed previously. However, to be consistent with the notation, we will denote by ADB the strategy in a method that includes binarization, such as Canny’s and the multi-scale method, when it is applied to the gray level image resulting from the aggregation of the images from the HH, HV and VV polarizations. Experimental Results {#sec:expresults} ==================== The output of the Lopez-Molina gravitational method is an image with values in $[0,1]$. In order to obtain binary indicators of edges, the authors use a hysteresis transformation. Here, we use a simple threshold and search for the value in the $[0.05,0.15]$ interval which produces the best BDM. For Canny’s method, we search for the best value for the noise standard deviation parameter $\sigma$ in the interval $[0.3, 1.5]$. The intervals above for both Canny and Lopez-Molina are the ones that presented the best results by trial-and-error. The following parameters were used for the Lopez-Molina multi-scale method, as suggested in [@lopez-molina-13-multiscale]: $\delta_\sigma = 0.25$, $\sigma \in$ $\{0.50, 0.75, 1.00,$ $1.25, 1.50, 1.75, 2.00,$ $2.25, 2.50, 2.75, 3.00, $ $3.25, 3.50, 3.75, 4.00\}$. We applied two filters (Torres [@TorresPolarimetricFilterPatternRecognition] and Enhanced Lee [@Lopes90]) on intensity values, which were then transformed in amplitude before further processing. Strategy No filter Torres filter Enh. Lee filter ---------- -------------- --------------- ----------------- DB-HH 26.72 (1.22) 23.40 (1.92) 28.85 (1.86) DB-HV 30.70 (1.39) 26.74 (1.33) 66.40 (0.42) DB-VV 29.81 (2.17) 26.28 (1.09) 24.83 (1.98) ADB 27.87 (1.16) 48.16 (0.003) 30.24 (1.20) : Average BDM results for Canny’s method, with standard deviation ​​inside parentheses[]{data-label="aba:tab_BDM_Canny"} Strategy No filter Torres filter Enh. Lee filter ---------- -------------- --------------- ----------------- DB-HH 28.23 (0.89) 28.00 (0.93) 25.36 (2.11) DB-HV 28.34 (1.13) 24.55 (2.97) 19.56 (2.36) DB-VV 25.62 (1.10) 28.42 (0.56) 28.89 (2.25) ADB 25.15 (3.23) 24.37 (1.52) 20.71 (3.97) : Average BDM results for the multi-scale method, with standard deviation ​​inside parentheses[]{data-label="aba:tab_BDM_multi"} Strategy No filter Torres filter Enh. Lee filter ---------- -------------- --------------- ----------------- DB-HH 33.89 (1.98) 26.61 (2.00) 38.97 (0.79) DB-HV 31.95 (0.46) 27.14 (1.22) 32.26 (2.78) DB-VV 32.35 (1.47) 28.95 (0.95) 43.65 (1.13) DAB 29.26 (1.62) 25.91 (1.55) 27.71 (2.62) ADB 31.50 (0.82) 26.63 (1.27) 18.24 (3.41) : Average BDM results for the gravitational method; with standard deviation ​​inside parentheses[]{data-label="aba:tab_BDM_lopez_molina"} Strategy No filter Torres filter Enh. Lee filter ---------- -------------- --------------- ----------------- DB-HH 25.27 (0.76) 22.18 (0.48) 17.79 (3.05) DB-HV 26.48 (1.00) 24.21 (0.65) 18.40 (5.75) DB-VV 21.41 (1.97) 18.14 (0.77) 17.83 (2.54) DAB 22.67 (2.29) 18.97 (1.62) 5.43 (1.68) ADB 23.80 (2.23) 22.74 (0.50) 5.16 (0.36) : Average BDM results for the gravitational method modified by Fu’s neighbourhood, with standard deviation ​​inside parentheses[]{data-label="aba:tab_BDM_lopez_molina_Fu"} Tables \[aba:tab\_BDM\_Canny\], \[aba:tab\_BDM\_multi\], \[aba:tab\_BDM\_lopez\_molina\], and \[aba:tab\_BDM\_lopez\_molina\_Fu\] show the results for BDM mean and standard deviation after applying four methods to twenty simulated mosaic images: Canny’s method, Lopez-Molina et al’s multi-scale method, Lopez-Molina et al’s original gravitational method, and Lopez-Molina et al’s method modified using Fu’s $9 \times 9$ neighbourhood. We see that the best BDM average values were obtained with the use of Lopez-Molina et al’s gravitational method modified by Fu’s neighbourhood, using the ADB and DAB strategies, both preprocessed with the Enhanced Lee filter. Both are significantly higher than the other procedures. In Table \[aba:tab\_BDM\_Canny\], we see that filtering did not have a significative impact on Canny’s detector. The same is true, to a lesser degree, for most results of the multi-scale and (unmodified) gravitational methods, as can be seen in Tables \[aba:tab\_BDM\_multi\] and \[aba:tab\_BDM\_lopez\_molina\]. In these methods, there is a slight advantage in preprocessing the images using the Enhanced Lee filter. However, filtering has an impressive effect on the Lopez-Molina gravitational method modified with Fu’s neighbourhood. In particular, the best results are obtained for strategies DAB and ADB with preprocessing with the Enhanced Lee filter. Figure \[fig:best-meth\] shows the negative images corresponding to the best results, according to BDM, obtained by the edge detection methods and the filtering strategies with the best average values; note that the image boundaries are depicted only for illustrative purposes. We see that according to BDM the best binary image (depicted in Figure \[fig:best-meth\]b) presents little noise and most of the regions are separated, even though the lines are rather thick. We also see that BDM was able to distinguish the best image from the others. a\) ![Best BDM results obtained from the best methods (average): a) Canny (DB-HH, with Torres filtering, BDM = 18.51), b) Multi-scale (DB-HV, with Enh. Lee filtering, BDM = 14.94), c) Gravitational (ADB with Enh. Lee filtering, BDM = 10.96) and d) Gravitational and Fu (ADB with Enh. Lee filtering, BDM = 3.05) []{data-label="fig:best-meth"}](fig/cBest_CAN.png "fig:") b) ![Best BDM results obtained from the best methods (average): a) Canny (DB-HH, with Torres filtering, BDM = 18.51), b) Multi-scale (DB-HV, with Enh. Lee filtering, BDM = 14.94), c) Gravitational (ADB with Enh. Lee filtering, BDM = 10.96) and d) Gravitational and Fu (ADB with Enh. Lee filtering, BDM = 3.05) []{data-label="fig:best-meth"}](fig/cBest_multi_13C2.png "fig:")\ c) ![Best BDM results obtained from the best methods (average): a) Canny (DB-HH, with Torres filtering, BDM = 18.51), b) Multi-scale (DB-HV, with Enh. Lee filtering, BDM = 14.94), c) Gravitational (ADB with Enh. Lee filtering, BDM = 10.96) and d) Gravitational and Fu (ADB with Enh. Lee filtering, BDM = 3.05) []{data-label="fig:best-meth"}](fig/cBest_BUS.png "fig:") d) ![Best BDM results obtained from the best methods (average): a) Canny (DB-HH, with Torres filtering, BDM = 18.51), b) Multi-scale (DB-HV, with Enh. Lee filtering, BDM = 14.94), c) Gravitational (ADB with Enh. Lee filtering, BDM = 10.96) and d) Gravitational and Fu (ADB with Enh. Lee filtering, BDM = 3.05) []{data-label="fig:best-meth"}](fig/cBest_NBU.png "fig:") Figure \[fig:best-meth\]b shows the best results from the methods come from filtered images, which raises the question of how important preprocessing by filtering is. In what follows we discuss the details of the gravitational method using the original $3 \times 3$ and Fu’s $9 \times 9$ neighbourhood in relation to filtering. We take the simulation that obtained the best BDM results for each type of neighbourhood. We see in these examples, that filtering does, indeed, ameliorate the results for all methods. a\) ![Results for the gravitational method with original 3$\times$3 window, on a single simulated image from ADB: a) no filtering (BDM = 31.90), b) Torres (BDM = 27.73) and c) Enh. Lee (BDM = 10.96) []{data-label="fig:test-bustince"}](fig/bus_nof.png "fig:") b) ![Results for the gravitational method with original 3$\times$3 window, on a single simulated image from ADB: a) no filtering (BDM = 31.90), b) Torres (BDM = 27.73) and c) Enh. Lee (BDM = 10.96) []{data-label="fig:test-bustince"}](fig/bus_snl.png "fig:") c) ![Results for the gravitational method with original 3$\times$3 window, on a single simulated image from ADB: a) no filtering (BDM = 31.90), b) Torres (BDM = 27.73) and c) Enh. Lee (BDM = 10.96) []{data-label="fig:test-bustince"}](fig/bus_lee.png "fig:") a\) ![Results for the gravitational method modified with Fu’s neighbourhood, on a single simulated image from ADB: a) no filtering (BDM = 23.80), b) Torres (BDM = 23.76) and c) Enh. Lee (BDM = 3.05) []{data-label="fig:test-bustincefu"}](fig/nbu_nof.png "fig:") b) ![Results for the gravitational method modified with Fu’s neighbourhood, on a single simulated image from ADB: a) no filtering (BDM = 23.80), b) Torres (BDM = 23.76) and c) Enh. Lee (BDM = 3.05) []{data-label="fig:test-bustincefu"}](fig/nbu_snl.png "fig:") c) ![Results for the gravitational method modified with Fu’s neighbourhood, on a single simulated image from ADB: a) no filtering (BDM = 23.80), b) Torres (BDM = 23.76) and c) Enh. Lee (BDM = 3.05) []{data-label="fig:test-bustincefu"}](fig/nbu_lee.png "fig:") Figures \[fig:test-bustince\].a), \[fig:test-bustince\].b) and \[fig:test-bustince\].c) respectively show that: using the $3 \times 3$ neighbourhood for the original gravitational method, the unfiltered image is very noisy; the Torres filter reduced the noise and separated the regions; and the Lee filter detected false edges. In Figure \[fig:test-bustincefu\], we see that Fu’s $9 \times 9$ neighbourhood detected almost all the edges, especially when using Lee’s filter. Filtering for the modified method presented a larger trade-off between detection of edges and reduction of noise (some edges were detected using Torres filter with an increase of noise when compared to the unfiltered image). When we compare the results in Figures \[fig:test-bustince\] and \[fig:test-bustincefu\] we see that the gravitational method modified with Fu’s $3\times3$ neighbourhood clearly produced better results than the method with the original $3\times3$ window, which agrees with the BDM evaluation. ![ HV Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-hv"}](fig/selecao_canny_bebedouro_L_HV.png "fig:") ![ HV Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-hv"}](fig/selecao_mult_det_bebedouro_L_HV.png "fig:") ![ HV Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-hv"}](fig/selecao_gedto_bebedouro_L_HV2.png "fig:") ![ HV Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-hv"}](fig/selecao_gedt_bebedouro_L_HV2.png "fig:") ![ADB Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-adb"}](fig/selecao_canny_bebedouro_L_ADB.png "fig:") ![ADB Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-adb"}](fig/selecao_mult_det_bebedouro_L_ADB.png "fig:") ![ADB Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-adb"}](fig/selecao_gedto_bebedouro_L_ADB2.png "fig:") ![ADB Bebedouro binary images, with Enh. Lee filter: the first row depicts results obtained using the Canny and the Multi-scale methods; and the second row depicts results obtained using the original Gravitational method and its modification with Fu’s neighbourhood, (the latter two methods use binarization threshold=.2) []{data-label="fig:bebedouro-adb"}](fig/selecao_gedt_bebedouro_L_ADB2.png "fig:") ![ DAB Bebedouro binary images, with Enh. Lee filter: the first and second rows respectively depict results using the original Gravitational and Gravitational modified with Fu’s neighbourhood, the first and second columns respectively depict results obtained with binarization thresholds .1 and .2 []{data-label="fig:bebedouro-dab"}](fig/selecao_gedto_bebedouro_L_DAB1.png "fig:") ![ DAB Bebedouro binary images, with Enh. Lee filter: the first and second rows respectively depict results using the original Gravitational and Gravitational modified with Fu’s neighbourhood, the first and second columns respectively depict results obtained with binarization thresholds .1 and .2 []{data-label="fig:bebedouro-dab"}](fig/selecao_gedto_bebedouro_L_DAB2.png "fig:") ![ DAB Bebedouro binary images, with Enh. Lee filter: the first and second rows respectively depict results using the original Gravitational and Gravitational modified with Fu’s neighbourhood, the first and second columns respectively depict results obtained with binarization thresholds .1 and .2 []{data-label="fig:bebedouro-dab"}](fig/selecao_gedt_bebedouro_L_DAB1.png "fig:") ![ DAB Bebedouro binary images, with Enh. Lee filter: the first and second rows respectively depict results using the original Gravitational and Gravitational modified with Fu’s neighbourhood, the first and second columns respectively depict results obtained with binarization thresholds .1 and .2 []{data-label="fig:bebedouro-dab"}](fig/selecao_gedt_bebedouro_L_DAB2.png "fig:") Figures \[fig:bebedouro-hv\], \[fig:bebedouro-adb\], and \[fig:bebedouro-dab\] show the best edge detection results obtained for the Bebedouro SAR image in terms of visual analysis, with the application of the Enhanced Lee filter, for the parameterizations used here. The figures present the results for HV polarization; in general, HH (respec. VV) polarization produced images with more noise (respec. less information) than HV. Figures \[fig:bebedouro-adb\] and \[fig:bebedouro-dab\], respectively, show the results of the application of ADB and DAB aggregation strategies. ADB in general produced binary images with very little information for all methods; the decrease of noise in relation to the individual polarizations does not compensate the lack of information. In general, the DBA aggregation method produced results with less noise for the gravitational method, with and without modification, than the results obtained with the individual polarizations. The best results for the gravitational method, both with and without modification with Fu’s neighbourhood, were obtained with thresholds around the same interval that produced the best results using the mosaics. Conclusions {#sec:conclusions} =========== Contrary to what happens with optical imagery, few algorithms are specifically dedicated to PolSAR image edge detection [@Fu2012]. One interesting means to create edge detection algorithms for SAR images is to modify those created for optical images, in such a way as to reduce the non-Gaussian noise. Here we have investigated the modification of a method issued from Computational Intelligence for optical imagery, the gravitational edge detection method extension proposed in [@LopezMolina2010] (see also [@Sun2007]), to synthetic aperture radar imagery. In order to deal with speckle, we modified the gravitational method with a non-standard $9 \times 9$ neighbourhood configuration proposed by Fu et al [@Fu2012]: considering a $3 \times 3$ window centered around a given pixel, the value of any pixel in the window becomes the average value of the region associated to that pixel in the non-standard neighbourhood configuration. Considering that SAR imagery has different polarizations, and that their joint use may compensate for the presence of speckle, we also proposed a typology of experiments regarding aggregation of these images. In particular, we addressed two procedures: DAB (edge Detection on non-binary images, Aggregation of the resulting non-binary images, Binarization) and ADB (Aggregation of non-binary images, edge Detection on the resulting non-binary image, Binarization). For means of comparison, we also addressed the use of two other edge detector methods stemming from the realm of optical images: the traditional method proposed by Canny [@CANNY1986] and a recent multi-scale one coming from Computational Intelligence, based on Sobel operators for edge extraction and the concept of Gaussian scale-space [@lopez-molina-13-multiscale]. We studied the effect of filtering the images prior to edge detection by two procedures: Enhanced Lee [@Lopes90] and Torres [@TorresPolarimetricFilterPatternRecognition] filters. The methods were applied on twenty samples of a scene, which were simulated using Wishart distributions derived from a fully polarimetric image [@Barrreto2013]. Using both visual inspection and the Baddeley Delta metric [@Baddeley1992] we verified that the combination with the Lopez-Molina technique with the $9 \times 9$ neighbourhood proposed by Fu et al [@Fu2012] and preprocessing with the Enhanced Lee filter produced the best results. This paper is an extended version of [@SilvaFLINS14]; together, these studies represent a first step towards investigating the use of edge detection methods derived from Computational Intelligence techniques for use in SAR images. The main implication of our results is that the joint use filtering and neighbourhood modification on those methods, as well as the use of aggregation of the individual polarization images, are able to deal with speckle, which is crucial when detecting edges in radar imagery. Future work includes modifying the Lopez-Molina method with other types of neighbourhoods, such as Nagao-Matsuyama [@Nagao1979]; to verify the performance of other T-norms than the product to calculate the gravitational forces; and to perform preprocessing with other filters. We also intend to investigate the use of the proposed procedure with other edge detection methods, such as the one described in [@LopezMolina2011], involving fuzzy sets. We would like to better address the issue of aggregation. Here we have dealt exclusively with the aggregation of non-binary images, using the arithmetic means in strategies DAB and ADB. In the future, we intend to explore aggregation of the images considering families of operators in general, such as weighted means, ordered weighted means (OWA), T-norms, and T-conorms [@DP88]. Also, we intend to study other operators than the average to perform aggregation of pixel values in regions of a non-standard neighbourhood. Moreover, we intend to assess the results using other methods than BDM, such as the one proposed recently by Frery et al [@Buemi2014]. We would also like to draw comparisons with other edge detection algorithms, such as the one proposed by Fu in 2012 [@Fu2012]. Last but not least, we intend to verify the use of the approach considering fully polarimetric images (PolSAR), instead of just intensity images. In this case, Torres filter, designed specifically for PolSAR images, can be more adequately employed. Acknowledgements {#acknowledgements .unnumbered} ================ The authors are grateful for Wagner Barreto Silva, Leonardo Torres and Corina Freitas for help in the preparation of this manuscript. They are also thankful for the editor and reviewers for comments and suggestions. The Brazilian authors acknowledge support from CNPq (Projeto Universal 487032/2012-8). The Spanish authors have been supported by project TIN2013-40765-P of the Spanish Government. References {#references .unnumbered} ========== [^1]: Triangular norm operators are mappings from $[0, 1]^2$ to $[0, 1]$, that are commutative, associative, monotonic, and have $1$ as neutral element.

--- abstract: 'We introduce a class of conformal versions of the previously introduced quasi-conformal carpet cloak, and show how to construct such conformal cloaks for different cloak shapes. Our method provides exact refractive-index profiles in closed mathematical form for the usual carpet cloak as well as for other shapes. By analyzing their asymptotic behavior, we find that the performance of finite-size cloaks becomes much better for metal shapes with zero average value, [*e.g.*]{}, for gratings.' address: | $^1$Departement Physik, Universität Basel, Switzerland\ $^2$Max-Planck-Institut für Quantenoptik, Garching, Germany\ $^3$Fakultät für Physik, Ludwig-Maximilians-Universität, München, Germany\ $^4$Institut für Angewandte Physik, DFG Center for Functional Nanostructures (CFN), and Institut für Nanotechnologie, Karlsruhe Institute of Technology, Germany author: - 'Roman Schmied,$^{1,*}$ Jad C. Halimeh,$^{2,3}$ and Martin Wegener$^4$' bibliography: - 'MPQ.bib' title: Conformal carpet and grating cloaks --- Introduction ============ Soon after the introduction of the so-called carpet cloak by Li and Pendry in 2008 [@Li2008], broadband invisibility cloaking has become experimental reality in two [@Liu2009; @Valentine2009; @Gabrielli2009; @Lee2009] and three [@Ergin2010; @Ma2010] dimensions from microwaves [@Liu2009; @Ma2010] to the optical regime [@Valentine2009; @Gabrielli2009; @Lee2009; @Ergin2010]. In essence, the carpet or ground-plane cloak makes a bump (more generally a corrugation) in a metallic carpet appear flat and hence undetectable. Objects may be hidden in the space underneath the bump. However, Ref. [@Zhang2010] has highlighted lateral beam displacements as an inherent limitation originating from approximating the locally anisotropic optical properties, which arise from the quasi-conformal mappings employed in the construction of these finite-size cloaks, by locally *isotropic* ones. This artifact contributes to the Ostrich effect [@Nicorovici2008]: one cannot see the cloaked bump, but one can see that *something* is there. These limitations can be traced back to the finite size of the cloak, since for an infinite carpet cloak the quasi-conformal map becomes strictly conformal, in which case cloaking becomes perfect in both wave and ray optics (for reviews on transformation optics see Refs. [@Shalaev2008; @Chen2010]). Recently, a specific example for a strictly conformal map has been given and discussed [@Zhang2010b]. Here we introduce an entire class of strictly conformal maps and discuss how the performance of *finite-size* carpet cloaks can be systematically improved with respect to Refs. [@Zhang2010] and [@Zhang2010b]. Conformal mapping ================= Let us start by emphasizing that our approach follows a somewhat different spirit than the one by Li and Pendry [@Li2008]. They start with a predefined shape of the bump. In principle, its shape as well as its aspect ratio can be chosen arbitrarily. Furthermore, they fix the boundaries of the finite-size cloaking structure (with slipping boundary conditions). Numerically minimizing the modified-Liao functional [@Li2008], they arrive at a quasi-conformal map – the closest one can get to a conformal map under the given constraints. We rather introduce a class of *strictly conformal* transformations. For each transformation of this class, the shape of the bump results automatically, [*i.e.*]{}, it can generally *not* be chosen arbitrarily. However, in the special and rather important limit of shallow bumps (which *all* of the aforementioned experiments have used [@Liu2009; @Valentine2009; @Gabrielli2009; @Lee2009; @Ergin2010; @Ma2010]), the connection between bump shape and conformal transformation is mathematically simple and intuitive. In this case, the bump shape can again be chosen arbitrarily. Furthermore, in our approach the ideal cloaking structure is infinitely extended; it has no intrinsic boundaries. One can, however, just truncate the refractive-index profile to obtain a finite-size cloak. In this paper we show that under certain constraints the effect of this truncation can be minimized. Mathematically, we start from the conformal map $z \mapsto f(z)$ given by $$\label{eq:conformalmap} f(z) = z + \int_0^{\infty} c_k e^{{\ensuremath{\text{i}}}k z}\text{d}k,$$ where $z=x+{\ensuremath{\text{i}}}y$; $x\in\mathbbm{R}$ and $y\in\mathbbm{R}^+$ are the coordinates of points in the Cartesian two-dimensional half-space above the horizontal axis, assuming translational invariance in the third dimension. They are mapped onto the transformed coordinates $(u(x,y),v(x,y))\in\mathbbm{R}^2$ given by $f(x+{\ensuremath{\text{i}}}y)=u+{\ensuremath{\text{i}}}v$. The right-hand side of Eq.  contains a truncated (lower integral bound is zero) Fourier transform of the coefficients $c_k$. This truncation reflects the fact that we consider only light propagating in the half-space above the metallic bump. The refractive indices of the *virtual* space $n_0(x,y)$ ($=1$ in our work) and of the *physical* space $n(u,v)$ are related by [@Leonhardt2006] $$\label{eq:refindex} n(f(z))= \frac{n_0(z)}{| \text{d}f/\text{d}z |}.$$ This form has a known [@Leonhardt2006] and intuitive physical interpretation: The conformal map $z\mapsto f(z)$ locally stretches (or compresses) space while preserving angles and the shapes of infinitesimally small figures. The factor by which space is stretched by the map is given by the modulus of the spatial derivative of the map, $\mathcal{C}=|f'(z)|$. But if physical space is stretched by a linear scale factor $\mathcal{C}$ with respect to virtual space, the refractive index in physical space has to be multiplied by $1/\mathcal{C}$ such that the optical path lengths are identical in the virtual and physical spaces (by Fermat’s principle). ------------------------------------------------------------------------ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from the cloak of Eq.  for $w=1.66$ and $h=0.77$ (matching the width and height of the cloak in Ref. [@Zhang2010b], see Fig. \[fig:ZhangBump\]). All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at $y=0$, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.86 to 2.10.[]{data-label="fig:GaussBump"}](rays_Gaussian_legend-crop.pdf "fig:"){width="\legwidth"}\ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from the cloak of Eq.  for $w=1.66$ and $h=0.77$ (matching the width and height of the cloak in Ref. [@Zhang2010b], see Fig. \[fig:ZhangBump\]). All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at $y=0$, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.86 to 2.10.[]{data-label="fig:GaussBump"}](map_Gaussian-crop.pdf "fig:"){width="49.00000%"} ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from the cloak of Eq.  for $w=1.66$ and $h=0.77$ (matching the width and height of the cloak in Ref. [@Zhang2010b], see Fig. \[fig:ZhangBump\]). All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at $y=0$, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.86 to 2.10.[]{data-label="fig:GaussBump"}](rays_Gaussian-crop.png "fig:"){width="49.00000%"} The shape of the bump giving rise to the refractive-index profile following from Eq.  is defined by the implicit form $(u(x,0),v(x,0))$. In general, it is difficult to obtain a closed expression for the parametric dependence of $v(x,0)$ on $u(x,0)$. However, for shallow bumps an explicit expression can be obtained. To see this, let us consider the example of a Gaussian for the coefficients $c_k$ in Eq. , $$\label{eq:GaussianCoeff} c_k = \frac{{\ensuremath{\text{i}}}h w}{\sqrt{\pi}} e^{-(k w/2)^2}\,.$$ The conformal map is then $$\label{eq:GaussianBumpCM} f(z) = z + {\ensuremath{\text{i}}}h e^{-(z/w)^2} \left[ 1+\operatorname{erf}({\ensuremath{\text{i}}}z/w) \right]\,,$$ where $\operatorname{erf}$ is the error function (see the left panel of Fig. \[fig:GaussBump\]). For shallow bumps ($h \ll w$ in this example) we obtain $u(x,0) \approx x$, and derive the explicit form for the bump shape $$v(u)\approx h e^{-(u/w)^2}\,.$$ The parameter $h$ is therefore the height of the Gaussian bump, and $w$ is its width. In general, in this limit of shallow bumps, the $c_k=a_k+ {\ensuremath{\text{i}}}b_k$ are the Fourier coefficients of the bump shape since $$v(x,0) = \int_0^{\infty} \left[ a_k \sin(k x) + b_k \cos(k x) \right]\text{d}k$$ and hence, since $u(x,0)\approx x$, $$\label{eq:vofu} v(u) \approx \int_0^{\infty} \left[ a_k \sin(k u) + b_k \cos(k u) \right]\text{d}k\,.$$ Thus, in this limit the coefficients $c_k$ for any desired conformal map (and refractive-index profile) can be obtained by Fourier transformation of the real-space bump shape $v(u)$. Outside of the shallow-bump approximation, the same coefficients can still be used for perfect cloaking, but these coefficients and the corresponding conformal map $z\mapsto f(z)$ refer to a different bump shape $v(u)$ than in the shallow-bump limit. In the above example , at the critical ratio $h/w=\sqrt{\pi}/2$ the maximum of the bump in the metal carpet develops into a sharp tip and the refractive index becomes singular. We have extensively tested the above analytical results by numerical ray-tracing calculations in two dimensions. For the artificial and experimentally irrelevant case of an infinitely extended structure, cloaking is perfect – as can be expected from the fact that the underlying transformation is strictly conformal. Experimentally relevant finite-size refractive-index profiles can be obtained by simply truncating the exact refractive-index profile, [*i.e.*]{}, by setting $n=1$ outside of the finite-size cloak. This procedure delivers cloaking results which are similar to those obtained for the quasi-conformal carpet cloak [@Zhang2010]. In particular, as can be seen on the right-hand side of Fig. \[fig:GaussBump\], we obtain the same lateral beam shifts as discussed in Ref. [@Zhang2010]. While the performances of quasi-conformal and conformal cloaks are qualitatively and quantitatively similar, we emphasize that our refractive-index profiles are given in closed mathematical form, whereas those of quasi-conformal carpet cloaks are derived from a nontrivial numerical minimization of the modified-Liao functional [@Li2008]. To further investigate the cloaking imperfections arising from the spatial truncation, it is interesting to study the asymptotic decay of the refractive-index profile towards its vacuum value $n=1$ far away from the bump. At large distances the refractive index due to Eq.  behaves like $$n(u+{\ensuremath{\text{i}}}v=\rho e^{{\ensuremath{\text{i}}}\varphi}) = 1 - \frac{h w \cos(2\varphi)}{\sqrt{\pi}\rho^2} + \mathcal{O}(\rho^{-4})\,.$$ This decay is polynomial, which means that $n(u,v)$ approaches the vacuum limit $n=1$ rather slowly – necessitating undesirably large cloaking structures. This slow decay, which corresponds to small spatial-frequency components in $f(z)$, is connected to the small spatial-frequency components of the bump shape $v(u)$ itself. ------------------------------------------------------------------------ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from Eq.  with $c={\ensuremath{\text{i}}}/12$ and $k=2\pi$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a height of one normalized unit. Above it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.66 to 2.10.[]{data-label="fig:GaussianBumpwiggle"}](rays_wiggle_legend-crop.pdf "fig:"){width="\legwidth"}\ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from Eq.  with $c={\ensuremath{\text{i}}}/12$ and $k=2\pi$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a height of one normalized unit. Above it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.66 to 2.10.[]{data-label="fig:GaussianBumpwiggle"}](map_wiggle-crop.pdf "fig:"){width="49.00000%"} ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from Eq.  with $c={\ensuremath{\text{i}}}/12$ and $k=2\pi$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a height of one normalized unit. Above it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.66 to 2.10.[]{data-label="fig:GaussianBumpwiggle"}](rays_wiggle-crop.png "fig:"){width="49.00000%"} To exemplify this observation, we consider a mathematically much simpler case which uses only a single non-zero spatial-frequency component $k$ in Eq. , $$\label{eq:wiggleCM} f(z)=z+c e^{{\ensuremath{\text{i}}}k z}$$ with $c=a+{\ensuremath{\text{i}}}b$. The resulting transformation is illustrated in the left panel of Fig. \[fig:GaussianBumpwiggle\]. Its refractive-index profile is shown in the right panel, $$n(\zeta=u+{\ensuremath{\text{i}}}v) = \left| 1 + W_0\left({\ensuremath{\text{i}}}c k e^{{\ensuremath{\text{i}}}k \zeta} \right) \right|^{-1} = 1+[a\sin(ku)+b\cos(ku)]k e^{-kv}+\mathcal{O}(e^{-2kv}),$$ where $W_0$ is the Lambert function.[^1] The refractive index approaches the vacuum limit $n=1$ *exponentially* fast with increasing vertical coordinate $v$. The absence of zero and small spatial-frequency components means that the average value of the “carpet” shape $v(u)$ is zero. This implies that the shape $v(u)$ no longer only exhibits values *above* the fictitious ground plane (maxima), but also values *below* that ground plane (minima) – in sharp contrast to the usual carpet [@Li2008] with a single maximum. We have rather found a cloak for a corrugated metal surface, [*i.e.*]{}, for a one-dimensional metal grating. More generally, we can introduce a spatial cutoff frequency $\kappa>0$ such that $c_k=0$ for all $k<\kappa$. In this case, the refractive index will approach the vacuum limit $n=1$ according to $e^{-\kappa v}$ for $v\to \infty$. Hence, the metal surface $v(u)$ can be almost perfectly cloaked using a finite-size refractive-index profile with an extent comparable to only $2\pi/\kappa$. ------------------------------------------------------------------------ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from Eq.  with $w=2$, $h=0.5$, and $\kappa=0.5$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.78 to 2.14.[]{data-label="fig:GaussianBumpExp"}](rays_exp2_legend-crop.pdf "fig:"){width="\legwidth"}\ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from Eq.  with $w=2$, $h=0.5$, and $\kappa=0.5$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.78 to 2.14.[]{data-label="fig:GaussianBumpExp"}](map_exp2-crop.pdf "fig:"){width="49.00000%"} ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from Eq.  with $w=2$, $h=0.5$, and $\kappa=0.5$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane indicated in gray, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.78 to 2.14.[]{data-label="fig:GaussianBumpExp"}](rays_exp2-crop.png "fig:"){width="49.00000%"} As an example we apply this cutoff to the Gaussian map resulting from Eq. , with which we had started our discussion. Interpreting the cutoff as a shift on the spatial modes, this leads to $$\label{eq:GaussianBumpCM2} f(z) = z+\int_{0}^{\infty} c_k e^{{\ensuremath{\text{i}}}(k+\kappa) z}\text{d}k = z + {\ensuremath{\text{i}}}h e^{-(z/w)^2} \left[ 1+\operatorname{erf}({\ensuremath{\text{i}}}z/w) \right] e^{i \kappa z}\,.$$ Far from the bump, the refractive-index profile derived from this transformation decays according to $$n(u+{\ensuremath{\text{i}}}v=\rho e^{{\ensuremath{\text{i}}}\varphi}) = 1 + \frac{h w \kappa \sin(\varphi-\kappa\rho\cos\varphi)}{\sqrt{\pi}} \times \frac{e^{-\kappa\rho\sin\varphi}}{\rho} + \mathcal{O}\left(\frac{e^{-\kappa\rho\sin\varphi}}{\rho^2}\right).$$ The behavior of this modified Gaussian bump is illustrated in Fig. \[fig:GaussianBumpExp\]. In particular we note in the left panel that the transformed coordinates $(u,v)$ converge rapidly towards the Cartesian ones $(x,y)$ when moving away from the mirror plane. In the right panel this finding translates into much smaller beam displacements than those discussed in Ref. [@Zhang2010] (and in our Fig. \[fig:GaussBump\]) between the green rays (vacuum, reflected off of a planar mirror) and the red rays (reflected off of the curved mirror shown in black) when using such a finite-size cloak. With a further increase in cloak size these shifts disappear *exponentially*, as opposed to the polynomial decrease observed for $\kappa=0$. We have found the same behavior for rays impinging under different angles (not depicted). ------------------------------------------------------------------------ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from the cloak of Ref. [@Zhang2010b], [*i.e.*]{}, for the conformal map $z \mapsto f(z)=z-1/(z+{\ensuremath{\text{i}}}\delta\!y)$ with $\delta\!y=1.3$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at $y=0$, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.92 to 2.45.[]{data-label="fig:ZhangBump"}](rays_Zhang_legend-crop.pdf "fig:"){width="\legwidth"}\ ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from the cloak of Ref. [@Zhang2010b], [*i.e.*]{}, for the conformal map $z \mapsto f(z)=z-1/(z+{\ensuremath{\text{i}}}\delta\!y)$ with $\delta\!y=1.3$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at $y=0$, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.92 to 2.45.[]{data-label="fig:ZhangBump"}](map_Zhang-crop.pdf "fig:"){width="49.00000%"} ![Left: illustration of the original coordinates $(x,y)$ (gray) and the transformed coordinates $(u,v)$ (red) resulting from the cloak of Ref. [@Zhang2010b], [*i.e.*]{}, for the conformal map $z \mapsto f(z)=z-1/(z+{\ensuremath{\text{i}}}\delta\!y)$ with $\delta\!y=1.3$. All coordinates are in normalized units. Right: refractive-index profile and selected rays. The green rays correspond to vacuum and the mirror plane at $y=0$, the red ones to the finite cloaking structure and the ground-plane shape shown in black. The cloak has a size of $9\times4.5$ normalized units. Outside of it, we assume vacuum (white). The color scale is logarithmic, ranging from 0.92 to 2.45.[]{data-label="fig:ZhangBump"}](rays_Zhang-crop.png "fig:"){width="49.00000%"} It is instructive to compare our results with those of Ref.[@Zhang2010b], which has used the conformal map $z \mapsto f(z)=z-1/(z+{\ensuremath{\text{i}}}\delta\!y)$ and a mirror plane at fixed height $\delta\!y=1.3$; in our notation this cloak results from $c_k={\ensuremath{\text{i}}}\exp(-k \delta\!y)$. Corresponding results for a finite-size cloaking structure with a height of 4.5 normalized units are shown in Fig. \[fig:ZhangBump\] – allowing for direct comparison with Figs. \[fig:GaussBump\] and \[fig:GaussianBumpExp\]. The lateral beam displacement highlighted in Ref. [@Zhang2010] is very pronounced for this cloak, which, as in Fig. \[fig:GaussBump\], is due to the presence of strong components $c_k$ at small spatial frequencies $k$ and thus a slowly decaying refractive-index profile: $$n(u+{\ensuremath{\text{i}}}v=\rho e^{i\varphi}) = 1-\frac{\cos(2\varphi)}{\rho^2}+\mathcal{O}(\rho^{-3}).$$ Finally, we note that refractive-index profiles for cloaking further ground-plane shapes $v(u)$ can easily be obtained with our analytical approach. Textbook examples of conformal maps representing circular, rectangular, and triangular bumps, however, lead to infinities in the required refractive-index profiles. In contrast, as long as the shapes $v(u)$ are smooth and do not exhibit any kinks, the resulting conformal maps and refractive-index profiles are smooth as well and represent realistic proposals for experimental cloaks. Conclusion ========== To summarize, we have introduced a class of conformal versions of the quasi-conformal carpet cloak previously introduced by Li and Pendry. We have obtained exact analytical mathematical forms for the refractive-index profiles of usual bumps as well as of other shapes, [*e.g.*]{}, of gratings. Our analytical formulas can simply replace nontrivial numerical calculations along the lines of the quasi-conformal mapping. This step considerably eases working with these refractive-index profiles in practice. The analytical forms also allow us to study the asymptotic behavior of the refractive-index profiles ([*e.g.*]{}, polynomial or exponential). This aspect is important for assessing and optimizing the performance of [*finite-size*]{} cloaks as blueprints for experiments. In this regard, we obtain much smaller lateral beam displacements for certain metal profiles than previous quasi-conformal [@Zhang2010] as well as previous conformal maps [@Zhang2010b]. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Nicolas Stenger and Tolga Ergin (Karlsruhe) for discussions and for a critical reading of the manuscript. M.W. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) and the State of Baden-Württemberg through the DFG Center for Functional Nanostructures (CFN) within subproject A1.5. The project PHOME acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 213390. The project METAMAT is supported by the Bundesministerium für Bildung und Forschung (BMBF). [^1]: The Lambert function (or product logarithm) $W_0(z)$ is the principal solution for $w$ in $z=w e^w$.

--- abstract: 'We study some effects the inclusion of the radiation energy component in the universe, $\Omega_{\rm r}$, can have on several quantities of interest for the large-scale structure of the universe in a $\Lambda$CDM cosmological simulation; started at a very high redshift ($z \!=\! 500$). In particular we compute the power spectrum density, the halo mass function, and the concentration-mass relation for haloes. We find that $\Omega_{\rm r}$ has an important contribution in the long-term nonlinear evolution of structures in the universe. For instance, a lower matter density power, by $\approx \!50$%, in all scales is obtained when compared with a simulation without the radiation term. Also, haloes formed with the $\Omega_{\rm r}$ taken into account are $\approx \! 20$% less concentrated than when not included in the Hubble function.' author: - | Héctor Aceves[^1]\ Instituto de Astronomía, Universidad Nacional Autónoma de México. Apdo. Postal 106, Ensenada B. C. 22800 México date: 'Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11' title: 'The radiation energy component of the Hubble function and a $\Lambda$CDM cosmological simulation ' --- \[firstpage\] methods: numerical, $N$-body simulations –cosmology: theory Introduction {#sec:intro} ============ In the current cosmological paradigm, structures form in the Universe by amplification of primordial fluctuations driven by a gravitational instability in the expanding universe (e.g. Peacock 1999, Weinberg 2008, Mo, van den Bosh and White 2010). The growth of the instability can be studied analytically and to some extent an exploration of the weak non-linear regime may be done. However, the full non-linear evolution is essentially studied by means of numerical simulations. The formation of non-linear structures in the universe has been studied by means of cosmological simulations, since some of the first works of, for example, Mellot et al. (1983) and Davis et al. (1985). Recent simulations have reached a high degree of complexity, both those including only dark matter (e.g. Springel et al. 2005, Diemand et al. 2008, Boylan-Kolchin et al. 2009, Heitmann et al. 2010, Klypin et al. 2011, Prada et al. 2012) and those with baryonic physics included (e.g. Wise et al. 2012, Bird et al. 2013). Cosmological simulations considering physics outside the standard $\Lambda$CDM cosmology, such as including warm dark matter (e.g. Colín et al. 2000) and different kinds of equations for the dark matter and energy (e.g. Klypin et al. 2003, Linder & Jenkins 2003, Dolag et al. 2004, Grossi & Springel 2009, Rocha et al. 2012) have also been increasing in complexity and in the physics explored to understand the universe. Cosmological simulations with modified dynamics have also been done (e.g. Angus & Diaferio 2011). In all of the simulations up to now, to our knowledge, there has been no consideration of the radiation energy density contribution to the equation of motion of particles in a cosmological setting. This is in part understandable since the current cosmic background temperature of photons is $T_{\gamma 0}=2.725\,$K (e.g. Weinberg 2008), leading to an energy density of $\rho_{\gamma 0} = a_{\rm B} T_{\gamma 0}^4= 4.64 \times 10^{-34}\,{\rm g}\,{\rm cm}^{-3}$, and a density parameter of photons $\Omega_{\gamma} = \rho_{\gamma 0 }/\rho_{0 \rm c} = 2.47 \times 10^{-5} h^{-2}$; a value much smaller than the current matter density parameter $\Omega_{\rm m} \approx 0.3$ and the vacuum energy density parameter $\Omega_\Lambda \approx 0.7$. Even if we consider that an additional contribution to the radiation component of the universe comes from the neutrinos from the era of ${\rm e}^- + {\rm e}^+$ pair annihilation the situation does not change by much; the total total density of radiation (assuming massless neutrinos) becomes $ \rho_{{\rm r} 0} = \left[ 1 + 3 \left(7/8 \right) \left(4/11 \right)^{4/3} \right] \rho_{\gamma 0 } = 7.80 \times 10^{-34}\,{\rm g}\,{\rm cm}^{-3} $ leading to a radiation parameter $ \Omega_{\rm r} = \rho_{{\rm r} 0 } / \rho_{0 \rm c} = 4.15 \times 10^{-5} h^{-2}$. Nonetheless, the effect of the radiation energy density becomes more important towards higher redshifts. There are several problems, such as the mass function at high redshifts (e.g. Reed et al. 2007, Lukić et al. 2007), that demand a treatment as accurate as possible of the evolution of structures in the universe. For example, Reed et al. start some of their simulations at $z\approx 300$, and Lukić et al. run simulations going to as high as $z=500$ in their study. At those starting redshifts for the simulations the radiation energy density is not negligible, due to its $(1+z)^4$ dependence. Also, due to the nonlinear way matter clusters, the effect of changing at high redshift the rate of expansion of the universe by including a radiation energy term can be significant in structures we see today. In this [*Letter*]{} we present results of two cosmological simulations done within the standard cosmological scenario, but one including the radiation energy density term in the equation of motion of dark matter particles. We quantify differences between both cases in regard to the matter power spectrum, the mass function and the concentration-mass relation for halos. Other properties of the clustering of dark matter or haloes themselves are not considered here, nor a detailed study of each part is considered; such work is postponed for future communications. The objective is to point out the need to include the radiation energy term in cosmological simulations, specially those starting at high redshift, for better [*consistency*]{} with the theoretical framework of standard cosmology. The outline of this work is as follows. In § \[sec:methods\] we describe the model used and describe some numerical matters. In § \[sec:results\] we show some of the results of both of our simulations. Finally, in § \[sec:final\] we provide a summary and final comments on this work. Models and Numerical Methods {#sec:methods} ============================ In cosmological simulations the expansion of the universe has to be considered (e.g. Hockney & Eastwood 1981). In an $N$-body simulation with periodic boundary conditions the equation of motion of particle $i$ is (e.g. Bertschinger 1998, Springel et al. 2001): $${\ddot \mathbf{x}}_i + 2 \frac{\dot a}{a} {\dot \mathbf{x}_i} = - \frac{G }{a^3} \sum_{j\ne i} \frac{m_j ( \mathbf{x}_i - \mathbf{x}_j )}{ | \mathbf{x}_i - \mathbf{x}_j |^3 } \,, \label{eq:motion}$$ where the summation goes over all periodic images of the particles $j$, and $a$ is the scale factor of the Universe. The evolution of the latter follows from Friedman equation, $$H (a) = H_0 \left( \frac{\Omega_{\rm m}}{a^3} + \frac{\Omega_{\rm r}}{a^4} + \Omega_\Lambda \right)^{1/2} \,, \label{eq:termsH}$$ with $H={\dot a}/a$, $\Omega_{\rm m}$ the current epoch matter energy density parameter, $\Omega_{\rm r}$ the present day radiation energy density parameter, and $\Omega_{\Lambda}$ the vacuum energy density parameter. A flat universe has been assumed in the preceding equations. Solving the coupled set of equations (\[eq:motion\]) and (\[eq:termsH\]), along the corresponding Poisson’s equation, determines the dynamics of the $N$-body simulation of the Universe. We performed two cosmological simulations using the publicly available parallel Tree-code [Gadget2]{} (Springel 2005). This code uses the Hubble function (\[eq:termsH\]), including a curvature term, but does not include the radiation component. In [Gadget2]{} such function is required for computing the time steps in the advancement of the motion of particles, thru drift and kick factors. We essentially modified subprograms [driftfac.c]{} and [timestep.c]{} of the [Gadget2]{} code in order for the simulation to account for the $\Omega_{\rm r}$ contribution. We will denote by $\Lambda$CDM, as is customary, the standard cosmological simulation with $\Omega_{\rm r}\!=\!0$, and with $\Lambda$rCDM the one including the radiation term in the Hubble function. Our two simulations take as cosmological parameters those of the mean values of the [Wmap7]{} results (Komatsu et al. 2011), where the matter density $\Omega_m=0.275$, spectral index $n_s=0.968$, mass fluctuation $\sigma_8=0.816$ and the Hubble parameter $h=0.702$, and we take the vacuum parameter as $\Omega_\Lambda=1-\Omega_{\rm m} -\Omega_{\rm r}$. The value of $\Omega_{\rm r}$ used is that indicated in Section $\S$\[sec:intro\]. Initial conditions were generated, using a 2nd-order Lagrangian perturbation code (Crocce, Pueblas & Scoccimarro 2006), at a redshift of $z\!=\!500$. The initial linear power spectrum density is calculated using the transfer function from the cosmic microwave background code [camb]{} (Lewis, Challinor & Lasenby 2000), normalized to the above $\sigma_8$ value at $z=0$. The spectrum is evolved back in time to $z=500$, using the linear growth factor $D_+$ (\[eq:Dplus\]) given by (e.g. Carroll et al. 1992, Mo et al. 2010), $$D_+ = \frac{5\Omega_{\rm m}}{2} \frac{H(z)}{H_0} \int_z^\infty \frac{(1+z)}{[H(z)/H_0]^3} \, {\rm d}z \,, \label{eq:Dplus}$$ to generate our initial conditions. Computing (\[eq:Dplus\]) at $z\!=\!500$ for the $\Lambda$CDM case gives $1.93\times 10^{-3}$ while for the $\Lambda$rCDM one $1.53\times 10^{-3}$; the radiation energy reduces the growth factor by $\approx 20$% at that redshift. Such difference will be reflected in the displacements and peculiar velocities of particles at the initial condition. This approach to modifying the initial conditions can only be considered approximate, but serves our purpose of elucidating differences when including or not the radiation term in (\[eq:motion\]). Each simulation box has a comoving length of $L\!=\! 100\,h^{-1}$Mpc with $N_p \!=\! 512^3$ dark matter particles, leading to each particle having a mass of $m_p \!=\! 5.5\times 10^8 h^{-1} \rm {M}_\odot$. The smallest halo we are able to resolve with some confidence has a mass of $M=100 m_p \approx 6\times 10^{10} h^{-1} \rm {M}_\odot$. The gravitational (Plummer equivalent) softening length was kept at the fixed value of $\epsilon \!= \! 5\, h^{-1}\,$kpc in comoving coordinates. Halos were identified with the [AHF]{} public code (Gill, Knebe & Gibson 2004 and Knollmann & Knebe 2009). Results {#sec:results} ======= [*Qualitative structure*]{}. The large scale structure in our box appears rather similar for both kind of simulations, however important differences appear at smaller scales. In Figure \[fig:evolutionZ\] we show snapshots at different times ($z=2,1$ and $z\!=\!0$, from [*left*]{} to [*right*]{}) of the distribution of dark matter particles around the most massive halo ($M =4.5 \times 10^{14}\, h^{-1}\,{\rm M}_\odot$) in our $\Lambda$CDM cosmology ([*top*]{}) and the same region for that of the $\Lambda$rCDM (the halo has $M= 3.5 \times 10^{14}\, h^{-1}\,{\rm M}_\odot$). At $z\!=\!0$ we find for the $\Lambda$CDM cosmology a total of 47,728 halos, while for the $\Lambda$rCDM a total of 46,473 is found. [*Power spectrum*]{}. In Figure \[fig:pks\] we show the power spectrum density $P(k)$ computed at different redshifts from $z\!=\!5$ to $z\!=\!0$, for both kinds of cosmological evolution considered in this work. It is readily noticeable that more power density $\ga 50$%, in more than 2 decades in $k$, is deposited by the $\Lambda$CDM model than the $\Lambda$rCDM. The difference tends to increase at higher redshifts as shown in the bottom of Figure \[fig:pks\]. It is worth noticing that the minimum discrepancy in power density is $\approx 50$% even at the smallest scale of our simulations. Not including the effect of $D_+$ in the initial conditions, just the $\Omega_{\rm r}$ in the Hubble function, leads to a discrepancy of $\,\approx 15$% at $z\!=\!0$. [*Mass function*]{}. The effect of the $\Omega_{\rm r}$ term in (\[eq:motion\]) can also be seen in the halo mass function, $F(M)=N_{\rm h}/V \Delta \log M$ with $N_{\rm h}$ the number of halos in a log-bin of mass $M$ and volume $V$; we computed $F(M)$ as in Lukić et al. (2007). The mass function of our haloes for different redshifts is shown in Figure \[fig:massfunc\]. As noted, the mass function at $z\!=\!0$ is somewhat similar with and without the radiation energy density term, but the effect of the latter is stronger toward higher redshifts; as was also indicated by the behaviour of the $P(k)$. Including the $\Omega_{\rm r}$ term leads to a lowering of the formation of halos at higher redshifts. This can have important consequences for the demographics of haloes that form galaxies, and one may speculate that at much higher redshifts in the abundance of primordial haloes that would host the first stars in the Universe. [*Haloes concentration-mass relation*]{}. As a preliminary result on the properties of halos formed under the two cosmologies explored here, we computed the mean concentration–mass relation, $c(M)$, for the halos found in our simulations. Haloes are assumed to follow a NFW profile (Navarro, Frenk & White 1997) and concentrations are computed by the AHF code. We computed the mean $c(M)$ relation as in Kwan et al. (2012). In Figure \[fig:CM\] we show the mean $c\,$–$M$ relation found. Halos at $z\!=\!0$ formed in the $\Lambda$rCDM cosmology tend to be $\approx 20$% less concentrated than in the standard $\Lambda$CDM, for most of the mass range of our halos; at higher masses ($\sim 10^{14}\,h^{-1} {\rm M}_\odot$) the situation is not that clear. The $\Lambda$CDM simulations yields results which are consistent with those, for example, of Duffy et al. (2008). A simple fit to the $\Lambda$CDM results at $z=0$ yields $c(M) = 5.2 (M_{\rm vir}/M_{14})^{-0.81}$, with $M_{14}=10^{14} h^{-1} {\rm M}_\odot$, while for the $\Lambda$rCDM halos the coefficient in former expression is $4.7$ with essentially the same slope. At higher redshifts the slope tends to flatten, as found in other works. [*Substructure*]{}. The impact of the radiation term also is noticed in the number of subhaloes detected by the [AHF code]{}. In the $\Lambda$CDM model the number of subhaloes for the six most massive halos are $\{94,47,64,36,54,51\}$ while for the $\Lambda$rCDM we obtained $\{55,35,14,18,19,24\}$; i.e. the inclusion of $\Omega_{\rm r}$ tends to reduce the number of subhaloes. In a future work we explore the substructure differences in more detail. Summary and Final Comments {#sec:final} ========================== We have carried out two numerical experiments on the evolution of $N$-body dark matter cosmological simulation, one with the usual neglect of the radiation energy density $\Omega_{\rm r}$ term in the Hubble function and one that includes it. Different diagnostics, such as the power density spectrum $P(k)$, the halo mass function, and the $c\,$–$M$ relation, were used to quantify differences that occur when neglecting the radiation energy density. Other tests, such as the subhalo velocity distribution function are not considered in this work, in one part due to lacking the mass resolution for an adequate treatment and on other due reduced space and nature of this [*Letter*]{}. Future works will address some of these topics. Including the radiation term makes the structure formation of the universe to lag in time in comparison to when it is not included. This may be understood, at least in the linear regime, from recalling the growing mode $D_+$ (\[eq:Dplus\]). Including $\Omega_{\rm r}$ makes the expansion rate larger, hence the perturbation growth is reduced. The same kind of effect occurs when, for instance, the $\Omega_{\Lambda}$ is included -for instance- in a Einstein-de Sitter universe; both terms $\Omega_{\Lambda}$ and $\Omega_{\rm r}$ enter the Hubble function (\[eq:termsH\]) with the same sign. This enhancement of the Hubble drag in comparison with the standard $\Lambda$CDM treatment has also the effect of reducing in average the concentration of the halos formed; see Figure \[fig:CM\]. We have shown that including the radiation energy term in a high redshift ($z\!=\!500$) simulation leads to important differences in the structures of the universe than when not including it. The power spectrum density “deposited” at all scales tend to be lower by $\approx 50$% in the $\Lambda$rCDM cosmology than in the $\Lambda$CDM. Also, the effect of the radiation density reflects itself in the mean concentration of halos, by lowering it $\approx 20$% at the current epoch. All the results presented in this work point toward the necessity to include in simulations the $\Omega_{\rm r}$ term in the Hubble function. This is particularly important for questions regarding the first structures formed in the universe and their evolution. Also it may bear importance in problems at galactic scale such as the “missing satellites” (Klypin et al. 1999, Bullock 2010) or the “too big to fail” (Boylan-Kolchin et al. 2011). Implications of the $\Omega_{\rm r}$ on such problems are the subject of future works, as well as comparisons to observations. Acknowledgments {#acknowledgments .unnumbered} =============== This research was funded by CONACyT Research Project 179662. We thank both Volker Springel and Alex Knebe for making [Gadget-2]{} and [AHF]{}, respectively, publicly available. [99]{} Angus, G. W., & Diaferio, A. 2011, , 417, 941 Bertschinger, E. 1998, , 36, 599 Bird, S., Vogelsberger, M., Sijacki, D., et al. 2013, , 429, 3341 Boylan-Kolchin, M., Springel, V., White, S. D. M., Jenkins, A., & Lemson, G. 2009, , 398, 1150 Boylan-Kolchin, M., Bullock, J. S., & Kaplinghat, M. 2011, , 415, L40 Bullock, J. S. 2010, arXiv:1009.4505 Carroll, S. M., Press, W. H., & Turner, E. L. 1992, , 30, 499 Col[í]{}n, P., Avila-Reese, V., & Valenzuela, O. 2000, , 542, 622 Crocce, M., Pueblas, S., & Scoccimarro, R. 2012, Astrophysics Source Code Library, 1005 Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, , 292, 371 Diemand, J., Kuhlen, M., Madau, P., et al. 2008, , 454, 735 Dolag, K., Bartelmann, M., Perrotta, F., et al. 2004, , 416, 853 Duffy, A. R., Schaye, J., Kay, S. T., & Dalla Vecchia, C. 2008, , 390, L64 Gill, S. P. D., Knebe, A., & Gibson, B. K. 2004, , 351, 399 Grossi, M., & Springel, V. 2009, , 394, 1559 Heitmann, K., White, M., Wagner, C., Habib, S., & Higdon, D. 2010, , 715, 104 Hockney, R.W., Eastwood, J.W. 1981. Computer Simulation Using Particles. McGraw-Hill, New York. Klypin, A., Kravtsov, A. V., Valenzuela, O., & Prada, F. 1999, , 522, 82 Klypin, A., Macci[ò]{}, A. V., Mainini, R., & Bonometto, S. A. 2003, , 599, 31 Klypin, A. A., Trujillo-Gomez, S., & Primack, J. 2011, , 740, 102 Knollmann, S. R., & Knebe, A. 2009, , 182, 608 Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, , 192, 18 Kwan, J., Bhattacharya, S., Heitmann, K., & Habib, S. 2012, arXiv:1210.1576 Lewis, A., Challinor, A., & Lasenby, A. 2000, , 538, 473 Linder, E. V., & Jenkins, A. 2003, , 346, 573 Luki[ć]{}, Z., Heitmann, K., Habib, S., Bashinsky, S., & Ricker, P. M. 2007, , 671, 1160 Melott, A. L., Einasto, J., Saar, E., et al. 1983, Physical Review Letters, 51, 935 Mo, H., van den Bosch, F. C., & White, S. 2010, Galaxy Formation and Evolution. Cambridge University Press, 2010. Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 Peacock, J. A. 1999, Cosmological Physics, by John A. Peacock. Cambridge, UK: Cambridge University Press Prada, F., Klypin, A. A., Cuesta, A. J., Betancort-Rijo, J. E., & Primack, J. 2012, , 423, 3018 Reed, D. S., Bower, R., Frenk, C. S., Jenkins, A., & Theuns, T. 2007, , 374, 2 Rocha, M., Peter, A. H. G., Bullock, J. S., et al. 2012, arXiv:1208.3025 Springel, V., Yoshida, N., & White, S. D. M. 2001, , 6, 79 Springel, V. 2005, , 364, 1105 Springel, V., White, S. D. M., Jenkins, A., et al. 2005, , 435, 629 Weinberg, S. 2008, Cosmology, Oxford University Press, Oxford, UK, 2008., Wise, J. H., Turk, M. J., Norman, M. L., & Abel, T. 2012, , 745, 50 \[lastpage\] [^1]: E-mail: aceves@astro.unam.mx

--- abstract: 'Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A'' \cdot B \subsetneq G$ for any $\emptyset \neq A'' \subsetneq A$ and $A\cdot B'' \subsetneq G$ for any $\emptyset \neq B'' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, the group ${\ensuremath{\mathbb{Z}}}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $\sigma$, there exists a subset $A$ of ${\ensuremath{\mathbb{Z}}}^{2d}$ such that $A$ and $\sigma(A)$ form a co-minimal pair. The existence and construction of co-minimal pairs in the integers with both the subsets $A$ and $B$ ($A\neq B$) of infinite cardinality was unknown. We show that such pairs exist and explicitly construct these pairs satisfying a number of algebraic properties.' address: - 'Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel' - 'Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India' author: - Arindam Biswas - Jyoti Prakash Saha title: 'Infinite co-minimal pairs in the integers and integral lattices' --- [^1] [^2] Introduction ============ Let $(G,+)$ be an abelian group and $W\subseteq G$ be a nonempty subset. A nonempty set $W'\subseteq G$ is said to be an *additive complement* to $W$ if $W + W' = G.$ Additive complements have been studied since a long time in the context of representations of the integers, e.g., they appear in the works of Erdős, Hanani, Lorentz and others. See [@Lorentz54], [@Erdos54], [@ErdosSomeUnsolved57] etc. In [@NathansonAddNT4], Nathanson introduced the notion of *minimal additive complements* for nonempty subsets of groups. An additive complement $W'$ to $W$ is said to be minimal if no proper subset of $W'$ is an additive complement to $W$, i.e., $$W + W' = G \,\text{ and }\, W + (W'\setminus \lbrace w'\rbrace)\subsetneq G \,\,\, \forall w'\in W'.$$ Nathanson was interested in a number of questions from metric geometry arising in the context of discrete groups. As an example, the existence of minimal nets in groups is strongly related to the existence of minimal additive complements of generating sets, see [@NathansonAddNT4 §1] (see also [@NathansonAddNT4 Problem 1]). A notion stronger to minimal additive complements is that of *additive co-minimal pairs*. Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. Thus, they are pairs $(A\in G,B\in G)$ such that each element in a pair is a minimal additive complement to the other. The notion of co-minimal pairs was considered in a prior work of the authors see [@CoMin1 Definition 1.2]. Henceforth, by a complement we shall mean an additive complement. If we mean set-theoretic complement, we shall explicitly state it. It is a challenging task to classify the co-minimal pairs in a given group. Even in the context of the group of integers ${\ensuremath{\mathbb{Z}}}$, they are not completely understood. In [@CoMin1], it was shown that non-empty finite subsets in free abelian groups (not necessarily of finite rank) belong to co-minimal pairs. However, the existence and the construction of infinite co-minimal pairs in $\mathbb{Z}$, i.e., co-minimal pairs $(A,B)$ where both $A$ and $B$ are infinite ($A\neq B$) has been unknown. One of our motivations in this article is to show the existence and give explicit constructions of these pairs. Further, we also study co-minimal pairs in higher rank integer lattices. Our constructions satisfy certain nice combinatorial and group theoretic properties. They are mainly motivated by the following questions: \[Qn:BddBelowBddAbove\] Does there exist infinite subsets $A, B$ of ${\ensuremath{\mathbb{Z}}}$ which form a co-minimal pair and one of them is bounded below and the other is bounded above? \[Qn:OneIsSymm\] Does there exist infinite subsets $A, B$ of ${\ensuremath{\mathbb{Z}}}$ which form a co-minimal pair and at least one of them is a symmetric subset of ${\ensuremath{\mathbb{Z}}}$? In [@Kwon] and [@CoMin1], it was established that $(A,A)$ is a co-minimal pair in an abelian group $G$, if and only if $A + A = G$ and $A$ avoids $3$-term arithmetic progressions. This motivates the following two questions when we note that the trivial automorphism $\sigma(g) = g, \forall g\in G$ fixes any subset $A\in G$. \[Qn:AExists\] Given an automorphism $\sigma$ of a group $G$, does there exist subsets $A$ in $G$ such that $(A, \sigma (A))$ is a co-minimal pair? \[Qn:AExistsQuadrant\] Given an automorphism $\sigma$ of ${\ensuremath{\mathbb{Z}}}^d$, does there exist subsets $A$ in $G$ such that $A$ is contained in a quadrant [^3] and $(A, \sigma (A))$ is a co-minimal pair? Note that Questions \[Qn:AExists\], \[Qn:AExistsQuadrant\] are related to Questions \[Qn:OneIsSymm\], \[Qn:BddBelowBddAbove\]. Indeed, having affirmative answers to Questions \[Qn:OneIsSymm\], \[Qn:BddBelowBddAbove\] allows to answer Questions \[Qn:AExists\], \[Qn:AExistsQuadrant\] for certain automorphisms of free abelian groups. For instance, if ${\ensuremath{\mathcal{U}}}, {\ensuremath{\mathcal{V}}}$ are infinite subsets of ${\ensuremath{\mathbb{Z}}}$ forming a co-minimal pair and ${\ensuremath{\mathcal{V}}}$ is symmetric, then taking $A = {\ensuremath{\mathcal{U}}}\times {\ensuremath{\mathcal{V}}}, {\ensuremath{\mathcal{V}}}\times {\ensuremath{\mathcal{U}}}$, we obtain an affirmative answer to Question \[Qn:AExists\] when $G = {\ensuremath{\mathbb{Z}}}^2$ and $\sigma = \left( \begin{smallmatrix} 0 & 1\\ -1 & 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 & -1\\ 1 & 0 \end{smallmatrix} \right)$. Moreover, if ${\ensuremath{\mathcal{S}}}, {\ensuremath{\mathcal{T}}}$ are infinite subsets of ${\ensuremath{\mathbb{Z}}}$ forming a co-minimal pair and ${\ensuremath{\mathcal{S}}}$ (resp. ${\ensuremath{\mathcal{T}}}$) is bounded above (resp. below) by $0$, then taking $A = (-{\ensuremath{\mathcal{S}}}) \times {\ensuremath{\mathcal{T}}}$, we obtain an affirmative answer to Question \[Qn:AExistsQuadrant\] when $d = 2$ and $\sigma = \left( \begin{smallmatrix} 0 & -1\\ -1 & 0 \end{smallmatrix} \right)$. In the following, for $?\in \{<, \leq, > , \geq\}$ and $x\in {\ensuremath{\mathbb{Z}}}$, the set $\{n\in {\ensuremath{\mathbb{Z}}}\,|\, n?x\}$ is denoted by ${\ensuremath{\mathbb{Z}}}_{?x}$. Let $X$ be a subset of ${\ensuremath{\mathbb{Z}}}$ of the form $\{n \in {\ensuremath{\mathbb{Z}}}\,|\, a\leq n \leq b\}$ for some $a, b\in {\ensuremath{\mathbb{Z}}}$. If $X$ contains an even number of elements (i.e., if $b-a + 1$ is even), then the subset $\{x\in X\,|\, x\leq \frac{b + a -1}{2}\}$ (resp. $\{x\in X\,|\, x\geq \frac{b + a +1}{2}\}$) of $X$ is called the left half (resp. right half) of $X$. If $b-a + 1$ is divisible by $4$, then the subsets $\{x\in X\,|\, x\leq a - 1 + \frac{b - a + 1}{4}\}$, $\{x\in X\,|\, a - 1 + \frac{b - a + 1}{4} < x \leq a - 1 + \frac{b - a + 1}{2}\}$, $\{x\in X\,|\, a - 1 + \frac{b - a + 1}{2} < x \leq a - 1 + \frac{3(b - a + 1)}{4}\}$, $\{x\in X\,|\, a - 1 + \frac{3(b - a + 1)}{4} < x \}$ of $X$ are called the first quarter, the second quarter, the third quarter, and the fourth quarter of $X$ respectively. The first quarter (resp. the fourth quarter) of $X$ is also called the left quarter (resp. the right quarter) of $X$. Statement of results -------------------- In Theorems \[Thm:CoMinST\], \[Thm:CoMinUV\], we prove that Questions \[Qn:BddBelowBddAbove\], \[Qn:OneIsSymm\] admit answers in the affirmative. Theorems \[Thm:CoMinST\], \[Thm:CoMinUV\] follow from Theorems \[Thm:CoMinS\], \[Thm:CoMinU\]. \[Thm:CoMinST\] Let ${\ensuremath{\mathcal{T}}}$ denote the subset $\{1, 2, 2^2, \cdots\}$ of ${\ensuremath{\mathbb{Z}}}$. Then there exists an infinite subset ${\ensuremath{\mathcal{S}}}$ of ${\ensuremath{\mathbb{Z}}}_{\leq -1}$ such that $({\ensuremath{\mathcal{S}}}, {\ensuremath{\mathcal{T}}})$ is a co-minimal pair. \[Thm:CoMinUV\] Let ${\ensuremath{\mathcal{V}}}$ denote the subset of ${\ensuremath{\mathbb{Z}}}$ defined as $${\ensuremath{\mathcal{V}}}: = \{1, 2, 2^2, 2^3, \cdots\} \cup \{-1, -2, -2^2, -2^3, \cdots\}.$$ Then there exists an infinite subset ${\ensuremath{\mathcal{U}}}$ of ${\ensuremath{\mathbb{Z}}}_{\leq -1}$ such that $({\ensuremath{\mathcal{U}}}, {\ensuremath{\mathcal{V}}})$ is a co-minimal pair. Using these two results, we establish Theorems \[Thm:AExists\], \[Thm:AExistsQuadrant\], which prove that Questions \[Qn:AExists\], \[Qn:AExistsQuadrant\] admit answers in the affirmative for an infinite class of automorphisms of free abelian groups. An immediate consequence of Theorem \[Thm:AExists\] is stated below. \[Thm:Z2d\] If $\sigma$ is an automorphism of ${\ensuremath{\mathbb{Z}}}^2$, i.e., an element of $ {\ensuremath{\operatorname{GL}}}_2({\ensuremath{\mathbb{Z}}})$ having exactly two nonzero entries, then there exists a subset $A$ of ${\ensuremath{\mathbb{Z}}}^2$ such that $(A, \sigma(A))$ forms a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^2$. For any $d\geq 1$, the group ${\ensuremath{\mathbb{Z}}}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $\sigma$, there exists a subset $A$ of ${\ensuremath{\mathbb{Z}}}^{2d}$ such that $A$ and $\sigma(A)$ form a co-minimal pair. A co-minimal pair involving a bounded below subset ================================================== In this section, we establish Theorem \[Thm:CoMinST\], which follows from Theorem \[Thm:CoMinS\]. Consider the subsets $\{J_n\}_{n\geq 0}, \{K_n\}_{n\geq 0}, \{I_n\}_{n\geq 0}, \{{\ensuremath{\mathcal{I}}}_n\}_{n\geq 0}, S$ of ${\ensuremath{\mathbb{Z}}}$ defined by $$\begin{aligned} J_n & = \begin{cases} \{1\} & \text{ if } n = 0, \\ \{1\} & \text{ if } n = 1, \\ \{1, 2, \cdots, 2^{n-2}\} \cup (2^{n-1} + J_{n-1}) & \text{ if } n\geq 2, \end{cases} \\ K_n & = \begin{cases} J_0 & \text{ if } n = 0, \\ J_1 & \text{ if } n = 1, \\ J_2 \setminus\{1\} & \text{ if } n = 2, \\ \{2^{n-3}+1, 2^{n-3}+2, 2^{n-3}+3, \cdots, 2^{n-2}\} \cup (2^{n-1} + 2^{n-2} + J_{n-2}) & \text{ if }n\geq 3, \end{cases} \\ I_n & = K_n - (1 + 2^{n+1})\quad \text{ if } n\geq 0, \\ {\ensuremath{\mathcal{I}}}_n & = \begin{cases} \{-2, -1\} & \text{ if } n = 0, \\ \{1, 2, 3, \cdots, 2^n\} - (1+2^{n+1}) & \text{ if } n \geq 1, \end{cases} \\ S & = \cup _{n\geq 0} I_n.\end{aligned}$$ The set ${\ensuremath{\mathcal{T}}}$ is an additive complement of $S$ in ${\ensuremath{\mathbb{Z}}}$. For any $m\geq 3$, $$\cup_{k = 3}^m (I_k + 2^{k+1}) \supseteq \cup_{k = 3}^m \{2^{k-3}, \cdots, 2^{k-2}-1\} = \{1, 2, \cdots, 2^{m-2}-1\}$$ holds, which implies that $S + {\ensuremath{\mathcal{T}}}$ contains ${\ensuremath{\mathbb{Z}}}_{\geq 1}$. Note that $I_n$ is contained in ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 0$, and the union of the sets $\{{\ensuremath{\mathcal{I}}}_n\}_{n\geq 0}$ is equal to ${\ensuremath{\mathbb{Z}}}_{\leq -1}$ and these sets lie next to each other in the sense that $\min {\ensuremath{\mathcal{I}}}_n = 1 + \max {\ensuremath{\mathcal{I}}}_{n+1}$ for all $n\geq 0$. Thus, to prove that $S+{\ensuremath{\mathcal{T}}}$ contains ${\ensuremath{\mathbb{Z}}}$, it suffices to show that $S + {\ensuremath{\mathcal{T}}}$ contains $\{0\}$ and ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 0$, which we establish by proving the following statements. 1. $S + {\ensuremath{\mathcal{T}}}$ contains the left half of ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 1$, 2. $S + {\ensuremath{\mathcal{T}}}$ contains the left quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 3$, 3. $S + {\ensuremath{\mathcal{T}}}$ contains the second quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 3$, 4. $S + {\ensuremath{\mathcal{T}}}$ contains the right quarter of ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 4$, 5. $S + {\ensuremath{\mathcal{T}}}$ contains $\{0\},{\ensuremath{\mathcal{I}}}_0, {\ensuremath{\mathcal{I}}}_1$, the right half of ${\ensuremath{\mathcal{I}}}_2$ and the right quarter of ${\ensuremath{\mathcal{I}}}_3$. Note that the sets $I_0 + \{1, 2\}$, $I_1 + \{1, 2\}$, $I_2 + \{1, 2\}$, $I_3 + 2^2$ contain $\{-1, 0\}$, $\{-3, -2\}$, $\{-5, -4\}$, $\{-6\}$ respectively. So, $S + {\ensuremath{\mathcal{T}}}$ contains ${\ensuremath{\mathcal{I}}}_1 \cup {\ensuremath{\mathcal{I}}}_0 \cup \{0\}$ and the right half of ${\ensuremath{\mathcal{I}}}_2$. Since $I_3 + 1$ contains $\{-9\}$ and $I_4 + 2^3$ contains $\{-10\}$, the set $S + {\ensuremath{\mathcal{T}}}$ contains the right quarter of ${\ensuremath{\mathcal{I}}}_3$. This establishes the fifth statement. Note that for any $n\geq 1$, the left half of ${\ensuremath{\mathcal{I}}}_n$ is contained in $I_{n+3}+2^{n+3}$, i.e., the inclusions $$\begin{aligned} I_{n+3} + 2^{n+3} & = K_{n+3} - ( 1 + 2^{n+4}) + 2^{n+3} \\ &= K_{n+3} - (1 + 2^{n+3})\\ & \supseteq (2^{n+2} + 2^{n+1} + J_{n+1}) - (1 + 2^{n+3})\\ &= J_{n+1}- (1 + 2^{n+1})\\ & \supseteq \{1, 2, 3, \cdots, 2^{n-1}\} - (1 + 2^{n+1})\end{aligned}$$ hold, thus establishing the first statement. Note that for any $n\geq 3$, the left quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ is contained in $I_{n+1} + 2^{n}$, i.e., the inclusions $$\begin{aligned} I_{n+1} + 2^{n} & = K_{n+1} - ( 1 + 2^{n+2}) + 2^{n} \\ &= K_{n+1} - (1 + 2^{n+2}) + 2^n\\ & \supseteq (2^{n} + 2^{n-1} + J_{n-1}) - (1 + 2^{n+2})+2^n\\ &= (2^{n-1} + J_{n-1})- (1 + 2^{n+2}) + 2^{n+1}\\ &= (2^{n-1} + J_{n-1})- (1 + 2^{n+1}) \\ & \supseteq \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^{n-1}+2^{n-3}\} - (1 + 2^{n+1})\end{aligned}$$ hold, the second quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ is contained in $I_n + 2^{n-1}$, i.e., the inclusions $$\begin{aligned} I_n + 2^{n-1} & = K_n - (1 + 2^{n+1}) + 2^{n-1} \\ & \supseteq \{2^{n-3}+1, 2^{n-3}+2, 2^{n-3}+3, \cdots, 2^{n-2}\} - (1 + 2^{n+1}) + 2^{n-1} \\ & = \{2^{n-1}+2^{n-3}+1, 2^{n-1}+2^{n-3}+2, 2^{n-1}+2^{n-3}+3, \cdots, 2^{n-1}+2^{n-2}\} - (1 + 2^{n+1}) \end{aligned}$$ hold. This establishes the second and the third statement. For $n\geq 3$, the points in the right quarter of ${\ensuremath{\mathcal{I}}}_n$ that lie $I_n$ are contained in $I_{n+1}+2^n$, i.e., the inclusions $$\begin{aligned} I_{n+1} + 2^{n} &= (2^{n-1} + J_{n-1})- (1 + 2^{n+1}) \\ & \supseteq (2^{n-1}+2^{n-2}+J_{n-2}) - (1 + 2^{n+1})\end{aligned}$$ hold. Note that for any $m\geq 2$, $$J_m + \{0, 1, 2, 2^2, \cdots, 2^{m-2}\} \supseteq \{1, 2, 3, \cdots, 2^m\}$$ holds. Indeed, it holds for $m = 2$, and assuming that $$J_k + \{0, 1, 2, 2^2, \cdots, 2^{k-2}\} \supseteq \{1, 2, 3, \cdots, 2^k\}$$ holds for some integer $k\geq 2$, it follows that the inclusions $$\begin{aligned} & J_{k+1} + \{0, 1, 2, 2^2, \cdots, 2^{k-2}, 2^{k-1}\} \\ & \supseteq (\{1, 2, \cdots, 2^{k-1}\} \cup (2^{k} + J_{k})) + \{0, 1, 2, 2^2, \cdots, 2^{k-2}, 2^{k-1}\} \\ & \supseteq (\{1, 2, \cdots, 2^{k-1}\} + \{0, 2^{k-1}\}) \cup ((2^{k} + J_{k}) + \{0, 1, 2, 2^2, \cdots, 2^{k-2}\}) \\ & = \{1, 2, \cdots, 2^{k}\} \cup (2^{k} + (J_{k} + \{0, 1, 2, 2^2, \cdots, 2^{k-2}\})) \\ & \supseteq \{1, 2, \cdots, 2^{k}\} \cup (2^{k} + \{1, 2, 3, \cdots, 2^k\}) \\ & = \{1, 2, \cdots, 2^{k}\} \cup \{2^{k} +1,2^{k} + 2, 2^{k} +3, \cdots, 2^{k+1}\} \\ & = \{1, 2, \cdots, 2^{k+1}\} \end{aligned}$$ hold. Consequently, for any $n\geq 4$, the points in the right quarter of ${\ensuremath{\mathcal{I}}}_n$ are contained in $(I_{n+1} + 2^n) \cup ( I_n + \{1, 2, 2^2, \cdots, 2^{n-4}\})$, i.e., the inclusions $$\begin{aligned} & (I_{n+1} + 2^n) \cup ( I_n + \{1, 2, 2^2, \cdots, 2^{n-4}\}) \\ & \supseteq ((2^{n-1}+2^{n-2}+J_{n-2}) - (1 + 2^{n+1})) \cup (((2^{n-1}+2^{n-2}+J_{n-2}) - (1 + 2^{n+1}))+ \{1, 2, 2^2, \cdots, 2^{n-4}\})\\ & = ((2^{n-1}+2^{n-2}+J_{n-2}) - (1 + 2^{n+1}))+ \{0, 1, 2, 2^2, \cdots, 2^{n-4}\}\\ & = (2^{n-1}+2^{n-2}+(J_{n-2}+ \{0, 1, 2, 2^2, \cdots, 2^{n-4}\})) - (1 + 2^{n+1})\\ & \supseteq (2^{n-1}+2^{n-2}+\{1, 2, 3, \cdots, 2^{n-2}\}) - (1 + 2^{n+1})\\ & = \{2^{n-1}+2^{n-2}+1, 2^{n-1}+2^{n-2}+2, 2^{n-1}+2^{n-2}+3, \cdots, 2^{n}\} - (1 + 2^{n+1})\end{aligned}$$ hold. This proves the fourth statement. So, the set $S + {\ensuremath{\mathcal{T}}}$ contains ${\ensuremath{\mathbb{Z}}}$. In the above proof, we established that for any $n\geq 3$, the left quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ is contained in $I_{n+1} + 2^{n}$, and the second quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ is contained in $I_n + 2^{n-1}$. In fact, the points of $I_{n+1}$ (which are precisely the points in the left quarter of the right quarter of ${\ensuremath{\mathcal{I}}}_{n+1}$) that yield the left quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ and the points of $I_n$ (which are precisely the points in the second quarter of the left half of ${\ensuremath{\mathcal{I}}}_n$) that yield the second quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ cannot be removed from $S$ in a sense made precise in the following result. \[Thm:RequiredClusters\] Let ${\ensuremath{\mathscr{S}}}, {\ensuremath{\mathscr{T}}}$ be nonempty subsets of $S, {\ensuremath{\mathcal{T}}}$ respectively such that ${\ensuremath{\mathscr{S}}}+ {\ensuremath{\mathscr{T}}}= {\ensuremath{\mathbb{Z}}}$. Then for $n\geq 3$, the set ${\ensuremath{\mathscr{T}}}$ contains $2^{n-1}$, ${\ensuremath{\mathscr{S}}}$ contains the points in the second quarter of the left half of ${\ensuremath{\mathcal{I}}}_n$, i.e., $$\label{Eqn:LeftHalf} {\ensuremath{\mathscr{S}}}\supseteq \{2^{n-3} + 1, 2^{n-3} + 2, 2^{n-3} + 3, \cdots, 2^{n-2}\} - (1 + 2^{n+1}) \quad \text{ for } n\geq 3,$$ and the points in the left quarter of the right quarter of ${\ensuremath{\mathcal{I}}}_{n+1}$, i.e., $$\label{Eqn:LeftQuarDeRightQuar} {\ensuremath{\mathscr{S}}}\supseteq \{2^{n}+ 2^{n-1} + 1, 2^{n}+ 2^{n-1} + 2, 2^{n}+ 2^{n-1} + 3, \cdots, 2^{n}+ 2^{n-1} + 2^{n-3}\} - (1 + 2^{n+2}) \quad \text{ for } n\geq 3.$$ Moreover, ${\ensuremath{\mathscr{T}}}$ contains $1, 2$, ${\ensuremath{\mathscr{S}}}$ contains $-2, -4$. Consequently, ${\ensuremath{\mathcal{T}}}$ is a minimal complement of $S$. We claim that for any $n\geq 2$, no point in the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $\cup _{m \neq n, n+1}I_m + {\ensuremath{\mathcal{T}}}$, i.e., $$\label{Eqn:RightHalfNotObtained} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup _{m \neq n, n+1}I_m + {\ensuremath{\mathcal{T}}}\right) = \emptyset \quad \text{ for } n\geq 2.$$ Since the inclusions $$\begin{aligned} (\cup _{0 \leq m < n} I_m) + {\ensuremath{\mathcal{T}}}& \subseteq (\cup _{0 \leq m < n} {\ensuremath{\mathcal{I}}}_m) + {\ensuremath{\mathcal{T}}}\\ & \subseteq \{- 2^n,-2^n+1, -2^n+2, -2^n + 3, \cdots, -1\} + {\ensuremath{\mathcal{T}}}\end{aligned}$$ hold and the elements of ${\ensuremath{\mathcal{T}}}$ are positive, it follows that the smallest element of $(\cup _{0 \leq m < n} I_m) + {\ensuremath{\mathcal{T}}}$ is greater than $-2^n$, which is greater than the largest element of $\{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1})$, and hence $$\label{Eqn:RightHalfNotObtainedRHS} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup _{0 \leq m < n}I_m + {\ensuremath{\mathcal{T}}}\right) = \emptyset \quad \text{ for } n\geq 1.$$ For any $n\geq 2$, the inclusion $I_n \subseteq {\ensuremath{\mathcal{I}}}_n = \{1, 2, 3, \cdots, 2^n\}- (1 + 2^{n+1})$ yields that $I_n \cap {\ensuremath{\mathbb{Z}}}_{\geq 1} = \emptyset, I_n \cap {\ensuremath{\mathbb{Z}}}_{\leq - 2^{n+1} - 1} = \emptyset$, and for any $m\geq n+2$, the inclusions $$\begin{aligned} I_m + 2^k & \subseteq {\ensuremath{\mathcal{I}}}_m + 2^k \\ & \subseteq (\{1, 2, 3, \cdots, 2^m\} - (1 + 2^{m+1})) + 2^k \\ & = \{1, 2, 3, \cdots, 2^m\} + 2^k- 2^{m+1} - 1\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1}\end{aligned}$$ hold for any $k\geq m+1$, and the inclusions $$\begin{aligned} I_m + 2^k & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1})} + 2^k \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1}) + 2^k } \\ & = {\ensuremath{\mathbb{Z}}}_{\leq -1-2^m + 2^k} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 1 - 2^m + 2^{m-1}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 1 - 2^{m-1}} \\ &\subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 2^{n+1} - 1}\end{aligned}$$ hold for $0 \leq k < m$, and hence $$\label{Eqn:RightHalfNotObtainedLHSSans1pt} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup_{m \geq n+2} (I_m + ({\ensuremath{\mathcal{T}}}\setminus \{2^m\})) \right) = \emptyset \quad \text{ for } n\geq 2.$$ For any $m\geq 3$, the inclusions $$\begin{aligned} I_m + 2^m & \subseteq (\{2^{m-3}+1, 2^{m-3}+2, 2^{m-3}+3, \cdots, 2^{m-2}\} \cup (2^{m-1} + 2^{m-2} + J_{m-2})) - (1 + 2^{m+1}) + 2^m \\ & \subseteq (J_m \cup (2^{m-1} + 2^{m-2} + J_{m-2})) - (1 + 2^m) \\ &\subseteq (J_m \cup (2^{m-1} + J_{m-1})) - (1 + 2^m) \\ &= J_m - (1 + 2^m) \\ & = (\{1, 2, \cdots, 2^{m-2}\} - (1 + 2^m)) \cup \cdots \cup (\{1, 2, 3, 2^2\} - (1+2^4)) \cup (\{1, 2\} - (1+2^3)) \\ & \qquad \cup (\{1\} - (1+2^2)) \cup (\{1\} - ( 1+ 2)) \\ & = \left( \cup_{k=0}^{m-2} (\{1, 2, 3, \cdots, 2^k\} - (1 + 2^{k+2})) \right) \cup (\{1\}-(1+2))\end{aligned}$$ hold, which implies that for $n\geq 2$ and $m\geq n+1$, the inclusions $$\begin{aligned} (I_m + 2^m) \cap {\ensuremath{\mathcal{I}}}_n & = (I_m + 2^m) \cap (\{1, 2, 3, \cdots, 2^n\} - (1 + 2^{n+1}))\\ & \subseteq \left( \left( \cup_{k=0}^{m-2} (\{1, 2, 3, \cdots, 2^k\} - (1 + 2^{k+2})) \right) \cup (\{1\}-(1+2)) \right) \cap {\ensuremath{\mathcal{I}}}_n \\ & = \left( \left( \cup_{k=1}^{m-1} (\{1, 2, 3, \cdots, 2^{k-1}\} - (1 + 2^{k+1})) \right) \cup (\{1\}-(1+2)) \right) \cap {\ensuremath{\mathcal{I}}}_n \\ & = \{1, 2, 3, \cdots, 2^{n-1}\} - (1 + 2^{n+1}),\end{aligned}$$ hold, which yields $$\label{Eqn:RightHalfNotObtainedLHS1pt} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup_{m \geq n+1} (I_m + 2^m) \right) = \emptyset \quad \text{ for } n\geq 2.$$ Consequently, Equation follows from Equations , , . We claim that for any $n\geq 3$, no point in the second quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $I_{n+1} + {\ensuremath{\mathcal{T}}}$, i.e., $$\label{Eqn:2ndQuarDeRightHalfNotObtained} \left( \{ 2^{n-1}+2^{n-3}+1, 2^{n-1}+2^{n-3}+2, 2^{n-1}+2^{n-3}+3,\cdots, 2^{n-1} + 2^{n-2} \} - (1 + 2^{n+1}) \right) \cap \left( I_{n+1} + {\ensuremath{\mathcal{T}}}\right) = \emptyset.$$ This claim follows since for $n\geq 3$, the inclusions $$\begin{aligned} I_{n+1} + 2^k & \subseteq {\ensuremath{\mathcal{I}}}_{n+1} + 2^k\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n+1} - (1 + 2^{n+2}) + 2^k} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^k- (1 + 2^{n+1})}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1}- (1 + 2^{n+1})}\end{aligned}$$ hold for $0 \leq k \leq n-1$, the inclusions $$\begin{aligned} I_{n+1} + 2^n & = K_{n+1} - (1 + 2^{n+2}) + 2^n \\ & \subseteq \left( \{2^{n-2}+1, 2^{n-2}+2, \cdots, 2^{n-1}\} \cup (2^n + 2^{n-1} + J_{n-1}) \right) - (1 + 2^{n+2}) + 2^n \\ & = \left( \{2^{n-2}+1, 2^{n-2}+2, \cdots, 2^{n-1}\} - (1 + 2^{n+2}) + 2^n \right) \cup \left( (2^n + 2^{n-1} + J_{n-1}) - (1 + 2^{n+2}) + 2^n \right) \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} - (1 + 2^{n+2}) + 2^n} \cup \left( (2^n + 2^{n-1} + J_{n-1}) - (1 + 2^{n+2}) + 2^n \right) \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n} - (1 + 2^{n+2}) + 2^n} \cup \left( (2^n + 2^{n-1} + \left( \{1, 2, \cdots, 2^{n-3}\} \cup (2^{n-2} + J_{n-2}) \right) ) - (1 + 2^{n+2}) + 2^n \right) \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - (1 + 2^{n+1}) } \cup {\ensuremath{\mathbb{Z}}}_{\leq 2^n + 2^{n-1} + 2^{n-3} - (1 + 2^{n+2}) + 2^n} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+2}) + 2^n} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n} - (1 + 2^{n+2}) + 2^n} \cup \left( (2^n + 2^{n-1} + \left( \{1, 2, \cdots, 2^{n-3}\} \cup (2^{n-2} + J_{n-2}) \right) ) - (1 + 2^{n+2}) + 2^n \right) \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - (1 + 2^{n+1}) } \cup {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-3} - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})}\end{aligned}$$ hold, and the inclusions $$\begin{aligned} I_{n+1} + 2^k & \subseteq {\ensuremath{\mathcal{I}}}_{n+1} + 2^k \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+2}) + 2^k}\\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^k - 2^{n+2}}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 0}\end{aligned}$$ hold for $k \geq n+2$. We claim that for any $n\geq 3$, no point in the second quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $I_n + ({\ensuremath{\mathcal{T}}}\setminus \{2^{n-1}\})$, i.e., $$\label{Eqn:2ndQuarDeRightHalfNotObtainedIn} \left( \{ 2^{n-1}+2^{n-3}+1, 2^{n-1}+2^{n-3}+2, \cdots, 2^{n-1} + 2^{n-2} \} - (1 + 2^{n+1}) \right) \cap \left( I_n + ({\ensuremath{\mathcal{T}}}\setminus \{2^{n-1}\}) \right) = \emptyset.$$ This claim follows since for $n\geq 3$, $$I_n = (\{ 2^{n-3}+1, 2^{n-3}+2, 2^{n-3}+3, \cdots, 2^{n-2} \} - (1 + 2^{n+1})) \cup ((2^{n-1}+ 2^{n-2} + J_{n-2}) - (1 + 2^{n+1})),$$ the inclusions $$\begin{aligned} ((2^{n-1}+ 2^{n-2} + J_{n-2}) - (1 + 2^{n+1})) + {\ensuremath{\mathscr{T}}}& \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1}+2^{n-2}+ 1 - (1 + 2^{n+1})} + {\ensuremath{\mathbb{Z}}}_{\geq 1}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1}+2^{n-2}+ 1 - (1 + 2^{n+1})} \end{aligned}$$ hold, the inclusions $$\begin{aligned} & (\{2^{n-3}+1, 2^{n-3}+2, 2^{n-3}+3, \cdots, 2^{n-2}\} - (1 + 2^{n+1}))+ 2^k \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3} + 1 - (1 + 2^{n+1}) + 2^k}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3} + 1 - (1 + 2^{n+1}) + 2^n}\\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3} + 1 - (1 + 2^{n+1}) + 2^{n-1} + 2^{n-2} + 2^{n-2} }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})} \end{aligned}$$ hold for $k\geq n$, and the inclusions $$\begin{aligned} & (\{2^{n-3}+1, 2^{n-3}+2, 2^{n-3}+3, \cdots, 2^{n-2}\} - (1 + 2^{n+1}))+ 2^k \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1}) + 2^k}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1}) + 2^{n-2}}\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} - (1 + 2^{n+1}) }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-3} - (1 + 2^{n+1}) }\end{aligned}$$ hold for $k\leq n-2$. We now show that Equation holds. From Equations , , it follows that $$({\ensuremath{\mathscr{S}}}\cap I_n) + {\ensuremath{\mathscr{T}}}\supseteq \{ 2^{n-1}+2^{n-3}+1, 2^{n-1}+2^{n-3}+2, 2^{n-1}+2^{n-3}+3,\cdots, 2^{n-1} + 2^{n-2} \} - (1 + 2^{n+1}).$$ Using Equation , it follows that $2^{n-1} \in {\ensuremath{\mathscr{T}}}$ and $$({\ensuremath{\mathscr{S}}}\cap I_n) + 2^{n-1} \supseteq \{ 2^{n-1}+2^{n-3}+1, 2^{n-1}+2^{n-3}+2, 2^{n-1}+2^{n-3}+3,\cdots, 2^{n-1} + 2^{n-2} \} - (1 + 2^{n+1}).$$ Thus Equation holds. We claim that for any $n\geq 3$, no point in the left quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $I_n+ {\ensuremath{\mathcal{T}}}$, i.e., $$\label{Eqn:1stQuarDeRightHalfNotObtainedIn} \left( \{ 2^{n-1} + 1, 2^{n-2} + 2, \cdots, 2^{n-1}+2^{n-3} \} - (1 + 2^{n+1}) \right) \cap \left( I_n + {\ensuremath{\mathcal{T}}}\right) = \emptyset \quad \text{ for } n\geq 3.$$ This claim follows since for $n\geq 3$, the inclusions $$\begin{aligned} I_n + 2^k & \subseteq K_n - (1 + 2^{n+1}) + 2^k \\ & \subseteq \left( \{2^{n-3} + 1, 2^{n-3} + 2, \cdots, 2^{n-2} \} \cup (2^{n-1} + 2^{n-2} + J_{n-2} ) \right) - (1 + 2^{n+1}) + 2^k \\ & \subseteq \left( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2}} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1} \right) - (1 + 2^{n+1}) + 2^k \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1}) + 2^k} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1}) + 2^k} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1}) + 2^{n-2}} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1}) } \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} - (1 + 2^{n+1}) } \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1}) } \end{aligned}$$ hold for $k\leq n-2$, and the inclusions $$\begin{aligned} I_n + 2^k & = K_n - (1 + 2^{n+1}) + 2^k \\ & \subseteq \left( \{2^{n-3} + 1, 2^{n-3} + 2, \cdots, 2^{n-2} \} \cup (2^{n-1} + 2^{n-2} + J_{n-2} ) \right) - (1 + 2^{n+1}) + 2^k \\ & \subseteq \left( {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3}+1} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1} \right) - (1 + 2^{n+1}) + 2^k \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3}+1} - (1 + 2^{n+1}) + 2^k \\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3}+1 - (1 + 2^{n+1}) + 2^k} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3}+1 - (1 + 2^{n+1}) + 2^{n-1}} \\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-3}+1 - (1 + 2^{n+1}) } \end{aligned}$$ hold for $k\geq n-1$. We claim that for any $n\geq 3$, no point in the left quarter of the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $I_{n+1} + ({\ensuremath{\mathcal{T}}}\setminus \{2^n\})$, i.e., $$\label{Eqn:1stQuarDeRightHalfNotObtained} \left( \{ 2^{n-1} + 1, 2^{n-1} + 2, \cdots, 2^{n-1}+2^{n-3} \} - (1 + 2^{n+1}) \right) \cap \left( I_{n+1} + ({\ensuremath{\mathcal{T}}}\setminus \{2^n\}) \right) = \emptyset.$$ This claim follows since for $n\geq 3$, the inclusions $$\begin{aligned} I_{n+1} + 2^k & \subseteq {\ensuremath{\mathcal{I}}}_{n+1} + 2^k \\ & = (\{1, 2, 3, \cdots, 2^{n+1} \} - ( 1+ 2^{n+2} ) ) + 2^k \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n+2} ) + 2^k} \\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^k - 2^{n+2} }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 0}\end{aligned}$$ hold for $k \geq n+2$, the inclusions $$\begin{aligned} I_{n+1} + 2^{n+1} & = K_{n+1} - (1 + 2^{n+2}) + 2^{n+1} \\ & = \left( \{2^{n-2} + 1, 2^{n-2} + 2, \cdots, 2^{n-1} \} \cup (2^{n} + 2^{n-1} + J_{n-1} ) \right) - (1 + 2^{n+2}) + 2^{n+1} \\ & = \left( \{2^{n-2} + 1, 2^{n-2} + 2, \cdots, 2^{n-1} \} \cup (2^{n} + 2^{n-1} + J_{n-1} ) \right) - (1 + 2^{n+1}) \\ & \subseteq \left( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} } \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 2^{n-1} + 1} \right) - (1 + 2^{n+1}) \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} - (1 + 2^{n+1}) } \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n +1 - (1 + 2^{n+1}) }\end{aligned}$$ hold, and the inclusions $$\begin{aligned} I_{n+1} + 2^k & \subseteq {\ensuremath{\mathcal{I}}}_{n+1} + 2^k \\ & = (\{1, 2, 3, \cdots, 2^{n+1} \} - ( 1+ 2^{n+2} ) ) + 2^k \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n+1} - ( 1+ 2^{n+2} ) + 2^k} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^k - ( 1+ 2^{n+1} )} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} - ( 1+ 2^{n+1} )} \end{aligned}$$ hold for $k\leq n-1$. We now show that Equation holds. From Equations , , it follows that $$({\ensuremath{\mathscr{S}}}\cap I_{n+1}) + {\ensuremath{\mathscr{T}}}\supseteq \{ 2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3,\cdots, 2^{n-1} + 2^{n-3} \} - (1 + 2^{n+1}).$$ Using Equation , it follows that $2^n \in {\ensuremath{\mathscr{T}}}$ and $$({\ensuremath{\mathscr{S}}}\cap I_{n+1}) + 2^n \supseteq \{ 2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3,\cdots, 2^{n-1} + 2^{n-3} \} - (1 + 2^{n+1}).$$ So the inclusions $$\begin{aligned} {\ensuremath{\mathscr{S}}}& \supseteq \{ 2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3,\cdots, 2^{n-1} + 2^{n-3} \} - (1 + 2^{n+1}) - 2^n\\ & = \{ 2^n + 2^{n-1}+1, 2^n + 2^{n-1}+2, 2^n + 2^{n-1}+3,\cdots, 2^n + 2^{n-1} + 2^{n-3} \} - (1 + 2^{n+1}) - 2^n - 2^n \\ & = \{ 2^n + 2^{n-1}+1, 2^n + 2^{n-1}+2, 2^n + 2^{n-1}+3,\cdots, 2^n + 2^{n-1} + 2^{n-3} \} - (1 + 2^{n+2}).\end{aligned}$$ hold, which establishes Equation . Note that $S + ({\ensuremath{\mathcal{T}}}\setminus \{1\})$ does not contain $-1$. It follows that $a = 0$. So $1\in {\ensuremath{\mathscr{T}}}$ and $-2\in {\ensuremath{\mathscr{S}}}$. The integer $-5 - (1 + 2^5)$ does not lie in $S + ({\ensuremath{\mathcal{T}}}\setminus\{2\})$. Hence ${\ensuremath{\mathscr{T}}}$ contains $2$. Note that $(S \setminus \{-4\}) + {\ensuremath{\mathcal{T}}}$ does not contain $-3$. It follows that $-4\in {\ensuremath{\mathscr{S}}}$. Consequently, ${\ensuremath{\mathcal{T}}}$ is a minimal complement of $S$. By Theorem \[Thm:RequiredClusters\], it follows that ${\ensuremath{\mathcal{T}}}$ is a minimal complement of $S$. However, it turns out that $S$ is not a minimal complement of ${\ensuremath{\mathcal{T}}}$. In fact, $$S \setminus\{2^{2n} + 2^{2n-1} + 2^{2n-2} + \cdots + 2^4 + 2^3 + \frac{2^2}{2} - (1 + 2^{2n+2})\,|\, n\geq 3\}$$ is also an additive complement of ${\ensuremath{\mathcal{T}}}$. In the following result, we prove that $S$ contains a subset ${\ensuremath{\mathcal{S}}}$ such that ${\ensuremath{\mathcal{S}}}$ is a minimal complement of ${\ensuremath{\mathcal{T}}}$. \[Thm:CoMinS\] Let $S = \{ -x_1, - x_2, -x_3, \cdots\}$ where $x_1 < x_2 < x_3< \cdots$. Define $S_1 = S$ and for each positive integer $i\geq 1$, define $$S_{i+1} : = \begin{cases} S_i \setminus \{-x_i\} & \text{ if $S_i \setminus \{-x_i\}$ is a complement to ${\ensuremath{\mathcal{T}}}$,}\\ S_i & \text{ otherwise.} \end{cases}$$ Let ${\ensuremath{\mathcal{S}}}$ denote the subset $\cap _{i \geq 1} S_i$ of ${\ensuremath{\mathbb{Z}}}$. The subsets ${\ensuremath{\mathcal{S}}}$ and ${\ensuremath{\mathcal{T}}}$ of ${\ensuremath{\mathbb{Z}}}$ form a co-minimal pair. By Theorem \[Thm:RequiredClusters\], for $i\geq 1$, $S_i$ contains the points in the second quarter of the left half of ${\ensuremath{\mathcal{I}}}_n$, i.e., $$S_i \supseteq \{2^{n-3} + 1, 2^{n-3} + 2, 2^{n-3} + 3, \cdots, 2^{n-2}\} - (1 + 2^{n+1}) \quad \text{ for } n\geq 3,$$ thus ${\ensuremath{\mathcal{S}}}$ contains these points and hence ${\ensuremath{\mathbb{Z}}}_{\geq 1}$ is contained in ${\ensuremath{\mathcal{S}}}+ {\ensuremath{\mathcal{T}}}$. By Theorem \[Thm:RequiredClusters\], for any $i\geq 1$, $S_i$ contains $-2, -4$. So $\{-3, -2, -1, 0\} \in {\ensuremath{\mathcal{S}}}+ {\ensuremath{\mathcal{T}}}$. By Theorem \[Thm:RequiredClusters\], for any $i\geq 1$, $S_i$ contains the points in the left quarter of the right quarter of ${\ensuremath{\mathcal{I}}}_{n}$, i.e., $$S_i \supseteq \{2^{n-1}+ 2^{n-2} + 1, 2^{n-1}+ 2^{n-2} + 2, 2^{n-1}+ 2^{n-2} + 3, \cdots, 2^{n-1}+ 2^{n-2} + 2^{n-4}\} - (1 + 2^{n+1}) \quad \text{ for } n\geq 4,$$ thus ${\ensuremath{\mathcal{S}}}$ contains these points and hence the left half of ${\ensuremath{\mathcal{I}}}_n$ is contained in ${\ensuremath{\mathcal{S}}}+ {\ensuremath{\mathcal{T}}}$ for any $n\geq 1$. It remains to show that the second half of ${\ensuremath{\mathcal{I}}}_n$ is also contained in ${\ensuremath{\mathcal{S}}}+ {\ensuremath{\mathcal{T}}}$ for any $n\geq 2$. For any element $y\in {\ensuremath{\mathbb{Z}}}_{\leq -1}$ that lie in the right half of some ${\ensuremath{\mathcal{I}}}_n$ for some $n\geq 2$ and for any $i\geq 1$, there exist elements $s_{y, i}\in S_i , t_{y, i}\in {\ensuremath{\mathcal{T}}}$ such that $y = s_{y, i} + t_{y, i}$. By Equation , it follows that for some element $s_y \in S$, the equality $s_y = s_{y, i}$ holds for infinitely many $i$ and for such integers $i$, we have $t_y = t_{y, i}$ where $t_y: = y - s_y\in {\ensuremath{\mathcal{T}}}$. Thus $y - t_y = s_{y, i}$ holds for infinitely many $i$. Hence, for each integer $i\geq 1$, there exists an integer $m_i \geq i$ such that $y - t_y = s_{y, m_i}$, which yields $y\in t_y + S_{m_i} \subseteq t_y + S_i$. Thus $y$ lies in $t_y + {\ensuremath{\mathcal{S}}}$. This proves that the second half of ${\ensuremath{\mathcal{I}}}_n$ is also contained in ${\ensuremath{\mathcal{S}}}+ {\ensuremath{\mathcal{T}}}$ for any $n\geq 2$. Hence ${\ensuremath{\mathcal{S}}}$ is an additive complement of ${\ensuremath{\mathcal{T}}}$. It follows that ${\ensuremath{\mathcal{S}}}$ is a minimal complement to ${\ensuremath{\mathcal{T}}}$. By Theorem \[Thm:RequiredClusters\], it follows that ${\ensuremath{\mathcal{T}}}$ is a minimal complement to ${\ensuremath{\mathcal{S}}}$. This proves that $({\ensuremath{\mathcal{S}}}, {\ensuremath{\mathcal{T}}})$ is a co-minimal pair. A co-minimal pair involving an infinite symmetric subset ======================================================== In this section, we establish Theorem \[Thm:CoMinUV\], which follows from Theorem \[Thm:CoMinU\]. Consider the subsets $\{{\ensuremath{\mathcal{I}}}_n\}_{n\geq 0}, \{U_n\}_{n\geq 0}$ of ${\ensuremath{\mathbb{Z}}}$ defined by $$\begin{aligned} {\ensuremath{\mathcal{I}}}_n & = \begin{cases} \{-2, -1\} & \text{ if } n = 0, \\ \{1, 2, 3, \cdots, 2^n\} - (1+2^{n+1}) & \text{ for } n \geq 1, \end{cases} \\ U_n & = \begin{cases} \{-2, -1\} & \text{ if } n = 0,\\ \emptyset & \text{ if } n = 1, 2, \\ \{6\} - ( 1+ 2^4) & \text{ if } n = 3,\\ \{3, 4, 14\} - ( 1+ 2^5) & \text{ if } n = 4,\\ ((2^{n-3} + \{1,2, 3, \cdots, 2^{n-3}\}) \cup (2^{n-1}+2^{n-2}+\{1,2, 3, \cdots, 2^{n-4}\}))-(1 + 2^{n+1}) & \text{ if } n\geq 5. \end{cases} \end{aligned}$$ Denote the union $\cup _{n\geq 0} U_n$ by $U$. \[Prop:VAddComp\] The set ${\ensuremath{\mathcal{V}}}$ is an additive complement of $U$ in ${\ensuremath{\mathbb{Z}}}$. Since the inclusion $$U_0 + \{2\} \supseteq \{0, 1\}$$ holds and the inclusions $$\begin{aligned} \cup_{k = 4}^m (U_k + 2^{k+1}) & \supseteq \cup_{k = 4}^m (2^{k-3} + \{0, 1,2, \cdots, 2^{k-3}-1\})\\ & \supseteq \{2, 3, 4, \cdots, 2^{m-2}-1\}\end{aligned}$$ hold for any $m\geq 4$, it follows that $U + {\ensuremath{\mathcal{V}}}$ contains ${\ensuremath{\mathbb{Z}}}_{\geq 0}$. Note that $$U_n \subseteq {\ensuremath{\mathcal{I}}}_n\quad \text{ for all }n\geq 0,$$ and $$\bigcup_{n\geq 0} {\ensuremath{\mathcal{I}}}_n = {\ensuremath{\mathbb{Z}}}_{\leq -1},$$ and the sets ${\ensuremath{\mathcal{I}}}_n$ lie next to each other in the sense that $$\min {\ensuremath{\mathcal{I}}}_n = 1 + \max {\ensuremath{\mathcal{I}}}_{n+1} \quad \text{ for all }n\geq 0.$$ Thus, to prove that $U+{\ensuremath{\mathcal{V}}}$ is equal to ${\ensuremath{\mathbb{Z}}}$, it remains to show that $U + {\ensuremath{\mathcal{V}}}$ contains ${\ensuremath{\mathcal{I}}}_n$ for all $n\geq 0$. The left half of ${\ensuremath{\mathcal{I}}}_n$ is contained in $U_{n+3} + 2^{n+3}$ for any $n\geq 1$, i.e., the inclusions $$\begin{aligned} U_{n+3} + 2^{n+3} & \supseteq (2^{n+2} + 2^{n+1} + \{1, 2, 3, \cdots, 2^{n-1}\}) - ( 1 + 2^{n+4}) + 2^{n+3}\\ & = (\{1, 2, 3, \cdots, 2^{n-1}\} - (1 + 2^{n+1})) + (1 + 2^{n+1}) + 2^{n+2} + 2^{n+1}- ( 1 + 2^{n+4}) + 2^{n+3}\\ & = \{1, 2, 3, \cdots, 2^{n-1}\} - (1 + 2^{n+1})\end{aligned}$$ hold for any $n\geq 1$. The third quarter of ${\ensuremath{\mathcal{I}}}_n$ is contained in the set $(U_{n+1} + 2^n )\cup (U_n + 2^{n-1})$ for $n\geq 4$, i.e., the inclusions $$\begin{aligned} (U_{n+1} + 2^n )\cup (U_n + 2^{n-1}) & \supseteq \left((2^n + 2^{n-1} + \{1, 2, 3, \cdots, 2^{n-3}\} ) - (1 + 2^{n+2})+ 2^n \right) \\ & \qquad \cup \left( (2^{n-3} + \{1, 2, 3, \cdots, 2^{n-3}\}) - (1 + 2^{n+1}) + 2^{n-1} \right)\\ & \supseteq ((2^{n-1} + \{1, 2, 3, \cdots, 2^{n-3}\} ) - (1 + 2^{n+1})) \\ & \qquad \cup ( (2^{n-1}+2^{n-3} + \{1, 2, 3, \cdots, 2^{n-3}\}) - (1 + 2^{n+1}))\\ & \supseteq (2^{n-1} + \{1, 2, 3, \cdots, 2^{n-2}\} ) - (1 + 2^{n+1})\end{aligned}$$ hold for $n\geq 4$. The left half of the right quarter of ${\ensuremath{\mathcal{I}}}_4$ is contained in the set $(U_3 + (-2^3))\cup (U_4 + (-1))$, and the left half of the right quarter of ${\ensuremath{\mathcal{I}}}_n$ is contained in the set $(U_n + (-2^{n-5})) \cup (U_n + 2^{n-5}) \cup (U_n + 2^{n-4})$ for $n\geq 5$, i.e., the inclusions $$\begin{aligned} & (U_n + (-2^{n-5})) \cup (U_n + 2^{n-5}) \cup (U_n + 2^{n-4})\\ & \supseteq ((2^{n-1} + 2^{n-2} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1})) + \{-2^{n-5}, 2^{n-5}, 2^{n-4} \} \\ & \supseteq \left( ((2^{n-1} + 2^{n-2} + \{2^{n-5}+1,2^{n-5}+ 2,2^{n-5}+ 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1})) -2^{n-5} \right)\\ & \qquad \cup \left( ((2^{n-1} + 2^{n-2} + \{1, 2, 3, \cdots, 2^{n-5}\}) - (1 + 2^{n+1})) + 2^{n-5} \right)\\ & \qquad \cup \left( ((2^{n-1} + 2^{n-2} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1})) + 2^{n-4} \right)\\ & \supseteq ((2^{n-1} + 2^{n-2} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1})) \\ & \qquad \cup ((2^{n-1} + 2^{n-2} + 2^{n-4} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1})) \\ & \supseteq (2^{n-1} + 2^{n-2} + \{1, 2, 3, \cdots, 2^{n-3}\}) - (1 + 2^{n+1}),\end{aligned}$$ hold for $n\geq 5$. The right half of the right quarter of ${\ensuremath{\mathcal{I}}}_4$ is contained in the set $U_0 + (-2^4)$, and the right half of the right quarter of ${\ensuremath{\mathcal{I}}}_n$ is contained in the set $(U_n + 2^{n-3}) \cup (U_{n-1} - 2^{n-3} )$ for $n\geq 5$, i.e., the inclusions $$\begin{aligned} & (U_n + 2^{n-3}) \cup (U_{n-1} - 2^{n-3} ) \\ & \supseteq \left( ((2^{n-1} + 2^{n-2} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1})) + 2^{n-3} \right) \\ & \qquad \cup \left( ((2^{n-4} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^n) ) - 2^{n-3} \right) \\ & = \left(( (2^{n-1} + 2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1}) \right) \\ & \qquad \cup \left( ((2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^n) ) - 2^{n-3} - 2^{n-1} - 2^{n-2} - 2^{n-3} \right)\\ & = \left(( (2^{n-1} + 2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1}) \right) \\ & \qquad \cup \left( (2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} + \{1, 2, 3, \cdots, 2^{n-4}\}) - (1 + 2^{n+1}) \right)\\ & = (2^{n-1} + 2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-3}\}) - (1 + 2^{n+1})\end{aligned}$$ hold for $n\geq 5$. This proves that ${\ensuremath{\mathcal{I}}}_n$ is contained in $U + {\ensuremath{\mathcal{V}}}$ for $n\geq 4$, and hence $U + {\ensuremath{\mathcal{V}}}$ contains ${\ensuremath{\mathbb{Z}}}_{\leq 2^4 - (1 + 2^5)} = {\ensuremath{\mathbb{Z}}}_{\leq - 1 - 2^4}$. Also note that the left half of ${\ensuremath{\mathcal{I}}}_1, {\ensuremath{\mathcal{I}}}_2, {\ensuremath{\mathcal{I}}}_3$ is contained in $U + {\ensuremath{\mathcal{V}}}$. From the inclusions $$\begin{aligned} & (U_4 \cup U_3 \cup U_0) + \{-4, -2, -1, 1, 2, 4, 8\}\\ & \supseteq \left( (U_4 + \{8\}) \cup (U_3 + \{-1, 1, 2\} ) \right) \cup (U_0 + \{-4\}) \cup (U_0 + \{-2\}) \cup (U_0 + \{-1, 1\}) \\ & \supseteq (\{5, 6, 7, 8\} - ( 1 + 2^4)) \cup (\{3, 4\} - (1+ 2^3)) \cup (\{1, 2\} - ( 1+ 2^2)) \cup \{-2, -1\}, \end{aligned}$$ it follows that the subsets ${\ensuremath{\mathcal{I}}}_0, {\ensuremath{\mathcal{I}}}_1$ and the right half of ${\ensuremath{\mathcal{I}}}_2, {\ensuremath{\mathcal{I}}}_3$ is contained in $U + {\ensuremath{\mathcal{V}}}$. Hence ${\ensuremath{\mathcal{V}}}$ is an additive complement of $U$. \[Thm:RequiredClustersV\] The set ${\ensuremath{\mathcal{V}}}$ is a minimal complement of $U$ in ${\ensuremath{\mathbb{Z}}}$. By Proposition \[Prop:VAddComp\], ${\ensuremath{\mathcal{V}}}$ is an additive complement of $U$. To prove that ${\ensuremath{\mathcal{V}}}$ is a minimal complement of $U$ in ${\ensuremath{\mathbb{Z}}}$, it suffices to prove that if ${\ensuremath{\mathscr{V}}}$ is nonempty subset of ${\ensuremath{\mathcal{V}}}$ satisfying $U + {\ensuremath{\mathscr{V}}}= {\ensuremath{\mathbb{Z}}}$, then ${\ensuremath{\mathscr{V}}}= {\ensuremath{\mathcal{V}}}$, which follows from the nine claims below. Indeed, the third claim below implies that $2^n\in {\ensuremath{\mathscr{V}}}$ for all $n\geq 2$, the fourth (resp. fifth, sixth, seventh, eighth) claim implies that ${\ensuremath{\mathscr{V}}}$ contains $2$ (resp. $1, -1, -2, -4$), and the ninth claim implies that ${\ensuremath{\mathscr{V}}}$ contains $-2^n$ for all $n\geq 3$. Thus, from the following claims, it follows that ${\ensuremath{\mathscr{V}}}= {\ensuremath{\mathcal{V}}}$ and hence ${\ensuremath{\mathcal{V}}}$ is a minimal complement to $U$. We claim the following. 1. For any $n\geq 2$, no point in the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $\cup _{m \geq n+2}U_m + {\ensuremath{\mathcal{V}}}$, i.e., $$\label{Eqn:RightHalfNotObtainedUV} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup _{m \geq n+2}U_m + {\ensuremath{\mathcal{V}}}\right) = \emptyset \quad \text{ for } n\geq 2.$$ 2. For any $n\geq 3$, no point in the right half of ${\ensuremath{\mathcal{I}}}_n$ lie in $U_{n+1} + ({\ensuremath{\mathcal{V}}}\setminus\{2^n\})$, i.e., $$\label{Eqn:RightHalfNotObtainedLHS1ptUVbis} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( U_{n+1} + ({\ensuremath{\mathcal{V}}}\setminus \{ 2^n\} ) \right) = \emptyset \quad \text{ for } n\geq 3.$$ 3. For any $n\geq 5$, the largest element of the third quarter of the right quarter of ${\ensuremath{\mathcal{I}}}_n$ does not lie in $U + ({\ensuremath{\mathcal{V}}}\setminus \{2^{n-3} \})$, more precisely, $$\label{Eqn:VContainsPositifPowersOf2} 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - (1 + 2^{n+1}) \notin (U_n + ({\ensuremath{\mathcal{V}}}\setminus\{2^{n-3}\})) \cup (\cup_{m \geq 0, m \neq n} U_m + {\ensuremath{\mathcal{V}}}) \quad \text{ for } n\geq 5.$$ 4. $$\label{Eqn:VContains2} 1 \notin ((U \setminus \{-1\}) + {\ensuremath{\mathcal{V}}}) \cup (-1 + ({\ensuremath{\mathcal{V}}}\setminus \{2\})) .$$ 5. $$\label{Eqn:VContains1} 26 - (1 + 2^6) \notin ((U \setminus \{25 - (1+2^6)\}) + {\ensuremath{\mathcal{V}}}) \cup (25 - (1 + 2^6) + ({\ensuremath{\mathcal{V}}}\setminus \{1\})).$$ 6. $$\label{Eqn:VContainsMinus1} 25 - (1 + 2^6) \notin ((U \setminus \{26 -(1+2^6)\}) + {\ensuremath{\mathcal{V}}}) \cup (26 - (1 + 2^6) + ({\ensuremath{\mathcal{V}}}\setminus \{-1\})).$$ 7. $$\label{Eqn:VContainsMinus2} -4 \notin ((U \setminus \{-2\}) + {\ensuremath{\mathcal{V}}}) \cup (-2 + ({\ensuremath{\mathcal{V}}}\setminus \{-2\})) .$$ 8. $$\label{Eqn:VContainsMinus4} -6 \notin ((U \setminus \{-2\}) + {\ensuremath{\mathcal{V}}}) \cup (-2 + ({\ensuremath{\mathcal{V}}}\setminus \{-4\})) .$$ 9. For any $n\geq 6$, the third largest element of ${\ensuremath{\mathcal{I}}}_n$, i.e., the integer $2^n - 2 - (1 + 2^{n+1})$ does not belong to $U + ({\ensuremath{\mathcal{V}}}\setminus \{-2^{n-3}\})$, i.e., $$\label{Eqn:VContainsNegativePowersOf2} 2^n - 1 - (1 + 2^{n+1}) \notin U + ({\ensuremath{\mathcal{V}}}\setminus \{-2^{n-3}\}).$$ First, we establish Equation . For any $n\geq 2$, the inclusion $U_n \subseteq {\ensuremath{\mathcal{I}}}_n = \{1, 2, 3, \cdots, 2^n\}- (1 + 2^{n+1})$ yields that $$U_n \cap {\ensuremath{\mathbb{Z}}}_{\geq 0} = \emptyset, U_n \cap {\ensuremath{\mathbb{Z}}}_{\leq - 2^{n+1} - 1} = \emptyset,$$ and for any $m\geq n+2$ with $n\geq 2$, the inclusions $$\begin{aligned} U_m + v & \subseteq {\ensuremath{\mathcal{I}}}_m + v \\ & \subseteq (\{1, 2, 3, \cdots, 2^m\} - (1 + 2^{m+1})) + v \\ & = \{1, 2, 3, \cdots, 2^m\} + v- 2^{m+1} - 1\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 0}\end{aligned}$$ hold for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \geq 2^{m+1}$, and the inclusions $$\begin{aligned} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1})} + v \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1}) + v } \\ & = {\ensuremath{\mathbb{Z}}}_{\leq -1-2^m + v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 1 - 2^m + 2^{m-1}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 1 - 2^{m-1}} \\ &\subseteq {\ensuremath{\mathbb{Z}}}_{\leq - (1+2^{n+1})}\end{aligned}$$ hold for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \leq 2^{m-1}$, and hence $$\label{Eqn:RightHalfNotObtainedLHSSans1ptUV} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup_{m \geq n+2} (U_m + ({\ensuremath{\mathcal{V}}}\setminus \{2^m\})) \right) = \emptyset \quad \text{ for } n\geq 2.$$ For any $m\geq 4$, the inclusions $$\begin{aligned} U_m + 2^m & \subseteq ((2^{m-3} + \{1,2, 3, \cdots, 2^{m-3}\}) \cup (2^{m-1}+2^{m-2}+\{1,2, 3, \cdots, 2^{m-4}\}))-(1 + 2^{m+1}) + 2^m\\ & = ((2^{m-3} + \{1,2, 3, \cdots, 2^{m-3}\}) \cup (2^{m-1}+2^{m-2}+\{1,2, 3, \cdots, 2^{m-4}\}))-(1 + 2^m)\\ & \subseteq (\{1,2, 3, \cdots, 2^{m-2}\} - (1 + 2^m)) \cup (2^{m-1}+2^{m-2}+\{1,2, 3, \cdots, 2^{m-4}\}-(1 + 2^m))\\ & = (\{1,2, 3, \cdots, 2^{m-2}\} - (1 + 2^m)) \cup (\{1,2, 3, \cdots, 2^{m-4}\}-(1 + 2^{m-2}))\end{aligned}$$ hold, which implies that for $n\geq 3$ and $m\geq n+1$, the set $U_m + 2^m$ is contained in the union of the left half of ${\ensuremath{\mathcal{I}}}_{m-1}$ and the left half of ${\ensuremath{\mathcal{I}}}_{m-3}$, and hence $$\label{Eqn:RightHalfNotObtainedLHS1ptUV} \left( \{2^{n-1}+1, 2^{n-1}+2, 2^{n-1}+3, \cdots, 2^n\} - (1 + 2^{n+1}) \right) \cap \left( \cup_{m \geq n+1} (U_m + 2^m) \right) = \emptyset \quad \text{ for } n\geq 3.$$ From Equations , , Equation follows. Now, we prove that Equation holds. For any $m\geq 4$, the inclusions $$\begin{aligned} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{m+1})} + v \\ & = {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{m+1}) + v }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{m+1}) + 2^{m+1} }\\ & = {\ensuremath{\mathbb{Z}}}_{\geq 0}\end{aligned}$$ hold for $v\in {\ensuremath{\mathcal{V}}}$ with $v\geq 2^{m+1}$, and the inclusions $$\begin{aligned} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1})} + v \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1}) + v }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1}) + 2^{m-2} }\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{m-2} -(1 + 2^{m})}\end{aligned}$$ hold for $v\in {\ensuremath{\mathcal{V}}}$ with $v \leq 2^{m-2}$, and thus Equation yields Equation . Now, we show that Equation holds. Note that the inclusions $$\begin{aligned} {\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m < n} U_m + v) & \subseteq {\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m < n} {\ensuremath{\mathcal{I}}}_m + v)\\ & \subseteq {\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\leq -1} +v )\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1})}\cap ({\ensuremath{\mathbb{Z}}}_{\leq -(1 + 2^{n+1})}) \\ & \subseteq \emptyset\end{aligned}$$ hold for any $n\geq 5$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v\leq - 2^{n+1}$. The inclusions $$\begin{aligned} & (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \cap (\cup_{0 \leq m < n}U_m - 2^n )\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap (((U_{n-1} \cup U_{n-2} \cup U_{n-3} )\cup U_{n-4}\cup (\cup_{0 \leq m \leq n-5}U_m)) - 2^n )\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap ((({\ensuremath{\mathcal{I}}}_{n-1} \cup {\ensuremath{\mathcal{I}}}_{n-2} \cup {\ensuremath{\mathcal{I}}}_{n-3} )\cup U_{n-4} \cup(\cup_{0 \leq m \leq n-5}{\ensuremath{\mathcal{I}}}_m)) - 2^n )\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap ((({\ensuremath{\mathbb{Z}}}_{\leq 2^{n-4}-1 - (1 + 2^{n-3})})\cup ({\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n-4}) })) - 2^n )\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap ( ({\ensuremath{\mathbb{Z}}}_{\leq 2^{n-4}-1 - (1 + 2^{n-3})} - 2^n)\cup (({\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n-4}) }) - 2^n))\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap ( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-4} -1- (1 + 2^{n-3})- 2^n}\cup {\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n-4})- 2^n })\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap ( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - 1 - (1 + 2^{n+1})}\cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} + 1 - ( 1+ 2^{n+1})})\\ & \subseteq (\{2^{n-1} + 2^{n-2} + 2^{n-3}+k\,|\,1 \leq k \leq 2^{n-4}\} - (1 + 2^{n+1}) ) \\ & \qquad \cap {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - 1 - (1 + 2^{n+1})}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - 1 - (1 + 2^{n+1})}\end{aligned}$$ hold for $n\geq 5$. The inclusions $$\begin{aligned} & {\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m < n}U_m - 2^{n-1} )\\ & \subseteq (U_{n-1} - 2^{n-1} ) \cup ({\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m \leq n-2}U_m - 2^{n-1} ))\\ & \subseteq \left( ( ((2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) \cup (2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-5}\})) - (1 + 2^n) )- 2^{n-1} \right) \\ & \qquad \cup ({\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m \leq n-2}{\ensuremath{\mathcal{I}}}_m - 2^{n-1} ))\\ & \subseteq \bigl( ((2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) - ( 1+ 2^n + 2^{n-1})) \\ & \qquad \cup ((2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-5}\}) - ( 1+ 2^n + 2^{n-1}))) \bigr) \cup ({\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n-1})} - 2^{n-1}))\\ & = \bigl( ((2^{n-1} + 2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) - ( 1+ 2^{n+1})) \\ & \qquad \cup ((2^{n-1} + 2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-5}\}) - ( 1+ 2^{n+1}))) \bigr) \cup ({\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\geq -2^n}))\\ & = \left( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-4} + 2^{n-4 } - ( 1+ 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3}+ 2^{n-5} - ( 1+ 2^{n+1})} \right) \cup ({\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\geq 2^n + 1 - ( 1 + 2^{n+1})}))\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-3} - ( 1+ 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3}+ 2^{n-5} - ( 1+ 2^{n+1})}\end{aligned}$$ hold for any $n\geq 5$. The inclusions $$\begin{aligned} & {\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m < n}U_m - 2^{n-2} )\\ & \subseteq (U_{n-1} - 2^{n-2} ) \cup ({\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m \leq n-2}U_m - 2^{n-2} ))\\ & \subseteq \left( ( ((2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) \cup (2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-5} + 1\})) - (1 + 2^n) )- 2^{n-2} \right) \\ & \qquad \cup ({\ensuremath{\mathcal{I}}}_n \cap (\cup_{0 \leq m \leq n-2}{\ensuremath{\mathcal{I}}}_m - 2^{n-2} ))\\ & \subseteq \bigl( ((2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) - ( 1+ 2^n + 2^{n-2})) \\ & \qquad \cup ((2^{n-2} + 2^{n-3} + \{1, 2, 3, \cdots, 2^{n-5}+1\}) - ( 1+ 2^n + 2^{n-2}))) \bigr) \cup ({\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n-1})} - 2^{n-2}))\\ & = \bigl( ((2^{n-1} + 2^{n-2} + 2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) - ( 1+ 2^{n+1})) \\ & \qquad \cup ((2^{n-3} + \{1, 2, 3, \cdots, 2^{n-5}+1\}) - ( 1+ 2^n))) \bigr) \cup ({\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\geq -2^{n-1} - 2^{n-2}}))\\ & = \bigl( ((2^{n-1} + 2^{n-2} + 2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) - ( 1+ 2^{n+1})) \\ & \qquad \cup ((2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} + \{1, 2, 3, \cdots, 2^{n-5}+1\}) - ( 1+ 2^{n+1}))) \bigr) \\ & \qquad \cup ({\ensuremath{\mathcal{I}}}_n \cap ({\ensuremath{\mathbb{Z}}}_{\geq 2^n + (1 + 2^{n-2}) - (1 + 2^{n+1}) }))\\ & = \bigl( ((2^{n-1} + 2^{n-2} + 2^{n-4} + \{1, 2, 3, \cdots , 2^{n-4} \}) - ( 1+ 2^{n+1})) \\ & \qquad \cup ((2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} + \{1, 2, 3, \cdots, 2^{n-5}+1\}) - ( 1+ 2^{n+1}))) \bigr)\end{aligned}$$ hold for any $n\geq 5$. The inclusions $$\begin{aligned} \cup_{0 \leq m < n}U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-4}+1 - (1+2^{n}) } + v \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-4}+1 - (1+2^{n}) - 2^{n-3}}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} + 1- ( 1+ 2^{n+1})}\end{aligned}$$ hold for any $n\geq 5$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v\geq - 2^{n-3}$. These inclusions prove that for $n\geq 5$, the largest element of the third quarter of the right quarter of ${\ensuremath{\mathcal{I}}}_n$ does not belong to $\cup_{0 \leq m < n} U_m + {\ensuremath{\mathcal{V}}}$, i.e., $$2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - (1 + 2^{n+1}) \notin \cup_{0 \leq m < n} U_m + {\ensuremath{\mathcal{V}}}.$$ Combining the above with Equations , , we obtain $$2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - (1 + 2^{n+1}) \notin \left(\cup_{m \geq 0, m \neq n, n+1} U_m + {\ensuremath{\mathcal{V}}}\right) \cup (U_{n+1} + ({\ensuremath{\mathcal{V}}}\setminus\{2^n\})) \quad \text{ for } n\geq 5.$$ Since the inclusions $$\begin{aligned} U_{n+1} + 2^n & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n + 2^{n-1} + 2^{n-3} - (1 + 2^{n+2})} + 2^n\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n + 2^{n-1} + 2^{n-3} - (1 + 2^{n+2}) + 2^n}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-3} - (1 + 2^{n+1}) }\\\end{aligned}$$ hold for $n\geq 5$, it follows that $$2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-4} - (1 + 2^{n+1}) \notin \left(\cup_{m \geq 0, m \neq n} U_m + {\ensuremath{\mathcal{V}}}\right) \quad \text{ for } n\geq 5.$$ Note that the inclusions $$\begin{aligned} U_n + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1})} + v \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1}) + v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1}) + 2^n} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n}) } \\\end{aligned}$$ hold for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \geq 2^n$, the inclusions $$\begin{aligned} U_n + 2^{n-1} & \subseteq ({\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} + 2^{n-3} - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})}) + 2^{n-1} \\ & \subseteq ({\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})}) + 2^{n-1} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1}) + 2^{n-1}} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})+ 2^{n-1}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1}+2^{n-2} - (1 + 2^{n+1}) } \cup {\ensuremath{\mathbb{Z}}}_{\geq 1 + 2^{n-2} - (1 + 2^{n})}\end{aligned}$$ hold, the inclusions $$\begin{aligned} U_n + 2^{n-2} & \subseteq ({\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} + 2^{n-3} - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})}) + 2^{n-2} \\ & \subseteq ({\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})}) + 2^{n-2} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} - (1 + 2^{n+1}) + 2^{n-2}} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})+ 2^{n-2}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} - (1 + 2^{n+1}) } \cup {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n})}\end{aligned}$$ hold, and the inclusions $$\begin{aligned} U_n + v & \subseteq {\ensuremath{\mathbb{Z}}}_{2^{n-1} + 2^{n-2} + 2^{n-4} - (1 + 2^{n+1})} + v\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{2^{n-1} + 2^{n-2} + 2^{n-4} - (1 + 2^{n+1}) + v}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{2^{n-1} + 2^{n-2} + 2^{n-4} - (1 + 2^{n+1}) + 2^{n-4}}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{2^{n-1} + 2^{n-2} + 2^{n-3} - (1 + 2^{n+1})}\\\end{aligned}$$ hold for any $v\in {\ensuremath{\mathcal{V}}}$ with $v\leq 2^{n-4}$. This yields Equation . Now, we show that Equation holds. The inclusions $$\begin{aligned} U_n + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n - ( 1+ 2^{n+1})} + v \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n - ( 1+ 2^{n+1})+ v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n - ( 1+ 2^{n+1})+ 2^n} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq -1}\end{aligned}$$ hold for any $n\geq 1$ and $v\in {\ensuremath{\mathcal{V}}}$ with $v \leq 2^n$. The inclusions $$\begin{aligned} U_n + 2^{n+1} & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 3 - (1 + 2^{n+1})} + 2^{n+1} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 3 - (1 + 2^{n+1}) + 2^{n+1}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2} \\\end{aligned}$$ hold for any $n\geq 1$. The inclusions $$\begin{aligned} U_n + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n+1})} + v \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n+1})+ v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - ( 1+ 2^{n+1})+ 2^{n+2}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n+1}}\end{aligned}$$ hold for any $n\geq 1$ and $v\in {\ensuremath{\mathcal{V}}}$ with $v \geq 2^{n+2}$. This proves that $$1 \notin (U\setminus U_0) + {\ensuremath{\mathcal{V}}}= (U \setminus \{-2, -1\}) + {\ensuremath{\mathcal{V}}}.$$ Note that $$\begin{aligned} -2 + {\ensuremath{\mathcal{V}}}& \subseteq (-2 + \{1\}) \cup (-2 + ({\ensuremath{\mathcal{V}}}\setminus \{1\})) \\ & \subseteq \{-1\} \cup (-2 + ({\ensuremath{\mathcal{V}}}\setminus \{1\})) \end{aligned}$$ and the elements of $-2 + ({\ensuremath{\mathcal{V}}}\setminus \{1\})$ are even. So $1$ does not belong to $-2 + {\ensuremath{\mathcal{V}}}$. So $1 \notin (U \setminus \{-1\}) + {\ensuremath{\mathcal{V}}}$, and hence Equation holds. Now, we prove that Equations , hold. By Equations , , we obtain $$(\{25, 26\} - ( 1+ 2^6) ) \cap \left( (U_6 + ({\ensuremath{\mathcal{V}}}\setminus \{2^5\})) \cup (\cup_{m\geq 7} U_m + {\ensuremath{\mathcal{V}}}) \right) = \emptyset.$$ It follows that $$(\{25, 26\} - ( 1+ 2^6) ) \cap (U_6 + 2^5) = \emptyset,$$ $$(\{25, 26\} - ( 1+ 2^6) ) \cap ((U_0 \cup \cdots \cup U_4) + {\ensuremath{\mathcal{V}}}) = \emptyset,$$ and hence $$(\{25, 26\} - ( 1+ 2^6) ) \cap (\cup_{m\neq 5} U_m + {\ensuremath{\mathcal{V}}}) = \emptyset.$$ It also follows that $$(\{ 26\} - ( 1+ 2^6) ) \cap ((U_5 \setminus \{25 - ( 1 + 2^6)\}) + {\ensuremath{\mathcal{V}}}) = \emptyset,$$ $$(\{ 26\} - ( 1+ 2^6) ) \cap (25 - ( 1 + 2^6) + ({\ensuremath{\mathcal{V}}}\setminus \{1\})) = \emptyset,$$ $$(\{ 25\} - ( 1+ 2^6) ) \cap ((U_5 \setminus \{26 - ( 1 + 2^6)\}) + {\ensuremath{\mathcal{V}}}) = \emptyset,$$ $$(\{ 25\} - ( 1+ 2^6) ) \cap (26 - ( 1 + 2^6) + ({\ensuremath{\mathcal{V}}}\setminus \{-1\})) = \emptyset.$$ Consequently, Equations , hold. By considering the representation of the integer $-4$ as a sum of an element of $U$ and an element of ${\ensuremath{\mathcal{V}}}$, Equation follows. Using Equations , , we obtain $$\{-6\} \cap \left( (U_4 + ({\ensuremath{\mathcal{V}}}\setminus \{2^3\}) \cup (\cup_{m\geq 5} U_m + {\ensuremath{\mathcal{V}}}) \right) = \emptyset.$$ Then Equation follows from considering the representation of $-6$. Now, we prove that Equation holds. Note that the inclusions $$\begin{aligned} \cup_{0 \leq m \leq n-2} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 +2^{n-1})} + v \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 +2^{n-1}) + v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 +2^{n-1}) - 2^{n-1}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 +2^{n})} \end{aligned}$$ hold for $n\geq 6$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v\geq -2^{n-1}$. The inclusions $$\begin{aligned} \cup_{0 \leq m \leq n-2} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq -1} + v \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq -1 + v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq -1 -2^{n+1}} \end{aligned}$$ hold for $n\geq 6$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v\leq -2^{n+1}$. The inclusions $$\begin{aligned} \cup_{0 \leq m \leq n-2} U_m - 2^n & \subseteq (\cup_{3 \leq m \leq n-2} {\ensuremath{\mathcal{I}}}_m - 2^n) \cup (\{-2, -1\} - 2^n) \\ & \subseteq ({\ensuremath{\mathbb{Z}}}_{\leq -9} - 2^n) \cup (\{-1, 0\} - (1 + 2^n)) \\ & \subseteq ({\ensuremath{\mathbb{Z}}}_{\leq -8 - (1 + 2^n)}) \cup (\{-1, 0\} - (1 + 2^n)) \\ & = ({\ensuremath{\mathbb{Z}}}_{\leq 2^n-8 - (1 + 2^{n+1})}) \cup (\{2^n-1,2^n\} - (1 + 2^{n+1}))\end{aligned}$$ hold for $n\geq 6$. These inclusions prove that for $n\geq 6$, the third largest element of ${\ensuremath{\mathcal{I}}}_n$ does not belong to $\cup_{0 \leq m \leq n-2} U_m + {\ensuremath{\mathcal{V}}}$, i.e., $$2^n - 2 - (1 + 2^{n+1}) \notin \cup_{0 \leq m \leq n-2} U_m + {\ensuremath{\mathcal{V}}}.$$ Combining the above with Equations , , we obtain $$2^n - 2 - (1 + 2^{n+1}) \notin \left(\cup_{m \geq 0, m \neq n-1, n, n+1} U_m + {\ensuremath{\mathcal{V}}}\right) \cup (U_{n+1} + ({\ensuremath{\mathcal{V}}}\setminus\{2^n\})) \quad \text{ for } n\geq 6.$$ Since the inclusions $$\begin{aligned} U_{n+1} + 2^n & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n + 2^{n-1} + 2^{n-3} - (1 + 2^{n+2})} + 2^n \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n + 2^{n-1} + 2^{n-3} - (1 + 2^{n+2}) + 2^n }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-3} - (1 + 2^{n+1}) }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-1}-3 - (1 + 2^{n+1}) }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n-3 - (1 + 2^{n+1}) }\end{aligned}$$ hold for $n\geq 6$, it follows that $$2^n - 2 - (1 + 2^{n+1}) \notin \left(\cup_{m \geq 0, m \neq n-1, n} U_m + {\ensuremath{\mathcal{V}}}\right) \quad \text{ for } n\geq 6.$$ The inclusions $$\begin{aligned} U_n + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1})} + v\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1}) + v}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^{n+1}) + 2^n}\\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 1 - (1 + 2^{n+1})}\end{aligned}$$ hold for any $n\geq 5$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \geq 2^n$. The inclusions $$\begin{aligned} U_n + 2^{n-1} & \subseteq \left( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} + 2^{n-3} - ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})} \right) + 2^{n-1} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} + 2^{n-3} + 2^{n-1}- ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 + 2^{n-1}- (1 + 2^{n+1})} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} + 2^{n-1}- ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 2^{n-2} + 1 - (1 + 2^{n+1})} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^n - 2^{n-2}- ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n +2^{n-2} + 1 - (1 + 2^{n+1})} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n - 3 - ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 2^{n-2} + 1 - (1 + 2^{n+1})}\end{aligned}$$ hold for any $n\geq 6$. The inclusions $$\begin{aligned} U_n + 2^{n-2} & \subseteq \left( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} + 2^{n-3} - ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 - (1 + 2^{n+1})} \right) + 2^{n-2} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} + 2^{n-3} + 2^{n-2}- ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-1} + 2^{n-2} + 1 + 2^{n-2}- (1 + 2^{n+1})} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1}- ( 1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n +1- (1 + 2^{n+1})}\end{aligned}$$ hold for any $n\geq 6$. The inclusions $$\begin{aligned} U_n + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-4} - (1 + 2^{n+1})} + v\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-4} - (1 + 2^{n+1}) + v}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-4} - (1 + 2^{n+1}) + 2^{n-3}}\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} +2^{n-3} + 2^{n-4} - (1 + 2^{n+1}) }\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^n - 2^{n-4} - (1 + 2^{n+1}) }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^n - 3 - (1 + 2^{n+1}) }\\\end{aligned}$$ hold for any $n\geq 6$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \leq 2^{n-3}$. Consequently, we obtain $$2^n - 2 - (1 + 2^{n+1}) \notin \left(\cup_{m \geq 0, m \neq n-1} U_m + {\ensuremath{\mathcal{V}}}\right) \quad \text{ for } n\geq 6.$$ The inclusions $$\begin{aligned} U_{n-1} + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-4} + 1 - (1 + 2^n)} +v\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-4} + 1 - (1 + 2^n) - 2^{n-4} }\\ & = {\ensuremath{\mathbb{Z}}}_{\geq 1 - (1 + 2^n) }\\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 1 - (1 + 2^{n+1}) }\end{aligned}$$ hold for $n\geq 6$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \geq -2^{n-4}$. The inclusions $$\begin{aligned} U_{n-1} - 2^{n-2} & \subseteq ( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-4} + 2^{n-4} - (1 + 2^n) } \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-2} + 2^{n-3} + 1 - (1 + 2^n)} ) - 2^{n-2} \\ & = ( {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} - (1 + 2^n) } \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-2} + 2^{n-3} + 1 - (1 + 2^n)} ) - 2^{n-2} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} - (1 + 2^n) - 2^{n-2}} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-2} + 2^{n-3} + 1 - (1 + 2^n)- 2^{n-2}} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-3} - (1 + 2^n) - 2^{n-2}} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3} + 1 - (1 + 2^n)} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3} - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3} + 1 - (1 + 2^n)} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-2} - 4 - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3} + 1 - (1 + 2^n)} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^n - 4 - (1 + 2^{n+1})} \cup {\ensuremath{\mathbb{Z}}}_{\geq 2^n + 2^{n-3} + 1 - (1 + 2^{n+1})}\end{aligned}$$ hold for $n\geq 6$. The inclusions $$\begin{aligned} U_{n-1} + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} + 2^{n-3} + 2^{n-5} - (1 + 2^n)} +v\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} + 2^{n-3} + 2^{n-5} - (1 + 2^n) +v}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-2} + 2^{n-3} + 2^{n-5} - (1 + 2^n) - 2^{n-1}}\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-5} - (1 + 2^{n+1})}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{n-1} + 2^{n-2} + 2^{n-3} + 2^{n-3} -4 - (1 + 2^{n+1})}\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{n} -4 - (1 + 2^{n+1})}\end{aligned}$$ hold for $n\geq 6$ and for any $v\in {\ensuremath{\mathcal{V}}}$ with $v \leq -2^{n-1}$. This yields $$2^n - 2 - (1 + 2^{n+1}) \notin \left(\cup_{m \geq 0} U_m + {\ensuremath{\mathcal{V}}}\right) \cup (U_{n-1} + ({\ensuremath{\mathcal{V}}}\setminus \{ - 2^{n-3} \})) \quad \text{ for } n\geq 6.$$ So Equation follows. This establishes all of the nine claims, and hence ${\ensuremath{\mathcal{V}}}$ is a minimal complement of $U$. \[Thm:CoMinU\] Let $U = \{ -u_1, - u_2, -u_3, \cdots\}$ where $u_1 < u_2 < u_3< \cdots$. Define $U_1 = U$ and for each positive integer $i\geq 1$, define $$U_{i+1} : = \begin{cases} U_i \setminus \{-u_i\} & \text{ if $U_i \setminus \{-u_i\}$ is a complement to ${\ensuremath{\mathcal{V}}}$,}\\ U_i & \text{ otherwise.} \end{cases}$$ Let ${\ensuremath{\mathcal{U}}}$ denote the subset $\cap _{i \geq 1} U_i$ of ${\ensuremath{\mathbb{Z}}}$. The subsets ${\ensuremath{\mathcal{U}}}$ and ${\ensuremath{\mathcal{V}}}$ of ${\ensuremath{\mathbb{Z}}}$ form a co-minimal pair. Note that for any $n\geq 4$ and any $m\geq n$, the inclusions $$\begin{aligned} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1})} + v\\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1}) + v}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^m - (1 + 2^{m+1}) + 2^{m-1}}\\ & = {\ensuremath{\mathbb{Z}}}_{\leq - (1 + 2^{m-1}) }\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 2^{m-3} -1}\\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 2^{n-3}-1 }\\\end{aligned}$$ hold for any $v\leq 2^{m-1}$, the inclusions $$\begin{aligned} U_m + 2^m & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{m-1} + 2^{m-2} + 2^{m-4} - (1 + 2^{m+1})} + 2^m \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{m-1} + 2^{m-2} + 2^{m-4} - (1 + 2^{m+1}) + 2^m} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq 2^{m-1} + 2^{m-2} + 2^{m-3} - (1 + 2^{m+1}) + 2^m} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq 2^{m-1} + 2^{m-2} + 2^{m-3} + 2^{m-3} - 2^{m-3} - (1 + 2^{m+1}) + 2^m} \\ & = {\ensuremath{\mathbb{Z}}}_{\leq - 2^{m-3} -1 } \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\leq - 2^{n-3} -1} \end{aligned}$$ hold, the inclusions $$\begin{aligned} U_m + v & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{m-3} + 1 - (1 + 2^{m+1})} + v \\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^{m-3} + 1 - (1 + 2^{m+1}) + v} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{m-3} + 1 - (1 + 2^{m+1}) + 2^{m+1}} \\ & = {\ensuremath{\mathbb{Z}}}_{\geq 2^{m-3}} \\ & \subseteq {\ensuremath{\mathbb{Z}}}_{\geq 2^{n-3}}\end{aligned}$$ hold for any $v\geq 2^{m+1}$. It follows that $$\label{Eqn:UVFiniteness} \{- 2^{n-3}, - 2^{n-3} +1, - 2^{n-3} + 2, \cdots, -2, -1, 0, 1, 2, 3, \cdots, 2^{n-3}-1\} \cap (\cup_{m\geq n} U_m + {\ensuremath{\mathcal{V}}}) = \emptyset$$ for any $n\geq 4$. Consequently, for any element $y\in {\ensuremath{\mathbb{Z}}}$, the equation $y = u +v$ holds for finitely many pairs $(u, v)\in U \times {\ensuremath{\mathcal{V}}}$. For any $y\in {\ensuremath{\mathbb{Z}}}$ and for any $i\geq 1$, there exist elements $u_{y, i}\in U_i , v_{y, i}\in {\ensuremath{\mathcal{V}}}$ such that $y = u_{y, i} + v_{y, i}$. It follows that for some element $u_y \in U$, the equality $u_y = u_{y, i}$ holds for infinitely many $i$ and for such integers $i$, we have $v_y = v_{y, i}$ where $v_y: = y - u_y\in {\ensuremath{\mathcal{V}}}$. Thus $y - v_y = u_{y, i}$ holds for infinitely many $i$. Hence, for each integer $i\geq 1$, there exists an integer $m_i \geq i$ such that $y - v_y = u_{y, m_i}$, which yields $y\in v_y + U_{m_i} \subseteq v_y + U_i$. Thus $y$ lies in $v_y + {\ensuremath{\mathcal{U}}}$, i.e., $y\in {\ensuremath{\mathcal{U}}}+ {\ensuremath{\mathcal{V}}}$. Hence ${\ensuremath{\mathcal{U}}}$ is an additive complement of ${\ensuremath{\mathcal{V}}}$. It follows that ${\ensuremath{\mathcal{U}}}$ is a minimal complement to ${\ensuremath{\mathcal{V}}}$. By Theorem \[Thm:RequiredClustersV\], it follows that ${\ensuremath{\mathcal{V}}}$ is a minimal complement to ${\ensuremath{\mathcal{U}}}$. This proves that $({\ensuremath{\mathcal{U}}}, {\ensuremath{\mathcal{V}}})$ is a co-minimal pair. Co-minimal pairs in the integral lattices ========================================= \[Prop:ABCD\] Let $H$ be a subgroup of an abelian group $G$. Let $(A, B)$ be a co-minimal pair in $H$, and $(C, D)$ be subsets of $G$ whose images form a co-minimal pair in $G/H$, and the restrictions of the map $G \to G/H$ on $C$ and $D$ induces bijections onto respective images. Then $(A+C, B+D)$ is a co-minimal pair in $G$. Note that $A+C + B+ D = G$. Let $a\in A, c\in C$ be such that $(A+C)\setminus \{a+c\}$ is a complement to $B+D$. Let $b\in B, d\in D$ be such that $a +b \notin (A \setminus \{a\}) \cdot B$, $c+ d {\ensuremath{\mathrm{\;mod\;}}}H \notin (C \setminus \{c\}) \cdot D{\ensuremath{\mathrm{\;mod\;}}}H$. Since $a+c+b+d\in ((A+C)\setminus \{a+c\}) +B +D$, it follows that $a+c+b+d = a'+c'+b'+d'$ for some $a'\in A, c'\in C, b'\in B, d'\in D$ with $a'+c'\notin (A+C) \setminus \{a+c\}$. This implies $c+d {\ensuremath{\mathrm{\;mod\;}}}H = c'+d'{\ensuremath{\mathrm{\;mod\;}}}H$, and hence $c \equiv c' {\ensuremath{\mathrm{\;mod\;}}}H, d \equiv d' {\ensuremath{\mathrm{\;mod\;}}}H$, and thus $c = c', d = d'$, which yields $a+b = a'+b'$. Since $a + b \notin (A \setminus \{a\}) + B$, we obtain $a = a', b = b'$. This contradicts the fact that $a'+c'\notin (A+C) \setminus \{a+c\}$. Hence $A+C$ is a minimal complement to $B+D$. Similarly, it follows that $B+D$ is a minimal complement to $A+C$. As an application of the above result, we obtain the following result. Let ${\ensuremath{\mathcal{S}}}= \{s_1, s_2, \cdots \}$. For any two sequences $\{x_n\}_{n\geq 1}$ and $\{y_n\}_{n\geq 1}$, each of $$( \{ (x_i + s_j , s_i)\,|\, i, j \geq 1 \}, \{ (y_u + 2^{v-1} , 2^{u-1})\,|\, u, v \geq 1 \}),$$ $$( \{ (x_i + 2^{j-1} , s_i)\,|\, i, j \geq 1 \}, \{ (y_u + s_v , 2^{u-1})\,|\, u,v \geq 1 \})$$ is a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^2$. For any two sequences $\{x_n\}_{n\geq 1}$ and $\{y_n\}_{n\geq 1}$, the subsets $$({\ensuremath{\mathcal{S}}},0) + \{(x_1, s_1), (x_2, s_2) , (x_3, s_3), \cdots \} = \{ (x_i + s_j , +s_i)\,|\, i, j \geq 1 \},$$ $$({\ensuremath{\mathcal{T}}},0) + \{(y_1, 1), (y_2, 2) , (y_3, 2^2), \cdots \} = \{ (y_u + 2^{v-1} , 2^{u-1})\,|\, u,v \geq 1 \}$$ of ${\ensuremath{\mathbb{Z}}}^2$ form a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^2$, and the subsets $$({\ensuremath{\mathcal{T}}},0) + \{(x_1, s_1), (x_2, s_2) , (x_3, s_3), \cdots \} = \{ (x_i + 2^{j-1} , s_i)\,|\, i, j \geq 1 \},$$ $$({\ensuremath{\mathcal{S}}},0) + \{(y_1, 1), (y_2, 2) , (y_3, 2^2), \cdots \} = \{ (y_u + s_v , 2^{u-1})\,|\, u,v \geq 1 \}$$ of ${\ensuremath{\mathbb{Z}}}^2$ form a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^2$ by Proposition \[Prop:ABCD\]. \[Thm:AExists\] Let $\sigma$ be an automorphism of ${\ensuremath{\mathbb{Z}}}^{n}$ such that there exists an increasing chain of subgroups $$0 = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_r = {\ensuremath{\mathbb{Z}}}^{n}$$ of ${\ensuremath{\mathbb{Z}}}^{n}$ such that each of them is stable under the action of $\sigma$ and the successive quotients $M_i/M_{i-1}$ are free of rank at most two and for any such quotient, the restriction of $\sigma$ to it is the identity map if the quotient is of rank one, or conjugate to some element of ${\ensuremath{\operatorname{GL}}}_2({\ensuremath{\mathbb{Z}}})$ having exactly two nonzero entries, i.e., some of $$\label{Eqn:8Matrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} , \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} , \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} , \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \footnote{ Note that the above eight matrices form the subgroup $$ \left \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\ensuremath{\operatorname{GL}}}_2({\ensuremath{\mathbb{Z}}})\,|\, \text{ exactly two of $a, b, c, d$ are equal to zero} \right \} $$ of ${\ensuremath{\operatorname{GL}}}_2({\ensuremath{\mathbb{Z}}})$. }$$ if the quotient is of rank two. Then there exists a subset $A$ of ${\ensuremath{\mathbb{Z}}}^{n}$ such that $(A, \sigma(A))$ is a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^n$. If $n = 1$, then $\sigma$ is the identity map, and thus in this case, $A$ can taken to be the subset $W$ of ${\ensuremath{\mathbb{Z}}}$ as in [@Kwon Proposition 3]. If $n = 2$, then one of the following conditions hold. 1. $M_0$ is a subgroup of ${\ensuremath{\mathbb{Z}}}^2$ of rank one and $\sigma$ is conjugate to $$\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix},$$ 2. $M_0 = {\ensuremath{\mathbb{Z}}}^2$ and $\sigma$ is conjugate to one of the matrices in Equation . For simplicity, we will assume that $\sigma$ is equal to the above matrix, or equal to one of the matrices in Equation . In the first case, we can take $A = W \times W$ (by Proposition \[Prop:ABCD\]) and in the second case, we can take $A$ to be - $W \times W$, - $\{(x, 0)\,|\, x\in {\ensuremath{\mathbb{Z}}}\} \cup \{(0, y)\,|\, y\in {\ensuremath{\mathbb{Z}}}_{\geq 1}\}$, - $\{(x, 0)\,|\, x\in {\ensuremath{\mathbb{Z}}}_{\geq 1}\} \cup \{(0, y)\,|\, y\in {\ensuremath{\mathbb{Z}}}\}$, - $\{(x, 0)\,|\, x\in {\ensuremath{\mathbb{Z}}}\} \cup \{(0, y)\,|\, y\in {\ensuremath{\mathbb{Z}}}_{\geq 1}\}$, - ${\ensuremath{\mathcal{S}}}\times {\ensuremath{\mathcal{T}}}$, - ${\ensuremath{\mathcal{V}}}\times {\ensuremath{\mathcal{U}}}$, - ${\ensuremath{\mathcal{U}}}\times {\ensuremath{\mathcal{V}}}$, - $(-{\ensuremath{\mathcal{S}}}) \times {\ensuremath{\mathcal{T}}}$ according as $\sigma$ is equal to the matrices in Equation . Thus Theorem \[Thm:AExists\] holds for $n= 1, 2$. Let $n\geq 3$ be an integer such that Theorem \[Thm:AExists\] holds for free abelian groups of rank $<n$. Let $M'$ be a subgroup of ${\ensuremath{\mathbb{Z}}}^n$ such that $M'$ surjects onto $M_r /M_{r-1}$. Denote the restriction of $\sigma$ to $M_r/M_{r-1}$ by $\bar \sigma$. Since the result holds for $n=1, 2$ and $M_r/M_{r-1}$ has rank at most two, it follows that there is a subset $C$ of $M_r/M_{r-1}$ such that $(C, \bar \sigma (C))$ is a co-minimal pair in $M_r/M_{r-1}$. Let $C'$ be a subset of $M'$ such that the map $M \to M_r/M_{r-1}$ yields a bijection between $C'$ and $C$. Note that $M_{r-1}$ is stable under the action of $\sigma$. By the induction hypothesis, there is a subset $A$ of $M_{r-1}$ such that $(A, \sigma(A))$ is a co-minimal pair $M_{r-1}$. By Proposition \[Prop:ABCD\], $(A + C', \sigma (A) + \sigma(C'))$ is a co-minimal pair in $M_r = {\ensuremath{\mathbb{Z}}}^n$. Thus Theorem \[Thm:AExists\] follows. \[Thm:AExistsQuadrant\] Let $\sigma$ be an automorphism of ${\ensuremath{\mathbb{Z}}}^{2d}$ such that its matrix with respect to the standard basis of ${\ensuremath{\mathbb{Z}}}^{2d}$ has a block upper triangular form having the matrices $$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$ along the diagonal. Then there exists a subset $A$ of ${\ensuremath{\mathbb{Z}}}^{2d}$ contained in a quadrant such that $(A, \sigma(A))$ is a co-minimal pair. If $d = 1$, then $A$ can be taken to be ${\ensuremath{\mathcal{S}}}\times {\ensuremath{\mathcal{T}}}, (-{\ensuremath{\mathcal{S}}}) \times {\ensuremath{\mathcal{T}}}$. Let $d\geq 2$ be an integer and assume that the result holds for free groups of rank $2(d-1)$. Let $\sigma$ be an automorphism of ${\ensuremath{\mathbb{Z}}}^{2d}$ satisfying the given condition. Let $M$ denote the subgroup of ${\ensuremath{\mathbb{Z}}}^{2d}$ generated by $e_1, \cdots, e_{2(d-1)}$ and $M'$ denote the subgroup of ${\ensuremath{\mathbb{Z}}}^{2d}$ generated by $e_{2d-1}, e_{2d}$. Note that $M'$ surjects onto ${\ensuremath{\mathbb{Z}}}^{2d}/M$. Note that $M$ is stable under the action of $\sigma$. By the induction hypothesis, there is a subset $A$ of $M$ contained in a quadrant such that $(A, \sigma(A))$ is a co-minimal pair in $M$. Since the result holds for $d = 1$, it follows that there exists a subset $C$ of $M'$ contained in a quadrant such that the images of $C$ and $\sigma(C)$ in ${\ensuremath{\mathbb{Z}}}^{2d}/M$ form a co-minimal pair. By Proposition \[Prop:ABCD\], $(A+ C, \sigma(A) + \sigma(C))$ is a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^{2d}$. Since $A+C$ is contained in a quadrant, the result follows by induction. \[Proof of Theorem \[Thm:Z2d\]\] The first part follows from Theorem \[Thm:AExists\]. For $d = 1$, the group ${\ensuremath{\mathbb{Z}}}^{2d}$ admits infinitely many automorphisms of the form $$\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$$ with $*\in {\ensuremath{\mathbb{Z}}}$. So by Theorem \[Thm:AExists\], for each such automorphism $\sigma$ of ${\ensuremath{\mathbb{Z}}}^{2d}$, there is a subset $A$ of ${\ensuremath{\mathbb{Z}}}^{2d}$ such that $(A, \sigma(A))$ is a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^{2d}$. To establish the second part for $d\geq 2$, note that there are infinitely many automorphisms of ${\ensuremath{\mathbb{Z}}}^{2d}$ which are block upper triangular where the blocks are of size $2\times 2$ and the matrices lying along the diagonal blocks are equal to some of $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} , \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} , \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} , \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} , \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix},$$ then by Theorem \[Thm:AExists\], for each such automorphism $\sigma$ of ${\ensuremath{\mathbb{Z}}}^{2d}$, there is a subset $A$ of ${\ensuremath{\mathbb{Z}}}^{2d}$ such that $(A, \sigma(A))$ is a co-minimal pair in ${\ensuremath{\mathbb{Z}}}^{2d}$. Acknowledgements ================ The first author would like to thank the Department of Mathematics at the Technion where a part of the work was carried out. The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India. \#1[7 71000017 10000 -17100007]{} \#1[7 71000017 10000 -17100007]{} \#1[0=]{} \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{} [Kwo19]{} Arindam Biswas and Jyoti Prakash Saha, *On additive co-minimal pairs*, Preprint available at <https://arxiv.org/abs/1906.05837>, 2019. Paul Erdős, *Some results on additive number theory*, Proc. Amer. Math. Soc. **5** (1954), 847–853. , *Some unsolved problems*, Michigan Math. J. **4** (1957), 291–300. Andrew Kwon, *A note on minimal additive complements of integers*, Discrete Math. **342** (2019), no. 7, 1912–1918. G. G. Lorentz, *On a problem of additive number theory*, Proc. Amer. Math. Soc. **5** (1954), 838–841. Melvyn B. Nathanson, *Problems in additive number theory, [IV]{}: [N]{}ets in groups and shortest length [$g$]{}-adic representations*, Int. J. Number Theory **7** (2011), no. 8, 1999–2017. [^1]: [^2]: [^3]: A subset $X$ of ${\ensuremath{\mathbb{Z}}}^d$ is said to be *contained in a quadrant* if for any given $1\leq i \leq d$, the $i$-th coordinate of all the points of $X$ is either positive or negative.

--- abstract: 'We have studied the diffusion of excess quasiparticles in a current-biased superconductor strip in proximity to a metallic trap junction. In particular, we have measured accurately the superconductor temperature at a near-gap injection voltage. By analyzing our data quantitatively, we provide a full description of the spatial distribution of excess quasiparticles in the superconductor. We show that a metallic trap junction contributes significantly to the evacuation of excess quasiparticles.' author: - 'Sukumar Rajauria$^{1,2,3}$' - 'L. M. A. Pascal$^{1}$' - 'Ph. Gandit$^{1}$' - 'F. W. J. Hekking$^{4}$' - 'B. Pannetier$^{1}$' - 'H. Courtois$^{1}$' title: Efficient Quasiparticle Evacuation in Superconducting Devices --- In a normal metal - insulator - superconductor (N-I-S) junction, charge transport is mainly governed by quasiparticles [@Rowell]. The presence of the superconducting energy gap $\Delta$ induces an energy selectivity of quasiparticles tunneling out of the normal metal [@nahum; @giazotto]. The quasiparticle tunnel current is thus accompanied by a heat transfer from the normal metal to the superconductor that is maximum at a voltage bias just below the superconducting gap (V $\leq \Delta$/e). For a double junction geometry (S-I-N-I-S), electrons in the normal metal can typically cool from 300 mK down to about 100 mK [@leivo; @giazotto; @sukumar07PRL]. However, in all experiments so far the electronic cooling is less efficient than expected [@leivo; @sukumar07PRL]. It has been proposed that this inefficiency is mostly linked to the injected quasiparticles accumulating near the tunnel junction area. This out-of-equilibrium electronic population, injected at an energy above the superconductor energy gap $\Delta$, relaxes by slow processes such as recombination and pair-breaking processes. The accumulation of quasiparticles is aggravated in sub-micron devices, where the relaxation processes are restricted by the physical dimensions of the device, leading to an enhanced density of quasiparticles close to the injection point. These quasiparticles can thereafter tunnel back into the normal metal [@Jochum], generating a parasitic power proportional to the bias current [@sukumar08arxiv; @VOutilainen]. The same phenomenon is relevant to other superconductor-based devices such as qubits [@Lang], single electron transistors [@Court] and low temperature detectors [@Booth]. In hybrid superconducting devices fabricated by multiple angle evaporation, a normal metal strip in tunnel contact with the superconducting electrode acts as a trap for excess near-gap quasiparticles, which removes them from the superconductor. This mechanism is usually not fully efficient due to the tunnel barrier between the normal metal and the superconductor [@Pekolatrap]. A detailed theory of non-equilibrium phenomena in a superconductor in contact with normal metal traps has been developed [@VOutilainen]. However, a quantitative comparison between experiments and theoretical predictions is so far still missing. ![(a) Schematic of the sample design with a trap junction on each S-strip of the superconducting microcooler. The curve shows the spatial profile of the excess quasiparticles $\delta N_{qp}$ along the S-strip. The two injectors are located at a$_{1}$ = 5 $\mu$m or a$_{2}$ = 21 $\mu$m; the detector (cooler junction island) is at x = 0 $\mu$m and the length of the S-strip is $L_{S}$ = 27 $\mu$m. (b) Calculated spatial decay of excess quasiparticles density $\delta$N$_{qp}$ along the S-strip. The two injectors are biased at eV/$\Delta$=4. []{data-label="fig:1"}](Fig-1-10.eps){width="1\linewidth"} In this Letter, we present an experimental investigation of the diffusion of out-of-equilibrium quasiparticles in a superconducting strip covered with a trap junction. A N-metal is used to inject quasiparticles in the S-strip. The local superconductor temperature is inferred from the heating of the central N-island of a S-I-N-I-S junction. We quantitatively compare our experimental data with a recently discussed theoretical model [@sukumar08arxiv]. We have used a S-I-N-I-S cooler device with a trap geometry similar to the one studied in Ref. [@sukumar08PRL], see Fig. 1. These devices are fabricated using electron beam lithography, two-angle shadow evaporation and lift-off on a silicon substrate having 500 nm thick SiO$_{2}$ on it. The central normal metal Cu electrode is 0.3 $\mu$m wide, 0.05 $\mu$m thick and 4 $\mu$m long. The 27 $\mu$m long symmetric S-strips of Al are then partially covered, through a tunnel barrier, by a Cu strip acting as a trap junction. At their extremity, the S-trips are connected to a contact pad acting as a reservoir. In addition to the cooler island, we added two normal metal Cu tunnel injector junctions of area around 0.09 $\mu$m$^2$ on one S-strip. Injector 1 and 2 are at a distance of $a$ = 5 $\mu$m and 21 $\mu$m respectively from the central Cu island. The Al tunnel barrier is assumed to be identical in the cooler, probe and trap junctions, since they have similar specific tunnel resistance. The normal state resistance of the double-junction S-I-N-I-S cooler is 1.9 k$\Omega$. The normal state resistance of N-I-S injector junctions 1 and 2 are respectively 2.5 k$\Omega$ and 2.3 k$\Omega$. The diffusion coefficient of the Al S-strip film was measured at 4.2 K to be 30 cm$^{2}$/s. N-I-S tunnel junctions are known to enable controlled quasiparticle injection in a superconductor [@Tinkham; @Yagi; @Hubler]. The tunnel current through a N-I-S junction is given by: $$I(V)= \frac{1}{eR_{N}}\int_{0}^{\infty}n_{S}(E)[f_{N}(E-eV)-f_{N}(E+eV)]dE \label{eq:1}$$ where $R_{N}$ is the normal state resistance, $f_{N}$ is the electron energy distribution in the normal metal and $n_{S}$ is the normalized density of states in the superconductor. In a superconducting wire undergoing quasiparticle injection, the superconductor gap $\Delta(T_{S})$ is suppressed locally. As this gap can be extracted from a N-I-S junction current-voltage characteristic, such a junction can be used for quasiparticles detection. Usually, an effective superconductor temperature $T_{\textrm{S}}$ is inferred from the superconductor gap-temperature dependence. Fig. 2(a) displays the differential conductance $dI/dV$ of a N-I-S probe junction (similar to an injector in Fig. 1(a)) located on a S-strip at different cooler bias voltages. At high injection, the gap suppression appears clearly, and enables a good determination of the superconductor effective temperature. This approach was used in numerous previous studies [@Tinkham; @Hubler; @Yagi]. At lower injection with a voltage closer to the gap voltage, the tunnel characteristic becomes little sensitive to quasiparticle injection. For instance, in Fig. 2a the probe junction characteristic at 1 mV injection voltage almost overlaps the equilibrium characteristic (at 0 mV). This limitation comes naturally from the saturation of the superconducting gap at low temperature $T_{\textrm{S}}$$\ll T_{\textrm{c}}$, where $T_{\textrm{c}}$ is the superconductor critical temperature. So far, the insensitivity of the N-I-S junction characteristic at low injection bias has been a major roadblock in investigating the decay of quasiparticles injected at energies just above the gap [@Yagi; @Hubler]. Instead of measuring directly the superconductor, a better detection sensitivity can be achieved by measuring the temperature of a small N-island connected to superconductor through a tunnel barrier [@Ullom]. In the absence of excess quasiparticles in the superconductor, the N-island is in thermal equilibrium with it. When the superconductor is under injection, some of the excess quasiparticles population will escape by tunneling (even at zero bias) from the superconductor to the central N-island. The injected quasiparticles population will then reach a quasi-equilibrium in the N-island with a electronic temperature $T_{\textrm{N}}$ different from the cryostat temperature $T_{\textrm{bath}}$. ![(a) Probe junction differential conductance under different injection bias voltage from the S-I-N-I-S cooler junction. (b) Low bias cooler junction data (solid black lines) at different injector 2 voltage level compared to calculated isotherms (red dashed lines) as obtained from Eq. 1 with $\Delta$ = 0.22 meV at different T$_{N}$.[]{data-label="fig:1"}](Fig-2-11.eps){height="0.68\linewidth" width="1\linewidth"} In this study, N-I-S junctions located on one S-strip of a S-I-N-I-S junction are used as quasiparticle injectors by current-biasing them. This leads to a spatial distribution of the excess quasiparticles density along the S-strip (see Fig 1 (b)). The central N-metal in the S-I-N-I-S cooler geometry is used as a detector for the quasiparticle density (at $x$ = 0) in the superconductor. In the N-island, the phase coherence time of about 200 ps (measured from a weak localization experiment in a wire from the same material) is much shorter than the mean escape time from the island estimated to about 100 ns. The N-island electronic population is then at quasi-equilibrium. Its temperature $T_{\textrm{N}}$ can be extracted from the zero bias conductance level of the S-I-N-I-S junction [@sukumar07PRL]. Further, the superconductor temperature is inferred from the N-metal temperature by considering its heat balance. As demonstrated below, this scheme is highly sensitive down to about 200 mK, where the S-I-N-I-S junction I-V becomes dominated by the Andreev current [@sukumar08PRL]. Fig. 2(b) shows the differential conductance of the cooler junction (full black lines) at different injector-2 bias voltages along with isotherms (dotted red lines) calculated from Eq. 1. Fig. 3(a) displays the central N-metal temperature extracted from the zero-bias conductance as a function of injector bias voltage. As the injector bias increases above the bath temperature $T_{\textrm{bath}}$, the temperature $T_{\textrm{N}}$ increases, indicating that more quasiparticles tunnel from the S-trip to the N-island. In order to obtain the superconductor temperature $T_{\textrm{S}}$ at the cooler edge ($x$ = $0$), we need to consider the heat balance in the normal metal. The heat flow across a N-I-S junction with different quasiparticle distribution on either side of the tunnel barrier is given by: $$P_{heat}(T_{N},T_{S})= \frac{1}{e^{2}R_{N}}\int_{-\infty}^{\infty}E n_{S}(E)[f_{N}(E)-f_{S}(E)]dE \label{eq:2}$$ where $f_{S}$ is the energy distribution function in the superconductor at temperature $T_{\textrm{S}}$. It is compensated by electron-phonon coupling power $P_{e-ph}$ so that $P_{heat}$ + $P_{e-ph}$ = 0. Here, we have used the usual expression for the electron-phonon coupling $P_{e-ph}=\Sigma U (T_{N}^{5}-T_{ph}^{5})$, where $\Sigma$ = 2 $nW \cdot \mu m^{-3}\cdot K^{-5}$ in Cu is a material-dependent constant and $U$ is the metal volume. For the normal metal phonons, the electron-phonon coupling power is compensated by the Kapitza power $P_{K}(T_{ph};T_{bath})= KA(T_{bath}^{4} - T_{ph}^{4})$, where K is an interface-dependent parameter and A the contact area. The inset of Fig. 3(a) displays the correspondence between the superconductor temperature $T_{\textrm{S}}$ and the central N-metal temperature $T_{\textrm{N}}$. We took the fitted Kapitza coupling parameter value $K.A$ = 144 $pW \cdot K^{-4}$ found in Ref. [@sukumar08PRL] in a very similar sample. The grey area shows the calculated temperature $T_{\textrm{S}}$ at different Kaptiza coupling coefficient $K$ ranging from 120 $W \cdot m^{-2} \cdot K^{-4}$ to infinity. The uncertainty in $T_{\textrm{S}}$ due to uncertainty in $K$ is negligible below 400 mK and at $T_{\textrm{S}}$ at 600 mK it is only 15 mK. ![(a) Central N-metal electronic temperature T$_{N}$ dependence on the injector 2 bias voltage at the bath temperature of 304 mK. Error in T$_{N}$ is smaller than the symbol size. Inset: calibration of the extracted superconductor temperature T$_{S}$ to the measured normal metal temperature T$_{N}$. The grey area shows the uncertainty in T$_{S}$ for different values of the Kapitza coupling coefficient. (b) Corresponding extracted S-metal temperature T$_{S}$ at x = 0. Dotted lines are fits using D$_{qp}$ = 35 cm$^{2}$/s.[]{data-label="fig:1"}](Fig-3-8.eps){height="0.65\linewidth" width="1\linewidth"} Fig. 3(b) and Fig. 4 show the extracted superconductor temperature $T_{S}$ at the cooler edge ($x$ = 0) for different injection bias, in the two injectors at three different bath temperatures $T_{\textrm{bath}}$ = 100 mK, 304 mK and 500 mK respectively. We have succeeded in obtaining accurately the superconductor temperature for injection bias voltage close to the gap voltage. For a given injection bias in the two injectors, the temperature $T_{\textrm{S}}$ of the detector at $x$ = 0 is higher for a closer injector. This confirms qualitatively the diffusion-based relaxation of hot quasiparticles in the superconductor. In an out-of-equilibrium superconductor, the quasiparticle density $N_{qp}$ and the phonon density of $2\Delta$ energy $N_{2\Delta}$ are coupled to each other by the well-known Rothwarf-Taylor (R-T) equations [@Rothwarf]. In a recent work [@sukumar08arxiv], some of us have extended the R-T model to include the influence of the trap junction on the quasiparticle diffusion. We considered a superconducting strip covered by a second normal metal separated by a tunnel barrier, which is in practice equivalent to a device fabricated with a shadow evaporation technique [@giazotto]. The normalized excess spatial quasiparticle density z(x) in the S-strip is given by the solution of the differential equation: $$\begin{aligned} D_{qp}\frac{d^2z}{dx^{2}}=\frac{z}{\tau_{0}}+\frac{z+z^{2}/2}{\tau_{eff}}.\end{aligned}$$ From this equation, one can find the quasiparticle decay length $\lambda$ = $\sqrt{D_{qp}\tau_{eff}/\alpha}$, where $\tau_{eff}$ is the material dependent effective recombination time and $\alpha$ = 1 + $\tau_{eff}/\tau_{0}$ ($>$1) is the enhancement ratio of the quasiparticle decay rate due to the presence of the trap junction. When the trapping effect is dominant, the quasiparticle decay length reduces to $\sqrt{D_{qp}\tau_{0}}$. The trap characteristic time $\tau_{0}$ describes the rate of quasiparticles escaping to the N-metal trap. It is defined as $\tau_{0} = e^{2}R_{NN}N(E_{F})d_{S}$, where $R_{NN}$ is the specific resistance of the trap junction and $d_{S}$ is the thickness of S. At equilibrium, the density of quasiparticles $N_{qp0}$ in the superconductor at temperature $T_{\textrm{S}}$ ($<T_{\textrm{c}}$) decays exponentially: $N_{qp0}(T_{S})=N(E_{F})\Delta(\pi k T_{S}/2\Delta)^{1/2}\exp[-\Delta/kT_{S}]$, where N$(E_{F})$ is the density of states at the Fermi level. In our experiment, the injectors are biased just above the gap voltage. Here, we assume that the injected quasiparticles relax fast (in comparison to other recombination processes) to the superconductor gap energy level and thus can be afterwards adequately described by the coupled R-T equations. To compare our experimental result with the theoretical model, we have extended the description shown in Ref. [@sukumar08arxiv]. We solved Eq. 3 numerically with boundary conditions so as to include the injection: $\frac{dz}{dx}|_{+} - \frac{dz}{dx}|_{-} = \frac{I_{inj}}{\lambda}$ at x = a; the detection: $\frac{dz}{dx} = 0$ at $x$ = 0; and the finite length of the S-strip: $z = 0$ at $x$ = L$_{S}$ into the model. The last boundary condition $x$ = L$_{S}$ provides an additional path for the excess quasiparticles to thermalize in addition to the N-trap. Further, the locally enhanced quasiparticle density $N_{qp}(x)$ is described by an equilibrium quasiparticle density $N_{qp0}$ such that $N_{qp}(x=0) = N_{qp0}(T =T_{S})$ to obtain the theoretical superconductor temperature $T_{\textrm{S}}$ [@Parker]. ![Extracted superconductor temperature T$_{S}$ as a function of injector bias voltage at a cryostat temperature of 100 mK (square dots) and 500 mK (circle dots). (a) and (b) correspond to an injection from injector 1 and 2 respectively. Dotted lines show the fits with the parameter D$_{qp}$ = 35 cm$^{2}$/s. Red and blue dashed lines are the calculated curve with the same D$_{qp}$ = 35 cm$^{2}$/s and for $\tau_{0}$ = 12 ns and $\infty$ respectively.[]{data-label="fig:1"}](Fig-4-12.eps){height="0.65\linewidth" width="1\linewidth"} In the fit procedure, we used the calculated values of $\tau_{eff}$ and $\tau_{0}$. The calculated effective recombination time for Al is $\tau_{eff}$ = 14 $\mu$s, [@kaplan] which is close to the experimental value in Ref. [@Gray]. For our sample parameters, the calculated trap characteristic time $\tau_{0}$ is equal to 0.3 $\mu$s. The model has then only one free parameter, which is the quasiparticle diffusion coefficient $D_{qp}$. We obtain a quantitative agreement between the theoretical predictions for $D_{qp}$ = 35 cm$^{2}$/s and the experiment (dotted lines in Fig. 3(b) and Fig. 4) on the two injectors, for every injection voltage, and for a bath temperature of 100 mK to 500 mK. The fit-derived value of D$_{qp}$ is comparable to the measured diffusion coefficient for Al at 4.2 K, and corresponds to $\lambda$ = 30 $\mu$m. This excellent agreement demonstrates that we quantitatively understand the diffusion of quasiparticles in the superconductor in proximity with the metallic trap junction. The quasiparticles relax mostly through two channels: N-metal trap (decay length $\lambda$) and absorption in the reservoir (decay length L$_{S}$). In our device, the decay length of both channels is around  30 $\mu$m, thus they act with a similar efficiency. Fig. 4 red and blue dashed lines shows the calculated superconductor temperature $T_{\textrm{S}}$ at $T_{\textrm{bath}}$ = 100 mK for a more transparent trap junction $\tau_{0}$ = 12 ns and $\infty$, in parallel to the experiment and its fit. For $\tau_{0} = \infty$, the excess quasiparticles relax at the reservoir $x = L_{S}$. For $\tau_{0}$ = 12 ns, the decay length is $\lambda$ = 6.5 $\mu$m, so that trapping then dominates the quasiparticles absorption. At such transparency the influence of proximity effect cannot be ignored [@VOutilainen]. In conclusion, we have studied experimentally the diffusion of quasiparticles injected in a superconductor with an energy close to the gap voltage $\Delta$ in the presence of normal metal trap traps. Our study demonstrates that in such devices quasiparticle trapping competes with relaxation in the reservoir. This new knowledge is of great importance in improving the geometry of the future cooling devices and other superconductor, based low-temperature devices. The authors are grateful to Nanofab-CNRS. SR acknowledges the support of CNST NIST during preparation of manuscript. This work is funded by MICROKELVIN, the EU FRP7 low temperature infrastructure grant 228464. [1]{} J. M. Rowell and T. S. Sui, [Phys. Rev. B $\textbf{14}$, 2456 (1976).]{} F. Giazotto, T. T. Heikkilä, A. Luukanen, A. M. Savin, and J. P. Pekola, [Rev. Mod. Phys. $\textbf{78}$, 217 (2006).]{} M. Nahum, T. M. Eiles, and J. M. Martinis, [Appl. Phys. Lett. $\textbf{65}$, 3123 (1994).]{} M. M. Leivo, J. P. Pekola, and D. V. Averin, [Appl. Phys. Lett. $\textbf{68}$, 1996 (1996).]{} S. Rajauria, P. S. Luo, T. Fournier, F. W. Hekking, H. Courtois, and B. Pannetier, [Phys. Rev. Lett. $\textbf{99}$, 047004 (2007). ]{} J. Jochum, C. Mears, S. Golwala, B. Sadoulet, J. P. Castle, M. F. Cunningham, O. B. Drury, M. Frank, S. E. Labov, F. P. Lipschultz, H. Netel and B. Neuhauser, [J. Appl. Phys. $\textbf{83}$, 3217 (1998).]{} S. Rajauria, H. Courtois, and B. Pannetier, [Phys. Rev. B $\textbf{80}$, 214521 (2009).]{} K. M. Lang, S. Nam, J. Aumentado, C. Urbina and J. M. Martinis, [IEEE Trans. on Appl. Superconduct. $\textbf{13}$, 2 (2003)]{}; R. Lutchyn, L. Glazman and A. Larkin, [Phys. Rev. B $\textbf{72}$, 014517 (2008).]{} N. A. Court, A. J. Ferguson, Roman Lutchyn and R. G. Clark, [Phys. Rev. B $\textbf{77}$, 100501(R) (2008).]{} N. E. Booth, [Appl. Phys. Lett. $\textbf{50}$, 2993 (1987)]{}; C. M. Wilson and D. E. Prober, [Phys. Rev. B $\textbf{69}$, 094524 (2004).]{} J. P. Pekola, D. V. Anghel, T. I. Suppula, J. K. Suoknuuti, A. J. Manninen, and M. Manninen, [Appl. Phys. Lett. $\textbf{76}$, 2782 (2000).]{} J. Voutilainen, T. T. Heikkilä and N. B. Kopnin, [Phys. Rev. B $\textbf{72}$, 054505 (2005)]{}; A.S. Vasenko and F. W. Hekking, [J. Low Temp Phys $\textbf{154}$, 221-232 (2009).]{} S. Rajauria, P. Gandit, T. Fournier, F. W. Hekking, B. Pannetier and H. Courtois, [Phys. Rev. Lett. $\textbf{100}$, 207002 (2008). ]{} M. Tinkham, [Phys. Rev. B $\textbf{6}$, 1747 (1972).]{} R. Yagi, [Phys. Rev. B $\textbf{73}$, 134507 (2006)]{}; K. Yu. Arutyunov, H. P. Auraneva and A. S. Vasenko, [Phys. Rev. B $\textbf{83}$, 104509 (2011).]{} A. Rothwarth and B. N. Taylor, [Phys. Rev. Lett. $\textbf{19}$, 61 (1967).]{} W. H. Parker, [Phys. Rev. B $\textbf{12}$, 3667 (1975).]{} S. B. Kaplan, C. C. Chi, D. N. Langenberg, J. J. Chang, S. Jafarey and D. J. Scalapino, [Phys. Rev. B $\textbf{14}$, 4854 (1976).]{} K. E. Gray, [J. Phys. F: Met. Phys. $\textbf{1}$, 290 (1971).]{} J. N. Ullom, P. A. Fisher, and M. Nahum, [Phys. Rev. B $\textbf{61}$, 14839 (2000).]{} F. Hubler, J. Camirand Lemyre, D. Beckmann and H. v. Lohneysen, [Phys. Rev. B $\textbf{81}$, 184524 (2010).]{}

--- abstract: 'Considering ultracold atoms traversing a high-$Q$ Fabry-Perot cavity, we theoretically demonstrate a quantum nondemolition measurement of the photon number. This fully quantum mechanical approach may be understood utilizing concepts as effective mass and group velocity of the atom. The various photon numbers induce a splitting of the atomic wave packet, and a time-of-flight measurement of the atom thereby reveals the photon number. While repeated atomic measurements increase the efficiency of the protocol, it is shown that by considering long interaction times only a few atoms are needed to resolve the photon number with almost perfect accuracy.' author: - Jonas Larson$^1$ - 'Mahmoud Abdel-Aty$^2$' title: Cavity QED nondemolition measurement scheme using quantized atomic motion --- Introduction ============ Any measurement of a quantity $\hat{A}$ of a quantum system has an impact on the system itself. For example, exactly determining $\hat{A}$ implies that any knowledge of a conjugate variable is lost. Moreover, the detection might even be destructive by nature, [*e.g.*]{} the standard way of measuring the electromagnetic field is by photocounting detectors where photons are actually absorbed. However, there exist situations where a measurement of some quantity $\hat{A}$ does not induce any back-action quantum noise on $\hat{A}$. That is, provided $\hat{A}$ is a constant of motion, then a second measurement of $\hat{A}$, after some time-delay relative the first measurement, would reveal the same value as obtained in the first measurement. This is called a quantum nondemolition (QND) measurement [@qnd]. By now, QND measurements of the optical field have been demonstrated both for an optical fiber [@qndexp1] and in cavity QED setups [@qndexp2]. In the cavity experiments to date, an atomic Ramsey interferometer technique has been utilized. The different phases aquired for an atom interacting dispersively with the cavity field either in its excited state $|e\rangle$ or ground state $|g\rangle$ contain information about the photon number $n$. Repeating the atomic QND measurements sufficiently many times determines the photon number definitely. The kinetic energy of the atoms in these experiment greatly exceeds the atom-field interaction energy, and thereby they can be safely treated by classical means. For ultracold atoms on the other hand, mechanical effects induced by the light fields become important. Such actions are indeed the building blocks for cooling and trapping of neutral atoms [@atcol]. In the cavity QED community, it has long been known that the atom-field dynamics may be considerably modified by treating the atomic motion quantum mechanically together with taking spatial mode variations into account [@mazer]. The system now includes additional degrees of freedom that become correlated with the cavity field. Already back in 1989 it was shown that deflection of a beam of ultracold atoms resonantly interacting and transversely scattered from a quantized standing wave field will depend on the actual photon distribution of the field [@stig]. For a classical field, the same effect, named optical Stern-Gerlach, was presented in Ref. [@optSG]. These observations led to the idea of performing QND measurments of the cavity field using ultracold atoms dispersively interacting with the field. Reference [@qndquant1] considered a beam of ultracold atoms transversely passing a Fabry-Perot cavity. In the Raman-Nath regime, valid for very short interaction times, it was shown that the deflection of the atomic beam after interacting with the cavity field depends on the number of photons. Thus, by recording the atomic positions of a sequence of atoms having interacted with the cavity field, a QND measurement of the photon number is possible. In this paper we address a different QND measurement where the time-of-flight of the atom is recorded. Instead of studying the deflection of transversely passing atoms, we consider atoms traversing the Fabry-Perot cavity along its axial direction. Our treatment is fully quantum mechanical, taking into account for; the atomic scattering effects occurring as the atom enters and exits the cavity, the quantum pressure arising from the atomic kinetic energy term (hence going beyond the Raman-Nath regime), as well as the uncertainty of the velocity of the incoming atom. The scheme is first studied in a semi-analytical model which relies on the concepts of effect mass and group velocity for the atom. By imposing a sort of single-band approximation, the semi-analytical model demonstrates that a single atom is sufficient for performing the QND measurement in the ideal situation provided the effective interaction time is long enough. The full system, going beyond single-band and taking into account for a finite cavity, is considered numerically, and it is found that typically around ten atoms is enough for achieving a highly efficient QND measurement. Model system {#sec2} ============ We consider an ultracold two-level atom sent through a Fabry-Perot cavity along its axial axis. The internal ground and excited atomic states are labeled $|g\rangle$ and $|e\rangle$, and their energy difference $\hbar\Omega$. The atom interacts with a single cavity mode with frequency $\omega$. Moving to an interaction frame and imposing the rotating wave approximation, the Hamiltonian reads $$\hat{H}'=\frac{\hat{P}^2}{2m}+\frac{\hbar\tilde{\Delta}}{2}\hat{\sigma_z}+\hbar \left[g(\hat{X})\hat{a}^\dagger\hat{\sigma}^-+g^*(\hat{X})\hat{\sigma}^+\hat{a}\right].$$ Here, $\hat{X}$ and $\hat{P}$ are the atomic center-of-mass position and momentum respectively, $\tilde{\Delta}=\Omega-\omega$ is the atom-field detuning, $g(\hat{X})$ the effective position dependent coupling, $\hat{a}^\dagger$ ($\hat{a}$) the creation (annihilation) operators for the field, and the Pauli-operators are $\hat{\sigma}_z=|e\rangle\langle e|-|g\rangle\langle g|$, $\hat{\sigma}^+=|e\rangle\langle g|$, and $\hat{\sigma}^-=|g\rangle\langle e|$. For a cavity of length $L$ we have $$\label{pot1} g(\hat{X})=\left\{\begin{array}{lll}\tilde{g}_0\cos(k\hat{X}) & 0\leq \hat{X}\leq L\\ 0 & \mathrm{othervice}.\end{array}\right.$$ The length $L$ of the cavity is assumed to be much larger than the wavelength of the field, and hence, the Hamiltonian is quasi-periodic. $k$ is the wave number and $\tilde{g}_0$ the effective atom-field coupling. Letting $E_{2r}=\frac{\hbar^2k^2}{m}$ define the characteristic energy, we scale the parameters accordingly $$\begin{array}{ll} g_0=\tilde{g}_0/E_{2r}, & \Delta=\tilde{\Delta}/E_{2r}, \\ \\ t=\tilde{t}E_{2r}/\hbar, & \hat{x}=k\hat{X}, \end{array}$$ where $\tilde{t}$ is the unscaled time. Using the fact that within the rotating wave approximation, the number of excitations $\hat{N}=\hat{a}^\dagger\hat{a}+\hat{\sigma}_z/2$ is preserved, the Hamiltonian may be written on block form within the states $|n,g\rangle$ and $|n-1,e\rangle$, $|n\rangle$ being the Fock state with $n$ photons. Thus, we have $\hat{H}'=\hat{H}_0'\otimes\hat{H}_1'\otimes\hat{H}_2'\otimes...$ where $\hat{H}_0'=-d^2/2d\hat{x}^2-\Delta/2$ and $$\label{ham1} \hat{H}_n'=-\frac{1}{2}\frac{d^2}{d\hat{x}^2}+\left[\begin{array}{cc} \frac{\Delta}{2} & g_0\cos(\hat{x})\sqrt{n} \\ \\ g_0\cos(\hat{x})\sqrt{n} & -\frac{\Delta}{2}\end{array}\right].$$ Here we have taken $g_0$ to be real. For ultracold atoms in the dispersive regime, $\Delta\gg g_0\sqrt{n}$, we adiabatically eliminate the excited atomic level $|e\rangle$ to obtain a Hamiltonian describing the dynamics for the atomic ground state alone. Following standard procedures [@adel], one derives $$\label{ham2} \hat{H}_n=-\frac{1}{2}\frac{d^2}{d\hat{x}^2}+Un\cos^2(\hat{x}),$$ where $U=\frac{g_0^2}{\Delta}$ is the amplitude of the single photon dipole induced potential. Semi-analytical analysis {#sec3} ======================== In the previous section we introduced the model Hamiltonian. Furthermore, it was assumed that $L\gg2\pi k^{-1}$, making sure that the system is quasi periodic. Neglecting boundary effects arising from having a finite cavity length, the corresponding eigenvalue problem relaxes to the Mathiue equation $$\hat{H}_n|\phi_\nu^n(q)\rangle=E_\nu^n(q)|\phi_\nu^n(q)\rangle.$$ As a periodic operator, the spectrum $E_\nu^n(k)$ for given $n$ is characterized by a band index $\nu=1,2,3,...$ and a quasi momentum $q$ defined within the first Brillouin zone ($-1\leq q<1$). The corresponding eigenstates $|\phi_\nu^n(q)\rangle$ are the so called Bloch functions. A typical spectrum is envisaged in Fig. \[fig1\]. The figure shows the first three energy Bloch bands $E_\nu^n(q)$. ![The first three Bloch bands $E_\nu^n(q)$ ($\nu=1,\,2,\,3$). The amplitude $Un=0.5$. All parameters are dimensionless. []{data-label="fig1"}](fig1.eps){width="8cm"} Typically, incoming atoms are not in pure momentum eigenstates. The atomic velocity selection is never perfect causing an uncertainty in the mean momentum. This is taken into account by considering initial Gaussian states $$\label{inat} \psi(p,0)=\frac{1}{\sqrt[4]{2\pi\Delta_p^2}}e^{-\frac{(p-p_0)^2}{4\Delta_p^2}},$$ where $\Delta_p$ is its width determined by the velocity uncertainty in the state preparation, and $p_0$ the mean initial momentum. The time evolution of this state is rendered by the Hamiltonian (\[ham2\]). For a small coupling $Un$ and moderate spreading $\Delta_p\ll1$, the atomic state will predominantly populate a single Bloch band $\nu'$ with average quasi momentum $q_0=p_0-\nu'$. By expanding the corresponding energy around $q_0$ $$E_{\nu'}^n(q)\approx E(q_0)+v_g^n(q_0)(q-q_0)+\frac{1}{2}\frac{1}{m_n^*(q_0)}(q-q_0)^2,$$ where $v_g^n=dE_{\nu'}^n(q)/dq|_{q=q_0}$ and $m_n^*(q_0)=\left(d^2E_{\nu'}^n(q)/dq^2\right)^{-1}$ it follows that, for a given $n$, the time evolved state in $x$-representation approximates [@effmass] $$\psi_n(x,t)\approx\frac{1}{\sqrt[4]{8\pi\Delta_p^2\Delta_x^4}}e^{-\frac{(x-v_g^nt)^2}{4\Delta_x^2}},$$ where $$\Delta_x^2=\frac{1}{4\Delta_p^2}+\frac{it}{2m_n^*},$$ and $v_g^n\equiv v_g^n(q_0)$ and $m_n^*\equiv m_n^*(q_0)$. Within these approximations, the wave packet preserves its Gaussian form, moves with a group velocity $v_g^n$, and spreads according to the effective mass $m_n^*$. For a general initial state of the cavity field $|\phi\rangle=\sum_{n=0}^\infty c_n|n\rangle$, we obtain the single-band approximated time evolved atom-field state $$\label{timestate1} \Psi(x,t)\approx\sum_{n=0}^\infty \psi_n(x,t)c_n|n\rangle,$$ and the corresponding atomic density $$\label{dens} \rho_{at}(x,t)=\sum_{n=0}^\infty|c_n|^2|\psi_n(x,t)|^2.$$ Due to the uncertainty of photon numbers, it follows that the atomic state will split into a set of Gaussians. Hence, whenever the distances between consecutive Gaussian wave packets exceeds their widths, a measurement of the atomic position will reveal the photon number $n$. Equivalent to a position measurement is a time-of-flight measurement, where the time it takes the atom to traverse the cavity is recorded. The atomic density $\rho_{at}(x,t=400)$ is depicted in Fig. \[fig2\]. The initial cavity field is a coherent state $$c_n=e^{-|\alpha|^2/2}\frac{\alpha^n}{\sqrt{n!}}$$ with an average number of photons $\bar{n}=|\alpha|^2=4$. For the current set of parameters, the wave packet evolves on the third Bloch band. The inset numbers give the corresponding number of photons $n$. For $\bar{n}=4$, the population of the Fock state with $n=10$ is less than one percent and therefore only the first ten Fock states are seen. ![The atomic density (\[dens\]) at time $t=400$. The field is initially in a coherent state with $\bar{n}=4$ and the inset numbers indicate the corresponding number of photons. The parameters are $p_0=2.58$, $\Delta_p=p_0/50$, and $U=0.7$. The initial velocity $p_0$ implies that the atomic wave packet evolves on the third Bloch band. []{data-label="fig2"}](fig2.eps){width="8cm"} For the example of Fig. \[fig2\], a single atomic detection is most likely sufficient for determining the photon number. However, for shorter interaction times the Gaussians overlap and a single measuremnt cannot resolve the photon number. Nonetheless, by repeating the measurement procedure for a second and third atom and so on, the photon distribution will finally pick a single Fock state $|n\rangle$ despite the non-zero Gaussian overlaps. Naturally, the less resolved the peaks are the more atoms are needed. Stated in other words, a strong atom-field correlation implies fewer atomic detections. Considering pure initial states and a closed system, we employ the von Neumann entropy as an estimate of the amount of atom-field correlation/entanglement. For the reduced field density operator $\rho_f(t)=\mathrm{Tr}_{at}[\rho(x,t)]$, where $\rho(x,t)$ is the full system density operator and the trace is over the atomic motional degrees of freedom, we have the von Neumann entropy $$\label{entropy} S_f(t)=-\mathrm{Tr}_f\big[\rho_f(t)\ln[\rho_f(t)]\big].$$ The trace is over field degrees of freedom. The maximum entropy is given by $S_f^{(max)}=-\sum_n|c_n|^2\ln[|c_n|^2]$, with $c_n$ the initial photon amplitudes. Using the same parameters as in Fig. \[fig2\], we present the corresponding entropy in Fig. \[fig3\]. The dashed line gives the maximum entropy, and it is clear that $S_f(t)$ approaches this value in the long time limit. On the other hand, for times $t$ less than say 100, a series of atomic detection is most likely required for an efficient QND measurement. ![Time evolution of the von Neumann entropy (\[entropy\]). Parameters are the same as in Fig. \[fig2\]. The amount of entanglement is rapidly increasing and asymptotically reaches its maximum value $S_f^{(max)}$ indicated by the dashed line. []{data-label="fig3"}](fig3.eps){width="8cm"} Numerical analysis {#sec4} ================== Complications due to a realistic system --------------------------------------- In realistic experimental setups, several complicating effects arise that were not fully addressed in the previous section. In this section we will take them into account numerically. By entering the cavity, we assume that the atom feels a fairly sudden turn on of the dipole induced cavity potential. To model the potential (\[pot1\]) for the whole $x$-axis, we introduce an envelope function such that $$V(\hat{x})\!=\!\frac{U}{2}\!\left[\tanh\!\left(\!\frac{\hat{x}\!-\!L/2}{X_s}\right)\!-\tanh\!\left(\!\frac{\hat{x}\!+\!L/2}{X_s}\right)\!\right]\!,$$ where $X_s$ determines the slope of the envelope function around $x=\pm L/2$ and we naturally chose $X_s\ll L$. The rapid change in $V(\hat{x})$ in the vicinity of $x=\pm L/2$ will induce some backward scattering of both the incoming and outgoing atomic wave packets. Thereby, especially for large amplitudes $Un$, part of the wave packet will not reach the detector. Experimentally, backward scattered atoms may be ignored and we will focus only on the forward scattered atoms, those who are detected. The measured time $t_n$ is presumably the time between atomic state preparation and the detection. As no backwarded scattered atoms are recorded, $t_n$ includes the free space time propagation $t_{fr}$ plus the time $\tau_n$ spent inside the cavity; $t_n=\tau_n+t_{fr}$. Since the atom-field interaction is dispersive, the atomic velocity before and after the cavity are the same and independent of $n$, and consequently $t_n$ is an indirect measure of the group velocity $v_g^n(q_0)\approx L/\tau_n$. Yet another complication is the fact that the true state of the atom (\[inat\]) is not restricted to a single Bloch band $\nu'$. From the symmetry of the Hamiltonian, and given $n$, it follows that the atomic state expressed in Bloch functions is $$\label{blochin} \psi(p,0)=\sum_\nu\int_{-1}^{+1}d_\nu^n\psi(q,0)|\phi_\nu^n(q)\rangle dq.$$ Here, $\psi(q,0)$ is $\psi(p,0)$ with the real momentum replaced by the quasi momentum [@com1], and the $d_\nu^n$ are the proper weights given $Un$. For small $Un$, the coefficient $|d_{\nu'}^n|\approx1$ with $\nu'=p_0-q_0$. For the parameters employed in the previous section one finds that all $|d_{\nu'}^n|$ is noticeable smaller than 1, and the atomic wave packet will therefore split up into sub-packets as it enters the cavity. The different parts deriving from atomic propagation according to the corresponding Bloch bands. It is clear that it is a trade-off between having a large separation of group velocities $v_g^n$, and small atomic scattering and splitting. The first favors large amplitudes $U$, while effects originating from scattering and splitting are decreased for small values of $U$. Despite this, we will now demonstrate that by repeated measurements, the efficiency can be made asymptotically close to unity. Results ------- The Schrödinger equation (\[ham1\]) is solved using the split-operator method [@split]. The atmic intial condition is a Gaussian with width $\Delta_x=15\gg2\pi k^{-1}$, initial position $x_0=-L/2-70$, and initial momentum $p_0=3.75$. The cavity length $L$ is varied, while $X_s=0.2$ and $U=0.7$ are kept fixed throughout. The initial cavity field is as before, [*i.e.*]{} a coherent state with $\bar{n}=4$. Note that the atomic wave packet is initialized in the regime where the amplitude of the cavity field is approximately zero; $V(-L/2-70)\approx0$. The time propagation $t_f$ is performed till the various fowardly scattered atomic wave packet components have left the interaction region. At this instant $t_f$, due to the back scattering we renormalize $\rho_{at}(x,t_f)$ accordingly $$\rho_{at}'(x,t_f)\equiv\left\{\begin{array}{ll} 0, & x<L/2 \\ \frac{\rho_{at}(x,t_f)}{\int_{L/2}^\infty\rho_{at}(x,t_f)dx}, & x\geq L/2. \end{array}\right.$$ In Fig. \[fig4\] (a) we present the renormalized atomic density for $x\geq L/2$. The cavity length for this example is fairly long, $L=1400$. Compared with the ideal situation of Fig. \[fig2\], we note that in this more realistic situation the various atomic wave packet components $\psi_n(x,t_f)$ are less resolved from each other. In Fig. \[fig4\] (b) we give the corresponding components $\psi_n'(x,t_f)$ renormalized within the interval $x\in[L/2,\infty]$. The numbers represent the photon number $n$. The peaks around $x\approx2000$, corresponding to photon numbers $n=6,\,7,\,8$, originate from the wave packet component propagating on higher excited Bloch bands. Expectedly, this component becomes significantly populated only for strong field amplitudes $Un$. ![The renormalized atomic density $\rho_{at}'(x,t_f)$ (a) and the different atomic wave packet components $\psi_n'(x,t_f)$ (b). The cavity length $L=1400$ and final time $t_f=660$. Other dimensionless parameters are $\Delta_x=15$, $x_0=-770$, $p_0=3.75$, $\bar{n}=4$, $U=0.7$, and $X_s=0.2$.[]{data-label="fig4"}](fig4.eps){width="8cm"} For the example of Fig. \[fig4\], a single atomic measurement will not reveal the photon number with very high efficiency. However, as mentioned in the previous section, repeated atomic measurements will improve the scheme considerably. First we note that numerically, instead of considering a time measurement we can equally well freeze the time evolution and make a position measurement. After the first atom has been recorded with a corresponding position $x_r$, the photon distribution becomes $|c_n^{(1)}|^2=|\psi_n'(x_r,t_f)|^2|c_n|^2/N$, where $N=\sum_n|c_n^{(1)}|^2$. For the second atom traversing the cavity, its atomic density $\rho_{at}'^{(1)}(x,t)$ is given by Eq. (\[dens\]) with $c_n$ replaced by $c_n^{(1)}$. Each atomic measurement acts as a “photon filter" [@jonas1], modifying the photon distribution with the weights $\psi_n'(x_r,t_f)$. For the numerical simulation, the position $x_r$ of atom $j$ is randomly picked according to the corresponding probability distribution $\rho_{at}'^{(j)}(x,t_f)$. Once the position has been determined, the photon distribution is adjusted accordingly and a new atomic density for atom $j+1$ is calculated. The process is repeated until only a single photon component survives the filtering. The number of iterations needed depends on; the randomly picked numbers and on how well separated the atomic wave packet components are. In general, large $L$ implies less atomic measurements. ![Simulations of the QND measurement using successive atomic measurements. The three plots correspond to $L=1400$, $L=600$, and $L=200$ respectively. For longer cavities, less atomic measurements are required to have an efficient QND measurement. The rest of the parameters are the same as in Fig. \[fig4\] except $x_0=-L/2-70$. []{data-label="fig5"}](fig5.eps){width="8cm"} The results of three simulations for different cavity lengths $L$ are presented in Fig. \[fig5\]. In the first case (a), the parameters are the same as for Fig. \[fig4\]. Already after three atomic measurements the photon distribution has collapsed to approximately a single Fock state. For the cases with $L=600$ (b) and $L=200$ (c), considerably more atoms are needed in order to single out a photon number. ![Evolution of the $Q$-function (\[qfun\]) after the positions of $j=0,\,1,\,5,\,10$ atoms have been recorded. For the initial state (a), the $Q$-function is a Gaussian centered around $(\alpha_r,\alpha_t)=(2,0)$. Even after 1 atomic measurement (b), the phase space quasi distribution does show a circular structure characterizing the Fock number state. After 5 measurements (c), the circular structure has actually declined, while after 10 atoms (d) the distribution is almost perfectly circular with a radius $|\alpha|\approx1.7$ corresponding to the $n=3$ Fock state. The parameters are as in Fig. \[fig5\] (b), but the sequence of random numbers $x_r$ is not the same as those used for that plot.[]{data-label="fig6"}](fig6.eps){width="8cm"} Each atomic measurement is projective, leaving the field in a pure state $|\phi\rangle_j=\sum_n\psi_n'^{(j)}(x_r,t_f)c_n^{(j)}|n\rangle/N_j$, $N_j$ being the proper normalization constant. The filtering projection onto a single Fock state seen in Fig. \[fig5\] can also be demonstrated via the phase space distributions, [*e.g.*]{} the Husimi $Q$-function [@mandel] $$\label{qfun} Q^{(j)}(\alpha)=\frac{\langle\alpha|\phi\rangle_{\!j\,j}\!\langle\phi|\alpha\rangle}{\pi}.$$ Here, $|\alpha\rangle$ is a coherent state with amplitude $\alpha$. For a coherent state, the $Q$-function is Gaussian, while for a Fock state it is a circle with radius $|\alpha|=\sqrt{n}$. Figure \[fig6\] gives four examples of $Q^{(j)}(\alpha)$ for $j=0$ (a), $j=1$ (b), $j=5$ (c), and $j=10$ (d). The initial state and the parameters are the same as those of Fig. \[fig5\] (b). In plots (b) and (c), there are two photon numbers dominating, $n=4,\,5$. In (d), almost all population resides in the $n=3$ state. Discussion and concluding remarks {#sec5} ================================= The idea behind our scheme is different from that of Ref. [@qndquant1]. In [@qndquant1], due to virtual exchange of photons with the cavity field, atoms are deflected perpendicularly with respect to their initial velocity. A position measurement is therefore an indirect measure of the number of photons that has been exchanged. Even for short interaction times (imposing the Raman-Nath approximation), the atom can acquire a certain number of $2\hbar k$ momentum kicks, where the factor 2 comes from the fact that the interaction is dispersive and absorption of a photon by the atom is always accompanied by emittance of a photon. The measurement then projects onto any of the momentum eigenstates $p=2r\hbar k$, $r=0,\,1,\,2,\,...\,$. In the present scheme, on the other hand, the time-of-flight measurement indirectly gives the wave packet velocity which for weak couplings approximate the group velocity $v_g$, characterizing the average velocity $\langle\hat{p}\rangle/m$ for the corresponding Bloch state. To make these arguments more transparent, in Fig. \[fig7\] we display the time evolution of $\langle\hat{p}\rangle$ for the $L=600$ cavity and $n=0,\,1,\,2,\,3,\,4$. The plot makes clear that the velocity decreases for increasing photon numbers, and also that the atom regains its initial velocity after exiting the interaction region. ![The average momentum for an atomic wave packet traversing the $L=600$ cavity. The different $n$’s give the number of photons. The parameters are the same as those of Fig. \[fig5\] (b). []{data-label="fig7"}](fig7.eps){width="8cm"} The measurement efficiency is enhanced by considering long cavities. However, long cavities naturally implies long interaction times and cavity losses may become significant. For short cavities, the individual atomic interaction times are shorter but, on the other hand, more atomic measurements are needed. This again might cause long total operational times. One way of decreasing the total process time is by using feedback-techniques [@feedback]. Nonetheless, even when cavity losses become important the scheme can be useful. For a lossy cavity, one typically considers an external pumping of the cavity, keeping the field in a coherent state with amplitudes determined from balancing the pump and loss rates. If the time-scale for the field is much shorter than that of repeated atomic measurements, the field attains its steady state between each measurement. The result of the measurements will then reveal the steady state photon distribution. As a summary, we have introduced a QND measurement scheme of the photon numbers in a cavity. It relies on time-of-flight measurements of atoms traversing a Fabry-Perot cavity along its axial direction. The field intensity $n$ directly affects the atomic velocity while traversing the cavity; the velocity drop is increased for larger photon numbers $n$. This can be explained using the language of group velocities for particles moving in periodic potentials. Employing this concept, we argued that for a sufficiently long cavity a single atom can resolve the photon number non-destructively. By decreasing the cavity length, repeated atomic measurements are required. Our analytical results were numerically verified considering more realistic cavity QED setups. In particular, we found that typically more than single atomic measurements are needed for reliable QND measurements. We further explained that even for lossy cavities the scheme is of interest as it can be used as a tool to measure the full steady state photon distribution. Getting access to the full state of the field would require more sophisticated approaches like tomography measurements [@tomo]. JL acknowledges support from the MEC program (FIS2005-04627). [999]{} W, G. Unruh, Phys. Rev. D [**18**]{}, 1764 (1978); D. F. Walls and G. J. Milburn, [*Quantum Optics*]{}, (Springer-Verlag, Berlin 2008). M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, Phys. Rev. Lett. [**57**]{}, 2473 (1986); A. LaPorta, R. E. Slusher, and B. Yurke, Phys. Rev. Lett. [**62**]{}, 28 (1989). M. Brune, E Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J.-M. Raimond, and S. Haroche, Phys. Rev. Lett. [**77**]{}, 4887 (1996); G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, nature [**400**]{}, 239 (1999); C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.-M. Raimond, and S. Haroche, Nature [**448**]{}, 889 (2007). P. Meystre, [*Atom Optics*]{}, (Springer-Verlag, Berlin 2001). B. G. Englert, J. Schwinger, A. O. Barut, and M. O. Scully, Europhys. Lett. [**14**]{}, 25 (1991); M. O. Scully, G. M. Meyer, and H. Walther, Phys. Rev. Lett. [**76**]{}, 4144 (1996). P. Meystre, E. Schumacher, and S. Stenholm, Opt. Commun. [**73**]{}, 443 (1989). T. Sleator, T. Pfau, V. Balykin, O. Carnal, and J. Mlynek, Phys. Rev. Lett. [**68**]{}, 1996 (1992). M. J. Holland, D. F. Walls, and P. Zoller, Phys. Rev. Lett. [**67**]{}, 1716 (1991). J. Larson, J. Mod. Opt. [**53**]{}, 1867 (2006). J. Larson, J. Salo, and S. Stenholm, Phys. Rev. A [**72**]{}, 013814 (2005). For the scaled parameters, the quasi momentum $q$ is related to the real momentum $p$ via $q=\mod(p,1)$ and the band index is $\nu=p-q$. M. D. Fleit, J. A. Fleck, and A. Steiger, J. Comput. Phys. [**47**]{}, 412 (1982). J. Larson and S. Stenholm, J. Mod. Opt. [**50**]{}, 1663 (2003). L. Mandel and E. Wolf, [*Optical Coherence and Quantum Optics*]{} (Cambridge University Press, Cambridge 1995). I. Dotsenko, M. Mirrahimi, M. Brune, S. Haroche, J.-M. Raimond, and P. Rouchon, Phys. Rev. A [**80**]{}, 013805 (2009). M. Freyberger and A. M. Herkommer, Phys. Rev. Lett. [**72**]{}, 1952 (1994); S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, Nature [**455**]{}, 510 (2008); M. Brune, J. Bernu, C. Guerlin, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, I. Dotsenko, J.-M. Raimond, and S. Haroche, Phys. Rev. Lett. [**101**]{}, 240402 (2008).

--- abstract: | Serverless computing is an excellent fit for big data processing because it can scale quickly and cheaply to thousands of parallel functions. Existing serverless platforms, however, isolate functions in ephemeral, stateless containers. This means that functions cannot efficiently share memory, forcing users to serialise data repeatedly when composing functions. We observe that container-based isolation is ill-suited for serverless big data processing. Instead it requires a lightweight isolation approach that allows for efficient state sharing. We introduce *[Faaslets]{}*, a new isolation abstraction for serverless big data computing. [Faaslets]{}isolate the memory of executed functions using *software-fault isolation* (SFI), as provided by *WebAssembly*, while allowing memory regions to be shared between functions in the same address space. [Faaslets]{}can thus avoid expensive data movement when functions are co-located on the same machine. Our runtime for [Faaslets]{}, [<span style="font-variant:small-caps;">Faasm</span>]{}, isolates other resources, CPU and network, using standard Linux *cgroups*, and provides a low-level POSIX host interface for networking, file system access and dynamic loading. To reduce initialisation times, [<span style="font-variant:small-caps;">Faasm</span>]{} restores [Faaslets]{}from already-initialised snapshots. We compare [<span style="font-variant:small-caps;">Faasm</span>]{}to a standard container-based platform and show that, when training a machine learning model, it achieves a 2$\times$ speed-up with 10$\times$ less memory; for serving machine learning inference, [<span style="font-variant:small-caps;">Faasm</span>]{}doubles the throughput and reduces tail latency by 90%. author: - | [Simon Shillaker]{}\ Imperial College London - | [Peter Pietzuch]{}\ Imperial College London bibliography: - 'main.bib' title: '[<span style="font-variant:small-caps;">Faasm</span>]{}: Lightweight Isolation for Efficient Stateful Serverless Computing' ---

--- abstract: | The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of this graph for codimension one subgroups leads to splittings. We also reprove some results of Peter Kropholler on splittings of groups over malnormal subgroups and variants of them. AMS subject classification : 20F67(Primary), 22E40, 57M50(Secondary) author: - | Mahan Mj\ Department of Mathematics, RKM Vivekananda University,\ Belur Math, WB-711 202, India\ email:mahan.mj@gmail.com - | Peter Scott\ Mathematics Department\ University of Michigan\ Ann Arbor, Michigan 48109, USA.\ email: pscott@umich.edu - | Gadde Swarup\ 718, High Street Road,\ Glen Waverley,\ Victoria 3150, Australia\ email: anandaswarupg@gmail.com bibliography: - 'cxsplit.bib' title: 'Splittings and C-Complexes' --- Introduction ============ Let $M$ be a closed $3$–manifold and $f:S\rightarrow M$ an immersed least area surface such that not all complementary regions in $M$ are handlebodies. Thickening $f(S)$ in $M$ and filling in all compressing disks and balls, we obtain a codimension zero submanifold with incompressible boundary $F$. Then $\pi_{1}(M)$ splits over $\pi_{1}(F)$. An interesting special case occurs when $M$ admits two immersed least area surfaces which are disjoint, as the condition on complementary components of each of the surfaces is then automatically satisfied. The aim of this paper is to obtain group-theoretic analogues of these and related facts using the theory of algebraic regular neighbourhoods developed by Scott and Swarup [@ss-book]. Statement of Results -------------------- Let $G$ be a group and $H$ an infinite subgroup. A simplicial complex (termed $C$–complex) can be constructed from the incidence relations determined by the cosets of $H$ as follows (see [@mahan-agt]). The vertices of $C(G,H)$ are the cosets $gH$ and the $(n-1)$–cells are $n$–tuples $\{g_{1}H,\cdots,g_{n}H\}$ of distinct cosets such that $\cap_{1}^{n}g_{i}Hg_{i}^{-1}$ is infinite. When $G$ is hyperbolic and $H$ quasiconvex, this is equivalent to the incidence complex where vertices are limit sets and $(n-1)$–cells are $n$–tuples of limit sets with non-empty intersection. (See [@sageev-th] and [@GMRS] for related material.) Let $e(G)$ denote the number of ends of a group $G$, and let $e(G,H)$ denote the number of ends of a group pair $(G,H)$. Our main Theorem states: **Theorem \[split\]** Suppose that $G$ is a finitely generated group and $H$ a finitely generated subgroup. Further, suppose that $e(G) = e(H) = 1$ and $e(G,H)\geq2$. If $C(G,H)$ is disconnected, then $G$ splits over a subgroup (that may not be finitely generated). Since we are only interested in the connectivity of $C(G,H)$, it is enough to consider the connectivity of its $1$–skeleton $C_{1}(G,H)$ which has the following simple description: the vertices of $C_{1}(G,H)$ are the essentially distinct cosets $gH$ of $H$ in $G$ and two vertices $gH$ and $kH$ are joined by an edge if and only if $gHg^{-1}$ and $kHk^{-1}$ intersect in an infinite set. The principal technique used to prove Theorem \[split\] is the theory of algebraic regular neighbourhoods developed by Scott and Swarup [@ss-book] and a lemma on crossings (in the sense of Scott [@scott-cross]) which may be of independent interest. Our results have some thematic overlap with results of Kropholler [@krop] and Niblo [@niblo], and this is discussed at the end of the paper. We also prove a slight generalization of a theorem of Kropholler [@krop] and the following variant of that theorem: **Theorem \[variant of Kropholler\]** Let $G$ be a finitely generated, one-ended group and let $K$ be a subgroup which may not be finitely generated. Suppose that $e(G,K)\geq2$, and that $K$ is contained in a proper subgroup $H$ of $G$ such that $H$ is almost malnormal in $G$ and $e(H)=1$. Then $G$ splits over a subgroup of $K$. Crossing\[crossing\] -------------------- We recall certain basic notions from [@scott-cross] and [@ss-book]. We say that a subset $A$ of $G$ is $H$*–finite* if $A$ is contained in a finite number of right cosets $Hg$ of $H$ in $G$. Two subsets $X$ and $Y$ of $G$ are said to be $H$*–almost equal* if their symmetric difference $(X-Y)\cup(Y-X)$ is $H$–finite. A subset $X$ of $G$ is said to be $H$*–almost invariant* if $HX=X$, and $X$ and $Xg$ are $H$–almost equal, for all $g$ in $G$. We may also say that $X$ is *almost invariant over* $H$. Such a set $X$ is said to be *nontrivial* if both $X$ and its complement $X^{\ast}$ are not $H$–finite. The number of ends, $e(G,H)$, of the pair $(G,H)$ is $\geq2$ if and only if $G$ has nontrivial $H$–almost invariant subsets. The following simple result will be needed later. It is Lemma 2.13 of [@ss-book]. \[L2.13ofssbook\] Let $G$ be a group with subgroups $H\ $and $K$. Suppose that $Xg$ is $K$–almost equal to $X$ for all $g$ in $G$, and that $X$ is $H$–finite. Then either $X$ is $K$–finite or $H$ has finite index in $G$. We do not assume that $KX=X$, so that $X$ need not be $K$–almost invariant. We shall use the notion of crossing of almost invariant sets in the sense of Scott [@scott-cross]. Let $G$ be a finitely generated group, let $H$ and $K$ be subgroups of $G$, and let $X$ and $Y$ be almost invariant subsets of $G$ over $H$ and $K$ respectively. Let $X^{\ast}$ and $Y^{\ast}$ denote their complements. Given two subsets $X\ $and $Y$ of a group $G$, it will be convenient to use the terminology *corner* for any one of the four sets $X\cap Y$, $X^{\ast}\cap Y$, $X\cap Y^{\ast}$ and $X^{\ast}\cap Y^{\ast}$. Thus any pair $(X,Y)$ has four corners. \[defnofcrossing\]Let $X$ be a $H$–almost invariant subset of $G$ and let $Y$ be a $K$–almost invariant subset of $G$. We will say that $Y$ *crosses* $X$ if each of the four corners of the pair $(X,Y)$ is $H$–infinite. Thus each of the corners of the pair projects to an infinite subset of $H\backslash G$. It is shown in [@scott-cross] that if $X$ and $Y$ are nontrivial, then $X\cap Y$ is $H$–finite if and only if it is $K$–finite. It follows that crossing of nontrivial almost invariant subsets of $G$ is symmetric, i.e. that $X$ crosses $Y$ if and only if $Y$ crosses $X$. Next we recall some material from [@ss-book]. Let $G$ be a group with subgroups $H$ and $K$, and let $X$ and $Y$ be nontrivial almost invariant subsets of $G$ over $H$ and $K$ respectively. We will denote the unordered pair $\{X,X^{\ast}\}$ by $\overline{X}$, and will say that $\overline{X}$ crosses $\overline{Y}$ if $X$ crosses $Y$. Now let $H_{i}$ be a subgroup of $G$ and let $X_{i}$ be a nontrivial $H_{i}$–almost invariant subset of $G$. Let $E=\{gX_{i},gX_{i}^{\ast}:g\in G,1\leq i\leq n\}$, and let $\overline{E}=\{g\overline{X_{i}}:g\in G,1\leq i\leq n\}$. Thus $G$ acts on the left on $E$ and on $\overline{E}$. Define an equivalence relation on $\overline{E}$ to be generated by the relation that two elements $A$ and $B$ of $\overline{E}$ are related if they cross. We call an equivalence class of this relation a *cross-connected component* (CCC) of $\overline{E}$, and denote the equivalence class of $A$ by $[A]$. We will denote the collection of all CCC’s of $\overline{E}$ by $P$. Note that the action of $G$ on $\overline{E}$ induces an action of $G$ on $P$. We will first introduce a partial order on $E$. If $U$ and $V$ are two elements of $E$ such that $U\subset V$, then our partial order will have $U\leq V$. But we also want to define $U\leq V$ when $U$ is nearly contained in $V$. If $U$ is $L$–almost invariant and $V$ is $M$–almost invariant, we will say that a corner of the pair $(U,V)$ is *small* if it is $L$–finite (and hence $M$–finite). We want to define $U\leq V$ if $U\cap V^{\ast}$ is small. Clearly there will be a problem with such a definition if the pair $(U,V)$ has two small corners, but this can be handled if we know that whenever two corners of the pair $(U,V)$ are small, then one of them is empty. Thus we consider the following condition on $E$: Condition (\*): If $U$ and $V$ are in $E$, and two corners of the pair $(U,V)$ are small, then one of them is empty. If $E$ satisfies Condition (\*), we will say that the family $X_{1},\ldots,X_{n}$ is in *good position*. Assuming that this condition holds, we can define a relation $\leq$ on $E$ by saying that $U\leq V$ if and only if $U\cap V^{\ast}$ is empty or is the only small set among the four corners of the pair $(U,V)$. Then $\leq$ turns out to be a partial order on $E$. If $U\leq V$ and $V\leq U$, it is easy to see that we must have $U=V$, using the fact that $E$ satisfies Condition (\*). It is proved in [@ss-book] that $\leq$ is transitive. We note here that the argument that $\leq$ is transitive does not require that the $H_{i}$’s be finitely generated. Now there is a natural idea of betweenness on the set $P$ of all CCC’s of $\overline{E}$. Given three distinct elements $A$, $B$ and $C$ of $P$, we say that $B$ *lies between* $A$ and $C$ if there are elements $U$, $V$ and $W$ of $E$ such that $\overline{U}\in A$, $\overline {V}\in B$, $\overline{W}\in C$ and $U\leq V\leq W$. Note that the action of $G$ on $P$ preserves betweenness. For the remainder of this discussion we will assume that $G$ and the $H_{i}$’s are all finitely generated. An important point is that if one is given a family $X_{1},\ldots,X_{n}$ of almost invariant subsets of $G$, the family need not be in good position, but it was shown in [@nsss], using the finite generation of $G$ and the $H_{i}$’s, that there is a family $Y_{1},\ldots,Y_{n}$ of almost invariant subsets of $G$, such that $X_{i}$ and $Y_{i}$ are equivalent, and the $Y_{i}$’s are in good position. A *pretree* consists of a set $P$ together with a ternary relation on $P$ denoted $xyz$ which one should think of as meaning that $y$ is strictly between $x$ and $z$. The relation should satisfy the following four axioms: - (T0) If $xyz$, then $x\neq z$. - (T1) $xyz$ implies $zyx$. - (T2) $xyz$ implies not $xzy$. - (T3) If $xyz$ and $w\neq y$, then $xyw$ or $wyz$. A pretree is said to be *discrete*, if, for any pair $x$ and $z$ of elements of $P$, the set $\{y\in P:xyz\}$ is finite. In [@ss-book], Scott and Swarup showed that if $G$ and the $H_{i}$’s are all finitely generated, then the set $P$ of all CCC’s of $\overline{E}$ with the above idea of betweenness is a discrete pretree. We say that two elements $x$ and $y$ of $P$ are *adjacent* if $xzy$ does not hold for any $z$ in $P$. We define a *star* in $P$ to be a maximal subset of $P$ which consists of mutually adjacent elements. It is a standard result that a discrete pretree $P$ can be embedded in a natural way into the vertex set of a tree $T$, and that an action of $G$ on $P$ which preserves betweenness will automatically extend to an action without inversions on $T$. Also $T$ is a bipartite tree with vertex set $V(T)=V_{0}(T)\cup V_{1}(T)$, where $V_{0}(T)$ equals $P$, and $V_{1}(T)$ equals the collection of all stars in $P$. It follows that the quotient $G\backslash T$ is naturally a bipartite graph of groups $\Theta$ with $V_{0}$–vertex groups conjugate to the stabilisers of elements of $P$ and $V_{1}$–vertex groups conjugate to the stabilisers of stars in $P$. When this construction is applied to the pretree $P$ of all CCC’s of $\overline{E}$, the points of $P$ form the $V_{0}$–vertices of the bipartite $G$–tree $T$ (Theorem 3.8 of [@ss-book]) with $V_{1}$–vertices corresponding to stars of $V_{0}$–vertices. The tree $T$ is minimal (Theorem 5.2 of [@ss-book]), and if $T$ has more than one $V_{0}$–vertex, i.e. if $\overline{E}$ has more than one CCC, then $G\backslash T$ does not reduce to a point, so that edges of $G\backslash T$ correspond to splittings of $G$. C–complexes ----------- The notion of height of a subgroup was introduced by Gitik, Mitra, Rips and Sageev in [@GMRS] and further developed in [@mitra-ht]. Let $H$ be a subgroup of a group $G$. We say that the elements $g_{1},\ldots,g_{n}$ of $G$ are essentially distinct if $g_{i}g_{j}^{-1}\notin H$ for $i\neq j$. Conjugates of $H$ by essentially distinct elements are called essentially distinct conjugates. Note that we are abusing terminology slightly here, as a conjugate of $H$ by an element belonging to the normalizer of $H$ but not belonging to $H$ is still essentially distinct from $H$. Thus in this context a conjugate of $H$ records (implicitly) the conjugating element. We now proceed to define the simplicial complex $C(G,H)$ for a group $G$ and $H$ a subgroup. Let $G$ be a group with an infinite subgroup $H$. Then the simplicial complex $C(G,H)$ has vertices ($0$–cells) which are the cosets $gH$ of $H$ (or equivalently the conjugates $gHg^{-1}$ of $H$ by essentially distinct elements), and the $(n-1)$–cells of $C(G,H)$ are $n$–tuples $\{g_{1}H,\cdots,g_{n}H\}$ of distinct cosets such that $\cap_{1}^{n}g_{i}Hg_{i}^{-1}$ is infinite. We shall refer to the complex $C(G,H)$ as the **C–complex** for the pair $(G,H)$. (**C** stands for coarse“ or Čech” or cover", since $C(G,H)$ is like a coarse nerve of a cover, reminiscent of constructions in Čech cochains.) If $G$ is a word hyperbolic group and $H$ is a quasiconvex subgroup, we give below two descriptions of $C(G,H)$ which are equivalent to the above definition. In this case, let $\partial G$ denote the boundary of $G$, let $\Lambda$ denote the limit set of $H$, and let $J$ denote the ‘convex hull’ (or join, strictly speaking) of $\Lambda$ in the Cayley graph $\Gamma_G$. 1) Vertices ($0$–cells) of $C(G,H)$ are translates of $\Lambda$ by essentially distinct elements, and $(n-1)$–cells are $n$–tuples $\{g_{1}\Lambda,\cdots,g_{n}\Lambda\}$ of distinct translates such that $\cap_{1}^{n}g_{i}\Lambda\neq\emptyset$.2) Vertices ($0$–cells) are translates of $J$ by essentially distinct elements, and $(n-1)$–cells are $n$–tuples $\{g_{1}J,\cdots ,g_{n}J\}$ of distinct translates such that $\cap_{1}^{n}g_{i}J$ is infinite. Non-crossing and splittings =========================== The Cayley graph $\Gamma_{G}$ of a group $G$ with respect to a finite generating set $S$, such that $S=S^{-1}$, will play a key role in our arguments. The vertex set of $\Gamma_{G}$ equals $G$, and elements $g$ and $h$ of $G$ are joined by an edge if $g=hs$ for some $s$ in $S$. Thus the action of $G$ on itself by left multiplication extends to a free action of $G$ on $\Gamma_{G}$ on the left. In particular, we will regard an almost invariant subset of $G$ as a set of vertices of $\Gamma_{G}$. We define the distance $d$ between two vertices $v$ and $w$ of $\Gamma_{G}$ to be the least number of edges among all paths joining $v$ and $w$. For the proof of Lemma \[nocross\] below, instead of using the notion of coboundary as in [@ss-book] we use terminology introduced by Guirardel in a different context. Let $A$ be a subset of $G$ (the vertex set of $\Gamma_{G}$). Define $\partial A=\{a\in A|$ there exists $a^{\prime}\in A^{\ast},d(a,a^{\prime })=1\}.$ Then $$\partial(A\cap B)=(\partial A\cap B)\cup(A\cap\partial B).$$ By a *connected component* of $A$ we mean a maximal subset of $A$ whose elements (vertices of $\Gamma_{G}$) can be joined by edge paths of $\Gamma _{G}$, none of whose vertices lie in $A^{\ast}$. If $B$ is finite, $G\setminus B$ has finitely many components. It is a beautiful fact that a subset $X$ of $G$ is $H$–almost invariant if and only if $\partial X$ is $H$–finite. This was first proved by Cohen [@cohen], but in the coboundary setting. Now suppose that $X$ and $Y$ are nontrivial almost invariant subsets of $G$ over subgroups $H$ and $K$ respectively, and that they are $H$–almost equal. Thus the corners $X\cap Y^{\ast}$ and $X^{\ast}\cap Y$ are both $H$–finite. As discussed immediately after Definition \[defnofcrossing\] this implies that both these corners are $K$–finite, so that $X$ and $Y$ are also $K$–almost equal. In this situation, we will simply say that $X$ and $Y$ are *equivalent*. This is indeed an equivalence relation on nontrivial almost invariant subsets of $G$. The following simple fact will be used in this paper. \[XequivalenttoYimpliesHandKarecommensurable\]Let $G$ be a finitely generated group, let $H$ and $K$ be subgroups of $G$, and let $X$ and $Y$ be nontrivial almost invariant subsets of $G$ over $H$ and $K$ respectively. If $X\ $and $Y$ are equivalent, then $H$ and $K$ are commensurable subgroups of $G$, i.e. $H\cap K$ has finite index in $H$ and in $K$. As $X$ is $H$–almost invariant and $Y$ is $K$–almost invariant, we know that $\partial X$ is $H$–finite and $\partial Y$ is $K$–finite. As $X$ is equivalent to $Y$, it follows that $X$, and hence $\partial X$, is contained in a bounded neighbourhood of $Y$. Similarly $\partial X^{\ast}$ is contained in a bounded neighbourhood of $Y^{\ast}$. It follows that $\partial X$ must be contained in a bounded neighbourhood of $\partial Y$. As $\partial Y$ is $K$–finite, $\partial X$ must also be $K$–finite. As $\partial X$ is $H$–finite, it follows that $\partial X$ is $(H\cap K)$–finite, so that $H\cap K$ must have finite index in $H$. By reversing the roles of $X$ and $Y$, the same argument shows that $H\cap K$ must have finite index in $K$. Thus $H$ and $K$ are commensurable subgroups of $G$, as required. We also need a simple lemma on the crossings of almost invariant sets; arguments similar to those in the following lemma occur in Kropholler’s paper [@krop]. We give a topological argument which is also used later. A non-crossing Lemma -------------------- Let $G$ be a finitely generated group with finitely generated subgroups $H$ and $K$. Let $X$ and $Y$ be nontrivial almost invariant subsets of $G$ over $H$ and $K$ respectively. Suppose that $e(G)=e(H)=e(K)=1$, and that $H\cap K$ is finite. Then $X$ and $Y$ do not cross. \[nocross\] **Proof:** Let $\Gamma_{G}$ be the Cayley graph of $G$ with respect to some finite generating set. Thus the vertex set of $\Gamma_{G}$ equals $G$. Our first step is to thicken $X$, $X^{\ast}$, $Y$ and $Y^{\ast}$ in $\Gamma_{G}$ to make them connected. For any subset $A$ of $\Gamma_{G}$, we let $N_{R}(A)$ denote the $R$–neighbourhood of $A$ in $\Gamma_{G}$. As $X$ is $H$–almost invariant, $\partial X$ is $H$–finite. Thus the image of $\partial X$ in $H\backslash\Gamma$ is finite. Hence we can choose an $R$–neighbourhood $W$ of this image which is connected and such that the natural map from $\pi_{1}(W)$ to $H$ is surjective. Thus the inverse image of $W$ in $\Gamma$, which equals $N_{R}(\partial X)$, is also connected. Since $N_{R}(\partial X)\subset N_{R}(X)$ and since any point of $X$ can be connected to a point of $\partial X$ by an edge path all of whose vertices lie in $X$, it follows that $N_{R}(\partial X)\cup X=N_{R}(X)$ is connected. Similarly, there is $S$ such that $N_{S}(\partial X^{\ast})$, and hence $N_{S}(X^{\ast})$, is also connected. Hence for any $T\geq\max\{R,S\}$, $N_{T}(X)$, $N_{T}(X^{\ast})$, $N_{T}(\partial X)$ and $N_{T}(\partial X^{\ast})$ are all connected. Similar arguments apply to $Y$ and $Y^{\ast}$. In what follows we will consider only sets $N_{R}(A)$, where $A$ is one of the sets $\partial X$, $\partial Y$, $X$, $X^{\ast}$, $Y$ or $Y^{\ast}$ in $G$, and $R$ is fixed so that each $N_{R}(A)$ is connected. Thus for notational simplicity we will denote $N_{R}(A)$ by $N(A)$. Now $N(\partial X)\cap N(\partial Y)$ is the intersection of an $H$–finite set with a $K$–finite set, and is therefore $(H\cap K)$–finite. As $H\cap K$ is finite, it follows that $N(\partial X)\cap N(\partial Y)$ is finite. Let $U$ denote this intersection. Then $N(\partial X)$ can be expressed as the union of $U$, $\left( N(\partial X)\cap N(Y)\right) \setminus U$ and $\left( N(\partial X)\cap N(Y^{\ast})\right) \setminus U$. Since $U$ is finite, $\left( N(\partial X)\cap N(Y)\right) \setminus U$ and $\left( N(\partial X)\cap N(Y^{\ast})\right) \setminus U$ have finitely many components. As $e(H)=1$, it follows that $N(\partial X)$ also has one end, so that only one of these components can be infinite. Thus one of $N(\partial X)\cap N(Y)$ and $N(\partial X)\cap N(Y^{\ast})$ must be finite. Without loss of generality, we can suppose that $N(\partial X)\cap N(Y)$ is finite. Similarly, by reversing the roles of $X$ and $Y$, one of $N(X)\cap N(\partial Y)$ and $N(X^{\ast})\cap N(\partial Y)$ must be finite. If $N(X)\cap N(\partial Y)$ is finite, then $\partial(X\cap Y)=(\partial X\cap Y)\cup(X\cap\partial Y)\subset(N(\partial X)\cap N(Y))\cup(N(X)\cap N(\partial Y))$, which is finite. Thus $\partial(X\cap Y)$ is finite. Since $(X\cap Y)^{\ast}=X^{\ast}\cup Y^{\ast}$ is infinite and $e(G)=1$, we see that $X\cap Y$ must itself be finite which shows that $X$ and $Y$ do not cross. Similarly if $N(X^{\ast})\cap N(\partial Y)$ is finite, then $X^{\ast}\cap Y$ must be finite, which again shows that $X$ and $Y$ do not cross. We conclude that in all cases $X$ and $Y$ cannot cross, as required. $\Box$ Splitting Theorem ----------------- We will now apply the preceding non-crossing result and the material from [@ss-book] discussed in subsection \[crossing\] to prove the following splitting results. Suppose that $G$ is a finitely generated group and $H$ a finitely generated subgroup. Further, suppose that $e(G)=e(H)=1$ and that $e(G,H)\geq2$. If $C(G,H)$ is disconnected, then $G$ splits over some subgroup (that may not be finitely generated). \[split\] **Proof:** Note that the assumption that $e(H)=1$ implies that $H\ $is infinite. As $e(G,H)\geq2$, there is a nontrivial $H$–almost invariant subset $X$ of $G$. By Lemma \[nocross\] applied to $X$ and $gX$, we see that if $H\cap gHg^{-1}$ is finite, then $X$ and $gX$ do not cross. Hence if $X$ and $gX$ cross, then $H\cap gHg^{-1}$ is infinite, and $H$ and $gH$ must lie in the same component of the $C$–complex $C(G,H)$. As $C(G,H)$ is not connected, we must have more than one CCC. Thus the tree $T$ constructed from the pretree of CCC’s does not reduce to a point, is a minimal $G$–tree, and each edge of $T$ induces a non-trivial splitting of $G$. This completes the proof that $G\ $splits over some subgroup. Note, however, that though $V_{0}$–vertices have finitely generated stabilizers, the edges and $V_{1}$–vertices need not. Thus the splitting may be over an infinitely generated subgroup.$\Box$  In the above proof, let $K$ denote the stabilizer of the CCC $v$ which contains $\overline{X}$. Now any edge incident to the $V_{0}$–vertex $v$ has stabilizer which is a subgroup of $K$. Thus $G$ splits over some subgroup of $K$, so that we do have slightly more information than stated in the above theorem. Essentially the same techniques show Suppose that $H$ and $K$ are finitely generated subgroups of a finitely generated group $G$, and suppose that $e(G)=e(H)=e(K)=1$; $e(G,H)\geq2$; $e(G,K)\geq2$. If all the conjugates of $K$ intersect $H$ in finite groups, then $G$ admits a splitting. \[2subgroups\] The graph considered here is reminiscent of the transversality graph considered by Niblo [@niblo], and Corollary \[2subgroups\] is similar to his Theorem D. The transversality graph considered by Niblo is dependent on the $H$–almost invariant set chosen, but if one chooses a set in very good position as in [@nsss], one obtains the regular neighbourhood graph considered above. Similarly, once we have the non-crossing lemma, by choosing almost invariant sets in very good position one can deduce Corollary \[2subgroups\] here from Theorem D of Niblo [@niblo]. See also the discussion on page 95 of [@ss-book]. Some other applications ======================= For a subgroup $H$ of a group $G$, and $g\in G$, we will denote the conjugate $gHg^{-1}$ by $H^{g}$. We recall that a subgroup $H$ of a group $G$ is said to be *almost malnormal* if whenever $H^{g}\cap H$ is infinite, it follows that $g$ lies in $H$. In Theorem \[split\], if we assume in addition that $H$ is almost malnormal, then the graph $C(G,S)$ is totally disconnected, and $X$ and $gX$ do not cross for any $g$ in $G$. Further we claim that $G$ splits over a subgroup of $H$. Note that as $X$ is $H$–almost invariant, $gX$ is $H^{g}$–almost invariant. In the proof of Theorem \[split\], the CCC $v$ of $\overline{E}$ which contains $[\overline{X}]$ consists of $[\overline{X}]$ only. Hence if $g$ in $G$ stabilizes $v$, we must have $gX$ equal to $X$ or to $X^{\ast}$. In particular, $gX$ is equivalent to $X$ or to $X^{\ast}$. Thus Lemma \[XequivalenttoYimpliesHandKarecommensurable\] tells us that $H$ and $H^{g}$ are commensurable. As $H$ is infinite, so is $H^{g}\cap H$. Thus, as $H$ is almost malnormal in $G$, it follows that the stabilizer of the CCC $v$ equals $H$. Hence the stabilizer of the vertex $v$ of $T$ equals $H$, so that the stabilizer of any edge of $T$ which is incident to $v$ must be a subgroup of $H$. Hence $G$ splits over a subgroup of $H$, as claimed. However in this case we can do slightly better by more elementary arguments. First we recall the following criterion of Dunwoody [@dunwoody]: Let $E$ be a partially ordered set with an involution $e\rightarrow \overline{e}$ where $e\neq\overline{e}$ such that: (D1) If $e,f\in E$ and $e\leq f$, then $\overline{f}\leq\overline{e}$, (D2) If $e,f\in E$, there are only finitely many $g\in E$ such that $e\leq g\leq f$, (D3) If $e,f\in E$, then at least one of the four relations $e\leq f$, $\overline{e}\leq f$, $e\leq\overline{f}$, $\overline{e}\leq\overline{f}$ holds, and (D4) If $e,f\in E$, one cannot have both $e\leq f$ and $e\leq\overline{f}$. Then there is an abstract tree $T$ with edge set equal to $E$ such that $e\leq f$ if and only if there is an oriented path in $T$ that starts with $e$ and ends with $f$. \[Dunwoody’s Criterion\] Next we recall the following result of Kropholler, which is Theorem 4.9 of [@krop]. We will discuss the definition of the invariant $\widetilde{e}(G,H)$ below. \[Krophollertheorem\] Suppose that $G$ is a finitely generated group with a finitely generated subgroup $H$, such that $e(G)=1=e(H)$. 1. If $H$ is malnormal in $G$, and $e(G,H)\geq2$, then $G$ splits over $H$. 2. If $H$ is malnormal in $G$, and $\widetilde{e}(G,H)\geq2$, then $G$ splits over a subgroup of $H$. Our methods allow us to extend this result. First we give the following slight generalization of the first part of Kropholler’s theorem. The only difference is that we have replaced malnormality by the weaker condition of almost malnormality. Later we will slightly generalize the second part in the same way, and will also prove a variant of Kropholler’s result. \[FirstKropholler Theorem\]Suppose that $G$ is a finitely generated group with a finitely generated subgroup $H$, such that $e(G)=1=e(H)$. If $H$ is almost malnormal in $G$, and $e(G,H)\geq2$, then $G$ splits over $H$. As $e(G,H)\geq2$, there is a nontrivial $H$–almost invariant subset $X$ of $G$. To prove this result, we will apply Dunwoody’s criterion to the set $E=\{gX,gX^{\ast},g\in G\}$, with the partial order $\leq$ discussed in subsection \[crossing\]. Recall that this partial order can only be defined if $X$ is in good position. We will show that this is automatic in the present setting. Let $g$ be an element of $G$ such that two corners of the pair $(X,gX)$ are finite. Thus $gX$ must be equivalent to $X$ or to $X^{\ast}$. Again Lemma \[XequivalenttoYimpliesHandKarecommensurable\] tells us that $H$ and $H^{g}$ are commensurable subgroups of $G$. As $H$ is infinite and almost malnormal in $G$, this can only occur if $g$ lies in $H$, so that $gX$ equals $X$ or $X^{\ast}$, and the two small corners are both empty. Thus $X$ is in good position, as required. Next we observe that with this partial order on $E$, conditions (D1) and (D4) of Dunwoody’s criterion are trivial. Condition (D3) holds, because our non-crossing lemma implies that for any $e,f\in E$ one of the corners of the pair $(e,f)$ is finite. Finally, as in the proof of Lemma B.1.15 of [@ss-book], condition (D2) holds because the set of $g\in G$ for which $X$ and $gX$ are not nested is contained in a finite number of double cosets $HgH$. This crucially uses the fact that $H$ is finitely generated and will be discussed in more detail in the proofs of the next theorems. Now Dunwoody’s criterion gives us a tree $T$ on which $G$ acts and which is minimal. Since the stabilizer of $X$ is $H$, we see that $G$ splits over $H$. This completes the proof of Theorem \[FirstKropholler Theorem\]. Even though, the condition $e(G)=1$ in the above result is generic, the hypotheses of almost malnormality and having one end are not generic for the subgroup $H$ and we would like to slightly weaken this condition. The statement of the second part of Kropholler’s theorem involves the notion of the number of relative ends $\widetilde{e}(G,H)$ of a pair of groups $(G,H)$, due to Kropholler and Roller [@krop-roll]. As discussed on pages 31-33 of [@ss-book], this is the same as the number of coends of the pair, as defined by Bowditch [@bowditch-split]. The following lemma (Lemma 2.40 of [@ss-book]) contains the only facts we will need about relative ends. \[comparingeandetwiddle\]Let $G$ be a finitely generated group and let $H$ be a finitely generated subgroup of infinite index in $G$. Then $\widetilde{e}(G,H)\geq2$ if and only if there is a subgroup $K$ of $H$ with $e(G,K)\geq2$. The subgroup $K$ need not be finitely generated. Let $\Gamma$ be the Cayley graph of $G$ with respect to a finite system of generators. The number of coends of the pair $(G,H)$ can be defined in terms of the number of $H$-infinite components of $\Gamma-A$ for a connected $H$–finite subset $A$ of $\Gamma$. So we have \[etwiddle\]Let $G$ be a finitely generated group and $H$ a finitely generated subgroup of $G$. Then $\widetilde{e}(G,H)\geq2$ if and only if there is a connected $H$-finite subcomplex $A$ of $\Gamma$ such that $\Gamma-A$ has at least two $H$–infinite components. Moreover, we may assume that $A$ is $H$-invariant. We now proceed to the statement and proof of a slight generalization of the second part of Kropholler’s theorem (\[Krophollertheorem\]), in which malnormal is again replaced by almost malnormal. Suppose that $G$ is a finitely generated group with a finitely generated subgroup $H$, such that $e(G)=1=e(H)$, and suppose that $\widetilde{e}(G,H)\geq2$. If $H$ is almost malnormal in $G$, then $G$ splits over a subgroup of $H$. \[Second Kropholler Theorem\] As $\widetilde{e}(G,H)\geq2$, there is a $H$–invariant, connected subcomplex $B$ of $\Gamma$ which is also $H$–finite, and such that $\Gamma-B$ has at least two $H$–infinite components. Since $H$ is almost malnormal in $G$, this implies that the stabilizer of $B$ is equal to $H$. Denote one of the $H$–infinite components of $\Gamma-B$ by $Q$ and let $K$ be the stabilizer of $Q$. Thus $K$ is a subgroup of $H$. We will denote by $X$ the set of vertices in $Q$. Thus $K$ is also the stabilizer of $X$. The frontier of $Q$ and the set $\partial X$ are in a $1$-neighbourhood of each other. Since the frontier of $Q$ is contained in $B$, we see that $\partial X$ is contained in the $1$-neighbourhood of $B$. We denote this $1$-neighbourhood by $A$. Note that $A$ is also $H$–invariant, connected and $H$–finite. We will show that $E=\{gX,gX^{\ast}:g\in G\}$, equipped with the partial order $\leq$ described earlier, satisfies the four conditions of Dunwoody’s Criterion (Theorem \[Dunwoody’s Criterion\]) and thus $G$ splits over $K$. First we observe that $\Gamma-Q$ must be connected, since $B$ is connected. As $H$ preserves $B$ it must also preserve the components of $\Gamma-B$, so that, for all $h$ in $H$, we have $hX=X$ or $hX\cap X=\emptyset$. Thus the pair $(hX,X)$ is nested, for each $h$ in $H$. Now suppose that $g$ is an element of $G$ such that the pair $(gX,X)$ is not nested, so that $g$ must lie in $G-H$. Thus each of the four corners of the pair $(gX,X)$ is non-empty. We note that $\partial X$ must intersect both $gX$ and $gX^{\ast}$, and that $\partial gX$ must intersect both $X$ and $X^{\ast}$. As $\partial X$ and $\partial gX$ are contained in $A$ and $gA$ respectively, we see that $A$ and $gA$ must also intersect. As $A$ is $H$–finite, $gA$ must be $H^{g}$–finite, and $A\cap gA$ must be $H\cap H^{g}$–finite. As $H$ is almost malnormal in $G$, and $g\in G-H$, it follows that $A\cap gA$ is finite. Now recall that $e(H)=1$. As $A$ is $H$–finite, it follows that $A$, and hence also $gA$, is one-ended. Thus one of $A\cap gX$ and $A\cap gX^{\ast}$ is finite, and one of $X\cap gA$ and $X^{\ast}\cap gA$ is finite. If the first of each pair is finite, we have $\partial(X\cap gX)=(\partial X\cap gX)\cup(X\cap\partial gX)\ \subseteq(A\cap gX)\cup(X\cap gA)$ is finite. As $e(G)=1$, and the complement of $X\cap gX$ in $G$ is clearly infinite, it follows that $X\cap gX$ is finite. Thus one of the corners of the pair $(gX,X)$ is finite, and two of them cannot be finite since $H$ is almost malnormal in $G$, and $g\notin H$. Similarly if one of the three other possibilities holds, then a different corner of the pair $(gX,X)$ will be finite and will be the only finite corner. Hence $X$ is in good position, and we have the partial order $\leq$ on the set $E=\{gX,gX^{\ast};g\in G\}$. All the conditions in Dunwoody’s Criterion (Theorem \[Dunwoody’s Criterion\]) are immediate except the finiteness condition (D2). Let $L$ denote $\{g\in G:$ the pair $(gX,X)$ is not nested$\}$. We saw above that if $g\in L$, then $gA$ and $A$ have nonempty intersection. As $A$ is $H$–finite, it follows that $L$ is contained in a finite number of double cosets $HgH$. We want to show that $L$ is actually contained in a finite number of double cosets $KgK$. To see this, consider $l\in L$. The preceding argument shows that $lA$ and $A$ have nonempty finite intersection. Since $A\cap lA$ is finite, $lA-A$ is contained in a finite number of components of $\Gamma-B$. Thus $lA$ meets only finitely many translates $hX$ of $X$ with $h\in H$. Since $\partial lX$ is contained in $lA$ it follows that $lX$ and $hX$ can be not nested, for only finitely many translates $hX$ of $X$ with $h\in H$, and hence that $hlX\ $and $X$ are not nested, for only finitely many translates $hlX$ of $X$ with $h\in H$. As $l^{-1}$ also lies in $L$, the same argument shows that $hl^{-1}X$ and $X$ are not nested, for only finitely many translates $hl^{-1}X$ of $X$ with $h\in H$, and hence that $X$ and $lhX$ are not nested, for only finitely many translates $lhX$ of $X$ with $h\in H$. As the stabilizer of $X$ is $K$, it follows that the intersection $L\cap HlH$ consists of finitely many double cosets $KgK$. Hence $L$ itself is contained in finitely many double cosets $KgK$. Choose $g_{1},...,g_{n}$ such that $L$ is contained in $\cup Kg_{i}K$. Consider $Y$ in $E$ with $Y\leq X$, so that $Y\cap X^{\ast}$ is $K$–finite. If $Y\cap X^{\ast}$ is not empty, so that $X$ and $Y$ are not nested, then $Y$ must be of the form $kg_{i}k^{\prime}X$ or $kg_{i}k^{\prime}X^{\ast}$. Now $kg_{i}k^{\prime}X^{(\ast)}\cap X^{\ast}=kg_{i}X^{(\ast)}\cap X^{\ast}=k(g_{i}X^{(\ast)}\cap X^{\ast})$. Choose $D$ such that the finite number of finite sets $(g_{i}X^{(\ast)}\cap X^{\ast})$ all lie in a $D$–neighbourhood of $X$. Then $Y$ also must lie in a $D$–neighbourhood of $X$. Thus every element $Y$ of $E$ such that $Y\leq X$ lies in a $D$–neighbourhood of $X$. Similarly every element $Y$ of $E$ such that $Y\leq X^{\ast}$ lies in a bounded neighbourhood of $X^{\ast}$. By increasing $D$ if necessary, we can assume that this neighbourhood is also of radius $D$. Now we can verify condition (D2) of Dunwoody’s criterion. Suppose that $U$ and $V$ are elements of $E$. We claim that there are only finitely many $W\in E$ with $U\leq W\leq V$. The first inequality implies that $W^{\ast}\leq U^{\ast }$, so that $W^{\ast}$ lies in a $D$-neighbourhood of $U^{\ast}$. Hence we can choose $x\in U$ which does not belong to any such $W^{\ast}$. Similarly the inequality $W\leq V$ implies that $W$ lies in a $D$–neighbourhood of $V$, so that we can choose $y\in V^{\ast}$ which does not belong to any such $W$. If $\omega$ is a path from $x$ to $y$, then $\omega$ should intersect $\partial W$. Since $G$ is finitely generated, there can be only finitely many such $W$. This completes the verification of Dunwoody’s Criterion and thus completes the proof of the theorem. Finally we give our variant of Kropholler’s theorem (\[Krophollertheorem\]). Let $G$ be a finitely generated, one-ended group and let $K$ be a subgroup which may not be finitely generated. Suppose that $e(G,K)\geq2$, and that $K$ is contained in a proper subgroup $H$ of $G$ such that $H$ is almost malnormal in $G$ and $e(H)=1$. Then $G$ splits over a subgroup of $K$. \[variant of Kropholler\] Lemma \[comparingeandetwiddle\] shows that the hypotheses imply that $\widetilde{e}(G,H)\geq2$. So we regard this result as a refinement of the second part of Kropholler’s theorem (\[Krophollertheorem\]). We start by observing that the assumptions that $H$ is proper and almost malnormal in $G$ imply that $H$ has infinite index in $G$. As $e(G,K)\geq2$, there is a nontrivial $K$–almost invariant subset $Y$ of $G$. As usual, we let $\Gamma$ denote a Cayley graph for $G$ with respect to some finite generating set. As $Y$ is $K$–almost invariant, $\partial Y$ is $K$–finite. Thus the image of $\partial Y$ in $H\backslash\Gamma$ must be finite. As $H$ is finitely generated, we can find a finite connected subgraph $W$ of $H\backslash\Gamma$ such that $W$ contains the image of $\partial Y$ and the natural map from $\pi_{1}(W)$ to $H$ is surjective. Thus the pre-image $A$ of $W$ in $\Gamma$ is connected, $H$-invariant and $H$–finite, and contains $\partial Y$. As $W$ is finite, the complement of $W$ in $H\backslash\Gamma$ has only a finite number of components. In particular it has only a finite number of infinite components. We consider the components of their inverse images in $\Gamma$. Each such component has vertex set contained in $Y$ or $Y^{\ast}$, since $\partial Y$ is contained in $A$. As $Y$ is $K$–infinite and $K$–almost invariant, and $H$ has infinite index in $G$, Lemma \[L2.13ofssbook\] implies that $Y$ must also be $H$–infinite. Hence at least one component of $\Gamma-A$ is $H$-infinite and has vertex set $X$ contained in $Y$. The stabilizer of $X$ is a subgroup of $K$ since $Y-A$ is preserved by $K$. Now we have the set up in the proof of Theorem \[Second Kropholler Theorem\]. The stabilizer of $X$ is a subgroup $K^{\prime}$ of the group $K$ in the hypotheses of this theorem. Thus $K^{\prime}$ replaces $K$ in the proof of Theorem \[Second Kropholler Theorem\]. In that proof we used only the almost malnormality of $H$, and that $K$ is contained in $H$. Thus nesting with respect to $H-K^{\prime}$ is automatic as before. Almost nesting with respect to elements of $G-H$ and verification of Dunwoody’s second condition follow exactly as in the previous theorem. In many of the above proofs, the hypotheses are used in two steps. The hypotheses on the subgroup $H$ ensure that one of the corners of the pair $(X,gX)$ has very small boundary and then the hypotheses on $G$ ensure that the corner set is small. Another hypothesis which ensures one of the corner sets has a relatively small boundary is formulated in a conjecture of Kropholler and Roller (discussed on pages 224-225 of [@ss-book]). We give our formulation of the conjecture: \[Conjecture of Kropholler and Roller\]Let $X$ be a $H$–almost invariant subset of $G$ with both $G$ and $H$ finitely generated. Suppose that $g\partial X$ is contained in a bounded neighbourhood of $X$ or $X^{\ast}$ for every $g\in G$. Then $G$ splits over a subgroup commensurable with a subgroup of $H$. This time the hypotheses ensure that if $g$ does not commensurise $H$, then, one of the corners of the pair $(X,gX)$ is an almost invariant set over a subgroup of infinite index in $H$. Dunwoody and Roller showed that one can get almost nesting with respect to the elements that commensurise $H$ by changing the almost invariant set, and changing the subgroup up to commensurability. (See Theorem B.3.10 of [@ss-book]. Note that almost nesting can be improved to nesting by using almost invariant sets in very good position.) This proof is one of the key steps in the proof of the algebraic torus theorem. Thus the obstructions to splitting $G$ over $H$ lie in almost invariant sets over subgroups of infinite index in $H$. One can wish away such sets by hypothesis, or can try to repeat the construction and look for conditions under which such repetitions must stop. A useful fact is that the corners obtained are invariant under the right action of $H$. This was originally used by Kropholler in the proof of Theorem \[Krophollertheorem\] when $H$ is malnormal in $G$, to obtain nesting. Nesting ensures the finiteness property required in the use of Dunwoody’s Criterion. In our proofs, we obtained almost nesting first and had to use the finiteness of double cosets to prove the finiteness property required in Dunwoody’s criterion. It is possible that a combination of these different techniques will give a bit more information about splittings. Acknowledgements ---------------- The first author would like to thank Michah Sageev for an extremely helpful conversation that led to the formulation of the problem we address in Theorem \[split\]. The last author thanks Vivekananda University, Belur Math for hospitality during the preparation of this paper. Research of the first author is supported in part by a Department of Science and Technology research grant. Finally all the authors thank the referee for spotting numerous minor errors.

[**Some remarks on contact manifolds, Monge-Ampère equations and solution singularities**]{} [A.M.Vinogradov]{} $^{1}$ Levi-Civita Institute, 83050 Santo Stefano del Sole (AV), Italia. [**Abstract.**]{} We describe some natural relations connecting contact geometry, classical Monge-Ampère equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampère equations and sheds new light on some aspects of contact geometry. Introduction ============ In this note we describe some natural relations among subjects in the title. While contact geometry and Monge-Ampère equations are classical themes, studied in numerous works, their natural ties with the theory of singularities of solutions to nonlinear PDEs, as far as we know, were not yet explicitly established and, a consequence, duly exploited. The fact that a (nonlinear) PDE is surrounded by a “cloud" of subsidiary equations, which describe the behavior of singularities of its generalized solutions is also not well-known. These equations were introduced by the author in [@Vsing] and since that are waiting to be systematically studied. A detailed exposition of fundamentals of this theory will be published in [@Vin]. It is worth noticing that some particular subsidiary equations are *implicitly* well-known. For instance, such are equations describing wave front propagation in the theory of hyperbolic equations, or equations of geometrical optics, which, in fact, describe some kind singularities of solutions of Maxwell’s equations. We used “implicitly" to stress that these equations were not identified as a part of solution singularity theory. A fundamental problem in this theory is whether it is possible to reconstruct the original equation if the subsidiary equations are known. In other words, we are asking whether the laws describing behavior of singularities of a physical field (continuum media, etc), predetermine the equations describing this field, (media, etc). This problem will be called the *reconstruction problem*. In this note we show that, informally speaking, the classical Monge-Ampère equations are characterized by the fact that the reconstruction problem for them is a tautology. The exact formulation of this assertion requires various facts from geometry of jet spaces, including contact geometry, and theory of generalized solutions of nonlinear PDEs, which are collected and discussed along the paper. When being put in this perspective they bring into the light some structures and their interrelations whose importance was not duly understood even in the case they are *formally known*. For instance, the intrinsic definition of a contact structure (see below definition \[main\]) belongs to this list. These small things are, in fact, rather useful and enrich even such classical subject as contact geometry. The fact that the intrinsic definition of contact structures immediately leads to to the explicit description of contact vector fields (see subsection \[cf\]) of this point. For the composition of this note is see the “contents". Everything (manifolds, vector fields, etc) in it is assumed to be smooth. The $C^{\infty}(M)$–module of $k$-th order differential forms (resp., vector fields) on a manifold $M$ is denoted by $\Lambda^k(M)$ (resp., $D(M)$). Contact manifolds and generalized solutions =========================================== Let $M$ be an $n$–dimensional manifold. Recall that any projective $C^{\infty}(M)$–module $P$ of finite type is canonically identified with the module $\Gamma(\pi)$ of sections of a vector bundle $\pi$ (see [@N]). The fiber $\pi^{-1}(x), \,x\in M,$ of $\pi$ is the quotient module $P_x:=P/\mu_x\cdot P$, $\mu_x=\{f\in C^{\infty}(M)\,\mid\,f(x)=0\},$ considered as an ${\mathbb R}$–vector space. The *dimension* of the vector bundle $\pi$ is the *rank* of $P$. Accordingly, below we treat a regular distribution on a manifold $M$ as a projective submodule ${\mathcal D}$ of the $C^{\infty}(M)$–module $D(M)$ of vector fields on $M$. Since $D(M)_x$ is canonically identified with the tangent to $M$ at $x$ space $T_xM$, ${\mathcal D}_x$ may be viewed as a subspace of $T_xM$ and this way one recovers the standard definition of a distribution as a family of vector spaces $x\mapsto {\mathcal D}_x\subset T_xM$. Put $\vk:=D(M)/{\mathcal D}$. So, $\vk_x$ is identified with $T_xM/{\mathcal D}_x$ and this is the reason to call $\vk$ *normal* to ${\mathcal D}$ bundle. Obviously, $\mathrm{rank}\,\vk=n-\mathrm{rank}\,{\mathcal D}$. Recall also that the [*curvature*]{} of ${\mathcal D}$ is the $C^{\infty}(M)$–bilinear form on ${\mathcal D}$ with values in $\vk$ defined as $$R(X,Y):=[X,Y](\mod{\mathcal D}), \,X,Y\in{\mathcal D}.$$ If the rank of $\vk$ is 1, i.e., the distribution ${\mathcal D}$ is of codimension 1, then ${\mathcal D}$ will be called *nondegenerate*, if $${\mathcal D}\ni X\mapsto R(X,\cdot)\in\operatorname{Hom}_{C^{\infty}(M)}({\mathcal D},\vk)$$ is an isomorphism of $C^{\infty}(M)$–modules. Equivalently, a distribution ${\mathcal D}$ is nondegenerate if its curvature is a nondenerate $\vk$–valued $C^{\infty}(M)$–bilinear form on it. The following is an intrinsic definition of a contact manifold. \[main\] A *contact manifold* is a manifold $M$ supplied with a nondegenerate distribution ${\mathcal D}$ of codimension 1. Such a distribution ${\mathcal D}$ is called a *contact* structure on $M$. Comparison with the standard definition. ---------------------------------------- Recall that the fiber at $x\in M$ of the vector bundle associated with the $C^{\infty}(M)$–module $\operatorname{Hom}_{C^{\infty}(M)}(P,Q)$ with $P,Q$ being projective $C^{\infty}(M)$–modules of finite type is $\operatorname{Hom}_{{\mathbb R}}(P_x,Q_x)$ (see [@N]). In particular, this fiber for $\operatorname{Hom}_{C^{\infty}(M)}({\mathcal D},\vk$ is $\operatorname{Hom}_{{\mathbb R}}({\mathcal D}_x,\vk_x)$. But ${\mathbb R}$–vector space $\vk_x$ is 1-dimensional. So, it is identified with ${\mathbb R}$ by choosing a base vector in it. If ${\mathcal D}$ is locally given by the Pfaff equation $\omega=0, \,\omega\in \Lambda^1(M)$, $X\in D(M)$ is such that $\omega(X)=1$ and $\nu=X(\mathrm{mod}\,{\mathcal D})$, then $\nu_x\in{\mathcal D}_x$ is a base vector, and $\operatorname{Hom}_{C^{\infty}(M)}({\mathcal D},\vk)$ (resp., $\operatorname{Hom}_{{\mathbb R}}({\mathcal D}_x,\vk_x)$) is identified with ${\mathcal D}^*:=\operatorname{Hom}_{C^{\infty}(M)}({\mathcal D},C^{\infty}(M))$ (resp., ${\mathcal D}^*_x:=\operatorname{Hom}_{{\mathbb R}}({\mathcal D}_x,{\mathbb R})$). The following assertion is a direct consequence of the standard formula $$d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]), \;X,Y\in D(M).$$ With the above identifications the curvature $R$ of ${\mathcal D}$ is identified with $d\omega\mid_{{\mathcal D}}$, i.e., $R(X,Y)=-d\omega(X,Y), \;\forall X,Y\in{\mathcal D}$. This lemma shows that nondegeneracy of $R_x$ is equivalent to nondegeneracy of $(d\omega)_x, \,\forall x\in M$, i.e., that the bilinear form on the vector space ${\mathcal D}_x$ $(d\omega)_x$ is symplectic . This nondegeneracy property may be also expressed by saying that $d\omega$ is nondegenerate on ${\mathcal D}$, i.e., that $${\mathcal D}\ni X\quad\mapsto\quad d\omega(X,\cdot)\mid_{{\mathcal D}}\in{\mathcal D}^*$$ is an isomorphism of $C^{\infty}(M)$–modules, or, equivalently, that $\omega\wedge d\omega^m$ is a local volume form on $M$ assuming that $n=2m+1$. These observations show that definition \[main\] is equivalent to the standard one. Recall that $\omega\in\Lambda(M)$ is a *contact form* on $M$ if ${\mathcal D}:=\{\omega=0\}$ is a contact structure on $M$. Contact forms defining the same distribution differ one from another by a nowhere vanishing factor $f\in C^{\infty}(M)$. Moreover, contact forms associated with a contact distribution are, generally, defined only locally. By these two reasons definition \[main\] is more convenient that the standard one. An illustration of that is in the subsequent subsection. Contact transformatins and vector fields. {#cf} ----------------------------------------- A symmetry of a distribution ${\mathcal D}$ on $M$ is a diffeomorphism $F\colon M\rw M$ that preserves ${\mathcal D}$, i.e., $F(X)\in{\mathcal D}$ if $X\in{\mathcal D}$ with $F(X):=(F^*)^{-1}\circ X\circ F^*$ being the image of $X$ via $F$. Traditionally symmetries of a contact structure/manifold are called *contact transformations*. An *infinitesimal symmetry* of ${\mathcal D}$ is a vector field $Z\in D(M)$ such that $[X,Z]\in{\mathcal D}, \,\forall X\in{\mathcal D}$. Alternatively, infinitesimal symmetries of ${\mathcal D}$ are defined as vector fields whose flows consist of (local) symmetries of ${\mathcal D}$. Infinitesimal symmetries of ${\mathcal D}$ form a Lie subalgebra of $D(M)$, which we denote by $\operatorname{Sym}{\mathcal D}$. If ${\mathcal D}$ is contact, then infinitesimal symmetries of ${\mathcal D}$ are called *contact vector fields* and some authors use a slightly ambiguous $\mathrm{Cont}\,M$ for $\operatorname{Sym}{\mathcal D}$ if $M$ is a contact manifold (see [@KLR]). Now we shall show that definition \[main\] directly leads to a natural description of contact vector fields. To this end, we note that with a vector field $Z\in M$ a homomorphism $\phi_Z\colon {\mathcal D}\rw\vk$ of $C^{\infty}(M)$–modules is associated. Namely, if $X\in{\mathcal D}$, then $\phi_Z(X)=[X,Z] (\mathrm{mod}\,{\mathcal D})$. Since $[fX,Z]=f[X,Z]-Z(f)X$, one sees that $\phi_Z$ is a $C^{\infty}(M)$–homomorphism. \[cont\] If ${\mathcal D}$ is contact, then the ${\mathbb R}$–linear map $$\operatorname{Sym}{\mathcal D}\ni Z \quad\mapsto \quad (Z\;\mathrm{mod}\,{\mathcal D})\in\vk$$ is biunique. **Injectivity**. If $Z\in {\mathcal D}\cap(\operatorname{Sym}{\mathcal D})$, then $$R(Z,X)=0, \,\forall X\in{\mathcal D}\quad\Longleftrightarrow\quad R(Z,\cdot)=0.$$ Hence, by nondegeneracy of $R$, $Z=0$.\ **Surjectivity.** Let $\vk\ni\nu=Z'(\mathrm{mod}\,{\mathcal D})$. Since the curvature form $R$ is nondegenerate, there is $Y\in{\mathcal D}$ such that $\phi_{Z'}=R(Y,\cdot)$, i.e., $[X,Z']=[Y, X]\,(\mathrm{mod}\,{\mathcal D}), \,\forall X\in{\mathcal D}$. So, $[X,Y+Z']\in{\mathcal D}, \,\forall X\in{\mathcal D}$ and, therefore, $Z=Y+Z'$ is contact and $Z(\mathrm{mod\,{\mathcal D}})=Z'(\mathrm{mod}\,{\mathcal D})=\nu$. If the contact vector field $Z$ corresponds via proposition \[cont\] to $\nu\in\vk$, then $\nu$ is called the *generating function* of $Z$ and $Z$ will be denoted by $X_{\nu}$. In other words, if $Z\in D(M)$ is contact, then $Z=X_{\nu}$ with $\nu=Z(\mathrm{mod}\,{\mathcal D})$. Also, proposition \[cont\] allows to transfer the Lie algebra structure in $\operatorname{Sym}{\mathcal D}$ to $\vk$. Namely, the transfered Lie bracket, denoted by $\{\cdot,\cdot\}$, is defined by the relation $$X_{\{\mu,\nu\}}=[X_{\mu},X_{\nu}], \quad \mu,\nu\in\vk$$ The ${\mathbb R}$–linear map $\chi\colon\vk\rw D(M), \,\chi(\nu)=X_{\nu}$, is a 1-st order differential operator. To prove it we have to show that $$[f,[g,\chi]](\nu)=0, \; \forall f,g\in C^{\infty}(M), \,\nu\in\vk$$ (see [@N]). In view of proposition \[cont\] it suffices to show that the vector field $$Z=[f,[g,\chi]](\nu)=X_{fg\nu}-fX_{g\nu}-gX_{f\nu}+fgX_{\nu}$$ is contact and its generating function is trivial. But, obviously, $X_{h\nu}-hX_{\nu}\in{\mathcal D}, \,\forall h\in C^{\infty}(M)$. This proves that $Z\in{\mathcal D}$. Next, if $X\in {\mathcal D}$, then $$\begin{array}{l} [X,Z]=[X,X_{fg\nu}]-f[X,X_{g\nu}]-g[X,X_{f\nu}]+\\ \qquad fg[X,X_{\nu}]-X(f)X_{g\nu}-X(g)X_{f\nu}+X(fg)X_{\nu} \end{array}$$ Each of first 4 terms in this expression, obviously, belongs to ${\mathcal D}$. The remaining 3 terms of it may be rewritten in the form $$X(f)(gX_{\nu}-X_{g\nu})+X(g)(fX_{\nu}-X_{f\nu})),$$ and, as we have seen before, each of 2 summands in this expression also belongs to ${\mathcal D}$. Jets and generalized solutions of nonlinear PDEs {#jets} ------------------------------------------------ Fix a manifold $E^{n+m}$ of dimension $n+m, \,m,n\in\ {\mathbb Z}_+$. The $k$–jet of an $n$–dimensional submanifold $N^n\subset E^{n+m}$ at a point $a\in N$ is the equivalence class of $n$–dimensional submanifolds of $E$ passing through $a$, which are tangent to $N$ with order $k$. It will be denoted by $[N]_a^k$. The totality of such jets forms a smooth manifold, denoted by $J^k(E,n)=\bigcup_{N\subset E, a\in N}[N]_a^k$. There is a natural map $$\begin{aligned} j_k(N):N {\longrightarrow} & J^k(E,n), \quad a \longmapsto & [N]_a^k\end{aligned}$$ A function $f$ on $J^k(E,n)$ is defined to be *smooth* if for any $n$–dimensional submanifold $N\subset E$ the function $j_k(N)^*(f)$ is smooth. The so-defined smooth function algebra, denoted by ${\mathcal F}_k(E,n)$, supplies $J^k(E,n)$ with a smooth manifold structure. The minimal distribution on $J^k(E,n)$ such that any submanifold of the form $N_{(k)}=\operatorname{Im}j_k(N)\subset J^k(E,n)$ is integral for it is the *$k$–th order contact structure*, or *Cartan distribution* on $J^k(E,n)$. Denote it by $\cC_k(E,n)$. If $m=1$, then $\cC_1(E,n)$ is the (classiical) contact structure on $J^1(E,n)$. For $k\geq l$ the natural projection $$J^k(E,n) \stackrel{\pi_{k,l}}{\longrightarrow} J^l(E,n),\quad [N]_a^k \longmapsto [N]_a^l$$ sends $\cC_k(E,n)$ to $\cC_l(E,n)$. An integral submanifold $L$ of a distribution ${\mathcal D}$ is called *locally maximal* if it is not contained even locally in an integral submanifold of greater dimension. Submanifolds $N_{(k)}$ are locally maximal integral submanifolds of $\cC_k(E,n)$ of dimension $n$. But except the case $k=m=1$ (contact geometry) there are locally maximal integral submanifolds of other types. More exactly, the *type* of such a submanifold $U$ is the dimension of $\pi_{k,k-1}(U)$, which may vary from $0$ to $n$. For instance, fibers of projection $\pi_{k,k-1}$ are of type $0$. It should be stressed that the notion of type is *intrinsic*, i.e., can be defined exclusively in terms of distribution $\cC_k(E,n)$. In this sense the contact geometry on $J^1(E^{n+1},n)$ is the only exception to the general case. Recall that a system of (nonlinear) PDEs of order $k$ is geometrically interpreted as a submanifold $\cE$ of $J^k(E,n)$ for a suitable $E$. In this approach “usual” solutions of $\cE$ are interpreted as submanifolds $N\subset E$ such that $N_{(k)}\subset \cE$. Moreover, this interpretation allows one to define an analogue of the notion of generalized solutions in the theory of linear PDEs for nonlinear equations. This is achieved by enlarging the class of submanifolds of the form $N_{(k)}$ to maximal integral submanifolds of type $n$, called *R-manifolds*. They play the role of Legendrian submanifolds in contact geometry. While an R-manifold $U$ is, by definition, smooth, its singular point are defined to be singular points of the projection $\pi_{k,k-1}\mid_U$. Their totality, denoted by $U_{sing}$, is a union of submanifolds with singularities. If $\theta\in U_{sing}$, then the kernel of the differential $d_{\theta}(\pi_{k,k-1}\mid_U):T_{\theta}U\rw T_{\pi_{k,k-1}(\theta)}J^{k-1}(E,n)$ at $\theta\in U$ is not trivial and is called the *bend* of $U$ at $\theta$. Denote it by $\gimel_{\theta}=\gimel_{\theta,U}$. We shall use the term $s$-*bend* for a bend of dimension $s$. The notion of a bend is key in the solution singularities theory (see subsection \[sng\]). A (generalized) $s$–bend solution of an equation $\cE\subset J^k(E,n)$ is an R-manifold $U\subset \cE$ such that for any $\theta\in U_{sing}$ the dimension of $\gimel_{\theta,U}$ is $s$. PDEs differ each other by types of generalized solutions they admit. Hence the problem of (local) classification of singularities of $R$–manifolds is a central one in geometrical theory of PDEs. In particular, as we shall see below, MA-equations are distinguished by the structure of singularities of their generalized solutions. Monge-Ampère equations ====================== Classical Monge-Ampère equations -------------------------------- Recall that *classical* Monge-Ampère equations (MAEs) are PDEs in two independent variables of the form $$\label{MA} N(u_{xx}u_{yy}-u_{xy}^2)+Au_{xx}+Bu_{xy}+Cu_{yy}+D=0$$ with $N,A,B,C,D$ being some functions of variables $x,y,u,u_{x},u_{y}$. In the current literature the term “Monge-Ampère equation" may refer to one of various generalizations of classical MAEs.(see, for instance, [@KLR]). Definition (\[MA\]) is descriptive and as such does not reveal the true nature of this class of equations. Our main goal here is to discover the hidden meaning behind analytical expression (\[MA\]). First of all, it is important to stress that equations (\[MA\]) are, in their turn, (locally) subdivided into 3 classes, namely, elliptic, parabolic, or hyperbolic ones, according to $AC-B^2-4ND<0, =0, or >0$, respectively. The first step toward revealing this meaning was due to S. Lie, who observed that the class of elliptic (resp., parabolic, or hyperbolic) MAEs is invariant with respect to contact transformations. Much later some authors and, first of all, Lychagin and his collaborators, interpreted a MA-equation as the condition $\omega\,|_L=0$ imposed on Legendrian submanifolds $L$’s of a 5–dimensional contact manifold $M$ for a given *effective* 2-form $\omega$ (see [@Ly], [@Mori], [@KLR]). The same condition imposed on Legendrian submanifolds of an arbitrary contact manifold is a natural generalization of classical MA-equations to higher dimensions. Nevertheless, though being coordinate-free, this definition is still of a descriptive character. Jordan algebras of self adjoint operators ----------------------------------------- In this subsection $M$ stands for a contact manifold of dimension $2n+1$. The curvature form allows one to distinguish an important class of operators in $\operatorname{End}_{C^{\infty}(M)}{\mathcal D}$. Namely, a $C^{\infty}(M)$–homomorphism $A\colon {\mathcal D}\rw{\mathcal D}$ is called *self-adjoint* if $$R(AX,Y)=R(X,AY), \,\forall X,Y\in{\mathcal D}.$$ *Scalar*, i.e., multiplication by a function operators are, obviously, self-adjoint. Self-adjoint operators form a Jordan algebra, denoted by $\mathrm{Sad}(\vk)$, with respect to the Jordan product $$A*B\colon= \frac{1}{2}(AB+BA).$$ $\mathrm{Sad}(\vk)$ is, obviously, a $C^{\infty}(M)$–module, and in what follows we shall concentrate on *JS-subalgebras* of $\mathrm{Sad}(\vk)$. These are submodules of $\mathrm{Sad}(\vk)$ closed with respect to the Jordan product. Such a subalgebra is *unital* if it contains the identity endomorphism and hence the algebra $C^{\infty}(M)$ interpreted as a subalgebra of $\operatorname{End}_{C^{\infty}(M)}{\mathcal D}$. Two JS-algebras are *isomorphic* if there is an isomorphism of supporting them $C^{\infty}(M)$–modules preserving the Jordan product. By literally repeating the above definitions in the situation when an ${\mathbb R}$–vector space $V$ and a skew-symmetric bilinear form $\sigma=<\cdot,\cdot>$ on $V$ taking values in a 1-dimensional vector space $W$ substitute $M, R$ and $\vk$, respectively, one gets notions of ($\sigma$–) self-adjoint operators in $\operatorname{End}\,V$, JS-subalgebras in $\operatorname{End}\,V$, etc. If ${\mathcal A}\subset \mathrm{Sad}(\vk)$ is a JS-subalgebra, then its fiber ${\mathcal A}_x$ at $x\in M$ inherits a structure of a JS-subalgebra in $\operatorname{End}_{{\mathbb R}}{\mathcal D}_x$ with respect to the $\vk_x$–valued form $R_x$. By choosing a base vector in $R_x$ one may identify it with a *symplectic* (=skew-symmetric and nondegenerate) bilinear form on the vector space $V={\mathcal D}_x$. Recall that a *symplectic vector space* (over ${\mathbb R}$) is an ${\mathbb R}$–vector space supplied with a non-degenerate skew-symmetric form. A unital JS-subalgebra of dimension $n$ with respect to the structure symplectic form on a $2n$–dimensional symplectic vector space will be called *basic*. Importance of basic algebras is that they classify geometric solution singularities of (nonlinear) PDEs (see [@Vsing]). So, the problem of describing basic algebras is fundamental in the solution singularities theory. This problem is not trivial just because it includes the problem of describing finite-dimensional commutative algebras over ${\mathbb R}$. By sightly abusing the language we shall call *basic* also a JP-subalgebra ${\mathcal A}$ of $\mathrm{Sad}(\vk)$ such that ${\mathcal A}_x$ is basic in $\operatorname{End}_{{\mathbb R}}({\mathcal D}_x)$ for almost all $x\in M$, i.e., for all $x\in N$ where $N$ is an everywhere dense open in $M$. Almost all fibers ${\mathcal A}_x$ of ${\mathcal A}$ are of dimension $n$. By this reason we say that the dimension of ${\mathcal A}$ is $n$. 2-dimensional unital JS-algebras {#JS} -------------------------------- Here we shall illustrate the above-said in the simplest nontrivial case $n=2$. But before we shall list some elementary properties of symplectic self-adjoint operators for arbitrary $n$. \[elementary\] Let $A$ be an symplectic self-adjont operator on a $2n$–dimensional symplectic vector space $V$. Then 1. The symplectic form $<\cdot,\cdot>$ vanishes on any cyclic subspace $$C_v\colon=\mathrm{Span}\{A^k(v)\}_{k\geq 0}, \,v\in V.$$. 2. Root subspaces of $A$ corresponding to different eigenvalues of $A$ are symplectic orthogonal. 3. $<\operatorname{Ker}A,\operatorname{Im}A>=0$. \(1) This is an obvious consequence of $<Aw,w>=0$ for any $w\in V$. But $<Aw,w>=<w,Aw>=-<Aw,w>$. \(2) The root subspace of $A$ corresponding to a real (resp., complex) eigenvalue $\lambda$ of it is of the form $\operatorname{Ker}f(A)$ where $f(t)=(t-\lambda)^k$ (resp., $f(t)=(t^2-(\lambda+\bar{\lambda})t+\lambda\bar{\lambda})^m)$. If eigenvalues $\lambda_1$ and $\lambda_2$ are different, then the corresponding to them polynomials $f_(t)$ and $f_2(t)$ are relatively prime and hence there are polynomials $g_1(t)$ and $g_2(t)$ such that $$f_1(t)g_1(t)+f_2(t)g_2(t)=1 \quad \Longrightarrow\quad f_1(A)g_1(A)+f_2(A)g_2(A)=\operatorname{id}_V.$$ I follows from the last relation that $f_1(A)$ is invertible on $\operatorname{Ker}f_2(A)$ and vice versa. So, $$\begin{aligned} 0=<f_1(A)(\operatorname{Ker}f_1(A),\operatorname{Ker}f_2(A))>=<\operatorname{Ker}f_1(A),f_1(A)(\operatorname{Ker}f_2(A)>=\nonumber\\ <\operatorname{Ker}f_1(A),\operatorname{Ker}f_2(A)>. \nonumber\end{aligned}$$ \(3) If $v\in\operatorname{Ker}A$, then $<v,Aw>=<Av,w>=0.$ \[4-s-a\] Let $V$ be a symplectic vector space of dimension $4$ and $A$ be a non scalar, symplectic self-adjoint operator on $V$. Then the minimal polynomial $f(t)$ of $A$ is of second order and 1. if $f(t)$ has complex roots, then $A$ supplies $V$ with a complex structure, all its proper subspaces are complex lines and these lines understood as 2-dimensional real planes are Lagrangian *(elliptic case);* 2. if $f(t)$ has different real roots $\lambda_1, \lambda_2$, then eigenspaces $\operatorname{Ker}(A-\lambda_i\operatorname{id}_V)$ are symplectic orthogonal and non-Lagrangian planes *(hyperbolic case);* 3. if roots of $f(t)$ coincide, i.e., $f(t)=(t-\lambda)^2$, then the unique eigenspace $W=\operatorname{Ker}(A-\lambda\operatorname{id}_V)$ of $A$ is a Lagrangian plane, which coincides with $\operatorname{Im}(A-\lambda\operatorname{id}_V)$. Lagrangian planes intersecting $W$ by a line are cyclic subspaces of $A$ of dimension 2 and vice versa *(parabolic case).* First, note that a linear operator possesses a cyclic subspace whose dimension equals to the degree of its minimal polynomial. By lemma \[elementary\], (1), the symplectic form vanishes on a cyclic subspace and hence its dimension cannot be greater than 2. It cannot be 1-dimensional, since $A$ is not scalar. Hence the minimal polynomial of $A$ is of second order. \(1) If $\lambda_1=a+b\mathbf{i}$ and $B=b^{-1}(A-a\operatorname{id}_V)$, then $B^2=-\operatorname{id}_V$. So, $B$ supplies $V$ with a structure of ${\mathbb C}$–vector space and $A$ is ${\mathbb C}$-linear. In view of lemma \[elementary\], (1), other assertions directly follows from this fact. (2)In this case $V$ is the direct sum of eigenspaces. By lemma \[elementary\], (2), they are symplectic orthogonal, and, so, none of them could be of dimension 1. Indeed, the symplectic orthogonal to a line is a 3-dimensional subspace containing this line. So, the eigenspaces are 2-dimensional, and none of them can be Lagrangian, since they are symplectic orthogonal. \(3) If $B=A-\lambda\operatorname{id}_V$, then $B^2=0$, i.e., $\operatorname{Im}B\subset \operatorname{Ker}B, W=\operatorname{Ker}B$ and $B$ is symplectic self-adjoint. If $\operatorname{Im}B$ is 1-dimensional, then $\dim(\operatorname{Ker}B)=\dim W=3$. On the other hand, by lemma \[elementary\], (3), $W$ is symplectic orthogonal to $\operatorname{Im}B$ and, therefore, coincides with the symplectic orthogonal complement of $\operatorname{Im}B$. If $0\neq v\in \operatorname{Im}B$ and $v=Bu$, then $u\notin W$ and hence is not symplectic orthogonal to $v$. But this contradicts to the fact that $<u,Bu>=0$ (lemma \[elementary\], (1)). Hence $\dim W=2, W=\operatorname{Im}B$. Moreover, $W$ is Lagrangian ss a symplectic orthogonal to $\operatorname{Im}B$ subspace. Finally, let $L$ be a Lagrangan plane that intersects $W$ by a line $\ell$ and $u\in L\setminus \ell$. Then $0\neq Bu\in W$. But the span $L'$ of $u$ and $Bu$ is a Lagrangian plane. It is symplectic orthogonal to $\ell$, since such are $u$ and $Bu$. But any Lagrangian plane, which is symplectic orthogonal to a line, contains this line. In particular, $L'$ contains $\ell$ and, therefore, $L'\cap W=\ell$. So, $Bu\in\ell$ and hence $L=L'$ and is cyclic. The converse is obvious. It is curious to observe that a Lagrangian plane $W$ in a symplectic $V^4$ defines a symplectic self-adjoint operator $B$ such that $B^2=0$ and $\operatorname{Ker}B=W$. Such an operator is unique up to a scalar factor. Indeed, if $u\in V\setminus W$, then $Bu$ should belong to $W$ and be symplectic orthogonal to $u$. Since the symplectic orthogonal complement of $u$ intersects $W$ by a line, say, $\ell_u$, $Bu$ should belong to $\ell_u$ and hence is unique up to a factor. In order to construct one such operator choose a non-Lagrangian and complementary to $W$ plane $U$ and an isomorphism $h:U\rw W$ such that $h(u)\in \ell_u$. Then any $v\in V$ is uniquely presented in the form $v=u+w, \,u\in U, \,w\in W,$ and we put $Bv:=h(u)$. Recall that a 2-dimensional unital associative algebra is isomorphic to the algebra of *$\zeta$-complex* numbers whose elements are of the form $x+y\zeta$ with $\zeta^2=-1, 0$ or $1$ and called *double, dual* or *complex*, numbers, respectively. Accordingly, denote these algebras by ${\mathbb C}_-(={\mathbb C}), {\mathbb C}_0$ and ${\mathbb C}_+$. There are 3 isomorphism classes of basic algebras for $n=2$ represented by algebras ${\mathbb C}_{\pm}$ and ${\mathbb C}_0$. Any basic algebra for $n=2$ is generated by $\operatorname{id}_V$ and a non-scalar symplectic self-adjoint operator $A$. A linear combination of these operators is an operator $B$ such that $B^2=\pm\operatorname{id}_V \mbox{or} \;0$. It follows from proposition \[4-s-a\] that two such operators of the same type are symplectically isomorphic. There is a simple approach to description of symplectic self-adjoint operators based on following observations. 1. If $W$ is an $n$–dimensional ${\mathbb R}$–vector space, then $V=W\oplus W^*$ is symplectic with respect to the form $\lfloor(w_1,\varphi_1),(w_2,\varphi_2)\rceil=\varphi_1(w_2)-\varphi_2(w_1)$. 2. Let $W$ and $W'$ be Lagrangian subspaces of a symplectic vector space $V$, which are complementary to each other. Then $\imath: w'\mapsto <w',\cdot>, \,w'\in W'$, is an isomorphism between $W'$ and $W^*$, which extends to the symplectic isomorphism $v=w+w'\mapsto w\oplus\imath(w')$ between $V$ and $W\oplus W^*$. 3. If $F\in \operatorname{End}W$, then $F\oplus F^*\in \operatorname{End}W\oplus W^*$ is symplectic self-adjoint. Now it is easily follows from (1)-(3) and *lemma \[elementary\], (1)*, that any symplectic self-adjoint operator is of the form $F\oplus F^*$. The approach we have chosen above is motivated by the fact that it is better adapted to the solution singularities theory. Geometric singularities of solutions of PDEs {#sng} -------------------------------------------- Here we shall assemble some facts concerning classification of singularities of R-manifolds that are necessary to encode the meaning of classical MA-equations (see [@Vsing], [@Vin]). The classification group is that of symmetries of the Cartan distribution $\cC_k(E,n)$. According to the Lie-Bäcklund theorem this group consists of diffeomorphisms of $E$ lifted to $J^k(E,n)$ if $m>1$ and of lifted contact transformations of $J^1(E,n)$ if $m=1$. The simplest question here is the first order classification, i.e., classification of tangent spaces to R-manifolds at singular points. The first result in this direction is that this classifications is equivalent to classification of bends (see subsection \[jets\]). The second important fact is that the classification of bends depends only on the dimension, say, $s$, of bends, i.e., does not depends on $m\geq1, k>1$ and $n\geq s$. This reduces the problem to the case $m=1, k=2, n=s$. As it follows from the definition, the bend at $\theta$ of an R–manifold $U$ is a subspace of $T_{\theta}\Phi$ where $\Phi$ is the fiber of $\pi_{k,k-1}$ passing through $\theta$. If $m=1$, then $T_{\theta}\Phi$ is identified with the space $P_{k,n}$ of homogeneous polynomials of degree $k$ in $n$ variables over ${\mathbb R}$ (see [@KLV]). In these terms, the $s$–bend of an R–manifold passing through $\theta$ can be characterized as a special subspace $\gimel$ in $P_{k,s}$, which, by abusing the language, we shall also call a bend. It holds the following key assertion (see [@Vin]) \[bend\] If $\gimel\subset P_{k,s}$ is an $s$–bend, then the span of polynomials in $P_{k+1,s}$ whose derivatives belong to $\gimel$ is also an $s$–bend in $P_{k+1,s}$. In the case of MA–equations we have $m=1, k=n=2$. Peculiarity of this case is twofold. First, it is easy to see that any 2-dimensional subspace of $P_{2,2}$ in the condition of proposition \[bend\], and hence any 2-dimensional subspace tangent to a fiber of $\pi_{2,1}$ is a 2-bend. Then, a second order PDE $\cE\subset J^2(E^3,2)$, generally, intersects fibers of $\pi_{2,1}$ by 2-dimensional submanifolds. So, tangent spaces to these submanifolds are 2-bends. 2-bends, basic algebras and $\zeta$–holomorphic functions --------------------------------------------------------- Proposition \[bend\] establishes a natural relation between bends and basic algebras, which we shall make explicit in the case $n=2, m=1$ (see [@Vin] for the general case). Let $\gimel\subset P_{k,2}, \,k\geq 2,$ be a bend. Then, according to proposition \[bend\], there is a polynomial $f(x,y)\in P_{k+1,2}$ whose derivatives $f_x$ and $f_y$ form a basis of $\gimel$. By the same proposition there should be a non proportional to $f$ polynomial $g$ whose derivatives belong to $\gimel$. Hence $g_x=\alpha f_x+\beta f_y, \,g_y=\gamma f_x+\delta f_y, \,\alpha,\dots,\delta\in{\mathbb R}$, and $(\alpha f_x+\beta f_y)_y=(\gamma f_x+\delta f_y)_x$, or, equivalently, $$\label{lapl} \gamma f_{xx}+(\delta-\alpha)f_{xy}+\beta f_{yy}=0.$$ Since $g$ is not proportional to $f$, coefficients of equation \[lapl\] do not vanish simultaneously, and this equation can be brought to one of the forms $u_{xx}\pm u_{yy}=0$ or $u_{xx}=0$. Accordingly, one of basic basic algebras ${\mathbb R}_{\pm}$ or ${\mathbb R}_0$ is associated with $\gimel$. More exactly, this basic algebra is generated by the operator whose matrix is $$\left(\begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array}\right).$$ Normal forms of 2-bends are easily deduced from this fact and they are $$\mathrm{Span}(\mathrm{Re}\,z^k,\mathrm{Im}\,z^k), \quad z=x+\zeta y,$$ where $\mathrm{Re}$ and $\mathrm{Im}$ refer to the “real" and “imaginary" parts of a $\zeta$-complex number (see subsection \[JS\]), respectively. Examples of R-manifolds with one singular point, in which these normal 2–bends are realized, are easily constructed in terms of $\zeta$-holomorphic functions. Namely, let $N$ be a 2-dimensional manifold. An $C^{\infty}(N)$-homomorphism $A\colon D(N)\rw D(N)$ such that $A^2=\zeta^2\operatorname{id}_{D(N)}$ supplies $N$ with a structure of a *$\zeta$-complex curve*. A function $u\in C^{\infty}(N)$ is *$\zeta$-harmonic* if $A^*(du)=dv$ for a function $v\in C^{\infty}(N)$. Here $A^*\colon \Lambda^1(N)\rw \Lambda^1(N)$ is the dual to $A$ homomorphism. The function $v$ is unique up to a constant and is called ($\zeta$-)conjugate to $u$. It is also $\zeta$-harmonic and together with $u$ form a ${\mathbb C}_{\zeta}$-valued function $f=u+\zeta v$. Such a function is called *$\zeta$-holomorphic*. Generally, non-constant $\zeta$–holomorphic functions on $N$ exist only locally. A pair of $\zeta$-conjugate local functions $x$ and $y$ on $N$ forms a local $\zeta$-complex chart on $N$, and, locally, $\zeta$-holomorphic functions may be viewed as functions of $\zeta$-complex variable $z=x+\zeta y$. In particular, the transition function between two $\zeta$-complex charts is $\zeta$-holomorphic. Standard complex curves are, obviously, $\zeta$-complex ones for $\zeta^2=-1$. If $\zeta^2=1$ (resp., $\zeta^2=0$), then, as it is easy to see, a $\zeta$-complex curve is a 2-dimensional manifold supplied with two transversal to each other 1-dimensional distributions (resp., one 1-dimensional distribution). Locally, the condition $A^*(du)=dv$ is equivalent to $(d\circ A\circ d)(u)=0$. The operator $\Delta_{\zeta}=d\circ A\circ d$ will be called the *$\zeta$-Laplace operator*, since in a $\zeta$-complex chart it reads $\Delta_{\zeta}={\partial}^2/{\partial}x^2-\zeta^2{\partial}^2/{\partial}y^2$. Accordingly, $u_{xx}-\zeta^2\ u_{yy}=0$ is the *$\zeta$–Laplace equation*. So, in these terms, $\zeta$-harmonic functions are solutions of the $\zeta$-Laplace equation. This equation express the compatibility condition for the *$\zeta$-Cauchy-Riemann equations* $A^*(du)=dv$, which in a $\zeta$-complex chart read $u_x=-\zeta^2 v_y, \;u_y=\zeta^2 v_x$. In other words, a function $f=u+\zeta v$ is $\zeta$-holomorphic iff its *real* and *imaginary* parts $u=\mathrm{Re}\,f$ and $v=\mathrm{Im}\,f$, respectively, satisfy the $\zeta$-Cauchy-Riemann equations. Now consider the $(k-2)$-th prolongation $\cE_{(k-2)}\subset J^k(E,n)$ of the $\zeta$-Laplace equation $\cE\subset J^{2}(E,n)$. It is given by equations $$\label{norm} u_{2+r,s}-\zeta^2u_{r,s+2}=0, \quad r+s\leq k-2$$ with $u_{p,q}$ being the function on $J^{p+q}(E,n)$ representing the operator ${\partial}^{p+q}/{\partial}x^p{\partial}y^q$ in a standard local jet-chart. Intersections of $\cE_{(k-2)}$ with fibers of $\pi_{k,k-1}$ are given by equations $$u_{p,q}=\mathrm{const}, \;p+q\leq k-1, \quad u_{2+r,s}-\zeta^2u_{r,s+2}=0, \quad r+s=k-2$$ and hence are 2-dimensional. Vectors $$\nu_1=\sum_{r=0}^{[k/2]}\zeta^{2r}{\partial}/{\partial}u_{2r,k-2r}, \quad \nu_2=\sum_{r=0}^{[k-1/2]}\zeta^{2r}{\partial}/{\partial}u_{1+2r,k-1-2r}$$ form a basis of the tangent space to such an intersection. In coordinates the identification of tangent to fibers of $\pi_{k,k-1}$ vectors with homogeneous polynomials of degree $k$ (see subsecton \[sng\]) looks as $$\frac{{\partial}}{{\partial}u_{r,k-r}}\quad\longleftrightarrow \quad\frac{1}{r!(k-r!}x^ry^{k-r}$$ So, the vectors $\nu_1$ and $\nu_2$ are identified with $\frac{1}{k!}\mathrm{Re}(x+\zeta y)^k$ and $\frac{1}{k!}\mathrm{Im}(x+\zeta y)^k$, respectively. Let now $1<l\in {\mathbb Z}$ and consider the R-manifold $L_{k,l}^{\zeta}$ in $J^k(E,n)$ given by equations (\[norm\]) and equations: $$\begin{aligned} \label{Rl} x=\frac{1}{(k+\frac{1}{l})!}\mathrm{Re}(u_{(k,0)}+\zeta u_{(k-1,1)})^k, \quad y=\frac{1}{(k+\frac{1}{l})!}\mathrm{Im}(u_{(k,0)}+\zeta u_{(k-1,1)})^k, \nonumber \\ u_{(k-r,0)}=\frac{1}{(r+\frac{1}{l})![(k+\frac{1}{l})!]^{{lr}}}\mathrm{Re}(u_{k,0}+\zeta u_{k-1,1})^{lr+1}, \quad 1\leq r\leq k, \\ u_{(k-r-1,1)}=\frac{1}{(r+\frac{1}{l})![(k+\frac{1}{l})!]^{{lr}}}\mathrm{Im}(u_{k,0}+\zeta u_{k-1,1})^{lr+1}, \quad 1\leq r\leq k. \nonumber\end{aligned}$$ where $(s+\frac{1}{l})!:=(1+\frac{1}{l})(2+\frac{1}{l})\dots(s+\frac{1}{l}), \,s\in {\mathbb Z}_+$. This R-manifold has the unique singular point $u_{(r,s)}=0, \,r+s\leq k$, in which the bend is $\mathrm{Span}(\mathrm{Re}\,z^k,\mathrm{Im}\,z^k)$. It may be viewed as the real part of the $k$–th jet of the multivalued $\zeta$-holomorfic function $f(z)=z^{k+\frac{1}{l}}$. It is not difficult to show that R-manifolds $L_{k,l}^{\zeta}$ corresponding to different $l$’s are not equivalent, while they have the common bend. The problem of finding $s$-bend solutions for relevant equations of mathematical physics and differential geometry was not yet systematically studied even for $s=2$. Some 2-bend solutions of the vacuum Einstein equations were constructed in [@SVV]. Among them are *foam-like* solutions describing “parallel universes" separated by singularities. The *square root* of the Schwarzschild metric is the simplest of that kind. Intrinsic definition of classical Monge-Ampère equations -------------------------------------------------------- Now we are ready to formulate a *conceptual definition* of classical MA-equations. We say that a Legendrian submanifold $L\subset M$ is invariant with respect to a JS-subalgebra ${\mathcal A}$ in $\mathrm{Sad}(\vk)$ if $T_xL$ is ${\mathcal A}_x$-invariant for all $x\in M$. \[concept\] A classical *Monge-Ampère problem* is that of finding Legendrian submanifolds of a 5-dimensional contact manifold $M$ that are invariant with respect to a given basic algebra ${\mathcal A}\subset\mathrm{Sad}(\vk)$. ${\mathcal A}$–invariant Legendrian submanifolds are called *solutions* of this problem. In order to connect definitions \[MA\] and \[concept\], recall that according to the classical Darboux lemma, a given contact form $\omega$ on a $2n+1$–dimensional manifold locally admits a *Darboux chart* $(x_i,p_i,u), \,i=1,\dots,n,$, in which it takes the *canonical* form $\omega=du-\sum_{i=1}^{n}p_idx_i$. A regular with respect to this chart Legendrian submanifold is given by equations $$u=f(x), \;p_i=\frac{{\partial}f(x)}{{\partial}x_i}, \,\;i=1,\dots,n, \;\mbox{with} \; x=(x_1,\dots,x_n).$$ To underline that such a submanifold is described by a function $f(x)$ we denote it by $L_f$. Solutions of equation *(\[MA\])* are solutions of a classical Monge-Ampère problem for Legendrian submanifolds of the form $L_f$ and vice versa. A natural basis of ${\mathcal D}$ in the Darboux chart $(x_i,p_i,u)$ is $${\partial}_{x_1}+p_1{\partial}u, \quad{\partial}_{x_2}+p_2{\partial}u, \quad{\partial}_{p_1}, \quad{\partial}_{p_2},$$ and the operator $\mathfrak{A}$ whose matrix in this basis is $$\left(\begin{array}{cccc} B & -2A & 0 & -2N \\ 2C & -B & 2N & 0 \\ 0 & 2D & B & 2C \\ -2D & 0 & -2A & -B \end{array}\right)$$ is such that $\mathfrak{A}^2= \Delta I$ where $I$ is the unit matrix and $\Delta=B^2-4AC+4ND$. Recall that equation (\[MA\]) is elliptic (resp., parabolic, or hyperbolic) if $\Delta<0$ (resp., $=0$, or $>0$). $\mathfrak{A}$ and $\operatorname{id}_{{\mathcal D}}$ span a basic algebra ${\mathcal A}$. Obviously, solutions of the corresponding Monge-Ampère problem are $\mathfrak{A}$–invariant Legendrian submanifolds. Vector fields $$Z_1={\partial}_{x_1}+p_1{\partial}u+f_{x_1x_1}{\partial}_{p_1}+f_{x_1x_2}{\partial}_{p_2}, \quad Z_2={\partial}_{x_2}+p_2{\partial}u+f_{x_1x_2}{\partial}_{p_1}+f_{x_2x_2}{\partial}_{p_2}$$ are tangent to $L_f$, while $$\mathfrak{A}(Z_1)=(B-2f_{x_1x_2}N)Z_1+2(C+f_{x_1x_1}N)Z_2-2E{\partial}p_2$$ where $E=N(f_{x_1x_1}f_{x_2x_2}-f_{x_1x_2}^2)+Af_{x_1x_1}+Bf_{x_1x_2}+Cf_{x_2x_2}+D$, and similarly for $\mathfrak{A}(Z_2)$. This shows that $L_f$ is $\mathfrak{A}$–invariant iff $E=0$. Thus this proposition establishes a one-to-one correspondence between MA-equations and 2-dimensional basic algebras on contact 5-folds. This correspondence put in evidence the fact that any solution of an MA-equation $\cE$ is a bi-dimensional manifold $L$ supplied with an algebra of endomorphisms of $D(L)$. Namely, this algebra, denoted by ${\mathcal A}_L$, is composed from restricted to $L$ elements of the associated with $\cE$ basic algebra ${\mathcal A}$. If $\cE$ is elliptic (resp., parabolic or hyperbolic), then ${\mathcal A}_L$ is of type ${\mathbb C}_-$ (resp., ${\mathbb C}_ 0$ or ${\mathbb C}_+$), i.e., $L$ is a $\zeta$-complex curve for the corresponding $\zeta$. In fact, solutions of arbitrary second order PDE in two independent variables are naturally supplied with a such an algebra of endomorphisms. This is due to the fact that any 2-dimensional subspace tangent to a fiber of $\pi_{2,1}$ is a 2-bend. One of central questions in solution singularity theory is to what extent a given PDE $\cE$ is predetermined by behavior of singularities of its solutions. More exactly, a series of subsidiary equations $\cE_{\Sigma}$ where $\Sigma$ is a singularity type admitted by solutions of $\cE$ is associated with $\cE$. See [@LMSV] for some examples. The *reconstruction problem* is : whether $\cE$ can be reconstructed if all equations $\cE_{\Sigma}$ are known? A spectacular, though implicit, historical example of solution of this problem is Maxwell’s deduction of his famous equations from previously found elementary (Coulomb, ... , Faraday) laws of electricity and magnetism, which, according to the modern point of view, describe behavior of singularities of electromagnetic fields. In this context classical MA-equations are distinguished by the fact that the reconstruction problem for them is a tautology, namely, by definition *\[concept\]*. Concluding remarks ------------------ The above interpretation of classical MA-equations was enlightening in the development fundamentals of solution singularities theory as one of key examples. Indeed, the discussed in this section constructions and results can be generalized to dimensions $n>2$. For instance, definition \[concept\] in an obvious manner generalizes to higher dimensions. It is not difficult to see that $n$–dimensional Monge-Ampère problem is described by a system of $\frac{n(n-1)}{2}$ second order PDEs, which which may be viewed as another natural generalization of classical MA-equations. This interpretation is useful in the study of classical MA-equations themselves. In particular, it suggests more exact techniques for integrating concrete MA-equations, finding their classical symmetries, conservation laws, etc. For instance, classical infinitesimal symmetries of the MA-equation associated with a basic algebra ${\mathcal A}$ may be defined as contact fields $Z$ such that $[Z,{\mathcal A}]\subset {\mathcal A}$ and this interpretation directly leads to a simple computational procedure. Another example we would like to mention here is the classical problem of contact classification of MA-equations, which, essentially, is equivalent to the problem of finding a sufficient number of their scalar differential invariants. Most systematically this problem was studied by V. Lychagin and his collaborators (V. Rubtsov, B. Kruglikov, A. Kushner and others) in last three decades. These authors exploited Lychagin’s idea to represent MA-equations in terms of *effective* 2-forms. Most complete results in this approach were obtained by A. Kushner (see short note [@K]). Full details of his work are reproduced in monograph [@KLR] together with earlier results of these authors. In these works scalar differential invariants of hyperbolic and elliptic MA-equations are constructed indirectly as differential invariants of the associated $e$–structures. On the contrary, definition \[concept\] allows a direct construction of scalar differential invariants in terms of operators of the corresponding basic algebra, which leads to more complete and exact results (see [@MVY], [@DV], [@VD], [@Diego]). [99]{} D. Catalano Ferraioli, A. M. Vinogradov, Differential invariants of generic parabolic Monge-Ampère equations, J. Phys. A: Math. Theor., 2012, vol.45, no. 26, 24 pp. A. De Paris, A.M.Vinogradov, Scalar Differential Invariants of Symplectic Monge-Ampere Equations, 2011, Cent. Eur. J. Math., vol.9, no.4, pp. 731-751. A. Kushner, Monge-Ampère equations and $e$–structures, Doklady Mathematics, Vol. 58, No. 1, 1998, pp. 103Ð104. A. Kushner, V. Lychagin, and V. Rubtsov, Contact geometry and nonlinear differential equations. Encyclopedia of Mathematics and Its Applications 101. Cambridge: Cambridge University Press, xxi, 496 p., 2007. F. Lizzi, B. Marmo, G. Sparano, A. M. Vinogradov, Eikonal type equations for geometrical singularities of solutions in field theory, J. Geom. and Phys., 1994, vol. 14, pp. 211-235. V. V. Lychagin, Contact geometry and non-linear second order differential equations, [*Russian Math. Surveys*]{} [**34**]{} (1979) 149–180. M. Marvan, A. M. Vinogradov, V. Yumaguzhin, Invariants of Generic Hyperbolic Monge-Ampere Equations Cent. Eur. J. Math., 2007, vol. 5, no. 1, pp. 105-133. T.  Morimoto, Monge–Ampère equations viewed from contact geometry. in: [*Symplectic Singularities and Geometry of Gauge Fields*]{} (Warsaw, 1995), 105–121, Banach Center Publ., 39, Polish Acad. Sci., Warsaw, 1997 J.Nestruev, Smooth manifolds and observables, Springer, series: Graduate Texts in Mathematics, vol. 220, 2002. G. Sparano, G .Vilasi, A. M. Vinogradov, Vacuum Einstein Metrics with Bidimensional Killing Leaves. II. Global Aspects, Diiff. Geom. and Appl., 2002, vol. 17, pp. 15-35. I. S. Krasil’shchik, A. M. Vinogradov, Symmetries and conservation laws of differential equations in mathematical physics, AMS, Translations of Mathematical Monograph series, vol. 182, 1999. A. M. Vinogradov, Geometric singularities of solutions of nonlinear partial differential equations, Proc. Conf. Differerential geometry and its applications, Brno, 1986, Math. Appl. (East European Ser.), , Reidel, Dordrecht-Boston, MA, 1987, vol. 27, pp. 359-379,. A. M. Vinogradov, On geometry of second order parabolic differential equations in two independent variables, Doklady Mathematics, 2008, v.78, no.3, pp. 887-890. A. M. Vinogradov, Geometric singularities of solutions of nonlinear partial differential equations, in preparation. I. S. Krasil’shchik, V. V.  Lychagin, A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics, 1 (1986), Gordon and Breach, New York, London. xx+441 pp.

--- abstract: | Most of the baryons are exceedingly difficulty to observe, at all epochs. Theoretically, we expect that the majority of the baryonic matter is located in low-density, highly ionized gaseous envelopes of galaxies – the “circumgalactic medium” – and in the highly ionized intergalactic medium. Interactions with the CGM and IGM are thought to play crucial roles in galaxy evolution through accretion, which provides the necessary fuel to sustain on-going star formation, and through feedback-driven outflows and dynamical gas-stripping processes, which truncate and regulate star formation as required in various contexts (e.g., low-mass vs. high-mass galaxies; cluster vs. field). Due to the low density and highly ionized condition of these gases, quasar absorption lines in the rest-frame ultraviolet and X-ray regimes provide the most efficient observational probes of the CGM and IGM, but ultraviolet spectrographs offer vastly higher spectral resolution and sensitivity than X-ray instruments, and there are many more suitable targets in the UV, which enables carefully designed studies of samples of particular classes of objects. This white paper emphasizes the potential of QSO absorption lines in the rest-frame far/extreme UV at $500 \lesssim \lambda _{\rm rest} \lesssim 2000$ Å. In this wavelength range, species such as Ne VIII, Na IX, and Mg X can be detected, providing diagnostics of gas with temperatures $\gg 10^{6}$ K, as well as banks of adjacent ions such as O I, O II, O III, O IV, O V, and O VI (and similarly N I – N V; S II – S VI; Ne II – Ne VIII, etc.), which constrain physical conditions with unprecedented precision. A UV spectrograph with good sensitivity down to observed wavelengths of 1000 Å can detect these new species in absorption systems with redshift [$z_{\rm abs}$]{} $\gtrsim 0.3$, and at these redshifts, the detailed relationships between the absorbers and nearby galaxies and large-scale environment can be studied from the ground. By observing QSOs at $z = 1.0 - 1.5$, [ *HST*]{} has started to exploit extreme-UV QSO absorption lines, but [*HST*]{} can only reach a small number of these targets. A future, more sensitive UV spectrograph could open up this new discovery space. author: - 'Todd M. Tripp \[tripp@astro.umass.edu, (413)-545-3070\]' title: 'Quasar Absorption Lines in the Far Ultraviolet: An Untapped Gold Mine for Galaxy Evolution Studies' --- **1. QSO Absorption Lines at Wavelengths $<$ 912 Å** High-resolution ultraviolet spectroscopy provides a unique ability to study low-density gas/plasma in galaxy disks, halos, and the intergalactic medium (IGM), i.e., all harbors of present-epoch baryons. Since stars account for only a small fraction of the baryon inventory and most of the ordinary matter is in very low-density gases (Fukugita et al. 1998, ApJ, 503, 518), UV spectroscopy is a crucial technique for the study of galaxy ecosystems and the cycles of inflowing and outflowing matter and energy that regulate galaxy formation. As anticipated by Verner et al. (1994, ApJ, 430, 186), the deployment of the *Cosmic Origins Spectrograph* (COS, Green et al. 2012, ApJ, 744, 60) on the *Hubble Space Telescope* has demonstrated a particularly powerful new window for UV spectroscopy: the study of QSO absorption lines in the “extreme” ultraviolet (EUV) at $\lambda <$ 912 Å. Normally, we assume that EUV absorption lines cannot be observed because the Galactic ISM prevents observations of transitions at these wavelengths in the Milky Way. However, if gas in a quasar absorption system has a sufficiently high redshift, these lines are redshifted into the *HST* bandpass; for example, the Ne VIII doublet at 770.4,780.3 Å can be studied in QSO absorbers with redshift $z_{\rm abs} \geq 0.3$ with a spectrograph sensitive down to 1000 Å. Unfortunately, in very high-redshift QSO absorbers that can be observed from the ground, these EUV lines are ruined by blending with the thick Ly$\alpha$ forest. However, as illustrated in Figure \[specsample\], QSOs at $z \approx 1 - 1.5$ are in a “sweet spot” where the EUV lines can be detected but the line density is low enough so that blending is not severe. These COS data demonstrate the potential of EUV lines, but unfortunately, HST+COS can only access a small number of these targets in reasonable exposure times. Moreover, while the COS spectra have signal-to-noise $\approx 30-50$ per resel, higher S/N ($\gtrsim 100$) would greatly improve this technique because the key lines (e.g., Ne VIII) can be quite weak (see, e.g. Meiring et al. 2012, arXiv1201.0939). {width="18.3cm"} ![ Column densities of various metals in collisional ionization equilibrium, as a function of temperature, for an absorber with $N$(H$_{\rm total}) = 10^{20}$ cm$^{-2}$ and $Z = 0.1 Z_{\odot}$. [*Lower panel:*]{} Strength of resonance lines vs. the rest-frame wavelength of the transition from Verner et al. (1994, A&AS, 108, 287) based on the element abundance and atomic data (taller lines indicate intrinsically stronger transitions). Colors indicate tracers of difference gas phases (see legend). In addition to providing access to a larger number of sight lines, a future UV spectroscopic facility with greater sensitivity could more effectively exploit the rich diagnostics available at $\lambda <$ 1000 Å by detecting weaker lines with higher signal-to-noise spectra. \[euvdemo\]](ttripp_f2.ps){width="17.5cm"} Figure \[euvdemo\] demonstrates the following unique diagnostics provided by EUV absorption spectroscopy: First, EUV absorption lines include species such as Ne VIII, Mg X, and Si XII, and these species are detectable in plasmas at $T > 10^{6}$ K. Thus, in the EUV, [ *HST*]{} can compete with X-ray telescopes, but [*HST*]{} has much better spectral resolution, better sensitivity, and a substantially larger pool of sufficiently bright targets, which enables more optimal target selection. The Astro2010 decadal survey identified the *International X-ray Observatory* as a top priority for the next 20 years, and one of the prime science drivers of IXO is the study of missing baryons and hot gas in low-density gaseous halos and the IGM using absorption spectroscopy. By using species such as Ne VIII and Mg X, we can pursue this IXO science goal immediately. We have already successfully detected Ne VIII, Mg X, and other highly ionized hot-gas tracers (see below). Second, the EUV includes transitions of suites of adjacent ions such as O I, O II, O III, O IV, O V, and O VI or S II, S III, S IV, and S V (similar sets are available for C, N, etc). These adjacent ions span a wide range of temperatures/ionization conditions (see Fig. \[euvdemo\]), and limits on or detections of these species can constrain the physics and metallicity of QSO absorbers with unprecedented precision \[note that currently, we typically only have access to scattered ionization stages such as O VI, C III, and Si III in low-$N$(H I) absorbers\]. As shown in the lower panel of Fig. \[euvdemo\], the EUV is the richest region of the spectrum for QSO absorption spectroscopy. Third, the redshifts of the absorbers are sufficient to bring many H I Lyman series lines into the [*HST*]{} band (see examples in Figure \[specsample\]). This enables accurate H I column-density measurements because the higher Lyman series lines are less likely to be saturated. Observations of lower-redshift absorbers often detect only a few Lyman lines or even only Ly$\alpha$, and often these lines are badly saturated so $N$(H I) is highly uncertain. Uncertain $N$(H I) measurements lead to uncertain metallicity measurements. With good constraints on metallicity and physical conditions, key properties such as mass and mass flow rates can be estimated. **2. Proof of Concept: First Results from *HST*** **Galactic Winds Driven by Star Formation and AGN.** The role of galactic outflows and “feedback” is one of the most pressing issues of current galaxy evolution studies. Some observations of objects such as Lyman-break galaxies, ULIRGs, and post-starburst galaxies have revealed dramatic outflows (e.g., Rupke et al. 2005, ApJS, 160, 87; Tremonti et al. 2007, ApJ, 663, L77; Steidel et al. 2010, ApJ, 717, 289). However, since these studies usually use the ULIRG or the post-starburst galaxy itself as the continuum source, they suffer from an ambiguity regarding the spatial extent, and hence the mass, of the outflow. These investigations also have a limited suite of diagnostics, e.g., Mg II or Na I and nothing else. By using absorption lines imprinted on background QSOs, these limitations can be overcome, and the EUV lines turn out to be particularly interesting. Multiple examples of different types of outflows are present in the sample COS data shown in Figure \[specsample\]. For example, toward PG1206+459 we have clearly detected, at high significance, a doublet of Na IX at $z_{\rm abs} = 1.0281$ (see the lowest panel in Fig. \[specsample\]). Na IX has never been detected before, but this absorber is also detected in Ne VIII, Mg X, Ar VII, Ar VIII, and O V. A possibly even more interesting outflow is detected at $z_{\rm abs}$ = 0.9276 in the PG1206+459 spectrum (Tripp et al. 2011, Science, 334, 952). From Fig. \[specsample\] we see that there is a dramatic cluster of absorption lines at this redshift detected in species such as S III and O III (middle and lower panels). This system is notable for the following reasons (see Tripp et al. for full details): First, we detect the adjacent suites of ions, including O III, O IV, O V, and O VI; N III, N IV, and N V; and S III, S IV, and S V. Second, we detect Ne VIII at very high significance. Third, the Ne VIII and N V velocity centroids are strongly correlated with the centroids of low ions such as Mg II, Si II, and C II (see Fig.3 in Tripp et al.). Fourth, while this absorption cluster is clearly a Lyman-limit absorber with many higher Lyman-series lines, the Lyman limit (LL) is not black and excellent $N$(H I) measurements can be obtained. Finally, there is a post-starburst galaxy with an AGN at an impact parameter of 68 kpc from the sight line. These results have interesting implications: (1) The components in the cluster extend from $-400$ to +1100 km s$^{-1}$; with these velocities, some components must be exceeding the escape velocity of the galaxy. (2) Using the adjacent ions (e.g., SIII/SIV/SV) we can pin down the ionization state of each component and estimate their total column densities. Combined with the large impact parameter (70 kpc) to the galaxy, this implies that each component carries $\approx 10^{8}$ M$_{\odot}$ of mass in cool, photoionized gas, assuming a standard thin-shell model (e.g., Tremonti et al. 2007). Other geometries would give different masses, but an important mass component is implicated in any case. (3) However, the Ne VIII and N V must arise in hot gas that is correlated with the cool gas – Ne VIII/N V cannot originate in the cool photoionized gas. Moreover, this hot gas contains $10\times$ to $150\times$ more mass than the cool phase. In addition, the remarkable correspondence of the Ne VIII with lower ions suggests that the outflowing material is also interacting with a hotter (unseen) phase. How do species like Mg II and Si II survive at these outflow velocities embedded in such hot gas? **Cold Accretion of Pristine (Low-Metallicity) Gas.** These spectra also reveal the opposite process: absorbers that are most naturally explained as cold, [*inflowing*]{} material, an equally important topic that is even more poorly constrained by observations. The partial Lyman limit absorber that produces the Lyman series lines shown in the top panel of Figure \[specsample\] is an example of apparently infalling, very metal poor gas. We have analyzed the metals affiliated with this partial Lyman limit system (Ribaudo et al. 2011, ApJ, 743, 207), and we find that the logarithmic metallicity is only \[Mg/H\] = $-1.71 \pm 0.06$. Moreover, we have spectroscopically identified and studied a nearby galaxy at the redshift of the Lyman limit absorber at an impact parameter of 37 kpc. Interestingly, that galaxy has a metallicity that is almost two orders of magnitude higher, \[O/H\]$_{\rm galaxy} -0.20 \pm 0.15$. This absorber may represent nearly primordial material that is accreting onto the galaxy via cold-mode accretion (Kereš et al. 2005), but other explanations remain viable. Subsequently, we studied all LL absorbers (16.0 $<$ log $N$(H I) $<$ 19.) in our data combined with measurements from the *HST* archive and literature (Lehner et al. 2012, in prep.), and we find that 50% of LL have very low-metallicity ($Z \leq 0.03 Z_{\odot}$). Our survey has tripled the sample of LL absorbers with good metallicity measurements at $z < 1$, but the sample is still small (28 systems total). **Requirements for a Future UV Telescope.** Technical concepts are deferred to the second RFI, but the key technical requirements for this science can be briefly summarized. While *HST* has begun to observe QSO absorption lines at $\lambda _{\rm rest} < 912$ Å, the number of $z_{\rm QSO} = 1 - 2$ QSOs bright enough for *HST* is extremely small. To exploit this discovery space, a future UV spectrograph must have substantially better sensitivity than *HST*+COS, good spectral resolution (comparable to STIS and COS), and wavelength coverage down to at least 1150 Å and preferrably down to $\approx$ 1000 Å.

--- abstract: 'We introduce a cluster DMFT (Dynamical Mean Field Theory) approach to study the normal state of the iron pnictides and chalcogenides. In the regime of moderate mass renormalizations, the self-energy is very local, justifying the success of single site DMFT for these materials and for other Hunds metals. We solve the corresponding impurity model with CTQMC (Continuous Time Quantum Monte-Carlo) and find that the minus sign problem is not severe in regimes of moderate mass renormalization.' author: - Patrick Sémon - Kristjan Haule - Gabriel Kotliar title: 'Validity of the Local Approximation in Iron- Pnictides and Chalcogenides' --- The unexpected discovery of superconductivity in the iron pnictide based materials has opened a new era of research in the field of condensed matter physics.[@Hosono:2006] Multiple approaches, starting from weak coupling such as the random phase approximation (RPA) and strong coupling approaches using lessons learned from the t-J model, have been proposed, but there is not yet consensus in the community of what constitutes the proper theoretical framework for describing these systems.[@ChubukovHirschfeld:2014] It has been proposed that iron pnictides and chalcogenides are important not only because of their high temperature superconductivity, but because their normal state properties represent a new class of strongly correlated systems, the Hunds metals. They are distinct from doped Mott Hubbard systems, in that correlations effects in their physical properties derive from the Hunds rule coupling J, rather than the Hubbard U. [@HauleKotliar:2009; @YinHauleKotliar:2011] Many other interesting Hunds metals have been recognized, as for example Ruthenates [@Mravlje:2011] and numerous 3d and 4d compounds [@Antoine:2013]. Dynamical Mean Field Theory[@Georges:1996](DMFT) and its cluster extensions[@Kotliar:2001; @MaierRev:2005] have provided a good starting point for the description of Mott Hubbard physics. It is now established that it describes many puzzling properties of three dimensional materials such as Vanadium oxides near their finite temperature Mott transition.[@Deng:2014] In materials such as cuprates, as the temperature is lowered, the description in terms of single site DMFT gradually breaks down. New phenomena such as momentum space differentiation and the opening of a pseudogap takes place,[@Huscroft:2000; @Lichtenstein:2000; @Jarrell:2001a; @Jarrell:2001b; @Haule:2003; @Parcollet:2004; @Carter:2004; @Civelli:2005; @Stanescu:2006; @Kyung:2006a; @Macridin:2006; @Haule:2007b; @Zhang:2007; @Civelli:2008; @Civelli:2009; @Liebsch:2009; @Sakai:2009; @Werner:2009; @Gull:2010; @Lin:2010; @Sordi:2012; @Sordi:2013; @Gull:2013a; @Gull:2013b; @Imada:2013] and cluster DFMT is essential. How different cluster sizes and methods captures these effects has been explored intensively.[@Jarrell:2001b; @Biroli:2002; @Aryanpour:2005; @Biroli:2005; @Maier:2005b; @Kyung:2006b; @Gull:2010; @Sakai:2012] The iron pnictides and chalcogenides have been extensively studied using LDA+DMFT by several groups.[@HauleKotliar:2009; @YinHauleKotliar:2011; @Yin:2014; @Valenti:2015; @Aichhorn:2010] It has been argued using the GW method, that the frequency dependence of low order diagrams in perturbation theory in these materials is very local.[@Tomczak:2012] However, because of the difficulties posed by the multiorbital nature of these compounds, the accuracy of the local approximation beyond the GW level has not been examined and is the main goal of this paper. Building on the work of Ref. , we introduce a cluster extension for the treatment of iron pinctides, which is numerically tractable using CTQMC. By comparing single site and cluster DMFT, we establish that in a broad range of parameters where the mass renormalizations are of the order of 2 to 3, which corresponds to the experimental situation in many iron pnictides and chalcogenides, the local approximation is extraordinarily accurate, justifying the success of a very large body of work. For simplicity, we use in this work a tight-binding hamiltonian $\mathbf{h}_0(\mathbf{k})$ of $FeAs$ layers with $As$ treated in second order perturbation theory, as presented by M. J. Calderón et al.[@Calderon:2009] For the hopping amplitudes the values suggested for $LaOFeAs$ are taken and scaled such that the bandwidth is $\approx 4eV$.[^1] However, the main conclusions of this work should not be very sensitive to the parametrization used. The wave vectors $\mathbf{k}$ label the irreducible representations of a glide-mirror symmetry group instead of the usual translation symmetry group, so that the Brillouin zone contains 1 $Fe$ atom instead of 2 $Fe$ atoms, with hole pockets at the $M$ and $\Gamma$ points and electron pockets at the $X$ and $Y$ points. Notice here that this unfolding, which is exact in two dimensions, is not exact when the $FeAs$ layers are coupled, i.e., a translation operation perpendicular to the layers does not commute with a glide mirror operation along the layers, and the corresponding symmetry group is not abelian. The correlations of the electrons within a $d$-shell are captured by adding a local Coulomb interaction, parametrized by the Hubbard repulsion $U$ and the Hund’s rule coupling $J$, see supplementary information Sec. D for more details. We solve this model using DMFT and Dynamical Cluster Approximation (DCA). DMFT starts by approximating the lattice self-energy locally with that of a single site impurity model. This neglects all $\mathbf{k}$-dependence of the lattice self-energy. DCA retains some of the momentum dependence by first cutting the Brillouin zone into patches of equal size, each patch $\mathcal{P}_\mathbf{K}$ enclosing a coarse grained momentum $\mathbf{K}$. The lattice self-energy is then approximated by a piecewise constant function over the patches and identified with that of a cluster impurity model written $\mathbf{K}$-space. In this work, we choose a minimal patching[@Ferrero:2009] which takes into account both the symmetries and the electron-hole pocket structure of the Brillouin zone, see Fig. \[fig:BrillouinPatches\]. ![Left panel: Orbital character of the Fermi surface in the unfolded Brillouin zone of the tight-binding hamiltonian[@Calderon:2009] used in this work. Right panel: Tiling of the Brillouin zone in two patches $\mathcal{P}_+$ and $\mathcal{P}_-$, enclosing the holes pockets at $\Gamma$ and $M$ and the electron pockets at $X$ and $Y$ respectively. This patching is compatible with the lattice symmetries.[]{data-label="fig:BrillouinPatches"}](bandstructure.jpg){width="1.0\columnwidth"} One patch ($\mathcal{P}_+$) encloses the holes at $(0,0)$ and $(\pi,\pi)$ and the other patch ($\mathcal{P}_-$) encloses the electrons at $(\pi,0)$ and $(0,\pi)$. The (cluster) impurity model is solved by continuous-time Monte-Carlo sampling of its partition function, written as a power series in the hybridization between impurity and bath (CT-HYB) [@GullRev:2011; @Haule:2007a; @Semon:2014]. This solver is well suited for strong and/or complex interactions as arising in the context of realistic material simulations. The price to pay is a complexity that scales with the dimension of the Hilbert space of the impurity. The 5 d-orbitals split into $e_g=\{ 3z^2 - r^2, x^2 - y^2 \}$ and $t_{2g} = \{yz, zx, xy \}$ degrees of freedom. Since the latter contribute the dominant character of the bands near the Fermi level, an idea to obtain a cluster impurity problem amenable for CT-HYB is to apply DCA only to the $t_{2g}$ orbitals, while the $e_g$ orbitals are treated within DMFT. To make this idea more specific, it is convenient to consider DMFT and DCA as approximations of the Luttinger-Ward functional[@LuttingerWard:1960] $\Phi_{UJ}[\mathbf{G}]$, a functional of the dressed Green function $\mathbf{G}$ which depends on the interacting part of the problem only, that is $U$ and $J$ in our case. Its derivative is the self-energy, and together with the Dyson equation $$\label{equ:Dyson} \mathbf{G}_0^{-1} - \mathbf{G}^{-1} = \mathbf{\Sigma}[\mathbf{G}]= \frac{1}{kT}\frac{\delta \Phi_{UJ}[\mathbf{G}]}{\delta \mathbf{G}},$$ the (approximate) Luttinger-Ward functional determines hence the (approximate) solution of the problem with bare Green function $\mathbf{G}_{0}$. Diagrammatically, the Luttinger-Ward functional is the sum of all vacuum-to-vacuum skeleton diagrams, and DMFT keeps only the diagrams with support on a site. In momentum space, this corresponds to neglect conservation of momentum at the vertices, which is partially restored in DCA by conserving at least the coarse grained momentum $\mathbf{K}$. We call the corresponding functionals $\Phi^\text{loc}_{UJ}[\mathbf{G}]$ and $\Phi^\text{cl}_{UJ}[\mathbf{G}]$, respectively. In this functional formulation, the mixed DMFT-DCA treatment of the orbitals that we propose consists in approximating the lattice functional as $$\label{equ:LWApprox} \Phi_{UJ}[\mathbf{G}] = \Phi^{\text{loc}}_{UJ}[\mathbf{G}] + \Phi^{\text{cl}}_{\tilde{U}\tilde{J}}[\hat{P}_{t_{2g}}\mathbf{G}] - \Phi^\text{loc}_{\tilde{U}\tilde{J}}[\hat{P}_{t_{2g}}\mathbf{G}],$$ where $\hat{P}_{t_{2g}}$ is the projector on $t_{2g}$ orbitals. One can think of this as a selective improvement of the diagrammatic summation by going from single site to cluster DCA for the $t_{2g}$ orbitals which is corrected by subtracting the double counting of the single site DMFT $t_{2g}$ diagrams. The use of $\tilde{U}$ and $\tilde{J}$ reflects the screening of the bare interactions by the elimination of the $e_g$ degrees of freedom in the cluster corrections. In the supplementary information, we show how the screening is determined and how the mixed DMFT-DCA scheme is solved in practice. For the sake of completeness, the solution of the DMFT equations and the impurity models are detailed as well. In the following, all energies are given in units of $eV$ and the filling is constrained to 6 electrons per $Fe$ atom. The upper panel in Fig. \[fig:DCAvsDMFT\] shows the $t_{2g}$ the self-energies obtained by DMFT and DCA at $T=174K$, $(U,\tilde{U})=(4.5,4.5)$ and $(J,\tilde{J})=(0.45,0.375)$. The DCA self-energy is shown in a “real-space site basis" with local part $(\mathbf{\Sigma}_{\mathbf{K}=+} + \mathbf{\Sigma}_{\mathbf{K}=-})/2$ ![(Color online) Comparison of the $t_{2g}$ self-energy obtained by DMFT (real/imaginary part with thin/bold lines) and DCA (local/non-local part in red/blue and real/imaginary part with crosses/circles). All self-energies are diagonal in orbital space, see supplementary information Sec. A. The left panels show the degenerate $yz,zx$ entries and the right panel shows the $xy$ entry. The temperature is $T=174K$. In the upper panels $(U,\tilde{U})=(4.5,4.5)$ and $(J,\tilde{J})=(0.45,0.375)$ while in the lower panels $(U,\tilde{U})=(10.125,9)$ and $(J,\tilde{J})=(0,0)$. []{data-label="fig:DCAvsDMFT"}](selfDCAvsDMFT.pdf){width="1\columnwidth"} and non-local part $(\mathbf{\Sigma}_{\mathbf{K}=+} - \mathbf{\Sigma}_{\mathbf{K}=-})/2$. The non-local self-energy is essentially zero and the local self-energy is in excellent agreement with DMFT. The quasiparticle weight is $(Z_{yz/zx},Z_{xy})=(0.4,0.3)$ and the filling of the $t_{2g}$-filling per $Fe$ atom is $N_{t_{2g}}=3.186$. To address the question ![(Color online) Impurity spin susceptibility (see Eq. \[equ:susceptibility\]) for DMFT (red squares) and DCA (blue circles) for parameters $(U,\tilde{U})=(4.5,4.5)$ and $(J,\tilde{J})=(0.45,0.375)$. The upper panel plots the susceptibility as a function of Matsubara frequencies for $T=174K$. The lower panel plots the susceptibility as function of the temperature for $i\nu_n=0$. The black diamonds display the ratio between the DMFT and DCA spin susceptibility, and its scale is displayed in the right y-axis.[]{data-label="fig:Susc"}](chi.pdf){width="1.0\columnwidth"} wether this is due to the Hund’s rule coupling or the orbital degeneracy, we set $J=0$ but increase $U$ in order to stay in a correlated regime, see lower panel in Fig. \[fig:DCAvsDMFT\]. The self-energies are local as well, $(Z_{yz/zx},Z_{xy})=(0.41,0.39)$ and $N_{t_{2g}}=2.77$. Another question that arises is the locality at the two particle level. To this end, we measure the impurity spin susceptibility defined as $$\label{equ:susceptibility} \chi_{z}(i \nu_n) = \frac{1}{N_{\mathcal{P}}}\int_{0}^\beta e^{i\nu_n \tau}\langle S^{t_{2g}}_z(\tau) S^{t_{2g}}_z\rangle d\tau,$$ where $S_z^{t_{2g}}$ is the total spin along the $z$ direction of the $t_{2g}$ degrees of freedom on the impurity, for both DMFT ($N_{\mathcal{P}}=1$) and DCA ($N_{\mathcal{P}}=2$), see Fig. \[fig:Susc\]. We also plot the ratio of the DCA and DMFT susceptibility which is $\approx 0.9$, meaning that even at the two particle level, our coarse graining does not show momentum space differentiation. This is very different from the cuprate case. Fig. \[fig:Sign\] shows the average sign in the CT-HYB simulations for the DCA impurity model for different temperatures and Hund’s rule couplings. The sign rapidly drops with increasing Hund’s rule coupling. This makes cluster simulations of materials with large mass renormalizations expensive, in particular at low temperatures. ![Average sign in the CT-HYB simulations for the DCA impurity model, in the left panel for $(J,\tilde{J})=(0.45,0.375)$ as function of the temperature and in the right panel for $T=174K$ as a function of the Hund’s rule coupling (the values are $(J,\tilde{J})=(0.45,0.375)$, $(0.488,0.441)$, $(0.506,0.469)$ and $(0.525,0.51)$ from left to right). The Coulomb repulsion is $(U,\tilde{U})=(4.5,4.5)$ in both panels. The CT-HYB simulations are carried out in the $\mathbf{K}$-space single particle basis, see supplementary information Sec. D.[]{data-label="fig:Sign"}](sign.pdf){width="1.0\columnwidth"} To conclude, we have demonstrated that the local approximation describes well Hunds metals, such as many iron-pnictides and chalcogenides, in their normal state. In the region of large mass renormalizations, relevant to materials such as $FeTe$, there is an onset of a severe minus sign problem. In itself this does not prove non-locality of the self-energies, but the investigation of this region will require other impurity solvers and is outside the scope of this work. We have solved the same model Hamiltonian with other two site tiling of the Brillouin zone. The results support our conclusion that the self energy is local, with little tendency towards momentum space differentiation in the parameter range explored in this paper. Recently a three band Hamiltonian with nearest neighbors on the square lattice and strong Hunds and Hubbard interactions was studied.[@Nomura:2015] Strong momentum space differentiation was found for much larger values of the Hunds coupling and the Hubbard U. Comparing our results with Ref.  raises the question of what are the essential ingredients (dispersion relation, filling or interaction strength) needed to obtain momentum space differentiation in multi-orbital problems. The technical advances introduced in this paper make possible the investigation of symmetry breaking phases. Future work will apply this formalism to address nematic, magnetic and superconducting order in the iron pnictides and chalcogenides. This work was supported by NSF. NSF-DMR1308141 (P.S. and G.K.) and NSF-DMR1405303 (K.H.). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC05-00OR22725. [54]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1021/ja063355c) @noop [ ()]{},  [****,  ()](http://stacks.iop.org/1367-2630/11/i=2/a=025021) [**** (), 10.1038/nphys1923](\doibase 10.1038/nphys1923) [****,  ()](\doibase 10.1103/PhysRevLett.106.096401) [****,  ()](\doibase 10.1146/annurev-conmatphys-020911-125045) [****,  ()](\doibase 10.1103/RevModPhys.68.13) [****,  ()](\doibase 10.1103/PhysRevLett.87.186401) [****,  ()](\doibase 10.1103/RevModPhys.77.1027) [****,  ()](\doibase 10.1103/PhysRevLett.113.246404) [****,  ()](\doibase 10.1103/PhysRevLett.86.139) [****,  ()](\doibase 10.1103/PhysRevB.62.R9283) [****,  ()](http://stacks.iop.org/0295-5075/56/i=4/a=563) [****,  ()](\doibase 10.1103/PhysRevB.64.195130) [****,  ()](\doibase 10.1103/PhysRevB.68.155119) [****,  ()](\doibase 10.1103/PhysRevLett.92.226402) [****,  ()](\doibase 10.1103/PhysRevB.70.045107) [****,  ()](\doibase 10.1103/PhysRevLett.95.106402) [****,  ()](\doibase 10.1103/PhysRevB.74.125110) [****,  ()](\doibase 10.1103/PhysRevB.73.165114) [****,  ()](\doibase 10.1103/PhysRevLett.97.036401) [****,  ()](\doibase 10.1103/PhysRevB.76.092503) [****,  ()](\doibase 10.1103/PhysRevB.76.045108) [****,  ()](\doibase 10.1103/PhysRevLett.100.046402) [****,  ()](\doibase 10.1103/PhysRevB.79.195113) [****, ()](\doibase 10.1103/PhysRevB.80.165126) [****,  ()](\doibase 10.1103/PhysRevLett.102.056404) [****,  ()](\doibase 10.1103/PhysRevB.80.045120) [****,  ()](\doibase 10.1103/PhysRevB.82.155101) [****,  ()](\doibase 10.1103/PhysRevB.82.045104) [**** (), 10.1038/srep00547](\doibase 10.1038/srep00547) [****,  ()](\doibase 10.1103/PhysRevB.87.041101) [****, ()](\doibase 10.1103/PhysRevLett.110.216405) [****,  ()](\doibase 10.1103/PhysRevB.88.075127) [****,  ()](http://stacks.iop.org/1742-6596/449/i=1/a=012005) [****,  ()](\doibase 10.1103/PhysRevB.65.155112) [****,  ()](\doibase 10.1103/PhysRevB.71.037101) [****,  ()](\doibase 10.1103/PhysRevB.71.037102) [****,  ()](\doibase 10.1103/PhysRevLett.95.237001) [****,  ()](\doibase 10.1103/PhysRevB.73.205106) [****,  ()](\doibase 10.1103/PhysRevB.85.035102) [****,  ()](http://dx.doi.org/10.1038/nphys3116) [****,  ()](\doibase 10.1103/PhysRevB.91.140503) [****,  ()](\doibase 10.1103/PhysRevB.82.064504) [****,  ()](\doibase 10.1103/PhysRevLett.109.237010) [****,  ()](\doibase 10.1103/PhysRevB.80.064501) [****,  ()](\doibase 10.1103/PhysRevB.80.094531) [****,  ()](\doibase 10.1103/RevModPhys.83.349) [****, ()](\doibase 10.1103/PhysRevB.75.155113) [****,  ()](\doibase 10.1103/PhysRevB.90.075149) [****, ()](\doibase 10.1103/PhysRev.118.1417) [****,  ()](\doibase 10.1103/PhysRevB.91.235107) Supplementary Information ========================= We begin here by writing down the equations for the the mixed DMFT-DCA scheme as defined by Eqs. \[equ:Dyson\] and \[equ:LWApprox\] by means of impurity models. We then show how the effective interactions $\tilde{U}$ and $\tilde{J}$ are determined and the equations are solved in practice. Finally, we detail the impurity models. A. Mixed DMFT-DCA equations --------------------------- The functional derivative of Eq. \[equ:LWApprox\] yields the approximation $$\label{equ:SelfApprox} \mathbf{\Sigma}(\mathbf{k})= \left ( \begin{array}{cc} \mathbf{\Sigma}^\text{loc}_{t_{2g}} + \mathbf{\tilde{\Sigma}}^\text{cl}_\mathbf{K} - \mathbf{\tilde{\Sigma}}^\text{loc}& 0 \\ 0 & \mathbf{\Sigma}_{e_g}^\text{loc} \end{array} \right )$$ for the lattice self-energy written in $\mathbf{k}$-space, where $\mathbf{K}=+$ $(\mathbf{K}=-)$ if $\mathbf{k}$ lies in the patch $\mathcal{P}_+$ ($\mathcal{P}_-$). The self-energies on the right hand side of Eq. \[equ:SelfApprox\] are identified with those of impurity models as follows: - $\mathbf{\Sigma}^\text{loc}$, a diagonal $5\times 5$ matrix in $d$-shell orbital space with components $\mathbf{\Sigma}^\text{loc}_{t_{2g}}$ and $\mathbf{\Sigma}^\text{loc}_{e_g}$, is the self-energy of a single site $d$-shell impurity model with interactions $U$ and $J$. - $\mathbf{\tilde{\Sigma}}^\text{loc}$, a diagonal $3\times 3$ matrix in $t_{2g}$ orbital space, is the self-energy of a single site $t_{2g}$-orbital impurity model with effective interactions $\tilde{U}$ and $\tilde{J}$. - $\mathbf{\tilde{\Sigma}}^\text{cl}_\mathbf{K}$, a diagonal $3\times 3$ matrix in $t_{2g}$ orbital space for each $\mathbf{K}$, is the self-energy of a two-site $t_{2g}$-orbital cluster impurity model with effective interactions $\tilde{U}$ and $\tilde{J}$. The non-interacting part of these impurity models is encapsulated in the Weiss-Fields $\mathbf{G}_0^\text{loc}$, $\mathbf{\tilde{G}}_0^\text{loc}$ and $\mathbf{\tilde{G}}^\text{cl}_{0\mathbf{K}}$, which relate the self-energies with the interacting Greens functions $\mathbf{G}^\text{loc}$, $\mathbf{\tilde{G}}^\text{loc}$ and $\mathbf{\tilde{G}}^\text{cl}_\mathbf{K}$ through the Dyson equations \[equ:ImpurityDyson\] $$\begin{aligned} (\mathbf{G}^\text{loc})^{-1}&=(\mathbf{G}_0^\text{loc})^{-1} - \mathbf{\Sigma}^\text{loc}\\ (\mathbf{\tilde{G}}^\text{loc})^{-1}&=(\mathbf{\tilde{G}}_{0}^\text{loc})^{-1} - \mathbf{\tilde{\Sigma}}^\text{loc}\\ (\mathbf{\tilde{G}}^\text{cl}_\mathbf{K})^{-1}&=(\mathbf{\tilde{G}}_{0\mathbf{K}}^\text{cl})^{-1} - \mathbf{\tilde{\Sigma}}^\text{cl}_\mathbf{K}.\end{aligned}$$ Eq. \[equ:SelfApprox\] yields the approximate lattice Green function $$\label{equ:LatticeDyson} \mathbf{G}^{-1}(\mathbf{k}) = \mathbf{G}_0^{-1}(\mathbf{k}) - \mathbf{\Sigma}(\mathbf{k}),$$ where $\mathbf{G}_0^{-1}(i\omega_n,\mathbf{k}) = i\omega_n + \mu - \mathbf{h}_0(\mathbf{k})$ is the bare lattice Green function. The DMFT and DCA approximations of the Luttinger-Ward functional then require \[equ:SelfConsistency\] $$\begin{aligned} \mathbf{G}^\text{loc}&=\frac{1}{|\text{BZ}|}\int_\text{BZ} d\mathbf{k} \mathbf{G}(\mathbf{k})\\ \mathbf{\tilde{G}}^\text{loc} &= \frac{1}{|\text{BZ}|} \int_{\text{BZ}} d\mathbf{k}\hat{P}_{t_{2g}}\mathbf{G}(\mathbf{k})\\ \mathbf{\tilde{G}}_\mathbf{K}^\text{cl} &= \frac{1}{|\mathcal{P}_\mathbf{K}|}\int_{\mathcal{P}_\mathbf{K}} d\mathbf{k} \hat{P}_{t_{2g}} \mathbf{G}(\mathbf{k}).\end{aligned}$$ Fixing the chemical potential by imposing 6 electrons per atom, above equations determine the Weiss-Fields and hereby the solution of the mixed DMFT-DCA scheme. The interactions $U$ and $J$ are taken as external parameters, and what remains to be determined are the effective interactions $\tilde{U}$ and $\tilde{J}$ which take into account the screening of the $t_{2g}$ degrees of freedom in the cluster corrections. Notice here that, in the normal phase, the DMFT self-energies $\mathbf{\Sigma}^\text{loc}$ and $\mathbf{\tilde{\Sigma}}^\text{loc}$ (and also $\mathbf{G}_0^\text{loc}$, $\mathbf{\tilde{G}}_0^\text{loc}$, $\mathbf{G}^\text{loc}$ and $\mathbf{\tilde{G}}^\text{loc}$) are diagonal in the orbital space. This comes from the $D_{2d}$ point symmetry group of an $Fe$ atom. Furthermore, the patches are invariant under this symmetry group, so that the DCA self-energy $\mathbf{\tilde{\Sigma}}_\mathbf{K}^\text{cl}$ (and also $\mathbf{\tilde{G}}_{0\mathbf{K}}^\text{cl}$ and $\mathbf{\tilde{G}}_{\mathbf{K}}^\text{cl}$) is diagonal in the orbital space as well. B. Effective interactions ------------------------- To determine the effective interactions, we define an effective problem where correlations are applied only to the $t_{2g}$ orbitals. These effective correlations, which are identified with $\tilde{U}$ and $\tilde{J}$, are then determined by requiring that this model reproduces at low energies the results of the five band calculation (with correlations $U$ and $J$), *when both models are solved via single site DMFT*. We use the following algorithm: - The five band model is solved with single site DMFT for a filling of 6 $d$-shell electrons per $Fe$ atom and interactions $U$ and $J$. This yields a local lattice self-energy $\mathbf{\Sigma}^{\text{loc}}$ (with components $\mathbf{\Sigma}_{e_g}^\text{loc}$ and $\mathbf{\Sigma}_{t_{2g}}^\text{loc}$), a filling of the $t_{2g}$ orbitals and a chemical potential. - We determine the low energy model by defining the effective bare lattice propagator $$\label{equ:effective_propagator} \mathbf{\tilde{G}}_0^{-1} := \mathbf{G}_0^{-1}-\left ( \begin{array}{cc} \Sigma^{\text{HF}}\cdot \mathbf{1}_{t_{2g}}& 0 \\ 0 & \mathbf{\Sigma}_{e_g}^\text{loc} \end{array} \right ).$$ Here, $\mathbf{\Sigma}^\text{loc}_{e_g}$ is the $e_g$ part of the self-energy obtained in (i), and $\Sigma^{\text{HF}}$ (which can be thought as an average Hartree-Fock contribution to the $t_{2g}$ self-energy coming from the $e_g$ orbitals) will be determined in the next step. The chemical potential is fixed to the value obtained in (i). - To determine $\Sigma^{\text{HF}}$, $\tilde{U}$ and $\tilde{J}$, we solve the problem with propagator Eq. \[equ:effective\_propagator\] and the effective interactions applied to the $t_{2g}$ orbitals with single-site DMFT. The resulting self-energy is denoted by $\mathbf{\tilde{\Sigma}}^\text{loc}$. $\Sigma^{\text{HF}}$ is determined by requiring that the $t_{2g}$ filling is the same as in (i). Requiring that $\mathbf{\tilde{\Sigma}}^\text{loc}+\Sigma^{\text{HF}}$ matches $\mathbf{\Sigma}_{t_{2g}}^\text{loc} $ at the lowest Matsubara frequencies determines the effective interactions $\tilde{U}$ and $\tilde{J}$. It is remarkable that these requirements give us very good matching of the self-energies at all energies, as shown in Fig. \[fig:t2gApproximation\]. ![(Color online) Self-energies (real/imaginary part with open/filled symbols) of the $t_{2g}$ orbitals obtained with DMFT for the model with interactions applied to all $d$-shell orbitals (red circles) and for the effective $t_{2g}$ model (blue diamonds). In the latter case, the Hartree-Fock constant $\Sigma^{\text{HF}}$ is added. The parameters are $T=174K$, $(U,\tilde{U})=(4.5,4.5)$ and $(J,\tilde{J})=(0.45,0.375)$.[]{data-label="fig:t2gApproximation"}](cut_eg.pdf){width="\columnwidth"} C. Solving the mixed DMFT-DCA equations and the DMFT equation ------------------------------------------------------------- The good agreement of the self-energies in Fig. \[fig:t2gApproximation\] suggests to solve the mixed DMFT-DCA scheme in a simplified manner. Instead of simultaneously solving the three impurity models in Sec. A, we just apply DCA to the effective $t_{2g}$ model used to determine the screened interactions $\tilde{U}$ and $\tilde{J}$ . $\Sigma^{\text{HF}}$ is slightly readjusted to preserve the $t_{2g}$ filling found in Sec. B (i). This simplified solution is justified *if the cluster corrections to local quantities is small*. Indeed, in this case we can start by ignoring the cluster corrections when solving the mixed DMFT-DCA scheme and solve the model with DMFT (which corresponds to step (i) in Sec. B). We then solve the model with cluster corrections (where $\tilde{U}$ and $\tilde{J}$ have been determined as discussed in Sec. B), keeping however the $e_g$ self-energy $\mathbf{\Sigma}_{e_g}^\text{loc}$ and the chemical potential obtained without cluster corrections fixed. Further, the contribution to the $t_{2g}$ self-energy from $\mathbf{\Sigma}^\text{loc}_{t_{2g}} - \mathbf{\tilde{\Sigma}}^\text{loc}$ is replaced by a constant proportional to the identity, which is justified by Fig. \[fig:t2gApproximation\]. Choosing this constant $\Sigma^{\text{HF}}$ to preserve the $t_{2g}$ filling found in Sec. B (i), this amounts just to solve the effective $t_{2g}$ model with DCA as mentioned above. Compared to the exact solution of the DMFT-DCA scheme, this simplified solution avoids stability issues and guaranties causality (c.f. nested cluster schemes in Ref. ). When comparing results from the mixed DMFT-DCA scheme with DMFT results, the latter is applied to the effective $t_{2g}$ problem for the sake of coherence. The DMFT self-energy is thus $\mathbf{\tilde{\Sigma}}^\text{loc}$ from Sec. B (iii), while the mixed DMFT-DCA self-energy is $\mathbf{\Sigma}_\mathbf{K}^\text{cl}$, obtained in the above approximation. D. Impurity Models ------------------ The aim here is to write down the action for the impurity models in Sec. A. To this end, we begin by detailing the interaction used in this work. With respect to the $d$-shell single particle basis $|\sigma m \rangle$, where $\sigma$ is the spin and the angular part is encapsulated in the spherical harmonics $Y_{l=2}^m$, the local interaction $$\label{equ:interaction} \hat{V} = \frac{1}{2}\sum_{\sigma \sigma'}\sum_{\{m_i\}}V_{m_1m_2m_3m_4}c^\dagger_{\sigma m_1}c^\dagger_{\sigma' m_2}c_{\sigma' m_3}c_{\sigma m_4}$$ is given by the tensor $$\label{equ:tensor} \begin{split} &V_{m_1m_2m_3m_4} = \sum_{k=0,2,4} \frac{4\pi}{2k+1}F^k \\ &\quad \times \sum_{m=-k}^k \langle Y_{2}^{m_1} | Y_k^{m*} | Y_2^{m_4}\rangle \langle Y_{2}^{m_2} | Y_k^m | Y_2^{m_3}\rangle. \end{split}$$ The Slater-Condon parameters $F^0$, $F^2$ and $F^4$ encapsulate both the radial part of the single particle basis (which is the same for all $|\sigma m\rangle$) and the interaction. In this work we use $F^0=U$, $F^2=14\cdot J /1.625$ and $F^4=0.625 \cdot F^2$, where $U$ is the Coulomb repulsion and $J$ the Hund’s rule coupling. In solids, it is more convenient to work in the basis of real spherical harmonics $|\sigma \alpha\rangle$ with $\alpha \in \{yz,zx,xy,3z^2-r^2,x^2-y^2\}$, and we denote the corresponding creation operators by $d_{\sigma \alpha}^\dagger$. We slightly simplify the interaction tensor $V_{\alpha_1\alpha_2\alpha_3\alpha_4}$ in this basis by setting all elements which are not of the form $V_{\alpha\alpha\alpha'\alpha'}$,$V_{\alpha\alpha'\alpha'\alpha}$ or $V_{\alpha\alpha'\alpha\alpha'}$ to zero. While this truncation preserves the spin $SU(2)$ invariance of the interaction Eqs. \[equ:interaction\] and \[equ:tensor\], the orbital $SO(3)$ invariance is lifted. However, the truncated interaction is still $D_{2d}$ invariant and the crystal fields in the present case lift the $SO(3)$ degeneracy anyway. In the basis of real spherical harmonics, the action for the single-site $d$-shell impurity model Sec. A (i) reads $$\label{equ:DMFTAction} \begin{split} S =& -\sum_\sigma \sum_\alpha \iint_0^\beta d^\dagger_{\sigma \alpha}(\tau)G^{-1}_{0\alpha \alpha }(\tau - \tau')d_{\sigma \alpha}(\tau')d\tau d\tau' \\ &+\frac{1}{2}\sum_{\sigma\sigma'}\sum_{\{\alpha_i\}} V^{UJ}_{\alpha_1\alpha_2\alpha_3\alpha_4}\int_0^\beta d_{\sigma \alpha_1}^\dagger(\tau) d_{\sigma'\alpha_2}^\dagger(\tau) \\ &\times d_{\sigma'\alpha_3}(\tau)d_{\sigma \alpha_4}(\tau) d\tau, \end{split}$$ where the superscript of the interaction tensor indicates that $U$ and $J$ enter the Slater-Condon parameters. Restricting in the action Eq. \[equ:DMFTAction\] the orbital sums to $t_{2g}$ orbitals and replacing $U$, $J$ and $\mathbf{G}_0$ by the effective $\tilde{U}$, $\tilde{J}$ and $\mathbf{\tilde{G}}_0$ respectively yields the single-site $t_{2g}$ impurity model Sec. A (ii). The non-interacting part of the cluster impurity model action Sec. A (iii) reads $$\begin{split} S_0 = -\sum_\sigma \sum_{\mathbf{K}=\pm}\sum_{\alpha \in t_{2g}}&\iint_0^\beta d^\dagger_{\sigma\alpha\mathbf{K}}(\tau)\tilde{G}^{-1}_{0\mathbf{K}\alpha \alpha }(\tau - \tau')\\ &\times d_{\sigma \alpha \mathbf{K}}(\tau')d\tau d\tau', \end{split}$$ where $d_{\sigma \alpha \mathbf{K}}^\dagger$ creates an electron with spin $\sigma$ and coarse grained momentum $\mathbf{K}$ in the orbital $\alpha$. The interacting part, written in a “real-space site basis" $d_{\alpha\sigma 1}:=(d_{\sigma\alpha +} + d_{\sigma \alpha -})/\sqrt 2$ and $d_{\alpha\sigma 2}:=(d_{\sigma\alpha +} - d_{\sigma\alpha -})/\sqrt 2$, reads $$\label{equ:DCAinteraction} \begin{split} S_I=&\frac{1}{2}\sum_{\sigma\sigma'}\sum_{a=1,2}\sum_{\lbrace \alpha_i\rbrace \in t_{2g}}V^{\tilde{U}\tilde{J}}_{\alpha_1 \alpha_2 \alpha_3 \alpha_4} \\ &\times \int_0^\beta d^\dagger_{\sigma \alpha_1 a}(\tau)d^\dagger_{\sigma' \alpha_2 a} (\tau)d_{\sigma' \alpha_3 a}(\tau)d_{\sigma \alpha_4 a}(\tau) d\tau. \end{split}$$ For the CT-HYB simulations, the $\mathbf{K}$-space single particle basis $d_{\sigma \mathbf{K} \alpha}^\dagger$ is used. [53]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/PhysRevB.69.205108) [^1]: This corresponds to choose $(pd\sigma)^2/|\epsilon_d - \epsilon_p|=0.75eV$ in Ref. .

--- abstract: 'The dynamics of vortex ring pairs in the homogeneous nonlinear Schrödinger equation is studied. The generation of numerically-exact solutions of traveling vortex rings is described and their translational velocity compared to revised analytic approximations. The scattering behavior of co-axial vortex rings with opposite charge undergoing collision is numerically investigated for different scattering angles yielding a surprisingly simple result for its dependence as a function of the initial vortex ring parameters. We also study the leapfrogging behavior of co-axial rings with equal charge and compare it with the dynamics stemming from a modified version of the reduced equations of motion from a classical fluid model derived using the Biot-Savart law.' author: - 'R. M. Caplan' - 'J. D. Talley' - 'R. Carretero-González' - 'P.G. Kevrekidis' title: Scattering and leapfrogging of vortex rings in a superfluid --- Introduction {#s:intro} ============ One of the most widespread and interesting models for studying the emergence, dynamics and interactions of coherent structures is the nonlinear Schr[ö]{}dinger equation (NLSE) [@sulem99]. The NLSE is a paradigm for the evolution of coherent structures since it is a universal model describing the evolution of complex field envelopes in nonlinear dispersive media [@dodd83]. This universality stems from the fact that the NLSE is the prototypical, lowest-order, nonlinear partial differential equation (PDE) that describes the dynamics of modulated envelope waves in a nonlinear medium [@debnath05]. As such, it appears in a wide variety of physical contexts, ranging from optics [@hasegawa90; @abdullaev93; @hasegawa95; @kivshar03] to fluid dynamics and plasma physics [@infeld90] to matter waves [@kevrekidis08; @carretero08b], while it has also attracted much interest mathematically [@sulem99; @ablowitz04; @bourgain99; @ablowitz81; @zakharov84; @newell85]. The general form of the NLSE (with the lowest order nonlinearity) can be written, in non-dimensional units, as $$\label{nlse} i\frac{\partial \Psi}{\partial t}+a\nabla^2\Psi +s|\Psi|^2\Psi=0$$ where $a$ is a parameter and $s$ represents the strength of the nonlinear interaction and determines whether the NLSE is attractive ($s>0$) or repulsive ($s<0$). The NLSE is a model for the evolution of the mean-field wavefunction in Bose-Einstein condensates (BECs) at low temperatures [@Dalfovo99; @Pethick2002], namely a superfluid. In the BEC setting, the NLSE also contains an external trapping potential term that is often spatially parabolic or periodic in nature and stems from the magnetic or optical confinement of the atoms. The NLSE allows for the prediction and description of a wide range of (nonlinear) excitations depending on the sign of the nonlinearity and the dimensionality of the system, the latter of which that can be tuned by the ratio of trapping strengths along the different spatial directions [@kevrekidis08]. In this manuscript we are interested in the fully three-dimensional regime (3D) with repulsive nonlinearity ($s<0$) whose basic nonlinear excitations correspond to vortex lines and vortex rings. A vortex line is the 3D extension of a two-dimensional (2D) vortex by (infinitely and homogeneously) extending the solution into the axis perpendicular to the vortex plane. Vortex lines might be rendered finite in length if the background where they live is bounded by the externally confining potential. In that case, vortex lines become vorticity “tubes” that are straight across the background or bent in [U]{} and [S]{} shapes, depending on the aspect ratio of the background [@VR:USshapedVLs1; @VR:USshapedVLs2]. If a vortex line is bent enough to close on itself, or if two vortex lines are close enough to each other, they can produce a vortex ring [@VR:CrowInstab]. Vortex rings are 3D structures whose core is a closed loop with vorticity around it [@donnelly] (i.e., a vortex line that loops back into itself). Vortex rings have been observed experimentally in superfluid helium [@Rayfield64; @Gamota73] as well as in the context of BECs in the decay of dark solitons in two-component BECs [@Anderson01], direct density engineering [@Shomroni09] (see also the complementary theoretical proposals of Refs. ), and in the evolution of colliding symmetric defects [@Ginsberg05]. They have also been argued to be responsible for unusual experimental collisional outcomes of structures that may appear as dark solitons in cigar-shaped traps [@sengstock]. Numerical studies of experimentally feasible vortex ring generation in BECs have also been explored in the context of flow past an obstacle [@Jackson99; @Rodrigues09], Bloch oscillations in an optical trap [@Scott04], collapse of bubbles [@Berloff04a], instability of 2D rarefaction pulses [@Berloff02], flow past a positive ion [@Berloff00; @Berloff00a] or an electron bubble [@Berloff01], crow instability of two vortex pairs [@Berloff01a] and collisions of multiple BECs [@Carretero08; @Carretero08a]. Vortex rings have an intrinsic velocity [@Roberts71], which can be overcome by counteracting the velocity with a trapping potential [@VR-BEC-STRUCTGOOD], adding Kelvin mode perturbations [@maggioni10; @Helm11; @Helm10], or by placing the vortex ring in a co-traveling frame (see Sec. \[s:1VR\]). In the present manuscript, motivated by the above abundant interest in this theme both from a theoretical perspective and from that of ongoing experimental efforts in BECs, we are interested in the dynamical evolution of vortex rings and in particular in their interactions. In an effort to exclusively capture the vortex ring dynamics and not the influence of the external potential (the latter is a natural subject for future work), we assume the absence of any trapping potential and thus consider the background supporting the vortex rings as homogeneous. On this homogeneous background we revisit the intrinsic velocity of a single vortex ring that serves as the self-interacting term that will be supplemented by the interaction terms between different vortex rings. The main goal of our work is to describe vortex ring interactions and scattering collisions in a reduced (i.e., “effective”) manner. The motivation for this approach stems from the fact that full numerical simulations in the 3D NLSE can be very time consuming especially when examining scenarios involving multiple vortex rings in large domains. The first dynamical scenario that we study corresponds to the scattering of colliding vortex rings of opposite charge. Using a large set of full 3D numerical simulations (based on efficient GPU codes, see below), we are able to distill a very simple, phenomenological, rule describing the scattering angle of vortex rings as a function of the vortex ring radii and the initial collisional offset. Then, based on the interaction of vortex filaments from classical fluids, we revisit a reduced, ordinary differential equation, model for the interaction of co-axial vortex rings based on the Biot-Savart law, adapting it to the superfluid (BEC) case. The ensuing reduced dynamics for the leapfrogging of vortex ring pairs is then favorably compared to full 3D simulations. Our manuscript is structured as follows. In Sec. \[s:1VR\] we describe a procedure to generate numerically-exact traveling vortex ring solutions in a homogeneous background, and test the resulting ring’s translational velocity against suitably revised known analytic approximations. Section \[s:scattering\] is devoted to describing scattering scenarios from the collisions of vortex rings, where we derive a very simple phenomenological relationship for the scattering angle as a function of the initial offset distance and radii of the colliding rings. In Sec. \[s:leapfrog\] we use effective equations of motion for the interaction of co-axial vortex rings to describe the leapfrogging behavior of two vortex rings and compare the results with direct simulations of the NLSE. Finally, Sec. \[s:conclu\] summarizes our results and prompts a few avenues for future exploration. Before we embark on the relevant analysis, we provide some details on our computational methods which pertain to all the numerical simulations given below. For all 3D NLSE simulations, we use high-order explicit finite-difference schemes on a uniform grid with cell-spacing $h=\Delta x=\Delta y=\Delta z$ and constant time-step $k=\Delta t$. Time-stepping is accomplished with the standard 4th-order Runge-Kutta method (RK4), while spatial differencing is performed with the 4th-order 2-step high-order compact scheme described in Ref. . The time-step $k$ is set based on the stability bounds of the overall scheme as discussed in Ref. . Since the stability forces the time-step to be proportional to $h^2$, the overall error of the scheme is $O(h^4)$. Due to the constant density background of the problem, we utilize a recently developed modulus-squared Dirichlet boundary condition [@Caplan14]. The simulations are computed using the NLSEmagic code package [@NLSEmagic] [^1] which contains algorithms written in C and CUDA, and are primarily run on NVIDIA GeForce GTX 580 and GeForce GT 650M GPU cards. Generation of a numerically-exact non-stationary vortex ring {#s:1VR} ============================================================ In order to achieve accurate results for simulating the interactions of multiple vortex rings in a homogeneous background, it is necessary to first be able to generate numerically “exact” solutions of the individual rings. Since vortex rings have an intrinsic transverse velocity due to their topological structure, standard steady-state methods (such as imaginary time integration [@VR-BEC-STRUCTGOOD] and nonlinear equation solvers) for obtaining solutions of coherent structures cannot be directly applied, unless the rings are in a configuration where they are at steady-state (such as in a magnetic trap [@VR-BEC-STRUCTGOOD]). In order to be able to generate rings in a homogeneous background, we apply a co-moving background velocity to render the vortex ring stationary. In order to apply the back-flow, we need a very accurate value of the vortex ring’s transverse velocity. The transverse velocity of a single vortex ring in the NLSE has been studied analytically [@Amit66; @Fetter66; @Roberts71] and the results were shown to be consistent with numerical simulations of the NLSE [@Koplik96; @Helm10]. For the NLSE in the form of Eq. (\[nlse\]), with $s<0$ and $V=0$, an asymptotically approximate velocity for the ring is given by [@Roberts71] $$\label{vrvel} c \approx -\frac{am}{d}\left(\ln \frac{8d}{r_c} + L_0(m) - 1\right),$$ where $d$ is the ring’s radius, $m$ is its the charge, and $r_c$ is the vortex ring’s core radius defined as $$r_c=|m| \xi,$$ where $\xi$ is the healing length given by $$\xi=\sqrt{-\frac{\Omega}{a}}.$$ $\Omega$ plays the role of the frequency and is tantamount to the system’s chemical potential. The value $L_0(m)$ is referred to as the vortex core parameter and is defined as the convergent part of the energy-per-unit-length of a vortex line in the NLSE [@Roberts71]. While $L_0(m)$ depends on the vortex ring charge $m$, it is independent of the NLSE parameters. In order to determine the velocity of the vortex ring as precisely as possible, $L_0(m)$ must be computed accurately. Since there are various values of $L_0(m)$ reported in the literature (cf. values given in Refs. ), we briefly show in the Appendix how $L_0(m)$ is numerically computed and give its values for $m\in[1,10]$. Since we are only focusing on vortex rings of charge $|m|=1$, (indeed, stable higher-charge vortex rings are not known to exist [@VLINE-L0-61]), we use the value $L_0(1)\approx 0.380868$. ![ Cylindrical coordinates used to describe the initial condition for a vortex ring. []{data-label="f:cylindrical"}](cylindricalN.jpg.ps){width="7.5cm"} The initial condition for a single vortex ring solution in cylindrical coordinates (see Fig. \[f:cylindrical\]) with radius $d$ centered at $(r,z)=(0,0)$ can be described by $$\Psi_0 = \Psi(r,z,\theta,0) = g(r,z)\,\exp[im(\phi^{-}(r,z)-\phi^{+}(r,z))],$$ where $$\phi^{\pm}(r,z) = \mbox{arctan}\left(\frac{z}{r\mp d}\right),$$ and the function $g(r,z)$ is an axisymmetric 2D profile which may contain additional phase information of the solution. This 2D profile can be approximated by the modulus of the radial solution of a 2D NLSE vortex in the $(r,\phi^{+})$ plane and is described by $$g(r,z)=f(\rho(r,z)),$$ where $\rho(r,z) = \sqrt{(r-d)^2 + z^2}$ and $f(\rho)$ is the numerical solution to the radial steady-state equation $$-\left(\Omega + \frac{am^2}{\rho^2}\right)f(\rho) + a\left(\frac{1}{\rho}\frac{df}{d\rho} + \frac{d^2f}{d\rho^2}\right) + s\,f^3(\rho)=0,$$ which is derived from inserting the form of a vortex solution $\Psi=f(\rho)\,\exp[i(m\phi^{+}+\Omega t)]$ into the 2D NLSE in polar coordinates. In order to obtain a numerically-exact vortex ring initial condition, we would like to solve a steady-state equation for $\Psi_0$ using the above approximation as an initial seed. This can be accomplished by noting that a steady-state solution to the NLSE of Eq. (\[nlse\]) in cylindrical coordinates in a co-moving frame in the $z$-direction with velocity $c$ is given by [@RMC-DISS] $$\Psi = U(r,Z,\theta)\,\exp\left[i\left( \frac{c}{2a}z + \left[\Omega - \frac{c^2}{4a}\right]t \right)\right],$$ where $Z = z-ct$ and $U(r,Z,\theta)$ solves the time-independent NLSE $$\label{comoveSSnlse} -\Omega U + a\left(\frac{1}{r}\frac{\partial U}{\partial r} + \frac{\partial^2 U}{\partial r^2} + \frac{\partial^2 U}{\partial z^2}\right) + s|U|^2U = 0.$$ Therefore, by imposing a counter-flow with a velocity equal and opposite to that of the intrinsic velocity of the vortex ring by the following Galilean boost: $$U_0(r,z) = \Psi_0\,\exp\left(-i\frac{c}{2a}z\right),$$ where $c$ is the analytical approximation of the vortex ring velocity given by Eq. (\[vrvel\]), we can then solve Eq. (\[comoveSSnlse\]) for $U$. We use the nonlinear Newton-Krylov solver [nsoli]{} [@OPT-NSOLI-BOOK] to find the solution, using a central-difference discretization of the spatial derivatives, along with the modulus-squared Dirichlet boundary conditions mentioned in Sec. \[s:intro\]. The initial condition of the vortex ring solution is then found by removing the added counter-flow velocity: $$\Psi(r,z,\theta,0) = U\,\exp\left(i\frac{c}{2a}z\right).$$ ![(Color online) Velocity for a vortex rings of charge $m=1$ as a function of its radius. The dots are the velocities computed from direct 3D simulations averaged over a $t\in[0,50]$, while the line is the predicted velocity of Eq. (\[vrvel\]). The NLSE parameters used are $a=1$, $s=-1$, and $\Omega=-1$, while the numerical parameters are $h=0.5$ (grid spacing) and $k=0.025$ (time step).\[f:VRvel\]](VRzvel_d_T5005_m2_h05_d2-10){width="7cm"} {width="\linewidth"} The numerically exact vortex ring solutions can be used to compare the analytical approximations of the vortex ring’s transverse velocity Eq. (\[vrvel\]) to direct simulations of the NLSE. In order to track the vortex rings during the simulations, a 2D $(y,z)$ cut of the computational grid is extracted, and a center-of-mass calculation of the modulus of the vorticity in the region of the ring’s intersection points in the cut are tracked. In Fig. \[f:VRvel\] we show the results of integrating vortex rings until $t=50$ for radii $d \in [2,10]$ and comparing their tracked velocity to Eq. (\[vrvel\]). It is seen that the direct velocity results match the analytical approximation very accurately; the disagreement is never more than $1.5\%$ in our studies. This illustrates that values obtained from the asymptotic velocity equation, Eq. (\[vrvel\]), match direct integration even for rings with relatively small radii. We note that although the velocity $c$ is used to generate the vortex rings initially, the fact that the rings maintain the velocity over a long integration time with no noticeable noise or counter-flow in the solution confirms that the simulations demonstrate the validity of Eq. (\[vrvel\]). We note here that the transverse velocity discussed in this section is that of an [*unperturbed*]{} vortex ring. It is possible to perturb a vortex ring (or line) to produce oscillations along the ring called Kelvin modes (or Kelvons) [@fetter04; @simula08; @horng06; @chevy03], one natural method being the merger and scattering of two vortex rings (discussed in the next section). These Kelvin modes not only have their own dynamics, interactions and decay properties [@garcia01; @Barenghi:arXiv13], but they can also self-interact within a vortex ring, resulting in a reduction or even reversal of its transverse velocity [@maggioni10; @Helm11; @Helm10]. Scattering vortex rings {#s:scattering} ======================= We now present a quantitative analysis of the scattering of two unit-charge ($|m|=1$) vortex rings in the NLSE. An initial qualitative study of the scattering of such rings was performed in Ref. , where the authors focused on head-on collisions of the rings at different angled orientations. In contrast, we focus here exclusively on offset co-planar collisions, not studied in Ref. . As shown in Ref. , two colliding vortex rings with zero offset (axisymmetric) expand and annihilate each other’s topological charge resulting in an axisymmetric decaying rarefaction pulse. An example of such a case is shown in Fig. \[f:headoncol\] where two vortex rings of radius $d=6$ (one with charge $m=1$ and one with $m=-1$) are positioned a distance of $6d$ apart from each other in the $z$-direction and allowed to collide. We study here the scattering that results when the colliding vortex rings are offset from each other in the planar direction. Our numerical experiments show that, typically, the two rings merge and then separate into two new vortex rings moving away from each other at an angled trajectory, each exhibiting large quadrupole Kelvin oscillations. ![ (a) Schematic of a scattering scenario between two co-planar vortex rings of initial radius $d$ offset by a distance $q$ (impact parameter). The initial, pre-scattering, configuration is depicted with dashed lines while the post-scattering configuration is depicted with solid lines. The scattering angle is defined as the, signed, angle between incoming and outgoing axes as shown in the schematic. (b) Schematic of the variables to describe the reduced dynamics for the leapfrogging between two co-axial vortex rings located at $Z_1$ and $Z_2$ with respective radii $R_1$ and $R_2$. []{data-label="f:VR_Drawing"}](VR_drawing_flip.jpg.ps){width="7.5cm"} {width="\linewidth"} ![ (Color online) Scattering angle versus the normalized offset distance $q/d$ for two colliding co-planar unitary-charge vortex rings. The different regions correspond to the three qualitatively scenarios depicted in Fig. \[f:scatqual\]. The (blue) circles are the results for vortex rings of radius $d=6$, the (red) squares for $d=8$ and the (black) diamonds for $d=10$. The (green) dashed line is the approximation of Eq. (\[scatpulse\]) and the (cyan) solid line is the approximation of Eq. (\[scatscat\]). The NLSE and numerical parameters used in the simulations are the same as in Fig. \[f:headoncol\]. \[f:scatangvsos\]](SAvsq_dN){width="8cm"} To quantify the relationship between offset distance and scattering angle, the vortex rings are positioned at a distance of $6d$ apart in the $z$-direction, and in the $y$-direction they are placed with a planar offset distance (impact parameter) defined as $q$ (see Fig. \[f:VR\_Drawing\](a)). The scattering angle $\theta_s$ is defined as the angle away from vertical so that if the vortex rings travel in a straight trajectory, the angle is $0$, while if they scatter at right-angles to each other, the angle is $\pi/2$ (see Fig. \[f:VR\_Drawing\](a)). Multiple simulations of colliding rings were performed with planar offsets ranging from $q/d=0$ to $q/d=3$ in increments of $0.05$ for vortex rings of radius $d=6,8,10$. We found three different qualitative scenarios as depicted and described in Fig. \[f:scatqual\]. The center of the vortex rings in the $(y,z)$ plane are tracked and the scattering angle recorded. The results are shown in Fig. \[f:scatangvsos\]. It is clear from the results that there are three distinct sections in the relationship between scattering angle and offset, which correspond to qualitatively distinct topological outcomes as follows: - For offsets with $q/d\in[0,1/2)$ the two rings annihilate each other’s topological charge, sending out a rarefaction pulse, see Fig. \[f:scatqual\](a), similar to the cases of head-on collision shown in Fig. \[f:headoncol\]. However in this case, as the rings approach each other, they rotate causing the collisions to occur at the scattering angle. Another distinction of these collisions is that the resulting rarefaction pulse is not axi-symmetric as in the head-on collision case, but more concentrated in the scattering direction. - When the offset has $q/d\in(1/2,2)$, the two rings undergo merging immediately followed by splitting into internally excited rings with quadrupole oscillations, (see Fig. \[f:scatqual\](b)). Interestingly, when the offset is set to $q/d=3/2$ or larger, the scattering angle becomes negative, but the qualitative behavior remains the same. - For larger offset such that $q/d>2$, the two rings no longer merge and separate into new rings, but are merely perturbed by each other in a fly-by scenario causing their trajectories to travel at a small negative angle from incidence (i.e., get weakly deflected “inward”), see Fig. \[f:scatqual\](c). As the offset increases, this interaction decreases until approximately $q/d=3$ where the rings essentially ignore each other’s presence and travel unperturbed. Noting that the scattering angle as a function of offset in Fig. \[f:scatangvsos\] appears linear in the first two qualitatively distinct intervals, simple phenomenological approximations of the relationships can be formulated. For the annihilation section \[$q/d\in(0,1/2)$\], the angle of the rarefaction pulse’s trajectory can be approximated by $$\label{scatpulse} \theta_s \approx -2\frac{q}{d} + \frac{\pi}{2},$$ while for the scattering rings \[$q/d\in(1/2,2)$\], the angle of the resulting rings can be approximated as $$\label{scatscat} \theta_s \approx -\frac{1}{2}\frac{q}{d} + \frac{\pi}{4}.$$ These approximations are shown along with the simulation data in Fig. \[f:scatangvsos\]. It is interesting to note that, despite the complex dynamics involved in the scattering of vortex rings, we are able to formulate a very simple, phenomenological scattering rule for co-planar collisions. It is interesting to contrast the 3D scenario of co-planar vortex ring scattering to its 2D counterpart. If one focuses on the dynamics restricted on the plane defined by the vortex ring axes, each vortex ring induces a vortex pair in 2D. The case of scattering between 2D (point-) vortex pairs has been studied in some detail [@Aref-PTRSA-88; @Price-PF-93; @Aref-PF-08]. The most notable differences between the ensuing dynamics for 2D vortex pairs and that of 3D vortex rings scattering is that the dynamics in 2D is apparently richer due to the fact that truly 2D vortices are more “free” to move (respecting the vortex-vortex interactions) while the vortex pairs produced by the cut of the vortex ring in the plane are more tightly linked to each other through the ring. This extra freedom for 2D vortices allows for a rich variety of dynamical evolution scenarios. For instance, depending of the initial configuration (initial positions, orientations and vortex pair internal distances) the collision dynamics of two vortex pairs can display: (a) transient bound vortex pairs where the two vortex pairs interact by orbiting around each other and exchange partners for a few periods until scattering away from each other as two distinct vortex pairs, (b) transient interactions between the four vortices that are chaotic and lead to the eventual expulsion of two distinct vortex pairs resulting in chaotic scattering, or (c) transient bound states where three of the four vortices lock into a translational and precessional motion, while the fourth one moves separately, before the two pairs again scatter away, separating indefinitely. ![ (Color online) Example of two vortex rings undergoing leapfrogging. The rings had an initial radius of $d=1$ and were initially separated in the $z$-direction by a distance $Q=5$. Times correspond to $t=0,16,27,44,58,68,85,101,110$ (left-to-right, top-to-bottom). The parameters are the same as in Fig. \[f:headoncol\] with $h=2/3$ and $k=0.045$.[]{data-label="f:VRlooploop"}](VRlooploopmod2_crop.ps){width="\linewidth"} Leapfrogging Vortex Rings {#s:leapfrog} ========================= Let us now consider the interaction between two same sign unit-charge vortex rings ($m_1=m_2=1$) aligned along the $z$ axis. In classical fluids, the analogous setup exhibits leapfrogging behavior [@Konstantinov94; @Shashikanth03]. In Fig. \[f:VRlooploop\], we show an example of two equal-charge vortex rings of radius $d=10$ and initially separated by a distance of $Q=Z_2(0)-Z_1(0)=5$ in the $z$-direction undergoing leapfrogging dynamics. In the course of our numerical investigation we have found that when $Q$ is small (around two or three times the vortex ring’s core radius), the resulting dynamics can be quite different. For example, instead of leapfrogging, the rings can form a bound pair and travel together maintaining their radii and common speed. While such dynamical features are intriguing in their own right, for the scope of the current work, we only use results from vortex rings separated with large enough values of $Q$ to result in leapfrogging dynamics. In order to formulate a reduced model of the leapfrogging interaction of the rings, we start with reviewing the situation in the classical fluid case. There, a vorticity field $\boldsymbol\omega$ will induce a velocity field according to the Biot-Savart law [@Batchelor67; @Saffman95] $$\label{vfield1} \textbf{u}_v(\textbf{x})=\frac{1}{4\pi}\int{\frac{\boldsymbol\omega(\textbf{x}^\prime)\times(\textbf{x}-\textbf{x}^\prime)}{\lvert\textbf{x}-\textbf{x}^\prime\rvert^3}d\textbf{x}^\prime},$$ where the integral is computed over all of space. When considering a vorticity distribution of infinite strength but having a constant circulation $\kappa$ along a closed curve $\textbf{R}$ parametrized by $\ell$, Eq. (\[vfield1\]) becomes $$\textbf{u}_v(\textbf{x})=\frac{\kappa}{4\pi}\oint{\frac{\textbf{s}\times(\textbf{x}-\textbf{R}(\ell))}{\lvert\textbf{x}-\textbf{R}(\ell)\rvert^3}\,d\ell} \label{biotsavart}$$ where $\textbf{s}$ is the unit tangent vector. Equation (\[biotsavart\]) can, in principle, be evaluated to find the effect of a vortex ring on another vortex ring. However, when trying to find a ring’s self-induced velocity, the integral diverges. To circumvent this singularity for the self-induced term the localized-induction approximation (LIA) [@DaRios-1906; @Betchov-JFM-65; @Arms65; @Siggia-PhysFluids-85] is employed by ignoring long-distance effects, yielding the expression for the self-interacting term: $$\textbf{u}_v=\frac{(\partial\textbf{R}/\partial \ell)\times(\partial^2\textbf{R}/\partial \ell^2)}{\lvert(\partial\textbf{R}/\partial \ell)\rvert^3}. \label{localinduction}$$ Ultimately, using Eq. (\[biotsavart\]) through a similar LIA for pairwise interactions and (\[localinduction\]) for self-interactions, the following reduced ordinary differential equations (ODEs) for the effective motion of a collection of $N$ co-axial vortex rings in a classical fluid are obtained [@Konstantinov94; @Shashikanth03]: $$\begin{aligned} \dot Z_i &= \frac{\kappa_i}{4\pi R_i}\left(\ln\frac{8R_i}{r_{c,i}}-C\right)+\frac{1}{\kappa_i R_i}\frac{\partial U}{\partial R_i}\label{vrvelz},\\ \dot R_i &= -\frac{1}{\kappa_i R_i}\frac{\partial U}{\partial Z_i}, \label{vrvelr}\end{aligned}$$ where $$U=\frac{1}{2\pi}\sum\limits_{i=1}^N\,\sum\limits_{j>i}^N \kappa_i \kappa_j I_{ij},$$ and $$I_{ij}=\int\limits_0^\pi \frac{R_i R_j \cos \theta\, d\theta}{\sqrt{\left(Z_i-Z_j\right)^2+R_i^2+R_j^2-2R_i R_j \cos\theta}},$$ $R_i$ and $Z_i$ are the radius and $z$-coordinate of the $i$-th ring, $\kappa_i$ is its circulation or “vortex strength” which, in our case, is a constant for each ring, $r_{c,i}$ is its core radius which is time-dependent and follows the relation $r_{c,i}^2(t)R_i(t)=\mbox{constant}$, and $C$ is a constant, which for the case of a classical fluid is given as $C=1/4$ (see Fig. \[f:VR\_Drawing\](b)). We note that the terms involving $U$ relate to the interaction of the rings, while the first term of Eq. (\[vrvelz\]) represents the innate translational velocity of each ring. If one compares this velocity with that of the quantum vortex ring velocity of Eq. (\[vrvel\]), we see that they are quite similar. We suggest that it is possible to use the results of Eqs. (\[vrvelz\]) and (\[vrvelr\]) for the vortex rings in the NLSE by simply substituting the quantum fluid values such that $C=L_0(m)-1$, $\kappa=4\pi a m$, and $r_c$ from Eq. (\[vrvel\]) into Eqs. (\[vrvelz\]) and (\[vrvelr\]). We point out, that in contrast to the classical fluid leapfrogging picture, in a quantum fluid, the healing length being constant forces $r_c$ to be, approximately, time-independent. The conservation of the core width during vortex ring evolution will be briefly addressed below. ![ (Color online) Leapfrogging orbits for two vortex rings in the co-moving $(R,Z)$ plane. The (black) solid curves depict the results from the effective, reduced, equations of motion (\[vrvelz\]) and (\[vrvelr\]), while the (red) dashed curves depict the corresponding results from those obtained from full 3D integrations of Eq. (\[nlse\]). The rings are initialized with radii $d=11$ and initial separation distances of $6$, $8$, $10$, $12$, and $14$ (from inner to outer curves). Only one period of the leapfrogging motion is shown in a reference frame moving at the average velocity of the vortex ring pair (see Fig. \[f:leapfrog\_v\]), i.e., a center-of-mass frame. []{data-label="f:rzorbits"}](rz11_new2){width="7.5cm"} In Figure \[f:rzorbits\], we show a series of $(R, Z)$ orbits —in the center-of-mass frame— as predicted by Eqs. (\[vrvelz\]) and (\[vrvelr\]) for different initial separation distances along with those obtained through tracking the vortex rings in the integration of the full 3D NLSE system (\[nlse\]). As it is clear from the figure, the $(R,Z)$ orbits from the reduced ODE system are in very good agreement with the ones obtained from the full PDE. These results stress the usefulness of the reduced system towards capturing the interaction dynamics of co-axial vortex rings. We note that while the ODE trajectories are periodic to numerical accuracy, the PDE results are only approximately periodic (only one “period” is shown here). The deviation from periodicity, in the co-moving reference frame, in the full PDE numerics might be attributed to several factors including: internal mode (Kelvin) excitations and stability of the actual leapfrogging orbits [@Barenghi:arXiv14]. Figure \[f:rzorbits\] suggests the existence of a stable, co-moving, equilibrium at the center of the periodic orbits for $Z=0$ and $R$ being constant. This point corresponds precisely to two overlapping ($Z_1=Z_2$) vortex rings of the same radius, namely a vortex ring of charge two. In fact the linearization about this co-moving equilibrium would yield an approximation for the period of the vortex ring leapfrogging in the limit of small initial separations. However, as vortex rings of higher charge are unstable and break up into two unit charge vortex rings, we will not study them further here —with the exception of using the theoretically predicted velocity of the doubly-charged vortex ring as an approximation for the average leapfrogging velocity. ![ (Color online) Time dependence for the sum of vortex radii for a pair of leapfrogging rings. The (blue) solid curves depict the results from the effective, reduced, equations of motion (\[vrvelz\]) and (\[vrvelr\]), while the (red) dashed curves depict the corresponding results from those obtained from full 3D integrations of Eq. (\[nlse\]). The different panels correspond to vortex rings initially separated by (a) $Q=Z_2(0)-Z_1(0)=7$, (b) $Q=Z_2(0)-Z_1(0)=9$, and (c) $Q=Z_2(0)-Z_1(0)=11$ units away starting with equal radii $R_1(0)=R_2(0)=11,10,9,8,7$ (respective curves from top to bottom). \[f:r1r2\] ](r1r2N.ps){width="\linewidth"} It is interesting to note that the vortex core size in the NLSE (\[nlse\]) should be approximately constant provided the vortex core is not significantly perturbed. In fact, if the vortex core size is invariant during evolution then the total length of the vorticity tube should be conserved due to conservation of angular momentum. Therefore, by measuring the total vortex tube length, it is possible to monitor the degree to which the vortex core width deviates from its unperturbed value. In the case of leapfrogging vortex rings, where one vortex is forced inside the other ring, there is a strong mutual perturbation that can affect the vortex core width. In Fig. \[f:r1r2\] we depict the sum of the vortex radii for different case examples of two vortex rings undergoing leapfrogging for both the effective ODE system and the original PDE. The sum $R_1+R_2$ is proportional to the total vortex tube length $L=2\pi(R_1+R_2)$. As it is clear from the figure, the total vortex length is not conserved but it oscillates around a mean value with a relatively small amplitude. For the effective ODE model, the total vortex tube length varies between 4% and 12%, while for the full 3D model it varies between 3% and 8% for the cases depicted in Fig. \[f:r1r2\]. This result is a clear indication that the vortex core is indeed perturbed by the strong mutual interaction between vortex rings during the leapfrogging dynamics. It is also worth mentioning that the results in Figs. \[f:rzorbits\] and \[f:r1r2\] suggest that although the reduced ODEs are able to capture very well the shape of the leapfrogging orbits in the $(R,Z)$ plane, there appears to be a systematic offset (between the ODEs of Eqs. (\[vrvelz\])-(\[vrvelr\]) and the full 3D NLSE) as regards the period for these orbits. Figure \[f:r1r2\] shows that for small initial vortex ring separations \[see panel (a)\] the offset in the period is larger than for large initial vortex ring separations \[see panel (c)\]. This can be qualitatively understood from the fact that an initial small separation between vortices induces stronger interactions between vortex rings bringing them closer to each other and thus strongly perturbing the vortex core which was assumed to have zero width in the Biot-Savart law yielding the ODE approximation. ![ (Color online) Average velocity (top panel) and looping period (bottom panel) of leapfrogging vortex rings as a function of the ring radii ($d$) for several separation distances ($Q$). The lines correspond to the results obtained through numerical integration of the effective equations of motion (\[vrvelz\]) and (\[vrvelr\]) and the symbols correspond to the full numerical integration of the NLSE model (\[nlse\]). For comparison we also include (thin black line) the velocity of a vortex ring of charge $m=2$ using Eq. (\[vrvel\]). The horizontal axis corresponds to the initial radius of the pair of identical vortex rings, while the different colors for the thick lines correspond (top to bottom) to initial separation distances of 7 (red solid line and + symbols), 9 (green dotted line and $\times$ symbols), and 11 (blue dashed line and \* symbols).[]{data-label="f:leapfrog_v"}](leapfrog_vvd2 "fig:"){width="7cm"} ![ (Color online) Average velocity (top panel) and looping period (bottom panel) of leapfrogging vortex rings as a function of the ring radii ($d$) for several separation distances ($Q$). The lines correspond to the results obtained through numerical integration of the effective equations of motion (\[vrvelz\]) and (\[vrvelr\]) and the symbols correspond to the full numerical integration of the NLSE model (\[nlse\]). For comparison we also include (thin black line) the velocity of a vortex ring of charge $m=2$ using Eq. (\[vrvel\]). The horizontal axis corresponds to the initial radius of the pair of identical vortex rings, while the different colors for the thick lines correspond (top to bottom) to initial separation distances of 7 (red solid line and + symbols), 9 (green dotted line and $\times$ symbols), and 11 (blue dashed line and \* symbols).[]{data-label="f:leapfrog_v"}](leapfrog_tvd2 "fig:"){width="7cm"} Two relevant quantities of the leapfrogging rings that we can use to further validate the reduced model are the average ring-pair $z$-velocity and the rings’ period of oscillation. We measure the average ring-pair $z$-velocity by tracking the $z$ value of the center position of a $(y,z)$ cut of the rings over time and then perform a least-squares linear fit. This is done for both the full NLSE simulations, and for the ODE integration of the reduced system. In Fig. \[f:leapfrog\_v\], the results are presented as a function of $d$ for a few values of the initial $z$ separation distance, $Q$. We observe that the average velocity of the leapfrogging rings obtained in the PDE model is always lower than that obtained in the effective ODE model, the percent difference ranges between $1.6\%$ and $7.2\%$. It is interesting to compare the average velocity of two leapfrogging rings with the velocity predicted for a vortex ring of charge $m=2$ from Eq. (\[vrvel\]). As depicted by the black thin line in the top panel of Fig. \[f:leapfrog\_v\], the velocity of a vortex ring of charge $m=2$ (evaluated at the initial radii for the leapfrogging vortex ring pair) closely resembles the average velocity of a vortex ring pair under leapfrogging evolution. This stems from the fact that, in the far field, a vortex ring pair can be approximated by a vortex ring of charge $m=2$. In fact, after closer inspection of the results in the top panels of Fig. \[f:leapfrog\_v\], the velocity of the doubly-charged vortex ring is closer to the velocity for the vortex pairs when their separation is small, i.e., when they are closer to each other and thus closer to a doubly-charged vortex ring. While, this relation could be further explored by taking the average radius along one leapfrogging period instead of the initial radii, the main phenomenology does not change significantly. In the bottom panel of Fig. \[f:leapfrog\_v\] we show the results of measuring the period of oscillation of the leapfrogging rings from both the full NLSE simulations and the ODE reduced model for the same ring separations and radii. We see that the period of oscillation for the PDE model is always longer than that of the ODE model, and the percent difference in period ranges from $4.2\%$ to $28.5\%$ percent, which, while clearly worse than the velocity comparison, is not unreasonable. It is interesting to note that while the period of the leapfrogging rings changes depending on the $z$-distance between the rings, it remains fairly constant for vortex ring pairs of [*different*]{} radii at the same separation distance. From the velocity and period results, we see that there are obviously some differences between the quantum and classical fluid regimes including the modified values that result in lower average pair velocities and longer periods of oscillation than those predicted by our current reduced equations of motion. These are worthy of further investigation, in order to improve the accuracy of the reduced equations of motion. Nevertheless, the trends of the two sets of results are very similar, and their magnitudes are comparable, which suggests that our effective description provides a valuable means of examining the numerical (and, presumably, also the experimental) data and, as such, is useful in its own right as well as worthy of further investigation. Finally, it is worth pointing out that there has been recent efforts in describing the leapfrogging dynamics of vortex ring bundles in classical fluids using the vortex filament method [@Barenghi:arXiv14]. In particular, it is found that the leapfrogging dynamics is weakly unstable for small number of vortex rings. In our numerics we have found similar instabilities, that depending on the separation and radii of the vortex rings, may not be visible until tens of leapfrogging periods. Therefore, although weakly unstable, the time required for the instability to manifest itself might be long enough to be able to observe leapfrogging during the typical lifetime of current BEC experiments. It is also important to mention that the 2D equivalent of leapfrogging has been studied in the case of point vortices [@Acheson-EJP-00; @Aref-PF-13]. This 2D leapfrogging corresponds to two vortex dipoles (vortex and anti-vortex pairs) and it is found to display three distinct regions depending on the distance between pairs (relative to the internal distance between vortices in each pair) as follows. (a) The small separation region does not yield leapfrogging. (b) Intermediate separations yield [*unstable*]{} leapfrogging. (c) Large separation distances yield [*stable*]{} leapfrogging. The 2D vortex dipole leapfrogging is strongly connected to 3D vortex ring leapfrogging since any 2D cut passing through the axes of the vortex rings will produce a 2D pair of vortex dipoles. Therefore, 2D vortex dipole dynamics is the basic ingredient responsible for vortex ring leapfrogging. Therefore, it would be interesting to test if the vortex ring leapfrogging also displays the three regions described above and whether the weak instability that we have observed can in fact be eliminated by using larger vortex ring radii. This avenue of research is currently being examined and a systematic investigation will be reported in a future publication. Conclusions {#s:conclu} =========== We studied the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation as a model for a homogeneous Bose-Einstein condensate. Upon corroborating the well-known single vortex ring evolution in our 3D NLSE framework, we focused on two dynamical scenarios: scattering and leapfrogging. For the former, we considered the scattering collision of two co-planar, opposite charge, vortex rings. Upon their collision, these scatter at an angle depending on the initial conditions. We found a surprisingly simple phenomenological rule for the scattering angle as a function of the vortex ring radii and the initial distance (impact parameter) between the vortex rings. For the second scenario, we followed the dynamics of a pair of co-axial vortex rings of the same charge. We proposed a modification of the effective equations of motion for the vortex ring interactions in a classical superfluid. The modification was based on correcting the self-induced velocity using previous results for NLSE settings in a quantum superfluid. We compared the resulting effective equations of motion against full 3D simulations of the original NLSE model for several cases of leapfrogging evolution and found good agreement between the two. Finally, we found a monotonic decrease for the average velocity of the leapfrogging as the radii of the ring increase and/or as their initial separation increases. We also found that the leapfrogging period remains approximately constant for vortex ring pairs of different radii starting at the same separation distance, while the period increases with initial separation distance for fixed radii. Future work will be directed towards refining the reduced equations of motion to better account for the differences between classical and quantum fluids (especially in the context of the term encoding the interaction between the different rings), and expanding them to describe more general configurations of vortex rings, as opposed to only those which are co-axial. It would also be interesting to understand and quantify the instability (and its range of relevance over initial conditions and system parameters) observed in the leapfrogging dynamics, not only for a pair of vortex rings but also, for vortex ring bundles [@Barenghi:arXiv14]. Another avenue for future exploration consists of using classical fluid-inspired effective equations of motion (Biot-Savart law) to obtain reduced ODEs describing the effects of slow-down created by internal excitation (Kelvin) modes [@maggioni10; @Helm11; @Helm10] and, in more general terms, the effects of these modes on the interactions and scattering between vortex rings. We are currently exploring these directions and will report on them in future works. Appendix: Computation of the vortex core parameter {#appendix-computation-of-the-vortex-core-parameter .unnumbered} ================================================== The “vortex core parameter” $L_0(m)$ is defined as the convergent part of the energy per unit length of a vortex line. In this appendix, we simply state the equations used to compute $L_0(m)$. For full details of the derivation of the equations, we refer the reader to Refs. . For the NLSE in the form of Eq. (\[nlse\]), the vortex core parameter is found by numerically integrating $$\begin{aligned} {2} \label{L0} L_0(m) &= \ln(r_c)+\frac{s}{am^2\Omega}\left(a \int_0^{\infty}\!\left(\frac{df}{dr}\right)^2 r\,dr \right. \\ &-\frac{s}{2}\int_0^\infty\! \left(\frac{\Omega}{s}-f^2(r)\right)^2 r\,dr \notag \\ &-\left.2am^2\int_0^\infty\! f(r)\frac{df}{dr}\ln(r)\,dr\right), \notag\end{aligned}$$ where $f(r)$ is found numerically by solving the ODE $$-\left(\Omega + \frac{am^2}{r^2}\right)f(r) + a\left(\frac{1}{r}\frac{df}{dr} + \frac{d^2f}{dr^2}\right) + sf^3(r)=0,$$ and where we note that for a given $m$, $L_0(m)$ is invariant over the parameters $a$, $s$, and $\Omega$. $m$ $L_0(m)$ $m$ $L_0(m)$ ----- -------------- ------ -------------- $1$  $0.380868$  $6$  $0.022954$  $2$ $0.133837$ $7$ $0.017785$ $3$ $0.070755$ $8$ $0.014247$ $4$ $0.044567$ $9$ $0.011713$ $5$ $0.030981$ $10$ $0.009833$ : Values of the vortex core parameter $L_0$ for charges $m\in[1,10]$. The values are computed through numerical integration of Eq. (\[L0\]).[]{data-label="t:l0ofm"} ![ (Color online) Computed vortex core parameters for charges $m=1$ to $m=10$. The dots are the results of integrating Eq. (\[L0\]). a) The dashed line is the fitted curve of Eq. (\[l0ofm\]). b) Same as panel a) but in log-log plot and using the best linear fit to the log-log of data giving the power law $L_0 \propto m^{-1.5956}$. \[f:l0\] ](L0_vs_m.ps){width="7.5cm"} We compute the integrals in Eq. (\[L0\]) using a simple trapezoid integration on a numerically-exact $f(r)$ computed on a domain of $r=[0,500]$, with a spatial step size of $\Delta r=0.0075$ and NLSE parameters $a=1$, $s=-1$, and $\Omega=-1$. The values of $L_0$ for charges $m=1,...,10$ are given in Table. \[t:l0ofm\]. We note that the values of $L_0$ for $|m|>3$ have not been previously reported, and those for $m=2$ and $m=3$ are given with more accuracy than reported in Ref. . It is intriguing to note that one can produce a very close power-law fit yielding an approximation of $L_0(m)$ given by $$\label{l0ofm} L_0(m) \approx L_0(1)\,m^{-3/2},$$ which may indicate the possibility that a closed-form expression for $L_0(m)$ is achievable. In Fig. \[f:l0\] we show a comparison of the values of $L_0$ given in Table \[t:l0ofm\] to those given by Eq. (\[l0ofm\]). [10]{} C. Sulem and P. L. Sulem. . Springer-Verlag, New York, 1999. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris. . Academic, New York, 1983. L. Debnath. . Birkhauser Boston, New York, 2005. A. Hasegawa. . Springer-Verlag, Heidelberg, 1990. A. Kh. Abdulaev, S. A. Darmanyan and P. K. Khabibullaev. . Springer-Verlag, Heidelberg, 1993. A. Hasegawa and Y. Kodama. . Clarendon Press, Oxford, 1993. Yu. S. Kivshar and G. P. Agrawal. . Academic Press, 2003. E. Infeld and G. Rowlands. . Cambridge University Press, Cambridge, 1990. , D. J. Frantzeskakis and [R. Carretero-González]{}. , volume 45. Springer Series on Atomic, Optical, and Plasma Physics, 2008. , D. J. Frantzeskakis and [P. G. Kevrekidis]{}. Nonlinearity [**21**]{}, R139–R202 (2008). M. J. Ablowitz, B. Prinari and A. D. Trubatch. . Cambridge University Press, Cambridge, 2004. J. Bourgain. . American Mathematical Society, Providence, 1999. M. J. Ablowitz and H. Segur. . SIAM, Philadelphia, 1981. V. E. Zakharov, S. V. Manakov, S. P. Nonikov and L. P. Pitaevskii. . Consultants Bureau, New York, 1984. A. C. Newell. . SIAM, Philadelphia, 1985. L. P. Pitaevskii F. Dalfovo, S. Giorgini and S. Stringari. Rev. Mod. Phys. [**71**]{}, 463–512 (1999). C. J. Pethick and H. Smith. . Cambridge University Press, 2002. A. Aftalion and I. Danaila. Phys. Rev. A [**68**]{}, 023603 (2003). N.R. Cooper S. Komineas an and N. Papanicolaou. Phys. Rev. A [**72**]{}, 053624 (2005). T.P. Simula. Phys. Rev. A [**84**]{}, 021603(R) (2011). R.J. Donnelly. . Cambridge University Press, 1991. G. W. Rayfield and F. Reif. Phys. Rev. [**136**]{}, A1194–A1208 (1964). G. Gamota. Phys. Rev. Lett. [**31**]{}, 517–520 (1973). B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark and E. A Cornell. Phys. Rev. Lett. [**86**]{}, 2926–2929 (2001). I. Shomroni, E. Lahoud, S. Levy and J. Steinhauer. Nat. Phys. [**5**]{}, 193–197 (2009). J. Ruostekoski and Z. Dutton. Phys. Rev. A [**72**]{}, 063626 (2005). J. Ruostekoski and J. R. Anglin. Phys. Rev. Lett. [**86**]{}, 3934–3937 (2001). N. S. Ginsberg, J. Brand and L. V. Hau. Phys. Rev. Lett. [**94**]{}, 040403 (2005). C. Becker, K. Sengstock, P. Schmelcher, P.G. Kevrekidis and R. Carretero-Gonz[á]{}lez. New J. Phys. [**15**]{}, 113028 (2013). J. F. McCann B. Jackson and C. S. Adams. Phys. Rev. A [**60**]{}, 4882–4885 (1999). A. S. Rodrigues, P. G. Kevrekidis, R. Carretero-Gonz[á]{}lez, D. J. Frantzeskakis, P. Schmelcher, T. J. Alexander and Yu. S. Kivshar. Phys. Rev. A [**79**]{}, 043603 (2009). R. G. Scott, A. M. Martin, S. Bujkiewicz, T. M. Fromhold, N. Malossi, O. Morsch, M. Cristiani and E. Arimondo. Phys. Rev. A [**69**]{}, 033605 (2004). N. G. Berloff and C. F. Barenghi. Phys. Rev. Lett. [**93**]{}, 090401 (2004). N. G. Berloff. Phys. Rev. B [**65**]{}, 174518 (2002). N. G. Berloff. Phys. Lett. A [**277**]{}, 240–244 (2000). N. G. Berloff and P. H. Roberts. J. Phys. A: Math. Gen. [**33**]{}, 4025 (2000). N. G. Berloff and P. H. Roberts. J. Phys. A: Math. Gen. [**34**]{}, 81 (2001). N. G. Berloff and P. H. Roberts. J. Phys. A [**34**]{}, 10057 (2001). R. Carretero-Gonz[á]{}lez, B. P. Anderson, P. G. Kevrekidis, D. J. Frantzeskakis and N. Whitaker. Phys. Rev. A [**77**]{}, 033625 (2008). R. Carretero-Gonz[á]{}lez, N. Whitaker, P. G. Kevrekidis and D. J. Frantzeskakis. Phys. Rev. A [**77**]{}, 023605 (2008). P. H. Roberts and J. Grant. J. Phys. A: Gen. Phys. [**4**]{}, 55–72 (1971). C. H. Hsueh, S. C. Gou, T. L. Horng and Y. M. Kao. J. Phys. B [**40**]{}, 4561–4571 (2007). F. Maggioni, S. Alamri, C. F. Barenghi and R. L. Ricca. Phys. Rev. E [**82**]{}, 026309 (2010). J. L. Helm, C. F. Barenghi and A. J. Youd. Phys. Rev. A [**83**]{}, 045601 (2011). J. L. Helm. Technical report, Newcastle University, http:${\mathbin{/\mkern-4mu/}}$research.ncl.ac.uk/quantum-fluids/files/helm-mmath.pdf, April 2010. R. M. Caplan and R. Carretero-Gonz[á]{}lez. J. Comp. and Appl. Math [**251**]{}, 33–46 (2013). R. M. Caplan and R. Carretero-Gonz[á]{}lez. Appl. Num. Math. [**71**]{}, 24–40 (2013). R. M. Caplan and R. Carretero-Gonz[á]{}lez. SIAM J. on Sci. Comp. [**36**]{}, A1–A19 (2014). R. M. Caplan. Comput. Phys. Commun. [**184**]{}, 1250–1271 (2013). D. Amit and E. P. Gross. Phys. Rev. [**145**]{}, 130–136 (1966). A. L. Fetter. Phys. Rev. [**151**]{}, 100–104 (1966). J. Koplik and H. Levine. Phys. Rev. Lett. [**76**]{}, 4745–4748 (1996). V. L. Ginzburg and L. P. Pitaevskii. Sov. Phys. JETP [**7**]{}, 858–861 (1958). P. Roberts and N. Berloff. In C. Barenghi, R. Donnelly and W. Vinen, editors, [*Quantized Vortex Dynamics and Superfluid Turbulence*]{}, volume 571 of [*Lecture Notes in Physics*]{}, pages 235–257. Springer Berlin / Heidelberg, 2001. L. P. Pitaevskii. Sov. Phys. JETP [**13**]{}, 451–454 (1961). R. M. Caplan. . PhD thesis, Claremont Graduate University and San Diego State University, 2012. C. T. Kelley. . Fundamentals of Algorithms. SIAM, 2003. A. L. Fetter. Phys. Rev. A [**69**]{}, 043617 (2004). T P. Simula, T. Mizushima and K. Machida. Phys. Rev. A [**78**]{}, 053604 (2008). T.-L. Horng, S.-C. Gou and T.-C. Lin. Phys. Rev. A [**74**]{}, 041603 (2006). F. Chevy and S. Stringari. Phys. Rev. A [**68**]{}, 053601 (2003). J. J. García-Ripoll and V. M. Pérez-García. Phys. Rev. A [**64**]{}, 053611 (2001). D. Proment, C. F. Barenghi and M. Onoroto. arXiv:1308.0852. B. Eckhardt and H. Aref. Phil. Trans. R. Soc. Lond. A [**326**]{}, 655–696 (1988). T. Price. Phys. Fluids [**5**]{}, 2479 (1993). L. Toph[ø]{}j and H. Aref. Phys. Fluids [**20**]{}, 093605 (2008). M. Konstantinov. Phys. Fluids [**6**]{}, 1752–1767 (1994). B. N. Shashikanth and J. E. Marsden. Fld. Dyn. Res. [**33**]{}, 333–356 (2003). G. K. Batchelor. . Cambridge University Press, 1967. P. G. Saffman. . Cambridge Monographs on Mechanics. Cambridge University Press, 1995. L. S. Da Rios. Rend. Circ. Mat. Palermo [**22**]{}, 117–135 (1906). R. Betchov. J. Fluid Mech. [**22**]{}, 471–479 (1965). R. J. Arms and F. R. Hama. Phys. Fluids [**8**]{}, 553–559 (1965). E. D. Siggia. Phys. Fluids [**28**]{}, 794–805 (1985). D. H. Wacks, A. W. Baggaley and C. F. Barenghi. arXiv:1401.6468. D. J. Acheson. Eur. J. Phys. [**21**]{}, 269–273 (2000). L. Toph[ø]{}j and H. Aref. Phys. Fluids [**25**]{}, 014107 (2013). [^1]: [URL:]{} http:${\mathbin{/\mkern-4mu/}}$www.nlsemagic.com

--- abstract: 'We present state sums for quantum link invariants arising from the representation theory of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$. We investigate the case of the $N$th exterior power of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{N|1})$ and explicit the relation with Kashaev invariants.' address: - 'Université de Genève, rue du lièvre 2–4, 1227 Genève, Switzerland' - 'Univ Paris Diderot, IMJ-PRG, UMR 7586 CNRS, F-75013, Paris, France' author: - 'Louis-Hadrien Robert' - Emmanuel Wagner bibliography: - 'biblio.bib' title: State sums for some super quantum link invariants --- Introduction {#sec:introduction .unnumbered} ============ In its 1990 ICM paper [@MR1159255], Turaev emphazided the proeminent role of state sum models in low dimensional topology. Models of this kind were and remain an important part of what is called quantum topology. They are closely related to mathematical physics, quantum algebra and statistical mechanics. The most famous example of state sum model in low dimensional topology is probably the Kauffman bracket [@MR899057]. Not only demonstrates it how methods of statistical mechanics and ideas of quantum field theory can be relevant in this subject as Witten explained [@MR990772], but it is also the first step in a recent major development of quantum topology: categorification of quantum invariants. Indeed, Khovanov homology [@MR1740682] which categorifies the Jones polynomial is clearly inspired by the combinatorial definition of the Jones polynomial given by the Kauffman bracket.\ In this paper, we give state sum models computing quantum invariants arising from the representation theory of the super quantum group $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$. We believe these state sums will be useful in the quest of categorifying these quantum invariants. Some of the invariants described by these state sums have a non-semi-simple nature. The prototypical example of such non-semi-simple invariants is the Alexander–Conway polynomial.\ Let us discuss—with a slightly biased point of view—some of the state sums and categorifications of the Alexander–Conway polynomial. The first categorification of the Alexander provided by the knot Floer homology of Ozsváth–Szabó and Rasmussen [@OS; @Rasmussen] uses its interpretation as Reidemeister torsion. In a combinatorial version of Manolescu–Ozsváth–Szabó–Thurston using grid diagrams, they used explicitely a determinantal description [@MR2372850 Appendix]. They were many attempts to have a direct combinatorial approach using the state model developped by Kauffman in [@MR712133], but finally Ozsváth–Szabó made the connection in [@MR2574747] using a twisted version of the knot Floer homology. In [@RW3], the authors of this paper provided a combinatorial categorification of the Alexander–Conway polynomial of knots starting from a representation theoric interpretation (see [@EPV] for the connection between knot Floer homology and representation theory). The state sum model underlying this last categorification is one of the example of the present paper (see also [@Viro; @MR3319619]).\ The paper [@MR1659228] by Murakami–Ohtsuki–Yamada is another prototypical example of a state sum model which found its full relevance and importance in the realm of the categorification process. They give an elementary description of Reshetikhin–Turaev invariant associated with exterior powers of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{N})$. The description of these quantum invariants by Murakami–Ohtsuki–Yamada proceeds in two steps. First, a link diagram is expressed as a formal linear combination of planar trivalent graphs (see also Kuperberg [@MR1403861]), then a positive state sum is provided for these graphs. Both of these steps were key ingredients for the categorification of $U_q({\ensuremath{\mathfrak{gl}}}_N)$-quantum invariants by Khovanov–Rozansky [@MR2391017] (see also Mackaay–Sto[š]{}ić–Vaz [@MR2491657] and Wu [@pre06302580], Yonezawa [@yonezawa2011], Mazorchuk–Stroppel[@zbMATH05656519] and Sussan [@MR2710319] for colored versions).\ State sum models should continue to be of importance and influencal in quantum topology. For instance, they play a direct role in another version of the categorification of the previous invariants by the authors in [@RW1]. A state sum model for closed 2-dimensional CW complexes called foams is given to produce a trivalent TQFT. This state sum model was reinvested by Khovanov and the first author [@KR1] to investigate the four color theorem.\ The present paper is firstly concerned with state sum models for exterior powers of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$. In particular the case $N=0$, corresponds to invariants associated with symmetric powers of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{M})$. These invariants have been categorified in [@RW2; @queffelec2018annular; @MR3709661]. As reproved in the present paper, the invariants only depend on $N-M$, providing in fact different state sum models for the same invariants. It is conjectured in [@GGS] that these various state sums could be categorified yielding potentially non-equivalent homological invariants.\ Secondly, the paper investigates further the invariants associated with the $N$th exterior power of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{N|1})$ which have vanishing quantum dimensions. These invariants can be renormalized, see [@MR2640994; @2015arXiv150603329Q]. This paper provides a direct proof that they can be renormalized giving explicitly a state sum model. It also explains how these invariants are related to Kashaev invariants [@MR1341338] (see [@MR2468374] for a different proof of this latter fact). As ADO invariants [@MR1164114], this family of non-semisimple invariants provides interesting family of invariants generalizing the Alexander polynomial containing Kashaev invariants. The Alexander polynomial has been categorified in this framework [@RW3] and the present state sum models provide a first step in the categorification of this family of invariants.\ Outline of the paper {#outline-of-the-paper .unnumbered} -------------------- - Section \[sec:qi\] is devoted to some combinatorics of $q$-binomials used in the rest of the paper. - Section \[sec:colorings-moy-graphs\] gives the state sum models for planar graphs. - Section \[sec:link-invariants\] extend the state sum models to link diagrams. - Section \[sec:non-semi-simple\] investigates the non-semisimple case of the $N$-th exterior power of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{N|1})$ and makes the connection with the Kashaev invariants. - Appendix \[sec:moy-graphs-an\] provides the representation theoric background; in particular the explicit maps providing the corresponding Reshetikhin–Turaev functors. Some $q$-identities {#sec:qi} =================== In this section, we give some useful identities on quantum integers and quantum binomials. See [@MR1865777] for a more detailed account. The first two lemmas and three next corollaries are obtained by direct computations or easy inductions and we omit their proofs. Unless otherwise specified, $q$ is a formal parameter. \[dfn:quantum-numbers\] Let $n$ be an integer, define the *quantum integer $[n]$* by the following formula: $$[n]:=\frac{q^n -q^{-n}}{q-q^{-1}}= \begin{cases} \sum_{i=1}^{n} q^{-1-n +2i} & \textrm{if $n> 0$,} \\ 0 & \textrm{if $n=0$ and} \\ \sum_{i=-n}^{-1} -q^{1+n +2i} & \textrm{if $n<0$.} \end{cases}$$ If $k$ and $n$ are two integers, define the *quantum binomial $ \begin{bmatrix} n \\ k \end{bmatrix} $* by the following formula: $$\begin{bmatrix} n \\ k \end{bmatrix} = \begin{cases} \frac{\prod_{i=0}^{k-1}[n-i]}{\prod_{i=1}^k[i]} & \textrm{if $k\geq 0$,} \\ 0 &\textrm{else.} \end{cases}$$ For all integers $k$ and $n$, one has: $$[-n] = -[n] \qquad \textrm{and} \qquad \begin{bmatrix} -n \\ k \end{bmatrix} = (-1)^k \begin{bmatrix} n+k -1 \\ k \end{bmatrix}.$$ \[lem:add-integers\] Let $n$ and $m$ be two integers, the following identity holds: $$[m+n] = q^{-n}[m] + q^{m}[n] = q^{n}[m] + q^{-m}[n].$$ \[cor:sum-integers\] Let $k$ be an integer and $(a_h)_{1\leq h \leq k}$ be a collection of $k$ integers, then the following identity holds: $$\left[\sum_{h=1}^k a_h\right]= \sum_{h=1}^k q^{\sum_{i=1}^{h-1} a_i - \sum_{i=h+1}^k a_i} [a_h] = \sum_{h=1}^k q^{-\sum_{i=1}^{h-1} a_i + \sum_{i=h+1}^k a_i} [a_h].$$ \[lem:diff-prod\] Let $k$, $m$ and $n$ be three integers, then the following identity holds: $$[m+k][n] -[m][n+k] = [k][n -m].$$ \[lem:pascal\] Let $n$ and $k$ be two integers, then the following identity holds: $$\begin{aligned} \begin{bmatrix} n \\k \end{bmatrix} = q^k \begin{bmatrix} n - 1 \\ k \end{bmatrix} + q^{k-n} \begin{bmatrix} n - 1 \\ k -1 \end{bmatrix}.\end{aligned}$$ \[cor:anti-pascal\] Let $n$ and $k$ be two integers with $k\geq 0$, then the following identity holds: $$\begin{aligned} \begin{bmatrix} n \\k \end{bmatrix} = \sum_{i=0}^{k}(-1)^{k-i}q^{(i-k)(n-1) - k} \begin{bmatrix} n +1 \\i \end{bmatrix}.\end{aligned}$$ A version of the next proposition appears in [@MR1865777 page 23] with a different proof only valid for $n\geq 0$ and $m\geq 0$. Hence, we provide a complete proof. \[prop:dec-binomial\] Let $n_1$, $n_2$ and $k$ be three integers. The following identity holds: $$\begin{aligned} \begin{bmatrix} n_1 + n_2 \\k \end{bmatrix} = \sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}} q^{n_1k_2 - n_2k_1} \begin{bmatrix} n_1 \\ k_1 \end{bmatrix} \begin{bmatrix} n_2 \\ k_2 \end{bmatrix}. \end{aligned}$$ This statement is proved by induction on $n_1$. If $n_1 =0$ this is obvious. One needs to consider the case $n_1\geq 0$ and the case $n_1 \leq 0$. Suppose that $n_1\geq 0$ and that the statement holds for $n_1$. We use Lemma \[lem:pascal\] to compute: $$\begin{aligned} &\sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}} q^{(n_1+1)k_2 - n_2k_1} \begin{bmatrix} n_1 +1 \\ k_1 \end{bmatrix} \begin{bmatrix} n_2 \\ k_2 \end{bmatrix} \\ &\qquad= \sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}}q^{(n_1+1)k_2 - n_2k_1} \left(q^{k_1} \begin{bmatrix} n_1 \\ k_1 \end{bmatrix} +q^{k_1 - (n_1 +1)} \begin{bmatrix} n_1 \\ k_1 -1 \end{bmatrix} \right) \begin{bmatrix} n_2 \\ k_2 \end{bmatrix} \\ & \qquad = q^{k}\sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}}q^{n_1k_2- n_2k_1} \begin{bmatrix} n_1 \\ k_1 \end{bmatrix} \begin{bmatrix} n_2 \\ k_2 \end{bmatrix} \\ & \hspace{2cm}+ q^{k-n_1-n_2-1}\sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}}q^{n_1k_2 - n_2(k_1-1)} \begin{bmatrix} n_1 \\ k_1 -1 \end{bmatrix} \begin{bmatrix} n_2 \\ k_2 \end{bmatrix} \\ &\qquad = q^k \begin{bmatrix} n_1 + n_2 \\k \end{bmatrix} + q^{k-(n_1+n_2+1)} \begin{bmatrix} n_1 + n_2 \\ k-1 \end{bmatrix} \\ &\qquad = \begin{bmatrix} n_1 + n_2 +1 \\k \end{bmatrix}. \end{aligned}$$ Hence the statement holds for $n_1+1$, and therefore for all $n_1 \in {\ensuremath{\mathbb{Z}}}_\geq0$. Suppose that $n_1 < 0$ and that the statement holds for $n_1+1$. We use Corollary \[cor:anti-pascal\] to compute: $$\begin{aligned} &\sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}} q^{n_1k_2 - n_2k_1} \begin{bmatrix} n_1 \\ k_1 \end{bmatrix} \begin{bmatrix} n_2 \\ k_2 \end{bmatrix} \\ &\qquad= \sum_{\substack{k_1 + k_2 = k \\ k_1, k_2 \geq 0}} q^{n_1k_2 - n_2k_1} \sum_{i=1}^{k_1}(-1)^{k_1-i}q^{(i-k_1)(n_1-1) - k_1} \begin{bmatrix} n_1 +1 \\ i \end{bmatrix} \begin{bmatrix} n_2 \\ k_2 \end{bmatrix} \\ &\qquad= \sum_{j=0}^k\sum_{\substack{j_1 + j_2 = j \\ j_1, j_2 \geq 0}} (-1)^{k-j_1 -j_2}q^{(n_1j_2- n_2k + n_2j_2) + (j-k)(n_1-1) - k +j_2} \begin{bmatrix} n_1 +1 \\ j_1 \end{bmatrix} \begin{bmatrix} n_2 \\ j_2 \end{bmatrix} \\ &\qquad= \sum_{j=0}^k (-1)^{k-j} q^{(j-k)(n-1) -k}\sum_{\substack{j_1 + j_2 = j \\ j_1, j_2 \geq 0}} q^{(n_1+1)j_2- n_2j_1} \begin{bmatrix} n_1 +1 \\ j_1 \end{bmatrix} \begin{bmatrix} n_2 \\ j_2 \end{bmatrix} \\ &\qquad= \sum_{j=0}^k (-1)^{k-j} q^{(j-k)(n_1 + n_2 -1) -k} \begin{bmatrix} n_1 + n_2 +1 \\j \end{bmatrix}\\ & \qquad = \begin{bmatrix} n_1 + n_2 \\k \end{bmatrix}. \end{aligned}$$ Hence, the statement holds for $n_1$ and therefore for all negative integers. Let $X$ be a set. A *multi-subset of $X$* is an application $Y: X\to {\ensuremath{\mathbb{Z}}}_{\geq 0}$. If $\sum_{x\in X}Y(x)< \infty$, the multi-subset $Y$ is said to be *finite* and the sum is its *cardinal* (denoted by $\#Y$). If $x$ is an element of $X$, the number $Y(x)$ is the *multiplicity of $x$ in $Y$*. An element $x$ of $X$ is *in* $Y$, if its multiplicity in $Y$ is greater than or equal to $1$ (we write $x \in Y$). Let $Y_1$ and $Y_2$ be two multi-subsets of a set $X$. - The *disjoint union* of $Y_1$ and $Y_2$ (denoted $Y_1\sqcup Y_2$) is the multi-subset $Y_1+Y_2$. - The *union* of $Y_1$ and $Y_2$ (denoted $Y_1\cup Y_2$) is the multi-subset $\max (Y_1,Y_2)$. - The *intersection* of $Y_1$ and $Y_2$ (denoted $Y_1\cap Y_2$) is the multi-subset $\min (Y_1,Y_2)$. A subset $Y$ of $X$ can be thought of as a multi-subset of $X$ by identifying it with its characteristic function. Let $X$ be a set. The power set of $X$ is denoted by ${\ensuremath{\mathcal{P}_{{}}(X)}}$. The set of finite multi-subsets of $X$ is denoted by ${\ensuremath{\mathcal{M}_{{}}(X)}}$. For any non-negative integer $k$, the set of subsets ([resp. ]{}multi-subsets) of $X$ of cardinal $k$ is denoted by ${\ensuremath{\mathcal{P}_{k}(X)}}$ ([resp. ]{}${\ensuremath{\mathcal{M}_{k}(X)}}$). Let $X$ be an ordered set and $Y$ a multi-subset of $X$. The *degree of $Y$ in $X$* is the integer ${\ensuremath{\mathrm{deg}_{X}(Y)}}$ given by the following formula: $$\begin{aligned} {\ensuremath{\mathrm{deg}_{X}(Y)}}:= \sum_{\substack{x<y \\x\in X\\y \in Y \\ x \notin Y}}Y(y) - \sum_{\substack{y<x \\x\in X\\y \in Y \\ x \notin Y}}Y(y). \end{aligned}$$ For $N$ a non-negative integer, denote by ${\ensuremath{\llbracket N \rrbracket}}$ the ordered set of integers between $1$ and $N$. A version of the next proposition can be found in [@MR1865777] (Theorem 6.1 on page 19). Let $N$ and $k$ be two non-negative integers. The following identities hold: $$\begin{aligned} \sum_{Y \in {\ensuremath{\mathcal{P}_{k}({\ensuremath{\llbracket N \rrbracket}})}}} q^{{\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket N \rrbracket}}}(Y)}}} &= \begin{bmatrix} N \\k \end{bmatrix} \quad \textrm{and} \\ (-1)^k \sum_{Y \in {\ensuremath{\mathcal{M}_{k}({\ensuremath{\llbracket N \rrbracket}})}}} q^{{\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket N \rrbracket}}}(Y)}}} &= \begin{bmatrix} -N \\k \end{bmatrix}. \end{aligned}$$ This is an easy induction. The statement is clear when $N=0$, and one can prove that the left-hand sides of these two identities satisfy the induction formulas given in Lemma \[lem:pascal\] and Corollary \[cor:anti-pascal\]. Colorings of MOY graphs {#sec:colorings-moy-graphs} ======================= The aim of this section is to present a state sum à la Murakami–Ohtsuki–Yamada [@MR1659228] computing the Reshetikhin–Turaev invariants of links associated with the exterior powers of the standard representation of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$. These invariants are denoted $P_{N|M}$. As in [@MR1659228], this state sum is a combinatorial interpretation of the underlying representation theory. Details about the algebraic definition is provided in Appendix \[sec:moy-graphs-an\]. Note however that consistency of the MOY calculus presented here follows from its definition as a state sum and does not rely on the algebraic interpretation. The ${\ensuremath{\mathfrak{sl}}}_N$-invariants associated with exterior powers correspond to setting $M=0$. Symmetric powers are covered by this state sum by essentially swaping $M$ and $N$. Indeed, the Hopf algebra $U_q({\ensuremath{\mathfrak{gl}}}_{M|N})$ is isomorphic to $U_{q}({\ensuremath{\mathfrak{gl}}}_{N|M})$ (see Remark \[rmk:swapMN\]). The standard representation ${\ensuremath{\mathbb{C}}}^{M|N}_q$ of the first is isomorphic to the standard representation ${\ensuremath{\mathbb{C}}}^{N|M}_{q}$ tensored with a trivial representation of odd super degree denoted $\boldsymbol{1}_{\mathrm{odd}}$. It induces isomorphisms between ${\ensuremath{\mathrm{Sym}}}_{q}^k{\ensuremath{\mathbb{C}}}^{M|N}_q$ and $\Lambda^k\left(_{q} {\ensuremath{\mathbb{C}}}^{N|M}_{q}\otimes \boldsymbol{1}_{\mathrm{odd}}\right)$. The later being isomorphic to $$\begin{aligned} \begin{cases} \Lambda^k_{q}\left({\ensuremath{\mathbb{C}}}^{N|M}_{q}\right) & \text{$k$ even,} \\[5pt] \Lambda^k_{q}\left({\ensuremath{\mathbb{C}}}^{N|M}_{q}\right) \otimes \boldsymbol{1}_{\mathrm{odd}} & \text{$k$ odd.} \end{cases}\end{aligned}$$ Since the $U_{q}({\ensuremath{\mathfrak{gl}}}_{N|M})$-invariant of a link with $\ell$ components associated with $\boldsymbol{1}_{\mathrm{odd}}$ is equal to $(-1)^\ell$, we obtain that if $L$ is an oriented link with $\ell$ components labeled by positive integers $k_1, \dots, k_\ell$, interpreted as exponents of symmetric powers, then its $U_q({\ensuremath{\mathfrak{gl}}}_{M|N})$-invariant is equal to $$(-1)^{\sum_ik_i}P_{N|M}(\overline{L}),$$ where $\overline{L}$ denote the mirror image of $L$ and the integers $k_1, \dots, k_\ell$ are now interpreted as exponents exterior powers. The mirror image appears because the isomorphism between $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ and $U_{q}({\ensuremath{\mathfrak{gl}}}_{M|N})$ maps the universal $R$-matrix of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ onto the inverse of that of $U_q({\ensuremath{\mathfrak{gl}}}_{M|N})$. The most famous invariant associated with symmetric powers is the colored Jones polynomial of links. For a link $L$ with $\ell$ components, denote $J_n(L)$ the un-normalized[^1] colored Jones polynomial associated with the $(n+1)$-dimensional irreducible representation of $U_q({\ensuremath{\mathfrak{sl}}}_2)$. The discussion above gives $$P_{0|2}(L_{n, \dots,n}) = (-1)^{n\ell}J_n(\overline{L}) = J_n(L, -q^{-1}),$$ where $L_{n, \dots,n}$ means that every component of $L$ is labeled by $n$. For further reference, denote by $J'_n(L):= \frac{J_n(L)}{[n+1]}$ the normalized colored Jones polynomial. \[dfn:abstractMOYgraph\] A *MOY graph* is a finite oriented planar trivalent graph $\Gamma = (V(\Gamma),E(\Gamma))$ with a labeling of its edges $\ell : E_\Gamma \to {\ensuremath{\mathbb{Z}}}_{\geq0}$ such that the flow given by labels and orientations is preserved at each vertex, meaning that every trivalent vertex follows one of the two models $${\ensuremath{\vcenter{\hbox{\tikz{\begin{scope} \draw[->] (0,0) -- (0,0.5) node [at end, above] {$a+b$}; \draw[>-] (-0.5, -0.5) -- (0,0) node [at start, below] {$a$}; \draw[>-] (+0.5, -0.5) -- (0,0) node [at start, below] {$b$}; \begin{scope}[xshift = 5cm, yscale = -1] \draw[-<] (0,0) -- (0,0.5) node [at end, below] {$a+b$}; \draw[<-] (-0.5, -0.5) -- (0,0) node [at start, above] {$a$}; \draw[<-] (+0.5, -0.5) -- (0,0) node [at start, above] {$b$}; \end{scope} \end{scope} }}}}}.$$ The first is a *merge* vertex, the second is a *split* vertex. For each vertex, there is a *thick* edge (the one labeled by $a+b$ in the depicted models) and two *thin* edges: the one labeled by $a$ in the previous models is the *left* thin edge, the one labeled by $b$ is the *right* thin edge. Let $\Gamma$ be a MOY graph, then the graph $\Gamma'$ obtained by erasing the edges of $\Gamma$ with label $0$ is still a MOY graph. Moreover in all what follows the two graphs can be considered as the same. The label $0$ is introduced only to make some definitions and proofs easier and more natural. \[dfn:cabling-and-numbers\] Let $\Gamma$ be a MOY graphs. The *cabling* of $\Gamma$ is the oriented simple multi-curve in the plane ${\ensuremath{\mathbb{R}}}^2$ obtained from $\Gamma$ by replacing each $e$ edge of $\Gamma$ by $\ell(e)$ parallel oriented intervals and joining these intervals at every vertex of $\Gamma$ by the only compatible oriented crossing-less matching. Let $\gamma$ be a simple multi-curve in the plane. The *rotational* of $\gamma$ is the integer $\rho(\gamma)$ equal to the number of circles of $\gamma$ oriented counterclockwise minus the number of circles of $\gamma$ oriented clockwise. A circle oriented counterclockwise is said *positively oriented* and a circle oriented clockwise is said *negatively oriented*. The *rotational* of a MOY graph $\Gamma$ is the integer $\rho(\Gamma)$ equal to the rotational of its cabling. \[dfn:sym1\] The *binomial weight* of a MOY graph $\Gamma$ is the Laurent polynomial $b(\Gamma)$ defined by the following formula: $$b(\Gamma) = \prod_{v\text{ split vertex}} \begin{bmatrix} v_l + v_r \\ v_r \end{bmatrix},$$ where for each split vertex $v$, $v_l$ ([resp. ]{}$v_r$) denotes the label of the left thin edge ([resp. ]{}right thin edge) of $v$. \[dfn:subMOY\] Let $\Gamma= (V(\Gamma), E(\Gamma), \ell)$ be a MOY graph. A *sub-MOY graph of $\Gamma$* is a MOY graph $\Gamma'=(V(\Gamma), E(\Gamma), \ell') $ such that for all $e$ in $E(\Gamma)$, $\ell'(e) \leq \ell(e)$. A sub-MOY graph $\Gamma'$ is *cyclic* if for all $e \in E(\Gamma)$, $\ell'(e) \leq 1$. Let $I$ be a finite set and consider a collection $\Gamma_i= \left(V(\Gamma), E(\Gamma), \ell_i)\right)_{i\in I}$ of sub-MOY graphs of $\Gamma$, the *sum* of the sub-MOY graphs $\left(\Gamma_i\right)_{i\in I}$ is the MOY graph $(V(\Gamma), E(\Gamma), \sum_{i\in I} \ell_i)$ and is denoted by $\sum_{i \in I}\Gamma_i$. \[dfn:u\] Let $\Gamma= (V(\Gamma), E(\Gamma), \ell)$ be a MOY graph and $\Gamma_1= \left(V(\Gamma), E(\Gamma), \ell_1)\right)$ and $\Gamma_2= \left(V(\Gamma), E(\Gamma), \ell_2)\right)$ be two sub-MOY graphs of $\Gamma$. For each split vertex $s$ of $\Gamma$, define $u_s(\Gamma_1, \Gamma_2)$ to be the integer defined by the formula: $$u_s(\Gamma_1, \Gamma_2):= a_2b_1 - a_1b_2,$$ where $a_1$ ([resp. ]{}$b_1$) is the label of $\Gamma_1$ of the left ([resp. ]{}right) small edge at the vertex $s$ and $a_2$ ([resp. ]{}$b_2$) is the label of $\Gamma_2$ of the left ([resp. ]{}right) small edge at the vertex $s$. Define $u(\Gamma_1, \Gamma_2)$ to be the integer given by the formula: $$u(\Gamma_1, \Gamma_2):= \sum_{\textrm{$s$ split vertex of $\Gamma$}} u_s(\Gamma_1, \Gamma_2).$$ Note that the sum of sub-MOY graphs of $\Gamma$ is a MOY graph but not necessarily a sub-MOY graph of $\Gamma$, since it is not required that for all $e\in E(\Gamma)$, $\sum_{i\in I} \ell_i(e)\leq \ell(e)$. For the rest of the paper, fix two non-negative integers $N$ and $M$. \[dfn:coloring\] Let $\Gamma$ be a MOY graph, an *$(N|M)$-coloring* (or simply *coloring*) of $\Gamma$ is a pair $(\underline{\Gamma^E}, \underline{\Gamma^S})$ where $\underline{\Gamma^E} = (\Gamma^E_1, \dots, \Gamma^E_N)$ is an $N$-tuple of cyclic sub-MOY graph of $\Gamma$ and $\underline{\Gamma^S} = (\Gamma^S_1, \dots, \Gamma^S_M)$ is an $M$-tuple of sub-MOY graph of $\Gamma$ such that: $$\sum_{j=1}^N \Gamma^E_{j} + \sum_{i=1}^M \Gamma^S_{i} = \Gamma.$$ The set of $(N|M)$-colorings of $\Gamma$ is denoted by ${\ensuremath{\mathrm{col}_{N|M}(\Gamma)}}$ (or simply ${\ensuremath{\mathrm{col}_{{}}(\Gamma)}}$). \[rmk:col2multiset\] Let $\Gamma$ be a MOY graph and $c = (\underline{\Gamma^E}, \underline{\Gamma^S})$ a coloring of $\Gamma$. If $e$ is an edge of $\Gamma$, consider the pair $(c^E(e), c^S(e))$: $c^E(e)$ is the subset of ${\ensuremath{\llbracket N \rrbracket}}$ which contains $j$ if and only if $\ell_{\Gamma^E_j}(e)=1$ and $c^S(e)$ is the sub-multiset of ${\ensuremath{\llbracket M \rrbracket}}$ where $i$ appears with multiplicity $\ell_{\Gamma^S_i}(e)$. We have $\#c^E(e) + \#c^S(e) = \ell(e)$. A coloring can be defined by such a collection $(c^E(e), c^S(e))_{e\in E(\Gamma)}$ satisfying a flow condition at every vertex. \[dfn:weight\] Let $\Gamma$ be a MOY graph and $c=(\underline{\Gamma^E}, \underline{\Gamma^S})$ an $(N|M)$-coloring of $\Gamma$. The *$s$-weight of $c$* is the integer denoted by $w_s(c)$ and defined by the following formula: $$\begin{aligned} w_s(c):=& \sum_{1\leq i <j \leq M} u(\Gamma^S_j,\Gamma_i^S) -\sum_{1\leq i <j \leq N} u(\Gamma^E_j,\Gamma_i^E) + \sum_{i=1}^N\sum_{j=1}^M u(\Gamma^E_i,\Gamma_j^S).\end{aligned}$$ The *$\rho$-weight of $c$* is the integer denoted by $w_\rho(c)$ and defined by the following formula: $$\begin{aligned} w_\rho(c):=& \sum_{i=1}^N(N+M-2i+1) \rho(\Gamma^E_i) + \sum_{j=1}^M (N+M-2j+1)\rho(\Gamma^S_j). \end{aligned}$$ The *multiplicity of $c$* is the Laurent polynomial $m(c)$ with positive coefficients and symmetric in $q$ and $q^{-1}$ given by the following formula: $$m(c) = \prod_{i=1}^M b(\Gamma_i^S).$$ The *parity of $c$* is the integer $s(c)$ given by the following formula: $$s(c) = {\sum_{i=1}^M \rho(\Gamma_i^S)},$$ as the name suggests, we are only interested in the value of $s(c)$ modulo $2$. Equality modulo $2$ is denoted by $\equiv$ in what follows. The *combinatorial $(N|M)$-evaluation of the coloring $c$* is the Laurent polynomial ${\left\langle \Gamma,c \right\rangle}_{N|M}$ given by the following formula: $${\left\langle \Gamma,c \right\rangle}_{N|M} = (-1)^{s(c)} q^{w_s(c)+w_\rho(c)}m(c).$$ Finally, the *combinatorial $(N|M)$-evaluation of $\Gamma$* is the Laurent polynomial ${\left\langle \Gamma \right\rangle}_{N|M}$ given by the following formula: $${\left\langle \Gamma \right\rangle}_{N|M} = \sum_{c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}}} {\left\langle \Gamma,c \right\rangle}_{N|M}.$$ We give in Appendix A a detailed account of the Reshetikhin–Turaev invariant for MOY graphs for the quantum group $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$. For any MOY graph $\Gamma$ it is denoted ${\left\llangle \Gamma \right\rrangle}_{N|M}$. The aim of the rest of this section is to prove the next theorem. \[thm:maintheorem\] For any MOY graph $\Gamma$, the algebraic and the combinatorial evaluation of $\Gamma$ agree. This means: $${\left\llangle \Gamma \right\rrangle}_{N|M} = {\left\langle \Gamma \right\rangle}_{N|M}.$$ In particular, the combinatorial evaluation depends only on $N-M$ and is symmetric in $q$ and $q^{-1}$. Following Proposition \[prop:completeness\], one only needs to check that ${\left\langle \bullet \right\rangle}_{N|M}$ satisfies the multiplicativity property and the local relations of Proposition \[prop:rel-kups\]. This is the object of the rest of this section. \[prop:mirror-image\] Let $\Gamma$ be a MOY graph, and $\overline{\Gamma}$ be its image under the transformation $\left(\begin{smallmatrix} x \\ y \end{smallmatrix}\right)\mapsto \left(\begin{smallmatrix} -x \\ y \end{smallmatrix}\right)$ Then ${\left\langle \Gamma \right\rangle}_{N|M}(q) = {\left\langle \overline{\Gamma} \right\rangle}_{N|M}(q^{-1})$. There is a canonical one-to-one correspondence between colorings of $\Gamma$ and colorings of $\overline{\Gamma}$. Let $c$ be a coloring of $\Gamma$ and $\overline{c}$ the corresponding coloring of $\overline{\Gamma}$. The following identities hold: $$\begin{aligned} m(c)(q) &= m(c)\left(q^{-1}\right) = m\left(\overline{c}\right)(q) = m\left(\overline{c}\right)\left(q^{-1}\right),\\ s(c) &\equiv s\left(\overline{c}\right), \\ w_s(c) &= - w_s\left(\overline{c}\right) \quad\textrm{and} \\ w_\rho(c) &= -w_\rho\left(\overline{c}\right).\end{aligned}$$ From this, one immediately obtains ${\left\langle \Gamma \right\rangle}_{N|M}(q) = {\left\langle \overline{\Gamma} \right\rangle}_{N|M}(q^{-1})$. \[lem:disjointunion\] Let $\Gamma$ and $\Upsilon$ be two MOY graphs, then $${\left\langle \Gamma \sqcup \Upsilon \right\rangle}_{N|M} = {\left\langle \Gamma \right\rangle}_{N|M}{\left\langle \Upsilon \right\rangle}_{N|M}.$$ First of all, there is a canonical one-to-one correspondence between the set of colorings of $\Gamma \sqcup \Upsilon$ and the Cartesian product of sets of colorings of $\Gamma$ and $\Upsilon$. If $c$ and $d$ are colorings of $\Gamma$ and $\Upsilon$, denote by $(c,d)$ the corresponding coloring of $\Gamma\sqcup\Upsilon$. It is enough to prove that for all colorings $(c,d)$, $$(-1)^{s(c)+s(d)} q^{w_s(c)+w_\rho(d)+w_s(c) + w_\rho(d)}m(c)m(d) = (-1)^{s((c,d))} q^{w_s((c,d)) + w_\rho((c,d))}m((c,d)).$$ Let $c=(\underline{\Gamma^E}, \underline{\Gamma^S})$ ([resp. ]{}$d=(\underline{\Upsilon^E}, \underline{\Upsilon^S})$) be a coloring of $\Gamma$ ([resp. ]{}$\Upsilon$). One has $s((c,d))=s(c)+s(d)$ and $m((c,d)) = m(c)m(d).$ Hence, it remains to prove that $w((c,d)) = w(c) + w(d)$. Let us denote by $\Phi$ the MOY graph $\Gamma \sqcup \Upsilon$. One has $$\begin{aligned} w_s((c,d))&= \sum_{1\leq i <j \leq M} u(\Phi^S_j,\Phi_i^S) -\sum_{1\leq i <j \leq N} u(\Phi^E_j,\Phi_i^E) + \sum_{i=1}^N\sum_{j=1}^M u(\Phi^E_i,\Phi_j^S) \\ &= \sum_{1\leq i <j \leq M} u(\Gamma^S_j,\Gamma_i^S) + u(\Upsilon^S_j,\Upsilon_i^S) -\sum_{1\leq i <j \leq N} u(\Gamma^E_j,\Gamma_i^E) + u(\Upsilon^E_j,\Upsilon_i^E) \\ & + \sum_{i=1}^N\sum_{j=1}^M u(\Gamma^E_i,\Gamma_j^S) +u(\Upsilon^E_i,\Upsilon_j^S). \\ =& w_s(c) + w_s(d)\end{aligned}$$ and $$\begin{aligned} w_\rho((c,d))=& \sum_{i=1}^M(N+M-2i+1)\rho(\Phi^S_i) + \sum_{i=1}^N(N+M-2i+1)\rho(\Phi^E_i) \\ = & \sum_{i=1}^M (N+M-2i+1)\left(\rho(\Gamma^S_i) + \rho(\Upsilon^S_i)\right) \\ & + \sum_{i=1}^N (N+M-2i+1)\left(\rho(\Gamma^E_i) + \rho(\Upsilon^E_i)\right) \\ =& w_\rho(c) + w_\rho(d). \qedhere\end{aligned}$$ \[lem:circles\] For any non-negative integer $k$, the following identity holds: $${\left\langle {\ensuremath{\vcenter{\hbox{\tikz{\draw[->] (0,0) arc(0:360:0.4) node[scale=0.6, right] {$k$};}}}}}\! \right\rangle}_{N|M} = \begin{bmatrix} N-M \\ k \end{bmatrix} = {\left\langle {\ensuremath{\vcenter{\hbox{\tikz{\draw[<-] (0,0) arc(0:360:0.4) node[scale=0.6, right] {$k$};}}}}}\! \right\rangle}_{N|M}.$$ Let us start with the first identity. Denote by $\Gamma$ the positively oriented circle of label $k$. A coloring $c=(\underline{\Gamma^E}, \underline{\Gamma^S})$ of $\Gamma$ is completely encoded by a pair consisting of a subset $X_c$ of ${\ensuremath{\llbracket N \rrbracket}}$ with $k^E_c$ elements a multi-subset $Y_c$ of ${\ensuremath{\llbracket M \rrbracket}}$ with $k^S_c$ elements such that $k_c^E+ k_c^S =k$. One has $s(c) = k^S_c$ and $m(c)= 1$. Since there is no vertex, $w_s(\Gamma)=0$. One has $$\begin{aligned} w_\rho(c) =& \sum_{i=1}^M (N+M-2i+1)\rho(\Gamma^S_i) + \sum_{i=1}^N(N+M-2i+1) \rho(\Gamma^E_i) \\ =& Nk^S_c + Mk^E_c + \sum_{1\leq i <j \leq M} \rho (\Gamma^S_j) - \rho(\Gamma_i^S) + \sum_{1\leq i <j \leq N} \rho (\Gamma^E_j) - \rho(\Gamma_i^E). \end{aligned}$$ Hence, if one fixes two integers $k^S$ and $k^E$ such that $k= k^S + k^E$, one gets: $$\begin{aligned} \sum_{\substack{c \in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} \\ k^S_c = k^S \\ k^E_c= k^E}} q^{w_\rho(c)}&= q^{Mk^E + Nk^S} \!\!\!\!\! \sum_{\substack{(\underline{\Gamma^E}, \underline{\Gamma^S}) \in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} \\ k^S_c = k^S \\ k^E_c= k^E}} \!\!\!\!\! q^{\sum_{1\leq i <j \leq M} \rho (\Gamma^S_j) - \rho(\Gamma_i^S)+\sum_{1\leq i <j \leq N} \rho (\Gamma^E_j) - \rho(\Gamma_i^E)}\\ &= q^{Mk^E + Nk^S} \sum_{\substack{ X \in {\ensuremath{\mathcal{P}_{k^E}({\ensuremath{\llbracket N \rrbracket}})}}\\Y\in {\ensuremath{\mathcal{M}_{k^S}({\ensuremath{\llbracket M \rrbracket}})}}}} q^{{\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket M \rrbracket}}}(X)}}} q^{{\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket N \rrbracket}}}(Y)}}} \\ &= q^{Mk^E_c + Nk^S_c} \begin{bmatrix} M + k^S -1 \\ k^S \end{bmatrix} \begin{bmatrix} N \\ k^E \end{bmatrix}.\end{aligned}$$ Hence, using Proposition \[prop:dec-binomial\], one gets: $$\begin{aligned} {\left\langle \Gamma \right\rangle}_{N|M}&= \sum_{k^S + k^E= k} (-1)^{k^S} q^{Mk^E + Nk^S} \begin{bmatrix} M + k^S -1 \\ k^S \end{bmatrix} \begin{bmatrix} N \\ k^E \end{bmatrix}\\ &=\sum_{k^S + k^E= k} q^{Mk^E + Nk^S} \begin{bmatrix} - M \\ k^S \end{bmatrix} \begin{bmatrix} N \\ k^E \end{bmatrix} \\ & = \begin{bmatrix} N - M \\k \end{bmatrix}.\end{aligned}$$ The second identity follows from the first one and Lemma \[prop:mirror-image\] (because quantum binomials are symmetric in $q$ and $q^{-1}$). \[lem:assoc\] The combinatorial $(N|M)$-evaluation satisfies the following two local identities: $$\begin{aligned} \label{eq:extrelass} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.3]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,1); \coordinate (V2) at (1,2); \coordinate (T1) at (-2,3); \coordinate (T2) at (0,3); \coordinate (T3) at (2,3); \draw[>-] (B) -- (V1) node [at start, below] {\tiny{$i+j+k$}}; \draw[->] (V1) -- (T1) node [at end, above] {\tiny{$i$}}; \draw[->] (V1) -- (V2) node[midway, right] {\tiny{$j+k$}}; \draw[->] (V2) -- (T2) node[at end, above] {\tiny{$j$}}; \draw[->] (V2) -- (T3) node[at end, above] {\tiny{$k$}}; }}}}}\right\rangle}_{N|M} &= {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.3]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,1); \coordinate (V2) at (-1,2); \coordinate (T1) at (-2,3); \coordinate (T2) at (0,3); \coordinate (T3) at (2,3); \draw[>-] (B) -- (V1) node [at start, below] {\tiny{$i+j+k$}}; \draw[->] (V1) -- (T3) node [at end, above] {\tiny{$k$}}; \draw[->] (V1) -- (V2) node[midway, left] {\tiny{$i+j$}}; \draw[->] (V2) -- (T1) node[at end, above] {\tiny{$i$}}; \draw[->] (V2) -- (T2) node[at end, above] {\tiny{$j$}}; }}}}}\right\rangle}_{N|M}, \\ {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.3, yscale = -1]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,1); \coordinate (V2) at (1,2); \coordinate (T1) at (-2,3); \coordinate (T2) at (0,3); \coordinate (T3) at (2,3); \draw[<-] (B) -- (V1) node [at start, above] {\tiny{$i+j+k$}}; \draw[-<] (V1) -- (T1) node [at end, below] {\tiny{$i$}}; \draw[-<] (V1) -- (V2) node[midway, right] {\tiny{$j+k$}}; \draw[-<] (V2) -- (T2) node[at end, below] {\tiny{$j$}}; \draw[-<] (V2) -- (T3) node[at end, below] {\tiny{$k$}}; }}}}}\right\rangle}_{N|M} &= {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.3, yscale = -1]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,1); \coordinate (V2) at (-1,2); \coordinate (T1) at (-2,3); \coordinate (T2) at (0,3); \coordinate (T3) at (2,3); \draw[<-] (B) -- (V1) node [at start, above] {\tiny{$i+j+k$}}; \draw[-<] (V1) -- (T3) node [at end, below] {\tiny{$k$}}; \draw[-<] (V1) -- (V2) node[midway, left] {\tiny{$i+j$}}; \draw[-<] (V2) -- (T1) node[at end, below] {\tiny{$i$}}; \draw[-<] (V2) -- (T2) node[at end, below] {\tiny{$j$}}; }}}}}\right\rangle}_{N|M}. \end{aligned}$$ We only prove the first one. Let us denote by $\Gamma$ and $\Upsilon$ respectively the MOY graph on the left and right-hand side of this identity. There is a canonical one-to-one correspondence between the $(N|M)$-colorings of $\Gamma$ and of $\Upsilon$. Let $c=(\underline{\Gamma^E}, \underline{\Gamma^S})$ be a $(N|M)$-coloring of $\Gamma$ and $c'=(\underline{\Upsilon^E}, \underline{\Upsilon^S})$ be the corresponding coloring of $\Upsilon$. The following holds: $$m(c) = m(c'), \quad s(c) = s(c'),\quad w_s(c) = w_s(c')\quad \textrm{and} \quad w_\rho(c) = w_\rho(c').$$ This implies ${\left\langle \Gamma,c \right\rangle}_{N|M} = {\left\langle \Upsilon, c' \right\rangle}_{N|M}$ and subsequently ${\left\langle \Gamma \right\rangle}_{N|M} = {\left\langle \Upsilon \right\rangle}_{N|M}$. \[lem:easy-digon\] The combinatorial $(N|M)$-evaluation satisfies the following identity: $$\begin{aligned} \label{eq:easy-digon} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m+1$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m+1$}}; \draw[->] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$1$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$m$}}; }}}}}\right\rangle}_{N|M} = [m+1]{\left\langle \,{\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m+1$}}; }}}}}\! \right\rangle}_{N|M}. \end{aligned}$$ Let $\Gamma$ and $\Upsilon$ be two MOY graphs which are identical except in a small ball where they are related by the local relation (\[eq:easy-digon\]), $\Gamma$ being on the left-hand side and $\Upsilon$ on the right-hand side. First note that any coloring of $\Gamma$ induces a coloring of $\Upsilon$. Let us fix a coloring $c'= (\underline{\Upsilon^E}, \underline{\Upsilon^S})$ of $\Upsilon$. If a coloring $c$ of $\Gamma$ induces $c'$ on $\Upsilon$, we write $c\to c'$. We will prove the following: $$\sum_{\substack{c\to c' \\ c \in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}}}} {\left\langle \Gamma, c \right\rangle}_{N|M} = [m+1] {\left\langle \Upsilon, c' \right\rangle}_{N|M},$$ which implies ${\left\langle \Gamma \right\rangle}_{N|M} = [m+1] {\left\langle \Upsilon \right\rangle}_{N|M}$. Denote by $e$ the edge of $\Upsilon$ which appears in the local relation. For $i$ in ${\ensuremath{\llbracket M \rrbracket}}$ and $j$ in ${\ensuremath{\llbracket N \rrbracket}}$, set $$\begin{aligned} k^S_i := \ell_{\Upsilon_i^S}(e), \quad k^E_j := \ell_{\Upsilon_j^E}(e), \quad k^S := \sum_{i=1}^M k^S_i, \quad \textrm{and} \quad k^E := \sum_{j=1}^N k^E_j. \end{aligned}$$ A coloring $c= (\underline{\Gamma^E}, \underline{\Gamma^S})$ of $\Gamma$ inducing $c'$ is totally determined by saying which MOY graph of the collection $\underline{\Gamma^E}\sqcup \underline{\Gamma^S}$ gives label $1$ to the right edge of the digon which appears in the local relation. Denote this graph by $\Gamma^A_h$. There are two possibilities: either this graph belongs to $\underline{\Gamma^S}$ ([i. e. ]{}$A=S$ and $h\in {\ensuremath{\llbracket M \rrbracket}}$) or to $\underline{\Gamma^E}$ ([i. e. ]{}$A=E$ and $h\in {\ensuremath{\llbracket N \rrbracket}}$). If $A = S$ and $h\in {\ensuremath{\llbracket M \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c')[k_h^S], &&\quad& s(c) &= s(c'),\\ w_\rho(c)&= w_\rho(c')&& \quad \textrm{and}\quad & w_s(c) &= w_s (c') - \sum_{i=1}^{h-1} k_i^S + \sum_{i=h+1}^M k_i^S - k^E. \end{aligned}$$ If $A = E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c')[k_h^E], \\ s(c) &= s(c'),\\ w_\rho(c)&= w_\rho(c') \quad \textrm{and}\\ w_s(c) &= w_s (c') + \sum_{i=1}^{h-1} k_i^E - \sum_{i=h+1}^N k_i^E + k^S. \end{aligned}$$ Note that for the coloring $c$ to exist, it is necessary that $k_h^E=1$. However, if $k_h^E=0$, the previous formula makes the contribution of this “virtual” coloring equal to $0$. This argument will be used throughout the proofs of this section. We will not repeat it. One can sum: $$\begin{aligned} \sum_{\substack{c\to c' \\ c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} }} {\left\langle \Gamma, c \right\rangle}_{N|M} &= \left(\sum_{h=1}^M q^{-k^E- \sum_{i=1}^{h-1} k_i^S + \sum_{i=h+1}^M k_i^S}[k_h^S] \right. \\ &\hspace{2cm} \left. + \sum_{h=1}^N q^{k^S + \sum_{i=1}^{h-1} k_i^E - \sum_{i=h+1}^N k_i^E}[k_h^E] \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M} \\ &= \left(q^{-k^E}[k^S] + q^{k^S}[k^E] \right){\left\langle \Upsilon, c' \right\rangle}_{N|M} \\ &= [k^S+k^E]{\left\langle \Upsilon, c' \right\rangle}_{N|M} = [m+1]{\left\langle \Upsilon, c' \right\rangle}_{N|M}. \qedhere \end{aligned}$$ \[cor:easy-digon2\] The combinatorial $(N|M)$-evaluation satisfies the following local identities and their mirror images: $$\begin{aligned} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m+1$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m+1$}}; \draw[->] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$m$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$1$}}; }}}}}\right\rangle}_{N|M}&= [m+1]{\left\langle \,{\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m+1$}}; }}}}}\! \right\rangle}_{N|M}, \\ {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m+n$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m+n$}}; \draw[->] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$n$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$m$}}; }}}}}\right\rangle}_{N|M}&= \begin{bmatrix} m+n \\ m \end{bmatrix} {\left\langle \,{\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m+n$}}; }}}}}\! \right\rangle}_{N|M},\\ {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[yscale=0.55, xscale=0.65]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1); \coordinate (D1) at (-1,2); \coordinate (C2) at (1,1); \coordinate (D2) at (1,2); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$k+s$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$k$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$k-r$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$l-s$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$l$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$l+r$}}; \draw[<-] (D2) -- (D1) node[midway, above] {\tiny{$r$}}; \draw[->] (C1) -- (C2) node[midway, below] {\tiny{$s$}}; }}}}}\right\rangle}_{N|M} &= \begin{bmatrix} r+s \\ r \end{bmatrix} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[yscale=0.55, xscale=0.65]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1.5); \coordinate (D1) at (-1,1.5); \coordinate (C2) at (1,1.5); \coordinate (D2) at (1,1.5); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$k+s$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$k-r$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$l-s$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$l+r$}}; \draw[->] (C1) -- (C2) node[midway, below] {\tiny{$r+s$}}; }}}}}\right\rangle}_{N|M}. \end{aligned}$$ The first identity is a consequence of Lemma \[lem:easy-digon\] and Lemma \[prop:mirror-image\]. The second is an easy induction. The third is a consequence of the second and of Lemma \[lem:assoc\]. \[lem:bad-digon\] The combinatorial $(N|M)$-evaluation satisfies the following local identity: $$\begin{aligned} \label{eq:bad-digon} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m$}}; \draw[<-] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$1$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$m+1$}}; }}}}}\right\rangle}_{N|M} = [N-M-m]{\left\langle \,{\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m$}}; }}}}}\! \right\rangle}_{N|M}. \end{aligned}$$ Let $\Gamma$ and $\Upsilon$ be two MOY graphs which are identical except in a small ball where they are related by the local relation (\[eq:bad-digon\]), $\Gamma$ being on the left-hand side and $\Upsilon$ on the right-hand side. First note that any coloring of $\Gamma$ induces a coloring of $\Upsilon$. Fix a coloring $c'= (\underline{\Upsilon^E}, \underline{\Upsilon^S})$ of $\Upsilon$. If a coloring $c$ of $\Gamma$ induces $c'$ on $\Upsilon$, we write $c\to c'$. We will prove the following: $$\sum_{\substack{c\to c'\\c \in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} }} {\left\langle \Gamma, c \right\rangle}_{N|M} = [N-M-m] {\left\langle \Upsilon, c' \right\rangle}_{N|M},$$ which implies ${\left\langle \Gamma \right\rangle}_{N|M} = [N-M-m] {\left\langle \Upsilon \right\rangle}_{N|M}$. Denote by $e$ the edge of $\Upsilon$ which appears in the local relation. For $i$ in ${\ensuremath{\llbracket M \rrbracket}}$ and $j$ in ${\ensuremath{\llbracket N \rrbracket}}$, set $$\begin{aligned} k^S_i := \ell_{\Upsilon_i^S}(e), \quad k^E_j := \ell_{\Upsilon_j^E}(e), \quad k^S := \sum_{i=1}^M k^S_i, \quad \textrm{and} \quad k^E := \sum_{i=1}^N k^E_i. \end{aligned}$$ A coloring $c= (\underline{\Gamma^S}, \underline{\Gamma^S})$ of $\Gamma$ inducing $c'$ is totally determined by saying which MOY graph of the collection $\underline{\Gamma^E}\sqcup \underline{\Gamma^S}$ gives label $1$ to the right edge of the digon which appears in the local relation. Denote this graph by $\Gamma^A_h$. There are two possibilities: either this graph belongs to $\underline{\Gamma^S}$ ([i. e. ]{}$A=S$ and $h\in {\ensuremath{\llbracket M \rrbracket}}$) or to $\underline{\Gamma^E}$ ([i. e. ]{}$A=E$ and $h\in {\ensuremath{\llbracket N \rrbracket}}$). If $A = S$ and $h\in {\ensuremath{\llbracket M \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c')[k_h^S+1], \\ s(c) &= s(c') +1,\\ w_\rho(c) &= w_\rho(c') + N+M-2h +1 \quad \textrm{and}\\ w_s(c) &= w_s(c')- \sum_{i=1}^{h-1} k_i^S + \sum_{i=h+1}^M k_i^S - k^E . \end{aligned}$$ Hence: $$\begin{aligned} w_\rho(c) + w_s(c)&= w_\rho(c') + w_s(c') + N - k^E- \sum_{i=1}^{h-1} (k_i^S +1) + \sum_{i=h+1}^M (k_i^S + 1). \end{aligned}$$ If $A = E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c')[1-k_h^E] \quad \textrm{note that one has $k^E_h=0$ }, \\ s(c) &= s(c'),\\ w_\rho(c)& = w_\rho(c') + N+M-2h +1 \quad \textrm{and}\\ w_s(c) &= w_s(c')+ \sum_{i=1}^{h-1} k_i^E - \sum_{i=h+1}^N k_i^E + k^S .\\ \end{aligned}$$ Hence: $$\begin{aligned} w_\rho(c) + w_s(c)&= w_\rho(c') + w_s(c') + M + k^S- \sum_{i=1}^{h-1} (1-k_i^E) + \sum_{i=h+1}^N (1-k_i^E ). \end{aligned}$$ Finally, one gets: $$\begin{aligned} \sum_{\substack{c\to c'\\ c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}}}} {\left\langle \Gamma, c \right\rangle}_{N|M} =&\left( -\sum_{h=1}^M q^{N -k^E - \sum_{i=1}^{h-1} (k_i^S +1) + \sum_{i=h+1}^M (k_i^S + 1)} [k_h^S+1] \right. \\ & \hspace{1cm}+ \left. \sum_{h=1}^N q^{M + k^S -\sum_{i=1}^{h-1} (1-k_i^E) + \sum_{i=h+1}^N (1-k_i^E)} [1-k_h^E] \right) {\left\langle \Upsilon, c' \right\rangle} \\ =& \left(-q^{N-k^E}[M+k^S] + q^{M+k^S}[N-k^E] \right){\left\langle \Upsilon, c' \right\rangle} \\ =& [N-M -m]{\left\langle \Upsilon, c' \right\rangle} . \end{aligned}$$ \[cor:bad-digon2\] The combinatorial $(N|M)$-evaluation satisfies the following local identities: $$\begin{aligned} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m$}}; \draw[<-] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$1$}}; \draw[->] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$m+1$}}; }}}}}\right\rangle}_{N|M}&= [N-M-m]{\left\langle \,{\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m$}}; }}}}}\! \right\rangle}_{N|M}, \\ {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m$}}; \draw[<-] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$n$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$m+n$}}; }}}}}\right\rangle}_{N|M}&= \begin{bmatrix} N-M-m \\ n \end{bmatrix} {\left\langle \,{\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m$}}; }}}}}\! \right\rangle}_{N|M} = {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m$}}; \draw[<-] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$n$}}; \draw[->] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$m+n$}}; }}}}}\right\rangle}_{N|M}. \end{aligned}$$ The first identity is a consequence of Lemma \[lem:bad-digon\] and Lemma \[prop:mirror-image\]. The second is an easy induction. \[lem:easy-square\] The combinatorial $(N|M)$-evaluation satisfies the following local identity: $$\begin{aligned} \label{eq:easy-square} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[xscale=0.65, yscale=0.55]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1.1); \coordinate (D1) at (-1,1.9); \coordinate (C2) at (1,0.9); \coordinate (D2) at (1,2.1); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$n$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$n+1 $}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$n$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$n+k$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$n+k-1$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$n+k$}}; \draw[->] (D1) -- (D2) node[midway, above] {\tiny{$1$}}; \draw[->] (C2) -- (C1) node[midway, below] {\tiny{$1$}}; }}}}}\right\rangle}_{N|M} = [k]{\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (T1) at (-1,3); \coordinate (B2) at (1,0); \coordinate (T2) at (1,3); \draw[->] (B1) -- (T1) node[midway, left] {\tiny{$n$}}; \draw[<-] (T2) -- (B2) node[midway, right] {\tiny{$n+k$}}; }}}}}\right\rangle}_{N|M} + {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[yscale=0.55, xscale=0.65]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,0.9); \coordinate (D1) at (-1,2.1); \coordinate (C2) at (1,1.1); \coordinate (D2) at (1,1.9); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$n$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$n-1$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$n$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$n+k$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$n+k+1$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$n+k$}}; \draw[->] (D2) -- (D1) node[midway, above] {\tiny{$1$}}; \draw[->] (C1) -- (C2) node[midway, below] {\tiny{$1$}}; }}}}}\right\rangle}_{N|M}. \end{aligned}$$ Let $\Gamma$, $\Upsilon$ and $\Phi$ be three MOY graphs which are identical except in a small ball $B$ where they are related by the local relation (\[eq:easy-square\]), $\Gamma$ being on the left-hand side and $\Upsilon$ and $\Phi$ on the right-hand side ($\Upsilon$ the first and $\Phi$ the second). Let $c$ be a coloring of $\Gamma$. There are two possibilities: either the two rungs receive the same color or they receive different colors. First consider a coloring $c=(\underline{\Gamma^E}, \underline{\Gamma^S})$ which gives different colors to the two rungs. There is no coloring of $\Upsilon$ which coincides with $c$ outside $B$. On the other hand, it induces a unique coloring $c''=(\underline{\Phi^E}, \underline{\Phi^S})$ of $\Phi$ which associates different colors to the two rungs of $\Phi$. One has: $$\begin{aligned} m(c) &= m(c''), && \quad\quad & s(c) &= s(c''),\\ w_\rho(c) &= w_\rho(c'') &&\quad \textrm{and}& w_s(c) &= w_s(c''). \end{aligned}$$ Hence $$\begin{aligned} \label{eq:easysquare-firstcase} {\left\langle \Gamma, c \right\rangle}_{N|M} = {\left\langle \Phi,c'' \right\rangle}_{N|M} \end{aligned}$$ Consider now the case where $c$ gives the same color to the two rungs. It induces a coloring $c'$ of $\Upsilon$. Similarly there are colorings of $\Phi$ which induce $c'$ on $\Upsilon$. Let us write $c\to c'$ to indicate that $c$ induces the coloring $c'$ on $\Upsilon$. We will show the following: $$\begin{aligned} & \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} \\ c\to c'}} {\left\langle \Gamma, c \right\rangle}_{N|M} - \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Phi)}} \\ c\to c'}} {\left\langle \Phi, c'' \right\rangle}_{N|M} = [k] {\left\langle \Upsilon, c' \right\rangle}_{N|M} \end{aligned}$$ which together with (\[eq:easysquare-firstcase\]) implies the lemma. Fix $c'= (\underline{\Upsilon^E}, \underline{\Upsilon^S})$ a coloring of $\Upsilon$. Let us denote by $e_l$ ([resp. ]{}$e_r$) the vertical edge on the left ([resp. ]{}right) of the part of $\Upsilon$ which is in $B$. We need a few extra notations. For $i \in {\ensuremath{\llbracket M \rrbracket}}$ and $j \in {\ensuremath{\llbracket N \rrbracket}}$, set: $$\begin{aligned} l_i^S &:= \ell_{\Gamma_i^S}(e_l), &\quad \quad&& r_i^S &:= \ell_{\Gamma_i^S}(e_r), &\quad \quad&& l_j^E &:= \ell_{\Gamma_j^E}(e_l), &\quad \quad&& r_j^E &:= \ell_{\Gamma_j^E}(e_r), \\ l^S &:= \sum_{i=1}^Ml_i^S, &\quad \quad&& r^S &:= \sum_{i=1}^Mr_i^S, &\quad \quad&& l^E &:= \sum_{j=1}^Nl_j^E, &\quad \quad&& r^E &:= \sum_{j=1}^Nr_j^E. \end{aligned}$$ Note that $k=r^E+ r^S- l^E-l^S$. A coloring of $\Gamma$ which induces $c'$ is totally determined by the color it gives to the two rungs. Let $c= (\underline{\Gamma^E}, \underline{\Gamma^S})$ and denote by $\Gamma^A_h$ the sub-MOY graph of $\Gamma$ which has label $1$ on the two rungs. There are two possibilities: either $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$ or $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$. In the case $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c') [l^S_h+1][r_h ^S] \\ s(c) &= s(c'),\\ w_\rho(c) &= w_\rho(c') \quad \textrm{and}\\ w_s(c) &= w_s(c') + r^E - l^E + \sum_{i=1}^{h-1} (l^S_i - r^S_i) - \sum_{i=h+1}^{M} (l^S_i - r^S_i). \end{aligned}$$ In the case $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c') [1-l^E_h][r_h ^E] \\ s(c) &= s(c'),\\ w_\rho(c) &= w_\rho(c') \quad \textrm{and}\\ w_s(c) &= w_s(c') + l^S - r^S + \sum_{i=1}^{h-1} (r^E_i - l^E_i) - \sum_{i=h+1}^{N} (r^E_i - l^E_i). \end{aligned}$$ Note that in order $c$ to exist it is necessary and sufficient that $r_h^E=1$ and $l^E_h=0$. We perform the same analysis with a coloring $c'' = (\underline{\Phi^E}, \underline{\Phi^S})$ inducing $c'$ on $\Upsilon$. Denote by $\Phi^A_h$ the sub-MOY graph of $\Phi$ which has label $1$ on the two rungs. There are two possibilities: either $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$ or $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$. In the case $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$, one has: $$\begin{aligned} m(c'') &= m(c') [l^S_h][r_h ^S+1] \\ s(c'') &= s(c'),\\ w_\rho(c'') &= w_\rho(c') \quad \textrm{and}\\ w_s(c'') &= w_s(c') + r^E - l^E + \sum_{i=1}^{h-1} (l^S_i - r^S_i) - \sum_{i=h+1}^{M} (l^S_i - r^S_i). \end{aligned}$$ In the case $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$, one has: $$\begin{aligned} m(c'') &= m(c') [l^E_h][1-r_h ^S] \\ s(c'') &= s(c'),\\ w_\rho(c'') &= w_\rho(c') \quad \textrm{and}\\ w_s(c'') &= w_s(c') + l_S - r_S + \sum_{i=1}^{h-1} (r^E_i - l^E_i) - \sum_{i=1}^{N} (r^E_i - l^E_i). \end{aligned}$$ We compute with help of Lemma \[lem:diff-prod\]: $$\begin{aligned} & \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}}\\ c\to c' }} {\left\langle \Gamma, c \right\rangle}_{N|M} - \sum_{\substack{c''\in {\ensuremath{\mathrm{col}_{{}}(\Phi)}}\\ c''\to c' }} {\left\langle \Phi, c'' \right\rangle}_{N|M} \\ &=\left(q^{r^E - l^{E}}\sum_{h=1}^M ([l^S_h+1][r_h ^S] -[l^S_h][r^S_h+1])q^{\sum_{i=1}^{h-1} (l^S_i - r^S_i) - \sum_{i=h+1}^{M} (l^S_i - r^S_i)} \right. \\ &\left. \hspace{0.5cm} + q^{l^S - r^{S}}\sum_{h=1}^N ([1-l^E_h][r_h ^E] -[l^E_h][1-r^E_h])q^{\sum_{i=1}^{h-1} (r^E_i - l^E_i) - \sum_{i=h+1}^{N} (r^E_i - l^E_i)} \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M} \\ & =\left(q^{r^E - l^{E}}\sum_{h=1}^M [r_h^S- l_h^S]q^{\sum_{i=1}^{h-1} (l^S_i - r^S_i) - \sum_{i=h+1}^{M} (l^S_i - r^S_i)} \right. \\ &\left. \hspace{2cm} + q^{l^S - r^{S}}\sum_{h=1}^N [r_h^E - l_h^E]q^{\sum_{i=1}^{h-1} (r^E_i - l^E_i) - \sum_{i=h+1}^{N} (r^E_i - l^E_i)} \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M} \\ &=\left( q^{r^E - l^{E}} [r^S - l^S] + q^{l^S - r^{S}}[r^E - l^E]\right){\left\langle \Upsilon, c' \right\rangle}_{N|M}\\ &=[k] {\left\langle \Upsilon, c' \right\rangle}_{N|M}.\qedhere\end{aligned}$$ \[cor:gen-square\] The combinatorial $(N|M)$-evaluation satisfies the following local identity and its mirror image: $${\left\langle {\ensuremath{\vcenter{\hbox{\tikz[xscale=0.65, yscale=0.55]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1.1); \coordinate (D1) at (-1,1.9); \coordinate (C2) at (1,0.9); \coordinate (D2) at (1,2.1); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$n$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$n+k $}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$m$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$m+l$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$m+l-k$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$n+l$}}; \draw[->] (D1) -- (D2) node[midway, above] {\tiny{$n+k-m$}}; \draw[->] (C2) -- (C1) node[midway, below] {\tiny{$k$}}; }}}}}\right\rangle}= \sum_{j=\max{(0,m-n)}}^l\begin{bmatrix}l \\ k-j \end{bmatrix} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[yscale=0.55, xscale=0.65]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,0.9); \coordinate (D1) at (-1,2.1); \coordinate (C2) at (1,1.1); \coordinate (D2) at (1,1.9); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$n$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$m-j$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$m$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$m+l$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$n+l+j$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$n+l$}}; \draw[->] (D2) -- (D1) node[midway, above] {\tiny{$j$}}; \draw[->] (C1) -- (C2) node[midway, below] {\tiny{$n+j-m$}}; }}}}}\right\rangle}.$$ This is an easy induction using Lemma \[lem:easy-square\] and Corollary \[cor:easy-digon2\]. \[lem:badsquare\] The combinatorial $(N|M)$-evaluation satisfies the following local identity: $$\begin{aligned} \label{eq:bad-square} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1); \coordinate (D1) at (-1,2); \coordinate (C2) at (1,1); \coordinate (D2) at (1,2); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$m$}}; \draw[-<-] (D1) -- (C1) node[midway, left ] {\tiny{$m+1$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$m$}}; \draw[->] (C2) -- (B2) node[at end, below] {\tiny{$m$}}; \draw[-<-] (C2) -- (D2) node[midway, right] {\tiny{$m+1$}}; \draw[->] (T2) -- (D2) node[at start, above] {\tiny{$m$}}; \draw[-<-] (D2) -- (D1) node[midway, above] {\tiny{$1$}}; \draw[-<-] (C1) -- (C2) node[midway, below] {\tiny{$1$}}; }}}}}\right\rangle}_{N|M} = [N-M-2m]{\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (T1) at (-1,3); \coordinate (B2) at (1,0); \coordinate (T2) at (1,3); \draw[->] (B1) -- (T1) node[midway, left] {\tiny{$m$}}; \draw[->] (T2) -- (B2) node[midway, right] {\tiny{$m$}}; }}}}}\right\rangle}_{N|M} + {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1); \coordinate (D1) at (-1,2); \coordinate (C2) at (1,1); \coordinate (D2) at (1,2); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$m$}}; \draw[-<-] (D1) -- (C1) node[midway, left ] {\tiny{$m-1$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$m$}}; \draw[->] (C2) -- (B2) node[at end, below] {\tiny{$m$}}; \draw[-<-] (C2) -- (D2) node[midway, right] {\tiny{$m-1$}}; \draw[->] (T2) -- (D2) node[at start, above] {\tiny{$m$}}; \draw[->-] (D2) -- (D1) node[midway, above] {\tiny{$1$}}; \draw[->-] (C1) -- (C2) node[midway, below] {\tiny{$1$}}; }}}}}\right\rangle}_{N|M}. \end{aligned}$$ Let $\Gamma$, $\Upsilon$ and $\Phi$ be three MOY graphs which are identical except in a small ball $B$ where they are related by the local relation (\[eq:bad-square\]), $\Gamma$ being on the left-hand side and $\Upsilon$ and $\Phi$ on the right-hand side ($\Upsilon$ the first and $\Phi$ the second). Let $c$ be a coloring of $\Gamma$. There are two possibilities: either the two rungs receive the different colors or the receive different colors. First, consider a coloring $c=(\underline{\Gamma^E}, \underline{\Gamma^S})$ which gives different colors to the two rungs. There is no coloring of $\Upsilon$ which coincides with $c$ outside $B$. On the other hand, it induces a unique coloring $c'=(\underline{\Phi^E}, \underline{\Phi^S})$ of $\Phi$ which associates different colors to the two rungs of $\Phi$. One has: $$\begin{aligned} m(c) &= m(c'), && \quad \quad & s(c) &= s(c'),\\ w_\rho(c) &= w_\rho(c') && \quad \textrm{and} & w_s(c) &= w_s(c'). \end{aligned}$$ Hence $$\begin{aligned} \label{eq:badsquare-firstcase} {\left\langle \Gamma,c \right\rangle}_{N|M}= (-1)^{s(c)}q^{w_\rho(c) + w_s(c)}m(c) = (-1)^{s(c')}q^{w_\rho(c') + w_s(c)}m(c')= {\left\langle \Phi,c' \right\rangle}_{N|M}.\end{aligned}$$ Consider now the case where $c$ gives the same color to the two rungs. It induces a coloring $c'$ of $\Upsilon$. Similarly there are colorings of $\Phi$ which induce $c'$ on $\Upsilon$. Let us write $c\to c'$ to indicate that $c$ induces the coloring $c'$ on $\Upsilon$. We will show the following: $$\begin{aligned} \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} \\ c\to c'}} {\left\langle \Gamma, c \right\rangle}_{N|M} - \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Phi)}} \\ c''\to c'}} {\left\langle \Phi, c'' \right\rangle}_{N|M} = [N-M-2m] {\left\langle \Upsilon, c' \right\rangle}_{N|M} \end{aligned}$$ which together with (\[eq:badsquare-firstcase\]) implies the lemma. Let us fix a coloring $c'= (\underline{\Upsilon^E}, \underline{\Upsilon^S})$ of $\Upsilon$. and denote by $e_l$ ([resp. ]{}$e_r$) the vertical edge on the left ([resp. ]{}right) of the part of $\Upsilon$ which is in $B$. We need a few extra notations. For $i \in {\ensuremath{\llbracket M \rrbracket}}$ and $j \in {\ensuremath{\llbracket N \rrbracket}}$, set: $$\begin{aligned} l_i^S &:= \ell_{\Gamma_i^S}(e_l), &\quad \quad&& r_i^S &:= \ell_{\Gamma_i^S}(e_r), &\quad \quad&& l_j^E &:= \ell_{\Gamma_j^E}(e_l), &\quad \quad&& r_j^E &:= \ell_{\Gamma_j^E}(e_r), \\ l^S &:= \sum_{i=1}^Ml_i^S, &\quad \quad&& r^S &:= \sum_{i=1}^Mr_i^S, &\quad \quad&& l^E &:= \sum_{j=1}^Nl_j^E, &\quad \quad&& r^E &:= \sum_{j=1}^Nr_j^E. \end{aligned}$$ A coloring of $\Gamma$ inducing $c'$ is totally determined by the color it gives to the two rungs. Let $c= (\underline{\Gamma^E}, \underline{\Gamma^S})$ and denote by $\Gamma^A_h$ the sub-MOY graph of $\Gamma$ which has label $1$ on the two rungs. There are two possibilities: either $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$ or $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$. In the case $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c') [l^S_h+1][r_h^S + 1] \\ s(c) &= s(c') -1 ,\\ w_\rho(c) &= w_\rho(c') + (N+M -2h +1) \quad \textrm{and}\\ w_s(c) &= w_s(c') - r^E - l^E - \sum_{i=1}^{h-1} (l^S_i + r^S_i) + \sum_{i=h+1}^{M} (l^S_i + r^S_i). \end{aligned}$$ In the case $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$, one has: $$\begin{aligned} m(c) &= m(c') [1-l^E_h][1-r_h ^E] \\ s(c) &= s(c'),\\ w_\rho(c) &= w_\rho(c') + (N+M -2h +1)\quad \textrm{and}\\ w_s(c) &= w_s(c') + l^S + r^S + \sum_{i=1}^{h-1} (r^E_i + l^E_i) - \sum_{i=1}^{N} (r^E_i + l^E_i). \end{aligned}$$ Note that in order $c$ to exist it is necessary and sufficient that $r_h^E=l_h^E=1$. We perform the same analysis with a coloring $c'' = (\underline{\Phi^E}, \underline{\Phi^S})$ inducing $c'$ on $\Upsilon$. Denote by $\Phi^A_h$ the sub-MOY graph of $\Phi$ which has label $1$ on the two rungs. There are two possibilities: either $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$ or $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$. In the case $A=S$ and $h \in {\ensuremath{\llbracket M \rrbracket}}$, one has: $$\begin{aligned} m(c'') &= m(c') [l^S_h][r_h ^S] \\ s(c'') &\equiv s(c') + 1,\\ w_\rho(c'') &= w_\rho(c') + (N+M-2h+1) \quad \textrm{and}\\ w_s(c'') &= w_s(c') - r^E - l^E - \sum_{i=1}^{h-1} (l^S_i + r^S_i)+ \sum_{i=h+1}^{M} (l^S_i + r^S_i). \end{aligned}$$ Note that in order $c''$ to exist it is necessary and sufficient for $r_h^S$ and $l_h^S$ to be positive. In the case $A=E$ and $h \in {\ensuremath{\llbracket N \rrbracket}}$, we have: $$\begin{aligned} m(c'') &= m(c') [l^E_h][r_h ^E] \\ s(c'') &\equiv s(c') ,\\ w_\rho(c'') &= w_\rho(c') + (N+M-2h+1)\quad \textrm{and}\\ w_s(c'') &= w_s(c') + l^S + r^S + \sum_{i=1}^{h-1} (r^E_i + l^E_i) - \sum_{i=1}^{N} (r^E_i + l^E_i). \end{aligned}$$ Hence, one has: $$\begin{aligned} & \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Gamma)}} \\ c\to c'}} {\left\langle \Gamma, c \right\rangle}_{N|M} - \sum_{\substack{c\in {\ensuremath{\mathrm{col}_{{}}(\Phi)}} \\ c''\to c'}} {\left\langle \Phi, c'' \right\rangle}_{N|M} \\ =& \left(\sum_{i=1}^M \left([l^S_h][r_h ^S] - [l^S_h+1][r_h ^S+1] \right) q^{N+M-2h +1 - r^E - l^E - \sum_{i=1}^{h-1} (l^S_i + r^S_i) + \sum_{i=h+1}^{M} (l^S_i + r^S_i)} \right. \\ &+ \left. \sum_{i=1}^N \left([1-l^E_h][1-r_h ^E] - [l^E_h][r_h ^E] \right) q^{N+M-2h +1 + r^S + l^S + \sum_{i=1}^{h-1} (l^E_i + r^E_i) - \sum_{i=h+1}^{N} (l^E_i + r^E_i)} \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M}\\ =&\left(\sum_{i=1}^M \left(-[l^S_h+r_h^S+1] \right) q^{N+M-2h +1 - r^E - l^E - \sum_{i=1}^{h-1} (l^S_i + r^S_i) + \sum_{i=h+1}^{M} (l^S_i + r^S_i)} \right. \\ &+ \left. \sum_{i=1}^N \left([1-l^E_h-r_h ^E]\right) q^{N+M-2h +1 + r^S + l^S + \sum_{i=1}^{h-1} (l^E_i + r^E_i) - \sum_{i=h+1}^{N} (l^E_i + r^E_i)} \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M}\\ =& \left(\sum_{i=1}^M \left(-[l^S_h+r_h^S+1] \right) q^{N - r^E - l^E - \sum_{i=1}^{h-1} (l^S_i + r^S_i+1) + \sum_{i=h+1}^{M} (l^S_i + r^S_i+1)} \right. \\ &+ \left. \sum_{i=1}^N \left([1-l^E_h-r_h ^E]\right) q^{M + r^S + l^S + \sum_{i=1}^{h-1} (l^E_i + r^E_i-1) - \sum_{i=h+1}^{N} (l^E_i + r^E_i-1)} \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M}\\ =& \left(-[M+ l^S + r^S] q^{N- l^E - r^E} + [N - l^E - r^E]q^{M + l^S +r^S} \right) {\left\langle \Upsilon, c' \right\rangle}_{N|M}\\ =& [N-M - 2m] {\left\langle \Upsilon, c' \right\rangle}_{N|M}.\qedhere\end{aligned}$$ Link invariants {#sec:link-invariants} =============== The aim of this section is to define link invariants using the combinatorics worked out in the previous section. The definition are really close from [@MR1659228]. The main point here is to fix normalization. \[dfn:colored-diag\] A *labeled* link diagram is an oriented link diagram whose components are labeled by non-negative integers. If $D$ is a labeled link diagram, denote by ${{\ensuremath{\vcenter{\hbox{\scalebox{1.3}{\ensuremath{\times}}}}}}}(D)$ (or simply ${{\ensuremath{\vcenter{\hbox{\scalebox{1.3}{\ensuremath{\times}}}}}}}$) the set of crossings of $D$. A diagram is *unlabeled* or *trivially labeled* if all its components are labeled by $1$. For each crossing $x$ in ${{\ensuremath{\vcenter{\hbox{\scalebox{1.3}{\ensuremath{\times}}}}}}}$, define $k_x$ and $e_x$ by the following formula: $$\begin{aligned} k_x &= \begin{cases} m(N-M-m +1) & \textrm{if $x$ is positive and the two strands have label $m$,} \\ -m(N-M-m +1) & \textrm{if $x$ is negative and the two strands have label $m$,} \\ 0 & \textrm{else;} \end{cases} \\ e_x &= \begin{cases} m & \textrm{if the two strands of $x$ have label $m$,} \\ 0 & \textrm{else.} \end{cases} \end{aligned}$$ Finally, define $k(D)$ ([resp. ]{}$e(D)$) to be the sum of the $k_x$ ([resp. ]{}$e_x$) for all $x$ in ${{\ensuremath{\vcenter{\hbox{\scalebox{1.3}{\ensuremath{\times}}}}}}}(D)$. Let $D$ be a labeled link diagram. The $(N|M)$-evaluation of $D$ is the Laurent polynomial in $q$ with integral coefficients denoted by ${\left\langle D \right\rangle}_{N|M}$ defined by the two following local relations: $$\begin{aligned} \label{eq:extcrossplus}{\left\langle \scriptstyle{{\ensuremath{\vcenter{\hbox{\tikz[scale=0.6]{\PandocStartInclude{cef_crossingplustalk.tex}\PandocEndInclude{input}{2800}{91}}}}}}} \right\rangle}_{N|M} &= \sum_{k= \max(0, m-n)}^m (-1)^{m-k}q^{k-m}{\left\langle \!\!{\ensuremath{\vcenter{\hbox{\tikz[scale=0.9]{\begin{scope} \coordinate (A) at (-1,-1); \coordinate (B) at (1,-1); \coordinate (C) at (1,1); \coordinate (D) at (-1,1); \coordinate (a) at (-.5,-.5); \coordinate (b) at (.5,-.5); \coordinate (c) at (.5,.5); \coordinate (d) at (-.5,.5); \draw[->] (A) -- (a) node[at start, below] {\tiny{$n$}}; \draw[->] (c) -- (C) node[at end, above ] {\tiny{$n$}}; \draw[->] (B) -- (b) node[at start , below ] {\tiny{$m$}}; \draw[->] (d) -- (D) node[at end, above] {\tiny{$m$}}; \draw[<-] (c) -- (d) node[midway, above] {\tiny{$n+k-m$}}; \draw[<-] (a) -- (b) node[midway, below] {\tiny{$k$}}; \draw[->] (a) -- (d) node[midway, left] {\tiny{$n+k$}}; \draw[->] (b) -- (c) node[midway, right] {\tiny{$m-k$}}; \end{scope} }}}}}\!\! \right\rangle}_{N|M},\\\label{eq:extcrossminus} {\left\langle \scriptstyle{{\ensuremath{\vcenter{\hbox{\tikz[scale=0.6]{\PandocStartInclude{cef_crossingminustalk.tex}\PandocEndInclude{input}{2820}{69}}}}}}} \right\rangle}_{N|M} &= \sum_{k= \max(0, m-n)}^m (-1)^{m-k}q^{m-k} {\left\langle \!\!{\ensuremath{\vcenter{\hbox{\tikz[scale=0.9]{\begin{scope} \coordinate (A) at (-1,-1); \coordinate (B) at (1,-1); \coordinate (C) at (1,1); \coordinate (D) at (-1,1); \coordinate (a) at (-.5,-.5); \coordinate (b) at (.5,-.5); \coordinate (c) at (.5,.5); \coordinate (d) at (-.5,.5); \draw[->] (A) -- (a) node[at start, below] {\tiny{$n$}}; \draw[->] (c) -- (C) node[at end, above ] {\tiny{$n$}}; \draw[->] (B) -- (b) node[at start , below ] {\tiny{$m$}}; \draw[->] (d) -- (D) node[at end, above] {\tiny{$m$}}; \draw[<-] (c) -- (d) node[midway, above] {\tiny{$n+k-m$}}; \draw[<-] (a) -- (b) node[midway, below] {\tiny{$k$}}; \draw[->] (a) -- (d) node[midway, left] {\tiny{$n+k$}}; \draw[->] (b) -- (c) node[midway, right] {\tiny{$m-k$}}; \end{scope} }}}}}\!\! \right\rangle}_{N|M}.\end{aligned}$$ In order to compute ${\left\langle D \right\rangle}_{N|M}$ one first expresses the diagram $D$ as a linear combination of MOY graphs and then uses the definition of ${\left\langle \bullet \right\rangle}_{N|M}$ for graphs given in Definition \[dfn:weight\]. \[thm:link-invariant\] For any non-negative integers $M$ and $N$, the Laurent polynomial $P_{N|M}(D) :=(-1)^{e(D)}q^{k(D)}{\left\langle D \right\rangle}_{N|M}$ depends only on the oriented labeled link represented by $D$. It is enough to check invariance under Reidemeister moves. This follows from the various identities satisfied by the $(N|M)$-evaluation given in Section \[sec:colorings-moy-graphs\]. The way to deduce invariance from these identities is given in [@MR1659228]. From Corollary \[cor:depends-N-M\], one deduces that for any link $L$, the polynomial $P_{N|M}(L)$ depends only on $L$ and $N-M$. The polynomial $P_{N|M}$ satisfies the following skein relation: $$q^{M-N} P_{N|M}\left( {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B2) -- (T1) node[pos=0.8, left] {$1$}; \fill[white] (0,0) circle (2mm); \draw [->] (B1) -- (T2) node[pos=0.8, right] {$1$}; }}}}}\right) - q^{N-M} P_{N|M}\left( {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B1) -- (T2) node[pos=0.8, right] {$1$}; \fill[white] (0,0) circle (2mm); \draw [->] (B2) -- (T1) node[pos=0.8, left] {$1$}; }}}}}\right) = (q^{-1}-q) P_{N|M}\left( {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B1) .. controls +( 0.3, 0.3) and +( 0.3, -0.3) .. (T1) node[pos=0.5, left] {$1$}; \draw [->] (B2) .. controls +(-0.3, 0.3) and +(-0.3, -0.3) .. (T2) node[pos=0.5, right] {$1$}; }}}}}\right).$$ This follows from the local definition of ${\left\langle \bullet \right\rangle}_{N|M}$ on unlabeled crossings: simpler $$\begin{aligned} {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B2) -- (T1) node[pos=0.8, left] {$1$}; \fill[white] (0,0) circle (2mm); \draw [->] (B1) -- (T2) node[pos=0.8, right] {$1$}; }}}}}\right\rangle}_{N|M} &= {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (MB) at ( 0,-0.5); \coordinate (MT) at ( 0, 0.5); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->-] (B1) -- (MB) node [font=\tiny, below, midway] {$1$}; \draw [->-] (B2) -- (MB) node [font=\tiny, below, midway] {$1$}; \draw [->-] (MB) -- (MT) node [font=\tiny, right, midway] {$2$}; \draw [-<-] (T1) -- (MT) node [font=\tiny, above, midway] {$1$}; \draw [-<-] (T2) -- (MT) node [font=\tiny, above, midway] {$1$}; }}}}}\right\rangle}_{N|M} -q^{-1}{\left\langle {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B1) .. controls +( 0.3, 0.3) and +( 0.3, -0.3) .. (T1) node[pos=0.5, left] {$1$}; \draw [->] (B2) .. controls +(-0.3, 0.3) and +(-0.3, -0.3) .. (T2) node[pos=0.5, right] {$1$}; }}}}}\right\rangle}_{N|M}, \\ {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B1) -- (T2) node[pos=0.8, right] {$1$}; \fill[white] (0,0) circle (2mm); \draw [->] (B2) -- (T1) node[pos=0.8, left] {$1$}; }}}}}\right\rangle}_{N|M} &= {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (MB) at ( 0,-0.5); \coordinate (MT) at ( 0, 0.5); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->-] (B1) -- (MB) node [font=\tiny, below, midway] {$1$}; \draw [->-] (B2) -- (MB) node [font=\tiny, below, midway] {$1$}; \draw [->-] (MB) -- (MT) node [font=\tiny, right, midway] {$2$}; \draw [-<-] (T1) -- (MT) node [font=\tiny, above, midway] {$1$}; \draw [-<-] (T2) -- (MT) node [font=\tiny, above, midway] {$1$}; }}}}}\right\rangle}_{N|M} - q{\left\langle {\ensuremath{\vcenter{\hbox{\tikz[font=\tiny, scale=0.4]{ \coordinate (B1) at (-1,-1); \coordinate (B2) at ( 1,-1); \coordinate (T1) at (-1, 1); \coordinate (T2) at ( 1, 1); \draw [->] (B1) .. controls +( 0.3, 0.3) and +( 0.3, -0.3) .. (T1) node[pos=0.5, left] {$1$}; \draw [->] (B2) .. controls +(-0.3, 0.3) and +(-0.3, -0.3) .. (T2) node[pos=0.5, right] {$1$}; }}}}}\right\rangle}_{N|M} . \end{aligned}$$ Taking in account the contribution of $k(D)$ and $(-1)^{e(D)}$ in the definition of $P_{N|M}$ gives the identity. The connection between $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-link invariants for links labeled by an arbitrary partition $\lambda$ and generic versions of them known as colored HOMFLY-PT polynomials was explored by Queffelec and Sartori [@2015arXiv150603329Q]. We refer the reader to this paper for a detailed account. In particular, they show that such invariants depend only on $N-M$ and the partition $\lambda$. The present paper provides an alternative proof for this well-known fact for partitions which are rows. Non semi-simple invariants {#sec:non-semi-simple} ========================== In this part we suppose that $1\leq M\leq N$. Let us consider a link $L$ and suppose that one of its component has label $n$ with $n >N-M$. One can show that in this case the ${\ensuremath{\mathfrak{gl}}}_{N|M}$-invariant is equal to $0$ (see for instance [@2015arXiv150603329Q]. This follows from the fact that the ${\ensuremath{\mathfrak{gl}}}_{N|M}$-invariant of the unknot labeled $n$ is equal to $\left[\begin{smallmatrix} M-N \\n\end{smallmatrix}\right]$ which happens to be $0$. The aim of this part is to directly re-normalize the ${\ensuremath{\mathfrak{gl}}}_{N|M}$ invariant in this case in order to get a non-trivial invariant. The theory of renormalized invariants is nowadays well developed but we refer to one the early paper by Geer and Patureau-Mirand which treats the case we are looking at [@MR2640994] and also to the treatment by Queffelec and Sartori [@2015arXiv150603329Q]. The interested reader could also consult the paper by Geer, Patureau-Mirand and Turaev [@MR2480500]. Here, we focus on the case $M=1$ and $n=N$. Incidentally, the re-normalized ${\ensuremath{\mathfrak{gl}}}_{1|1}$-invariant for links uniformly colored by $1$ equals the (one-variable) Alexander polynomial. Hence, this construction can be thought of a generalization of the Alexander polynomial. Let uspoint out that these invariants do not coincide with the colored Alexander polynomials (or ADO invariants) [@MR1164114]. \[dfn:markedMOY\] A *marked MOY graph $\Gamma_\star$* is a MOY graph $\Gamma$ with a base point $\star$ in the interior of one of its edges. If the marked point is on an edge of label $k$, we say that $\Gamma_\star$ has *type $k$.* \[dfn:eval-marked\] Let $\Gamma_\star$ be a marked MOY graph and denote $e$ the edge of $\Gamma$ which contains the base point. A *$(N|1)$-coloring* of $\Gamma_\star$ is a $(N|1)$-coloring $(\underline{\Gamma^E},\underline{\Gamma^S})$ of the underlying $\Gamma$ such that for all $j$ in ${\ensuremath{\llbracket N \rrbracket}}$, $\ell_{\Gamma^E_j}(e)=0$. The set of $(N|1)$-coloring of $\Gamma_\star$ is denoted by ${\ensuremath{\mathrm{col}_{N|1}(\Gamma_\star)}}$ (or simply ${\ensuremath{\mathrm{col}_{{}}(\Gamma_\star)}}$). If $\Gamma_\star$ is a marked MOY graph and $c$ is a $(N|1)$-coloring of $\Gamma_\star$, define ${\left\langle \Gamma_\star, c \right\rangle}_{N|1}:= {\left\langle \Gamma, c \right\rangle}_{N|1}$, and: $${\left\langle \Gamma_\star \right\rangle}_{N|1} = \sum_{c\in {\ensuremath{\mathrm{col}_{N|1}(\Gamma_\star)}}} {\left\langle \Gamma_\star,c \right\rangle}_{N|1}.$$ 1. In the previous definition, $\underline{\Gamma^S}$ contains exactly one sub-MOY graph: $\Gamma^S_1$. One has necessarily $\ell_{\Gamma^S_1}(e) = \ell_{\Gamma}(e)$. 2. Suppose that $\Gamma_\star$ is a marked circle of label $k$ with $k\geq N$, and denote by $\Gamma$ the underlying not marked circle. One has: $${\left\langle \Gamma_\star \right\rangle}_{N|1} = 1 \quad \textrm{and} \quad {\left\langle \Gamma \right\rangle}_{N|1} = 0.$$ The results of the previous section extend naturally: \[prop:marked-far-skein-relations\] The local relations given in Proposition \[prop:rel-kups\] are still valid for the evaluation of marked MOY graphs (provided the marked point is not in the ball where the relations take place). \[dfn:1n-invariant\] Let $\beta$ be a braid diagram, define ${\left\langle \beta_\star \right\rangle}_{N|1}$ by the following procedure: 1. Label all strands of $\beta$ by $N$. 2. Place a marked point on the left-most strand at the bottom of $\beta$. 3. Close $\beta$ on the right. 4. Use formulas (\[eq:extcrossplus\]) and (\[eq:extcrossminus\]) to get rid of crossings. 5. Evaluate the obtained marked MOY graphs with ${\left\langle \bullet \right\rangle}_{N|1}$ as given in Definition \[dfn:eval-marked\]. As in Section \[sec:link-invariants\], ${\left\langle \bullet \right\rangle}_{N|1}$ needs to be normalized to obtain a link invariant. Define $Q_{N|1}(\beta) = (-1)^{N|\beta|}{\left\langle \bullet \right\rangle}_{N|1}$, where $|\beta|$ is the number of crossings of $\beta$. Besides the appearances, the normalization used for $Q_{N|1}$ is the same as for $P_{N|1}$ (see Definition \[dfn:colored-diag\]) but is especially simple for the present choice of labels. \[prop:invariance-1n\] The Laurent polynomial $Q_{N|1}(\beta)$ depends only on the link represented by $\beta$. \[not:1n-link\] If a link $L$ can be obtained by closing a braid $\beta$, set $Q_{N|1}(L):= Q_{N|1}(\beta)$. The invariant $Q_{N|1}$ vanishes on split links. We need to prove invariance under - Braid relations. - Markov moves. Invariance under braid relations and stabilization follows from the general setting see [@MR1659228]. Invariance under conjugation by $\sigma_i$ for $i\geq 2$ is obvious. Hence the only thing to show is invariance under conjugation by $\sigma_1$. On the closure of $\beta$ it only changes the base point in the following way: $${\ensuremath{\vcenter{\hbox{\tikz[scale=0.6]{\begin{scope} \begin{scope} \draw[->] (-1,-1) -- (+1,+1) node[pos = 0.25, red] {${\star}$}; \fill[white] (0,0) circle (2mm); \draw[->] (+1,-1) -- (-1,+1); \end{scope} \node at (3, 0) {$\leftrightsquigarrow$}; \begin{scope}[xshift = 6cm] \draw[->] (-1,-1) -- (+1,+1); \fill[white] (0,0) circle (2mm); \draw[->] (+1,-1) -- (-1,+1) node[pos = 0.75, red] {${\star}$}; \end{scope} \end{scope} }}}}}$$ It is enough to show that $${\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.8]{\begin{scope}[scale=0.5] \begin{scope} \draw[->] (-1,-1) .. controls +(0, 0.5) and +(0, -0.5) .. (+1,+1) node[pos = 0.25, red] {${\star}$}; \fill[white] (0,0) circle (2mm); \draw[->] (+1,-1) .. controls +(0, 0.5) and +(0, -0.5) .. (-1,+1); \draw [densely dotted] (1.5,-1) -- (4.5, -1) -- (4.5, +1) -- (1.5, 1) --cycle; \node at (3,0) {$\Gamma$}; \draw (-1, 1) arc ( 180:0:2.5 and 0.75); \draw ( 1, 1) arc ( 180:0:0.5); \draw (-1, -1) arc (-180:0:2.5 and 0.75); \draw ( 1, -1) arc (-180:0:0.5); \end{scope} \end{scope} }}}}} \right\rangle}_{N|1}= {\left\langle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.8]{\begin{scope}[scale=0.5] \begin{scope} \draw[->] (-1,-1) .. controls +(0, 0.5) and +(0, -0.5) .. (+1,+1); \fill[white] (0,0) circle (2mm); \draw[->] (+1,-1) .. controls +(0, 0.5) and +(0, -0.5) .. (-1,+1) node[pos = 0.75, red] {${\star}$}; \draw [densely dotted] (1.5,-1) -- (4.5, -1) -- (4.5, +1) -- (1.5, 1) --cycle; \node at (3,0) {$\Gamma$}; \draw (-1, 1) arc ( 180:0:2.5 and 0.75); \draw ( 1, 1) arc ( 180:0:0.5); \draw (-1, -1) arc (-180:0:2.5 and 0.75); \draw ( 1, -1) arc (-180:0:0.5); \end{scope} \end{scope} }}}}} \right\rangle}_{N|1}$$ for any MOY graph $\Gamma$ with four ends. One can actually suppose (see [@2017arXiv171103333S Section 2.3.1]) that $\Gamma$ is equal to a linear combination of MOY graph of the form $${\ensuremath{\vcenter{\hbox{\tikz[scale=1]{\begin{scope}[font= \tiny] \draw[-<] (0, -2) -- +(0,2) node[pos= 0.5,left] {$N+j$} node[pos= 0,left] {$N$} node[pos= 1,left] {$N$} ; \draw[-<] (1, -2) -- +(0,2) node[pos= 0.5,right] {$N-j$} node[pos= 0,right] {$N$} node[pos= 1,right] {$N$}; \draw[-<-] (1,-1.5) -- +(-1, 0.25) node[midway, above] {$j$}; \draw[-<-] (0,-0.75) -- +(1, 0.25) node[midway, above] {$j$}; \end{scope} }}}}}$$ with $0\leq j\leq N$. Using relation (\[eq:extcrossplus\]), one can express ${\left\langle \beta_1 \right\rangle}$ and ${\left\langle \beta_2 \right\rangle}$ as: $$\begin{aligned} {\left\langle \beta_1 \right\rangle}_{N|1} &= \sum_{i=0}^n \sum_{j=0}^n \lambda_{ij} {\left\langle \Gamma_\star^{ij} \right\rangle}_{N|1} \quad \textrm{and} \\ {\left\langle \beta_2 \right\rangle}_{N|1} &= \sum_{i=0}^n \sum_{j=0}^n \lambda_{ij} {\left\langle \Phi_\star^{ij} \right\rangle}_{N|1} \end{aligned}$$ Where $$\Gamma_{\star}^{ij} = {\ensuremath{\vcenter{\hbox{\tikz[scale= 0.6]{\begin{scope}[font= \tiny] \draw[->-] (0, -2) -- +(0,4) node[pos= 0.2,left] {$N+i$} node[pos= 0.5,left] {$N$} node[pos= 0.8,left] {$N-j$} arc(180:0:2) -- +(0, -4) arc (0:-180:2); \draw[->-] (1, -2) -- +(0,4) node[pos= 0.2,right] {$N-i$} node[pos= 0.5,right] {$N$} node[pos= 0.8,right] {$N+j$} arc(180:0:1) -- +(0, -4) arc (0:-180:1); \draw[->-] (1,-2) -- +(-1, 0.25) node[midway, above] {$i$}; \draw[->-] (0,-0.75) -- +(1, 0.25) node[midway, above] {$i$}; \draw[->-] (0, 0.5) -- +(1, 0.25) node[midway, above] {$j$}; \draw[->-] (1, 1.75) -- +(-1, 0.25) node[midway, above] {$j$}; \node[red, font=\normalsize] at (0, -2) {$\star$}; \end{scope} }}}}} \qquad \text{and} \qquad \Phi_{\star}^{ij} = {\ensuremath{\vcenter{\hbox{\tikz[scale= 0.6]{\begin{scope}[font= \tiny] \draw[->-] (0, -2) -- +(0,4) node[pos= 0.2,left] {$N+i$} node[pos= 0.5,left] {$N$} node[pos= 0.8,left] {$N-j$} arc(180:0:2) -- +(0, -4) arc (0:-180:2); \draw[->-] (1, -2) -- +(0,4) node[pos= 0.2,right] {$N-i$} node[pos= 0.5,right] {$N$} node[pos= 0.8,right] {$N+j$} arc(180:0:1) -- +(0, -4) arc (0:-180:1); \draw[->-] (1,-2) -- +(-1, 0.25) node[midway, above] {$i$}; \draw[->-] (0,-0.75) -- +(1, 0.25) node[midway, above] {$i$}; \draw[->-] (0, 0.5) -- +(1, 0.25) node[midway, above] {$j$}; \draw[->-] (1, 1.75) -- +(-1, 0.25) node[midway, above] {$j$}; \node[red, font=\normalsize] at (0, 0) {$\star$}; \end{scope} }}}}}.$$ Using the skein relations, one can compute ${\left\langle \Gamma^{ij}_\star \right\rangle}_{N|1}$ and ${\left\langle \Phi_\star^{ij} \right\rangle}_{N|1}$: $$\begin{aligned} {\left\langle \Gamma_\star^{ij} \right\rangle}_{N|1} &= \begin{bmatrix} i+j \\ i \end{bmatrix} \begin{bmatrix} N-1 -(i+j) \\ N-i \end{bmatrix} \begin{bmatrix} N-1 -N \\ i \end{bmatrix} \begin{bmatrix} N \\ j \end{bmatrix}, \\ {\left\langle \Phi_\star^{ij} \right\rangle}_{N|1} &= \begin{bmatrix} i+j \\ i \end{bmatrix} \begin{bmatrix} N-1 -(i+j) \\ N-j \end{bmatrix} \begin{bmatrix} N-1 -N \\ j \end{bmatrix} \begin{bmatrix} N \\ i \end{bmatrix}.\end{aligned}$$ These two products of quantum binomials are equal. Hence ${\left\langle \beta_1 \right\rangle}_{N|1} = {\left\langle \beta_2 \right\rangle}_{N|1}$ and therefore $Q(\beta_1)_{N|1}= Q(\beta_2)_{N|1}$. \[thm:kashaev\] For any $N \in {\ensuremath{\mathbb{Z}}}_{\geq 0}$ and any link $L$, one has: $$Q_{N|1}(L,q= e^{\frac{i\pi}{N+1}}) = J'_N(\overline{L}, q=e^{\frac{i\pi}{N+1}}),$$ where $\overline{L}$ denotes the mirror image of $L$. Note that the sequence $\left(J'_n(\overline{L}, q=e^{\frac{i\pi}{n+1}})\right)_{n \in {\ensuremath{\mathbb{Z}}}_{>0}}$ appears in the formulation of the volume conjecture [@MR1341338; @MR1828373]. For proving this theorem we need a technical result about quantum binomials evaluated at root of unity. \[lem:qbin-root\] Let $N$, $a$ and $b$ be three integers. The following identity holds: $$\begin{bmatrix} N-1 - a \\ b \end{bmatrix}_{q= e^{\frac{i\pi}{N+1}}}= (-1)^b\begin{bmatrix} -2- a \\ b \end{bmatrix}_{q= e^{\frac{i\pi}{N+1}}}$$ We assume here that $q = e^{\frac{i\pi}{N+1}}$. $$\begin{aligned} \begin{bmatrix} N-1 - a \\ b \end{bmatrix} &=\left(\prod_{i=1}^b\frac{q^{N-1-a+i} - q^{-N-a+1-i}}{q^{i} - q^{-i}} \right) \\ &=\left(\prod_{i=1}^b\frac{-q^{-2-a +i} + q^{-2-a-i}}{q^{i} - q^{-i}} \right) \\ & = (-1)^b \begin{bmatrix} -2- a \\ b \end{bmatrix}. \qedhere \end{aligned}$$ Let us fix a braid diagram $\beta$ representing $L$ and denote $k$ the braid index of $\beta$. Once every strand of $\beta$ is labeled by $N$, the level of $\beta$ is $kN$. We will compare ${\left\langle \beta_\star \right\rangle}_{N|1}$ and $\frac{{\left\langle \beta \right\rangle}_{0|2}}{[-N]}$ evaluated at $q= e^{\frac{i\pi}{N+1}}$. The expansion of crossings is the same in both cases. Hence it is enough to compare $$\left.{\left\langle \Gamma_\star \right\rangle}_{N|1}\right|_{q= e^{\frac{i\pi}{N+1}}} \quad \text{and} \quad \left.\frac{{\left\langle \Gamma \right\rangle}_{0|2}}{[-N]}\right|_{q= e^{\frac{i\pi}{N+1}}}$$ for $\Gamma_\star$ a marked MOY graph appearing in the expansion of $\beta_\star$. Both these quantities can be computed using the relations in Proposition \[prop:rel-kups\] for $(N|1)$ and $(0|2)$ and the fact that ${\left\langle U_\star \right\rangle}_{N|1}= \frac{{\left\langle U \right\rangle}_{0|2}}{[-N]} =1$ for $U$ a circle of label $N$. More precisely, one has: $$\begin{aligned} {\left\langle \Gamma_\star \right\rangle}_{N|1} &= R_\Gamma(N-1, q){\left\langle U_\star \right\rangle}_{N|1} \quad \text{and} \quad \frac{{\left\langle \Gamma \right\rangle}_{0|2}}{[-N]} = R_\Gamma(-2,q) \frac{{\left\langle U \right\rangle}_{0|2}}{[-N]}, \end{aligned}$$ for a $R_\Gamma(n, q)$ a sum of product of quantum binomials of the form $\begin{bmatrix} a_i \\ b_i \end{bmatrix}$ and $\begin{bmatrix} n- c_j \\ d_j \end{bmatrix}$. The first binomials correspond to relations (\[eq:extrelbin1\]), (\[eq:extrelsquare2\]) and (\[eq:extrelsquare3\]) which preserve the level, the second correspond to relations (\[eq:extrelcircle\]) (\[eq:extrelbin2\]) (\[eq:extrelsquare1\]) which do not preserve the level. Note that a binomial $\begin{bmatrix}n- c_j \\ d_j \end{bmatrix}$ appears exactly when the level decreases by $d_j$. Since to go from $\Gamma$ to $U$, the level is decreases by $(k-1)N$, for each product of binomials in $R_\Gamma(n, q)$, the sum of the $d_j$’s equals $(k-1)N$. This gives, in view of Lemma \[lem:qbin-root\]: $$R_\Gamma\left(N-1,e^{\frac{i\pi}{N+1}}\right)=(-1)^{(k-1)N}R_\Gamma\left(-2, e^{\frac{i\pi}{N+1}}\right)$$ and therefore $${\left\langle \beta_\star, q= e^{\frac{i\pi}{N+1}} \right\rangle}_{N|1} = (-1)^{(k-1)N}\frac{{\left\langle \beta, q= e^{\frac{i\pi}{N+1}} \right\rangle}_{0|2}}{[-N]_{q= e^{\frac{i\pi}{N+1}}}}.$$ Since all strands are labeled by $N$, one has: $$\begin{aligned} P_{0|2}(L) &= (-1)^{N(c_+ + c_-)} q^{N(N+1)(c_- - c_+)} {\left\langle \beta \right\rangle}_{0|2} \quad \text{and} \\ Q_{N|1}(L) &= (-1)^{N(c_+ + c_-)} {\left\langle \beta_\star \right\rangle}_{N|1},\end{aligned}$$ where $c_+$ and $c_-$ are the number of positive and negative crossings of $\beta$ respectively. Hence, for $q = \exp(\frac{i\pi}{N+1})$, $$Q_{N|1}(L) = (-1)^{N(k-1 +c_- -c_+)} \frac{P_{0|2}(L)}{[-N]} = (-1)^{N(k +c_- -c_+)} \frac{P_{0|2}(L)}{[N]} .$$ It turns out that $k +c_--c_+$ has the same parity as the number $\ell$ of components of $L$. This gives: $$Q_{N|1}(L,q= e^{\frac{i\pi}{N+1}})=\left.(-1)^{N\ell} \frac{P_{0|2}(L)}{[N]}\right|_{q= e^{\frac{i\pi}{N+1}}} = J'_N(\overline{L},q= e^{\frac{i\pi}{N+1}}).\qedhere$$ The previous result can also be obtained in an indirect way from the work of Geer and Patureau-Mirand [@MR2468374]. In this paper, they prove that the generalized Links–Gould invariants specialize to the Kashaev invariants. These invariants are constructed using typical representations $V(\alpha)$ of $U_q({\ensuremath{\mathfrak{gl}}}_{N|1})$ depending on a complex parameter $\alpha$. For $\alpha=-1$, these are exactly the representations considered in the present paper and this specialization $\alpha=-1$ is compatible with the ones providing the Kashaev invariants. MOY graphs, an algebraic approach {#sec:moy-graphs-an} ================================= For a general introduction to the Reshetikhin–Turaev functors, we refer to Turaev’s book [@MR1292673]. For more details on the super setting, we refer to Geer and Patureau-Mirand [@MR2640994] and references therein. The later paper can also be consulted for the renormalization process (see also [@2015arXiv150603329Q]). Here, we follow and expand the diagrammatic presentation of Tubbenhauer–Vaz–Wedrich [@tubbenhauer2015super]. For the reader convenience, we made explicit the various pieces of the Reshetikhin–Turaev functor. All proofs are omitted since they are all direct verifications. In what follows, a *super* algebra is presented. It means that it is endowed with a ${\ensuremath{\mathbb{Z}}}/2$-grading and more importantly, its modules are objects of ${\ensuremath{\mathsf{Svect}}}$ the category of super vector spaces. Objects of this category are ${\ensuremath{\mathbb{Z}}}/2$-graded vector spaces, morphisms are linear maps preserving the ${\ensuremath{\mathbb{Z}}}/2$ grading. The ${\ensuremath{\mathbb{Z}}}/2$-grading, called *parity*, is denoted $|\!\bullet\!|$. Homogeneous elements with parity equal to $0$ ([resp. ]{}$1$) are *even* ([resp. ]{}*odd*). This category inherits from ${\ensuremath{\mathsf{vect}}}$ a monoidal structure and a duality. The braiding $c$ on ${\ensuremath{\mathsf{Svect}}}$ differs from that of ${\ensuremath{\mathsf{vect}}}$: $$\begin{array}{crcl} c_{V,W} \colon\thinspace & V\otimes W & \to & W\otimes V \\ & v\otimes w &\mapsto & (-1)^{|v||w|}w\otimes v. \end{array}$$ If $V$ is an object of ${\ensuremath{\mathsf{Svect}}}$ which has finite dimension as a vector space. Its *super dimension* ${\mathop{\mathrm{sdim}}\nolimits}V$ is the integer $\dim V_0 - \dim V_1$, where $V= V_0 \oplus V_1$ is the ${\ensuremath{\mathbb{Z}}}/2$-decomposition[^2]. The *classical dimension* of $V$ is its dimension as a vector space. Let $N$ and $M$ be two non-negative integers. The *quantum general linear superalgebra* $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ is the associative, unital, ${\ensuremath{\mathbb{Z}}}/2$-graded ${\ensuremath{\mathbb{C}}}(q)$-algebra generated by $L_i$, $L_i^{-1}$, $F_j$ and $E_j$, with $1\leq i \leq N+M$ and $1\leq j \leq N+M-1$ subject to the nonsuper relations $$\begin{gathered} L_iL_j = L_jL_i, \qquad L_iL_i^{-1}= L_i^{-1}L_i= 1, \\ L_iE_i = qE_iL_i, \qquad L_{i}E_{i-1} = q^{-1}E_{i-1}L_{i}, \qquad \text{for $i\leq N$},\\ L_iF_i = q^{-1}F_iL_i, \qquad L_{i}F_{i-1} = qF_{i-1}L_{i}, \qquad \text{for $i\leq N$},\\ L_iE_i = q^{-1}E_iL_i, \qquad L_{i}E_{i-1} = qE_{i-1}L_{i}, \qquad \text{for $i\geq N+1$},\\ L_iF_i = qF_iL_i, \qquad L_{i}F_{i-1} = q^{-1}F_{i-1}L_{i}, \qquad \text{for $i\geq N+1$},\\ L_iF_j = F_jL_i, \qquad L_iE_j = E_jL_i \qquad \text{for $j\neq i, i-1$,} \\ E_iF_j - F_jE_i = \delta_{ij}\frac{L_iL_{i+1}^{-1} - L_i^{-1}L_{i+1}}{q-q^{-1}}, \text{for $1\leq i\leq N-1$,} \\ E_iF_j - F_jE_i = -\delta_{ij}\frac{L_iL_{i+1}^{-1} - L_i^{-1}L_{i+1}}{q-q^{-1}}, \text{for $N+1\leq i <N+M-1$,} \\ [2]_qF_i F_j F_i = F_i^2F_j + F_j F_i^2 \qquad \text{if $|i − j| = 1$ and $i\neq N$,} \\ [2]_qE_i E_j E_i = E_i^2 E_j + E_j E_i^2 \qquad \text{if $|i − j| = 1$ and $i\neq N$,} \\ E_i E_j = E_j E_i, \qquad F_iF_j = F_jF_i \qquad \text{if $|i − j| > 1$}\end{gathered}$$ and the super relations $$\begin{aligned} & E_N^2= F_N^2 =0 \qquad E_NF_N + F_NE_N =\frac{L_NL_{N+1}^{-1} - L_N^{-1}L_{N+1}}{q-q^{-1}}\\ &[2]F_{N}F_{N-1}F_{N+1}F_{N}=\\ &F_{N}F_{N-1}F_{N}F_{N+1} + F_{N}F_{N+1}F_{N}F_{N-1} + F_{N-1}F_{N}F_{N+1}F_{N}+F_{N+1}F_{N}F_{N-1}F_{N}, \\ &[2]E_{N}E_{N-1}E_{N+1}E_{N}=\\ &E_{N}E_{N-1}E_{N}E_{N+1} + E_{N}E_{N+1}E_{N}E_{N-1} + E_{N-1}E_{N}E_{N+1}E_{N}+E_{N+1}E_{N}E_{N-1}E_{N}. \end{aligned}$$ All these generators are ${\ensuremath{\mathbb{Z}}}/2$-homogeneous and even, but $E_N$ and $F_N$ which are ${\ensuremath{\mathbb{Z}}}/2$-homogeneous and odd. Defining $\Delta:U_q({\ensuremath{\mathfrak{gl}}}_{N|M})\to U_q({\ensuremath{\mathfrak{gl}}}_{N|M})^{\otimes 2}$, $S:U_q({\ensuremath{\mathfrak{gl}}}_{N|M})^{\mathrm{op}} \to U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ and $\epsilon: U_q({\ensuremath{\mathfrak{gl}}}_{N|M})\to {\ensuremath{\mathbb{C}}}(q)$ to be the ${\ensuremath{\mathbb{C}}}(q)$ algebra maps defined by: $$\begin{aligned} &\Delta(L_i^{\pm 1}) = L_i^{\pm1}\otimes L_i^{\pm1} & \quad &S(L_i^{\pm1})= L_i^{\mp1} & \quad & \epsilon(L_i^{\pm 1}) =1 & \\ &\Delta(F_i) = F_i\otimes 1 + L_i^{-1}L_{i+1}\otimes F_i & \quad &S(F_i) = - L_iL_{i+1}^{-1}F_i & \quad & \epsilon(F_i)=0 & \\ &\Delta(E_i)= E_i\otimes 1 + L_iL_{i+1}^{-1}\otimes E_i &\quad & S(E_i) = - E_iL_i^{-1}L_{i+1} & \quad &\epsilon(E_i)=0& \end{aligned}$$ endows $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ with a structure of ${\ensuremath{\mathbb{Z}}}/2$-graded Hopf algebra with antipode. Furthermore the category of finite-dimensional $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-modules is braided. \[rmk:swapMN\] The Hopf algebras $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ and $U_q({\ensuremath{\mathfrak{gl}}}_{M|N})$ are isomorphic. Indeed, one easily checks that the map $$\begin{array}{crcl} \varphi :& U_q({\ensuremath{\mathfrak{gl}}}_{N|M}) &\to & U_q({\ensuremath{\mathfrak{gl}}}_{M|N})\\ & L_i &\mapsto &L_{M+N+1-i}^{- 1} \\ & L_i^{-1} &\mapsto &L_{M+N+1-i} \\ & E_i &\mapsto &F_{M+N-i} \\ & F_i &\mapsto &e_{M+N-i} \end{array}$$ induces an isomorphism of Hopf algebras. Let ${\ensuremath{\mathbb{C}}}_q^{N|M}$ be the ${\ensuremath{\mathbb{Z}}}/2$-graded ${\ensuremath{\mathbb{C}}}(q)$-vector space generated by the homogeneous basis $(b_i)_{i=1,\dots, N+M}$ (with $|b_i|=0$ if $i\leq N$ and $|b_i|=1$ if $i> N$). The formulas $$\begin{aligned} &L_i b_i = qb_i,& &L_i^{-1} b_{i} =q^{-1}b_{i},& &\text{if }1\leq i\leq N, \\ &L_j b_j = q^{-1}b_j,& &L_j^{-1} b_{j} =qb_{j},& &\text{if }N+1\leq j \leq N +M, \\ &L_i^{\pm 1} b_j = b_j && &&\text{if }i \neq j, &\\ &E_{i-1} b_i = b_{i-1}& &E_i b_{j} =0,& &\textrm{if $i\neq j-1$,} & \\ &F_i b_i = b_{i+1}& &F_i b_{j} =0,& &\textrm{if $i\neq j$}& \end{aligned}$$ endow ${\ensuremath{\mathbb{C}}}^{N|M}_q$ with a structure of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-module. Since $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$ is isomorphic to $U_q({\ensuremath{\mathfrak{gl}}}_{M|N})$, ${\ensuremath{\mathbb{C}}}_q^{M|N}$ inherits a structure of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-module. Denote $\boldsymbol{1}_{\mathrm{odd}}$ the trivial[^3] one-dimensional $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-module concentrated in odd parity and $1_{\mathrm{odd}}$ a generator of this module. The $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-modules ${\ensuremath{\mathbb{C}}}_q^{M|N}$ and $\boldsymbol{1}_{\mathrm{odd}}\otimes {\ensuremath{\mathbb{C}}}_q^{N|M}$ are isomorphic. Indeed, one can check that $$\begin{array}{crcl} \psi :& {\ensuremath{\mathbb{C}}}_q^{M|N} &\to & {\ensuremath{\mathbb{C}}}_q^{N|M} \otimes \boldsymbol{1}_{\mathrm{odd}} \\ & b_i &\mapsto &b_{N+M-i+1}\otimes 1_{\mathrm{odd}} \end{array}$$ induces an isomorphism of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-modules. Denote $T{\ensuremath{\mathbb{C}}}^{N|M}_q$ the ${\ensuremath{\mathbb{Z}}}/2\times {\ensuremath{\mathbb{Z}}}$-graded algebra generated by ${\ensuremath{\mathbb{C}}}^{N|M}_q$ and $\mathrm{Sym_q^2}{\ensuremath{\mathbb{C}}}^{N|M}_q$ the ideal generated by the set $$\{ b_i \otimes b_i | 1\leq i \leq N \} \cup \{ q^{-1}b_i \otimes b_j + (-1)^{|b_i||b_j|} b_j \otimes b_i | 1\leq i<j\leq N+M \}.$$ Denote $\Lambda_q {\ensuremath{\mathbb{C}}}^{N|M}_q$ the space $T{\ensuremath{\mathbb{C}}}^{N|M}_q\left/ \mathrm{Sym}_q^2{\ensuremath{\mathbb{C}}}^{N|M}_q\right.$ For $k \in {\ensuremath{\mathbb{Z}}}_{\geq 0}$, denote $\Lambda^k_q {\ensuremath{\mathbb{C}}}^{N|M}_q$ the $k$-degree (for the ${\ensuremath{\mathbb{Z}}}$-grading) part of $\Lambda_q {\ensuremath{\mathbb{C}}}^{N|M}_q$. For all $k$, $\Lambda^k_q {\ensuremath{\mathbb{C}}}^{N|M}_q$ inherits a $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-module structure. Its super dimension over ${\ensuremath{\mathbb{C}}}(q)$ is $ \begin{pmatrix} N-M \\k \end{pmatrix}$. Its classical dimension is $$\sum_{i=0}^k \begin{pmatrix} N \\k-i \end{pmatrix} \begin{pmatrix} M + i -1 \\ i \end{pmatrix}.$$ In what follows, we define some morphisms in the category of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-modules. For this, we need to introduce a few notations. The image of a pure tensor $x_1\otimes \dots \otimes x_k$ of $T {\ensuremath{\mathbb{C}}}_q^{N|M}$ in $\Lambda^k{\ensuremath{\mathbb{C}}}_q^{N|M}$ is denoted by $x_1\wedge \dots \wedge x_k$. In particular one has: $$\begin{aligned} &b_i\wedge b_i =0 \quad\text{if $i\leq N$,}\\ &b_i\wedge b_j= -(-1)^{|b_i||b_j|}qb_j\wedge b_i \quad \text{for $i<j$.}\end{aligned}$$ The ${\ensuremath{\mathbb{C}}}(q)$-vector space $\Lambda^k_q{\ensuremath{\mathbb{C}}}_q^{N|M}$ is spanned by the vectors $$\left(b_{i_1}\wedge \dots \wedge b_{i_{k_1}}\wedge b_{j_1}\wedge \dots \wedge b_{j_{k_2}} \right) _{\substack{ 1\leq i_1< \dots < i_{k^E}\leq N\\ N+1\leq j_1\leq \dots \leq j_{k^S}\leq N+M\\ k = k^E + k^S }}.$$ If $1\leq i_1< \dots < i_{k^E}\leq N$ and $N+1 \leq j_1 \leq \dots \leq j_{k^S} \leq N+M$, denote $b_{I,J} = b_{i_1}\wedge \dots \wedge b_{i_{k^E}}\wedge b_{j_1}\wedge \dots \wedge b_{j_{k^S}} $, where $I= \{i_1, \dots, i_{k^E}\}$ and $J = \{j_1, \dots, j_{k^S}\}$. Note that $I$ is a subset of $\{1, \dots, N\}$ while $J$ is a multi-subset of $\{N+1, \dots, N+M\}$. With these notations, $(b_{I,J})_{\#I + \# J = k}$ is an homogeneous basis of $\Lambda^k{\ensuremath{\mathbb{C}}}^{N|M}_q$. Denote $(b^{I,J})_{\#I + \# J = k}$ its dual basis and define the following morphisms[^4]: $$\begin{aligned} &\begin{array}{crcl} \Lambda_{k,\ell} :& \Lambda_q^k V_q\otimes \Lambda_q^\ell V_q &\to & \Lambda^{k+\ell}_q V_q \\ & b_{I_1, J_1} \otimes b_{I_2, J_2} & \mapsto & \begin{cases} q^{-\left|(I_2\cup J_2)< (I_1 \cup J_1) \right|} b_{I_1\sqcup I_2, J_1 \sqcup J_2} & \textrm{if $I_1\cap I_2 =\emptyset$,} \\ 0 & \textrm{else.} \end{cases} \end{array} \\ &\begin{array}{crcl} Y_{k,\ell} :& \Lambda^{k+\ell}_q V_q &\to & \Lambda_q^k V_q\otimes \Lambda_q^\ell V_q \\ & b_{I,J} &\mapsto & \displaystyle{\sum_{\substack{I_1\sqcup I_2 = I, \, J_1 \sqcup J_2 = J \\ \#I_1 + \#J_1 = k, \, \# I_2 + \# J_2 = \ell }} [J_1, J_2]q^{\left|(I_2\cup J_2)< (I_1 \cup J_1) \right|} b_{I_1, J_1}\otimes b_{I_2,J_2}} \end{array} \\ &\begin{array}{crcl} \stackrel{\leftarrow}{\cup}_{k} :& {\ensuremath{\mathbb{C}}}(q) &\to & \Lambda_q^a V_q\otimes (\Lambda_q^k V_q)^* \\ & 1 &\mapsto & \displaystyle{\sum_{\#I=k} b_{I,J}\otimes b^{I,J}} \end{array} \\ &\begin{array}{crcl} \stackrel{\leftarrow}{\cap}_{k} :& (\Lambda_q^k V_q)^*\otimes \Lambda_q^k V_q &\to & {\ensuremath{\mathbb{C}}}(q) \\ & f\otimes x &\mapsto & f(x) \end{array} \\ &\begin{array}{crcl} \stackrel{\rightarrow}{\cup}_{k} :& {\ensuremath{\mathbb{C}}}(q) &\to & (\Lambda_q^k V_q)^*\otimes \Lambda_q^k V_q \\ & 1 &\mapsto & \displaystyle{\sum_{\substack{k= k^E+ k^S\\ \#I=k^E,\, \#J= k^S}}{q^{{\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket M+N \rrbracket}}}(J)}} - {\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket M+N \rrbracket}}}(I)}}}}{(-1)^{k^S}}b^{I,J}\otimes b_{I,J}} \end{array} \\ &\begin{array}{crcl} \stackrel{\rightarrow}{\cap}_{k} :& \Lambda_q^k V_q\otimes( \Lambda_q^k V_q)^* &\to & {\ensuremath{\mathbb{C}}}(q) \\ & b_{I_1, J_1} \otimes b^{I_2, J_2} &\mapsto & \displaystyle{{q^{{\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket M+N \rrbracket}}}(I_1)}} - {\ensuremath{\mathrm{deg}_{{\ensuremath{\llbracket M+N \rrbracket}}}(J_1)}}}} {(-1)^{|J_1|}}\delta_{I_1,I_2}\delta_{J_1, J_2}} \end{array}\end{aligned}$$ We should explain what $|I<J|$ and $[I,J]$ mean. If $I$ and $J$ are two multi-subsets of an ordered set $X$, define $$\begin{aligned} &|I<J|= \prod_{x<y \in X}I(x)J(y) \quad \text{and} \\ &[I,J]=\prod_{x\in X} {\ensuremath \begin{bmatrix} I(x) + J(x) \\ I(x) \quad J(x) \end{bmatrix} }.\end{aligned}$$ These maps are morphisms of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-modules. Any MOY graph $\Gamma$ is isotopic to a MOY graph $\Gamma'$ *in good position*, this means that $\Gamma'$ can be obtained by vertically concatenating horizontal disjoint unions of the following elementary pieces: $${\ensuremath{\vcenter{\hbox{\tikz[scale=0.9]{\begin{scope}[font=\small] \begin{scope}[xshift= -0.5cm] \draw[->-] (0, -0.5) -- +(0, 1) node [midway, left] {$k$}; \end{scope} \begin{scope}[xshift=0.5cm] \draw[-<-] (0, -0.5) -- +(0, 1) node [midway, left] {$\ell$}; \end{scope} \begin{scope}[xshift=2cm] \draw[-<-] (-0.5, -0.25) arc (180:0:0.5cm) node [midway, above] {$k$}; \end{scope} \begin{scope}[xshift=4cm] \draw[->-] (-0.5, -0.25) arc (180:0:0.5cm) node [midway, above] {$\ell$}; \end{scope} \begin{scope}[xshift=6cm] \draw[-<-] (-0.5, 0.25) arc (-180:0:0.5cm) node [midway, below] {$k$}; \end{scope} \begin{scope}[xshift=8cm] \draw[->-] (-0.5, 0.25) arc (-180:0:0.5cm) node [midway, below] {$\ell$}; \end{scope} \begin{scope}[xshift=10cm] \draw[->] (0,0) -- (0,0.5) node [at end, above] {$k+\ell$}; \draw[>-] (-0.5, -0.5) -- (0,0) node [at start, below] {$k$}; \draw[>-] (+0.5, -0.5) -- (0,0) node [at start, below] {$\ell$}; \end{scope} \begin{scope}[xshift = 12cm, rotate= 180] \draw[-<] (0,0) -- (0,0.5) node [at end, below] {$k+\ell$}; \draw[<-] (-0.5, -0.5) -- (0,0) node [at start, above] {$\ell$}; \draw[<-] (+0.5, -0.5) -- (0,0) node [at start, above] {$k$}; \end{scope} \end{scope} }}}}}.$$ When a MOY graph $\Gamma'$ is in good position, interpreting these elementary pieces using the morphisms $${\mathrm{Id}}_{\Lambda^k_q{\ensuremath{\mathbb{C}}}_q^{N|M}},\,\, {\mathrm{Id}}_{(\Lambda^\ell_q{\ensuremath{\mathbb{C}}}_q^{N|M})^*},\,\, \stackrel{\leftarrow}{\cap}_{k},\,\, \stackrel{\rightarrow}{\cap}_{\ell},\,\, \stackrel{\leftarrow}{\cup}_{k},\,\, \stackrel{\rightarrow}{\cup}_{\ell},\,\, \Lambda_{k,\ell},\,\, Y_{k,\ell}$$ horizontal disjoint unions as tensor products and vertical concatenations as compositions, one obtains a morphism of $U_q({\ensuremath{\mathfrak{gl}}}_{N|M})$-modules ${\left\llangle \Gamma' \right\rrangle}_{N|M}:{\ensuremath{\mathbb{C}}}(q) \to {\ensuremath{\mathbb{C}}}(q)$. One sees ${\left\llangle \Gamma' \right\rrangle}_{N|M}$ as an element of ${\ensuremath{\mathbb{C}}}(q)$. Using the definition of the morphisms associated with the elementary pieces, one can show that ${\left\llangle \Gamma' \right\rrangle}_{N|M}$ is a Laurent polynomial in $q$ with integer coefficients and symmetric in $q$ and $q^{-1}$ If $\Gamma$ is an arbitrary MOY graph, define ${\left\llangle \Gamma \right\rrangle}_{N|M}$ to be ${\left\llangle \Gamma' \right\rrangle}_{N|M}$ for $\Gamma'$ a MOY graph in good position and isotopic to $\Gamma$. \[prop:RT-welldefined\] The Laurent polynomial ${\left\llangle \Gamma \right\rrangle}_{N|M}$ is well-defined [i. e. ]{}it does not depend on the MOY graph in good position and isotopic to $\Gamma$ chosen to compute it. \[prop:rel-kups\] The map ${\left\llangle \bullet \right\rrangle}_{N|M}: \{\text{MOY graphs} \} \to {\ensuremath{\mathbb{Z}}}[q, q^{-1}]$ is multiplicative under disjointe union [i. e. ]{} $$\begin{aligned} {\left\llangle \Gamma \sqcup \Upsilon \right\rrangle}_{N|M}={\left\llangle \Gamma \right\rrangle}_{N|M}{\left\llangle \Upsilon \right\rrangle}_{N|M} \quad \text{for all MOY graphs } \Gamma\mbox{ and } \Upsilon. \end{aligned}$$ and satisfies the following the local relations and their mirror images: $$\begin{aligned} \label{eq:extrelcircle} {\left\llangle \vcenter{\hbox{\tikz[scale= 0.5]{\draw[->] (0,0) arc(0:360:1cm) node[right] {\tiny{$\!k\!$}};}}} \right\rrangle}_{N|M}= \begin{bmatrix} N-M \\ k \end{bmatrix}\end{aligned}$$ $$\begin{aligned} \label{eq:extrelass} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.3]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,1); \coordinate (V2) at (1,2); \coordinate (T1) at (-2,3); \coordinate (T2) at (0,3); \coordinate (T3) at (2,3); \draw[>-] (B) -- (V1) node [at start, below] {\tiny{$i+j+k$}}; \draw[->] (V1) -- (T1) node [at end, above] {\tiny{$i$}}; \draw[->] (V1) -- (V2) node[midway, right] {\tiny{$j+k$}}; \draw[->] (V2) -- (T2) node[at end, above] {\tiny{$j$}}; \draw[->] (V2) -- (T3) node[at end, above] {\tiny{$k$}}; }}}}}\right\rrangle}_{N|M} = {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.3]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,1); \coordinate (V2) at (-1,2); \coordinate (T1) at (-2,3); \coordinate (T2) at (0,3); \coordinate (T3) at (2,3); \draw[>-] (B) -- (V1) node [at start, below] {\tiny{$i+j+k$}}; \draw[->] (V1) -- (T3) node [at end, above] {\tiny{$k$}}; \draw[->] (V1) -- (V2) node[midway, left] {\tiny{$i+j$}}; \draw[->] (V2) -- (T1) node[at end, above] {\tiny{$i$}}; \draw[->] (V2) -- (T2) node[at end, above] {\tiny{$j$}}; }}}}}\right\rrangle}_{N|M} \end{aligned}$$ $$\begin{aligned} \label{eq:extrelbin1} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m+n$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m+n$}}; \draw[->] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$n$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$m$}}; }}}}}\right\rrangle}_{N|M} = \arraycolsep=2.5pt \begin{bmatrix} m+n \\ m \end{bmatrix} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m+n$}}; }}}}}\right\rrangle}_{N|M}\end{aligned}$$ $$\begin{aligned} \label{eq:extrelbin2} \arraycolsep=2.5pt {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (V1) at (0,0.5); \coordinate (V2) at (0,2.5); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (V1) node[midway, left] {\tiny{$m$}}; \draw[->] (V2) -- (T) node[midway, left] {\tiny{$m$}}; \draw[<-] (V1) .. controls +(+0.5, +0.5) and +(+0.5, -0.5).. (V2) node[midway, right] {\tiny{$n$}}; \draw[->] (V1) .. controls +(-0.5, +0.5) and +(-0.5, -0.5).. (V2) node[midway, left] {\tiny{$m+n$}}; }}}}}\right\rrangle}_{N|M} = \begin{bmatrix} N-M-m \\ n \end{bmatrix} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B) at (0,0); \coordinate (T) at (0,3); \draw[white] (0, -0.5) -- (0, 3.5); \draw[->] (B) -- (T) node[midway, right] {\tiny{$m$}}; }}}}}\right\rrangle}_{N|M} \end{aligned}$$ $$\begin{aligned} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1); \coordinate (D1) at (-1,2); \coordinate (C2) at (1,1); \coordinate (D2) at (1,2); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$1$}}; \draw[->] (D1) -- (C1) node[midway, left ] {\tiny{$m$}}; \draw[->-] (D1) -- (T1) node[at end , above ] {\tiny{$1$}}; \draw[->] (C2) -- (B2) node[at end, below] {\tiny{$m$}}; \draw[->-] (C2) -- (D2) node[midway, right] {\tiny{$1$}}; \draw[->] (T2) -- (D2) node[at start, above] {\tiny{$m$}}; \draw[->-] (D2) -- (D1) node[midway, above] {\tiny{$m+1$}}; \draw[->-] (C1) -- (C2) node[midway, below] {\tiny{$m+1$}}; }}}}}\right\rrangle}_{N|M} = {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (T1) at (-1,3); \coordinate (B2) at (1,0); \coordinate (T2) at (1,3); \draw[->] (B1) -- (T1) node[midway, left] {\tiny{$1$}}; \draw[->] (T2) -- (B2) node[midway, right] {\tiny{$m$}}; }}}}}\right\rrangle}_{N|M} + [N-M-m-1]{\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (T1) at (-1,3); \coordinate (C) at (0,1); \coordinate (D) at (0,2); \coordinate (B2) at (1,0); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C) node[at start, below] {\tiny{$1$}}; \draw[->] (C) -- (B2) node[at end, below] {\tiny{$m$}}; \draw[->] (D) -- (C) node[midway, left] {\tiny{$m-1$}}; \draw[->] (T2) -- (D) node[at start, above] {\tiny{$m$}}; \draw[->] (D) -- (T1) node[at end, above] {\tiny{$1$}}; }}}}}\right\rrangle}_{N|M} \label{eq:extrelsquare1}\end{aligned}$$ $$\begin{aligned} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.55]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1); \coordinate (D1) at (-1,2); \coordinate (C2) at (1,1); \coordinate (D2) at (1,2); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$1$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$l+n$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$l$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$m+l-1$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$m-n$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$m$}}; \draw[->] (D1) -- (D2) node[midway, above] {\tiny{$n$}}; \draw[->] (C2) -- (C1) node[midway, below] {\tiny{$l+n-1$}}; }}}}}\right\rrangle}_{N|M}=\! \begin{bmatrix} m-1 \\ n \end{bmatrix} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (T1) at (-1,3); \coordinate (M1) at (-1,1.5); \coordinate (M2) at (1,1.5); \coordinate (B2) at (1,0); \coordinate (T2) at (1,3); \draw[->] (B1) -- (M1) node[at start, below] {\tiny{$1$}}; \draw[->] (B2) -- (M2) node[at start, below] {\tiny{$m+l-1$}}; \draw[->] (M2) -- (M1) node[midway, above] {\tiny{$l-1$}}; \draw[->] (M2) -- (T2) node[at end, above] {\tiny{$m$}}; \draw[->] (M1) -- (T1) node[at end, above] {\tiny{$l$}}; }}}}}\right\rrangle}_{N|M} + \!\begin{bmatrix} m-1 \\n-1 \end{bmatrix} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[scale=0.4]{ \coordinate (B1) at (-1,0); \coordinate (T1) at (-1,3); \coordinate (C) at (0,1); \coordinate (D) at (0,2); \coordinate (B2) at (1,0); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C) node[at start, below] {\tiny{$1$}}; \draw[<-] (C) -- (B2) node[at end, below] {\tiny{$m+l-1$}}; \draw[<-] (D) -- (C) node[midway, left] {\tiny{$l+m$}}; \draw[<-] (T2) -- (D) node[at start, above] {\tiny{$m$}}; \draw[->] (D) -- (T1) node[at end, above] {\tiny{$l$}}; }}}}}\right\rrangle}_{N|M} \label{eq:extrelsquare2}\end{aligned}$$ $$\begin{aligned} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[xscale=0.65, yscale=0.55]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,1.1); \coordinate (D1) at (-1,1.9); \coordinate (C2) at (1,0.9); \coordinate (D2) at (1,2.1); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$n$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$n+k $}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$m$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$m+l$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$m+l-k$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$n+l$}}; \draw[->] (D1) -- (D2) node[midway, above] {\tiny{$n+k-m$}}; \draw[->] (C2) -- (C1) node[midway, below] {\tiny{$k$}}; }}}}}\right\rrangle}_{N|M}= \sum_{j=\max{(0, m-n)}}^m\begin{bmatrix}l \\ k-j \end{bmatrix} {\left\llangle {\ensuremath{\vcenter{\hbox{\tikz[yscale=0.55, xscale=0.65]{ \coordinate (B1) at (-1,0); \coordinate (B2) at (1,0); \coordinate (C1) at (-1,0.9); \coordinate (D1) at (-1,2.1); \coordinate (C2) at (1,1.1); \coordinate (D2) at (1,1.9); \coordinate (T1) at (-1,3); \coordinate (T2) at (1,3); \draw[->] (B1) -- (C1) node[at start, below] {\tiny{$n$}}; \draw[->] (C1) -- (D1) node[midway, left ] {\tiny{$m-j$}}; \draw[->] (D1) -- (T1) node[at end , above ] {\tiny{$m$}}; \draw[->] (B2) -- (C2) node[at start, below] {\tiny{$m+l$}}; \draw[->] (C2) -- (D2) node[midway, right] {\tiny{$n+l+j$}}; \draw[->] (D2) -- (T2) node[at end, above] {\tiny{$n+l$}}; \draw[->] (D2) -- (D1) node[midway, above] {\tiny{$j$}}; \draw[->] (C1) -- (C2) node[midway, below] {\tiny{$n+j-m$}}; }}}}}\right\rrangle}_{N|M}\label{eq:extrelsquare3}\end{aligned}$$ \[prop:completeness\] The multiplicativity property and the local relations given in Proposition \[prop:rel-kups\] are enough to compute the value of ${\left\llangle \Gamma \right\rrangle}_{N|M}$ for any MOY graph $\Gamma$. From this statement we immediately get the following corollary which should be compared with [@2015arXiv150603329Q Theorem 4.7]. \[cor:depends-N-M\] The Laurent polynomial ${\left\llangle \Gamma \right\rrangle}_{N|M}$ only depends on $\Gamma$ and $N-M$. Two special cases are especially easy to compute: $N-M=\pm1$. Suppose $N-M=1$, ${\left\llangle \Gamma \right\rrangle}_{N|M}=0$ unless $\Gamma$ is a (maybe empty) collection of circles of label $1$ and ${\left\llangle \bigsqcup_i\vcenter{\hbox{\tikz[scale= 0.3]{\draw (0,0) arc(0:360:1cm) node[right] {\tiny{$\!\!1\!$}};}}} \right\rrangle}_{N|M}=1$. Suppose $N-M=-1$, then for any MOY graph $\Gamma$, the following identity holds: $${\left\llangle \Gamma \right\rrangle}_{N|M}= (-1)^{\rho(\Gamma)} b(\Gamma),$$ where $b(\Gamma)$ is given in Definition \[dfn:sym1\]. [^1]: This means that the value of the unknot is $[n+1]$. [^2]: The super dimension is actually the categorical dimension, the sign coming from the braiding on ${\ensuremath{\mathsf{Svect}}}$. [^3]: This means that the generators $L^{\pm1}_\bullet$ act by $1$ and the other generators act by $0$. [^4]: See [@RW2 Appendix A] for similar definitions in the symmetric case, [i. e. ]{}$N=0$.

--- abstract: 'Employing non-equilibrium quantum transport models, we investigate the details and operating conditions of nano-structured Peltier coolers embedded with an energy filtering barrier. Our investigations point out non-trivial aspects of Peltier cooling which include an inevitable trade-off between the cooling power and the coefficient of performance, the coefficient of performance being high at a low voltage bias and subsequently deteriorating with increasing voltage bias. We point out that there is an optimum energy barrier height for nanowire Peltier coolers at which the cooling performance is optimized. However, for bulk Peltier coolers, the cooling performance is enhanced with the height of the energy filtering barrier. Exploring further, we point out that a degradation in cooling performance with respect to bulk is inevitable as a single moded nanowire transitions to a multi-moded one. The results discussed here can provide theoretical insights for optimal design of nano Peltier coolers.' author: - Aniket Singha - Bhaskaran Muralidharan bibliography: - 'apssamp.bib' title: Performance analysis of nanostructured Peltier coolers --- Introduction ============ In the current nanotechnology era, the rise in operating temperatures of nanodevices as a result of increasing dissipated heat density has revived an interest in effective heat management and Peltier coolering. With the recent discovery of thermoelectric materials with high figures of merit [@highfom1; @highfom2; @thermoinnano; @aniket; @anifil], there has been a lot of theoretical and experimental effort in an attempt to meet the demand for high performance Peltier coolers [@snyder_thompson; @cooling_ref1; @cooling_ref2; @cooling_ref3; @cooling_ref4; @cooling_ref5; @cooling_ref6; @cooling_ref7; @cooling_ref8; @cooling_ref9; @cooling_ref10; @cooling_ref11]. Peltier cooling is facilitated by an energy selective disturbance in quasi-equilibrium among the electronic population via energy filtering. Such a disturabance in quasi-equilibrium, in conjugation with inelastic processes, initiates heat absorption from the lattice [@snyder_thompson; @cooling_ref2; @cooling_ref5; @cooling_ref6; @cooling_ref9; @cooling_ref10; @cooling_ref11; @whitney2; @whitney]. Despite attempts towards the theoretical and experimental realization of high performance Peltier coolers [@snyder_thompson; @cooling_ref1; @cooling_ref2; @cooling_ref3; @cooling_ref4; @cooling_ref5; @cooling_ref6; @cooling_ref7; @cooling_ref8; @cooling_ref9; @cooling_ref10; @cooling_ref11], an overall analysis of the functionality and optimum operating conditions of nano Peltier coolers is missing in the current literature. In this paper, we hence study the performance of various nanoscale Peltier coolers in order to analyze the optimum operating conditions.\ There are two major pathways to facilitate an overall performance improvement in Peltier coolers: (i) decreasing the lattice heat conductivity (ii) enhancing the cooling power via suitable energy filtering techniques. In the last few decades, approaches towards nano-structuring, hetero-structuring and density of states engineering have so far proven successful in the suppression of phonon mediated lattice thermal conductivity via scattering and confinement of long wavelength phonons [@phonon1; @phonon2; @phonon3; @phonon4; @phonon5; @phonon6; @phonon7; @superlattice1; @superlattice2; @nanoflake_heat; @nanowire_heat1; @nanowire_heat2]. Hence, in this paper, we explore the other aspect, that is, enhancing the cooling power in nanostructures. In particular, we explore Peltier cooling in nanostructures embedded with an energy filtering barrier.\ In the aspect of Peltier cooling, we believe that a few points deserve special attention. First of all, the few recent theoretical works [@ieeecool; @snyder_thompson; @cooling_ref1; @cooling_ref2; @cooling_ref6] in this aspect are based on a linear response analysis. Linear response analysis masks the essential combination of transport physics and scattering events that jointly determine the net cooling power as well as the coefficient of performance ($COP$). In addition, the linear response limit is broken in the regions which strongly deviate from quasi equilibrium, namely in the vicinity of the energy filtering barrier. Secondly, a generalized picture of the physics of cooling performance is unclear from the available literature. With respect to the first point, our analysis of cooling performance is based on the non-equilibrium Green’s function formalism which accounts for the non-equilibrium nature of transport to directly evaluate the charge and heat currents.\ This paper is organized as follows. First, we briefly elaborate the underlying physics of Peltier cooling in Sec. \[explain\] following which we briefly describe the transport formulation in Sec. \[transport\]. We next elaborate our study on Peltier cooling in nanostructures in Sec. \[results\] where we mainly analyze the *cooling power* and the *COP* at a chosen cooling power. We show that the cooling power increases while the $COP$ decreases as one increases the applied bias voltage depicting a trade-off between the the two. Exploring further, we demonstrate that a deterioration in cooling performance with respect to bulk is unavoidable as a single-moded nanowire transitions to a multi-moded regime. We end the paper with a general conclusion in Sec. \[conclude\]. The transport formalism used for the simulations is detailed in the Appendix at the end of this paper. ![Schematic diagram showing the phenomenon of Peltier cooling. The electrons at the source side of the barrier tend to equilibriate by absorbing heat from the lattice while those at the drain side of the barrier tend to equilibriate by releasing heat to the lattice.[]{data-label="fig:cooling_schematic"}](one.eps) Peltier cooling in semiconductor heterostructures {#explain} ================================================= Thermoelectric or Peltier cooling in semiconductor heterostructures is facilitated by giving rise to an energy selective lack of quasi-equilibrium among the electronic population via energy filtering. In the classical limit, when a potential is applied across the barrier, electrons above the energy barrier height $E_b$ tend to migrate from the source contact to the drain contact giving rise to a local non-equilibrium among the high energy electronic population. The electronic population below the energy barrier height $E_b$, however, remains in quasi-equilibrium with the respective contacts. This energy-selective lack of quasi-equilibrium initiates a heat absorption process from the lattice via inelastic scattering. In the case of a Peltier refrigerator embedded between macroscopic contacts, two equivalent phenomena give rise to cooling and heating at the two interfaces of the energy filtering barrier as demonstrated in Fig. \[fig:cooling\_schematic\] (a). (i) The high energy electrons at the source side of the barrier interface are driven out of equilibrium initiating heat absorption in that region. (ii) The high energy electrons migrating towards the drain side are out of equilibrium with the drain quasi-Fermi potential due to the external voltage bias ($\mu_D=\mu_S-V$). Hence, the electrons energetically relax giving up heat energy to the lattice. The efficacy of a Peltier refrigerator is measured by the $COP$ ($\zeta$) defined as: $$\zeta=J_C/P, \label{eq:COP}$$ where, $J_C$ is the rate of electronic heat extraction from the source side of the barrier interface, or equivalently, the rate of Peltier cooling and $P$ is the power consumed from the external source given by, $$P=V\times I,$$ $V$ being the applied bias and $I$ being the current flowing through the Peltier refrigerator. The rate of heat extracted from the source side of the barrier interface ($J_C$) as well as the the $COP$ ($\zeta$), depend on the position of the equilibrium Fermi potential ($\mu_0$) or equivalently, the electrochemical potential with respect to the height of the energy filtering barrier. Transport formulation and model {#transport} ================================ Transport formulation --------------------- To perform the calculations, we employ the NEGF transport formalism with inelastic scattering incorporated via the self-consistent Born approximation [@dattabook; @Datta_Green; @LNE] (details given in the Appendix). The single particle Green’s function $G(\overrightarrow{k_{m}},E)$, for each transverse sub-band $m$ [@dattabook], can be calculated from the device Hamiltonian $[H]$: $$\begin{gathered} \label{eq:negf_main} G(\overrightarrow{k_{m}},E)=[EI-H-U-E_m-\Sigma(\overrightarrow{k_{m}},E)]^{-1} ,\nonumber \\ \Sigma(\overrightarrow{k_{m}},E)=\Sigma_L(\overrightarrow{k_{m}},E)+\Sigma_R(\overrightarrow{k_{m}},E)+\Sigma_S(\overrightarrow{k_{m}},E), \end{gathered}$$ where $[H]$ is the device Hamiltonian matrix constructed with effective mass approach [@dattabook; @Datta_Green] and $I$ is the identity matrix of identical order as the Hamiltonian. The spatial profile of the conduction band minimum is described by the matrix $U$, while $E$ is the free variable representing the energy of electronic wavefunction. The sub-band energy of the $m^{th}$ sub-band is calculated assuming parabolic $E-k$ dispersion relation: $$E_m=\frac{\hslash^2k_m^2}{2m_t}.$$ The wavevector of the electron in the transverse direction for the $m^{th}$ sub-band is denoted by $\overrightarrow{k_{m}}$. The total scattering self-energy matrix $[\Sigma(\overrightarrow{k_{m}},E)]$ incorporates the effect of scattering of the electronic wavefunctions from the contacts into the active device region, which is represented by $\Sigma_L(\overrightarrow{k_{m}},E)+\Sigma_R(\overrightarrow{k_{m}},E)$ as well as the scattering of electronic wavefunctions inside the device due to inelastic processes, which is denoted by $\Sigma_S(\overrightarrow{k_{m}},E) $ (detailed in the Appendix). The scattering functions are calculated self-consistently with the transport calculations, (detailed in the Appendix), with the electron and the hole density operators $G^n(\overrightarrow{k_{m}},E)$, $G^p(\overrightarrow{k_{m}},E)$ given by $$\begin{aligned} G^n(\overrightarrow{k_{m}},E)=G(\overrightarrow{k_{m}},E)\Sigma^{in}(\overrightarrow{k_{m}},E)G^{\dagger}(\overrightarrow{k_{m}},E), \nonumber \\ G^p(\overrightarrow{k_{m}},E)=G(\overrightarrow{k_{m}},E)\Sigma^{out}(\overrightarrow{k_{m}},E)G^{\dagger}(\overrightarrow{k_{m}},E). \end{aligned}$$ On the convergence of the self-consistent calculations, the charge and heat currents propagating from the $j^{th}$ lattice point to the $(j+1)^{th}$ lattice point are computed as: $$\begin{gathered} I^{j\rightarrow j+1}_C =\underset{k_m}{\sum}i\frac{e}{\pi \hslash} \int[ G^n_{j+1,j}(\overrightarrow{k_{m}},E)H_{j,j+1}(E) \\ -H_{j+1,j}(E)G^n_{j,j+1}(\overrightarrow{k_{m}},E) ]dE, \nonumber \end{gathered}$$ $$\begin{gathered} I_Q^{j\rightarrow j+1} =\underset{k_m}{\sum}\frac{i}{\pi \hslash} \times \int E[ G^n_{j+1,j}(\overrightarrow{k_{m}},E) H_{j,j+1}(E) \\ -H_{j+1,j}(E)G^n_{j,j+1}(\overrightarrow{k_{m}},E) ]dE, \label{eq:heatcurrentnegf}\end{gathered}$$ where $M_{i,j}$, in the above set of equations, denotes a generic matrix element of the operator $M$ between two lattice points $i$ and $j$. In the nearest neighbour tight-binding approximation used here, we only consider the next nearest neighbor such that $j=i \pm 1$. The cooling power per unit volume ($\frac{1}{A} \frac{dJ_C}{dz}$) at the $j^{th}$ point along the transport direction is then calculated from the equation: $$\frac{1}{A}\frac{dJ_C}{dz}\Bigg|_j=\frac{1}{A}\frac{dI_Q}{dz}\Bigg|_j=\frac{1}{A}\frac{I_Q^{j\rightarrow j+1}-I_Q^{j-1\rightarrow j}}{a},$$ $a$ being the lattice constant used for simulation and $A$ is the cross-sectional area of the Peltier refrigerator. The total cooling power per unit area at the source side of the barrier interface is the given by: $$\frac{J_C}{A}=\frac{1}{A} \int \frac{dJ_C}{dz} \theta \left(\frac{dJ_C}{dz}\right)dz,$$ where $\theta(\iota)$ is the unit step function with argument $\iota$. Model ----- We perform a detailed analysis of Peltier cooling in nano-wires and bulk. The device structures considered here include nanowires whose transverse extent include only one sub-band and bulk whose transverse extent is infinite (schematic shown in Fig. \[fig:cooling\_schematic\] (b) and (c) respectively). We analyze the cooling power vs. $COP$ for a range of values of the reduced Fermi energy given by: $$\eta_f=\frac{E_c+E_b-\mu}{k_BT}, \label{eq:reduced_mu}$$ where $\mu$ is the equilibrium Fermi potential and $E_b$ is the height of the energy barrier. For the purpose of simulation, we use the parameters of $\Delta_2$ valley of lightly doped silicon [@book1], the longitudinal effective mass being $m_l=m_e$ and the transverse effective mass being $m_t=0.2m_e$ ($m_e$ being the free electron mass). The inelastic processes considered here are assumed to be local with an energy exchange given by $\hslash \omega=30meV$. The temperature of the entire device is considered to be $T=300K$. Under normal conditions, the device region at the source side of the barrier interface would be cooled while the same in close proximity to the drain side of the barrier interface would be heated. However, we assume that the difference between the maximum and the minimum temperature in the Peltier refrigerator is small compared to the average temperature and hence an assumption of constant temperature throughout the entire device is justified. For simplicity, the contacts are assumed to be reflection-less macroscopic bodies with electronic distribution in equilibrium at temperature $T$. Their respective quasi-Fermi potential, labeled as $\mu_S$ and $\mu_D$ respectively, are assumed to be $\mu_{S(D)}=\mu_0 \pm V/2$, where $V$ is the externally applied bias voltage. For the purpose of simulation, the devices are assumed to be embedded with a Gaussian potential barrier of the form: $$U=E_bexp\left[-\frac{(z-z_0)^2}{2\sigma _w^2}\right],$$ where $E_b$ and $\sigma_w$ define the energy filtering barrier height and width respectively and $z_0=L/2$ is the mid-point of the device, $L$ being the total length of the refrigerator in between the contacts.\  ![Maximum cooling power at a given voltage for (a) nanowire Peltier refrigerator (b) bulk Peltier refrigerator. Operating line characteristics (maximum $COP$ for a given cooling power ) for (c) nanowire Peltier coolers (d) bulk Peltier coolers. The coolers are assumed to be $27nm$ in length with an embedded Gaussian energy barrier ($\sigma_w=2.7nm$)[]{data-label="fig:op_line"}](op_line) ![The change in maximum cooling power density ($J_C^{MAX}$) and $COP$ at the maximum cooling power density as a single-moded nanowire makes transition towards the bulk regime.[]{data-label="fig:transit"}](scaling_effect) ![The density of modes per unit area of a multimoded nanowire ($16.2nm \times 16.2nm$) and bulk Peltier refrigerator. The density of modes per unit area in the multi-moded regime is less compared to the bulk regime resulting in a deterioration of the maximum cooling power.[]{data-label="fig:modal"}](modal_density) Although a more rigorous method would be to solve the potential profile along the device using information on position dependent doping concentration and hetero-junction band-offsets, we believe that our model captures the essential physics and hence, the trends noted in the simulation results would not deviate drastically with the shape of energy filtering barrier used for simulation. This is because Peltier cooling is dependent on the efficacy of electronic filtering as well as the rate inelastic scattering and not on the absolute nature of the energy filtering barrier being used. A list of parameters used for the simulation of the NEGF equations are given in Tab. I.\ **Parameters** **Values** ------------------------------ ---------------------------------- $T~~(k_BT)$ $300K~~(25.85~meV)$ Length of device $27nm$ $m_l$ $m_e$ $m_t$ $0.2m_e$ $D_O$ (Appendix) $0.1F~eV^2$ $a$ (lattice constant) $2.7 {\text{\normalfont\oldAA}}$ $E_c$ (conduction band-edge) $0eV$ $\hslash \omega$ $30meV$ : Parameters used for simulation in this chapter. [***Note:*** $F=\frac{1}{N_xN_y}$, where $N_x$ and $N_y$ are the number of lattice points in the $x$ and $y$ directions. $m_e$ is the free electron mass and $D_O$ is related to the acoustic deformation potential (See Appendix). ]{} Results ======= **Peltier cooling with inelastic scattering in nanowires:** We first explore Peltier cooling in a single-moded nanowire embedded with an energy filtering barrier. We plot in Fig. \[fig:energy\_current\] the spatially resolved average energy of the electronic current flowing through the nanowire Peltier refrigerator at a low voltage bias. The average energy is high near the energy barrier interface due to the absorption of lattice heat energy. The spatial cooling profile of a nanowire Peltier refrigerator is shown in Fig. \[fig:cooling\_comparable\]. Particularly, Fig. \[fig:cooling\_comparable\] (a) and (b) demonstrate the spatial cooling profile ($\frac{1}{A} \frac{dJ_C}{dz}$) when the applied voltage biases are low and high respectively compared to $k_BT$.\ When the applied voltage $V<<k_BT$, the amount of heat extracted from the source side is almost identical to the amount of heat dissipated at the drain side of the barrier interface (Fig. \[fig:cooling\_comparable\] a) resulting a high $COP$. The operating point in such a case is near the reversible limit. However, the net cooling power under such conditions is low. On the other hand, when the applied bias is high compared to $k_BT$, the heat extracted from the source side of the barrier interface is much less compared to the heat dissipated at the drain side of the barrier interface (Fig. \[fig:cooling\_comparable\] b) resulting in a strong deviation from the reversible regime. The $COP$ in such a case is low. Fig. \[fig:nanowire\] demonstrates the cooling characteristics of a nanowire Peltier cooler.\ In particular, Figs. \[fig:nanowire\] (a-d) depict the cooling power vs. applied bias characteristics while Figs. \[fig:nanowire\] (e-h) depict the cooling power vs. the $COP$ characteristics for the nanowire cooler for various heights of the energy filtering barrier. We note that the maximum cooling power increases with an increase in the height of the energy barrier upto a saturation point. Such a saturation occurs approximately beyond $E_b=150meV$. The optimized $COP$ at a given cooling power, on the other hand, is achieved when the height of the embedded energy filtering barrier is approximately $150meV$. The optimum position of the Fermi energy in such a case is given by $\eta_f=2$. We hence conclude that there is a particular height of the energy filtering barrier at which the performance of the nanowire Peltier cooler is optimized.\ Two competing phenomena can be responsible for such a behaviour as the energy barrier height is increased: (a) an increase in the charge current due to a decrease in the electronic scattering rate as a result of direct tunneling, and (b) a decrease in the rate of heat absorption per electron from the lattice due to the decrease in the scattering rate. These two competing phenomena result in a saturation of the cooling power beyond $E_b=150meV$. The slight decrease in the $COP$ at a given cooling power with the increase in barrier height beyond $E_b=150meV$ occurs due to a saturation in the cooling power despite an increase in the electronic current. We also note from Figs. \[fig:nanowire\] (e-h) that the $COP$ decreases with an increase in cooling power indicating a trade-off between the two. The deterioration in cooling power with the increase in the applied potential bias beyond a certain limit occurs as a result of lowering of the energy filtering barrier due to the external bias voltage. Such a lowering of the potential barrier causes an increase in the direct electronic tunneling rate.\ **Operating lines:** In the context of Peltier coolers, we define operating line as the locus of points in the $J_C-\zeta$ space where the maximum $\zeta$ is obtained for a given cooling power. The operating line is important for practical applications where the design or operating considerations mainly aim to maximize $\zeta$ for a given value of $J_C$. We plot in Figs. \[fig:op\_line\] (a) and (b) the maximum cooling power of nanowire and bulk Peltier coolers at a given voltage bias while Figs. \[fig:op\_line\] (c) and (d) demonstrate the operating lines of a nanowire and bulk Peltier cooler respectively for several heights of the energy filtering barrier. As stated previously, we note that the cooling power for nanowire Peltier coolers (Fig. \[fig:op\_line\] a) practically saturates beyond $E_b=150~meV$. The COP $\zeta$ at a given cooling power along the operating line of the nanowire rcooler (Fig. \[fig:op\_line\] c), however, is optimized for $E_b=150~meV$. However, we note that the cooling power as well as $\zeta$ in bulk Peltier coolers increase with the increase in energy barrier heights (Fig. \[fig:op\_line\] b and d). Such a trend occurs due to a monotonic increase in the density of states as well as inelastic scattering rates (assuming a parabolic dispersion relationship) with energy in the case of bulk Peltier coolers.\ **Nanowire to bulk transition in Peltier coolers:** The cooling performance in the transition regime between single-moded nanowire and bulk Peltier coolers is of particular interest. Two quantities which may be used to gauge the performance of Peltier coolers are $(i)$ the maximum cooling power density ($\frac{J_C^{MAX}}{A}$) and $(ii)$ the COP $\zeta$ at the maximum cooling power ($\zeta _{J_C^{MAX}}$). We plot in Figs. \[fig:transit\] (a) and (b), the maximum cooling power and the $\zeta$ at the maximum cooling power as a single-moded nanowire gradually transitions to the bulk regime. It is evident from Fig. \[fig:transit\] (a) that a single-moded nanowire provides an enhanced cooling performance compared to bulk due to greater conductance per unit area as well as efficient energy filtering due to the abrupt feature in the density of states (the Van Hove singularity). The variation in cooling performance as a single-moded nanowire transitions to the bulk regime is, however, of particular interest. The maximum cooling power density ($\frac{J_C^{MAX}}{A}$) as well as the $COP$ at the maximum power ($\zeta _{J_C^{MAX}}$) of a single-moded nanowire Peltier cooler, demonstrated in Figs. \[fig:transit\] (a) and (b) respectively, deteriorate compared to bulk as it transitions to the multi-moded regime and subsequently increases toward the bulk values as the multimoded nanowire gradually becomes equivalent to the bulk regime. For large cross-section, the separation between consecutive sub-bands in a nanowire becomes much less than $k_BT$ and the nanowire begins to exhibit bulk properties.\ Such a behaviour in the maximum power density can be well explained from the modal density profile of a multi-moded nanowire compared to bulk, as shown in Fig. \[fig:modal\]. The enhanced cooling power in bulk coolers compared to multi-moded nanowire coolers is a result of higher density of modes in bulk. The degradation in $\zeta _{J_C^{MAX}}$ in the multi-moded regime is not intuitive from a similar argument since $\zeta$ is the ratio between two quantities that are themselves dependent on the modal density profile. However, we noted that the maximum cooling power in multimoded nanowires occurs at a higher bias voltage which, we speculate, leads to a degradation in the $COP$. Conclusion {#conclude} ========== In this paper, we have analyzed the cooling performance in nanowire and bulk Peltier coolers. The two parameters we have focused on include the cooling power ($J_C$) and the $COP$ ($\zeta$). We have uncovered some crucial aspects in Peltier coolers which include: (i) there is a trade-off between the cooling power and the $COP$, (ii) there is an optimized energy barrier height in nanowires for which the cooling performance is optimized. For bulk coolers, on the other hand, the cooling power increases as one increases the energy barrier height. (iii) The cooling performance in nanowires deteriorates compared to bulk as a single-moded nanowire transitions to a multi-moded one. While exploring the cooling performance of the Peltier cooler, we have considered a parabolic dispersion relationship and assumed optical phonon scattering to be the dominant scattering mechanism. However, it remains to be explored how the theory is modified with different elastic and inelastic scattering mechanisms [@anifil] and non-parabolic dispersion relations. In particular, it remains an interesting problem to formulate a compact parameter that can be used to speculate the cooling performance based on the energy dependence of the electronic density of states, relaxation time and electronic transport velocity. This paper, however, sets the stage for an exploration of Peltier cooling in nanostructures. We believe that the conclusions presented here would establish a general viewpoint to understand the basics of the design of Peltier coolers and their optimization. NEGF equations for dissipative transport ======================================== In case of dissipative transport in nano devices, the generalized equations for non-equilibrium Green’s function formalism (NEGF) are given by [@dattabook; @Datta_Green; @LNE]: $$\begin{gathered} G(\overrightarrow{k_{m}},E)=[EI-H-U-E_m-\Sigma(\overrightarrow{k_{m}},E)]^{-1} \nonumber \\ \Sigma(\overrightarrow{k_{m}},E)=\Sigma_L(\overrightarrow{k_{m}},E)+\Sigma_R(\overrightarrow{k_{m}},E)+\Sigma_S(\overrightarrow{k_{m}},E) \nonumber \\ A(\overrightarrow{k_{m}},E)=i[G(\overrightarrow{k_{m}},E)-G^{\dagger}(\overrightarrow{k_{m}},E)] \nonumber \\ \Gamma(\overrightarrow{k_{m}},E)=[\Sigma(\overrightarrow{k_{m}},E)-\Sigma^{\dagger}(\overrightarrow{k_{m}},E)], \label{eq:negf}\end{gathered}$$ where $H$ is the discretized Hamiltonian matrix constructed using the nearest neighbour tight-binding approximation in an effective mass approach, $U$ denotes the modification in the conduction band minima due to the embedded energy barrier and $\Sigma_{L(R)}(\overrightarrow{k_{m}},E)$ and $\Sigma_S(\overrightarrow{k_{m}},E)$ describe the effect of coupling and scattering of the electronic wavefunction due to contacts and inelastic events (electron-phonon interaction) respectively. $\overrightarrow{k_{m}}$ in the above set of Eqs. denotes the transverse wavevector of the $m^{th}$ sub-band. $A(\overrightarrow{k_{m}},E)$ is the $1-D$ spectral function for the $m^{th}$ sub-band and $\Gamma(\overrightarrow{k_{m}},E)$ is the broadening matrix for the $m^{th}$ sub-band at energy $E$. For moderate electron-phonon interaction, it is generally assumed that the real part of $\Sigma_S=0$. Hence, $$\Sigma_S(\overrightarrow{k_{m}},E)=i\frac{ \Gamma_S(\overrightarrow{k_{m}},E)}{2}=\Sigma^{in}_S(\overrightarrow{k_{m}},E)+\Sigma^{out}_S(\overrightarrow{k_{m}},E) \label{eq:sigma_phonon}$$ $ \Sigma^{in}(\overrightarrow{k_{m}},E)$ and $ \Sigma^{out}(\overrightarrow{k_{m}},E)$ are the in-scattering and the out-scattering functions which model the rate of scattering of the electrons due to incoherence inside the device and external contacts. $$\begin{gathered} \Sigma^{in}(\overrightarrow{k_{m}},E)=\Sigma^{in}_L(\overrightarrow{k_{m}},E)+\Sigma^{in}_R(\overrightarrow{k_{m}},E) \\ +\Sigma^{in}_S(\overrightarrow{k_{m}},E) \nonumber \end{gathered}$$ $$\begin{gathered} \Sigma^{out}(\overrightarrow{k_{m}},E)=\Sigma^{out}_L(\overrightarrow{k_{m}},E)+\Sigma^{out}_R(\overrightarrow{k_{m}},E) \\ +\Sigma^{out}_S(\overrightarrow{k_{m}},E), \label{eq:sig}\end{gathered}$$ The in-scattering and out-scattering functions are related to the contact quasi-Fermi distribution functions via the equations: $$\begin{gathered} \Sigma^{in}(\overrightarrow{k_{m}},E)=\underbrace{\Gamma_L(\overrightarrow{k_{m}},E)f_L(E)}_{inflow~from~left~contact} \\+\underbrace{\Gamma_R(\overrightarrow{k_{m}},E)f_R(E)}_{inflow~from~right~contact}+\underbrace{\Sigma^{in}_S(\overrightarrow{k_{m}},E)}_{inflow~due~to~phonons}, \nonumber \end{gathered}$$ $$\begin{gathered} \Sigma^{out}(\overrightarrow{k_{m}},E)=\underbrace{\Gamma_L(\overrightarrow{k_{m}},E)\Big\{ 1-f_L(E)\Big \} }_{outflow~to~left~contact} \\+\underbrace{\Gamma_R(\overrightarrow{k_{m}},E)\Big \{ 1-f_R(E)\Big \}}_{outflow~to~right~contact}+\underbrace{\Sigma^{out}_S(\overrightarrow{k_{m}},E)}_{outflow~due~to~phonons}, \label{eq:sig1}\end{gathered}$$ where $f_{L(R)}$ represent the quasi-Fermi distribution of left(right) contact. For local scattering mechanisms, the rate of inelastic scattering of electrons is dependent on the electron and the hole correlation functions $(G^n$ and $G^p)$ via: $$\begin{gathered} \Sigma^{in}_S(\overrightarrow{k_{m}},E)=diag \Bigg\{ D_O \times \Big[ (N+1) \underset{\overrightarrow{q_{t}}}{\sum}G^n(\overrightarrow{k_{m}}+\overrightarrow{q_{t}},E+\hslash \omega) \\ N \underset{\overrightarrow{q_{t}}}{\sum}G^n(\overrightarrow{k_{m}}+\overrightarrow{q_{t}},E-\hslash \omega)\Big] \Bigg\} \nonumber \end{gathered}$$ $$\begin{gathered} \Sigma^{out}_S(\overrightarrow{k_{m}},E)=diag \Bigg\{D_O \times \Big[ (N+1) \underset{\overrightarrow{q_{t}}}{\sum}G^p(\overrightarrow{k_{m}}+\overrightarrow{q_{t}},E-\hslash \omega) \\ N \underset{\overrightarrow{q_{t}}}{\sum}G^p(\overrightarrow{k_{m}}+\overrightarrow{q_{t}},E+\hslash \omega)\Big] \Bigg\} \label{eq:sig_ph} \end{gathered}$$ In the above set of Eqs., $N$ denotes the average phonon number given by: $$N=\frac{1}{e^{\frac{\hslash\omega_o}{k_BT_L}}-1},$$ $D_O$ is related to the optical deformation potential ($D$) via the equation: $$D_O=\frac{\hslash D^2F}{2\rho\omega_o a^3},$$ and $\hslash \omega_o$ denotes the optical phonon energy, $\omega_o$ being the optical phonon radial frequency. $\{\overrightarrow{q_t}\}$ denotes the set of transverse phonon wave vectors.\ $G^n(\overrightarrow{k_{m}},E)$ and $G^p(\overrightarrow{k_{m}},E)$ are the electron and the hole correlation functions for the $m^{th}$ sub-band. The electron and the hole correlation functions are again related to the electron in-scattering and the electron out-scattering functions via the equations: $$\begin{gathered} G^n(\overrightarrow{k_{m}},E)=G(\overrightarrow{k_{m}},E)\Sigma^{in}(\overrightarrow{k_{m}},E)G^{\dagger}(\overrightarrow{k_{m}},E) \nonumber \\ G^p(\overrightarrow{k_{m}},E)=G(\overrightarrow{k_{m}},E)\Sigma^{out}(\overrightarrow{k_{m}},E)G^{\dagger}(\overrightarrow{k_{m}},E) \nonumber \\ \label{eq:correlation} \end{gathered}$$ Solving the dynamics of the entire system involves a self consistent solution of , , and . The electron density and current at the grid point $j$ can be calculated from the above equations as: $$n_j=\underset{m}{\sum}\int\frac{[G^n(\overrightarrow{k_{m}},E)dE]}{\pi aA}$$ $$\begin{gathered} I^{j\rightarrow j+1}=\underset{k_m}{\sum}i\frac{e}{\pi \hslash} \int[ G^n_{j+1,j}(\overrightarrow{k_{m}},E)H_{j,j+1}(E) \\ -H_{j+1,j}(E)G^n_{j,j+1}(\overrightarrow{k_{m}},E) ]dE, \label{eq:currentnegf}\end{gathered}$$ where $a$ is the distance between two adjacent grid points and $A$ is the cross sectional area of the device. $\hslash k_m$ denotes the transverse momentum of the electrons in the $m^{th}$ sub-band. The summations in run over all the sub-bands available for conduction. The heat current flowing through the device from the $j^{th}$ point to the $(j+1)^{th}$ point is given by: $$\begin{gathered} I_Q^{j\rightarrow j+1}=\underset{k_m}{\sum}\frac{i}{\pi \hslash} \times \int E[ G^n_{j+1,j}(\overrightarrow{k_{m}},E) \\ H_{j,j+1}(E)-H_{j+1,j}(E) G^n_{j,j+1}(\overrightarrow{k_{m}},E) ]dE, \label{eq:heatcurrentnegf}\end{gathered}$$

--- abstract: 'The $\gamma$-decay of the anti-analog of the giant dipole resonance (AGDR) has been measured to the isobaric analog state excited in the $p$($^{124}$Sn,$n$) reaction at a beam energy of 600 MeV/nucleon. The energy of the transition was also calculated with state-of-the-art self-consistent random-phase approximation (RPA) and turned out to be very sensitive to the neutron-skin thickness ($\Delta R_{pn}$). By comparing the theoretical results with the measured one, the $\Delta R_{pn}$ value for $^{124}$Sn was deduced to be 0.175 $\pm$ 0.048 fm, which agrees well with the previous results. The energy of the AGDR measured previously for $^{208}$Pb was also used to determine the $\Delta R_{pn}$ for $^{208}$Pb. In this way a very precise $\Delta R_{pn}$ = 0.181 $\pm$ 0.031 neutron-skin thickness has been obtained for $^{208}$Pb. The present method offers new possibilities for measuring the neutron-skin thicknesses of very exotic isotopes.' author: - 'A. Krasznahorkay' - 'L. Stuhl' - 'M. Csatlós' - 'A. Algora' - 'J. Gulyás' - 'J. Timár' - 'N. Paar' - 'D. Vretenar' - 'K. Boretzky' - 'M. Heil' - 'Yu.A. Litvinov' - 'D. Rossi' - 'C. Scheidenberger' - 'H. Simon' - 'H. Weick' - 'A. Bracco' - 'S. Brambilla' - 'N. Blasi' - 'F. Camera' - 'A. Giaz' - 'B. Million' - 'L. Pellegri' - 'S. Riboldi' - 'O. Wieland' - 'S. Altstadt' - 'M. Fonseca' - 'J. Glorius' - 'K. Göbel' - 'T. Heftrich' - 'A. Koloczek' - 'S. Kräckmann' - 'C. Langer' - 'R. Plag' - 'M. Pohl' - 'G. Rastrepina' - 'R. Reifarth' - 'S. Schmidt' - 'K. Sonnabend' - 'M. Weigand' - 'M.N. Harakeh' - 'N. Kalantar-Nayestanaki' - 'C. Rigollet' - 'S. Bagchi' - 'M.A. Najafi' - 'T. Aumann' - 'L. Atar' - 'M. Heine' - 'M. Holl' - 'A. Movsesyan' - 'P. Schrock' - 'V. Volkov' - 'F. Wamers' - 'E. Fiori' - 'B. Löher' - 'J. Marganiec' - 'D. Savran' - 'H.T. Johansson' - 'P. Diaz Fernández' - 'U. Garg' - 'D.L. Balabanski' title: 'Neutron-skin thickness from the study of the anti-analog giant dipole resonance' --- Recent progress in development of radioactive beams has made it possible to study the structure of nuclei far from stability. An important issue is the size of the neutron skin of unstable neutron-rich nuclei, because this feature may provide fundamental nuclear structure information. There is a renewed interest in measuring precisely the thickness of the neutron skin, because it constrains the symmetry-energy term of the nuclear equation of state. The precise knowledge of the symmetry energy is essential not only for describing the structure of neutron-rich nuclei, but also for describing the properties of the neutron-rich matter in nuclear astrophysics. The symmetry energy determines to a large extent, through the Equation of State (EoS), the proton fraction of neutron stars [@la01], the neutron skin in heavy nuclei [@fu02] and enters as input in the analysis of heavy-ion reactions [@li98; @ba02], etc. Furnstahl [@fu02] demonstrated that in heavy nuclei there exists an almost linear empirical correlation between the neutron-skin thickness and theoretical predictions for the symmetry energy of the EoS in terms of various mean-field approaches. This observation has contributed to a renewed interest in an accurate determination of the neutron-skin thickness in neutron-rich nuclei [@te08; @ta11; @ro11; @ab12]. In this work, we are suggesting a new precise method for measuring the neutron-skin thickness using both stable and radioactive beams. In our previous work on inelastic alpha scattering, excitation of the isovector giant dipole resonance was used to extract the neutron-skin thickness of nuclei [@kr91; @kr94]. The cross section of this process depends strongly on $\Delta R_{pn}$. Another tool used earlier for studying the neutron-skin thickness, is the excitation of the isovector spin giant dipole resonance (IVSGDR). The L=1 strength of the IVSGDR is sensitive to the neutron-skin thickness [@kr99; @kr04]. Vretenar et al. [@vr03] suggested another new method for determining the $\Delta R_{pn}$ by measuring the energy of the GTR. Constraints on the nuclear symmetry energy and neutron skin were also obtained recently from studies of the strength of the pygmy dipole resonance [@kl07]. In Ref. [@ca10], more nuclei were added and the theory was better constrained. The aim of the present work is to study the energy of the anti-analog of the giant dipole resonance (AGDR) [@st80], which depends strongly on the neutron-skin thickness. The non-energy-weighted sum rule (NEWSR) we used earlier [@kr99; @kr04] is valid (apart from a factor of 3) also for the giant dipole resonance excited in charge-exchange reactions and predicts increasing strengths as a function of the neutron-skin thickness. Auerbach et al. [@au81] derived an energy-weighted sum rule (EWSR) also for the dipole strengths excited in charge-exchange reactions. The corresponding energies are measured with respect to the RPA g.s. energy (IAS state) in the parent. The result of such EWSR is almost independent of the neutron-skin thickness [@au81] so the mean energy of the dipole strength should decrease with increasing dipole strengths and therefore with increasing neutron-skin thickness in consequence of NEWSR. The strong sensitivity of the AGDR energy on $\Delta R_{pn}$ is mentioned also by Krmpotić [@kr83], who preformed calculations with random-phase approximation (RPA). In the present work, we want to use such sensitivity of the energy of the AGDR on $\Delta R_{pn}$ to constrain the $\Delta R_{pn}$ of $^{124}$Sn. Due to the isovector nature of the (p,n) reaction, the strength of the E1 excitation is divided into T$_0$-1, T$_0$ and T$_0$+1 components, where T$_0$ is the ground state isospin of the initial nucleus, which is 12 for $^{124}$Sn. Because of the relevant Clebsch-Gordan coefficients [@os92], the T$_0$-1 component (AGDR) is favored compared to the T$_0$ and T$_0$+1 one by about a factor of T$_0$, and 2T$_0^2$, respectively. Dipole resonances were excited earlier in ($p$,$n$) reactions at E$_p$= 45 MeV by Sterrenburg et al. [@st80] in $^{92}$Zr, $^{93}$Nb, $^{94}$Mo, $^{120}$Sn and in $^{208}$Pb. Nishihara et al. [@ni85] measured also the dipole strength distributions at E$_p$ = 41 MeV. However, it was shown experimentally [@os81; @au01] that the observed $\Delta L$= 1 resonance was in general a superposition of all possible spin-flip dipole (SDR) modes and the non-spin-flip dipole GDR. According to the work of Osterfeld [@os92] the spin non-flip/spin-flip ratio is favored at low bombarding energy (below 50 MeV) and also at very high bombarding energies (above 600 MeV). The experiments, aiming at studying the neutron-skin thickness of $^{124}$Sn, were performed at GSI using 600 MeV/nucleon $^{124}$Sn relativistic heavy-ion beams on 2 and 5 mm thick CH$_2$ and 2 mm thick C targets. This allowed us to subtract the contribution of the C to the yield measured from the CH$_2$ target during the analysis. (Doing the experiment at 600 MeV/nucleon, in iverse kinematics we could increase the target thickness by a factor of 40 compared to 50 MeV/nucleon case without loosing the energy resolution.) According to the previous experimental studies [@st80; @ni85] the excitation energy of the AGDR is expected to be at E$_x$= 26 MeV. The differential cross section of the AGDR excited in ($p$,$n$) reaction was calculated at E$_p$ = 600 MeV with the computer code DW81 [@ra67; @sc81]. The wave functions used by the code were constructed using the normal-mode formalism [@ho95] with the code NORMOD [@normod]. The optical model parameters were taken from Ref. [@lo80]. According to such calculations the dipole cross section peaks at $\Theta_{CM} = 3^\circ$. The E$_x$= 26 MeV and $\Theta_{CM} =3^\circ$ correspond in inverse kinematics in the laboratory system to a scattered neutron with energy of E$_n\approx$ 2.4 MeV and $\Theta_{LAB} \approx 68^\circ$, respectively. The ejected neutrons were detected by a low-energy neutron-array (LENA) ToF spectrometer [@la11], which was developed in Debrecen and which was placed at 1 m from the target and covered a laboratory scattering-angle region of 65$^\circ\leq\Theta_{LAB}\leq75^{\circ}$. Similar neutron spectrometers have been built also by Beyer et al. [@be07] and by Perdikakis et al. [@pe09] and one of those was used recently as an effective tool for studying Gamow-Teller giant resonances in radioactive nuclear beams [@sa11]. The energy of de-exciting $\gamma$-transitions was measured by six large cylindrical ($3.5''\times 8''$) state-of-the-art LaBr$_3$ $\gamma$-detectors placed at 31 cm from the target and $\Theta_{LAB}$= 21$^\circ$ in order to use the advantage of the large Lorentz-boost. The large Doppler shift ($E_\gamma /E_0 = 2.33$) was taken into account in the analysis. The precise energy and efficiency calibrations of the detectors were performed after the experiments by using different radioactive sources and (p,$\gamma$) reactions on different targets [@ci09; @labr2012]. The response function of the detector was also checked up to 17.6 MeV and could be reproduced well with GEANT Monte-Carlo simulations. In order to make a correct energy calibration for the AGDR, the simulations were extended up to 40 MeV and convoluted with a Gaussian function with the width of the resonance. This convolution caused about 10% lowering of the positions of the peaks, which was then taken into account in the calibration of the detectors. The $\gamma$-ray energy spectrum measured in coincidence with the low-energy neutrons is shown in Fig. 1. ![The $\gamma$-ray energy spectrum measured in coincidence with the low-energy neutrons that fulfill the conditions of $1.0\leq E_n\leq 3.5$ MeV and 67$^\circ \leq\Theta_{LAB}\leq 70^\circ$. The calibrated energy scale was corrected already for the Doppler effect. The solid line shows the result of the fit described in the text. \[AGDR\]](agdr1.eps){width="90mm"} The width of the peak has a value of $\Gamma\approx$3.6 MeV, which is much larger than the width of the previously measured AGDR resonances [@st80; @ni85]. It can be explained by the Doppler broadening caused by the large solid angle of the detectors. The energy distribution of the $\gamma$-rays was fitted by a Lorentzian curve and a second order polynomial background, and the obtained parameters are shown in the figure. The contribution of the statistical error in the uncertainty of the position of the peak is 0.2 MeV, while the systematical error coming from the uncertainty of the energy calibration is about 0.25 MeV ($2.5\%$), which can be improved in the future. If we take into account the $E_\gamma^3$ dependence of the $\gamma$-transition probability, then the $E_{AGDR}-E_{IAS}$ = 10.70 $\pm$ 0.32 MeV. The direct $\gamma$-branching ratio of the AGDR to the IAS is expected to be similar to that of the GDR to the g.s. in the parent nucleus, which can be calculated from the parameters of the GDR [@kr94]. The branching ratio obtained for $^{124}$Sn is about 1%. In contrast, in the investigation of the electromagnetic decay properties of the SDR by Rodin and Dieperink [@ro02] the $\gamma$-decay branching ratio was in the range of 10$^{-4}$. This means that in the $n$-$\gamma$ coincidence spectrum the contribution of the SDR is suppressed by about a factor of 100. Therefore, the coincidence measurements deliver a precise energy for the AGDR, which in case of $^{124}$Sn agrees well with the results obtained by Sterrenburg et al. [@st80]. The theoretical analysis employed in this work was carried out with the fully self-consistent relativistic proton-neutron quasiparticle random-phase approximation (pn-RQRPA) based on the Relativistic Hartree-Bogoliubov model (RHB) [@VALR.05]. The RQRPA was formulated in the canonical single-nucleon basis of the RHB model in Ref. [@Paar2003] and extended to the description of charge-exchange excitations (pn-RQRPA) in Ref. [@Paar2004]. The RHB + pn-RQRPA model is fully self-consistent: in the particle-hole channel, effective Lagrangians with density-dependent meson-nucleon couplings are employed, and pairing correlations are described by the pairing part of the finite-range Gogny interaction [@BGG.91]. For the purpose of the present study, we employ a family of density-dependent meson-exchange (DD-ME) interactions, for which the constraint on the symmetry energy at saturation density has been systematically varied, $a_4 =$ 30, 32, 34, 36 and 38 MeV, and the model parameters are adjusted to accurately reproduce nuclear matter properties (the binding energy, the saturation density, the compression modulus) and the binding energies and charge radii of a standard set of spherical nuclei [@VNR.03]. These effective interactions were used to provide a microscopic estimate of the nuclear matter compressibility and symmetry energy in relativistic mean-field models [@VNR.03] and in Ref. [@kl07] to study a possible correlation between the observed pygmy dipole strength (PDS) in $^{130,132}$Sn and the corresponding values for the neutron-skin thickness. In addition to the set of effective interactions with $K_{\rm nm} =$ 250 MeV (this value reproduces the excitation energies of giant monopole resonances), and $a_4 =$ 30, 32, 34, 36 and 38 MeV, the relativistic functional DD-ME2 [@LNVR.05] will be used here to calculate the excitation energies of the AGDR with respect to the IAS, as a function of the neutron skin. Important for the present analysis is the fact that the relativistic RPA with the DD-ME2 effective interaction predicts the dipole polarizability for $^{208}$Pb, $\alpha_D$=20.8 fm$^3$, in very good agreement with the recently measured value: $\alpha_D = (20.1\pm 0.6)$ fm$^3$ [@ta11]. The results of the calculations for $^{124}$Sn are shown in Fig. 2. The difference in the excitation energy of the AGDR and the IAS, calculated with the pn-RQRPA based on the RHB self-consistent solution for the ground-state of the target nucleus, is plotted as a function of the corresponding RHB prediction for the neutron-skin thickness. For the excitation energy of the AGDR we take the centroid of the theoretical strength distribution, calculated in the energy interval above the IAS that corresponds to the measured spectrum of $\gamma$-ray energies: 6 to 14.8 MeV (cf. Fig. 2). A single peak is calculated for the IAS. For effective interactions with increasing value of the symmetry energy at saturation $a_4 =$ 30, 32, 34, 36 and 38 MeV (and correspondingly the slope of the symmetry energy at saturation [@VNPM.12]), we find an almost perfect linear decrease of $E(AGDR) - E(IAS)$ with the increase of the neutron skin $\Delta R_{np}$. The value calculated with DD-ME2 ($a_4 =32.3 $MeV) is denoted by the star symbol. The uncertainty of the theoretical predictions for the neutron-skin thicknesses is estimated to be 10 %. Such an uncertainty was used earlier for the differences between the neutron and proton radii for the nuclei $^{116}$Sn, $^{124}$Sn, and $^{208}$Pb in adjusting the parameters of the effective interactions [@VNR.03; @LNVR.05]. These effective interactions were also used to calculate the electric dipole polarizability and neutron-skin thickness of $^{208}$Pb, $^{132}$Sn and $^{48}$Ca, in comparison to the predictions of more than 40 non-relativistic and relativistic mean-field effective interactions [@Pie12]. From the results presented in that work one can also assess the accuracy of the present calculations. In comparison to the experimental result for $E(AGDR) - E(IAS)$ we deduce the value of the neutron skin thickness in $^{124}$Sn: $\Delta R_{np} = 0.175 \pm 0.048 $ fm (including the 10% theoretical uncertainty). In Table I, this value is compared to previous results obtained with a variety of experimental methods. Very good agreement has been obtained with the previous data, which supports the reliability of our method. ![The difference in the excitation energy of the AGDR and the IAS for the target nucleus $^{124}$Sn, calculated with the pn-RQRPA using five relativistic effective interactions characterized by the symmetry energy at saturation $a_4 =$ 30, 32, 34, 36 and 38 MeV (squares), and the interaction DD-ME2 ($a_4 =32.3$ MeV) (star). The theoretical values $E(AGDR) - E(IAS)$ are plotted as a function of the corresponding ground-state neutron-skin thickness $\Delta R_{pn}$, and compared to the experimental value $E(AGDR) - E(IAS) = 10.70 \pm 0.32 $ MeV. \[skin124\]](sn124-skin3.eps){width="90mm"} --------------------------------- ---------------- ------ ------------------- (fm) ($p$,$p$) 0.8 GeV [@ra79; @ba89] 1979 0.25 $\pm$ 0.05 ($\alpha, \alpha$’) GDR 120 MeV [@kr94] 1994 0.21 $\pm$ 0.11 ($^3$He,$t$) SDR+GDR 177 MeV [@kr04] 2004 0.27 $\pm$ 0.07 antiproton absorption [@tr01] 2001 0.19 $\pm$ 0.02 pygmy res. [@kl07] 2007 0.24 $\pm$ 0.04 ($p$,$p$) 295 MeV [@te08] 2008 0.185 $\pm$ 0.017 AGDR pres. res. 2012 0.175 $\pm$ 0.048 --------------------------------- ---------------- ------ ------------------- : \[tab:table1\] Neutron-skin thicknesses ($\Delta R_{pn}$) of $^{124}$Sn determined in the present work compared to previously measured values. As the $n$-$\gamma$ coincidence method delivered similar results in the case of $^{124}$Sn for the energy of the AGDR as obtained by Sterrenburg et al. [@st80] by a ToF method, it is reasonable to assume that their result for the energy of the AGDR obtained for $^{208}$Pb is correct. Fig. 3 displays the corresponding theoretical results for $E(AGDR) - E(IAS)$ for the target nucleus $^{208}$Pb, as a function of the corresponding ground-state neutron-skin thickness $\Delta R_{pn}$, and compared with the experimental result of Sterrenburg et al. [@st80]. In this case the $E(AGDR)$ is calculated as the centroid of the theoretical strength distribution in the interval between 5 and 15 MeV above the IAS. The deduced value of the neutron-skin thickness is compared with previous results in Table II. ![Same as described in the caption to Fig. 2 but for the target nucleus $^{208}$Pb. \[Pb208\]](pb208-skin3.eps){width="90mm"} --------------------------------- ------------ ------ ------------------- (fm) ($p$,$p$) 0.8 GeV [@ho80] 1980 0.14 $\pm$ 0.04 ($p$,$p$) 0.65 GeV [@sta94] 1994 0.20 $\pm$ 0.04 ($\alpha,\alpha$’) GDR 120 MeV [@kr94] 1994 0.19 $\pm$ 0.09 antiproton absorption [@tr01] 2001 0.18 $\pm$ 0.03 ($\alpha, \alpha$’) GDR 200 MeV [@kr04] 2003 0.12 $\pm$ 0.07 pygmy res. [@kl07] 2007 0.180 $\pm$ 0.035 pygmy res. [@ca10] 2010 0.194 $\pm$ 0.024 ($\vec p$,$\vec p\ '$) [@ta11] 2011 0.156 $\pm$ 0.025 parity viol. ($e$,$e$) [@ab12] 2012 0.33 $\pm$ 0.17 AGDR pres. res. 2012 0.181 $\pm$ 0.031 --------------------------------- ------------ ------ ------------------- : \[tab:table2\]Neutron-skin thicknesses of $^{208}$Pb determined in the present work compared to previously measured values. In conclusion, we have investigated the energy of the AGDR excited in the $^{124}$Sn($p$,$n$) reaction performed in inverse kinematics. Using the experimental results from this study, Ref. [@st80] for $^{208}$Pb, and RHB+pn-RQRPA model, we deduce the following values of the neutron skin: $\Delta R_{pn}$=(0.175 $\pm$ 0.048) fm in $^{124}$Sn and 0.181 $\pm$ 0.031 fm in $^{208}$Pb. The agreement between the $\Delta R_{pn}$ determined using measurements of the AGDR-IAS and previous methods is very good in both the studied cases. In particular, the present study supports the results from very recent high-resolution study of electric dipole polarizability $\alpha_D$ in $^{208}$Pb [@ta11], respective correlation analysis of $\alpha_D$ and $\Delta R_{pn}$ [@Pie12], as well as the Pb Radius Experiment (PREX) using parity-violating elastic electron scattering at JLAB [@ab12]. The method we have introduced provides not only stringent constraint to the neutron-skin thicknesses in nuclei under consideration, but it also offers new possibilities for measuring $\Delta R_{pn}$ in rare-isotope beams. [**Acknowledgments**]{} This work has been supported by the European Community FP7 - Capacities, contract ENSAR n$^\circ$ 262010, the Hungarian OTKA Foundation No. K72566, the Croatian Science Foundation, the Bulgarian Science Found, contacts DID-02/16 and DRNF-02/5, the U. S. National Science Foundation (Grant No. 1068192), by the TÁMOP-4.2.2/B-10/1-2010-0024 project, by the Alliance Program of the Helmholtz Association (HA216/EMMI) and partially supported by the Italian Istituto Nazionale di Fisica Nucleare. UG acknowledges the support of EMMI during his sojourn at GSI. [99]{} J. M. Lattimer and M. Prakash, Astrophys. J. [**550**]{}, 426 (2001); astro-ph/0002232. R. J. Furnstahl, Nucl. Phys. [**A706**]{}, 85 (2002). B. A. Li, C. M. Ko, and W. Bauer, Int. J. Mod. Phys. [**E 7**]{}, 147 (1998). Bao-An Li, Phys. Rev. Lett. [**88**]{}, 192701 (2002); Nucl. Phys. [**A708**]{}, 365 (2002). S. Terashima et al., Phys. Rev. [**C 77**]{}, 024317 (2008). A. Tamii et al., Phys. Rev. Lett. [**107**]{}, 062502 (2011). X. Roca-Maza and M. Centelles, X. Vinas, M. Warda, Phys. Rev. Lett. [**106**]{}, 252501 (2011). S. Abrahamyan et al., Phys. Rev. Lett. [**108**]{}, 112502 (2012). A. Krasznahorkay et al., Phys. Rev. Lett. [**66**]{}, 1287 (1991). A. Krasznahorkay et al., Nucl. Phys. [**A567**]{}, 521 (1994). A. Krasznahorkay et al., Phys. Rev. Lett. [**82**]{}, 3216 (1999). A. Krasznahorkay et al., Nucl. Phys. [**A731**]{}, 224 (2004). D. Vretenar, N. Paar, T. Niksic, P. Ring, Phys. Rev. Lett. [**91**]{}, 262502 (2003). A. Klimkiewicz et al., Phys. Rev. [**C 76**]{}, 051603 (2007). A. Carbone et al., Phys. Rev. [**C 81**]{}, 041301 (2010). W.A. Sterrenburg, S.M. Austin, R.P. DeVito, A. Galonsky, Phys. Rev. Lett. [**45**]{}, 1839 (1980). N. Auerbach, A. Klein, and Nguyen van Gai, Phys. Lett. [**B 106**]{} 347 (1981). F. Krmpotic et al., Nucl. Phys. [**A399**]{}, 478 (1983). F. Osterfeld, Rev. Mod. Phys. [**64**]{}, 491 (1992). S. Nishihara et al., Phys. Lett. [**B 160**]{}, 369 (1985). F. Osterfeld et al., Phys. Lett. [**B 105**]{}, 257 (1981). S. M. Austin et al., Phys. Rev. [**C 63**]{}, 034322 (2001). J. Raynal, Nucl. Phys. [**A97**]{}, 572 (1967). R. Schaeffer and J. Raynal, computer code DWBA70 (1971), unpublished, extended version DW81 by J.R. Comfort (1981). M. Hofstee et al., Nucl. Phys. [**A588**]{}, 729 (1995). S.Y. van der Werf, Program NORMOD, unpublished. W.G. Love and M.A. Franey, Bull. Am. Phys. Soc. [**25**]{} (1980) 730; W.G. Love private communication. C. Langer et al., Nucl. Instr. Meth. Phys. Res. [**A659**]{}, 411 (2011). R. Beyer et al., Nucl. Instr. Meth. Phys. Res. [**A575**]{}, 449 (2007). G. Perdikakis et al., IEEE Trans. Nucl. Sci. [**56**]{}, 1174 (2009). M. Sasano et al., Phys. Rev. Lett. [**107**]{}, 202501 (2011). M. Ciemala et al., Nucl. Instr. Meth. Phys. Res. [**A608**]{}, 76 (2009). A. Giaz et al., to be submitted to Nucl. Instr. Meth. Phys. Res. V. A. Rodin and A. E. L. Dieperink, Phys. Lett. [**B 541**]{}, 7 (2002). D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Phys. Rep. [**409**]{}, 101 (2005). N. Paar, P. Ring, T. Nikši' c, and D. Vretenar, Phys. Rev. [**C 67**]{}, 034312 (2003). N. Paar, T. Nik[š]{}i[ć]{}, D. Vretenar, and P. Ring, Phys. Rev. [**C 69**]{}, 054303 (2004). J. F. Berger, M. Girod, and D. Gogny, Comp. Phys. Comm.[**63**]{}, 365 (1991). D. Vretenar, T. Nik[š]{}i[ć]{}, and P. Ring, Phys. Rev.[**C 68**]{}, 024310 (2003). G. A. Lalazissis, T. Nik[š]{}i[ć]{}, D. Vretenar, and P. Ring, Phys. Rev. [**C 71**]{}, 024312 (2005). D. Vretenar, Y.F. Niu, N. Paar, J. Meng, Phys. Rev.[**C 85**]{}, 044317 (2012) L. Ray, Phys. Rev. [**C 19**]{}, 1855 (1979). C. J. Batty et al., Adv. Nucl. Phys. [**19**]{}, 1 (1989). A. Trzcinska et al., Phys. Rev. Lett. [**87**]{}, 082501 (2001). G.W. Hoffmann et al., Phys. Rev. [**C 21**]{}, 1488 (1980). V. E. Starodubsky and N. M. Hintz, Phys. Rev. [**C 49**]{}, 2118 (1994). J. Piekarewicz et al., Phys. Rev.[**C 85**]{}, 041302 (2012).

--- bibliography: - 'auto\_generated.bib' title: 'Constraints on parton distribution functions and extraction of the strong coupling constant from the inclusive jet cross section in pp collisions at $\sqrt{s} = 7$' --- =1 $Revision: 288504 $ $HeadURL: svn+ssh://svn.cern.ch/reps/tdr2/papers/SMP-12-028/trunk/SMP-12-028.tex $ $Id: SMP-12-028.tex 288504 2015-05-12 15:53:03Z alverson $ Introduction {#sec:intro} ============ Collimated streams of particles, conventionally called jets, are abundantly produced in highly energetic proton-proton collisions at the LHC. At high transverse momenta these collisions are described by quantum chromodynamics (QCD) using perturbative techniques (pQCD). Indispensable ingredients for QCD predictions of cross sections in  collisions are the proton structure, expressed in terms of parton distribution functions (PDFs), and the strong coupling constant , which is a fundamental parameter of QCD. The PDFs and both depend on the relevant energy scale $Q$ of the scattering process, which is identified with the jet for the reactions considered in this report. In addition, the PDFs, defined for each type of parton, depend on the fractional momentum $x$ of the proton carried by the parton. The large cross section for jet production at the LHC and the unprecedented experimental precision of the jet measurements allow stringent tests of QCD. In this study, the theory is confronted with data in previously inaccessible phase space regions of $Q$ and $x$. When jet production cross sections are combined with inclusive data from deep-inelastic scattering (DIS), the gluon PDF for $x \gtrsim 0.01$ can be constrained and can be determined. In the present analysis, this is demonstrated by means of the CMS measurement of inclusive jet production [@Chatrchyan:2012bja]. The data, collected in 2011 and corresponding to an integrated luminosity of 5.0, extend the accessible phase space in jet up to 2, and range up to $\yabs = 2.5$ in absolute jet rapidity. A PDF study using inclusive jet measurements by the ATLAS Collaboration is described in Ref. [@Aad:2013lpa]. This paper is divided into six parts. Section \[sec:measurement\] presents an overview of the CMS detector and of the measurement, published in Ref. [@Chatrchyan:2012bja], and proposes a modified treatment of correlations in the experimental uncertainties. Theoretical ingredients are introduced in Section \[sec:theory\]. Section \[sec:alphas\] is dedicated to the determination of at the scale of the -boson mass $M_\cPZ$, and in Section \[sec:herafitter\] the influence of the jet data on the PDFs is discussed. A summary is presented in Section \[sec:summary\]. The inclusive jet cross section {#sec:measurement} =============================== Overview of the CMS detector and of the measurement {#sec:measurementoverview} --------------------------------------------------- =1000 The central feature of the CMS detector is a superconducting solenoid of 6 internal diameter, providing a magnetic field of 3.8. Within the superconducting solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass/scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Muons are measured in gas-ionisation detectors embedded in the steel flux-return yoke outside the solenoid. Extensive forward calorimetry (HF) complements the coverage provided by the barrel and endcap detectors. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [@Chatrchyan:2008aa]. Jets are reconstructed with a size parameter of $R=0.7$ using the collinear- and infrared-safe anti-clustering algorithm [@Cacciari:2008gp] as implemented in the package [@Cacciari:2011ma]. The published measurements of the cross sections were corrected for detector effects, and include statistical and systematic experimental uncertainties as well as bin-to-bin correlations for each type of uncertainty. A complete description of the measurement can be found in Ref. [@Chatrchyan:2012bja]. The double-differential inclusive jet cross section investigated in the following is derived from observed inclusive jet yields via $$\frac{\rd^2\sigma}{\rd\pt\,\rd y} = \frac{1}{\epsilon\cdot\mathcal{L}_{\text{int}}} \frac{N_\text{jets}}{\Delta\pt\,\left(2\cdot\Delta\yabs\right)},$$ where $N_\text{jets}$ is the number of jets in the specific kinematic range (bin), $\mathcal{L}_{\text{int}}$ is the integrated luminosity, $\epsilon$ is the product of trigger and event selection efficiencies, and $\Delta \pt$ and $\Delta\yabs$ are the bin widths in and . The factor of two reflects the folding of the distributions around $y=0$. Experimental uncertainties {#sec:measurementuncertainties} -------------------------- The inclusive jet cross section is measured in five equally sized bins of $\Delta\yabs = 0.5$ up to an absolute rapidity of $\yabs = 2.5$. The inner three regions roughly correspond to the barrel part of the detector, the outer two to the endcaps. Tracker coverage extends up to $\yabs = 2.4$. The minimum imposed on any jet is 114. The binning in jet follows the jet resolution of the central detector and changes with . The upper reach in is given by the available data and decreases with . Four categories [@Chatrchyan:2012bja] of experimental uncertainties are defined: the jet energy scale (JES), the luminosity, the corrections for detector response and resolution, and all remaining uncorrelated effects. The JES is the dominant source of systematic uncertainty, because a small shift in the measured translates into a large uncertainty in the steeply falling jet spectrum and hence in the cross section for any given value of . The JES uncertainty is parameterized in terms of jet and pseudorapidity $\eta = -\ln\tan(\theta/2)$ and amounts to 1–2% [@CMS-DP-2012-006], which translates into a 5–25% uncertainty in the cross section. Because of its particular importance for this analysis, more details are given in Section \[sec:measurementjec\]. The uncertainty in the integrated luminosity is 2.2% [@CMS-PAS-SMP-12-008] and translates into a normalisation uncertainty that is fully correlated across and . The effect of the jet energy resolution (JER) is corrected for using the D’Agostini method [@D'Agostini:1994zf] as implemented in the package [@Adye:2011gm]. The uncertainty due to the unfolding comprises the effects of an imprecise knowledge of the JER, of residual differences between data and the Monte Carlo (MC) modelling of detector response, and of the unfolding technique applied. The total unfolding uncertainty, which is fully correlated across $\eta$ and , is 3–4%. Additionally, the statistical uncertainties are propagated through the unfolding procedure, thereby providing the correlations between the statistical uncertainties of the unfolded measurement. A statistical covariance matrix must be used to take this into account. Remaining effects are collected into an uncorrelated uncertainty of $\approx$1%. Uncertainties in JES {#sec:measurementjec} -------------------- The procedure to calibrate jet energies in CMS and ways to estimate JES uncertainties are described in Ref. [@Chatrchyan:2011ds]. To use CMS data in fits of PDFs or , it is essential to account for the correlations in these uncertainties among different regions of the detector. The treatment of correlations uses 16 mutually uncorrelated sources as in Ref. [@Chatrchyan:2012bja]. Within each source, the uncertainties are fully correlated in and $\eta$. Any change in the jet energy calibration (JEC) is described through a linear combination of sources, where each source is assumed to have a Gaussian probability density with a zero mean and a root-mean-square of unity. In this way, the uncertainty correlations are encoded in a fashion similar to that provided for PDF uncertainties using the Hessian method [@Pumplin:2001ct]. The total uncertainty is defined through the quadratic sum of all uncertainties. The full list of sources together with their brief descriptions can be found in Appendix \[sec:jessources\]. The JES uncertainties can be classified into four broad categories: absolute energy scale as a function of , jet flavour dependent differences, relative calibration of JES as a function of $\eta$, and the effects of multiple proton interactions in the same or adjacent beam crossings (pileup). The absolute scale is a single fixed number such that the corresponding uncertainty is fully correlated across and $\eta$. Using photon+jet and $Z$+jet data, the JES can be constrained directly in the jet range 30–600. The response at larger and smaller is extrapolated through MC simulation. Extra uncertainties are assigned to this extrapolation based on the differences between MC event generators and the single-particle response of the detector. The absolute calibration is the most relevant uncertainty in jet analyses at large . The categories involving jet flavour dependence and pileup effects are important mainly at small and have relatively little impact for the phase space considered in this report. The third category parameterizes $\eta$-dependent changes in relative JES. The measurement uncertainties within different detector regions are strongly correlated, and thus the $\eta$-dependent sources are only provided for wide regions: barrel, endcap with upstream tracking, endcap without upstream tracking, and the HF calorimeter. In principle, the $\eta$-dependent effects can also have a dependence. Based on systematic studies on data and simulated events, which indicate that the and $\eta$ dependence of the uncertainties factorise to a good approximation, this is omitted from the initial calibration procedure. However, experiences with the calibration of data collected in 2012 and with fits of reported in Section \[sec:alphas\] show that this is too strong an assumption. Applying the uncertainties and correlations in a fit of to the inclusive jet data separately for each bin in leads to results with values of that scatter around a central value. Performing the same fit taking all bins together and assuming 100% correlation in within the JES uncertainty sources results in a bad fit quality (high per number of degrees of freedom $n_\mathrm{dof}$) and a value of that is significantly higher than any value observed for an individual bin in . Changing the correlation in the JES uncertainty from 0% to 100% produces a steep rise in , and influences the fitted value of for correlations near 90%, indicating an assumption on the correlations in that is too strong. The technique of nuisance parameters, as described in Section \[sec:fitsetup\], helped in the analysis of this issue. To implement the additional $\eta$-decorrelation induced by the -dependence in the $\eta$-dependent JEC introduced for the calibration of 2012 data, the source from the single-particle response JEC2, which accounts for extrapolation uncertainties at large as discussed in Appendix \[sec:jessources\], is decorrelated versus $\eta$ as follows: 1. in the barrel region ($\yabs < 1.5$), the correlation of the single-particle response source among the three bins in is set to 50%, 2. in the endcap region ($1.5 \leq \yabs < 2.5$), the correlation of the single-particle response source between the two bins in is kept at 100%, 3. there is no correlation of the single-particle response source between the two detector regions of $\yabs < 1.5$ and $1.5 \leq \yabs < 2.5$. The additional freedom of -dependent corrections versus $\eta$ hence leads to a modification of the previously assumed full correlation between all $\eta$ regions to a reduced estimate of 50% correlation of JEC2 within the barrel region, which always contains the tag jet of the dijet balance method [@Chatrchyan:2011ds]. In addition, the JEC2 corrections are estimated to be uncorrelated between the barrel and endcap regions of the detector because of respective separate -dependences of these corrections. Technically, this can be achieved by splitting the single-particle response source into five parts (JEC2a–e), as shown in Table \[tab:cmsjets2011:nuisance\]. Each of these sources is a duplicate of the original single-particle response source that is set to zero outside the respective ranges of $\yabs < 1.5$, $1.5 \leq \yabs < 2.5$, $\yabs < 0.5$, $0.5 \leq \yabs < 1.0$, and $1.0 \leq \yabs < 1.5$, such that the original full correlation of $$\mathrm{corr}_\mathrm{JEC2,old} = \begin{pmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ \end{pmatrix}$$ is replaced by the partially uncorrelated version of $$\mathrm{corr}_\mathrm{JEC2,new} = \begin{pmatrix} 1 & 0.5 & 0.5 & 0 & 0\\ 0.5 & 1 & 0.5 & 0 & 0\\ 0.5 & 0.5 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1 & 1\\ \end{pmatrix}, \label{eqn:decorr}$$ which is more accurate as justified by studies based on 2012 data. For the proper normalisation of the five new correlated sources, normalisation factors of $1/\sqrt{2}$ (JEC2a, JEC2c–JEC2f) and $1$ (JEC2b) must be applied. With these factors, the sum of the five sources reproduces the original uncertainty for each , while the additional freedom gives the estimated level of correlation among the regions. All results presented in this paper are based on this improved treatment of the correlation of JES uncertainties. While some decorrelation of these uncertainties versus $\eta$ is important for the fits of described in Section \[sec:alphas\], the exact size of the estimated decorrelation is not. Varying the assumptions according to Eq. (\[eqn:decorr\]) from 50% to 20 or 80% in the barrel region, from 100 to 80% in the endcap region, or from 0 to 20% between the barrel and endcap regions leads to changes in the fitted value of that are negligible with respect to other experimental uncertainties. Theoretical ingredients {#sec:theory} ======================= The theoretical predictions for the inclusive jet cross section comprise a next-to-leading order (NLO) pQCD calculation with electroweak corrections (EW) [@Butterworth:2014efa; @Dittmaier:2012kx]. They are complemented by a nonperturbative (NP) factor that corrects for multiple-parton interactions (MPI) and hadronization (HAD) effects. Parton shower (PS) corrections, derived from NLO predictions with matched parton showers, are tested in an additional study in Section \[section:results\_a\_s\], but are not applied to the main result. Fixed-order prediction in perturbative QCD {#sec:fixedorder} ------------------------------------------ The same NLO prediction as in Ref. [@Chatrchyan:2012bja] is used, the calculations are based on the parton-level program version 4.1.3 [@Nagy:2001fj; @Nagy:2003tz] and are performed within the framework version 2.1 [@Britzger:2012bs]. The renormalization and factorisation scales, and respectively, are identified with the individual jet . The number of active (massless) flavours $N_f$ in has been set to five. Five sets of PDFs are available for a series of values of , which is a requisite for a determination of from data. For an overview, these PDF sets are listed in Table \[tab:pdfsets\] together with the respective references. The ABM11 PDF set employs a fixed-flavour number scheme with five active flavours, while the other PDF sets use a variable-flavour number scheme with a maximum of five flavours, $N_{f,\mathrm{max}} = 5$, except for NNPDF2.1 which has $N_{f,\mathrm{max}} = 6$. All sets exist at next-to-leading and next-to-next-to-leading evolution order. The PDF uncertainties are provided at $68.3\%$ confidence level (CL) except for CT10, which provides uncertainties at $90\%$ CL. For a uniform treatment of all PDFs, the CT10 uncertainties are downscaled by a factor of $\sqrt{2}\erf^{-1}{(0.9)} \approx 1.645$. \[tab:pdfsets\] Base set Refs. Evol.  $N_f$ $M_\cPqt$ () $M_\cPZ$ () range ------------ ---------------------------------- -------- ----------- -------------- ------------- -------- ------------- ABM11 [@Alekhin:2012ig] NLO 5 180 91.174 0.1180 0.110–0.130 ABM11 [@Alekhin:2012ig] NNLO 5 180 91.174 0.1134 0.104–0.120 CT10 [@Lai:2010vv] NLO ${\leq}5$ 172 91.188 0.1180 0.112–0.127 CT10 [@Lai:2010vv] NNLO ${\leq}5$ 172 91.188 0.1180 0.110–0.130 HERAPDF1.5 [@Aaron:2009aa] NLO ${\leq}5$ 180 91.187 0.1176 0.114–0.122 HERAPDF1.5 [@Aaron:2009aa] NNLO ${\leq}5$ 180 91.187 0.1176 0.114–0.122 MSTW2008 [@Martin:2009iq; @Martin:2009bu] NLO ${\leq}5$ $10^{10}$ 91.1876 0.1202 0.110–0.130 MSTW2008 [@Martin:2009iq; @Martin:2009bu] NNLO ${\leq}5$ $10^{10}$ 91.1876 0.1171 0.107–0.127 NNPDF2.1 [@Ball:2011mu] NLO ${\leq}6$ 175 91.2 0.1190 0.114–0.124 NNPDF2.1 [@Ball:2011mu] NNLO ${\leq}6$ 175 91.2 0.1190 0.114–0.124 The electroweak corrections to the hard-scattering cross section have been computed with the CT10-NLO PDF set for a fixed number of five flavours and with the of the leading jet, , as scale choice for and instead of the of each jet. At high jet and central rapidity, where the electroweak effects become sizeable, NLO calculations with either of the two scale settings differ by less than one percent. Given the small impact of the electroweak corrections on the final results in Sections \[sec:alphas\] and \[sec:herafitter\], no uncertainty on their size has been assigned. Theoretical prediction from MC simulations including parton showers and nonperturbative effects ----------------------------------------------------------------------------------------------- The most precise theoretical predictions for jet measurements are usually achieved in fixed-order pQCD, but are available at parton level only. Data that have been corrected for detector effects, however, refer to measurable particles, to colour-neutral particles with mean decay lengths such that $c\tau>10\unit{mm}$. Two complications arise when comparing fixed-order perturbation theory to these measurements: emissions of additional partons close in phase space, which are not sufficiently accounted for in low-order approximations, and effects that cannot be treated by perturbative methods. The first problem is addressed by the parton shower concept [@Marchesini:1987cf; @Knowles:1988hu; @Knowles:1988vs] within pQCD, where multiple parton radiation close in phase space is taken into account through an all-orders approximation of the dominant terms including coherence effects. Avoiding double counting, these parton showers are combined with leading-order (LO) calculations in MC event generators, such as  [@Sjostrand:2006za] and  [@Bahr:2008pv]. The second issue concerns NP corrections, which comprise supplementary parton-parton scatters within the same colliding protons, MPI, and the hadronization process including particle decays. The MPI [@Sjostrand:1987su; @Bahr:2008dy] model for additional soft-particle production, which is detected as part of the underlying event, is implemented in as well as . Hadronization describes the transition phase from coloured partons to colour-neutral particles, where perturbative methods are no longer applicable. Two models for hadronization are in common use, the Lund string fragmentation [@Andersson:1983ia; @Andersson:1983jt; @Sjostrand:1984iu] that is used in , and the cluster fragmentation [@Webber:1983if] that has been adopted by . Beyond LO combining fixed-order predictions with parton showers, MPI, and hadronization models is much more complicated. Potential double counting of terms in the perturbative expansion and the PS has to be avoided. In recent years programs have become available for dijet production at NLO that can be matched to PS MC event generators. In the following, one such program, the package [@Frixione:2007vw; @Alioli:2010xd] will be used for comparisons with dijet events [@Alioli:2010xa] to the LO MC event generators. NP corrections from PYTHIA6 and HERWIG++ {#sec:assumptions} ---------------------------------------- For the comparison of theoretical predictions to the measurement reported in Ref. [@Chatrchyan:2012bja], the NP correction was derived as usual [@Campbell:2006wx] from the average prediction of two LO MC event generators and more specifically from version 6.4.22 tune Z2 and version 2.4.2 with the default tune of version 2.3. Tune Z2 is identical to tune Z1 described in [@Field:2010bc] except that Z2 employs the CTEQ6L1 [@Pumplin:2002vw] PDF set, while Z1 uses the CTEQ5L [@Lai:1999wy] PDF set. The NP correction factor can be defined for each bin in and as $$C _\mathrm{LO}^{\text{NP}} = \frac{\sigma_{\mathrm{LO+PS+HAD+MPI}}}{\sigma_{\mathrm{LO+PS}}}\, \label{e:C_LO_NP}$$ where $\sigma$ represents the inclusive jet cross section and the subscripts “LO+PS+HAD+MPI” and “LO+PS” indicate which steps of a general MC event generation procedure have been run, see also Refs. [@Campbell:2006wx; @Buckley:2011ms]. The central value is calculated by taking the average of the two predictions from and . In applying these factors as corrections for NP effects to NLO theory predictions, it is assumed that the NP corrections are universal, they are similar for LO and NLO. NP and PS corrections from POWHEG + PYTHIA6 {#sec:powhegpythia} ------------------------------------------- Alternative corrections are derived, which use the revision 197 with the CT10-NLO PDF set for the hard subprocess at NLO plus the leading emission [@Nason:2004rx] complemented with the matched showering, MPI, and hadronization from version 6.4.26. The NLO event generation within the framework, and the showering and hadronization process performed by are done in independent steps. For illustration, Fig. \[fig:DataTheory\_comp4a\] shows the comparison of the inclusive jet data with the + tune Z2\* particle-level prediction complemented with electroweak corrections. The tune Z2\* is derived from the earlier tune Z2, where the parameters PARP(82) and PARP(90) that control the energy dependence of the MPI are retuned, yielding 1.921 and 0.227, respectively. The error boxes indicate statistical uncertainties. Ratio plots of this comparison for each separate region in can be found in Appendix \[theory\_data\]. The corrections to NLO parton-level calculations that are derived this way consist of truly nonperturbative contributions, which are optionally complemented with parton shower effects. They are investigated separately in the following two sections. A previous investigation can be found in Ref. [@Dooling:2012uw]. ![Measured inclusive jet cross section from Ref. [@Chatrchyan:2012bja] compared to the prediction by + tune Z2\* at particle level complemented with electroweak corrections. The boxes indicate the statistical uncertainty of the calculation.[]{data-label="fig:DataTheory_comp4a"}](InclusiveJets_xs_11_twiki_combineBornKt){width="\cmsFigWidth"} ### NP corrections from POWHEG + PYTHIA6 {#sec:npcorrection} The NP corrections using a NLO prediction with a matched PS event generator can be defined analogously as in Eq. (\[e:C\_LO\_NP\]): $$C_\mathrm{NLO}^{\text{NP}} = \frac{\sigma_{\text{NLO+PS+HAD+MPI}}}{\sigma_{\text{NLO+PS}}}, \label{e:C_NLO_NP}$$ the numerator of this NP correction is defined by the inclusive cross section, where parton showers, hadronization, and multiparton interactions are turned on, while the inclusive cross section in the denominator does not include hadronization and multiparton interactions. A NLO calculation can then be corrected for NP effects as $$\frac{\rd^2 \sigma_\mathrm{theo}}{\rd\pt\, \rd{}y} = \frac{\rd^2 \sigma_\mathrm{NLO}}{\rd\pt\, \rd{}y} \cdot C_\mathrm{NLO}^\mathrm{NP}.$$ =900 In contrast to the LO MC event generation with , the parameters of the NP and PS models, however, have not been retuned to data for the use with NLO+PS predictions by . Therefore two different underlying event tunes of for LO+PS predictions, P11 [@Skands:2010ak] and Z2\*, are used. In both cases a parameterization using a functional form of $a_0 + a_1 / \pt^{a_2}$ is employed to smoothen statistical fluctuations. For $\pt > 100\GeV$ the difference in the NP correction factor between the two tunes is very small such that their average is taken as $C_\mathrm{NLO}^{\text{NP}}$. Since procedures to estimate uncertainties inherent to the NLO+PS matching procedure are not yet well established and proper tunes to data for + are lacking, the centre of the envelope given by the three curves from , , and the + average of tunes Z2\* and P11 is adopted as the final NP correction for the central results in Sections \[sec:alphas\] and \[sec:herafitter\]. Half the spread among these three predictions defines the uncertainty. The NP correction, as defined for + , is shown in Fig. \[fig:np\_corrections\_powheg\_pythia\] together with the original factors from and , as a function of the jet for five ranges in absolute rapidity of size 0.5 up to $\yabs = 2.5$. The factors derived from both, LO+PS and NLO+PS MC event generators, are observed to decrease with increasing jet and to approach unity at large . Within modelling uncertainties, the assumption of universal NP corrections that are similar for LO+PS and NLO+PS MC event generation holds approximately above a jet of a few hundred . {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} ### PS corrections from POWHEG + PYTHIA6 {#sec:pscorrection} Similarly to the NP correction of Eq. (\[e:C\_NLO\_NP\]), a PS correction factor can be defined as the ratio of the differential cross section including PS effects divided by the NLO prediction, as given by , including the leading emission: $$C _\mathrm{NLO}^{\text{PS}} = \frac{\sigma_{\text{NLO+PS}}}{\sigma_{\text{NLO}}}. \label{e:C_NLO_PS}$$ The combined correction for NP and PS effects can then be written as $$\frac{\rd^2 \sigma_\mathrm{theo}}{\rd\pt\, \rd{}y} = \frac{\rd^2 \sigma_\mathrm{NLO}}{\rd\pt\, \rd{}y} \cdot C_\mathrm{NLO}^\mathrm{NP} \cdot C_\mathrm{NLO}^\mathrm{PS}.$$ The PS corrections derived with + are presented in Fig. \[fig:ps\_corrections\_powheg\_pythia\]. They are significant at large , particularly at high rapidity, where the factors approach $-20$%. However, the combination of + has never been tuned to data and the Z2\* tune strictly is only valid for a LO+PS tune with , but not with showers matched to . Moreover, employs the CT10-NLO PDF, while the Z2\* tune requires the CTEQ6L1-LO PDF to be used for the showering part. Therefore, such PS corrections can be considered as only an illustrative test, as reported in Section \[section:results\_a\_s\]. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} The maximum parton virtuality allowed in the parton shower evolution, $\mu_\mathrm{PS}^2$, is varied by factors of 0.5 and 1.5 by changing the corresponding parameter PARP(67) in from its default value of 4 to 2 and 6, respectively. The resulting changes in the PS factors are shown in Fig. \[fig:ps\_corrections\_powheg\_pythia\]. The + PS factors employed in an illustrative test later are determined as the average of the predictions from the two extreme scale limits. Again, a parameterization using a functional form of $a_0 + a_1 / \pt^{a_2}$ is employed to smoothen statistical fluctuations. Finally, Fig. \[fig:total\_corrections\_powheg\_pythia\] presents an overview of the NP, PS, and combined corrections for all five ranges in . ![NP correction (top left) obtained from the envelope of the predictions of tune Z2, tune 2.3, and + with the tunes P11 and Z2\*, PS correction (top right) obtained from the average of the predictions of + tune Z2\* with scale factor variation, and combined correction (bottom), defined as the product of the NP and PS correction, for the five regions in .[]{data-label="fig:total_corrections_powheg_pythia"}](Non_perturbative_Corrections_Uncertainties "fig:"){width="48.00000%"} ![NP correction (top left) obtained from the envelope of the predictions of tune Z2, tune 2.3, and + with the tunes P11 and Z2\*, PS correction (top right) obtained from the average of the predictions of + tune Z2\* with scale factor variation, and combined correction (bottom), defined as the product of the NP and PS correction, for the five regions in .[]{data-label="fig:total_corrections_powheg_pythia"}](Parton_Shower_Corrections_Uncertainties "fig:"){width="48.00000%"} ![NP correction (top left) obtained from the envelope of the predictions of tune Z2, tune 2.3, and + with the tunes P11 and Z2\*, PS correction (top right) obtained from the average of the predictions of + tune Z2\* with scale factor variation, and combined correction (bottom), defined as the product of the NP and PS correction, for the five regions in .[]{data-label="fig:total_corrections_powheg_pythia"}](Total_Corrections_Uncertainties "fig:"){width="48.00000%"} Determination of the strong coupling constant {#sec:alphas} ============================================= The measurement of the inclusive jet cross section [@Chatrchyan:2012bja], as described in Section \[sec:measurement\], can be used to determine , where the proton structure in the form of PDFs is taken as a prerequisite. The necessary theoretical ingredients are specified in Section \[sec:theory\]. The choice of PDF sets is restricted to global sets that fit data from different experiments, so that only the most precisely known gluon distributions are employed. Combined fits of and the gluon content of the proton are investigated in Section \[sec:combinedfits\]. In the following, the sensitivity of the inclusive jet cross section to is demonstrated. Subsequently, the fitting procedure is given in detail before presenting the outcome of the various fits of . Sensitivity of the inclusive jet cross section to alpha-S(M(Z))\[section:sens\_as\] ----------------------------------------------------------------------------------- Figures \[fig:DataTheory\_as\_ABM11nlo\]–\[fig:DataTheory\_as\_NNPDF21nlo\] present the ratio of data to the theoretical predictions for all variations in available for the PDF sets ABM11, CT10, MSTW2008, and NNPDF2.1 at next-to-leading evolution order, as specified in Table \[tab:pdfsets\]. Except for the ABM11 PDF set, which leads to QCD predictions significantly different in shape to the measurement, all PDF sets give satisfactory theoretical descriptions of the data and a strong sensitivity to is demonstrated. Because of the discrepancies, ABM11 is excluded from further investigations. The CT10-NLO PDF set is chosen for the main result on , because the value of preferred by the CMS jet data is rather close to the default value of this PDF set. As crosschecks fits are performed with the NNPDF2.1-NLO and MSTW2008-NLO sets. The CT10-NNLO, NNPDF2.1-NNLO, and MSTW2008-NNLO PDF sets are employed for comparison. {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} {width="47.00000%"} The fitting procedure\[section:fit\_proc\] ------------------------------------------ The value of is determined by minimising the between the $N$ measurements $D_i$ and the theoretical predictions $T_i$. The is defined as $$\chi^2 = \sum_{ij}^N \left(D_i - T_i\right) \mathrm{C}_{ij}^{-1} \left(D_j - T_j\right), \label{chi2_square}$$ where the covariance matrix $C_{ij}$ is composed of the following terms: and the terms in the sum represent 1. [$\cov_\text{stat}$: statistical uncertainty including correlations induced through unfolding]{}; 2. [$\cov_\text{uncor}$: uncorrelated systematic uncertainty summing up small residual effects such as trigger and identification inefficiencies, time dependence of the jet resolution, or the uncertainty on the trigger prescale factor]{}; 3. [$\cov_\mathrm{JES\,sources}$: systematic uncertainty for each JES uncertainty source]{}; 4. [$\cov_\text{unfolding}$: systematic uncertainty of the unfolding]{}; 5. [$\cov_\text{lumi}$: luminosity uncertainty]{}; and 6. [$\cov_\mathrm{PDF}$: PDF uncertainty]{}. All JES, unfolding, and luminosity uncertainties are treated as 100% correlated across the and bins, with the exception of the single-particle response JES source as described in Section \[sec:measurementjec\]. The JES, unfolding, and luminosity uncertainties are treated as multiplicative to avoid the statistical bias that arises when estimating uncertainties from data [@Lyons:1989gh; @D'Agostini:2003nk; @Ball:2009qv]. The derivation of PDF uncertainties follows prescriptions for each individual PDF set. The CT10 and MSTW PDF sets both employ the eigenvector method with upward and downward variations for each eigenvector. As required by the use of covariance matrices, symmetric PDF uncertainties are computed following Ref. [@Pumplin:2002vw]. The NNPDF2.1 PDF set uses the MC pseudo-experiments instead of the eigenvector method in order to provide PDF uncertainties. A hundred so-called replicas, whose averaged predictions give the central result, are evaluated following the prescription in Ref. [@Ball:2010de] to derive the PDF uncertainty for NNPDF. As described in Section \[sec:npcorrection\], the NP correction is defined as the centre of the envelope given by , , and the + average of tunes Z2\* and P11. Half the spread among these three numbers is taken as the uncertainty. This is the default NP correction used in this analysis. Alternatively, the PS correction factor, defined in Section \[sec:pscorrection\], is applied in addition as an illustrative test to complement the main results. The uncertainty in due to the NP uncertainties is evaluated by looking for maximal offsets from a default fit. The theoretical prediction $T$ is varied by the NP uncertainty $\Delta\mathrm{NP}$ as $T\cdot\mathrm{NP} \to T\cdot\left(\mathrm{NP} \pm \Delta\mathrm{NP}\right)$. The fitting procedure is repeated for these variations, and the deviation from the central values is considered as the uncertainty in . =900 Finally the uncertainty due to the renormalization and factorisation scales is evaluated by applying the same method as for the NP corrections: and are varied from the default choice of $\mur=\muf=\pt$ between $\pt/2$ and $2\pt$ in the following six combinations: $(\mur/\pt,\muf/\pt) = (1/2,1/2)$, $(1/2,1)$, $(1,1/2)$, $(1,2)$, $(2,1)$, and $(2,2)$. The minimisation with respect to is repeated in each case. The contribution from the and scale variations to the uncertainty is evaluated by considering the maximal upwards and downwards deviation of from the central result. The results on alpha-s(MZ) \[section:results\_a\_s\] ---------------------------------------------------- The values of obtained with the CT10-NLO PDF set are listed in Table \[tbl:CT10\_nlo\_as\_results\] together with the experimental, PDF, NP, and scale uncertainties for each bin in rapidity and for a simultaneous fit of all rapidity bins. To disentangle the uncertainties of experimental origin from those of the PDFs, additional fits without the latter uncertainty source are performed. An example for the evaluation of the uncertainties in a $\chi^{2}$ fit is shown in Fig. \[fig:chi2\_points\]. The NP and scale uncertainties are determined via separate fits, as explained above. For the two outer rapidity bins ($1.5<\yabs<2.0$ and $2.0<\yabs<2.5$) the series in values of of the CT10-NLO PDF set does not reach to sufficiently low values of . As a consequence the shape of the curve at minimum up to $\chisq+1$ can not be determined completely. To avoid extrapolations based on a polynomial fit to the available points, the alternative evolution code of the package [@Salam:2008qg] is employed. This is the same evolution code as chosen for the creation of the CT10 PDF set. Replacing the original evolution in CT10 by , can be set freely and in particular different from the default value used in a PDF set, but at the expense of losing the correlation between the value of and the fitted PDFs. Downwards or upwards deviations from the lowest and highest values of , respectively, provided in a PDF series are accepted for uncertainty evaluations up to a limit of $\abs{\Delta\alpsmz} = 0.003$. Applying this method for comparisons, within the available range of values, an additional uncertainty is estimated to be negligible. For comparison the CT10-NNLO PDF set is used for the determination of . These results are presented in Table \[tbl:CT10\_nnlo\_as\_results\]. -------------------- ------------- -------- --------------------------------------------------------------- ------------- No. of data points $\yabs<0.5$ 33 0.1189 $ 0.0024\,(\text{exp}) \pm 0.0030\,(\mathrm{PDF})$ $16.2/32$ $0.0008\,(\mathrm{NP}) ^{+0.0045}_{-0.0027}\,(\text{scale})$ $0.5\leq\yabs<1.0$ 30 0.1182 $ 0.0024\,(\text{exp}) \pm 0.0029\,(\mathrm{PDF})$ $25.4/29$ $0.0008\,(\mathrm{NP}) ^{+0.0050}_{-0.0025}\,(\text{scale})$ $1.0\leq\yabs<1.5$ 27 0.1165 $ 0.0027\,(\text{exp}) \pm 0.0024\,(\mathrm{PDF})$ $9.5/26$ $0.0008\,(\mathrm{NP}) ^{+0.0043}_{-0.0020}\,(\text{scale})$ $1.5\leq\yabs<2.0$ 24 0.1146 $ 0.0035\,(\text{exp}) \pm 0.0031\,(\mathrm{PDF})$ $20.2/23$ $0.0013\,(\mathrm{NP}) ^{+0.0037}_{-0.0020}\,(\text{scale})$ $2.0\leq\yabs<2.5$ 19 0.1161 $ 0.0045\,(\text{exp}) \pm 0.0054\,(\mathrm{PDF})$ $12.6/18$ $0.0015\,(\mathrm{NP}) ^{+0.0034}_{-0.0032}\,(\text{scale})$ $\yabs<2.5$ 133 0.1185 $ 0.0019\,(\text{exp}) \pm 0.0028\,(\mathrm{PDF})$ $104.1/132$ $0.0004\,(\mathrm{NP}) ^{+0.0053}_{-0.0024}\,(\text{scale})$ -------------------- ------------- -------- --------------------------------------------------------------- ------------- ![The minimisation with respect to using the CT10-NLO PDF set and data from all rapidity bins. The experimental uncertainty is obtained from the values for which is increased by one with respect to the minimum value, indicated by the dashed line. The curve corresponds to a second-degree polynomial fit through the available points.[]{data-label="fig:chi2_points"}](chi_all_ct10_nlo){width="\cmsFigWidth"} -------------------- ------------- -------- --------------------------------------------------------------- ------------- No. of data points $\yabs<0.5$ 33 0.1180 $0.0017\,(\text{exp}) \pm 0.0027\,(\mathrm{PDF})$ $15.4/32$ $0.0006\,(\mathrm{NP})^{+0.0031}_{-0.0026}\,(\text{scale})$ $0.5\leq\yabs<1.0$ 30 0.1176 $0.0016\,(\text{exp}) \pm 0.0026\,(\mathrm{PDF})$ $23.9/29$ $ 0.0006\,(\mathrm{NP}) ^{+0.0033}_{-0.0023}\,(\text{scale})$ $1.0\leq\yabs<1.5$ 27 0.1169 $ 0.0019\,(\text{exp}) \pm 0.0024\,(\mathrm{PDF})$ $10.5/26$ $ 0.0006\,(\mathrm{NP}) ^{+0.0033}_{-0.0019}\,(\text{scale})$ $1.5\leq\yabs<2.0$ 24 0.1133 $ 0.0023\,(\text{exp}) \pm 0.0028\,(\mathrm{PDF})$ $22.3/23$ $ 0.0010\,(\mathrm{NP}) ^{+0.0039}_{-0.0029}\,(\text{scale})$ $2.0\leq\yabs<2.5$ 19 0.1172 $ 0.0044\,(\text{exp}) \pm 0.0039\,(\mathrm{PDF})$ $13.8/18$ $ 0.0015\,(\mathrm{NP}) ^{+0.0049}_{-0.0060}\,(\text{scale})$ $\yabs<2.5$ 133 0.1170 $ 0.0012\,(\text{exp}) \pm 0.0024\,(\mathrm{PDF})$ $105.7/132$ $ 0.0004\,(\mathrm{NP}) ^{+0.0044}_{-0.0030}\,(\text{scale})$ -------------------- ------------- -------- --------------------------------------------------------------- ------------- The final result using all rapidity bins and the CT10-NLO PDF set is (last row of Table \[tbl:CT10\_nlo\_as\_results\]) where experimental, PDF, NP, and scale uncertainties have been added quadratically to give the total uncertainty. The result is in agreement with the world average value of $\alpsmz = 0.1185 \pm 0.0006$ [@Agashe:2014kda], with the Tevatron results [@Affolder:2001hn; @Abazov:2009nc; @Abazov:2012lua], and recent results obtained with LHC data [@Malaescu:2012ts; @Chatrchyan:2013txa; @CMS-PAPERS-TOP-12-022]. The determination of , which is based on the CT10-NLO PDF set, is also in agreement with the result obtained using the NNPDF2.1-NLO and MSTW2008-NLO sets, as shown in Table \[tbl:ALL\_nlo\_nnlo\_as\_results\]. For comparison this table also shows the results using the CT10, MSTW2008, and NNPDF2.1 PDF sets at NNLO. The values are in agreement among the different NLO PDF sets within the uncertainties. PDF set --------------- -------- ------------------------------------------------------------- ------------- CT10-NLO 0.1185 $ 0.0019\,(\text{exp}) \pm 0.0028\,(\mathrm{PDF})$ $104.1/132$ $0.0004\,(\mathrm{NP})^{+0.0053}_{-0.0024}\,(\text{scale})$ NNPDF2.1-NLO 0.1150 $ 0.0015\,(\text{exp}) \pm 0.0024\,(\mathrm{PDF})$ $103.5/132$ $0.0003\,(\mathrm{NP})^{+0.0025}_{-0.0025}\,(\text{scale})$ MSTW2008-NLO 0.1159 $ 0.0012\,(\text{exp}) \pm 0.0014\,(\mathrm{PDF})$ $107.9/132$ $0.0001\,(\mathrm{NP})^{+0.0024}_{-0.0030}\,(\text{scale})$ CT10-NNLO 0.1170 $ 0.0012\,(\text{exp}) \pm 0.0024\,(\mathrm{PDF})$ $105.7/132$ $0.0004\,(\mathrm{NP})^{+0.0044}_{-0.0030}\,(\text{scale})$ NNPDF2.1-NNLO 0.1175 $ 0.0012\,(\text{exp}) \pm 0.0019\,(\mathrm{PDF})$ $103.0/132$ $0.0001\,(\mathrm{NP})^{+0.0018}_{-0.0020}\,(\text{scale})$ MSTW2008-NNLO 0.1136 $ 0.0010\,(\text{exp}) \pm 0.0011\,(\mathrm{PDF})$ $108.8/132$ $0.0001\,(\mathrm{NP})^{+0.0019}_{-0.0024}\,(\text{scale})$ Applying the PS correction factor to the NLO theory prediction in addition to the NP correction as discussed in Section \[sec:pscorrection\], the fit using all rapidity bins and the CT10-NLO PDF set yields $\alpsmz = 0.1204 \pm 0.0018\,(\text{exp})$. This value is in agreement with our main result of Eq. (\[eqn:analytic\_result\]), which is obtained using only the NP correction factor. To investigate the running of the strong coupling, the fitted region is split into six bins of and the fitting procedure is repeated in each of these bins. The six extractions of are reported in Table \[tbl:as\_values\]. The values are evolved to the corresponding energy scale $Q$ using the two-loop solution to the renormalization group equation (RGE) within . The value of $Q$ is calculated as a cross section weighted average in each fit region. These average scale values $Q$, derived again with the framework, are identical within about 1for different PDFs. To emphasise that theoretical uncertainties limit the achievable precision, Tables \[tbl:as\_unc\] and \[tbl:as\_q\_unc\] present for the six bins in the total uncertainty as well as the experimental, PDF, NP, and scale components, where the six experimental uncertainties are all correlated. ---------- ------ -------------------------------- -------------------------------- ------------- ----------- range $Q$ No. of data () () points 114–196 136 $0.1172\,_{-0.0043}^{+0.0058}$ $0.1106\,^{+0.0052}_{-0.0038}$ 20 $6.2/19$ 196–300 226 $0.1180\,_{-0.0046}^{+0.0063}$ $0.1038\,^{+0.0048}_{-0.0035}$ 20 $7.6/19$ 300–468 345 $0.1194\,_{-0.0049}^{+0.0064}$ $0.0993\,^{+0.0044}_{-0.0034}$ 25 $8.1/24$ 468–638 521 $0.1187\,_{-0.0051}^{+0.0067}$ $0.0940\,^{+0.0041}_{-0.0032}$ 20 $10.6/19$ 638–905 711 $0.1192\,_{-0.0056}^{+0.0074}$ $0.0909\,^{+0.0042}_{-0.0033}$ 22 $11.2/21$ 905–2116 1007 $0.1176\,_{-0.0065}^{+0.0111}$ $0.0866\,^{+0.0057}_{-0.0036}$ 26 $33.6/25$ ---------- ------ -------------------------------- -------------------------------- ------------- ----------- ---------- ------ -------- --------------- --------------- --------------- ------------------------ range $Q$ () () 114–196 136 0.1172 $\pm{0.0031}$ $\pm{0.0018}$ $\pm{0.0007}$ $_{-0.0022}^{+0.0045}$ 196–300 226 0.1180 $\pm{0.0034}$ $\pm{0.0019}$ $\pm{0.0011}$ $_{-0.0025}^{+0.0048}$ 300–468 345 0.1194 $\pm{0.0032}$ $\pm{0.0023}$ $\pm{0.0010}$ $_{-0.0027}^{+0.0049}$ 468–638 521 0.1187 $\pm{0.0029}$ $\pm{0.0031}$ $\pm{0.0006}$ $_{-0.0027}^{+0.0052}$ 638–905 711 0.1192 $\pm{0.0034}$ $\pm{0.0032}$ $\pm{0.0005}$ $_{-0.0030}^{+0.0057}$ 905–2116 1007 0.1176 $\pm{0.0047}$ $\pm{0.0040}$ $\pm{0.0002}$ $_{-0.0020}^{+0.0092}$ ---------- ------ -------- --------------- --------------- --------------- ------------------------ ---------- ------ -------- --------------- --------------- --------------- ------------------------ range $Q$ () () 114–196 136 0.1106 $\pm{0.0028}$ $\pm{0.0016}$ $\pm{0.0006}$ $_{-0.0020}^{+0.0040}$ 196–300 226 0.1038 $\pm{0.0026}$ $\pm{0.0015}$ $\pm{0.0008}$ $_{-0.0019}^{+0.0037}$ 300–468 345 0.0993 $\pm{0.0022}$ $\pm{0.0016}$ $\pm{0.0007}$ $_{-0.0019}^{+0.0033}$ 468–638 521 0.0940 $\pm{0.0018}$ $\pm{0.0019}$ $\pm{0.0004}$ $_{-0.0017}^{+0.0032}$ 638–905 711 0.0909 $\pm{0.0019}$ $\pm{0.0018}$ $\pm{0.0003}$ $_{-0.0017}^{+0.0032}$ 905–2116 1007 0.0866 $\pm{0.0025}$ $\pm{0.0021}$ $\pm{0.0001}$ $_{-0.0011}^{+0.0048}$ ---------- ------ -------- --------------- --------------- --------------- ------------------------ Figure \[fig:as\_running\] presents the running of the strong coupling and its total uncertainty as determined in this analysis. The extractions of in six separate ranges of $Q$, as presented in Table \[tbl:as\_values\], are also shown. In the same figure the values of at lower scales determined by the H1 [@Aaron:2009vs; @Aaron:2010ac; @Andreev:2014wwa], ZEUS [@Abramowicz:2012jz], and D0 [@Abazov:2009nc; @Abazov:2012lua] collaborations are shown for comparison. Recent CMS measurements [@Chatrchyan:2013txa; @CMS-PAPERS-TOP-12-022], which are in agreement with the determination of this study, are displayed as well. The results on reported here are consistent with the energy dependence predicted by the RGE. ![The strong coupling (full line) and its total uncertainty (band) as determined in this analysis using a two-loop solution to the RGE as a function of the momentum transfer $Q=\pt$. The extractions of in six separate ranges of $Q$ as presented in Table \[tbl:as\_values\] are shown together with results from the H1 [@Aaron:2010ac; @Andreev:2014wwa], ZEUS [@Abramowicz:2012jz], and D0 [@Abazov:2009nc; @Abazov:2012lua] experiments at the HERA and Tevatron colliders. Other recent CMS measurements [@Chatrchyan:2013txa; @CMS-PAPERS-TOP-12-022] are displayed as well. The uncertainties represented by error bars are subject to correlations.[]{data-label="fig:as_running"}](As_running_ALL){width="\cmsFigWidth"} Study of PDF constraints with HERAFitter {#sec:herafitter} ======================================== The PDFs of the proton are an essential ingredient for precision studies in hadron-induced reactions. They are derived from experimental data involving collider and fixed-target experiments. The DIS data from the HERA-I  collider cover most of the kinematic phase space needed for a reliable PDF extraction. The  inclusive jet cross section contains additional information that can constrain the PDFs, in particular the gluon, in the region of high fractions $x$ of the proton momentum. The project [@Alekhin:2014rma; @HERAFitter:2013hf] is an open-source framework designed among other things to fit PDFs to data. It has a modular structure, encompassing a variety of theoretical predictions for different processes and phenomenological approaches for determining the parameters of the PDFs. In this study, the recently updated version 1.1.1 is employed to estimate the impact of the CMS inclusive jet data on the PDFs and their uncertainties. Theory is used at NLO for both processes, i.e. up to order $\alps^2$ for DIS and up to order $\alps^3$ for inclusive jet production in  collisions. Correlation between inclusive jet production and the PDFs {#sec:pdf_sensitivity} --------------------------------------------------------- The potential impact of the CMS inclusive jet data can be illustrated by the correlation between the inclusive jet cross section $\sigma_{\text{jet}}(Q)$ and the PDF $xf(x,Q^2)$ for any parton flavour $f$. The NNPDF Collaboration [@Ball:2008by] provides PDF sets in the form of an ensemble of replicas $i$, which sample variations in the PDF parameter space within allowed uncertainties. The correlation coefficient $\varrho_f(x,Q)$ between a cross section and the PDF for flavour $f$ at a point $(x,Q)$ can be computed by evaluating means and standard deviations from an ensemble of $N$ replicas as Here, the angular brackets denote the averaging over the replica index $i$, and $\Delta$ represents the evaluation of the corresponding standard deviation for either the jet cross section, $\Delta_{\sigma_{\text{jet}}(Q)}$, or a PDF, $\Delta_{xf(x,Q^2)}$. Figure \[fig:correlation\_pdf\_xs\_gqq\] presents the correlation coefficient between the inclusive jet cross section and the gluon, u valence quark, and d valence quark PDFs in the proton. {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} The correlation between the gluon PDF and the inclusive jet cross section is largest at central rapidity for most jet . In contrast, the correlation between the valence quark distributions and the jet cross section is rather small except for very high such that some impact can be expected at high $x$ from including these jet data in PDF fits. In the forward region the correlation between the valence quark distributions and the jet cross sections is more pronounced at high $x$ and smaller jet . Therefore, a significant reduction of the PDF uncertainties is expected by including the CMS inclusive jet cross section into fits of the proton structure. The fitting framework {#section:fittingframework} --------------------- ### The HERAFitter setup {#section:herafittersetup} The impact of the CMS inclusive jet data on proton PDFs is investigated by including the jet cross section measurement in a combined fit at NLO with the HERA-I inclusive DIS cross sections [@Aaron:2009aa], which were the basis for the determination of the HERAPDF1.0 PDF set. The analysis is performed within the framework using the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi [@Gribov:1972ri; @Altarelli:1977zs; @Dokshitzer:1977sg] evolution scheme at NLO as implemented in the package [@Botje:2010ay] and the generalised-mass variable-flavour number Thorne–Roberts scheme [@Thorne:1997ga; @Thorne:2006qt]. In contrast to the original HERAPDF fit, the results presented here require the DIS data to fulfill $Q^2 > Q_\text{min}^2 = 7.5 \GeVsq$ instead of $3.5\GeVsq$. The amount of DIS data left out by the increased $Q_\text{min}^2$ threshold is rather small and concerns a phase space where a perturbative description is less reliable. A similar, higher cutoff has been applied by the ATLAS Collaboration [@Aad:2012sb; @Aad:2014qja]. As a crosscheck all fits have been performed for a cutoff of $Q^2 > Q_\text{min}^2 = 3.5 \GeVsq$, and the results are consistent with the ones obtained using the more stringent cutoff. Differences beyond the expected reduction of uncertainties at low $x$ have not been observed. The following PDFs are independent in the fit procedure: $xu_v(x)$, $xd_v(x)$, $xg(x)$, and $x\overline{U}(x)$, $x\overline{D}(x)$, where $x\overline{U}(x) = x\overline{u}(x)$, and $x\overline{D}(x) = x\overline{d}(x) + x\overline{s}(x)$. Similar to Ref. [@Abramowicz:1900rp], a parameterization with 13 free parameters is used. At the starting scale $Q_0$ of the QCD evolution, chosen to be $Q_0^2 = 1.9 \GeVsq$, the PDFs are parameterized as follows: $$\begin{aligned} xg(x) &= A_g x^{B_g} (1-x)^{C_g} - A'_g x^{B'_g} (1-x)^{C'_g},\\ xu_v(x) &= A_{u_{v}} x^{B_{u_{v}}} (1-x)^{C_{u_{v}}} (1 + E_{u_{v}}x^2),\\ xd_v(x) &= A_{d_v} x^{B_{d_v}} (1-x)^{C_{d_{v}}},\\ x\overline U(x) &= A_{\overline U} x^{B_{\overline U}} (1-x)^{C_{\overline U}}, \text{and}\\ x\overline D(x) &= A_{\overline D} x^{B_{\overline D}} (1-x)^{C_{\overline D}}. \end{aligned}$$ The normalisation parameters $A_g$, $A_{u_{v}}$, and $A_{d_{v}}$ are constrained by QCD sum rules. Additional constraints $B_{\overline U}=B_{\overline D}$ and $A_{\overline U} = A_{\overline D}(1-f_s)$ are applied to ensure the same normalisation for the $\overline u$ and $\overline d$ densities for $x \to 0$. The strangeness fraction is set to $f_s = 0.31$, as obtained from neutrino-induced dimuon production [@Mason:2007zz]. The parameter $C'_g$ is fixed to 25 [@Martin:2009iq; @Thorne:2006qt] and the strong coupling constant to $\alpsmz= 0.1176$. ### Definition of the goodness-of-fit estimator {#sec:fitsetup} The agreement between the $N$ data points $D_i$ and the theoretical predictions $T_i$ is quantified via a least-squares method, where For fully correlated sources of uncertainty following a Gaussian distribution with a zero mean and a root-mean-square of unity as assumed here, this definition is equivalent to Eq. (\[chi2\_square\]) [@Stump:2001gu]. As a bonus, the systematic shift of the nuisance parameter $r_k$ for each source in a fit is determined. Numerous large shifts in either direction indicate a problem as for example observed while fitting with this technique and the old uncertainty correlation prescription. In the following, the covariance matrix is defined as $\mathrm{C} = \cov_{\text{stat}} + \cov_{\text{uncor}}$, while the JES, unfolding, and luminosity determination are treated as fully correlated systematic uncertainties $\beta_{ik}$ with nuisance parameters $r_k$. Including also the NP uncertainties, treated via the offset method in Section \[sec:alphas\], in the form of one nuisance parameter in total $K$ such sources are defined. Of course, PDF uncertainties emerge as results of the fits performed here, in contrast to serving as inputs, as they do in the fits of presented in Section \[sec:alphas\]. All the fully correlated sources are assumed to be multiplicative to avoid the statistical bias that arises from uncertainty estimations taken from data [@Lyons:1989gh; @D'Agostini:2003nk; @Ball:2009qv]. As a consequence, the covariance matrix of the remaining sources has to be re-evaluated in each iteration step. To inhibit the compensation of large systematic shifts by increasing simultaneously the theoretical prediction and the statistical uncertainties, the systematic shifts of the theory are taken into account before the rescaling of the statistical uncertainty. Otherwise alternative minima in can appear that are associated with large theoretical predictions and correspondingly large shifts in the nuisance parameters. These alternative minima are clearly undesirable [@HERAFitter:2013hf]. ### Treatment of CMS data uncertainties {#section:cmsdatauncertainties} The JES is the dominant source of experimental systematic uncertainty in jet cross sections. As described in Section \[sec:measurementjec\], the - and $\eta$-dependent JES uncertainties are split into 16 uncorrelated sources that are fully correlated in and $\eta$. Following the modified recommendation for the correlations versus rapidity of the single-particle response source as given in Section \[sec:measurementjec\], it is necessary to split this source into five parts for the purpose of using the uncertainties published in Ref. [@Chatrchyan:2012bja] within the fits. The complete set of uncertainty sources is shown in Table \[tab:cmsjets2011:nuisance\]. By employing the technique of nuisance parameters, the impact of each systematic source of uncertainty on the fit result can be examined separately. For an adequate estimation of the size and the correlations of all uncertainties, the majority of all systematic sources should be shifted by less than one standard deviation from the default in the fitting procedure. Table \[tab:cmsjets2011:nuisance\] demonstrates that this is the case for the CMS inclusive jet data. \[tab:cmsjets2011:nuisance\] ------- ------------------------------------------------------------------- --------- JEC0 absolute jet energy scale $ 0.09$ JEC1 MC extrapolation $0.00$ JEC2a single-particle response barrel $ 1.31$ JEC2b single-particle response endcap $-1.46$ JEC2c single-particle decorrelation $\yabs<0.5$ $0.20$ JEC2d single-particle decorrelation $0.5\leq\yabs<1.0$ $ 0.19$ JEC2e single-particle decorrelation $1.0\leq\yabs<1.5$ $ 0.92$ JEC3 jet flavor correction $ 0.04$ JEC4 time-dependent detector effects $-0.15$ JEC5 jet resolution in endcap 1 $ 0.76$ JEC6 jet resolution in endcap 2 $-0.42$ JEC7 jet resolution in HF $ 0.01$ JEC8 correction for final-state radiation $0.03$ JEC9 statistical uncertainty of $\eta$-dependent correction for endcap $-0.42$ JEC10 statistical uncertainty of $\eta$-dependent correction for HF $ 0.00$ JEC11 data-MC difference in $\eta$-dependent pileup correction $ 0.91$ JEC12 residual out-of-time pileup correction for prescaled triggers $-0.17$ JEC13 offset dependence in pileup correction $-0.03$ JEC14 MC pileup bias correction $ 0.39$ JEC15 jet rate dependent pileup correction $ 0.29$ $-0.26$ $-0.07$ $ 0.60$ ------- ------------------------------------------------------------------- --------- In contrast, with the original assumption of full correlation within the 16 JES systematic sources across all bins, shifts beyond two standard deviations were apparent and led to a re-examination of this issue and the improved correlation treatment of the JES uncertainties as described previously in Section \[sec:measurementjec\]. Determination of PDF uncertainties according to the HERAPDF prescription {#section:herapdf_pdf_uncertainties} ------------------------------------------------------------------------ The uncertainty in the PDFs is subdivided into experimental, model, and parameterization uncertainties that are studied separately. In the default setup of the framework, experimental uncertainties are evaluated following a Hessian method [@Stump:2001gu], and result from the propagated statistical and systematic uncertainties of the input data. For the model uncertainties, the offset method [@Botje:2001fx] is applied considering the following variations of model assumptions: 1. The strangeness fraction $f_s$, by default equal to $0.31$, is varied between $0.23$ and $0.38$. 2. The b-quark mass is varied by $\pm 0.25\GeV$ around the central value of $4.75\GeV$. 3. The c-quark mass, with the central value of $1.4\GeV$, is varied to $1.35\GeV$ and $1.65\GeV$. For the downwards variation the charm production threshold is avoided by changing the starting scale to $Q_0^2=1.8\GeVsq$ in this case. 4. The minimum $Q^2$ value for data used in the fit, $Q^2_\mathrm{min}$, is varied from $7.5\GeVsq$ to $5.0\GeVsq$ and $10\GeVsq$. The PDF parameterization uncertainty is estimated as described in Ref. [@Aaron:2009aa]. By employing the more general form of parameterizations for gluons and the nongluon flavours, respectively, it is tested whether the successive inclusion of additional fit parameters leads to a variation in the shape of the fitted results. Furthermore, the starting scale $Q_0$ is changed to $Q^2_0 = 1.5\GeVsq$ and $2.5\GeVsq$. The maximal deviations of the resulting PDFs from those obtained in the central fit define the parameterization uncertainty. The experimental, model, and parameterization uncertainties are added in quadrature to give the final PDF uncertainty according to the HERAPDF prescription [@Aaron:2009aa]. Using this fitting setup, the partial values per number of data points, , are reported in Table \[tab:fit:results\] for each of the neutral current (NC) and charged current (CC) data sets in the HERA-I DIS fit and for the combined fit including the CMS inclusive jet data. The achieved fit qualities demonstrate the compatibility of all data within the presented PDF fitting framework. The resulting PDFs with breakdown of the uncertainties for the gluon, the sea, u valence, and d valence quarks with and without CMS inclusive jet data are arranged next to each other in Figs. \[fit:cmsjets2011:gsea:fitscale:qcut75\] and \[fit:cmsjets2011:uvdv:fitscale:qcut75\]. Figure \[fit:cmsjets2011:hera:directcomparison:1\_4:all\] provides direct comparisons of the two fit results with total uncertainties. The parameterization and model uncertainties of the gluon distribution are significantly reduced for almost the whole $x$ range from $10^{-4}$ up to 0.5. When DIS data below $Q^2_\mathrm{min} = 7.5 \GeVsq$ are included in the fit, the effect is much reduced for the low $x$ region $x < 0.01$, but remains important for medium to high $x$. Also, for the u valence, d valence, and sea quark distributions some reduction in their uncertainty is visible at high $x$ ($x \gtrsim 0.1$). At the same time, some structure can be seen, particularly in the parameterization uncertainties that might point to a still insufficient flexibility in the parameterizations. Therefore, a comparison is presented in the next Section \[section:mcddr\_pdf\_uncertainties\], using the MC method with the regularisation based on data, which is also implemented within the framework. \[tab:fit:results\] -------------------------------------- ----- ----- ------ ----- ------ data set NC HERA-I H1-ZEUS combined $\Pem\Pp$ 145 109 0.75 109 0.75 NC HERA-I H1-ZEUS combined $\Pep\Pp$ 337 309 0.91 311 0.92 CC HERA-I H1-ZEUS combined $\Pem\Pp$ 34 20 0.59 22 0.65 CC HERA-I H1-ZEUS combined $\Pep\Pp$ 34 29 0.85 35 1.03 CMS inclusive jets 133 — — 102 0.77 data set(s) HERA-I data 537 468 0.87 — — HERA-I & CMS data 670 — — 591 0.88 -------------------------------------- ----- ----- ------ ----- ------ {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Determination of PDF uncertainties using the MC method with regularisation {#section:mcddr_pdf_uncertainties} -------------------------------------------------------------------------- =800 To study more flexible PDF parameterizations, a MC method based on varying the input data within their correlated uncertainties is employed in combination with a data-based regularisation technique. This method was first used by the NNPDF Collaboration and uses a more flexible parameterization to describe the $x$ dependence of the PDFs [@Ball:2008by]. To avoid the fitting of statistical fluctuations present in the input data (over-fitting) a data-based stopping criterion is introduced. The data set is split randomly into a “fit” and a “control” sample. The minimisation is performed with the “fit” sample while simultaneously the of the “control” sample is calculated using the current PDF parameters. It is observed that the of the “control” sample at first decreases and then starts to increase again because of over-fitting. At this point, the fit is stopped. This regularisation technique is used in combination with a MC method to estimate the central value and the uncertainties of the fitted PDFs. Before a fit, several hundred replica sets are created by allowing the central values of the measured cross section to fluctuate within their statistical and systematic uncertainties while taking into account all correlations. For each replica, a fit to NLO QCD is performed, which yields an optimum value and uncertainty for each parameter. The collection of all replica fits can then provide an ensemble average and root-mean-square. Moreover, the variations to derive the model dependence of the HERAPDF prescription do not lead to any further increase of the uncertainty. Similarly to Fig. \[fit:cmsjets2011:hera:directcomparison:1\_4:all\] for the HERAPDF method, a direct comparison of the two fit results with total uncertainties is shown in Fig. \[fit:cmsjets2011:ddr:directcomparison:1\_4:all\] for the MC method. The total uncertainty derived with the MC method is almost always larger than with the HERAPDF technique, and in the case of the gluon at low $x$, it is much larger. In both cases a significant reduction of the uncertainty in the gluon PDF is observed, notably in the $x$ range from $10^{-2}$ up to 0.5. Both methods also lead to a decrease in the gluon PDF between $10^{-2}$ and $10^{-1}$ and an increase for larger $x$. Although this change is more pronounced when applying the MC method, within the respective uncertainties both results are compatible. For the sea quark only small differences in shape are observed, but, in contrast to the HERAPDF method that exhibits reduced uncertainties for $x > 0.2$, this is not visible when using the MC method. Both methods agree on a very modest reduction in uncertainty at high $x > 0.05$ in the u valence quark PDF and a somehwat larger improvement for the d valence quark PDF, which is expected from the correlations, studied in Fig. \[fig:correlation\_pdf\_xs\_gqq\], where the quark distributions are constrained via the  contribution to jet production at high and . Changes in shape of the d valence quark PDF go into opposite directions for the two methods, but are compatible within uncertainties. All preceding figures presented the PDFs at the starting scale of the evolution of $Q^2 = 1.9 \GeVsq$. For illustration, Fig. \[fit:cmsjets2011:ddr:directcomparison:10\_4:all\] displays the PDFs derived with the regularised MC method after evolution to a scale of $Q^2 = 10^4 \GeVsq$. Finally, Fig. \[fit:cmsjets2011:overview\] shows an overview of the gluon, sea, u valence, and d valence distributions at the starting scale of $Q^2 = 1.9 \GeVsq$ for both techniques, the HERAPDF and the regularised MC method. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} ![Overview of the gluon, sea, u valence, and d valence PDFs before (dashed line) and after (full line) including the CMS inclusive jet data into the fit. The plots show the PDF fit outcome from the HERAPDF method () and from the MC method with data-derived regularisation (). The PDFs are shown at the starting scale $Q^2 = 1.9 \GeVsq$. The total uncertainty including the CMS inclusive jet data is shown as a band around the central fit result.[]{data-label="fit:cmsjets2011:overview"}](HERADISCMSJETS2011V2QED_13P_NLO_V301_EIG_1_9 "fig:"){width="48.00000%"} ![Overview of the gluon, sea, u valence, and d valence PDFs before (dashed line) and after (full line) including the CMS inclusive jet data into the fit. The plots show the PDF fit outcome from the HERAPDF method () and from the MC method with data-derived regularisation (). The PDFs are shown at the starting scale $Q^2 = 1.9 \GeVsq$. The total uncertainty including the CMS inclusive jet data is shown as a band around the central fit result.[]{data-label="fit:cmsjets2011:overview"}](HERAMCDR_DISCMSJETS_22P_NLO_QMIN75_V300_1_9 "fig:"){width="48.00000%"} Combined fit of PDFs and the strong coupling constant {#sec:combinedfits} ----------------------------------------------------- Inclusive DIS data alone are not sufficient to disentangle effects on cross section predictions from changes in the gluon distribution or simultaneously. Therefore was always fixed to 0.1176 in the original HERAPDF1.0 derivation. When the CMS inclusive jet data are added, this constraint can be dropped and and its uncertainty (without $Q$ scale variations) is determined to $\alpsmz = 0.1192\,^{+0.0023}_{-0.0019}\,\text{(all except scale)}$. Repeating the fit with the regularised MC method gives $\alpsmz = 0.1188\pm0.0041\,\text{(all except scale)}$. =1000 Since a direct correspondence among the different components of the uncertainty can not easily be established, only the quadratic sum of experimental, PDF, and NP uncertainties are presented, which is equivalent to the total uncertainty without scale uncertainty. For example, the HERA-I DIS data contribute to the experimental uncertainty in the combined fits, but contribute only to the PDF uncertainty in separate fits. The HERAPDF prescription for PDF fits tends to small uncertainties, while the uncertainties of the MC method with data-derived regularisation are twice as large. For comparison, the corresponding uncertainty in using more precisely determined PDFs from global fits as in Section \[sec:alphas\] gives a result between the two: $\alpsmz = 0.1185\pm0.0034\,\text{(all except scale)}$. The evaluation of scale uncertainties is an open issue, which is ignored in all global PDF fits given in Table \[tab:pdfsets\]. The impact is investigated in Refs. [@Martin:2009iq; @Gao:2012he; @Ball:2012wy; @Watt:2013oha], where scale definitions and $K$-factors are varied. Lacking a recommended procedure for the scale uncertainties in combined fits of PDFs and , two evaluations are reported here for the HERAPDF method. In the first one, the combined fit of PDFs and is repeated for each variation of the scale factors from the default choice of $\mur=\muf=\pt$ for the same six combinations as explained in Section \[section:fit\_proc\]. The scale for the HERA DIS data is not changed. The maximal observed upward and downward changes of with respect to the default scale factors are then taken as scale uncertainty, irrespective of changes in the PDFs: $\Delta\alpsmz =\,^{+0.0022}_{-0.0009}\,\mathrm{(scale)}$. The second procedure is analogous to the method employed to determine in Section \[sec:alphas\]. The best PDFs are derived for a series of fixed values of as done for the global PDF sets. Using this series of PDFs with varying values of , the combination of PDF and that best fits the HERA-I DIS and CMS inclusive jet data is found. The values determined both ways are consistent with each other. The fits are now repeated for the same scale factor variations, and the maximal observed upward and downward changes of with respect to the default scale factors are taken as scale uncertainty: $\Delta\alpsmz =\,^{+0.0024}_{-0.0039}\,\mathrm{(scale)}$. In contrast to the scale uncertainty of the first procedure, there is less freedom for compensating effects between different gluon distributions and values in the second procedure, and the latter procedure leads to a larger scale uncertainty as expected. In overall size the uncertainty is similar to the final results on reported in the last section: $\Delta\alpsmz =\,^{+0.0053}_{-0.0024}\,\mathrm{(scale)}$. Summary {#sec:summary} ======= An extensive QCD study has been performed based on the CMS inclusive jet data in Ref. [@Chatrchyan:2012bja]. Fits dedicated to determine have been performed involving QCD predictions at NLO complemented with electroweak and nonperturbative (NP) corrections. Employing global parton distribution functions (PDFs), where the gluon is constrained through data from various experiments, the strong coupling constant has been determined to be which is consistent with previous results. It was found that the published correlations of the experimental uncertainties adequately reflect the detector characteristics and reliable fits of standard model parameters could be performed within each rapidity region. However, when combining several rapidity regions, it was discovered that the assumption of full correlation in rapidity $y$ had to be revised for one source of uncertainty in the jet energy scale, which suggested a modified correlation treatment that is described and applied in this work. To check the running of the strong coupling, all fits have also been carried out separately for six bins in inclusive jet , where the scale $Q$ of is identified with . The observed behaviour of is consistent with the energy scale dependence predicted by the renormalization group equation of QCD, and extends the H1, ZEUS, and D0 results to the region. The impact of the inclusive jet measurement on the PDFs of the proton is investigated in detail using the tool. When the CMS inclusive jet data are used together with the HERA-I DIS measurements, the uncertainty in the gluon distribution is significantly reduced for fractional parton momenta $x \gtrsim 0.01$. Also, a modest improvement in uncertainty in the u and d valence quark distributions is observed. The inclusion of the CMS inclusive jet data also allows a combined fit of and of the PDFs, which is not possible with the HERA-I inclusive DIS data alone. The result is consistent with the reported values of obtained from fits employing global PDFs. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: the Austrian Federal Ministry of Science, Research and Economy and the Austrian Science Fund; the Belgian Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport, and the Croatian Science Foundation; the Research Promotion Foundation, Cyprus; the Ministry of Education and Research, Estonian Research Council via IUT23-4 and IUT23-6 and European Regional Development Fund, Estonia; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucléaire et de Physique des Particules / CNRS, and Commissariat à l’Énergie Atomique et aux Énergies Alternatives / CEA, France; the Bundesministerium für Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Innovation Office, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Republic of Korea; the Lithuanian Academy of Sciences; the Ministry of Education, and University of Malaya (Malaysia); the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Business, Innovation and Employment, New Zealand; the Pakistan Atomic Energy Commission; the Ministry of Science and Higher Education and the National Science Centre, Poland; the Fundação para a Ciência e a Tecnologia, Portugal; JINR, Dubna; the Ministry of Education and Science of the Russian Federation, the Federal Agency of Atomic Energy of the Russian Federation, Russian Academy of Sciences, and the Russian Foundation for Basic Research; the Ministry of Education, Science and Technological Development of Serbia; the Secretaría de Estado de Investigación, Desarrollo e Innovación and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the Ministry of Science and Technology, Taipei; the Thailand Center of Excellence in Physics, the Institute for the Promotion of Teaching Science and Technology of Thailand, Special Task Force for Activating Research and the National Science and Technology Development Agency of Thailand; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the National Academy of Sciences of Ukraine, and State Fund for Fundamental Researches, Ukraine; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation. Individuals have received support from the Marie-Curie programme and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the Compagnia di San Paolo (Torino); the Consorzio per la Fisica (Trieste); MIUR project 20108T4XTM (Italy); the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF; and the National Priorities Research Program by Qatar National Research Fund. Sources of uncertainty in the calibration of jet energies in CMS {#sec:jessources} ================================================================ In the following, the full list of uncertainty sources of the jet energy calibration procedure that were originally considered by CMS and that were used in Ref. [@Chatrchyan:2012bja] is presented including a short description. It is recommended to apply the procedure with updated correlations for the JEC2 source, as described in Section \[sec:measurementjec\]. A general description of the jet energy calibration procedure of CMS is given in Ref. [@Chatrchyan:2011ds]. When simulations were employed, the following event generators have been used: version 6.4.22 [@Sjostrand:2006za] tune Z2 and version 2.4.2 [@Bahr:2008pv] with the default tune of version 2.3. JEC0: : Absolute uncertainty.\ Using data with photon+jet and $Z$+jet events an absolute calibration of jet energies is performed in the jet range of 30–600. Uncertainties in the determination of electromagnetic energies in the ECAL, of the muon momenta from $Z\to\mu\mu$ decays, and of the corrections for initial- and final-state (ISR and FSR) radiation are propagated together with the statistical uncertainty to give the absolute JES uncertainty. JEC1: : High- and low-extrapolation uncertainty.\ Where an absolute calibration with data is not possible, events are produced with the event generators and and are subsequently processed through the CMS detector simulation based on  [@Agostinelli:2002hh]. Differences in particular in modelling the fragmentation process and the underlying event lead to an extrapolation uncertainty relative to the directly calibrated jet range of $30$–$600$. JEC2: : High-extrapolation uncertainty.\ This source accounts for a $\pm 3$% variation in the single-particle response that is propagated to jets using a parameterized fast simulation of the CMS detector [@Abdullin:2011zz]. JEC3: : Jet flavour related uncertainty.\ Differences in detector response to light, charm, and bottom quark as well as gluon jets relative to the mixture predicted by QCD for the measured processes are evaluated on the basis of simulations with and . JEC4: : Uncertainty caused by time dependent detector effects.\ This source considers residual time-dependent variations in the detector conditions such as the endcap ECAL crystal transparency. JEC5–JEC10: : $\eta$-dependent uncertainties coming from the dijet balance method: JEC5–JEC7: : Caused by the jet energy resolution. These three sources are assumed to be fully correlated for the endcap with upstream tracking detectors (JEC5), the endcap without upstream tracking detectors (JEC6), and the HF calorimeter (JEC7). JEC8: : $\eta$-dependent uncertainty caused by corrections for final-state radiation. The uncertainty is correlated from one region to the other and increases towards HF. JEC9–JEC10: : Statistical uncertainty in the determination of $\eta$-dependent corrections. These are two separate sources for the endcap without upstream tracking detectors (JEC9), and the HF calorimeter (JEC10). JEC11–JEC15: : Uncertainties for the pileup corrections: JEC11: : parameterizes differences between data and MC events versus $\eta$ in zero-bias data. JEC12: : estimates residual out-of-time pileup for prescaled triggers, if MC events are reweighted to unprescaled data. JEC13: : covers an offset dependence on jet (due to, e.g. zero-suppression effects), when the correction is calibrated for jets in the range of 20–30. JEC14: : accounts for differences in measured offset from zero-bias MC events and from generator-level information in a QCD sample. JEC15: : covers observed jet rate variations versus the average number of reconstructed primary vertices in the 2011 single-jet triggers after applying L1 corrections. Comparison to theoretical predictions by POWHEG + PYTHIA6 {#theory_data} ========================================================= Figure \[fig:DataTheory\_comp4\] presents ratios of data over theory predictions at NLO using the CT10-NLO PDF set multiplied by electroweak and NP corrections including PDF uncertainties. {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} The CMS Collaboration \[app:collab\] ==================================== =5000=500=5000

--- abstract: 'We report the discovery of a quasar pair at $z=5$ separated by 21. Both objects were identified as quasar candidates using simple color selection techniques applied to photometric catalogs from the CFHT Legacy Survey (CFHTLS). Spectra obtained with the MMT present no discernible offset in redshift between the two objects; on the other hand, there are clear differences in the emission line profiles and in the multiwavelength spectral energy distributions that strongly disfavor the hypothesis that they are gravitationally lensed images of a single quasar. Both quasars are surprisingly bright given their proximity (a projected separation of $\sim135$ kpc), with $i=19.4$ and $i=21.4$. Previous measurements of the luminosity function demonstrate that luminous quasars are extremely rare at $z=5$; the existence of this pair suggests that quasars have strong small-scale clustering at high redshift. Assuming a real-space correlation function of the form $\xi(r) \propto (r/r_0)^{-2}$, this discovery implies a correlation length $r_0 \ga 20$[$h^{-1}\,{\rm Mpc}$]{}, consistent with a rapid strengthening of quasar clustering at high redshift as seen in previous observations and predicted by theoretical models where feedback effects are inefficient at shutting down black hole growth at high redshift.' author: - 'Ian D. McGreer, Sarah Eftekharzadeh, Adam D. Myers, and Xiaohui Fan' title: A Constraint on Quasar Clustering at $z=5$ from a Binary Quasar --- [^1] Introduction {#sec:introduction} ============ The discovery that luminous quasars cluster strongly at redshifts approaching $z\sim 4$ [with a scale length of $r_0\sim25$[$h^{-1}\,{\rm Mpc}$]{}; @She07] potentially poses an interesting cosmological challenge. It could be that quasar clustering strongly declines with luminosity at high redshift, meaning that current samples only trace the most strongly clustered sources. But, faint quasars do not appear to cluster significantly more weakly than bright quasars at $z\sim2.5$ [@Whi12; @Eft15] or below [e.g. @daA08; @She09; @She13]. Thus, if quasar clustering is highly luminosity-dependent at $z\sim4.5$, then quasars (as a population) would have to alter rapidly over 10% of the Hubble Time, then change more quiescently over the final 80% of cosmic history. Further, most models invoke a narrow range of halo mass for a wide range of quasar luminosity in order to reproduce the quasar luminosity function (e.g., @Lid06; see also the discussion in Appendix B of @Whi12). Alternatively, quasars at high redshift could simply trace the growth of their parent dark matter halos while they are actively accreting. Such a scenario essentially represents the maximal possible increase in clustering amplitude with redshift; under scenarios other than this “maximal growth” model the correlation length of quasars should eventually diminish at high redshift. Fig. 13 of @Hop07 illustrates this point — quasar clustering should decrease at $z>4$ for scenarios in which quasars are efficiently quenched. Observations of quasar clustering at $z>4$ are currently limited to relatively small samples of highly luminous quasars [@She07]. Improving this situation is a significant challenge, given that it requires expensive spectroscopic campaigns targeting faint candidates at low sky density ($\sim$1 deg$^{-2}$). One promising avenue is the study of pairs of quasars that are separated by both a small angle on the plane of the sky and by a small velocity window in redshift space. Such pairs, often called “binary quasars” in the literature, are sufficiently rare to confirm with dedicated spectroscopic follow-up, but have a very strong clustering strength. Binary quasars can therefore be used to estimate the correlation length of quasar clustering even using small samples [e.g. @Hen06; @Mye08; @She10]. Of order a dozen $z\sim4$ binary quasars with proper separations of less than $\sim1$ Mpc are currently known [e.g. @Hen10]. The highest redshift binary quasar discovered to date is a quasar pair at $z=4.26$ separated by 33, or about 230kpc proper, on the plane of the sky [@Sch00]. The pair was discovered serendipitously—while spectroscopically confirming an $i=20.4$ quasar candidate a second $i=21.3$ quasar at the same redshift happened to be located in the slit. This single quasar pair was sufficient to ascertain that $z \sim4.25$ quasars cluster with a correlation length of $r_0\sim10$–30Mpc, an observation later confirmed using much larger samples by @She07. In this paper, we present a similar find. During a survey of $4.7~\la\,z~\la\,5.2$ quasar candidates, we have discovered a quasar pair separated by 21, or about 135kpc proper. In this paper, we discuss the discovery of this pair, our reasoning for why it is a binary quasar (rather than a gravitational lens) and the implications of such a pair for quasar clustering at $z\sim5$. All quoted magnitudes are on the AB system [@OkeGunn83] and corrected for Galactic extinction using the dust maps of @Sch98. We adopt a cosmology of ($\Omega_{\rm m}, \Omega_\Lambda, h\equiv H_0/100\,{\rm km\,s^{-1}\,Mpc^{-1}}) = (0.307,0.693,0.677)$ consistent with recent results from [*Planck*]{} [@Planck15]. Observations {#sec:observations} ============ Initial Selection from CFHTLS-W1 -------------------------------- In previous work, we measured the $z=5$ quasar luminosity function (QLF) using quasars selected from the SDSS Stripe 82 region to a depth of $i=22$ [@McGreer13]. We are extending this work to fainter quasars using the CFHTLS-Wide survey [@Gwyn12]. The full CFHTLS-Wide encompasses four fields with a total area of 150 deg$^2$ and includes five optical bands, $ugriz$. We downloaded the publicly available stacked images[^2] and generated object catalogs using [SExtractor]{} [@sextractor]. The catalogs include PSF photometry derived from the [PSFEx]{} models provided by the CFHTLS. We included two of the CFHT-Wide fields in the selection described here; W1 at 02:18 -07:00 and W3 at 14:18 +54:30. Full details of our faint $z\sim5$ quasar selection will be provided in a future work. Briefly, we use the difference between the elliptical Kron aperture magnitude (MAG\_AUTO) flux measurements and the PSF flux measurement (MAG\_PSF) from [SExtractor]{} to obtain a rough star/galaxy separation. Through various tests we found that requiring ${\rm MAG\_AUTO} - {\rm MAG\_PSF} > -0.15$ is highly complete to point sources to a limit of $i<23$, while greatly reducing contamination from compact galaxies. We further apply a number of quality cuts. First, we require clean photometry [SExtractor]{} FLAGS $<=$ 4. Second, we remove objects lying within the masked regions (generally due to bright stars) as provided by the CFHTLS; this reduces our effective area by $\sim1$%. Finally, we remove CFHTLS fields for which the stellar locus is poorly matched to a reference locus derived from the CFHTLS-Deep survey, indicating issues with the photometric calibration. This reduces the areas of both W1 and W3 to 45 deg$^2$ (the full areas are 72 deg$^2$ and 49 deg$^2$, respectively). After applying the morphological and quality cuts and a loose color cut of $r-i > 0.8$ the resulting density of objects is $\sim$130 deg$^{-2}$ in the two CFHTLS fields. In order to select $z\sim5$ quasar candidates we adapt the color criteria employed in @McGreer13 to account for the bandpass differences between the SDSS and CFHT photometric systems. This results in the following color cuts: $$\begin{aligned} S/N(u) &< 2.2 \\ S/N(g) &< 2.2 ~~ {\rm OR} ~~ g-r > 1.8 \\ r-i &> 1.3 \\ i-z &< 0.625((r-i) - 1.0) \\ i-z &< 0.55 \\\end{aligned}$$ The resulting set of objects were visually examined and those likely to be artefacts (e.g., diffraction spikes) were rejected. In the W1 (W3) field 26 (21) objects are identified as quasar candidates to a limit of $i=23$. When preparing our observations we noticed that two of the bright candidates had a very small separation on the sky. We examined the imaging and considered both to be viable high redshift quasar candidates, and thus prioritized them for observation. However, we emphasize that we did not search for binary candidates [*a priori*]{}; rather, we selected the objects simultaneously with identical criteria. MMT Observations {#sec:mmtspec} ---------------- We observed [CFHTLS J0221-0342]{} with the Red Channel spectrograph [@mmtred] on the MMT 6.5m telescope on 2014 Jan 9, 2014 Jan 10, and 2014 Aug 28. All observations utilized a 1$\times$180 longslit aligned at a position angle of $-12.2^{\circ}$ in order to capture both quasar candidates. The objects were dispersed with the 270 mm$^{-1}$ grating at a resolution of $R\sim640$. For the 2014 Jan 9 observations the central wavelength was set to 7500 Å, providing wavelength coverage from 5670Å to 9290Å, and the total integration time was 70min. For the 2014 Jan 10 and 2014 Aug 28 observations the central wavelength was 8500Å (6600Å $\la \lambda \la$ 1$\mu$m) and the total integration times were 45min. and 60min., respectively. In all cases the seeing was marginal (15 – 2) with non-photometric conditions. The spectra were processed in a standard fashion with Pyraf-based scripts; details of the processing method are given in @McGreer13. Wavelength calibration was provided by an internal HeNeAr lamp, and an approximate flux calibration was obtained from observations of the spectrophotometric standard star Feige 110. The calibrations were taken immediately before the science spectra. The processed spectra from each of the three nights were interpolated onto a common linear wavelength grid and combined using inverse-variance weighting. The final spectra are displayed in Figure \[fig:spectra\]. The spectroscopy immediately confirmed that both candidates are quasars at $z\sim5$. In Section \[sec:pairorlens\] we interpret the spectra and other available data in order to determine whether they represent two quasars at a similar redshift or gravitationally lensed images of a single source quasar. We have obtained a total of 19 MMT spectra out of the 47 candidates with $i<23$ in W1 and W3. A more complete analysis of this sample will be presented in a future work. Relevant to this work, we note that [*all*]{} of the observed objects are quasars at $z \ga 4.5$, indicating that the color selection is highly pure. In addition, our simulations show that the color selection is highly complete ($>90$%) in the range $4.75 < z \la 5.15$, with a tail to $\sim50\%$ completeness out to $z\sim5.4$ [see @McGreer13 for details on the simulation method]. Although we have spectra for only 40% of our candidates, we consider it highly likely that any similar pair of small-separation quasars would be included in our target list, and thus we conclude that only one such pair lies within the 90 deg$^2$ search area to the flux limit of $i<23$[^3]. This estimate of the area of our survey (90 deg$^2$) will be used in §\[sec:clustering\] to infer the clustering strength of quasars at $z\sim5$. Additional Observations ----------------------- [CFHTLS J0221-0342]{} lies within the XMM-LSS [@xmmlss] survey region, and thus has a wide array of multiwavelength observations. Table \[tab:photometry\] lists photometric observations of the quasar pair, including deep near-IR photometry from the UKIDSS Deep eXtragalactic Survey [DXS; @ukidss] and deep Spitzer photometry from the SWIRE survey [@swire]. The brighter quasar is also an X-ray source in the XMM-XXL survey and was included as an ancillary quasar target in the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 [@DR12]. This ancillary program[^4] targeted XMM-XXL sources for spectroscopy; [QSO-A]{} was the highest redshift X-ray source in the sample, with a redshift of $z=5.011$. It is important to note that although [CFHTLS J0221-0342]{} happens to lie within a deep extragalactic survey field, for our survey it was selected based on optical colors from the CFHTLS-Wide alone. [lcc]{} RA (J2000) & 02:21:12.613 & 02:21:12.315\ Dec (J2000) & -03:42:52.19 & -03:42:31.64\ $g$ & $ 24.104 \pm 0.029 $ & $ 25.822 \pm 0.142 $\ $r$ & $ 21.221 \pm 0.004 $ & $ 23.594 \pm 0.032 $\ $i$ & $ 19.383 \pm 0.001 $ & $ 21.423 \pm 0.005 $\ $z$ & $ 19.524 \pm 0.002 $ & $ 21.478 \pm 0.011 $\ $J$ & $ 19.452 \pm 0.012 $ & $ 21.588 \pm 0.047 $\ $K$ & $ 19.086 \pm 0.009 $ & $ 21.044 \pm 0.038 $\ $3.6\mu$m & $ 18.735 \pm 0.012 $ & $ 20.705 \pm 0.037 $\ $4.5\mu$m & $ 18.923 \pm 0.013 $ & $ 20.996 \pm 0.067 $\ $5.8\mu$m & $ 19.248 \pm 0.073 $ & -\ $8.0\mu$m & $ 18.825 \pm 0.046 $ & -\ $24\mu$m & $ 17.090 \pm 0.051 $ & - \[tab:photometry\] A Quasar Pair or a Lens? {#sec:pairorlens} ======================== We first consider whether [CFHTLS J0221-0342]{} represents a binary quasar or a pair of gravitationally lensed images of a single source at $z=5$. Because lensing is achromatic, if they are lensed images the two objects should present similar colors at all wavelengths. There are two important effects that can affect the observed colors: 1) differential reddening along the independent light paths to the two images, and 2) time delays between the lensed images combined with intrinsic source variability. Figure \[fig:seds\] presents the multiwavelength SEDs of [CFHTLS J0221-0342]{}, where deviations as large as $\sim$30% from the mean flux offset are present across a wide range of wavelengths. These differences are far greater than the photometric uncertainties, which are $<1$% from $g$-band to $K$-band. The statistical uncertainties are appropriate here since the photometry represents [*relative*]{} flux measurements between the two objects, as the same calibrations have been applied to both objects. In addition, because of their proximity, the photometry of the two objects is always simultaneous, minimizing any differences arising from intrinsic source variability. We apply the $\chi^2$ statistic given by @Hen06 [their equation 2] to the two SEDs as a test of the hypothesis that a simple flux scaling combined with photometric scatter accounts for their differences (i.e., they are lensed images) and rule this out at high significance, obtaining $\chi^2/\nu$ = 223/7 from the multiwavelength SEDs[^5]. Finally, there is no trend in the flux differences that would be consistent with reddening. The MMT spectra of the two objects are highly similar to the level of the $S/N$ and resolution available. We obtain a small redshift difference from fitting the emission line in the MMT spectra; Gaussian fits with the IRAF [*splot*]{} command return $z({\rm A})=5.016$ and $z({\rm B})=5.019$, a difference of $\approx160$[kms$^{-1}$]{}. However, the line is weakly detected in the [QSO-B]{} spectrum and the uncertainty on the line centroid is $\approx 300$[kms$^{-1}$]{}, thus this difference is not significant. The Ly$\alpha$ lines are detected at high $S/N$ in both spectra and the line profiles in the wings are nearly identical (Fig. \[fig:lyacompare\]), agreeing to within two spectral pixels, or $\la270$[kms$^{-1}$]{}. As it is difficult to conclusively state the velocity offset between the two spectra, in the rest of this work we adopt the difference obtained from the fits, $\Delta{z} \la 0.03$. There are differences between the two spectra that indicate they are not likely to originate from a single source quasar. [QSO-B]{} has a clear absorption feature just blueward of Ly$\alpha$ at $\sim$7286Å while [QSO-A]{} has a transmission peak at $\sim$7170Å; both of these features are significantly weaker in the opposite spectrum (Figure \[fig:lyacompare\]). Also, the emission from [QSO-A]{} is greater than that from [QSO-B]{}. Another consideration is that a configuration that produces a large separation (21) image pair of a $z=5$ quasar is highly unlikely; only three large-separation lensed quasar systems are known in the entire SDSS [@Inada03; @Dahle13; @Rusu13]. Large separations can arise from group- or cluster-scale lens masses. For this configuration, the peak in the expected lens redshift distribution is at $z\sim0.7$ and the probability of a source quasar at $z=5$ with an image separation $>20$ is extremely small [@Hen07; @Li07]. Furthermore, the CFHTLS imaging is sufficiently deep that a group-scale overdensity at $z<1$ should be immediately apparent in the optical imaging, but as Figure \[fig:colorim\] shows, there are no galaxies detected between [QSO-A]{} and [QSO-B]{}, and no obvious overdensity of galaxies in the vicinity. In conclusion, the mismatched SEDs of the two objects and the lack of any obvious foreground mass to generate a wide separation lens configuration strongly argue against the lens hypothesis for this pair of objects. We proceed to interpret them as a binary quasar. Implications for Quasar Clustering at $z > 5$ {#sec:clustering} ============================================= Quasars peak as a luminous population near $z\sim2.5$, and bright quasars at higher redshift are increasingly rare [@Ric06]. Combining the depth and area necessary to survey large numbers of $z\sim5$ quasars is therefore taxing. Due to the difficulty in studying a significant quantity of quasars at high redshift, quasar clustering has only been measured in a statistical fashion out to $z\sim4$, by @She07, who found a correlation length of $r_{\rm 0} \sim24$[$h^{-1}\,{\rm Mpc}$]{}. This level of clustering is considered “large,” in the sense that it is at the limits of what might be predicted by theoretical models that use the luminosity function to infer quasar clustering [e.g. @Hop07] and in the sense that clustering measurements at $z\sim2.5$ typically obtain significantly smaller values of $r_{\rm 0} \sim8$[$h^{-1}\,{\rm Mpc}$]{} [e.g. @Whi12; @Eft15]. That $r_{\rm 0}$ seems large at $z\sim4$ motivates further measurements of quasar clustering to determine if $r_{\rm 0}$ remains large at comparable or higher redshifts. A small number of close quasar pairs can be used as an alternative method for quantifying the clustering of high redshift quasars. In the absence of clustering it would be extraordinarily unlikely to find multiple quasars within a small cosmological volume, hence pairs can be used to infer the clustering strength required to increase the likelihood of companion quasars with small separations. An example of this approach is that of @Sch00, who used a single binary quasar at $z=4.25$ separated by $\Delta\theta=33.4$ on the plane of the sky to infer $r_{\rm 0} \sim12$–30Mpc for the correlation length of $z\sim4$ quasars, presaging the @She07 estimate of $r_{\rm 0} \sim24$[$h^{-1}\,{\rm Mpc}$]{}. In our chosen cosmology, the transverse projected separation of the @Sch00 quasar pair is 160[$h^{-1}\,{\rm kpc}$]{} compared to 90[$h^{-1}\,{\rm kpc}$]{} ($\Delta\theta=21$; $z=5.02$) for the pair we have discovered. In this section we use our binary quasar to estimate $r_{\rm 0}$ for quasars at $z\sim5$. It may be helpful to remember that at $z=5$, an angle of 1 subtends a transverse separation of 26[$h^{-1}\,{\rm kpc}$]{} comoving (4.35[$h^{-1}\,{\rm kpc}$]{} proper). The luminosity function {#qlf} ----------------------- The significance of observing a close pair of quasars relative to random chance is determined from the QLF. We adopt the recent measurement of the QLF at $z=5$ from @McGreer13 based on quasars drawn from the SDSS Stripe 82 region, extending to a depth of $i=22$. Although [CFHTLS J0221-0342]{} is drawn from a deeper survey ($i<23$) and thus requires extrapolation from the @McGreer13 results, it is worth noting that both quasars have $i<22$ and are thus within the range of the Stripe 82 measurement. The QLF is typically fit with a double power-law form, $$\label{phi} \Phi(M,z)=\frac{\Phi^{*}(z)}{10^{0.4(\alpha+1)(M-M^{*})}+10^{0.4(\beta+1)(M-M^{*})}} ~.$$ @McGreer13 estimate the characteristic luminosity to be $M^{*}=-27.21$ and the faint and bright end slopes to be $\alpha=-2.03$ and $\beta=-4.0$, respectively. The parameter $\Phi^{*}$ is best described by a term that evolves with redshift, $\log \Phi^{*}(z)=\log \Phi^{*}(z=6)+k(z-6)$, with $\log \Phi^{*}(z=6)=-8.94$ and $\rm k=-0.47$ (see §6 of @McGreer13 for a detailed discussion on the fitting procedure and redshift evolution of the QLF parameters). As noted in §\[sec:observations\], our survey of $z\sim5$ quasars has progressed such that it is reasonable to assume our binary quasar is drawn from a complete survey covering 90[deg$^{2}$]{} to a flux limit of $i < 23$. The number density of quasars brighter than $i'=23$ in our survey is calculated by taking the integral of the QLF between the $k$-corrected absolute magnitudes, $M_{i}^{\rm bright}$ and $M_{i'}$, corresponding to the bright and faint end of the apparent magnitude range of our survey: $$\label{phist} n(z,i<i')=\int_{M_{i}^{\rm bright}}^{M_{i'}} dM_{i} ~ \Phi(M_{i},z).$$ We use a constant $k$-correction of $k_{\rm corr}=-2.2$, which is reasonable over the full redshift range of interest [see Fig. 6 of @McGreer13]. The number densities obtained from the QLF are relatively insensitive to the bright limit (in apparent magnitude) adopted for the integration. We ignore the incompleteness due to our selection efficiency, which would reduce the observed number densities. We stress that this makes our measurements more conservative, in that any incompleteness in our survey would [*increase*]{} the inferred clustering signal, as we would be more likely to have missed additional close pairs. The QLF predicts $\sim0.9$[deg$^{2}$]{} quasars over the redshift range $4.7<z<5.2$, which already hints that a quasar pair separated by 21 at $z\sim5$ would be highly unusual if quasars were not significantly clustered at high redshift. Estimating the Correlation Length Using the @Sch00 Formalism {#clsres} ------------------------------------------------------------ Following @Sch00 we determine the correlation length of quasars by comparing the single pair we have found to the number of pairs we would expect to find in the volume enclosing our pair. The mean number expected in a given volume can be determined from the QLF. The odds of finding two quasars in that volume (corresponding to our binary quasar) can then be determined from the Poisson distribution, as Poisson statistics are an excellent model for quasar clustering on small scales where the pairs are independent [e.g. @Mye06]. The correlation length can be related to the excess clustering over random as $$\begin{gathered} \label{schr} \frac{N}{N_{\rm rand}}= \frac{\int_{0}^{R} 4\pi r^2 \xi(r)~dr}{\int_{0}^{R}4{\pi}r^2~dr} \\ = \frac{3}{3-\gamma}\left(\frac{R}{r_0}\right)^{-\gamma}|_{\gamma=2} = 3\left(\frac{R}{r_0}\right)^{-2} \ ,\end{gathered}$$ where we have adopted a power-law form of $\xi(r)=(r/r_0)^{-\gamma}$ with $\gamma=2$ for the slope of the correlation function, as used in many studies of the clustering of quasars at high redshift on large and small scales [e.g. @She10; @Whi12; @Eft15]. As it is unclear [*a priori*]{} to what degree the redshift difference between the components of our binary quasar is due to line-of-sight separation versus infall, we will calculate “minimum,” “medium” and “maximum” separations based on the transverse and line-of-sight comoving separations of our pair [again following @Sch00]. Our quasar pair is at $z=5.02$, is separated by 21 on the plane of the sky, and has a redshift difference of $\Delta z \la 0.03$. If the separation of the quasars is entirely in the transverse direction, then the components of our pair are separated by 810kpc comoving (the “minimum” separation). If the full redshift difference is also due to physical separation, then the components of our pair are separated by 16.1Mpc comoving (the “maximum” separation). If half of the redshift difference is attributable to physical separation, then the components of our pair are separated by 8.08Mpc comoving (the “medium” separation). Integrating the @McGreer13 QLF to a limit of $i=23$ over the redshift range $4.7<z<5.2$ results in a number density of $\approx 1.75\times10^{-7}\, \rm Mpc^{-3}$, ignoring selection completeness. Our “minimum” separation of 810kpc implies a quasar pair embedded in a volume of 2.25Mpc$^{3}$. Multiplying this by the number density yields an expectation of $3.9 \times 10^{-7}$ quasars in the volume of interest. Assuming a Poisson distribution, the probability of two quasars lying within this volume is $7.3\times 10^{-3}$ for an all-sky survey. As our survey only encompasses 90[deg$^{2}$]{}, the odds of finding the binary quasar within our survey are 1 in 62[,]{}400, implying that this discovery would have been extremely unlikely in the absence of clustering. Substituting $N/N_{\rm rand} = 62{,}400$ and $R = 810$kpc into Eqn. \[schr\] implies $r_0 = 117$Mpc, or $r_0 = 74$[$h^{-1}\,{\rm Mpc}$]{}. Similar logic implies $r_0 = 25$[$h^{-1}\,{\rm Mpc}$]{} for our “medium” separation case and $r_0 = 18$[$h^{-1}\,{\rm Mpc}$]{} for our “maximum” separation case. Thus our expectation is that $r_0 \sim 25$[$h^{-1}\,{\rm Mpc}$]{} for quasars at $z\sim5$, with a lower-bound of $r_0 \sim 18$[$h^{-1}\,{\rm Mpc}$]{}. If we adopt a shallower slope for the power law index the correlation length would need to be even greater; e.g., for $\gamma=1.8$ the “medium” separation case results in $r_0 \sim 33$[$h^{-1}\,{\rm Mpc}$]{}. Our measurement implies that the amplitude of quasar clustering at $z\sim5$ is similar to that measured at $z\sim4$ by @She07. Estimating the Correlation Length Using the @Hen06 Formalism {#estimating-the-correlation-length-using-the-formalism} ------------------------------------------------------------ The @Sch00 formalism is simple and straightforward. However, by selecting “minimum” and “maximum” extremes for the distribution of peculiar velocities in the redshift-space direction this method ignores our expectation for this distribution. In particular, the “minimum” case applies if the two quasars are at the same distance and any redshift difference is due to the local velocity field. This is a reasonable assumption in our case; however, it is useful to characterize the uncertainty on that difference, and for that we turn to the method of @Hen06. This method accounts for a realistic peculiar velocity distribution for the quasars so that we can place a more formal (Poisson) error on the correlation length we infer from the existence of the binary. Following the method described in @Hen06, we assume that binary quasars are well-described by a maximum possible peculiar velocity of $|v_{\rm max}| = 2000$[kms$^{-1}$]{}. We then project the redshift-space correlation function over this velocity interval, $$\label{wp} w_p(R,z)=\int_{-v_{\rm max}/aH(z)}^{v_{\rm max}/aH(z)} \xi_{s}(R, s, z) ds ~,$$ where $H(z)$ is the expansion rate at redshift $z$ and $\xi_{s}$ is the redshift-space quasar correlation function. We include the cosmological scale factor $a=1/(1+z)$ to convert distances to comoving units. Given that we are working with a single pair embedded in a relatively large volume, $w_p$ could be highly sensitive to changes in the model correlation function with scale and/or redshift. Again following @Hen06 we ameliorate this effect by measuring the volume-averaged correlation function $\bar W_{p}(z)$ over the entire radial bin of comoving distance that corresponds to the transverse separation of our binary quasar $[R_{\rm min}, R_{\rm max}]$. This results in $$\label{wpbar} \bar W_{p}(z) = \frac{\int_{-\frac{v_{\rm max}}{aH(z)}}^{\frac{v_{\rm max}}{aH(z)}} \int_{R_{\rm min}}^{R_{\rm max}} \xi_s(R, s, z) 2\pi R dR ds}{V_{\rm shell}} ~,$$ where $V_{\rm shell}$, the volume of a cylindrical shell in redshift space, is given by $$\label{vol} V_{\rm shell} = \pi \left(R_{\rm max}^{2}-R_{\rm min}^{2}\right) \left[\frac{2v_{\rm max}}{aH(z)}\right] ~.$$ Although the redshift-space correlation function is a convolution of the real-space correlation function with the distribution of peculiar velocities, we are projecting over a volume large enough to contain the full extent of this distribution function. Hence it is a reasonable approximation to replace the redshift-space correlation function $\xi_{s}(R,s,z)$ with its real-space counterpart $\xi(r,z)$ where $\xi(r)=(r/r_{0})^{-\gamma}$ and $r^2 = R^2 + x^2$. We adopt $\gamma=2$ for the real-space correlation function, as explained in §\[clsres\], and assume that this form remains valid for all redshifts of interest (i.e. $\xi(r,z) = \xi(r)$). The integral in eqn. \[wpbar\] is instead conducted along the line-of-sight distance $x$, $$\begin{gathered} \label{wpbarfinal} \bar W_{p}(R_{\rm min},R_{\rm max},z)= \\ \frac{\int_{-\frac{v_{\rm max}}{aH(z)}}^{\frac{v_{\rm max}}{aH(z)}} \int_{R_{\rm min}}^{R_{\rm max}} \left(\frac{x^2+R^2}{{r_0}^2}\right)^{-\frac{\gamma}{2}} 2\pi R dR dx} {V_{\rm shell}} ~.\end{gathered}$$ We adopt $[R_{\rm min},R_{\rm max}] = [25, 550]$[$h^{-1}\,{\rm kpc}$]{} (i.e. \[40, 810\] kpc). Here, $R_{\rm max}$ corresponds to the 21 separation of our binary quasar at $z=5.02$. We set $R_{\rm min}$ to correspond to 1, below which the seeing in the CFHTLS imaging we used for target selection would have precluded the selection of a pair of quasars. The number of expected companions of any individual quasar in our survey as a function of transverse separation and redshift is then $$\begin{gathered} \label{nc} N_{c} = n(4.7<z<5.2,i<23) \\ \times V_{\rm shell} ~[1+\bar W_{p}(25, 550, z)] ~,\end{gathered}$$ where $n$ is given by our adopted QLF (see Eqn. \[phist\]). By varying the correlation length in Eqn. \[wpbarfinal\] we obtain a range of model values for the number of companions we expect at a separation of 21 within our survey volume at $z=5.02$. We then compare the predicted number of companions to the discovery of a single binary out of a sample of 47 quasar candidates,[^6] i.e., within our sample of 47 quasars there are two objects within the cylindrical shell defined by $V_{\rm shell}$. Thus we are seeking a model for the correlation function that results in the expected number of companions to be $N_c = 2/47 = 0.04255$. We find the correlation length that best describes our binary quasar is $86$[$h^{-1}\,{\rm Mpc}$]{}, with a 1$\sigma$ lower bound of $25$[$h^{-1}\,{\rm Mpc}$]{}. The lower bound has been determined using the confidence interval for a single measurement provided by @Geh86. As with the analysis in §\[clsres\], this result depends on our adopted QLF and that our assumed form of the correlation function is valid and non-evolving across our redshift range of interest. In particular, in calculating an $r_0$ that is significantly larger than the scales probed by our pair, we are implicitly assuming that clustering at small scales can be extrapolated to large scales[^7] (as was found to be the case for low-$z$ quasars by @Kayo+12 and consistent with results at $z\sim3\mbox{--}4$ from @She10). Whether we employ the @Hen06 formalism or the @Sch00 formalism, we find that the existence of this binary quasar implies a correlation length $r_0 > 20$[$h^{-1}\,{\rm Mpc}$]{}, consistent with the $r_0 \sim 25$[$h^{-1}\,{\rm Mpc}$]{} measured at $z\sim4$ by @She07. This strongly suggests that quasars are at least as clustered at $z\sim5$ as has been found at $z\sim4$. Conclusions {#sec:conclusions} =========== We have discovered a pair of quasars with apparently identical redshifts of $z=5.02$ and a separation of 21 on the sky. A number of factors argue against the pair being gravitationally lensed images of a single source quasar. These include differences in spectral profiles and SED shapes, and the fact that no deflector is present between the two quasars in relatively deep optical imaging. Assuming the quasar pair is a binary, the small projected separation (135 kpc proper) and lack of a clear redshift offset implies their physical separation is quite small, within a factor of $\sim2\mbox{--}3$ of the virial radius of a typical quasar-hosting dark matter halo at high redshift [$\sim 10^{12}\mbox{--}10^{13}~M_\sun$, @Hop07; @She07; @Whi12; @Eft15]. This single detection of a binary at $z=5$ favors models where quasars are strongly clustered at high redshift, at least on small scales. The clustering of quasars is sensitive not only to the triggering mechanism(s), but also feedback effects that terminate black hole growth. Globally, the quasar population experiences a “downsizing” trend at $z\la3$, as activity shifts to lower mass and lower luminosity systems [e.g., @Ross+13]. This is often thought to be due to feedback, as the most massive systems form early but rapidly shut down after their quasar phase, freezing their black hole mass while the host halos continue to grow. At high redshift the picture is murkier, with few constraints on the black hole mass and Eddington ratio distributions. This is demonstrated by @Hop07, who compare three disparate models for the continued growth of black holes at high redshift after their luminous quasar phase. If feedback is efficient at high redshift, the correlation length should decrease strongly with increasing redshift. If feedback is inefficient such that the black holes grow continuously until $z\sim2$, the correlation length flattens out at high redshift. If quasars grow at the same rate as their host halos at $z>3$ (the “maximal” growth model), the correlation length rises sharply, implying that quasars at $z=5$ are more strongly clustered by a factor of a few compared to the measurements at $z\sim2.5$. While repeating the caveat that we have only measured small-scale clustering from a single, high-luminosity binary at $z=5$, this observation is most consistent with a large correlation length, favoring the models in which feedback is highly inefficient. Indeed, @Willott+10 find that the Eddington ratios of $z\sim6$ quasars are near unity across a range of luminosities, suggesting that fainter quasars are not in a “decaying” phase of black hole growth. It is surprising to have found two highly luminous quasars — presumably powered by $>10^8~M_\sun$ black holes and situated in massive dark matter halos — in such close proximity at this redshift. Previous searches have relied on wide-area surveys such as the SDSS, whereas we surveyed only $\sim0.1$% of the sky and yet discovered a $z=5$ binary quasar [*bright enough to have been selected from SDSS imaging*]{}. Whether this was simply a chance find will await a more comprehensive search for quasar pairs at $z \ga 5$. Measurements of high-redshift quasar clustering on large scales are crucial to discriminating between feedback models and better understanding the early growth of the most massive black holes in the universe. Such measurements are just possible today with wide-area, medium-depth fields such as SDSS Stripe 82 and the CFHTLS, and ongoing surveys such as the DES, the DESI Imaging Surveys (DECaLS, BASS, and MzLS), and KIDS also provide the requisite combination of depth and area. Obtaining a fully three-dimensional clustering measurement demands a considerable investment in spectroscopic follow-up given the low sky density; however, if quasars do cluster strongly at high redshift (as implied by our observations), a dense survey over a relatively small area could produce a statistically meaningful result. IDM and XF acknowledge support from NSF grants 11-06682 and NSF 15-15115. ADM and SE were supported in part by NASA ADAP award NNX12AE38G and by NSF awards 12-11112 and 15-15404. This research made use of Astropy, a community-developed core Python package for Astronomy [@astropy]. Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. Also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. The observations at the Canada-France-Hawaii Telescope were performed with care and respect from the summit of Maunakea which is a significant cultural and historic site. This work is based in part on data products produced at the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. [*Facilities:*]{} , [36]{} natexlab\#1[\#1]{} Alam, S. [et al.]{} 2015, , 219, 12 2013, , 558, A33 , E. & [Arnouts]{}, S. 1996, , 117, 393 , J., [Shanks]{}, T., [Croom]{}, S. M., [et al.]{} 2008, , 383, 565 , H., [Gladders]{}, M. D., [Sharon]{}, K., [et al.]{} 2013, , 773, 146 , S., [Myers]{}, A. D., [White]{}, M., [et al.]{} 2015, , 453, 2779 , N. 1986, , 303, 336 , S. D. J. 2012, , 143, 38 , J. F., [Dalal]{}, N., & [Bode]{}, P. 2007, , 654, 93 , J. F., [Myers]{}, A. D., [Shen]{}, Y., [et al.]{} 2010, , 719, 1672 , J. F., [Strauss]{}, M. A., [Oguri]{}, M., [et al.]{} 2006, , 131, 1 , P. F., [Lidz]{}, A., [Hernquist]{}, L., [Coil]{}, A. L., [Myers]{}, A. D., [Cox]{}, T. J., & [Spergel]{}, D. N. 2007, , 662, 110 , N., [Oguri]{}, M., [Pindor]{}, B., [et al.]{} 2003, , 426, 810 , I. & [Oguri]{}, M. 2012, , 424, 1363 , A., [Warren]{}, S. J., [Almaini]{}, O., [et al.]{} 2007, , 379, 1599 , G. L., [Mao]{}, S., [Jing]{}, Y. P., [Lin]{}, W. P., & [Oguri]{}, M. 2007, , 378, 469 , A., [Hopkins]{}, P. F., [Cox]{}, T. J., [Hernquist]{}, L., & [Robertson]{}, B. 2006, , 641, 41 , C. J. [et al.]{} 2003, , 115, 897 , I. D., [Jiang]{}, L., [Fan]{}, X., [et al.]{} 2013, , 768, 105 , A. D., [Brunner]{}, R. J., [Richards]{}, G. T., [et al.]{} 2006, , 638, 622 , A. D., [Richards]{}, G. T., [Brunner]{}, R. J., [et al.]{} 2008, , 678, 635 Oke, J. B. & Gunn, J. E. 1983, AJ, 266, 713 Pierre, M. [et al.]{} 2004, [\[@\]jnl[J. Cosmology Astropart. Phys.]{}]{}, 2004, 011 2015, ArXiv e-prints , G. T., [Strauss]{}, M. A., [Fan]{}, X., [et al.]{} 2006, , 131, 2766 , N. P., [McGreer]{}, I. D., [White]{}, M., [et al.]{} 2013, , 773, 14 , C. E., [Oguri]{}, M., [Iye]{}, M., [et al.]{} 2013, , 765, 139 , D. J., [Finkbeiner]{}, D. P., & [Davis]{}, M. 1998, , 500, 525 , G. D., [Weymann]{}, R. J., & [Foltz]{}, C. B. 1989, , 101, 713 , D. P., [Fan]{}, X., [Strauss]{}, M. A., [et al.]{} 2000, , 120, 2183 , Y., [Hennawi]{}, J. F., [Shankar]{}, F., [et al.]{} 2010, , 719, 1693 , Y., [McBride]{}, C. K., [White]{}, M., [et al.]{} 2013, , 778, 98 , Y., [Strauss]{}, M. A., [Oguri]{}, M., [et al.]{} 2007, , 133, 2222 , Y., [Strauss]{}, M. A., [Ross]{}, N. P., [et al.]{} 2009, , 697, 1656 , M., [Myers]{}, A. D., [Ross]{}, N. P., [et al.]{} 2012, , 424, 933 , C. J., [Albert]{}, L., [Arzoumanian]{}, D., [et al.]{} 2010, , 140, 546 [^1]: Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. [^2]: [<http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/megapipe/cfhtls/index.html>]{} [^3]: We searched our candidate list for additional pairs and found two quasars with a separation of 80; however, they have a redshift difference of $\Delta{z}=0.15$. [^4]: [<http://www.sdss.org/dr12/algorithms/ancillary/boss/xmmfollowup/>]{} [^5]: @Hen06 obtain a median value of $\chi^2/\nu$ = 33.1/4 for a sample of SDSS quasars at $2.4 < z < 2.45$; the expectation for lensed images in the absence of differential reddening is $\chi^2/\nu \sim 1$. To compute this value we use the photometry from $g$ through 4.5$\mu$m where both objects are well detected, hence the 7 degrees of freedom. If we restrict the data to the $griz$ bands to better compare with the SDSS data, we obtain $\chi^2/\nu$ = 212/3. [^6]: We ignore the fact that we do not have spectroscopic confirmation for all of our candidates. First, as mentioned in §\[sec:mmtspec\], we consider our survey to be highly complete to small-separation pairs. Second, our spectroscopy has shown that our color selection is highly pure, so that we expect nearly all of the 47 candidates to be $z\sim5$ quasars. It is more conservative to use the full candidate sample, as including only the objects with spectroscopy would greatly [*increase*]{} the implied clustering signal. In addition, the binary reported here was prioritized for observation, so it is more correct to adopt the full sample. [^7]: It is worth noting that the projected separation of our binary is just at the scale at which the two-halo term begins to contribute to the projected correlation function in the HOD models of @Kayo+12; see their Fig. 6.

--- author: - | Steven Weinberg[^1]\ Physics Department, University of Texas\ Austin, Texas 78712\ weinberg@physics.utexas.edu date: 'August 29, 1996' title: Theories of the Cosmological Constant --- [**Abstract**]{} — This is a talk given at the conference [*Critical Dialogues in Cosmology*]{} at Princeton University, June 24–27, 1996. It gives a brief summary of our present theoretical understanding regarding the value of the cosmological constant, and describes how to calculate the probability distribution of the observed cosmological constant in cosmological theories with a large number of subuniverses (i. e., different expanding regions, or different terms in the wave function of the universe) in which this constant takes different values. UTTG-10-96 Introduction ============ The problem of the cosmological constant looks different to astronomers and particle physicists. Astronomers may prefer the simplicity of a zero cosmological constant, but they are also prepared to admit the possibility of a cosmological constant in a range extending up to values that would make up most of the critical density required in a spatially flat Robertson–Walker universe. To a particle physicist, all the values in this observationally allowed range seem ridiculously implausible. To see why, it is convenient to consider the effective quantum field theory that takes into account only degrees of freedom with energy below about 100 GeV, with all higher energy radiative corrections buried in corrections to the various parameters in the effective Lagrangian. In this effective field theory, the vacuum energy density that serves as a source of the long-range gravitational field may be written as $$\rho_V=\frac{\Lambda}{8\pi G}+\frac{1}{2}\sum \hbar\omega\;,$$ where $\Lambda$ is the cosmological constant appearing in the Einstein field equations, and the second term symbolizes the contribution of quantum fluctuations in the fields of the effective field theory, cut off at particle energies equal to 100 GeV. Now, we know almost everything about this effective field theory — it is what particle physicists call the standard model — and we know that the quantum fluctuations do not cancel, so that on dimensional grounds, in units with $\hbar=c=1$, they yield $$\frac{1}{2}\sum \hbar\omega\approx (100\;{\rm GeV})^4\;$$ On the other hand, observations do not allow $\rho_V$ to be much greater than the critical density, which in these units is roughly $10^{-48}\;{\rm GeV}^4$. Not to worry — just arrange that the Einstein term $\Lambda$ has a value for which the two terms in Eq. (1) cancel to fifty-six decimal places. This is the cosmological constant problem: to understand this cancellation. Here I will consider three main directions for solving this problem[@Weinberg89]: -    Deep Symmetries -    Cancellation Mechanisms -    Anthropic Constraints By a ‘deep symmetry’ I mean some new symmetry of an underlying theory, which is [*not*]{} an unbroken symmetry of the effective field theory below 100 GeV (because we know all these symmetries), but which nevertheless requires $\rho_V$ to vanish. In other contexts supersymmetry can sometimes play the role of a deep symmetry, in the sense that some dimensionless bare constants that are required to vanish by supersymmetry can be shown to vanish to all orders in perturbation theory even though supersymmetry is spontaneously broken. Unfortunately the vacuum density is not a constant of this sort — it has dimensionality (mass)$^4$ instead of being dimensionless, and it is a renormalized coupling rather than a bare coupling. Recently Witten has proposed a highly imaginative and speculative mechanism by which some form of supersymmetry makes $\rho_V$ vanish[@Witten95]. I am grateful to the organizing committee of this conference for giving me only 15 minutes to talk, so that I don’t have to try to explain Witten’s idea. I turn instead to the other two approaches on my list. Cancellation Mechanisms ======================= The special thing about having $\rho_V=0$ is that it makes it possible to find spacetime-independent solutions of the Einstein gravitational field equations. For such solutions, we have $$\partial {\cal L}/\partial g_{\mu\nu}=0\;,$$ where ${\cal L}$ is the Lagrangian density for constant fields. The problem occurs only in the trace of this equation, which receives a contribution from $\rho_V$ which for $\rho_V\neq 0$ prevents a solution. Many theorists have tried to get around this difficulty by introducing a scalar field $\phi$ in such a way that the trace of $\partial {\cal L}/\partial g_{\mu\nu}$ is proportional to $\delta {\cal L}/\delta\phi$: $$g_{\mu\nu}\partial {\cal L}/\partial g_{\mu\nu}=f(\phi)\delta {\cal L}/\delta\phi\;,$$ with $f(\phi)$ arbitrary, except for being finite. Where this is done, the existence of a solution of the field equation $\delta {\cal L}/\delta\phi=0$ for a spacetime-independent $\phi$ implies that the trace $g_{\mu\nu}\partial {\cal L}/\partial g_{\mu\nu}=0$ of the Einstein field equation for a spacetime-independent metric is also satisfied. The trouble is that, with these assumptions, the Lagrangian has such a simple dependence on $\phi$ that it is not possible to find a solution of the field equation for $\phi$. This is because Eq. (4), together with the general covariance of the action $\int d^4x\, {\cal L}$, tells us that, when the action is stationary with respect to variations of all other fields, it has a symmetry under the transformations $$\delta g_{\lambda\nu}=2\epsilon g_{\lambda\nu}\;,~~~~~~\delta\phi=-\epsilon f(\phi)\;,$$ which requires the Lagrangian density for spacetime-independent fields $g_{\mu\nu}$ and $\phi$ to have the form $${\cal L}=c\;\sqrt{\det\,g}\;\exp\left(4\int^\phi\frac{d\phi'}{f( \phi')}\right)\;,$$ where $c$ is a constant whose value depends on the lower limit chosen for the integral. For $c\neq 0$, there is no solution at which this is stationary with respect to $\phi$. The literature is full of proposed solutions of the cosmological constant problem based on this sort of spontaneous adjustment of one or more scalar fields, but if you look at them closely, you will see that either they do not satisfy Eq. (4), in which case there may be a solution for $\phi$ but it does not imply the vanishing of $\rho_V$, or else they do satisfy Eq. (4), in which case a solution of the field equation for $\phi$ would imply a vanishing $\rho_V$, but there is no solution of the field equation for $\phi$. To the best of my knowledge, no one has found a way out of this impasse. Anthropic Considerations ======================== Suppose that the observed subuniverse is only one of many subuniverses, in which $\rho_V$ takes a variety of different values. This is the case for instance in theories of chaotic inflation[@Linde86], in which various scalar fields on which the vacuum energy depends take different values in different expanding regions of space. In a somewhat more subtle way, this can also be the case in some versions of quantum cosmology, where the wave function of the universe is a superposition of terms in which $\rho_V$ takes different values, either because of the presence of some vacuum field (like the antisymmetric tensor gauge field $A_{\mu\nu\lambda}$ introduced for this purpose by Hawking[@Hawking84]), or because of wormholes, as in the work of Coleman[@Coleman88a]. Some authors[@Hawking84], [@Baum84], [@Coleman88b] have argued that in quantum cosmology the distribution of values of $\rho_V$ is very sharply peaked at $\rho_V=0$, which would immediately solve the cosmological constant problem. This conclusion has been challenged[@Fischler89], and it will be assumed here that the probability distribution of $\rho_V$ is smooth at $\rho_V=0$, without any sharp peaks or dips. In any theory of this general sort the measured effective cosmological constant would be much smaller than the value expected on dimensional grounds in elementary particle physics, not because there is any physical principle that makes it small in all subuniverses, but because it is only in the subuniverses where it is sufficiently small that there would be anyone to measure it. For negative values of $\rho_V$, this limitation comes from the requirement that the subuniverse must survive long enough to allow for the evolution of life[@Barrow86]. For positive values of $\rho_V$ (which are observationally more promising) the limitation comes from the requirement that large gravitational condensations like galaxies must be able to form before the subuniverse begins its final exponential expansion[@Weinberg87]. If you don’t find this sort of anthropic explanation palatable, consider the following fable. You are an astronaut, sent out to explore a randomly chosen planet around some distant star, about which nothing is known. Shortly before you leave you learn that because of budget cuts, NASA has not been able to supply you with any life-support equipment to use on the planet’s surface. You arrive on the planet, and find to your relief that conditions are quite tolerable — the air is breathable, the temperature is about 300$^\circ$ K, and the surface gravity is not very different from what it is on earth. What would you conclude about the conditions on planets in general? It all depends on how many astronauts NASA has sent out. If you are the only one then it’s reasonable to infer that tolerable conditions must be fairly common, contrary to what planetologists would have naturally expected. On the other hand, if NASA has sent out a million astronauts, then all you can conclude about the statistics of planetary conditions is that the number of planets with tolerable conditions is probably not much less than one in a million — for all you know, almost all of the astronauts have arrived on planets that cannot support human life. Naturally, the only astronauts in this program that are in a position to think about the statistics of planetary conditions are those like you who are lucky enough to have landed on a planet on which they can live; the others are no longer worrying about it. In previous work[@Weinberg87] I calculated the anthropic [*upper bound*]{} on the cosmological constant, which arises from the condition that $\rho_V$ should not be so large as to prevent the formation of gravitational condensations on which life could evolve. This bound is naturally larger than the [*average*]{} value of the cosmological constant that would be measured by typical observers, which obviously gives a better estimate of what we might find in our subuniverse. (Vilenkin[@Vilenkin95] has advocated this point of view under the name of the ‘principle of mediocrity’, but did not attempt a detailed analysis of its consequences.) The difference is important, because the anthropic upper bound on $\rho_V$ is considerably larger than the largest value of $\rho_V$ allowed by observation. I will leave the observational limits on the cosmological constant to Dr. Fukugita’s talk, but without going into details, it seems that for a spatially flat (i.e., $k=0$) universe, $\rho_V$ is likely to be positive and somewhat larger than the present mass density $\rho_0$, but probably not larger than $3\rho_0$[@Ostriker96]. On the other hand, we know that some galaxies were already formed at redshifts $z\approx 4$, at which time the density of matter was larger than the present density $\rho_0$ by a factor $(1+z)^3\approx 125$. It therefore seems unlikely that a vacuum energy density much smaller than $125\rho_0$ could have completely prevented the formation of galaxies, so the anthropic upper bound on $\rho_V$ cannot be much less than about $125\rho_0$, which is much greater than the largest observationally allowed value of $\rho_V$. In contrast, we would expect the anthropic [*mean*]{} value of $\rho_V$ to be roughly comparable to the mass density of the universe at the time of the greatest rate for the accretion of matter by growing galaxies, because it is unlikely for $\rho_V$ to be much greater than this and there is no reason why it should be much smaller. (I will make this more quantitative soon.) Although there is evidence that galaxy formation was well under way by a redshift $z\approx 3$, it is quite possible that most accretion of matter into galaxies continues to lower redshifts, as seems to be indicated by cold dark matter models. In this case the anthropic mean value $<\!\rho_V\!>$ will be considerably less than the anthropic upper bound, and perhaps within the range allowed observationally. I would like to present an illustrative example of a calculation of the whole probability distribution of the cosmological constant that would be measured by observers, weighted by the likelihood that there are observers to measure it. Instead of the very simple model[@Peebles67] of galaxy formation from spherically symmetric pressureless fluctuations used previously[@Weinberg87], here I will rely on the well-known model of Gunn and Gott[@Gunn72], which also assumes spherical symmetry and zero pressure, but takes into account the infall of matter from outside the initially overdense core. This is still far from realistic, but it will allow me to make four points about such calculations, which should be more generally applicable. As shown in earlier work[@Weinberg87], the condition for a spherically symmetric fluctuation to recondense is that $$\frac{500\,(\Delta\rho)^3}{729\,\rho^2}\!>\rho_V\;.$$ where $\rho$ and $\Delta\rho$ are the average density and the overdensity in the fluctuation at some early initial time, say the time of recombination. Previously $\Delta\rho$ was assumed to be uniform within a spherical fluctuation, but Eq. (7) actually applies to any sphere, with $\Delta\rho$ understood to be the spatially averaged initial overdensity within the sphere. Suppose that the fluctuation at recombination consists of a finite spherical core of volume $V$ with positive average overdensity $\delta\rho$, outside of which the density takes its average value $\rho$. (This picture is appropriate for well separated fluctuations. The effects of crowding and underdense regions will be considered in a future paper.) Then the average overdensity within a larger volume $V'$ centered on this core is $\Delta\rho=\delta \rho V/V'$. Assuming that Eq. (7) is satisfied by the average overdensity $\delta\rho$ within the core, $$\left.\frac{500\,(\delta\rho)^3}{729\,\rho^2}\right|_{\rm recomb}\!>\rho_V\;,$$ the average overdensity $\Delta\rho$ will satisfy the condition (7) out to a volume $$V_{\rm max}=\left(\frac{500}{729\rho_V}\right)^{1/3}\rho^{- 2/3}\delta\rho\,V$$ so the total mass that will eventually collapse is $$M=\delta\rho V+\rho V_{\rm max}=V\,\delta \rho\left[1+\left(\frac{500 \rho}{729\rho_V}\right)^{1/3}\right]\;.$$ Once a galaxy forms, the subsequent evolution of stars and planets and life is essentially independent of the cosmological constant ([*this is point 1*]{}), so the number of independent observers arising from a given fluctuation at the time of recombination is proportional to the mass (9) for those fluctuations satisfying Eq. (8), and is otherwise zero. Of course, the value of the cosmological constant might be correlated with the values of other fundamental constants, on which the evolution of life does depend, but the range of anthropically allowed cosmological constants is so small compared with the natural scale (2) of densities in elementary particle physics that within this range it is reasonable to suppose that all other constants are fixed. ([*This is point 2.*]{}) The range of values of $\rho_V$ for which gravitational condensations are possible is also so much less than the average density at the time of recombination, that the number of fluctuations ${\cal N}(\delta\rho,V)\,dV\,d\delta\rho$ with volume between $V$ and $V+dV$ and average overdensity between $\delta\rho$ and $\delta\rho+d\delta\rho$ should be nearly independent of $\rho_V$. ([*This is point 3.*]{}) If ${\cal P}(\rho_V)\,d\rho_V$ is the [*a priori*]{} probability that a random subuniverse has vacuum energy density between $\rho_V$ and $\rho_V+d\rho_V$, then according to the principles of Bayesian statistics, the probability distribution for [*observed*]{} values of $\rho_V$ is $$\begin{aligned} {\cal P}_{\rm obs}(\rho_V)&\propto& {\cal P}(\rho_V)\int_0^\infty dV \int_{(729\rho_V\rho^2/500)^{1/3}}^\infty d\delta\rho \;{\cal N}(\delta\rho,V)\nonumber\\&&\times\; V\delta \rho\left[1+\left(\frac{500 \rho}{729\rho_V}\right)^{1/3}\right]\nonumber\\ &\propto& {\cal P}(\rho_V)\left[1+\left(\frac{500 \rho}{729\rho_V}\right)^{1/3}\right] \int_{(729\rho_V\rho^2/500)^{1/3}}^\infty d\delta\rho \;{\cal N}(\delta\rho)\delta\rho\end{aligned}$$ where $${\cal N}(\delta\rho)\equiv \int_0^\infty dV\;V\, {\cal N}(\delta\rho,V)\;.$$ Finally, the range of values of $\rho_V$ for which gravitational condensations are possible is so small compared with the natural scale of densities in elementary particle physics that within this range the [*a priori*]{} probability ${\cal P}(\rho_V)$ may be taken as constant. ([*This is point 4.*]{}) The factor ${\cal P}(\rho_V)$ may therefore be omitted in the probability distribution (10). Also, all anthropically allowed values of $\rho_V$ are much smaller than the mass density $\rho$ at recombination, so we may neglect the $1$ in the square brackets in Eq. (10), which now becomes $${\cal P}_{\rm obs}(\rho_V)\propto \rho_V^{- 1/3}\int_{(729\rho_V\rho^2/500)^{1/3}}^\infty d\delta\rho \;{\cal N}(\delta\rho)\delta\rho\;.$$ Strictly speaking, this gives the probability distribution only for $\rho_V>0$. For $\rho_V<0$ and $k=0$, all mass concentrations that are large enough to allow pressure to be neglected will undergo gravitational collapse. The number of astronomers is instead limited[@Barrow86] for $\rho_V<0$ by the fact that the subuniverse itself also collapses, in a time $$T(|\rho_V|)=\frac{2\pi}{3}\sqrt{\frac{3}{8\pi G |\rho_V|}}\;.$$ In contrast, the probability distribution for $\rho_V>0$ is weighted by an $\rho_V$-independent factor, the average time ${\cal T}$ in which stars provide conditions favorable for intelligent life. The probability distribution for negative values of $\rho_V$ is small except for values of $|\rho_V|$ that are small enough so that $T(|\rho_V|)$ is less than or of order ${\cal T}$. It will be assumed here that ${\cal T}$ is very large, so that ${\cal P}_{\rm obs}(\rho_V)$ is negligible for $\rho_V<0$ except in a small range near zero, and may therefore be neglected in calculating the mean value of $\rho_V$. Using the probability distribution (12) and interchanging the order of the integrals over $\delta\rho$ and $\rho_V$, we easily see that the mean value of [*observed*]{} values of $\rho_V$ is $$\langle\rho_V\rangle=\frac{200<\!\delta\rho^6\!>}{729<\! \delta\rho^3\!>\rho^2}\;,$$ with all quantities on the right-hand side evaluated at the time of recombination, and the brackets on the right-hand side (unlike those in $\langle \rho_V\rangle$) indicating averages over fluctuations: $$<\!f(\delta\rho)\!>\equiv\int_0^\infty d\delta\rho\;{\cal N}(\delta\rho)\,f(\delta\rho)\;.$$ It remains to use astronomical observations to calculate the fluctuation spectrum ${\cal N}(\delta\rho)$ for the density fluctuations at recombination, which can then be used in Eq. (12) to calculate the probability distribution for $\rho_V$. Here I will just give one example of how information about the time of formation of galaxies can put constraints on $<\!\rho_V\!>$. With a positive $\rho_V$, the core of a fluctuation with average overdensity $\delta\rho$ at recombination will collapse at a time when the average cosmic density $\rho_{\rm coll}$ is less than it would be at the time of core collapse for $\rho_V=0$:[@Weinberg87] $$\rho_{\rm coll}<\!\frac{500\,\delta\rho^3}{243\,\pi^2\rho^2}\;,$$ with $\rho$ and $\delta\rho$ on the right-hand side evaluated at recombination. Using this in Eq. (14) gives a mean vacuum density $$<\!\!\rho_V\!\!>\;>\;\frac{2\pi^2\langle \rho_{\rm coll}\rangle}{15}\;.$$ Even if we suppose for example that core collapse occurs for most galaxies at a redshift as low as $z\approx 1$, then $\rho_{\rm coll}\approx 8\rho_0$, so Eq. (19) gives $<\!\rho_V\!>\;>\; 10\rho_0$, which exceeds current experimantal bounds on $\rho_V$. On the other hand, the median value of $\rho_V$ is less than the mean value, so the discrepancy is less than this. Even so, it seems that most galaxies must be formed quite late in order for the value of $\rho_V$ in our universe to be close to the value that is anthropically expected. \* \* \*  At the meeting in Princeton I learned of an interesting paper by Efstathiou[@Efstathiou95], in which he calculated the effect of a cosmological constant on the present number density of L$_*$ galaxies, which he took as a measure of the distribution function ${\cal P}_{\rm obs}(\rho_V)$. In this calculation he adopted a standard cold dark matter model for matter density fluctuations, with amplitude at long wavelengths fixed by the measured anisotropy of the cosmic microwave background. Efstathiou found that for a spatially flat universe the galaxy density falls off rapidly (say, by a factor 10) for values of $\rho_V$ around 7 to 9 times the present mass density $\rho_0$, so that $<\!\rho_V\!>/\rho_0$ should be less than of order 7 to 9, giving a contribution $\Omega_0=\rho_0/(\rho_0+\rho_V)$ of matter to the total density somewhat greater than around 0.1, which is consistent with lower bounds on the present matter density. At first sight this seems encouraging, but there are a few problems with Efstathiou’s calculation. For one thing, as pointed out by Vilenkin[@Vilenkin95], the probability distribution of observed values of $\rho_V$ is related to the number of galaxies (or, more accurately, the amount of matter in galaxies) that [*ever*]{} form, rather than the number that have formed when the age of the universe is at any fixed value, as assumed by Efstathiou. However, this will not make much difference if most galaxy formation is complete in typical subuniverses when they are as old as our own subuniverse. Efstathiou also encountered another problem that is endemic to this sort of calculation. The cosmological parameters that can reasonably be assumed to be uncorrelated with the cosmological constant are the baryon–to–entropy ratio and the spectrum of density fluctuations at recombination, because these are presumably fixed by events that happened before recombination, when any anthropically allowed cosmological constant would have been negligible. But the only way we know about the spectrum of density fluctuations at recombination is to use observations of the present microwave background (or possibly the numbers of galaxies at various redshifts), and unfortunately the results we obtain from this for ${\cal N}(\delta\rho)$ depend on the value of the cosmological constant in [*our*]{} subuniverse. In calculating ${\cal P}_{\rm obs}(\rho_V)$ one should ideally make some assumption about the value of $\rho_V$ in our subuniverse, then use this value to infer a spectrum of density fluctuations at recombination from the observed microwave anisotropies, and then calculate the number of galaxies that ever form as a function of $\rho_V$, with the spectrum of density fluctuations at recombination held fixed. Instead, Efstathiou calculated the number of L$_*$ galaxies as a function of $\rho_V$, with the microwave anisotropies held fixed, which gave ${\cal P}_{\rm obs}(\rho_V)$ an additional spurious dependence on $\rho_V$. This problem was known to Efstathiou, and apparently did not produce large errors. There is one other problem, that did have a significant effect in Efstathiou’s calculation. He relied on the standard method[@Press74] of calculating the evolution of density fluctuations using linear perturbation theory, and declaring a galaxy to have formed when the fractional overdensity $\Delta\rho/\rho$ reaches a value $\delta_c$, which is taken as the fractional overdensity of the linear perturbation at a time when a nonlinear pressureless spherically symmetric fluctuation would recollapse to infinite density. He took the effective critical overdensity for spatially flat cosmologies as $\delta_c=1.68/\Omega_0^{0.28}$, with $\Omega_0\equiv 1- \rho_V/\rho_{\rm crit}$, so that $\delta_c=3.2$ for $\Omega_0=0.1$. But numerical calculations of Martel and Shapiro[@Martel96] show that for all fluctuations that result in gravitational recollapse, $\delta_c$ is in a range from 1.63 to 1.69. The upper bound 1.69 is the well-known result $\delta_c=(3/5)(3\pi/2)^{3/2}=1.6865$ for $\rho_V=0$. The lower bound 1.63 can also be understood analytically[@Weinberg87]: it is the critical overdensity for the case where $\rho_V$ has a value that just barely allows gravitational recollapse $$(\delta_c)_{\rm min}=\frac{2}{\sqrt{\pi}}\,\left(\frac{729}{500}\right)^{1/3 }\, \Gamma\left(\frac{11}{6}\right)\, \Gamma\left(\frac{2}{3}\right)=1.629\;.$$ With $\delta_c$ always between these bounds, it is impossible that the effective value of $\delta_c$ for any ensemble of fluctuations could be greater than 1.69. Overestimating $\delta_c$ biases the calculation toward late galaxy formation, with a corresponding increased sensitivity to relatively small values of $\rho_V$. Efstathiou has now re-done his calculations with $\delta_c$ given the constant value 1.68, which should be a good approximation, and, as I interpret his results, he finds that this change in $\delta_c$ roughly doubles the value of $\rho_V$ at which the present density of L$_*$ galaxies drops by a factor 10, with a corresponding reduction in the expected value of $\Omega_0$. It remains to be seen whether this change in his results will lead to a conflict with observational bounds on $\Omega_0$ and $\rho_V$. At present Martel and Shapiro are carrying out a numerical calculation of ${\cal P}_{\rm obs}$ using Eq. (12). I am grateful for helpful discussions with George Efstathiou and Paul Shapiro. For a discussion of these and other possibilities, see Weinberg, S. 1989, Rev. Mod. Phys. 61, 1 Witten, E. 1995, Int. J. Mod. Phys. 10, 1247; preprint IASNS-HEP-95-51, hep-th/9506101 Linde, A. D. 1986, Phys. Lett. B 175, 395 Hawking, S. W. 1983, in [*Shelter Island II — Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics*]{}, ed. R. Jackiw [*et al.*]{} (MIT Press, Cambridge, 1985); Phys. Lett. B 134, 403. Coleman, S. 1988, Nucl. Phys. B 307, 867 Baum, E. 1984 Phys. Lett. B133, 185 Coleman, S. 1988, Nucl. Phys. B 310, 643 Fischler, W., Klebanov, I., Polchinski, J., and Susskind, L. 1989, Nucl. Phys. B237, 157 Barrow, J. D., and Tipler, F. J. 1986, [*The Anthropic Cosmological Principle*]{} (Clarendon, Oxford) Weinberg, S. 1987, Phys. Rev. Lett. 59, 2607 Vilenkin, A. 1995, Phys. Rev. Lett. 74, 846; Tufts preprint gr-qc/9507018, to be published in the Proceedings of the 1995 International School of Astrophysics at Erice; Phys. Rev. D52, 3365; Tufts preprint gr-qc/9512031 For a review and earlier references, see Ostriker, J. P. and Steinhardt, P. J. 1995, Nature 377, 600 Peebles, P. J. E. 1967, Astrophys. J. 147, 859 Gunn, J. E. and Gott, J. R. 1972, Astrophys. J. 176, 1 Efstathiou, G. 1995, Mon. Not. R. Astron. Soc. 174, L73 Press, W. H. and Schechter, P. 1974, Astrophys. J. 187, 425 Martel, H. and Shapiro, P. R. 1996, paper in preparation [^1]: Research supported in part by the Robert A. Welch Foundation and NSF Grants PHY 9009850 and PHY 9511632.

--- abstract: 'We report on an integrated photonic transmitter of up to 100 MHz repetition rate, which emits pulses centered at 850 nm with arbitrary amplitude and polarization. The source is suitable for free space quantum key distribution applications. The whole transmitter, with the optical and electronic components integrated, has reduced size and power consumption. In addition, the optoelectronic components forming the transmitter can be space-qualified, making it suitable for satellite and future space missions.' author: - | M. Jofre, A. Gardelein, G. Anzolin, G. Molina-Terriza,\  J.P. Torres, M.W. Mitchell and V. Pruneri title: '100 MHz amplitude and polarization modulated optical source for free-space quantum key distribution at 850 nm' --- Introduction ============ In many applications, *free space optical* (FSO) communications is the technology of choice to transmit information, especially when fiber optical cabling is not easily achievable or its installation is too expensive [@Carbonneau1998]. Compared to *radio frequency* (RF) techniques, its main advantages lie in high data rates (up to several Gb/s), minimum free space losses due to the small optical beam divergence and absence of regulatory issues thanks to the low interference level [@Garlington2005; @O'Brien2003; @Davis2003]. Therefore FSO communication is favorable for high data-rate, long-range point-to-point links, where the terminal size, mass, and power consumption are subjected to strong limitations, such is the case of aeronautical or space platforms. An important issue in today’s information society is the security of data transmission against potential intruders, which always put at risk the confidentiality. Current methods to increase security require that two parties wishing to transmit information securely need to exchange or share one or more keys. Once the key has been exchanged, the information can be transferred in a provable secure way using a one-time pad. Therefore, the security of the information transmission is based exclusively on the security of the key exchange. Quantum cryptography, or more precisely *Quantum Key Distribution* (QKD), guarantees absolutely secure key distribution based on the principles of quantum physics, since it is not possible to measure or reproduce a state (eg. polarization or phase of a photon) without being detected [@RevModPhys.77.1225]. The key is generated out from the measurement of the information encoded into specific quantum states of a photon. In particular, if a two-dimensional quantum system is used, information is said to be encoded into qubits. For example, a qubit can be created using properties such as the polarization or the phase of a photon. The first QKD scheme, due to Bennett and Brassard [@Bennett1984; @Bennett1992], employs single photons sent through a quantum channel, plus classical communications over a public channel to generate a secure shared key. This scheme is commonly known as the BB84 protocol. Although single photon sources may be very useful for quantum computing, they are not strictly required for QKD. This, and the relative difficulty of generating true single photons, motivates new approaches based on conventional light sources [@scarani-2008]. Indeed, attenuated laser pulses or *faint pulse sources* (FPS), which in average emit less than one photon per pulse, are often used as signals in practical QKD devices. The performance limitations of attenuated pulse systems had initially led to believe that single photon sources would be indispensable for building efficient QKD systems. However, the introduction of the decoy-state protocol [@PhysRevLett.94.230504; @PhysRevA.72.012326] made possible a much tighter bound for the key generation rate, achieving an almost linear dependency of the latter on the channel transmittance. In this way, the technologically much simpler faint pulse systems can offer comparable QKD security with respect to single photon sources. Another key feature of QKD is that the security is linked to the one-time-pad transmission, i.e. the key has to be used once and has to be equal or similar in size to the information being transmitted. It is thus evident the importance of developing faint pulse sources and systems for QKD which can generate high key bit rates. The highest Secure Key Rate reported to date over 20 Km of optical telecom fiber is of 1.02 Mb/s [@Dixon:08] and 14.1 b/s over 200 Km [@Chen2009], while the achieved speed over 144 Km free space link is of 12.8 Kb/s [@Schmitt-Manderbach2007] and 50 Kb/s over 480 m [@Weier2006]. The goal of QKD is to allow to distant parties to share a common key in the presence of an eavesdropper. Therefore, the most important question of QKD is its security. Therefore, an important aim of this work is to demonstrate a system to generate pulses that differ only in polarization, while being indistinguishable in the other degrees of freedom that characterize the quantum state of photons, such as arrival time, optical frequency, and spatial mode. In other words, to generate pulses which contain no side-channel information correlated to the polarization. We note that previous implementations based on multiple lasers [@Schmitt-Manderbach2007; @Weier2006; @Kurtsiefer; @4528718] have attempted to achieve time-frequency indistinguishability by laser pre-selection, current and temperature adjustment, and temporal and spectral measurements. Apart from being expensive and cumbersome, this kind of tuning has limited stability due to the inevitable aging of laser diodes. It is worth noting that the temporal and spectral distributions reported to date indicate indistinguishability in the time and frequency bases, but leave open the question of distinguishability based on other pulse characteristics such as chirp. A related issue which arises in a decoy-state protocol is possible side-channel information indicating the pulse intensity. Intensity level modulation could be achieved rapidly and conveniently by modulating the laser current. This method of modulation, however, induces strong nonlinearities and causes strong phase modulation, which makes it difficult to control the temporal and spectral shape of the output pulses. In this paper we report the development of a novel integrated pulse source which can reach rates as high as 100 Mb/s at 850nm modulated in amplitude and polarization. For QKD applications, it has been simulated that the source could achieve a Secure Key Rate of the order of 500 Kb/s at 20 Km using decoy-state protocol. The source is capable to generate pulses at around 850 nm with at least three different intensity levels (i.e. number of photons per pulse) and four different polarization states. The proposed FPS ensures indistinguishability among the different intensity and polarization pulses and ensures phase incoherence of consecutive generated states. It is based on a single diode emitting a continuous pulse train externally modulated in amplitude and polarization. The wavelength, reduced power consumption, compactness and space qualifiable optoelectronic components constituting the source make it very suitable for space transmission, for free space quantum and classical communication links. One of the foreseen applications is its use to overcome the distance limit of QKD in optical fibers [@Chen2009; @Takesue2007], by creating a global security network among very distant places on earth through satellite communication [@PhysRevLett.91.057901; @Rarity2002; @Bonato2009]. The integrated faint pulse source {#ExperimentalSetup} ================================= In order to use it for space applications, the proposed integrated FPS source for FSO communication consists of commercially available space-qualified discrete components; single semiconductor laser diode emitting a continuous pulse train at 100 MHz followed by integrated (waveguide) amplitude and polarization lithium niobate (LiNbO$_{3}$) modulators (Figure \[Fig:FPS-IMPM\_Blocks\]). ![Schematic of the QKD source. \[LD\] denotes a laser diode, \[AM\] an amplitude modulator, \[PM\] a polarization modulator and \[VOA\] a variable optical attenuator.[]{data-label="Fig:FPS-IMPM_Blocks"}](FPS-IMPM_Blocks2){width="\columnwidth"} A *distributed feedback* (DFB) *laser diode* (LD) at around 850 nm is driven at 100 MHz train of electrical pulses. The optical pulse of about $400$ps is generated via a current pulse of about $1$ns duration. In fact, the laser is biased using a DC current of $24$mA, far below threshold $36$mA, and it is directly modulated using a strong RF current of $50$mA (peak value) so that the optical pulse is generated [@Petermann1991]. The generated optical pulses do not have any phase coherence among them due to the fact that the laser is set below and above threshold from one pulse to the subsequent one, thus producing a random phase for each pulse. The output mirror reflectivity ($R$) of the DFB structure is $30$%, the cavity length ($L_c$) $300\mu$m and the active medium refractive index ($n$) $3.6$ [@Sadovnikov1995; @Li1998]. The *round-trip-time* (RTT) given by $2L_c n/c_0$, where $c_0$ is the speed of light in vacuum, is $\approx 20$ps. In one pulse train period ($10$ns) the optical pulse power left in the laser cavity, after going below threshold, has bounced back and forth $\approx 500$ times. Between pulses, the laser is biased at only $66$% of threshold, so that transmission loss through the output mirror is much greater than the round-trip gain. Conservatively assuming a round-trip loss $\ge 1$dB, the $\ge 500$dB loss from $500$ passes will atenuate any coherence to a very low level. At the same time, incoherent spontaneous emission is generated, further obscuring any possible coherence between optical pulses [@Petermann1991]. In terms of partial coherence theory, it is expected a first-order degree of coherence $g^{(1)}(\tau)$ which drops rapidly to zero for $\tau$ larger than the pulse duration. This is consistent with the observed spectrum, which implies a coherence length of order $0.75$ m [*during*]{} the pulse. Note that the spectrophotometer response is produced by photons coming from the pulses and the coherence length of about $0.75$ m which can be calculated from the measured bandwidth, as it was explained above, decreases significantly when the pulse extinguishes, thus making the inter-pulses value much smaller than the distance between consecutive pulses (about $3$ m). In this way, phase incoherence of consecutive generated states, which otherwise would be detrimental for the link security, is achieved. Then, the pulse train is sent through a *polarization maintaining fiber* (PMF) into an *amplitude modulator* (AM) (eg. a Mach-Zehnder modulator in LiNbO$_{3}$) that will randomly generate the three different levels of intensity. Note that if the DFB laser diode were driven in pure *continuous wave* (CW) mode (no pulse train) and externally modulated to generate the pulses, two potential issues would occur: (i) pulses with different energies (number of photons) would unavoidably have different temporal and spectral shapes due to the nonlinear electro-optic response (optical output as a function of driving voltage) of the amplitude modulator; (ii) there would be phase coherence between the pulses due to the relatively long coherence time (narrow spectrum) of a DFB structure, thus increasing the vulnerability of the QKD transmission [@lo-2007-8]. After the AM, the pulses are injected into a *polarization modulator* (PM) through a PMF. The polarization modulator is in fact a waveguide LiNbO$_{3}$ phase modulator where the PMF input axis is oriented at 45with respect to the optical axis. In this way, the two orthogonal equal amplitude polarization components of the electromagnetic field that propagates in the crystal experience a refractive index difference, which is proportional to the voltage applied to the modulator. By applying different voltages one can thus change the state of the output polarization, in particular linear +45, -45, right-handed circular and left-handed circular. The optical pulses present a spectrum within the acceptance bandwidth of the two modulators, so that amplitude and polarization modulation can be achieved with high extinction ratio. A proper electronic control of the different intensities and polarization states generated, at Alice, for the different states is fundamental in order to perform a QKD transmission. The synchronization and setting of the different optical components of the source is implemented by an automatic control which is split into two working operations. The control system first synchronizes and calibrates the driving signals timings and amplitudes to the AM and PM, and secondly generates the appropriate driving signals for the BB84+Decoy protocol transmission. For the implementation of a QKD system using decoy-state protocol, besides four different polarization states, the FPS source should generate three intensity levels (optimally 1/2, 1/8 and 0 photons in average per pulse [@PhysRevA.72.012326]) using a *variable optical attenuator* (VOA) in order to operate in the single photon regime. The optical pulse duration $\approx 400$ps and the pulse peak power $3.5$mW which corresponds to $1.4$pJ energy per pulse, thus a number of photons per pulse $\approx 6\cdot 10^{6}$. In order to get a mean photon number for the signal state which is within an optimum range for the distances of interest [@PhysRevA.72.012326], the VOA has to introduce an optical attenuation of $\approx 70$dB. The chosen FPS wavelength (850 nm) is optimum for free space operation considering attenuation (due to scattering, absorption and diffraction) and single-photon detector’s quantum efficiency [@scarani-2008]. Description of generated states {#Sec:Description_generated_states} =============================== In a BB84 protocol scheme implementing the decoy-state protocol different pulses should differ in polarization and amplitude while remaining indistinguishable in other characteristics, including temporal shape and frequency spectrum. If the pulses differ in spectrum, for example, an eavesdropper could use spectral measurements to infer the sent polarization without actually measuring it. Removal of this kind of [*side-channel*]{} information is thus critical to the security of the protocol. Since the information is encoded in the polarization state, the statistical similarity between pulses of different polarizations but same intensity level is more relevant than that of different intensity level but same polarization to prevent information leakage from the quantum link. Here we consider the quantum optics of side-channel information, limiting the discussion to pure states and simple measurements. A full treatment including mixed states and generalized measurements will be the subject of a future publication. We consider a source that produces pulses with amplitudes ${{\cal E}}_l$, polarizations ${{\bf p}}_l$ and pulse shapes $\Pi_l(t)$. Without loss of generality we assume the polarizations and pulses shapes are normalized ${{\bf p}}^*_l \cdot {{\bf p}}_l = \int dt \, \Pi_l^*(t) \Pi_l(t) = 1$. In a classical description, the field envelopes are $$\mathbf{E}_{l}(t)={{\cal E}}_l {{\bf p}}_{l}\Pi_{l}(t) \label{Eq:ClassicalPulse}$$ The corresponding quantum state is a generalized coherent state $$\left|{\bf \alpha}_{l}\right> \equiv D_l({{\eta}}{{\cal E}}_l {{\bf p}}_{l}) \left| 0 \right>$$ where $ \left| 0 \right>$ is the vacuum state and $D_l({\bf x}) \equiv \exp[{\bf x}\cdot {\bf A}_l^\dagger - {\bf x}^*\cdot {\bf A}_l ]$ is a displacement operator, defined in terms of the mode operator ${\bf A}_{l} \equiv \int dt\,\Pi_{l}^*(t){\bf a}(t) $, ${\bf a} \equiv (a_x,a_y)$ is a vector of annihilation operators, with $[a_p(t),a_q^\dagger(t')] = \delta(t-t')\delta_{p,q}$ for $p,q \in \{x,y\}$. A scaling factor ${{\eta}}$ is included to convert from photon units to field units, chosen such that the positive-frequency part of the quantized electric field is $\hat{\mathbf{E}}(t) = {{\eta}}^{-1} {\bf a}(t)$. It is easy to check that $\left<\alpha_l\right| {\bf a}(t) \left|\alpha_l\right> = {{\eta}}{{\cal E}}_l {{\bf p}}_{l}\Pi_{l}(t)$, so that the average quantum field$\left<\alpha_l\right| \hat{\mathbf{E}}(t) \left|\alpha_l\right> ={{\cal E}}_l {{\bf p}}_{l}\Pi_{l}(t)$ in agreement with Equation \[Eq:ClassicalPulse\]. Quantum mechanics allows measurements on the pulse-shape $\Pi$ without measurement of the polarization ${{\bf p}}$. For example, the number operator $N_l \equiv {\bf A}^\dagger_l \cdot {\bf A}_l = A_{l,x}^\dagger A_{l,x}+A_{l,y}^\dagger A_{l,y}$ counts photons in the mode $\Pi_l$ independent of ${{\bf p}}_l$. If the modes $\left\{\Pi_l\right\}$ are different, an eavesdropper could use state-discrimination techniques [@Bergou2004; @Barnett09] to determine $l$ (and thus the secret key) [*without*]{} disturbing ${{\bf p}}$. This kind of eavesdropping would not be detected by Bob’s polarization measurements. For this reason, it is critical to guarantee that this kind of [*side channel*]{} information is not present in the sent optical pulses. The similarity between the various $\Pi_l$ can be quantified by an overlap integral: $[{A}_{l,p},{A}_{m,q}^\dagger] = \int dt \Pi_l^*(t) \Pi_m(t)[a_p,a^\dagger_q] \equiv S_{lm} \delta_{p,q}$, so that for example two states with equal amplitudes $|{{\cal E}}_l|=|{{\cal E}}_m|$, $\left<\alpha_m\right| N_l \left|\alpha_m\right>/\left<\alpha_l\right| N_l \left|\alpha_l\right> = |S_{lm}|^2$. Finally, we note that it is possible for pulses to have the same spectra and temporal shape but still be distinguishable, for example if they have different chirp. For this reason, establishing that two (or more) distinct sources produce indistinguishable pulses is not easy. Our strategy to eliminate side-channel information in the pulse shapes is to dissociate pulse generation from the setting of polarization and amplitude levels. As described in the previous section the FPS consists of a single laser diode emitting a continuous train of optical pulses followed by an AM, a PM and a VOA. Considering that the laser operation is the same for each pulse sent, and that both the AM and PM control voltages are held constant over the duration of the pulse, we can assume that the pulse shape does not depend on the sent amplitude and polarization. This assumption is confirmed by measurements shown in Section \[Experimental\_measurements\]. The complex expression of the pulsed electromagnetic field exiting the FPS can be written as $$\mathbf{E}(t)=\sum_i A \alpha_i e^{j\phi_i}e^{j\beta_i}\frac{\bf{\hat x}+e^{j\gamma_{i}}\bf{\hat y}}{\sqrt{2}}\Pi\left({t-iT}\right)$$ where $t$ is the time, $T$ is the pulse train period and $A,\phi_i,\Pi$ are the amplitude, phase, and shape, respectively, of the optical pulse generated by the LD. $\alpha_i, \beta_i$ describe the transmission and introduced phase, respectively, of the AM. $\gamma_{i}$ is the phase difference between $\bf{\hat x}$ and $\bf{\hat y}$ introduced by the polarization modulator in order to generate the different polarization states. Another security consideration is optical coherence between successive pulses, which could in principle be used for eavesdropping attacks [@lo-2007-8]. As the LD is taken below threshold between pulses, each new pulse will start up from vacuum fluctuations, and will have a random overall phase $\phi_i$, thus eliminating coherence between successive pulses and thus among states. Similarly, any information contained in the AM phase $\beta_i$ is washed out by the random $\phi_i$. Experimental measurements {#Experimental_measurements} ========================= Figure \[Fig:Laser\_output\_and\_Laser\_driver\_output\] (a) shows the train of optical pulses generated by the laser diode when driven by electrical pulses of 1 ns at 100 MHz. The resulting optical pulse duration is about 400 ps. Since the obtained cw train of optical pulses are all generated in the same way, they can be assumed to be indistinguishable thus having no side-channel information. Furthermore, the short optical pulse duration of 400 ps (small duty cycle) has the advantage to increase the signal to noise ratio since the measurement window (detection time) in the receiver can be reduced. The DFB laser diode is driven in direct modulation with a strong RF driving signal with $24$mA DC bias current, far below threshold $36$mA, thus producing highly similar optical pulses and jitter as low as $100$ps, rise time $65$ps and fall time $129$ps, as shown in Figure \[Fig:Laser\_output\_and\_Laser\_driver\_output\] (b). From Figure \[Fig:Laser\_output\_and\_Laser\_driver\_output\] (b) one can see that the ringing of the laser driver current is repeatable from pulse to pulse, thus producing the overlapped temporal profile of several optical pulses, captured in real-time. The traces are indistinguishable by eye, indicating a very small pulse-to-pulse variation of energy, timing, and wave-form. Furthermore, the optical pulse bandwidth is small enough to enter the acceptance bandwidth of the subsequent polarization modulator. Figure \[Fig:AM\_3\_levels\_and\_PM\_Poincare\] (a) shows the three different intensity optical pulses generated after the AM. The attenuations for the medium and low level of intensity pulses are about 4.65dB and 14.76dB, with respect to the high intensity pulse. While Figure \[Fig:AM\_3\_levels\_and\_PM\_Poincare\] (b) shows the four polarization states generated after the PM, as measured with a terminating rotating waveplate polarimeter. The RF modulating signal is driven at 100 MHz, in this way, chirp produced at the pulse edges of the RF driving voltage is avoided and intensity and polarization indistinguishability is obtained. Figure \[Fig:Intensity\_indistinguishability\_time\_and\_spectrum\] shows pulses with the same polarization but with different intensity levels with the aim of comparing its temporal and spectral indistinguishability. A 8 GHz amplified photodiode and a 4 MHz resolution Fabry-Perot interferometer were used for the temporal and spectral measurements, respectively. In order to compare pulses with different intensity levels, the different pulses are normalized to their own total intensity. Fig. \[Fig:Polarization\_indistinguishability\_time\_and\_spectrum\] shows a similar comparison, but this time pulses have the same intensity level and different polarization. QKD performance analysis ======================== Given the experimental data on the classical optical performance of the proposed source, a low *Quantum Bit Error Rate* (QBER) as well as a high Secure Key Rate in the order of Mb/s are expected. A simulation-based analysis of the expected rates and performances of a QKD BB84, implementing decoy-state protocol, is derived below as a demonstration of the potentials for applications of the proposed FPS. In a BB84 only single photon pulses contribute to the secure key while in a 3-state decoy protocol one can obtain a lower bound for the secure key generation rate as $$R\geq q \frac{N_\mu}{t}\left\{-Q_\mu f\left(E_\mu\right)H_2\left(E_\mu\right)+Q_1\left[1-H_2\left(e_1\right)\right]\right\}$$ where $q$ depends on the implementation (1/2 for the BB84 protocol), $N_{\mu}$ is the total number of detected signal pulses, $t$ is the time duration of the QKD transmission, $\mu$ represents the intensity of the signal states, $Q_\mu$ is the gain of the signal states, $E_\mu$ is the total QBER, $Q_1$ is the gain of single photon states, $e_1$ is the error rate of single photon states, $f\left(x\right)$ is the bi-direction error correction efficiency (taken as $1.16$ [@PhysRevA.72.012326], for an error rate of $1$%) as a function of error rate, and $H_2\left(x\right)$ is the binary Shannon information function, given by $$H_2\left(x\right)=-x\log_2\left(x\right)-\left(1-x\right)\log_2\left(1-x\right)$$ Figure \[Fig:QKD\_simulation\_result\] shows the free space link distance dependence of the Raw Key Rate, Secure Key Rate and QBER. The same parameters used for the 20 Km experiment are used for all the distances considered. ![QKD BB84 implementing decoy-states simulation results. Raw Key Rate (blue solid line), Secure Key Rate (red dashed line) and QBER (green dotted line). In the simulation the detectors efficiency was set to 50$\%$, free space loss 0.1dB/Km, 5dB were accounted for the loss due to the transmitting and receiving optical systems, background yield $1\times 10^{-5}$ and detector misalignment error of $1\%$. All the parameters used in the simulation are consistent with experimental values reported in [@PhysRevA.72.012326].[]{data-label="Fig:QKD_simulation_result"}](QKD_simulation_result){width="0.8\columnwidth"} Results and discussion ====================== Table \[Tab:Relevant\_parameters\_AM\_levels\] summarizes the characteristics of the driving RF and corresponding optical pulses for the three levels of intensity, suitable for a decoy-state protocol. We believe that, should they be needed, larger intensity attenuation could be achieved by improved DC voltage bias of the AM. The AM driving RF signal and the corresponding AM output quality largely demonstrate the 100 MHz and even beyond capability of the source. The modulator “ON” window has a duration of at least 5 ns, much larger than that of the optical pulse. Therefore, only the amplitude of the optical pulse changes, while the temporal and spectral shape remain unaltered. In addition low driving voltages are needed, making the design suitable for electronic integration with low electrical power consumption drivers. Table \[Tab:Relevant\_parameters\_PM\_states\] summarizes the RF voltages driving the PM generating the four orthogonal states. In the same table, cross *polarization extinction ratio* (PER) values for the four different polarization states are given. The PER values obtained ($>$25dB) are significantly higher than those required for a low QBER (20dB). As for the AM case, low driving voltages are needed, suitable for integration with low power consumption and inexpensive electronics. As expected, Figure \[Fig:Intensity\_indistinguishability\_time\_and\_spectrum\] and \[Fig:Polarization\_indistinguishability\_time\_and\_spectrum\] show the high degree of similarity of the pulses, independently of their polarization or intensity state, indicating minimal pulse distortion due to the AM and the PM. It has to be noticed that the small differences for the different intensity pulses are due to measurement errors. Nevertheless, as commented in section \[Sec:Description\_generated\_states\] polarization statistical similarity is more important than intensity statistical similarity. Furthermore, information on the absolute or relative phase between pulses is not contained in these four figures. However, by design, the phase of each pulse varies at random between pulses due to the fact that, as already mentioned, pulses are generated by taking continuously the laser diode above and below threshold, as explained in section \[Sec:Description\_generated\_states\]. In the simulation the detectors efficiency was set to 50$\%$, free space loss 0.1dB/Km, 5dB were accounted for the loss due to the transmitting and receiving optical systems, background yield $1\times 10^{-5}$ and detector misalignment error of $1\%$. Note that the background yield $Y_0$ includes the detector dark count and other background contributions from stray light, including scattered light from timing pulses [@PhysRevA.72.012326; @Schmitt-Manderbach2007], being for larger distances the major cause of secure key rate drop. The parameters, derived from the values presented above, and results for the simulated BB84 transmission, implementing the decoy-state protocol as well as for free space distance of 20 Km, are shown in Table \[Tab:Relevant\_parameters\_QKD\_simulation\]. The simulation has been completed using values taken from [@PhysRevA.72.012326; @Dixon:08], achieving a theoretical Secure Key Rate of 559.80 Kb/s, which is consistent with the free space achieved value of 50 Kb/s (over 480 m) reported for a 10 MHz source in [@Weier2006] taking into account that the presented source emits pulses with a repetition rate one order of magnitude larger. The laser diode is DC biased at $24$mA presenting a DC resistance of $3\Omega$ accounting for $1.7$mW. In addition, an impedance matching circuit has been designed to $50\Omega$ for RF modulation where the electrical pulses of $50$mA, $1$ns wide, at $100$MHz account for $12.5$mW. The modulators do not have a termination resistor, basically, they present an open-ended transmission line with an equivalent loss resistor and parallel capacitor of $5\Omega$ and $10$pF, respectively. Considering the worst case situation where the source is at maximum modulation speed and using the maximum driving voltages, the power consumption for the AM is $2.7$mW and for the PM is $7.7$mW. Thus, the overall power consumption of the integrated module is potentially very low. Conclusion {#Conclusion} ========== We have shown that, starting from commercially available and space-qualifiable components, it is possible to build an integrated transmitter capable of generating the several intensity and polarization states required for decoy-state QKD. The experimental demonstration has been carried out at 850 nm with 100 MHz modulation rates. However, taking into consideration that the modulators bandwidth can go well beyond 10 GHz and operate also at other wavelengths (e.g. 1550 nm), the source can be easily scalable to higher bit rates, the upper limit being probably given by the laser diode itself, and other transmission systems (e.g. optical fibers). Although we believe that the proposed source is of general use in polarization modulation optical systems, especially free-space links, we have focused our demonstration in preparation for a QKD experiment using decoy-state protocol, where the indistinguishability of the pulses, both in the frequency and time domain, is the key for the security of the link. Given the relatively low driving voltages of the modulators, the proposed transmitter is potentially low power consumption and also highly integrable. Acknowledgment {#acknowledgment .unnumbered} ============== This work was carried out with the financial support of the Ministerio de Educacion y Ciencia (Spain) through grants TEC2007-60185, FIS2007-60179, FIS2008-01051 and Consolider Ingenio CSD2006-00019. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} T. H. Carbonneau and D. R. Wisely, “Opportunities and challenges for optical wireless: the competitive advantage of free space telecommunications links in today’s crowded marketplace,” *Wireless Technologies and Systems: Millimeter-Wave and Optical, Proc. SPIE*, vol. 3232, pp. 119–128, 1998. T. Garlington, J. Babbitt, and G. Long, “[Analysis of Free Space Optics as a Transmission Technology]{},” Tech. rep., U.S. Army Information Systems Engineering Command (USAISEC), Transmission Systems Directorate, Tech. Rep., 2005. D. O’Brien, G. Faulkner, K. Jim, E. Zyambo, D. Edwards, M. Whitehead, P. Stavrinou, G. Parry, J. Bellon, M. Sibley, V. Lalithambika, V. Joyner, R. Samsudin, D. Holburn, and R. Mears, “High-speed integrated transceivers for optical wireless,” *Communications Magazine, IEEE*, vol. 41, no. 3, pp. 58–62, Mar 2003. C. Davis, I. Smolyaninov, and S. Milner, “Flexible optical wireless links and networks,” *Communications Magazine, IEEE*, vol. 41, no. 3, pp. 51–57, Mar 2003. V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, “Quantum cloning,” *Rev. Mod. Phys.*, vol. 77, no. 4, pp. 1225–1256, Nov 2005. C. H. Bennett and G. Brassard, “[Quantum cryptography: Public-key distribution and coin tossing]{},” *Proceedings of IEEE International Conference on Computers, Systems and Signal Processing*, pp. 175–179, 1984. C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” *Journal of Cryptology*, vol. 5, pp. 3–28, 1992. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” *Rev. Mod. Phys.*, vol. 81, 1301, 2008. H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” *Phys. Rev. Lett.*, vol. 94, 230504, no. 23, Jun 2005. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” *Phys. Rev. A*, vol. 72, no. 1, p. 012326, Jul 2005. A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, “Gigahertz decoy quantum key distribution with 1 mbit/s secure key rate,” *Opt. Express*, vol. 16, no. 23, pp. 18790–18979, 2008. T.-Y. Chen, J. Wang, Y. Liu, W.-Q. Cai, X. Wan, L.-K. Chen, J.-H. Wang, S.-B. Liu, H. Liang, L. Yang, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “[200km Decoy-state quantum key distribution with photon polarization]{},” *arXiv:0908.4063v1*, 2009. T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, A. Rarity, J. G. Zeilinger, and H. Weinfurter, “[Experimental Demonstration of Free-Space Decoy-State Quantum Key Distribution over 144 km]{},” *Phys. Rev. Lett.*, vol. 98, 010504, 2007. H. Weier, T. Schmitt-Manderbach, N. Regner, C. Kurtsiefer, and H. Weinfurter, “[Free space quantum key distribution: Towards a real life application]{},” *Fortschr. Phys.*, vol. 54, no. 8-10, pp. 840–845, 2006. C. Kurtsiefer, P. Zarda, M. Halder, P. M. Gorman, P. R. Tapster, J. G. Rarity, and W. H., “[Long Distance Free Space Quantum Cryptography]{},” *Proc. SPIE*, vol. 4917, p. 25, 2002. M. Furst, T. Schmitt-Manderbach, H. Weier, R. Ursin, F. Tiefenbacher, T. Scheidl, C. Barbieri, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Free-space decoy-state quantum key distribution,” *Optical Fiber communication/National Fiber Optic Engineers Conference, 2008. OFC/NFOEC 2008. Conference on*, pp. 1–3, Feb. 2008. H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-db channel loss using superconducting single-photon detectors,” *Nature Photonics*, vol. 1, pp. 343–348, 2007. W.-Y. Hwang, “Quantum key distribution with high loss: Toward global secure communication,” *Phys. Rev. Lett.*, vol. 91, 057901, no. 5, Aug 2003. J. G. Rarity, P. R. Tapster, P. M. Gorman, and P. Knight, “Ground to satellite secure key exchange using quantum cryptography,” *New Journal of Physics*, vol. 4, 2002. C. Bonato, A. Tomaello, V. D. Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” *New Journal of Physics*, vol. 11, 045017, April 2009. A. D. Sadovnikov, X. Li, and W.-P. Huang, “[A Two-Dimensional DFB Laser Model Accounting for Carrier Transport Effects]{},” *IEEE Journal of Quantum Electronics*, vol. 31, no. 10, pp. 1856–1862, October 1995. X. Li, A. D. Sadovnikov, W.-P. Huang, and T. Makino, “[A Physics-Based Three-Dimensional Model for Distributed Feedback Laser Diodes]{},” *IEEE Journal of Quantum Electronics*, vol. 34, no. 9, pp. 1545–1553, September 1998. K. Petermann, *Laser diode modulation and noise*, A. in Optoelectronics, Ed.1em plus 0.5em minus 0.4emKluwer Academic Publishers, 1991. H.-K. Lo and J. Preskill, “Security of quantum key distribution using weak coherent states with nonrandom phases,” *Quantum Information and Computation*, vol. 8, no. 5, pp. 431–458, 2007. J. A. Bergou, U. Herzog, and M. Hillery, *Discrimination of quantum states*, ser. [Lecture Notes in Physics Vol. 649]{}, Springer, Berlin, 2004. S. M. Barnett and S. Croke, “Quantum state discrimination,” *Adv. Opt. Photon.*, vol. 1, no. 2, pp. 238–278, 2009.

--- abstract: 'Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length $3$ in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive explicit formulae or generating functions for various occurrences of length $3$ patterns on multiply restricted permutations, as well as some combinatorial interpretations for non-trivial pattern relationships.' author: - | Alina F. Y. Zhao\ School of Mathematical Sciences and Institute of Mathematics\ Nanjing Normal University, Nanjing 210023, PR China\ [alinazhao@njnu.edu.cn]{} title: '**Pattern Count on Multiply Restricted Permutations**' --- **Mathematics Subject Classification:** 05A05, 05A15, 05A19 Introduction ============ Let $\sg=\sg_1\sg_2\cdots \sg_n$ be a permutation in the symmetric group $S_n$ written in one-line notation, and $\sg_i$ is said to be a left to right maximum (resp. right to left maximum) if $\sg_i>\sg_j$ for all $j<i$ (resp. $j>i$). For a permutation $q\in S_k$, we say that $\sg$ contains $q$ as a pattern if there exist $1\leq i_1\leq i_2\leq \cdots \leq i_k \leq n$ such that the entries $\sg_{i_1}\sg_{i_2}\cdots \sg_{i_k}$ have the same relative order as the entries of $q$, i.e., $q_j < q_l$ if and only if $\sg_{i_j} < \sg_{i_l}$ whenever $1\leq j,l\leq k$. We say that $\sg$ avoids $q$ if $\sg$ does not contain $q$ as a pattern. For a pattern $q$, denote by $S_n(q)$ the set of all permutations in $S_n$ that avoiding the pattern $q$, and for $R\subseteq S_k$, we denote by $S_n(R)= \bigcap_{q\in R}S_n(q)$, i.e., the set of permutations in $S_n$ which avoid every pattern contained in $R$. For two permutations $\sg$ and $q$, we set $f_q(\sg)$ as the number of occurrences of $q$ in $\sg$ as a pattern, and we further denote the number of occurrences of $q$ in a permutation set $\Omega$ by $f_q(\Omega)=\sum_{\sg\in \Omega}f_q(\sg)$. Recently Cooper [@Coo] raised the problem of determining the total number $f_q(S_n(r))$ of all $q$-patterns in the $r$-avoiding permutations of length $n$. Bóna [@Bona2010] discovered the generating functions of the sequence $f_q(S_n(132))$ for the monotone $q$, and Bóna [@Bona] further studied the generating functions for other length $3$ patterns in the set $S_n(132)$, and showed both algebraically and bijectively that $$f_{231}(S_n(132)) = f_{312}(S_n(132))=f_{213}(S_n(132)).$$ Rudolph [@Rud] proved equipopularity relations between general length $k$ patterns on $132$-avoiding permutations based on the structure of their corresponding binary plane trees. Moreover, Homberger [@Hom] also presented exact formulae for each length $3$ pattern in the set $S_n(123)$. Therefore, the singly restricted permutations have been well studied by previous work, whereas it remains open for multiply restricted permutations, e.g., $S_n(123,132)$. [|p[13mm]{}|c|c|p[50mm]{}|]{} & [$f_{213}(n)=(n-3)2^{n-2}+1 $]{} & &[$\sum f_{231}(n)x^n=\sum f_{312}(n)x^n=\frac{x^3(1+2x)}{(1-x-x^2)^3}$]{}\ & [$f_{231}(n)=f_{312}(n)=(n^2-5n+8)2^{n-3}-1$]{} & [$S_n(123,132,213)$]{} &[$\sum f_{321}(n)x^n= \frac{x^3(1+6x+12x^2+8x^3)}{(1-x-x^2)^4}$]{}\ & [$f_{321}(n)=(n^3/3-2n^2+14n/3-5)2^{n-2}+1$]{} & &[$f_{213}(n)=f_{312}(n)={n\choose 3}$]{}\ & [$f_{123}(n)=(n-4)2^{n-1}+n+2$]{} & & [$f_{321}(n)=(n-2){n\choose 3}$]{}\ &[$f_{231}(n)=f_{312}(n)=(\frac{n^2}{4}-\frac{7n}{4}+4)2^{n}-n-4$]{} & &[$f_{123}(n)=f_{312}(n)={n+1\choose 4}$]{}\ & [$f_{321}(n)=(\frac{1}{12}n^3-\frac{3}{4}n^2+\frac{38}{12}n-6)2^{n}+n+6$]{} & &[$f_{321}(n)=n(n-2)(n-1)^2/12$]{}\ & [$f_{123}(n)=f_{213}(n)=f_{312}(n)=f_{321}(n)=\frac{2^n}{8}{n\choose 3}$]{} & &[$f_{213}(n)=f_{231}(n)={n\choose 3}$]{}\ [$S_n(132,312)$]{}& [$f_{123}(n)=f_{213}(n)=f_{231}(n)=f_{321}(n)=\frac{2^n}{8}{n\choose 3}$]{} & &[$f_{321}(n)=(n-2){n\choose 3}$]{}\ & [$f_{213}(n)=f_{231}(n)=f_{312}(n)={n+2\choose 5}$]{}&&[$f_{132}(n)=f_{213}(n)={n+1\choose 4}$]{}\ & [$f_{123}(n)=\frac{7n^5}{120}-\frac{n^4}{3}+\frac{17n^3}{24}-\frac{2n^2}{3}+\frac{7}{30}$]{} &&[$f_{321}(n)=\frac{1}{12}n(n-2)(n-1)^2$]{}\ In this paper, we are interested in pattern count on multiply restricted permutations $S_n(R)$ for $R\subset S_3$, especially for double and triple restrictions. We derive explicit formulae or generating functions for the occurrences of each length $3$ pattern in multiply restricted permutations, and the detailed results are summarized in Table \[tab:sum\]. Also, we present some combinatorial interpretations for non-trivial pattern relationships. It is trivial to consider the restricted permutations of higher multiplicity since there are only finite permutations, as shown in [@Sim]. Therefore, this work presents a complete study on length $3$ patterns of multiply restricted permutations. Doubly Restricted Permutations ============================== For $\sg \in S_n$, the following three operations are very useful in pattern avoiding enumeration. The complement of $\sg$ is given by $\sg^c=(n+1-\sg_1)(n+1-\sg_2)\cdots (n+1-\sg_n)$, its reverse is defined as $\sg^r=\sg_n\cdots \sg_2\sg_1$ and the inverse $\sg^{-1}$ is the group theoretic inverse permutation. For any set of permutations $R$, let $R^c$ be the set obtained by complementing each element of $R$, and the set $R^r$ and $R^{-1}$ are defined analogously. \[oper\] Let $R\subseteq S_k$ be any set of permutations in $S_k$, and $\sg \in S_n$, we have $$\sg\in S_n(R)\Leftrightarrow \sg^c\in S_n(R^c)\Leftrightarrow \sg^r\in S_n(R^r)\Leftrightarrow \sg^{-1}\in S_n(R^{-1}).$$ From Lemma \[oper\] and the known results on $S_n(123)$ and $S_n(132)$, we can obtain each length $3$ pattern count in the singly restricted permutations $S_n(r)$ for $r=213,231,312$ and $321$. In this section, we focus on the number of length $3$ patterns in all doubly restricted permutations. A composition of $n$ is an expression of $n$ as an ordered sum of positive integers, and we say that $c$ has $k$ parts or $c$ is a $k$-composition if there are exactly $k$ summands appeared in a composition $c$. Denote by $\mathcal{C}_n$ and $\mathcal{C}_{n,k}$ the set of all compositions of $n$ and the set of $k$-compositions of $n$, respectively. It is known that $|\mathcal{C}_n|=2^{n-1}$ and $|\mathcal{C}_{n,k}|= {n-1\choose k-1}$ for $n\geq 1$ with $|\mathcal{C}_0|=1$. For more details, see [@Sta]. We begin with some enumerative results on compositions. \[com1\] For $n\geq 1$, we have $$\begin{aligned} a(n)&:=&\sum_{k\geq 1 \atop c_1+\cdots+c_{k-1}+c_k=n}c_k=2^n-1,\\ b(n)&:=&\sum_{k\geq 1 \atop c_1+\cdots+c_{k-1}+c_k=n}c_k(c_k-1)=2^{n+1}-2n-2,\end{aligned}$$ where the sum takes over all compositions of $n$. For $c_k=m$ , we can regard $c_1+c_2+\cdots+c_{k-1}$ as a composition of $n-m$. It is easy to get that the number of compositions of $n-m$ is $2^{n-m-1}$ for $1\leq m\leq n-1$ and the number of empty compositions is one. This follows that $$\begin{aligned} a(n)&=n+\sum_{m=1}^{n-1}m2^{n-m-1},\\ b(n)&=n(n-1)+\sum_{m=1}^{n-1}m(m-1)2^{n-m-1}.\end{aligned}$$ Let $g(x)=\sum_{i=0}^{n-1}x^{i}=\frac{1-x^n}{1-x}$, and we have $$\begin{aligned} &g'(x)=\sum_{i=1}^{n-1}ix^{i-1}=\frac{(n-1)x^n-nx^{n-1}+1}{(1-x)^2}, \\ &g''(x)=\sum_{i=1}^{n-1}i(i-1)x^{i-2}=\frac{(3n-n^2-2)x^n+(2n^2-4n)x^{n-1}+(n-n^2)x^{n-2}+2}{(1-x)^3}.&\end{aligned}$$ By setting $x=1/2$ in $g'(x)$ and $g''(x)$, we get $$\begin{aligned} g'(1/2)&=&2^2((n-1)2^{-n}-n2^{-n+1}+1),\\ g''(1/2)&=&2^3\left[(3n-n^2-2)2^{-n}+(2n^2-4n)2^{-n+1}+(n-n^2)2^{-n+2}+2\right].\end{aligned}$$ Observing $a(n)=2^{n-2}g'(1/2)+n$ and $b(n)=2^{n-3}g''(1/2)+n(n-1)$, this lemma follows as desired. \[com2\] For $n\geq 1$, we have $$\begin{aligned} c(n)&:=&\sum_{k\geq 1 \atop c_1+\cdots+c_{k-1}+c_k=n}k=(n+1)2^{n-2},\\ d(n)&:=&\sum_{k\geq 1 \atop c_1+\cdots+c_{k-1}+c_k=n}k(k-1)=(n^2+n-2)2^{n-3},\end{aligned}$$ where the sum takes over all compositions of $n$. Since the number of compositions of $n$ with $k$ parts is ${n-1\choose k-1}$, we have $$c(n)=\sum_{k=1}^{n}k{n-1\choose k-1} \text{ and } d(n)=\sum_{k=1}^{n}k(k-1){n-1\choose k-1}.$$ Let $h(x)=x\sum_{i=1}^{n}{n-1\choose i-1}x^{i-1}=x(1+x)^{n-1}$. Then $$\begin{aligned} h'(x) &=&\sum_{i=1}^n i{n-1\choose i-1} x^{i-1}=(nx+1)(1+x)^{n-2}, \\ h''(x)&=&\sum_{i=1}^n i(i-1){n-1\choose i-1}x^{i-2} =\big(n^2x+n(2-x)-2\big)(1+x)^{n-3}.\end{aligned}$$ We complete the proof by putting $x=1$ in the above formulae. Based on Lemma \[oper\], Simion and Schmidt [@Sim] showed that the pairs of patterns among the total ${6\choose 2}=15$ cases fall into the following $6$ classes. For every symmetric group $S_n$, 1. $|S_n(123,132)|= |S_n(123,213)|=|S_n(231,321)|= |S_n(312,321)|= 2^{n-1}$; 2. $|S_n(132,213)|=|S_n(231,312)|=2^{n-1}$; 3. $|S_n(132,231)|=|S_n(213,312)|=2^{n-1}$; 4. $|S_n(132,312)|=|S_n(213,231)|=2^{n-1}$; 5. $|S_n(132,321)|=|S_n(123,231)|=|S_n(123,312)|=|S_n(213,321)|={n\choose 2}+1$; 6. $|S_n(123,321)|=0$ for $n\geq 5$. Therefore, it is sufficient to consider the pattern count of the first set for each class in the subsequent sections, and the other sets can be obtained by taking the complement or reverse or inverse of the known results. Pattern Count on $(123,132)$-Avoiding Permutations -------------------------------------------------- We first present a bijection between $S_n(123,132)$ and $\mathcal{C}_n$ as follows: \[ta\] There is a bijection $\varphi_1$ between the sets $S_n(123,132)$ and $\mathcal{C}_n$. For any given $\sg\in S_n(123,132)$, let $\sg_{i_1},\sg_{i_2},\ldots,\sg_{i_k}$ be the $k$ right to left maxima with $i_1<i_2<\cdots<i_k$, which yields that $c=i_1+(i_2-i_1)+\cdots+(i_{k-1}-i_{k-2})+(i_{k}-i_{k-1})$ is a composition of $n$ since $i_k=n$. On the converse, let $m_i=n-(c_1+\cdots+c_{i-1})$ for any given $n=c_1+c_2+\cdots+c_k\in \C_n$, and we set $\tau_i=m_i-1,m_i-2,\ldots,m_i-c_i+1,m_i$ for $1\leq i \leq k$. Therefore, $\sg=\tau_1,\tau_2,\ldots,\tau_k \in S_n(123,132)$ is as desired. For example, we have $\sg=8\,9\,7\,5\,4\,3\,6\,1\,2$ for the composition $9=2+1+4+2$. For a pattern $q$, we denote by $f_{q}(n):=\sum_{\sigma\in S_n(123,132)}f_q(\sigma)$, i.e., the number of occurrences of the pattern $q$ in $S_n(123,132)$. For simplicity, we will use this notation in subsequent sections when the set in question is unambiguous. For convenience, we denote by $\tau_i>\tau_j$ if all the elements in the subsequence $\tau_i$ are larger than all the elements in subsequence $\tau_j$. Based on Lemma \[ta\], we have \[pa\] For $n\geq3$, $$\begin{aligned} f_{213}(n)&=&\sum_{k\geq1 \atop c_1+c_2+\cdots+c_k=n}\sum_{i=1}^k{c_i-1\choose 2},\label{pa1}\\ f_{231}(n)&=&\sum_{k\geq1 \atop c_1+c_2+\cdots+c_k=n}\sum_{i=1}^{k-1}\sum_{j= i+1}^kc_j(c_i-1) .\label{pa2}\end{aligned}$$ For each permutation $\sg\in S_n(123,132)$ with $\varphi_1(\sg)=c_1+ c_2+ \cdots+c_k$, we can rewrite $\sg$ as $\sg=\tau_1,\tau_2,\ldots,\tau_k$ from Lemma \[ta\]. For $j>i$, since $\tau_i>\tau_{j}$, and the elements except the last one are decreasing in $\tau_i$, the pattern $213$ can only occur in every subsequence $\tau_i$. Thus, we have ${c_i-1 \choose 2}$ choices to choose two elements in $\tau_i$ to play the role of $21$, and the last element of $\tau_i$ plays the role of $3$, summing up all the number of $213$-patterns in subsequences $\tau_1,\tau_2,\ldots,\tau_k$ gives the formula . For the pattern $231$, we have $c_i-1$ choices in the subsequence $\tau_i$ to choose one element playing the role of $2$ and one choice (always the last element of $\tau_i$) for $3$, and then, we have $c_{i+1}+\cdots+c_k$ choices to choose one element for the role of $1$ since all the elements after $\tau_i$ are smaller than those in $\tau_i$. Summing up all the number of $231$-patterns according to the position of $3$ gives the formula . Based on the previous analysis, we now present our first main results of the explicit formulae for pattern count in the set $S_n(123,132)$. \[ta1\] For $n\geq 3$, in the set $S_n(123,132)$, we have $$\begin{aligned} f_{213}(n)&=&(n-3)2^{n-2}+1, \label{fta1} \\ f_{231}(n)&=&f_{312}(n)=(n^2-5n+8)2^{n-3}-1,\label{fta2} \\ f_{321}(n)&=&(n^3/3-2n^2+14n/3-5)2^{n-2}+1.\label{fta4}\end{aligned}$$ From $S_3(123,132)= \{213,231,312,321\}$, it is obvious that $$f_{213}(3)=f_{231}(3)=1.$$ To prove formula , from Prop \[pa\], we observe that, for $n\geq3$ $$f_{213}(n+1)=\sum_{k\geq 1, c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i-1\choose2}+\sum_{k\geq 1, c_k\geq2 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i-1\choose2}.$$ If $c_k=1$, then $k\geq2$, and we further have $$\sum_{k\geq 1, c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i-1\choose2}=\sum_{k-1\geq 1\atop c_1+c_2+\cdots+c_{k-1}=n}\sum_{i=1}^{k-1} {c_i-1\choose 2}=f_{213}(n).$$ If $c_k\geq 2$, then we set $c_k=1+r_k$, and from Lemma \[com1\], it holds that $$\begin{aligned} \sum_{k\geq 1, c_k\geq2 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i-1\choose2}&=\sum_{k\geq 1 \atop c_1+\cdots+c_{k-1}+r_k=n}\left[\sum_{i=1}^{k-1}{c_i-1\choose 2}+{r_k-1\choose 2}+(r_k-1)\right]\\ &=f_{213}(n)+\sum_{k\geq 1\atop c_1+\cdots+c_{k-1}+r_k=n}(r_k-1) \\ &=f_{213}(n)+a(n)-\sum_{k\geq 1\atop c_1+\cdots+c_{k-1}+r_k=n}1 =f_{213}(n)+a(n)-2^{n-1}.\end{aligned}$$ Combining the above two cases, we have $$f_{213}(n+1)=2f_{213}(n)+2^{n-1}-1.$$ This proves formula  by solving the recurrence with initial value $f_{213}(3)=1$. To prove formula , first observe that from Lemma \[oper\], $\sg \in S_n(123,132)\Leftrightarrow \sg^{-1}\in S_n(123,132)$, this implies the first equality of formula  directly since $231^{-1}=312$. While for the second equality of formula , by Prop \[pa\], we have $$f_{231}(n+1)=\sum_{k\geq1,c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^{k-1}\sum_{j= i+1}^k c_j(c_i-1)+\sum_{k\geq1,c_k\geq2 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^{k-1}\sum_{j= i+1}^k c_j(c_i-1).$$ If $c_k=1$, then $k\geq 2$, and from Lemma \[com2\] we have $$\begin{aligned} \sum_{k\geq1,c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^{k-1}\sum_{j= i+1}^k c_j(c_i-1)&=\sum_{k-1\geq 1\atop c_1+\cdots+c_{k-1}=n}\sum_{i=1}^{k-2}\sum_{j= i+1}^{k-1} c_j(c_i-1) +\sum_{k-1\geq 1\atop c_1+\cdots+c_{k-1}=n}\sum_{i=1}^{k-1}(c_i-1)\\ &= f_{231}(n)+\sum_{k-1\geq 1 \atop c_1+\cdots+c_{k-1}=n}n-(k-1)\\ &= f_{231}(n)-c(n)+n2^{n-1}.\end{aligned}$$ If $c_k\geq 2$, then we set $c'_k=c_k-1$ and $c'_i=c_i$ for $1\leq i\leq k-1$. This holds that $$\begin{aligned} \sum_{k\geq1,c_k\geq2 \atop c_1+\cdots+c_k=n+1}\sum_{i=1}^{k-1}\sum_{j= i+1}^k c_j(c_i-1)&=\sum_{k\geq1 \atop c'_1+\cdots+c'_k=n}\sum_{i=1}^{k-1}\sum_{j= i+1}^k c'_j(c'_i-1)+\sum_{k\geq1 \atop c'_1+\cdots+c'_k=n}\sum_{i=1}^{k-1}(c'_i-1)\\ &= f_{231}(n)+\sum_{k\geq 1\atop c'_1+\cdots+c'_k=n}(n-c'_k-k+1)\\ &=f_{231}(n)-a(n)-c(n)+(n+1)2^{n-1},\end{aligned}$$ where the last equality holds from Lemma \[com1\] and Lemma \[com2\]. Therefore, after simplification, we have $$f_{231}(n+1)=2f_{231}(n)+(n-2)2^{n-1}+1,$$ which proves the second equality of formula  by using $f_{231}(3)=1$. Note that the total number of all length $3$ patterns in a permutation $\sg \in S_n$ is ${n\choose 3}$, for the set $S_n(123,132)$, this gives the relation $$f_{213}(n)+2f_{231}(n)+f_{321}(n)={n\choose 3}2^{n-1}.$$ Thus formula  is a direct computation of the above equation. This completes the proof. The first few values of $f_q(S_n(123,132))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $0$ $0$ $1$ $1$ $1$ $1$ $6$ $0$ $0$ $49$ $111$ $111$ $369$ $4$ $0$ $0$ $5$ $7$ $7$ $13$ $7$ $0$ $0$ $129$ $351$ $351$ $1409$ $5$ $0$ $0$ $17$ $31$ $31$ $81$ $8$ $0$ $0$ $321$ $1023$ $1023$ $4801$ \[2a\] Pattern Count on $(132,213)$-Avoiding Permutations -------------------------------------------------- We begin with the following correspondence between $(132,213)$-avoiding permutations and compositions of $n$. \[tb\] There is a bijection $\varphi_2$ between the sets $S_n(132,213)$ and $\mathcal{C}_n$. Given $\sg\in S_n(132,213)$, let $\sg_{i_1},\sg_{i_2},\ldots, \sg_{i_k}$ be the $k$ right to left maxima with $i_1<i_2<\cdots<i_k$. This follows that $c=i_1+(i_2-i_1)+\cdots+(i_{k-1}-i_{k-2})+(i_{k}-i_{k-1})$ is a composition of $n$ since $i_k=n$. On the converse, given $n=c_1+c_2+\cdots+c_k\in \C_n$, let $m_i=n-(c_1+\cdots+c_{i-1})$ and $\tau_i=m_i-c_i+1,m_i-c_i+2,\ldots,m_i-1,m_i$ for $1\leq i \leq k$. We set $\sg=\tau_1,\tau_2,\ldots,\tau_k$, and it is easy to check that $\sg \in S_n(132,213)$. For example, for the composition $9=3+3+1+2$, we get $\sg=7\,8\,9\,4\,5\,6\,3\,1\,2$. Based on the above lemma, we have \[pb\] For $n\geq3$, $$\begin{aligned} f_{123}(n)&=\sum_{k\geq 1 \atop c_1+c_2+\cdots+c_k=n} \sum_{i=1}^k{c_i\choose 3},\label{pb1}\\ f_{231}(n)&=\sum_{k\geq 1 \atop c_1+c_2+\cdots+c_k=n}\sum_{i=1}^{k-1} \sum_{j=i+1}^kc_j {c_i\choose 2}\label{pb2}.\end{aligned}$$ For a permutation $\sg\in S_n(132,213)$ with $\varphi_2(\sg)=c_1+c_2+\cdots+c_k$, we rewrite $\sg$ as $\sg=\tau_1,\tau_2,\ldots,\tau_k$ by Lemma \[tb\], and we see that the pattern $123$ can only occur in every subsequence $\tau_i$ since $\tau_i>\tau_{j}$ for $j>i$ and the elements in $\tau_i$ are increasing. Thus, we have ${c_i \choose 3}$ choices to choose three elements in $\tau_i$ to play the role of $123$, and formula follows by summing all $123$-patterns in subsequences $\tau_1,\tau_2,\ldots,\tau_k$. For pattern $231$, we have ${c_i \choose 2}$ choices in the subsequence $\tau_i$ to choose two elements to play the role of $23$, after this we have $c_{i+1}+\cdots+c_k$ choices to choose one element in $\tau_{i+1},\ldots,\tau_k$ for the role of $1$ since $\tau_j<\tau_i$ for all $j>i$. Summing up all the number of $231$-patterns according to the positions of $23$ gives the formula . \[tb1\] For $n\geq 3$, in the set $S_n(132,213)$, we have $$\begin{aligned} f_{123}(n)&=(n-4)2^{n-1}+n+2,\label{ftb1}\\ f_{231}(n)&=f_{312}(n)=(n^2-7n+16)2^{n-2}-n-4,\label{ftb2}\\ f_{321}(n)&=(n^3/3-3n^2+38n/3-24)2^{n-2}+n+6.\label{ftb3}\end{aligned}$$ From Prop \[pb\], we have $$f_{123}(n+1)=\sum_{k\geq 1, c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i\choose 3}+\sum_{k\geq 1, c_k\geq2 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i\choose 3}.$$ If $c_k=1$, then $k\geq 2$, and we have $$\begin{aligned} \sum_{k\geq 1 ,c_k=1 \atop c_1+c_2+\cdots+c_{k}=n+1}\sum_{i=1}^{k} {c_i\choose 3}=\sum_{k-1\geq 1 \atop c_1+c_2+\cdots+c_{k-1}=n}\sum_{i=1}^{k-1} {c_i\choose 3}=f_{123}(n).\end{aligned}$$ If $c_k\geq 2$, then we set $c_k=1+r_k$, where $r_k\geq 1$, and this follows $$\begin{aligned} \sum_{k\geq 1, c_k\geq2 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^k {c_i\choose 3}&=\sum_{k\geq 1 \atop c_1+c_2+\cdots+c_{k-1}+r_k=n} \left[\sum_{i=1}^{k-1}{c_i\choose 3}+{r_k\choose 3}+\frac{r_k(r_k-1)}{2}\right]\\ &=f_{123}(n)+b(n)/2.\end{aligned}$$ Combining the above two cases, we get that $$f_{123}(n+1)=2f_{123}(n)+2^n-n-1,$$ and formula  holds by solving the recurrence with initial value $f_{123}(3)=1$. From Lemma \[oper\], we see that $\sg \in S_n(132,213)\Leftrightarrow \sg^{-1}\in S_n(132,213)$, and this follows that $f_{231}(n)=f_{312}(n)$ from $231^{-1}=312$. To calculate $f_{231}(n)$, we have by using Prop \[pb\] again $$f_{231}(n+1)=\sum_{k\geq 1,c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^{k-1} \sum_{j=i+1}^kc_j {c_i\choose 2}+\sum_{k\geq 1,c_k\geq 2 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^{k-1} \sum_{j=i+1}^kc_j {c_i\choose 2}.$$ If $c_k=1$, then $k\geq 2$, and $$\begin{aligned} \sum_{k\geq 1,c_k=1 \atop c_1+c_2+\cdots+c_k=n+1}\sum_{i=1}^{k-1} \sum_{j=i+1}^kc_j{c_i\choose 2} &=\sum_{k-1\geq 1 \atop c_1+c_2+\cdots+c_{k-1}=n}\sum_{i=1}^{k-1} {c_i\choose 2}\left[\sum_{j=i+1}^{k-1}c_j+1\right]\\ &=f_{231}(n)+\alpha(n),\end{aligned}$$ where $\alpha(n)=\sum \limits_{k\geq 1\atop c_1+c_2+\cdots+c_k=n}\sum_{i=1}^{k} {c_i\choose 2}$. We further have $$\begin{aligned} \alpha(n)&=\sum_{k\geq 1 \atop c_1+\cdots+c_{k}=n}\sum_{i=1}^{k}\left[{c_i-1\choose 2}+c_i-1\right] =f_{213}(S_n(123,132))+\sum_{k\geq 1 \atop c_1+\cdots+c_{k}=n}(n-k)\\ &=f_{213}(S_n(123,132))-c(n)+n2^{n-1},\end{aligned}$$ in which we use the derived formula $f_{213}(S_n(123,132))=\sum\limits_{k\geq 1 \atop c_1+\cdots+c_{k}=n}\sum_{i=1}^{k}{c_i-1\choose 2}$.\ If $c_k\geq 2$, then we set $c'_k=c_k-1$ and $c'_i=c_i$ for $1\leq i\leq k-1$. This holds that $$\begin{aligned} \sum_{k\geq 1,c_k\geq 2 \atop c_1+\cdots+c_k=n+1} \sum_{i=1}^{k-1}\sum_{j=i+1}^kc_j{c_i\choose 2} &= \sum_{k\geq 1 \atop c'_1+\cdots+c'_k=n} \sum_{i=1}^{k-1}\sum_{j=i+1}^kc'_j{c'_i\choose 2}+ \sum_{k\geq 1 \atop c'_1+\cdots+c'_k=n} \sum_{i=1}^{k-1}{c'_i\choose 2}\\ &=f_{231}(n)+\beta(n),\end{aligned}$$ where $\beta(n)=\sum\limits_{k\geq 1 \atop c_1+\cdots+c_k=n} \sum_{i=1}^{k-1}{c_i\choose 2}$. Further, we can rewrite $\beta(n)$ as $$\begin{aligned} \beta(n)&=\sum_{k\geq 1 \atop c_1+\cdots+c_k=n} \sum_{i=1}^{k}{c_i\choose 2}-\sum_{k\geq 1 \atop c_1+\cdots+c_k=n} \frac{c_k (c_{k}-1)}{2}\\ &=\alpha(n)-b(n)/2=f_{213}(S_n(123,132))-c(n)-b(n)/2+n2^{n-1}.\end{aligned}$$ Substituting the known formulae for $f_{213}(S_n(123,132))$, $c(n)$ and $b(n)$, we get that $$f_{231}(n+1)=2f_{231}(n)+(2n-6)2^{n-1}+n+3,$$ and formula  holds by solving this recurrence with initial condition $f_{213}(3)=1$. Finally, formula follows from $f_{123}(n)+2f_{231}(n)+f_{321}(n)={n\choose 3}2^{n-1}$, and this completes the proof. The first few values of $f_q(S_n(132,213))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $1$ $0$ $0$ $1$ $1$ $1$ $6$ $72$ $0$ $0$ $150$ $150$ $268$ $4$ $6$ $0$ $0$ $8$ $8$ $10$ $7$ $201$ $0$ $0$ $501$ $501$ $1037$ $5$ $23$ $0$ $0$ $39$ $39$ $59$ $8$ $522$ $0$ $0$ $1524$ $1524$ $3598$ \[2b\] Pattern Count on $(132,231)$-Avoiding Permutations -------------------------------------------------- \[tc1\] For $n\geq 3$, in the set $S_n(132,231)$, we have $$\begin{aligned} \label{ftc1} f_{123}(n)=f_{213}(n)=f_{312}(n)=f_{321}(n)={n\choose 3}2^{n-3}.\end{aligned}$$ For each $\sg \in S_n(132,231)$, we observe that $n$ must lie in the beginning or the end of $\sg$, and $n-1$ must lie in the beginning or the end of $\sg \backslash \{n\}$, ..., and so on. Here $\sg \backslash \{n\}$ denotes the sequence obtained from $\sg$ by deleting element $n$. Based on such observation, suppose $abc$ is a length $3$ pattern in $S_n(132,231)$, and set $[n] \backslash \{a,b,c\}:=\{r_1>r_2>\cdots>r_{n-4}>r_{n-3}\}$. We can construct a permutation in the set $S_n(132,231)$ which contains an $abc$ pattern as follows: Start with the subsequence $\sg^0:=abc$, and for $i$ from $1$ to $n-3$, $\sg^i$ is obtained from $\sg^{i-1}$ by inserting $r_i$ into it. - if there are at least two elements in $\sg^{i-1}$ smaller than $r_i$, then choose the two elements $A$ and $B$ such that $A$ is the leftmost one and $B$ is the rightmost one. We put $r_i$ immediately to the left of $A$ or immediately to the right of $B$; - if there is only one element $A$ in $\sg^{i-1}$ such that $A<r_i$, then we can put $r_i$ immediately to the left or to the right of $A$; - if all the elements in $\sg^{i-1}$ are larger than $r_i$, then choose $A$ the least one, and put $r_i$ immediately to the left or to the right of $A$. Finally, we set $\sg:=\sg^{n-3}$ and $\sg\in S_n(132,231)$ from the above construction. Moreover, there are ${n\choose 3}$ choices to choose $abc$, and the number of permutations having $abc$ as a pattern is $2^{n-3}$ since each $r_i$ has $2$ choices. This completes the proof. Here we give an illustration of constructing a permutation in $S_8(132,231)$ which contains the pattern $abc=256$. Set $\sg^0:=256$, we may have $\sg^1=8\,2\,5\,6$, $\sg^2=8\,7\,2\,5\,6$, $\sg^3=8\,7\,2\,4\,5\,6$, $\sg^4=8\,7\,3\,2\,4\,5\,6$, $\sg:=\sg^5=8\,7\,3\,2\,1\,4\,5\,6$. We could also provide combinatorial proofs for the phenomenon $f_{123}(n)=f_{213}(n)=f_{312}(n)=f_{321}(n)$. From Lemma \[oper\], we have $\sg \in S_n(132,231)\Leftrightarrow \sg^{r}\in S_n(132,231)$, and this follows $f_{123}(n)=f_{321}(n)$ and $f_{213}(n)=f_{312}(n)$ from $123^{r}=321$ and $213^{r}=312$, respectively. It remains to give a bijection for $f_{213}(n)=f_{123}(n)$, and our following construction is motivated from Bóna [@Bona]. A binary plane tree is a rooted unlabeled tree in which each vertex has at most two children, and each child is a left child or a right child of its parent. For each $\sg\in S_n(132)$, we construct a binary plane tree $T(\sg)$ as follows: the root of $T(\sg)$ corresponds to the entry $n$ of $\sg$, the left subtree of the root corresponds to the string of entries of $\sg$ on the left of $n$, and the right subtree of the root corresponds to the string of entries of $\sg$ on the right of $n$. Both subtrees are constructed recursively by the same rule. For more details, see [@Bona2011; @Bona; @Rud]. A left descendant (resp. right descendant) of a vertex $x$ in a binary plane tree is a vertex in the left (resp. right) subtree of $x$. The left (resp. right) subtree of $x$ does not contain $x$ itself. Similarly, an ascendant of a vertex $x$ in a binary plane tree is a vertex whose subtree contains $x$. Given a tree $T$ and a vertex $v \in T$, let $T_v$ be the subtree of $T$ with $v$ as the root. Let $R$ be an occurrence of the pattern $123$ in $\sg \in S_n(132)$, and let $R_1,R_2,R_3$ be the three vertices of $T(\sg)$ that correspond to $R$, going left to right. Then, $R_1$ is a left descendant of $R_2$, and $R_2$ is a left descendant of $R_3$. From the above correspondence, we see that for $\sg \in S_n(132,231)$, $T(\sg)$ is a binary plane tree on $n$ vertices such that each vertex has at most one child from the forbiddance of the pattern $231$. For simplicity, denote by $\mathcal{T}_n$ the set of such binary plane trees on $n$ vertices. Let $Q$ be an occurrence of the pattern $213$ in $\sg \in S_n(132,231)$, and let $Q_2,Q_1,Q_3$ be the three vertices of $T(\sg)$ that correspond to $Q$, going left to right. From the characterization of trees in $\mathcal{T}_n$, $Q_2$ is a left descendant of $Q_3$, and $Q_1$ is a right descendant of $Q_2$. Let $\mathcal{A}_n$ be the set of binary plane trees in $\mathcal{T}_n$ where three vertices forming a $213$-pattern are colored black. Let $\mathcal{B}_n$ be the set of all binary plane trees in $\mathcal{T}_n$ where three vertices forming a 123-pattern are colored black. We will define a map $\rho: \mathcal{A}_n \rightarrow \mathcal{B}_n$ as follows. Given a tree $T\in \mathcal{A}_n$ with $Q_2,Q_1,Q_3$ being the three black vertices as a $213$-pattern, define $\rho(T)$ be the tree obtained by changing the right subtree of $Q_2$ to be its left subtree. See Figure \[fig1\] for an illustration. (50,55) (0,24)(6,18) (12,12)(18,0) (6,30) (12,36) (6,42) (0,54) (0,24)(3,21)(6,18) (6,18)(9,15)(12,12) (12,12)(15,6)(18,0) (0,24)(3,27)(6,30) (6,30)(9,33)(12,36) (12,36)(9,39)(6,42) (0,54)(3,48)(6,42) (17,12)[(0,0)[$Q_1$]{}]{} (-5,24)[(0,0)[$Q_2$]{}]{} (17,36)[(0,0)[$Q_3$]{}]{} (50,24)[(0,0)[$\Rightarrow$]{}]{} (70,0)(76,12) (82,18) (88,24) (94,30) (100,36) (94,42) (88,54) (70,0)(73,6)(76,12) (76,12)(79,15)(82,18)(82,18)(85,21)(88,24) (88,24)(91,27)(94,30) (94,30)(97,33)(100,36) (100,36)(97,39)(94,42) (94,42)(91,48)(88,54) (81,12)[(0,0)[$Q_1$]{}]{} (93,24)[(0,0)[$Q_2$]{}]{} (105,36)[(0,0)[$Q_3$]{}]{} (3,-2)[(18,23)]{} (4,4)[right subtree of $Q_2$]{} (67,-2)[(18,23)]{} (71,4)[left subtree of $Q_2$]{} In the tree $\rho(T)$, the relative positions of $Q_2$ and $Q_3$ keep the same, and $Q_1$ is a left descendant of $Q_2$. Therefore, the three black points $Q_1Q_2Q_3$ form a $123$-pattern in $\rho(T)$, and $\rho(T)\in \mathcal{B}_n$. It is easy to describe the converse and we omit here. The first few values of $f_q(S_n(132,231))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $1$ $0$ $1$ $0$ $1$ $1$ $6$ $160$ $0$ $160$ $0$ $160$ $160$ $4$ $8$ $0$ $8$ $0$ $8$ $8$ $7$ $560$ $0$ $560$ $0$ $560$ $560$ $5$ $40$ $0$ $40$ $0$ $40$ $40$ $8$ $1792$ $0$ $1792$ $0$ $1792$ $1792$ \[2c\] Pattern Count on $(132,312)$-Avoiding Permutations -------------------------------------------------- We begin with a correspondence between $(132,312)$-avoiding permutations and compositions of $n$. There is a bijection $\varphi_4$ between the sets $S_n(132,312)$ and $\mathcal{C}_n$. For $\sg\in S_n(132,312)$, let $\sg_{i_1},\sg_{i_2},\ldots, \sg_{i_k}$ be the $k$ left to right maxima with $i_1<i_2<\cdots<i_k$, and thus, $c=(i_2-i_1)+(i_3-i_2)+\cdots+(i_{k}-i_{k-1})+(n+1-i_{k})$ is a composition of $n$ from $i_1=1$. On the converse, let $n=c_k+c_{k-1}+\cdots+c_2+c_1\in \C_n$. For $1\leq i \leq k$, if $c_i=1$ then set $\tau_i=n-i+1$; otherwise, set $m_i=c_1+\cdots+c_{i-1}-i+2$ and $\tau_i=n-i+1,m_i+c_i-2,\ldots,m_i+1,m_i$. It is easy to get $\sg=\tau_k,\tau_{k-1},\ldots,\tau_2,\tau_1\in S_n(132,312)$ as desired. For example, if $9=3+1+2+3$, then $\sg=6\,5\,4\,7\,8\,3\,9\,2\,1$. \[pd\] For $n\geq3$, we have $$\begin{aligned} \label{pd1} f_{123}(n)=\sum_{k\geq 1 \atop c_1+c_2+\cdots+c_k=n} \sum_{i=1}^{k-2}c_i{k-i\choose 2}.\end{aligned}$$ Given a permutation $\sg=\tau_k,\ldots,\tau_2,\tau_1$ in $S_n(132,312)$ whose composition is given by $n=c_k+c_{k-1}+\cdots+c_2+c_1$, we see that the first element in $\tau_i$ is larger than all the elements in $\tau_{j}$, whereas the other elements in $\tau_i$ are smaller than the elements in $\tau_{j}$ for $i+1\leq j \leq k$. The left to right maxima form an increasing subsequence and the other elements form a decreasing subsequence. Thus we have $c_i$ choices to choose one element in $\tau_{k-i+1}$ to play the role of $1$, and then ${k-i\choose 2}$ choices to choose two left to right maxima in $\tau_j$ for $j<k-i+1$ to paly the role of $23$, summing up all the number of $123$ patterns in subsequences $\tau_k,\ldots,\tau_2,\tau_1$ gives the formula . \[td1\] For $n\geq 3$, in the set $S_n(132,312)$, we have $$\begin{aligned} \label{fd1} f_{123}(n)&=f_{321}(n)={n\choose 3}2^{n-3},\\\label{fd2} f_{213}(n)&=f_{231}(n)={n\choose 3}2^{n-3}.\end{aligned}$$ From Lemma \[oper\], we see that $\sg \in S_n(132,312)\Leftrightarrow \sg^{c}\in S_n(132,312)$, which follows $f_{123}(n)=f_{321}(n)$ and $f_{213}(n)=f_{231}(n)$ from $123^{c}=321$ and $213^{c}=231$, respectively. To calculate $f_{123}(n)$, we have by Prop \[pd\] $$f_{123}(n+1)=\sum_{k\geq 1,c_1=1 \atop c_1+c_2+\cdots+c_k=n+1} \sum_{i=1}^{k-2}c_i{k-i\choose 2}+\sum_{k\geq 1,c_1\geq 2 \atop c_1+c_2+\cdots+c_k=n+1} \sum_{i=1}^{k-2}c_i{k-i\choose 2}.$$ If $c_1=1$, then $k\geq 2$, and $$\begin{aligned} \sum_{k\geq 1,c_1=1 \atop c_1+c_2+\cdots+c_k=n+1} \sum_{i=1}^{k-2}c_i{k-i\choose 2}&=\sum_{k-1\geq 1 \atop c_2+\cdots+c_k=n} \sum_{i=2}^{k-2}c_i{k-i\choose 2}+\sum_{k-1\geq 1\atop c_2+\cdots+c_k=n} {k-1\choose 2}\\ &=f_{123}(n)+d(n)/2.\end{aligned}$$ If $c_1\geq 2$, let $c'_1=c_1-1$, $c'_i=c_i$ for $2\leq i \leq k$, then $c'_1\geq 1$, and $$\begin{aligned} \sum_{k\geq 1,c_1\geq 2 \atop c_1+c_2+\cdots+c_k=n+1} \sum_{i=1}^{k-2}c_i{k-i\choose 2} &=\sum_{k\geq 1\atop c'_1+c'_2+\cdots+c'_k=n} \sum_{i=1}^{k-2}c'_i{k-i\choose 2} +\sum_{k\geq 1\atop c'_1+c'_2+\cdots+c'_k=n} {k-1\choose 2}\\ &=f_{123}(n)+\sum_{k\geq 1\atop c'_1+c'_2+\cdots+c'_k=n}\left[{k\choose 2}+1-k\right]\\ &=f_{123}(n)+d(n)/2+2^{n-1}-c(n).\end{aligned}$$ Combining the above two cases, we get that $$f_{123}(n+1)=2f_{123}(n)+(n^2-n)2^{n-3},$$ and the formula  is derived by solving the recurrence with initial value $f_{123}(3)=1$. Formula  is a direct computation of the equality $2f_{123}(n)+2f_{213}(n)={n\choose 3}2^{n-1}$. In the subsequent section, we could also give a combinatorial interpretation for $f_{231}(n)=f_{123}(n)$ by using binary plane trees. For $\sg \in S_n(132,312)$, as in previous section, we could construct a binary plane tree $T(\sg)$ on $n$ vertices such that each vertex that is a right descendant of some vertex does not have a left descendant from the forbiddance of the pattern $312$. Denote by $\mathscr{T}_n$ the set of such trees on $n$ vertices. Let $Q$ be an occurrence of the pattern $231$ in $\sg \in S_n(132,312)$, and let $Q_2,Q_3,Q_1$ be the three vertices of $T(\sg)$ that correspond to $Q$, going left to right. Then, $Q_2$ is a left descendant of $Q_3$, and there exists a lowest ascendant $x$ of $Q_3$ or $x=Q_3$ so that $Q_1$ is a right descendant of $x$. Let $\mathscr{A}_n$ be the set of binary plane trees in $\mathscr{T}_n$ in which three vertices forming a $231$-pattern are colored black. Let $\mathscr{B}_n$ be the set of all binary plane trees in $\mathscr{T}_n$ in which three vertices forming a 123-pattern are colored black. It remains to construct a map $\varrho: \mathscr{A}_n \rightarrow \mathscr{B}_n$. Given a tree $T\in \mathscr{A}_n$ with $Q_2,Q_3,Q_1$ being the three black vertices as a $231$-pattern, denoted by $y$ the vertex that is the parent of $x$ if it exists. We can see that $x$ is the left child of $y$ from $T\in \mathscr{A}_n$. Let $T^u:=T-T_x$ be the tree obtained from $T$ by deleting the subtree $T_x$, and $T^d:=T_x-T_{Q_1}$ be the tree obtained from $T_x$ by deleting the subtree $T_{Q_1}$. Now we can define $\varrho(T)$ be the tree obtained from $T$ by first interchanging $Q_1$ as the left child of $y$, then adjoining the subtree $T^d$ as the left subtree of the vertex $Q_1$ and keeping all three black vertices the same if $y$ exits. Otherwise, we adjoin the subtree $T^d$ as the left subtree of the vertex $Q_1$ directly. An illustration is given in Figure \[fig2\]. (50,55) (0,0)(12,0) (6,12)(9,18) (12,24) (15,30) (18,36) (21,30) (24,24) (30,12) (21,42) (24,48)(27,42) (0,0)(3,6)(6,12)(12,0)(9,6)(6,12) (6,12)(7.5,15)(9,18) (9,18)(10.5,21)(12,24) (12,24)(13.5,27)(15,30) (15,30)(16.5,33)(18,36) (18,36)(19.5,33)(21,30) (21,30)(22.5,27)(24,24) (24,24)(27,18)(30,12) (18,36)(19.5,39)(21,42) (21,42)(22.5,45)(24,48) (24,48)(25.5,45)(27,42) (2,12)[(0,0)[$Q_2$]{}]{} (8,24)[(0,0)[$Q_3$]{}]{} (29,24)[(0,0)[$Q_1$]{}]{} (16,35)[(0,0)[$x$]{}]{} (23,42)[(0,0)[$y$]{}]{} (45,18)[(0,0)[$\Rightarrow$]{}]{} (60,0)(72,0) (66,12)(69,18) (72,24) (75,30) (78,36)(81,42) (81,30) (87,30) (84,48)(87,54) (90,48) (60,0)(63,6)(66,12)(72,0)(69,6)(66,12) (66,12)(67.5,15)(69,18) (69,18)(70.5,21)(72,24) (72,24)(73.5,27)(75,30) (75,30)(76.5,33)(78,36) (78,36)(79.5,33)(81,30) (78,36)(79.5,39)(81,42) (81,42)(84,36)(87,30) (81,42)(82.5,45)(84,48) (84,48)(85.5,51)(87,54) (87,54)(88.5,51)(90,48) (62,12)[(0,0)[$Q_2$]{}]{} (68,24)[(0,0)[$Q_3$]{}]{} (77,42)[(0,0)[$Q_1$]{}]{} (76,35)[(0,0)[$x$]{}]{} (86.5,48)[(0,0)[$y$]{}]{} (19,39)[(10,13)]{} (35,50)[$T_u$]{}(26,48)[$\nearrow$]{} (82,45)[(10,13)]{} (97,50)[$T_u$]{} (90,50)[$\rightarrow$]{} (-2,-2)[(24,40)]{} (27,4)[$T_d$]{} (20,10)[$\searrow$]{} (58,-2)[(24,40)]{} (90,4)[$T_d$]{} (80,4)[$\rightarrow$]{} In the tree $\varrho(T)$, the relative positions of $Q_2$ and $Q_3$ are unchanged, and $Q_3$ is a left descendant of $Q_1$, thus the three black points $Q_2Q_3Q_1$ form a $123$-pattern in $\varrho(T)$, and $\varrho(T)\in \mathscr{B}_n$. It is easy to describe the converse map and we omit here. The first few values of $f_q(S_n(132,312))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $1$ $0$ $1$ $1$ $0$ $1$ $6$ $160$ $0$ $160$ $160$ $0$ $160$ $4$ $8$ $0$ $8$ $8$ $0$ $8$ $7$ $560$ $0$ $560$ $560$ $0$ $560$ $5$ $40$ $0$ $40$ $40$ $0$ $40$ $8$ $1792$ $0$ $1792$ $1792$ $0$ $1792$ \[2d\] Pattern Count on $(132,321)$-Avoiding Permutations -------------------------------------------------- We begin with a correspondence from Simion and Schmidt [@Sim] as follows: \[2le\] There is a bijection $\varphi_5$ between the set $S_n(132,321)\backslash \{\text{identity}\}$ and the set of $2$-element subsets of $[n]$. For a permutation $\sg \in S_n(132,321)\backslash \{\text{identity}\}$, suppose that $\sg_k=m$ ($k<m$), and then we define $\varphi_5(\sg)=\{k,m\}$. On the converse, given two elements $1\leq k<m \leq n$, set $\tau_1=m-k+1,m-k+2,\ldots,m-1,m$, $\tau_2=1,2,\ldots,m-k$ and $\tau_3=m+1,m+2,\ldots,n-1,n$. Then define $\sg=\varphi_5^{-1}(k,m)=\tau_1,\tau_2,\tau_3$. For example, if $k=4,m=6$, then $\sg=3\,4\,5\,6\,1\,2\,7\,8$. From the above lemma, we have \[pe\] For $n\geq3$, $$\begin{aligned} f_{213}(n)&=\sum_{1\leq k<m \leq n}k(m-k)(n-m)\label{pe1},\\ f_{312}(n)&=\sum_{1\leq k<m \leq n}k{m-k\choose 2}.~~~~~~\label{pe2}\end{aligned}$$ Given a permutation $\sg=\tau_1,\tau_2,\tau_3$ with $\varphi_5(\sg)=\{k,m\}$ in the set $S_n(132,321)$, we see that the elements in $\tau_1$, $\tau_2$ and $\tau_3$ are increasing, and $\tau_2<\tau_1<\tau_3$. We have $k$ choices to select one element in $\tau_{1}$ to play the role of $2$, and then have $m-k$ choices to choose one element in $\tau_{2}$ to play the role of $1$, and $n-m$ choices to choose one element in $\tau_{3}$ to play the role of $3$. Summing up all possible $k$ and $m$ gives the formula . For the pattern $312$, we have $k$ choices in the subsequence $\tau_1$ to choose one element to play the role of $3$, after this we have ${m-k\choose 2}$ choices in the subsequence $\tau_2$ to choose two elements to play the role of $23$. Summing up all $k$ and $m$ gives the formula . Next, we exhibit some useful formulae for calculating $f_{213}(n)$ and $f_{312}(n)$ as follows: \[le\] For $n\geq 2$, $$\begin{aligned} \sum_{k=1}^{n-1}k&=\frac{n(n-1)}{2};~~~~~ \sum_{k=1}^{n-1}k^2=\frac{n(n-1)(2n-1)}{6};\\ \sum_{k=1}^{n-1}k^3&=\frac{n^2(n-1)^2}{4};~~~ \sum_{k=1}^{n-1}k^4=\frac{n(n-1)(2n-1)(3n^2-3n+1)}{30}.\end{aligned}$$ Based on previous analysis, we are ready to give the exact formulae for the pattern count in $S_n(132,321)$. \[te1\] For $n\geq 3$, in the set $S_n(132,321)$, we have $$\begin{aligned} f_{213}(n)&=f_{231}(n)=f_{312}(n)={n+2\choose 5},~~~~\\ f_{123}(n)&=n(7n^4-40n^3+85n^2-80n+28)/120.\end{aligned}$$ From Lemma \[oper\], we see that $\sg \in S_n(132,321)\Leftrightarrow \sg^{-1}\in S_n(132,321)$, it implies that $f_{312}(n)=f_{231}(n)$ since $312^{-1}=231$. By using Prop \[pe\], we have $$\begin{aligned} f_{312}(n)&=\sum_{1\leq k<m \leq n}k{m-k\choose 2}\\ &=\sum_{k=1}^{n-1}k\sum_{m=k+1}^n{m-k\choose 2}=\sum_{k=1}^{n-1}k{n-k+1\choose 3}\\ &=\sum_{k=1}^{n-1}\left[(n^3-n)k+(1-3n^2)k^2+2nk^3-k^4\right]={n+2\choose 5},\end{aligned}$$ where the last equality holds from Lemma \[le\]. By using Prop \[pe\] again, we have $$\begin{aligned} f_{213}(n)&=\sum_{1\leq k<m \leq n}k(m-k)(n-m)=\sum_{k=1}^{n-1}\sum_{m=k+1}^nk(m-k)(n-m)\\ &=\sum_{k=1}^{n-1}\sum_{m'=1}^{n-k}km'(n-m'-k)=\sum_{k=1}^{n-1}k(n-k)\sum_{m'=1}^{n-k}m'-\sum_{k=1}^{n-1}k\sum_{m'=1}^{n-k}m'^2\\ &=\sum_{k=1}^{n-1}\left[ \left(\frac{n^3}{6}-\frac{n}{6}\right)k+\left(\frac{1}{6}-\frac{n^2}{2}\right)k^2+ \frac{n}{2}k^3-\frac{1}{6}k^4\right]={n+2\choose 5},\end{aligned}$$ where the last equality holds by simple calculation from Lemma \[le\]. We complete the proof by employing the relation $2f_{231}(n)+f_{213}(n)+f_{123}(n)={n\choose 3}\left[{n\choose 2}+1\right]$. We also notice that the equality $f_{213}(n)=f_{231}(n)$ can be proved by using Bóna’s bijection [@Bona] directly on the set of binary plane trees on $n$ vertices such that the vertex which is a right descendant of some node has no right descendants. The first few values of $f_q(S_n(132,321))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $1$ $0$ $1$ $1$ $1$ $0$ $6$ $152$ $0$ $56$ $56$ $56$ $0$ $4$ $10$ $0$ $6$ $6$ $6$ $0$ $7$ $392$ $0$ $126$ $126$ $126$ $0$ $5$ $47$ $0$ $21$ $21$ $21$ $0$ $8$ $868$ $0$ $252$ $252$ $252$ $0$ \[2e\] Triply Restricted Permutations ============================== In this section, we study the pattern count in the simultaneous avoidance of any three patterns of length $3$. Based on Lemma \[oper\], Simion and Schmidt [@Sim] showed that the pairs of patterns among the total ${6\choose 3}=20$ cases fall into the following $6$ classes: For every symmetric group $S_n$,\ [(1) $|S_n(123,132,213)|=|S_n(231,312,321)|=F_{n+1}$;\ (2) $|S_n(123,132,231)|=|S_n(123,213,312)|=|S_n(132,231,321)|=|S_n(213,312,321)|=n$;\ (3) $|S_n(132,213,231)|=|S_n(132,213,312)|=|S_n(132,231,312)|=|S_n(213,231,312)|=n$;\ (4) $|S_n(123,132,312)|=|S_n(123,213,231)|=|S_n(132,312,321)|=|S_n(213,231,321)|=n$;\ (5) $|S_n(123,231,312)|=|S_n(132,213,321)|=n$;\ (6) $|S_n(R)|=0$ for all $R\supset \{123,321\}$ if $n\geq 5$,\ where $F_{n}$ is the Fibonacci number given by $F_0=0,F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for $n\geq 2$.]{} Pattern Count on $(123,132,213)$-Avoiding Permutations ------------------------------------------------------ It is known that $F_{n+1}$ counts the number of $0$-$1$ sequences of length $n-1$ in which there are no consecutive ones, see [@Com], and we call such a sequence a Fibonacci binary word for convenience. Let $B_n$ denote the set of all Fibonacci binary words of length $n$. Simion and Schmidt [@Sim] showed that There is a bijection $\psi_1$ between $S_n(123,132,213)$ and $B_{n-1}$. Let $w=w_1w_2\cdots w_{n-1} \in B_{n-1}$, and the corresponding permutation $\sg$ is determined as follows: For $1\leq i \leq n-1$, let $X_i=[n]-\{\sg_1,\ldots,\sg_{i-1}\}$, and then set [\_i=]{}   X\_i, &if $s_i=0$,\   X\_i, &if $s_i=1$. Finally, $\sg_n$ is the unique element in $X_n$. For example, if $w=01001010$, then $\psi_1(w)=9\,7\,8\,6\,4\,5\,2\,3\,1$. Given a word $w=w_1w_2\cdots w_{n} \in B_{n}$, we call $i$ ($1\leq i<n$) an ascent of $w$ if $w_i<w_{i+1}$, and denote by $\asc(w)=\{ i| w_i<w_{i+1} \}$ and $\maj(w)=\sum\limits_{i \in \asc(w)}i$. \[paa\] The total number of occurrences of the pattern $312$ in $S_n(123,132,213)$ is given by $$\begin{aligned} \label{paa1} f_{312}(n)=\sum_{w \in B_{n-1}}\maj(w).\end{aligned}$$ Suppose $\sg \in S_n(123,132,213)$ and $\psi_1(\sg)=w_1w_2\cdots w_{n-1}$. If $k$ is an ascent of $w$, then $w_kw_{k+1}=01$. We have $\sg_k>\sg_{j}$ for all $j>k$ since $\sg_{k}$ is the largest element in $X_{k}$. On the other hand, there exists a unique $l>k+1$ such that $\sg_{l}>\sg_{k+1}$ since $\sg_{k+1}$ is the second largest element in $X_{k+1}$. From the bijection $\psi_1$, we see that for all $i\in [n-1]$, there is at most one $j>i$ such that $\sg_j>\sg_i$; this implies that $\sg_i>\sg_{k+1}$ for all $i<k$. Thus we find that $\sg_i \sg_{k+1} \sg_{l}$ forms a $312$-pattern for all $i\leq k$, that is the ascent $k$ will produce $k$’s $312$-pattern in which $\sg_{k+1}$ plays the role of $1$. Summing up all the ascents, we derive that there are total $\maj(w)$ such patterns in $\sg$. \[Tta1\] For $n\geq 3$, in the set $S_n(123,132,213)$, we have $$\begin{aligned} \sum_{n\geq 3}f_{231}(n)x^n&=\sum_{n\geq 3}f_{312}(n)x^n=\frac{x^3(1+2x)}{(1-x-x^2)^3},\label{fTta1}\\ \sum_{n\geq 3}f_{321}(n)x^n&= \frac{x^3(1+6x+12x^2+8x^3)} {(1-x-x^2)^4}.\label{Tca2}\end{aligned}$$ From Lemma \[oper\], we have $f_{231}(n)=f_{312}(n)$ since $\sg \in S_n(123,132,213)\Leftrightarrow \sg^{-1}\in S_n(123,132,213)$ and $231^{-1}=312$. By Prop \[paa\], we write $$\sum_{n\geq 3}f_{312}(n)x^n=\sum_{n\geq 3}x^n\sum_{\sg \in B_{n-1}}\maj(w) =x\sum_{n\geq 3}\sum_{w \in B_{n-1}}\maj(w)x^{n-1}=xu(x),$$ where $u(x)=\sum\limits_{n\geq 2}\sum\limits_{w \in B_n}\maj(w)x^n$. Let $M_n(q)=\sum\limits_{w \in B_n} q^{\maj(w)}$ and $M(x,q)=\sum\limits_{n\geq 2}M_n(q)x^n$. Then $u(x)=\frac{\partial M(x,q)}{\partial q}\mid_{q=1}$. Given a word $w=w_1w_2\cdots w_n \in B_n$, if $w_n=0$, then $\maj(w)=\maj(w_1w_2\cdots w_{n-1})$; otherwise, $w_{n-1}w_n=01$ and $\maj(w)=\maj(w_1w_2\cdots w_{n-2})+(n-1)$. Hence, we have $$\begin{aligned} M_n(q)=M_{n-1}(q)+q^{n-1}M_{n-2}(q) \text{ for }n\geq 4,\end{aligned}$$ with $M_2(q)=2+q$ and $M_3(q)=2+q+2q^2$. Multiplying the recursion by $x^n$ and summing over $n\geq 4$ yields that $$M(x,q)-(2+q)x^2-(2+q+2q^2)x^3=x\left[M(x,q)-(2+q)x^2\right]+qx^2M(xq,q).$$ Therefore $$(1-x)M(x,q)=qx^2M(xq,q)+(2+q)x^2+2q^2x^3.$$ Differentiate both sides with respect to $q$, we get $$\begin{aligned} \label{ftaf} (1-x)\frac{\partial M(x,q)}{\partial q}=x^2\left[M(xq,q)+q\frac{\partial M(xq,q)}{\partial q} \right]+x^2+4qx^3.\end{aligned}$$ Setting $q=1$ in equation reads that $$(1-x)u(x)=x^2\left[M(x,1)+\frac{\partial M(xq,q)}{\partial q}\mid_{q=1}\right] +x^2+4x^3,$$ Employing the well-known generating $\sum_{n\geq 0}F_nx^n=\frac{x}{1-x-x^2}$, we have $$M(x,1)=\sum_{n\geq 2}|B_{n}|x^n=\sum_{n\geq 2}F_{n+2}x^n= \frac{x^2(3+2x)}{1-x-x^2}.$$ Further, $$\begin{aligned} \frac{\partial M(xq,q)}{\partial q}\mid_{q=1}&=\left(\sum_{n\geq 2}\sum_{w \in B_n} q^{n+\maj(w)}x^n \right)|_{q=1}\\ &=\sum_{n\geq 2}x^n\sum_{w \in B_n}(n+\maj(w))=\sum_{n\geq 2}nF_{n+2}x^n+u(x).\end{aligned}$$ From the generating function of $F_{n+2}$, we obtain $$\sum_{n\geq 2}nF_{n+2}x^n=x\left(\frac{x^2(3+2x)}{1-x-x^2}\right)' =\frac{x^2(6+3x-4x^2-2x^3)}{(1-x-x^2)^2},$$ which yields that $$(1-x)u(x)=x^2\left[\frac{x^2(3+2x)}{1-x-x^2}+\frac{x^2(6+3x-4x^2-2x^3)}{(1-x-x^2)^2}+u(x)\right] +x^2+4x^3.$$ Therefore, $$u(x)={x^2(1+2x)}/{(1-x-x^2)^3},$$ which gives the generating function for $f_{312}(n)$ as shown in formula . For formula , we first have $$\begin{aligned} \label{Tta2f} \sum_{n\geq 3}f_{321}(n)x^n=\sum_{n\geq 3}{n\choose 3}F_{n+1}x^n- 2\sum_{n\geq 3}f_{312}(n)x^n\end{aligned}$$ from $2f_{312}(n)+f_{321}(n)={n\choose 3}F_{n+1}$. Using the fact that $\sum_{n\geq 0}F_nx^n=\frac{x}{1-x-x^2}$, we get $$\begin{aligned} &\sum_{n\geq 3}F_{n+1}x^n=\frac{1}{x}\left(\frac{x}{1-x-x^2}-x-x^2-2x^3\right)=\frac{x^3(3+2x)}{1-x-x^2},\\ &\sum_{n\geq 3}{n\choose 3}F_{n+1}x^n=\frac{x^3}{6}\left(\sum_{n\geq 3}F_{n+1}x^n\right)^{'''}=\frac{x^3(3+8x+6x^2+4x^3)}{(1-x-x^2)^4}.\end{aligned}$$ Formula  follows by substituting the generating functions of ${n\choose 3}F_{n+1}$ and $f_{312}(n)$ into , and we complete the proof. The first few values of $f_q(S_n(123,132,213))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $0$ $0$ $0$ $1$ $1$ $1$ $6$ $0$ $0$ $0$ $40$ $40$ $180$ $4$ $0$ $0$ $0$ $5$ $5$ $10$ $7$ $0$ $0$ $0$ $95$ $95$ $545$ $5$ $0$ $0$ $0$ $15$ $15$ $50$ $8$ $0$ $0$ $0$ $213$ $213$ $1478$ \[3a\] Pattern Count on Other Triple Avoiding Permutations --------------------------------------------------- We begin with the pattern count on $(123,132,231)$-avoiding permutations. \[Ttb1\] For $n\geq 3$, in the set $S_n(123,132,231)$, we have $$\begin{aligned} f_{213}(n)&=f_{312}(n)={n\choose 3},\\ f_{321}(n)&=(n-2){n\choose 3}.\end{aligned}$$ We first give the following structure from Simion and Schmidt [@Sim] $$\label{eq:temp3} \sg\in S_n(123,132,231)\Leftrightarrow \sg=n,n-1,\ldots,k+1,k-1,k-2,\ldots,1,k \text{ for some } k.$$ Based on such structure, we can show $f_{213}(n)=f_{312}(n)$ by a direct bijection. Let $q=abc$ be a $213$-pattern of a permutation $\sg \in S_n(123,132,231)$. From $b<c$ and the fact that $\sg\in S_n(123,132,231)$ has only one ascent at position $n-1$, we have $\sg(n)=c$, and thus $\sg=n,n-1,\ldots,c+1,c-1,c-2,\ldots,2,1,c$ from the structure . Let $q'=cba$ (a $312$-pattern) and we set $\sg'=n,n-1,\ldots,a+1,a-1,a-2,\ldots,2,1,a$ as the desired permutation. For example, if $n=7$ and $q=326$, then $\sg=7\,5\,4\,3\,2\,1\,6$. Moreover, $q'=623$ and $\sg'=7\,6\,5\,4\,2\,1\,3$. To calculate $f_{312}(n)$, we suppose $\sg=n,n-1,\ldots,k+1,k-1,k-2, \ldots,2,1,k$ for some $k$ from structure . We can construct a $312$-pattern as follows: Choose one element from the first $n-k$ elements to play the role of $3$, then choose one element from the next $k-1$ elements to play the role of $1$, and the last element plays the role of $2$. Thus, summing up all possible $k$, we have $$f_{312}(n)=\sum_{k=1}^n (n-k)(k-1)=-n^2+(n+1)\sum_{k=1}^nk-\sum_{k=1}^nk^2=\frac{n(n-1)(n-2)}{6}={n\choose 3}.$$ The formula for $f_{321}(n)$ follows by using $f_{213}(n)+f_{312}(n)+f_{321}(n)=n{n\choose 3}$. The first few values of $f_q(S_n(123,132,231))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $0$ $0$ $1$ $0$ $1$ $1$ $6$ $0$ $0$ $20$ $0$ $20$ $80$ $4$ $0$ $0$ $4$ $0$ $4$ $8$ $7$ $0$ $0$ $35$ $0$ $35$ $175$ $5$ $0$ $0$ $10$ $0$ $10$ $30$ $8$ $0$ $0$ $56$ $0$ $56$ $336$ \[3b\] For $(132,213,231)$-avoiding permutations, we have \[Ttb1\] For $n\geq 3$, in the set $S_n(132,213,231)$, $$\begin{aligned} f_{123}(n)&=f_{312}(n)={n+1\choose 4},\\ f_{321}(n)&=\frac{n(n-2)(n-1)^2}{12}.\end{aligned}$$ We begin with the following structure by Simion and Schmidt [@Sim] $$\label{eq:tmp5} \sg\in S_n(132,213,231)\Leftrightarrow \sg=n,n-1,\ldots,k+1,1,2,\ldots,k-1,k \text{~for some~} k.$$ Based on such structure, we first prove $f_{123}(n)=f_{312}(n)$. For each $$\sg=n,n-1,\ldots,k+1,1,\ldots, \underline{a},a+1,\ldots, \underline{b},b+1,\ldots,c-1,\underline{c},c+1,\ldots,k-1,k$$ with $abc$ as a $123$-pattern, we set $$\sg'=n,n-1,\ldots,\underline{n-k+c},\ldots,c,1,2,\ldots, \underline{a},a+1,\ldots,\underline{b},b+1,\ldots,c-1,$$ and it is easy to check that $n-k+c,a,b$ is a $312$-pattern of $\sg'$. For example, if $\sg=9\,8\,7\,1\,\underline{2}\, \underline{3}\,4\, \underline{5}\,6$ then $\sg'=9\,\underline{8}\,7\,6\,5\,1\, \underline{2}\, \underline{3}\,4$. To calculate $f_{123}(n)$, we suppose $\sg=n,n-1,\ldots,k+1,1,2,\ldots,k-1,k$ for some $k$ from structure . Then, a $123$-pattern can be obtained by choosing three elements from the last $k$ elements to play the role of $123$. Thus, summing up all possible $k$ gives $$f_{123}(n)=\sum_{k=1}^n {k \choose 3}={n+1\choose 4}.$$ We complete the proof by using $f_{123}(n)+f_{312}(n)+f_{321}(n)=n{n\choose 3}$. The first few values of $f_q(S_n(132,213,231))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $1$ $0$ $0$ $0$ $1$ $1$ $6$ $35$ $0$ $0$ $0$ $35$ $50$ $4$ $5$ $0$ $0$ $0$ $5$ $6$ $7$ $70$ $0$ $0$ $0$ $70$ $105$ $5$ $15$ $0$ $0$ $0$ $15$ $20$ $8$ $126$ $0$ $0$ $0$ $126$ $196$ \[3c\] For $(123,132,312)$-avoiding permutations, we have \[Ttb1\] For $n\geq 3$, in the set $S_n(123,132,312)$, $$\begin{aligned} f_{213}(n)&=f_{231}(n)={n\choose 3},\\ f_{321}(n)&=(n-2){n\choose 3}.\end{aligned}$$ We begin with the following structure from Simion and Schmidt [@Sim] $$\label{eq:tmp4} \sg\in S_n(132,213,231)\Leftrightarrow \sg=n-1,n-2,\ldots,k+1,n,k,k-1,\ldots,1 \text{ for some } k.$$ Based on such structure, we prove $f_{213}(n)=f_{231}(n)$ by a direct correspondence. Let $\sg=n-1,\ldots,\underline{a},a+1,\ldots,\underline{b},b+1,\ldots,k+1,\underline{n}, k,k-1, \ldots,2,1$ with $abn$ as a $213$-pattern. Then, we set $$\sg'=n-1,\ldots,\underline{n-a+b},\ldots,n-a+k+1,\underline{n},n-a+k,n-a+k-1,\ldots, \underline{n-a},\ldots,2,1,$$ where $n-a+b,n,n-a$ is a $231$-pattern of $\sg'$. For example, if $\sg=8\,\underline{7}\,6\,\underline{5}\,4\,\underline{9}\,3\,2\,1$, then $\sg'=\sg=8\,\underline{7}\,6\,\underline{9}\,5\,4\,3\,\underline{2}\,1$. To calculate $f_{213}(n)$, we suppose that $\sg=n-1,n-2,\ldots,k+1,n,k,k-1,\ldots,2,1$ for some $k$. Then, a $213$-pattern can be obtained by choosing two elements from the first $n-k-1$ elements to play the role of $21$, and let $n$ play the role of $3$. Thus, summing up all possible $k$, we have $$f_{213}(n)=\sum_{k=0}^{n-1} {n-k-1 \choose 2}={n\choose 3}.$$ We complete the proof by using $f_{213}(n)+f_{231}(n)+f_{321}(n)=n{n\choose 3}$. The first few values of $f_q(S_n(123,132,312))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $0$ $0$ $1$ $1$ $0$ $1$ $6$ $0$ $0$ $20$ $20$ $0$ $80$ $4$ $0$ $0$ $4$ $4$ $0$ $8$ $7$ $0$ $0$ $35$ $35$ $0$ $175$ $5$ $0$ $0$ $10$ $10$ $0$ $30$ $8$ $0$ $0$ $56$ $56$ $0$ $336$ \[3d\] Finally, we study the pattern count on $(123,231,312)$-avoiding permutations. For $n\geq 3$, in the set $S_n(123,231,312)$, we have $$\begin{aligned} f_{132}(n)&=f_{213}(n)={n+1\choose 4},\label{Tte1}\\ f_{321}(n)&=\frac{n(n-2)(n-1)^2}{12}.\end{aligned}$$ From Lemma \[oper\], we see that $$\sg \in S_n(123,231,312)\Leftrightarrow \sg^{r}\in S_n(321,132,213)\Leftrightarrow (\sg^{r})^c\in S_n(123,231,312).$$ We have $f_{213}(n)=f_{132}(n)$ from $(213^{r})^c=312^c=132$. The following structure of the set $S_n(123,231,312)$ is given by Simion and Schmidt [@Sim] $$\label{s3e} \sg\in S_n(132,213,231)\Leftrightarrow \sg=k-1,k-2,\ldots,3,2,1,n,n-1\ldots,k \text{~for some~} k.$$ Suppose that $\sg=k-1,k-2,\ldots,3,2,1,n,n-1\ldots,k$ for some $k$, then a $213$-pattern can be obtained as follows: Choose two elements from the first $k-1$ elements to play the role of $21$, and choose one element from the last $n-k+1$ elements to play the role of $3$. Thus, summing up all possible $k$, we have $$\begin{aligned} f_{213}(n)&=\sum_{k=1}^n {k-1 \choose 2}(n-k+1)=\sum_{k=0}^{n-1}{k\choose 2}(n-k)={n+1\choose 4}.\end{aligned}$$ The formula for $f_{321}(n)$ is obtained by the relation $2f_{213}(n)+f_{321}(n)=n{n\choose 3}$. The first few values of $f_q(S_n(123,231,312))$ for $q$ of length $3$ are shown below. $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ $n$ $f_{123}$ $f_{132}$ $f_{213}$ $f_{231}$ $f_{312}$ $f_{321}$ ----- ----------- ----------- ----------- ----------- ----------- ----------- ----- ----------- ----------- ----------- ----------- ----------- ----------- $3$ $0$ $1$ $1$ $0$ $0$ $1$ $6$ $0$ $35$ $35$ $0$ $0$ $50$ $4$ $0$ $5$ $5$ $0$ $0$ $6$ $7$ $0$ $70$ $70$ $0$ $0$ $105$ $5$ $0$ $15$ $15$ $0$ $0$ $20$ $8$ $0$ $126$ $126$ $0$ $0$ $196$ \[3e\] [99]{} M. Bóna, A Walk Through Combinatorics, 3rd edition, World Scientific, 2011. M. Bóna, Surprising symmetries in objects counted by Catalan numbers, Electron. J. Combin. 19 (2012), P62. M. Bóna, The absence of a pattern and the occurrences of another, Discrete Math. Theor. Comput. Sci. 12 (2010), no. 2, 89–102. L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. J. Cooper, Combinatorial Problems I like, internet resource, available at http:// www.math.sc.edu/ cooper/combprob.html. C. Homberger, Expected Patterns in Permutation Classes, Electron. J. Combin. 19(3) (2012), P43. K. Rudolph, Pattern Popularity in $132$-avoiding Permutations, Electron. J. Combin. 20(1) (2013), P8. R. Simion and F.W. Schmidt, Restricted permutations, European J. Combin. 6 (1985), 383–406. R.P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge University Press, 1997.

[Boundary length of reconstructions in discrete tomography]{} [Birgit van Dalen]{} *Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands\ dalen@math.leidenuniv.nl* [**Abstract:** We consider possible reconstructions of a binary image of which the row and column sums are given. For any reconstruction we can define the length of the boundary of the image. In this paper we prove a new lower bound on the length of this boundary. In contrast to simple bounds that have been derived previously, in this new lower bound the information of both row and column sums is combined.]{} Introduction ============ An important problem in discrete tomography is to reconstruct a binary image on a lattice from given projections in lattice directions [@boek; @boeknieuw]. Each point of a binary image has a value equal to zero or one. The line sum of a line through the image is the sum of the values of the points on this line. The projection of the image in a certain direction consists of all the line sums of the lines through the image in this direction. Any binary image with exactly the same projections as the original image we call a *reconstruction* of the image. For any set of more than two directions, the problem of reconstructing a binary image from its projections in those directions is NP-complete [@gardner]. For exactly two directions, the horizontal and vertical ones, say, it is possible to reconstruct an image in polynomial time. Already in 1957, Ryser described an algorithm to do so [@ryser]. He also characterised the set of projections that correspond to a unique binary image. If there are multiple images corresponding to one set of line sums, it is interesting to reconstruct an image with a special property. In order to find reconstructions that look rather like a real object, two special properties in particular are often imposed on the reconstructions. The first is *connectivity* of the points with value one in the picture [@hv-convex2; @hv-convex3; @woeginger]. The second is *hv-convexity*: if in each row and each column, the points with value one form one connected block, the image is called *hv-convex*. The reconstruction of hv-convex images, either connected or not necessarily connected, has been studied extensively [@hv-convex1; @hv-convex2; @hv-convex3; @dahlflatberg; @woeginger]. Another relevant concept in this context is the *boundary* of a binary image. The boundary can be defined as the pairs consisting of two adjacent points, one with value 0 and one with value 1. Here we use 4-adjacency: that is, a point is adjacent to its two vertical and to its two horizontal neighbours [@connectivity]. The number of such pairs of adjacent points with two different values is called the *length of the boundary* or sometimes the *perimeter length* [@gray]. In this paper we will consider given line sums that may correspond to more than one binary image. Since the boundary of real objects is often small compared to the area, it makes sense to look for reconstructions of which the length of the boundary is as small as possible. In particular, if there exists an hv-convex reconstruction, then the length of the boundary of that image is the smallest possible. In that sense, the length of the boundary is a more general concept than hv-convexity. The question we are interested in in this paper is: given line sums, what is the smallest length of the boundary that a reconstruction fitting those line sums can have? We can give two straightforward lower bounds on the length of the boundary, given the row and column sums. Both are equivalent to bounds given by Dahl and Flatberg in [@dahlflatberg Section 2]. The first is that every column with a non-zero sum contributes 2 to the length of the horizontal boundary, while every row with non-zero sum contributes 2 to the length of the vertical boundary. So if there are $m$ non-zero row sums and $n$ non-zero column sums, then the total length of the boundary is at least $2n+2m$. For the second bound we use that if the row sums of two consecutive rows are different, the length of the horizontal boundary between those rows is at least the absolute difference between those row sums. A similar result holds for the column sums and the vertical boundary. So if an image has row sums $r_1$, $r_2$, …, $r_m$ and column sums $c_1$, $c_2$, …, $c_n$, then the length of the boundary is at least $$r_1 + \sum_{i=1}^{m-1} |r_i - r_{i+1}| + r_m + c_1 + \sum_{j=1}^{n-1} |c_j - c_{j+1}| + c_n.$$ Despite being simple, these bounds are sharp in many cases. For example, the first bound is sharp if and only if there exists a hv-convex image that satisfies the line sums. On the other hand it is clear that much information is disregarded in these bounds. The first bound does not use the actual value of the non-zero line sums at all, while the second bound only uses the column sums to estimate the length of the vertical boundary and only the row sums to estimate the length of the horizontal boundary. In this paper we prove a new lower bound on the length of the boundary that combines the row and column sums. After introducing some notation in Section \[notation\], we prove this bound in Section \[main\]. Some examples and a corollary are in Section \[examples\]. Finally, in Section \[variation\] we derive an extension of the bound that gives better results in certain cases. Definitions and notation {#notation} ======================== Let $F$ be a finite subset of $\mathbb{Z}^2$ with characteristic function $\chi$. (That is, $\chi(x,y) = 1$ if $(x,y) \in F$ and $\chi(x,y) = 0$ otherwise.) For $i \in \mathbb{Z}$, we define *row* $i$ as the set $\{(x,y) \in \mathbb{Z}^2: x = i\}$. We call $i$ the index of the row. For $j \in \mathbb{Z}$, we define *column* $j$ as the set $\{(x,y) \in \mathbb{Z}^2: y = j\}$. We call $j$ the index of the column. Following matrix notation, we use row numbers that increase when going downwards and column numbers that increase when going to the right. The *row sum* $r_i$ is the number of elements of $F$ in row $i$, that is $r_i = \sum_{j \in \mathbb{Z}} \chi(i,j)$. The *column sum* $c_j$ of $F$ is the number of elements of $F$ in column $j$, that is $c_j = \sum_{i \in \mathbb{Z}} \chi(i,j)$. We refer to both row and column sums as the *line sums* of $F$. We will usually only consider finite sequences $\mathcal{R} = (r_1, r_2, \ldots, r_m)$ and $\mathcal{C} = (c_1, c_2, \ldots, c_n)$ of row and column sums that contain all the nonzero line sums. Given sequences of integers $\mathcal{R} = (r_1, r_2, \ldots, r_m)$ and $\mathcal{C} = (c_1, c_2, \ldots, c_n)$, we say that $(\mathcal{R}, \mathcal{C})$ is *consistent* if there exists a set $F$ with row sums $\mathcal{R}$ and column sums $\mathcal{C}$. Define $b_i = \#\{j: c_j \geq i\}$ for $i = 1, 2, \ldots, m$. Ryser’s theorem [@ryser] states that if $r_1 \geq r_2 \geq \ldots \geq r_m$, the line sums $(\mathcal{R}, \mathcal{C})$ are consistent if and only if for each $k = 1, 2, \ldots, m$ we have $\sum_{i=1}^k b_i \geq \sum_{i=1}^k r_i$. From this we can conclude a similar result for the case of not necessarily non-increasing row sums: if the line sums $(\mathcal{R}, \mathcal{C})$ are consistent, then for all $k = 1, 2, \ldots, m$ we have $$\label{ryserconsequence} \sum_{i=1}^k b_i \geq \sum_{i=1}^k r_i.$$ The converse clearly does not hold. We can view the set $F$ as a picture consisting of cells with zeroes and ones. Rather than $(i,j) \in F$, we might say that $(i,j)$ has value 1 or that there is a one at $(i,j)$. Similarly, for $(i,j) \not\in F$ we sometimes say that $(i,j)$ has value zero or that there is a zero at $(i,j)$. We define the *boundary* of $F$ as the set consisting of all pairs of points $\big( (i,j), (i',j') \big)$ such that - $i=i'$ and $|j-j'| =1$, or $|i-i'| = 1$ and $j=j'$, and - $(i,j) \in F$ and $(i',j') \not\in F$. One element of this set we call *one piece of the boundary*. We can partition the boundary into two subsets, one containing the pairs of points with $i=i'$ and the other containing the pairs of points with $j=j'$. The former set we call the *vertical boundary* and the latter set we call the *horizontal boundary*. We define the *length of the (horizontal, vertical) boundary* as the number of elements in the (horizontal, vertical) boundary. The main theorem {#main} ================ \[altgrens\] Let row sums $\mathcal{R} = (r_1, r_2, \ldots, r_m)$ and column sums $\mathcal{C} = (c_1, c_2, \ldots, c_n)$ be given, where $r_1=n$, $r_m = 0$. Let $L_h$ be the total length of the horizontal boundary of an image with line sums $(\mathcal{R}, \mathcal{C})$. Define $b_i = \#\{j: c_j \geq i\}$ and $d_i = b_i - r_i$ for $i = 1, 2, \ldots, m$. For any integer $t \geq 0$ and any subset $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{1, 2, \ldots, m\}$ with $i_1 < i_2 < \ldots < i_{2t+1}$ we have $$\begin{aligned} L_h &\geq 2n + d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}}, \label{altgrens1} \\ L_h &\geq 2n - d_{i_{2t+1}} + d_{i_{2t}} - d_{i_{2t-1}} + \cdots + d_{i_2} - 2d_{i_1} \label{altgrens2}.\end{aligned}$$ First we prove (\[altgrens1\]) by induction on $n$. In the initial case $n=0$ we have $d_i = b_i = r_i = 0$ for all $i$, hence we have to prove that $L_h \geq 0$, which is obviously true. Now let $n \geq 1$ and consider a binary image $F$ with line sums $(\mathcal{R}, \mathcal{C})$. Let $I \subset \{1, 2, \ldots, m\}$ be the set of indices $i$ such that cell $(i,n)$ has value 1. Note that $\# I = c_n$. Let $F'$ be the binary image we obtain by deleting column $n$ from $F$. Let $(r_1', r_2', \ldots, r_m')$ be the row sums of $F'$. The column sums of $F'$ are $(c_1, c_2, \ldots, c_{n-1})$, and define $b_i' = \#\{ j \leq n-1 : c_j \geq i\}$ and $d_i' = b_i' - r_i'$ for $i=1,2,\ldots, m$. We have $$r_i' = \begin{cases} r_i & \text{if } i \not\in I, \\ r_i-1 & \text{if } i \in I, \end{cases}$$ $$b_i' = \begin{cases} b_i -1 & \text{if } i \leq c_n, \\ b_i & \text{if } i > c_n, \end{cases}$$ and therefore $$d_i' = \begin{cases} d_i - 1 & \text{if } i \not\in I \text{ and } i \leq c_n, \\ d_i & \text{if } i \notin I \text{ and } i > c_n, \text{ or } i \in I \text{ and } i \leq c_n, \\ d_i + 1 & \text{if } i \in I \text{ and } i > c_n. \end{cases}$$ As induction hypothesis we assume that (\[altgrens1\]) is true for the smaller image $F'$. So for the total length $L_h'$ of the horizontal boundary of $F'$ we have $$L_h' \geq 2(n-1) + d_{i_1}' - d_{i_2}' + d_{i_3}' - \cdots - d_{i_{2t}}' + 2d_{i_{2t+1}}'.$$ Let $2B$ be equal to the horizontal boundary in column $n$ of $F$. Then $L_h = L_h' + 2B$. We want to prove (\[altgrens1\]), hence it suffices to prove $$\label{altgrenstoprove} 2B - 2 \geq (d_{i_1} - d_{i_1}') - (d_{i_2} - d_{i_2}') + (d_{i_3} - d_{i_3}') - \cdots - (d_{i_{2t}} - d_{i_{2t}}') + 2(d_{i_{2t+1}} - d_{i_{2t+1}}').$$ Write the right-hand side as $$\sum_{s=1}^t \left( (d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}') \right) + 2(d_{i_{2t+1}} - d_{i_{2t+1}}').$$ Note that $$d_i - d_i' = \begin{cases} 1 & \text{if } i \not\in I \text{ and } i \leq c_n, \\ 0 & \text{if } i \notin I \text{ and } i > c_n, \text{ or } i \in I \text{ and } i \leq c_n, \\ -1 & \text{if } i \in I \text{ and } i > c_n. \end{cases}$$ The only possible values of $(d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}')$ are therefore $-1$, $0$, $1$ and $2$. If we have $i_{2s-1}, i_{2s} \leq c_n$ or $i_{2s-1}, i_{2s} > c_n$, then the value 2 is not possible and $$(d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}') = 1 \qquad \Leftrightarrow \qquad i_{2s-1} \not\in I \text{ and } i_{2s} \in I.$$ Furthermore note that of the $2B$ pieces of horizontal boundary in column $n$, one is above row 1 (as $r_1 = n$, so $1 \in I$) and exactly $B-1$ are between a pair of cells with row indices $i$ and $i+1$, such that $i \not\in I$ and $i+1 \in I$. We now distinguish between four cases. *Case 1.* Suppose $i_{2t+1} \leq c_n$ and $i_{2t+1} \not\in I$. Then $2(d_{i_{2t+1}} - d_{i_{2t+1}}') = 2$. In the first $c_n$ cells of column $n$, there is at least one cell (the one with row index $i_{2t+1}$) that has value 0, hence $B \geq 2$ and there is a cell with row index greater than $i_{2t+1}$ with value 1. This means that there are at most $B-2$ pairs $(i_{2s-1}, i_{2s})$ such that $i_{2s-1} \not\in I$ and $i_{2s} \in I$. Also, $i_{2s-1}, i_{2s} \leq c_n$ for all $s$. So $$\sum_{s=1}^t \left( (d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}') \right) + 2(d_{i_{2t+1}} - d_{i_{2t+1}}') \leq (B-2) + 2 = B \leq 2B-2.$$ *Case 2.* Suppose $i_{2t+1} \leq c_n$ and $i_{2t+1} \in I$. Then $2(d_{i_{2t+1}} - d_{i_{2t+1}}') = 0$. Now there are at most $B-1$ pairs $(i_{2s-1}, i_{2s})$ such that $i_{2s-1} \not\in I$ and $i_{2s} \in I$. Also, $i_{2s-1}, i_{2s} \leq c_n$ for all $s$. So $$\sum_{s=1}^t \left( (d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}') \right) + 2(d_{i_{2t+1}} - d_{i_{2t+1}}') \leq B-1 \leq 2B-2.$$ *Case 3.* Suppose $i_{2t+1} > c_n$ and $B \geq 2$. Then $2(d_{i_{2t+1}} - d_{i_{2t+1}}') \leq 0$. Again there are at most $B-1$ pairs $(i_{2s-1}, i_{2s})$ such that $i_{2s-1} \not\in I$ and $i_{2s} \in I$. If there does not exist an $u$ such that $i_{2u-1} \leq c_n$ and $i_{2u} > c_n$, then we are done, as in the previous case. If there does exist such an $u$, then $$(d_{i_{2u-1}} - d_{i_{2u-1}}') - (d_{i_{2u}} - d_{i_{2u}}') = 2 \qquad \Leftrightarrow \qquad i_{2u-1} \not\in I \text{ and } i_{2u} \in I.$$ If $(d_{i_{2u-1}} - d_{i_{2u-1}}') - (d_{i_{2u}} - d_{i_{2u}}') = 2$, then on the right-hand side of (\[altgrenstoprove\]) we have a 2 and at most $B-2$ times a 1. If not, then we have no 2 and at most $B$ times a 1. In both cases we find $$\sum_{s=1}^t \left( (d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}') \right) + 2(d_{i_{2t+1}} - d_{i_{2t+1}}') \leq B \leq 2B-2.$$ *Case 4.* Suppose $B=1$. Then $i \in I \Leftrightarrow i\leq c_n$, hence $$d_i' = d_i \qquad \text{for all } i.$$ Therefore $$\sum_{s=1}^t \left( (d_{i_{2s-1}} - d_{i_{2s-1}}') - (d_{i_{2s}} - d_{i_{2s}}') \right) + 2(d_{i_{2t+1}} - d_{i_{2t+1}}') = 0 = 2B-2.$$ In all possible cases we have now proved inequality (\[altgrenstoprove\]), which finishes the proof of (\[altgrens1\]). Now we prove (\[altgrens2\]). Let $F$ be a binary $m \times n$ image with row sums $\mathcal{R}$ and column sums $\mathcal{C}$. Define $\bar{F}$ as the binary $m \times n$ image that has zeroes where $F$ has ones and ones where $F$ has zeroes. Let $(\bar{r}_1, \ldots, \bar{r}_m)$ be the row sums of $\bar{F}$ and $(\bar{c}_1, \ldots, \bar{c}_n)$ the column sums. Define $\bar{b}_i = \# \{j: \bar{c}_j \geq i\}$ and $\bar{d}_i = \bar{b}_i - \bar{r}_{m+1-i}$ for $i = 1, 2, \ldots, m$. As $\bar{r}_i = n-r_i$ and $\bar{c}_j= m-c_j$ for all $i$ and $j$, we have $$\bar{b}_i = \# \{j : m-c_j \geq i\} = \# \{j : c_j \leq m-i\} = n - \# \{j: c_j \geq m+1-i\} = n-b_{m+1-i}.$$ Hence $$\bar{d}_i = \bar{b}_i - \bar{r}_{m+1-i} = n-b_{m+1-i} - n + r_{m+1-i} = -d_{m+1-i}.$$ As $\bar{r}_1 = 0$ and $\bar{r}_m = n$, we may apply (\[altgrens1\]) to the row sums $(\bar{r}_m, \bar{r}_{m-1}, \ldots, \bar{r}_1)$. We write the subset of the row indices we use as $(m+1-i_{2t+1}, m+1-i_{2t}, \ldots, m+1-i_1)$ with $i_1 < i_2 < \ldots < i_{2t+1}$. We find that for the total length $\bar{L}_h$ of the horizontal boundary of $\bar{F}$ holds: $$\begin{aligned} \bar{L}_h &\geq 2n + \bar{d}_{m+1-i_{2t+1}} - \bar{d}_{m+1-i_{2t}} + \bar{d}_{m+1-i_{2t-1}} - \cdots - \bar{d}_{m+1-i_2} + 2\bar{d}_{m+1-i_1} \\ &= 2n - d_{i_{2t+1}} + d_{i_{2t}} - d_{i_{2t-1}} + \cdots + d_{i_2} - 2d_{i_1}.\end{aligned}$$ In each column of $\bar{F}$, the number of horizontal pieces of boundary is equal to the number of pairs of neighbouring cells such that one cell has value 1 and the other has value 0, plus one for the boundary below row $m$. In each column of $F$, the number of horizontal pieces of boundary is equal to the number of pairs of neighbouring cells such that one cell has value 1 and the other has value 0, plus one for the boundary above row 1. As in each column the number of pairs of neighbouring cells such that one cell has value 1 and the other has value 0, is the same in $F$ and in $\bar{F}$, we have $\bar{L}_h = L_h$. Hence $$L_h \geq 2n - d_{i_{2t+1}} + d_{i_{2t}} - d_{i_{2t-1}} + \cdots + d_{i_2} - 2d_{i_1}.$$ Some examples and a corollary {#examples} ============================= To illustrate Theorem \[altgrens\], we apply it to two small examples. \[exaltgrens1\] Let $m=n=10$ and let row sums $(10, 7, 7, 5, 4, 3, 5, 6, 1, 0)$ and column sums $(8,8,8,8,6,3,2,2,2,1)$ be given. We compute $b_i$ and $d_i$, $i=1, 2, \ldots, 10$ as shown below. [c|\*[10]{}[c]{}]{} $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\ $b_i$ & 10 & 9 & 6 & 5 & 5 & 5 & 4 & 4 & 0 & 0\ $r_i$ & 10 & 7 & 7 & 5 & 4 & 3 & 5 & 6 & 1 & 0\ $d_i$ & $0$ & $+2$ & $-1$ & $0$ & $+1$ & $+2$ & $-1$ & $-2$ & $-1$ & $0$ We take $t=1$, $i_1 = 2$, $i_2 = 3$ and $i_3 = 6$. Now (\[altgrens1\]) tells us that $$L_h \geq 20 + 2 - (-1) + 2\cdot 2 = 27.$$ Alternatively, we take $t=2$, $i_1=2$, $i_2=3$, $i_3=6$, $i_4=8$ and $i_5 =10$. Now tells us that $$L_h \geq 20 + 2 - (-1) + 2 - (-2) + 2 \cdot 0 = 27.$$ As $L_h$ must be even, we conclude $L_h \geq 28$. This bound is sharp: in Figure \[figaltgrens1\] a binary image $F$ with the given row and column sums is shown, for which $L_h = 28$. \[exaltgrens2\] Let $m=n=10$ and let row sums $(10, 9, 7, 6, 8, 4, 5, 2, 3, 0)$ and column sums $(9, 8, 8, 6, 6, 4, 4, 4, 3, 2)$ be given. We compute $b_i$ and $d_i$, $i=1, 2, \ldots, 10$ as shown below. [c|\*[10]{}[c]{}]{} $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\ $b_i$ & 10 & 10 & 9 & 8 & 5 & 5 & 3 & 3 & 1 & 0\ $r_i$ & 10 & 9 & 7 & 6 & 8 & 4 & 5 & 2 & 3 & 0\ $d_i$ & $0$ & $+1$ & $+2$ & $+2$ & $-3$ & $+1$ & $-2$ & $+1$ & $-2$ & $0$ We take $t=2$, $i_1 = 5$, $i_2 = 6$, $i_3 = 7$, $i_4=8$ and $i_5 = 9$. Now (\[altgrens2\]) tells us that $$L_h \geq 20 - (-2) + 1 - (-2) + 1 - 2\cdot (-3) = 32.$$ This bound is sharp: in Figure \[figaltgrens2\] a binary image $F$ with the given row and column sums is shown, for which $L_h = 32$. In the Introduction we mentioned two simple bounds of the length of the boundary. We recall them here, just for the horizontal boundary. The first one uses that in every column, there are at least two pieces of boundary, so if there are $n$ columns with nonzero sums, then $$\label{simple1} L_h \geq 2n.$$ The other bound computes the sum of the absolute differences between consecutive row sums, which yields $$\label{simple2} L_h \geq r_1 + \sum_{i=1}^{m-1} |r_i - r_{i+1}| + r_m.$$ In order to compare the bounds in Theorem \[altgrens\] to these two simple bounds, we construct two families of examples. \[exaltgrens3\] Let the number of columns $n$ be even. Let $m=n+2$. Define line sums $$\mathcal{C} = (n, n, n-2, n-2, \ldots, 4, 4, 2, 2), \quad \mathcal{R} = (n, n-1, n-1, n-3, n-3, \ldots, 3, 3, 1, 1, 0).$$ We calculate $$(b_1, b_2, \ldots, b_m) = (n, n, n-2, n-2, \ldots, 2, 2, 0, 0),$$ $$(d_1, d_2, \ldots, d_m) = (0, +1, -1, +1, -1, \ldots, +1, -1, +1, -1, 0).$$ Now tells us that $$L_h \geq 2n + \frac{n}{2} \cdot (1 - - 1) + 2\cdot 0 = 3n.$$ On the other hand, says $L_h \geq 2n$, while gives $$L_h \geq n + 1 + \frac{n-2}{2} \cdot 2 + 1 = 2n.$$ So Theorem \[altgrens\] gives a much better bound in this family of examples. In fact, it is sharp: there exists a binary image with the length of the boundary equal to $3n$. Such an image is easy to construct; see for an example Figure \[figaltgrens3\]. \[exaltgrens4\] Let $m=n+2$. Define line sums $$\mathcal{C} = (2, 2, 2, \ldots, 2, 2, 2), \quad \mathcal{R} = (n, 1, 1, 1, \ldots, 1, 1, 1, 0).$$ We calculate $$(b_1, b_2, \ldots, b_m) = (n, n, 0, 0, 0, \ldots, 0, 0, 0),$$ $$(d_1, d_2, \ldots, d_m) = (0, +(n-1), -1, -1, -1, \ldots, -1, -1, -1, 0).$$ Now tells us that $$L_h \geq 2n + 2\cdot (n-1) = 4n-2.$$ On the other hand, says $L_h \geq 2n$, while gives $$L_h \geq n + (n-1) + 1 = 2n.$$ So again Theorem \[altgrens\] gives a much better bound. In fact, it is sharp: there exists a binary image with the length of the boundary equal to $4n-2$. Such an image is easy to construct; see for an example Figure \[figaltgrens4\]. We can easily generalise the result from Theorem \[altgrens\] to the case where the conditions $r_1 = n$ and $r_m = 0$ are not satisfied. \[corollaltgrens\] Let row sums $\mathcal{R} = (r_1, r_2, \ldots, r_m)$ and column sums $\mathcal{C} = (c_1, c_2, \ldots, c_n)$ be given. Let $L_h$ be the total length of the horizontal boundary of an image with line sums $(\mathcal{R}, \mathcal{C})$. Define $b_i = \#\{j: c_j \geq i\}$ and $d_i = b_i - r_i$ for $i = 1, 2, \ldots, m$. Also set $d_0 = d_{m+1} = 0$. For any integer $t \geq 0$ and any subset $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{0, 1, 2, \ldots, m, m+1\}$ with $i_1 < i_2 < \ldots < i_{2t+1}$ we have $$\begin{aligned} L_h &\geq 2r_1 + d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}}, \label{altgrens3} \\ L_h &\geq 2r_1 - d_{i_{2t+1}} + d_{i_{2t}} - d_{i_{2t-1}} + \cdots + d_{i_2} - 2d_{i_1} \label{altgrens4}.\end{aligned}$$ Let $F$ be a binary image with line sums $(\mathcal{R},\mathcal{C})$ and a horizontal boundary of total length $L_h$. Construct $F'$ by adding a row above row 1 with row sum $n$ and a row below row $m$ with row sum $0$. Let $L_h'$ be the length of the horizontal boundary of $F'$. We have $L_h' = L_h + 2(n-r_1)$. The column sums of $F'$ are $c_j' = c_j + 1$, $j = 1, 2, \ldots, n$. The row sums are $r_1' = n$, $r_i' = r_{i-1}$ for $i=2, 3, \ldots, m+1$ and $r_{m+2}' = 0$. Let $b_i' = \#\{j: c_j' \geq i\}$ and $d_i' = b_i' - r_i'$ for $i = 1, 2, \ldots, m$. Then for all $i = 2, 3, \ldots, m+1$ we have $$b_i' = \#\{j: c_j + 1 \geq i\} = \#\{j: c_j \geq i-1\} = b_{i-1},$$ so $d_i' = b_{i-1} - r_{i-1} = d_{i-1}$. Also, $d_1' = d_0 =0$ and $d_{m+2}' = d_{m+1}=0$. We apply Theorem \[altgrens\] to $F'$ with the set of indices $\{i_1+1, i_2 +1, \ldots, i_{2t+1}+1\}$ and we find $$\begin{aligned} L_h' &\geq 2n + d_{i_1+1}' - d_{i_2+1}' + d_{i_3+1}' - \cdots - d_{i_{2t}+1}' + 2d_{i_{2t+1}+1}' \\ &= 2n + d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}},\\ L_h' &\geq 2n - d_{i_{2t+1}+1}' + d_{i_{2t}+1}' - d_{i_{2t-1}+1}' + \cdots + d_{i_2+1}' - 2d_{i_1+1}' \\ &= 2n - d_{i_{2t+1}} + d_{i_{2t}} - d_{i_{2t-1}} + \cdots + d_{i_2} - 2d_{i_1},\end{aligned}$$ and therefore $$\begin{aligned} L_h &\geq 2r_1 + d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}}, \\ L_h &\geq 2r_1 - d_{i_{2t+1}} + d_{i_{2t}} - d_{i_{2t-1}} + \cdots + d_{i_2} - 2d_{i_1}.\end{aligned}$$ A variation {#variation} =========== \[thmsplits\] Let row sums $\mathcal{R} = (r_1, r_2, \ldots, r_m)$ and column sums $\mathcal{C} = (c_1, c_2, \ldots, c_n)$ be given, where $r_1=n$, $r_m = 0$. Suppose there exists an image $F$ with line sums $(\mathcal{R}, \mathcal{C})$ and let $L_h(F)$ be the total length of the horizontal boundary of this image. Define $b_i = \#\{j: c_j \geq i\}$ and $d_i = b_i - r_i$ for $i = 1, 2, \ldots, m$. Let $k$ be an integer with $2 \leq k \leq m-1$ such that $d_k < 0$ and $d_{k+1} \geq 0$. Let $\sigma = \sum_{i=1}^k d_k$. For any integers $t, s \geq 0$ and any sets $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{1, 2, \ldots, k-1, k, m\}$ with $i_1 < i_2 < \ldots < i_{2t+1}$ and $\{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\} \subset \{1, k+1, k+2, \ldots, m-1, m\}$ with $\tilde i_1 < \tilde i_2 < \ldots < \tilde i_{2s+1}$ we have $$\begin{aligned} L_h(F) \geq 2n &+ d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}} \notag \\ &+ d_{\tilde i_1} - d_{\tilde i_2} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}} - \sigma. \label{splits}\end{aligned}$$ We will prove the theorem by induction on $\sigma$. Note that by we have $\sigma \geq 0$, since the line sums are consistent. As we are only considering the horizontal boundary, we may for convenience assume that $c_1 \geq c_2 \geq \ldots \geq c_n$. Suppose $\sigma = 0$. Then $$\sum_{i=1}^k r_i = \sum_{i=1}^k b_i = \sum_{i=1}^k \#\{j: c_j \geq i\} = \sum_{j \mid c_j \leq k} c_j + \sum_{j \mid c_j>k} k.$$ So in any column $j$ with $c_j > k$ we must have $(i,j) \in F$ for $1 \leq i \leq k$, and in any column $j$ with $c_j \leq k$ we must have $(i,j) \not\in F$ for $k+1 \leq i \leq m$. This means that we can split the image $F$ into four smaller images, one of which contains only ones and one of which contains only zeroes. The other two parts we call $F_1$ and $F_2$ (see Figure \[figinductiebasis\]). In order to have images with the first row filled with ones and the last row filled with zeroes, we glue row $m$ to $F_1$ and row 1 to $F_2$. More precisely, let $F_1$ consist of rows $1, 2, \ldots, k-1, k$ and $m$ of $F$ and the columns $j$ with $c_j \leq k$; let $F_2$ consist of rows $1$ and $k+1, k+2, \ldots, m-1, m$ of $F$ and the columns $j$ with $c_j > k$. ![Splitting the image $F$ into four smaller images.[]{data-label="figinductiebasis"}](plaatje.3) The columns of $F$ with sum at most $k$ are exactly the columns with indices greater than $b_{k+1}$. Define $h = b_{k+1}$. Let $r_{1}^{(1)}$, $r_{2}^{(1)}, \ldots, r_k^{(1)}, r_{m}^{(1)}$ be the row sums of $F_1$, and let $r_{1}^{(2)}$, $r_{k+1}^{(2)}$, …, $r_{m-1}^{(2)}$, $r_{m}^{(2)}$ be the row sums of $F_2$. We have $$r_i^{(1)} = r_i - h, \quad \text{for $1 \leq i \leq k$, and} \quad r_m^{(1)} = r_m,$$ $$r_i^{(2)} = r_i \quad \text{for $k+1 \leq i \leq m$, and} \quad r_1^{(2)} = h = r_1 - (n-h).$$ Let $c_{h+1}^{(1)}$, $c_{h+2}^{(1)}$, …, $c_{n-1}^{(1)}$, $c_n^{(1)}$ be the column sums of $F_1$, and let $c_1^{(2)}$, $c_2^{(2)}$, …, $c_{h-1}^{(2)}$, $c_h^{(2)}$ be the column sums of $F_2$. We have $$c_j^{(1)} = c_j, \quad \text{and} \quad c_j^{(2)} = c_j - (k-1) \quad \text{for all $j$}.$$ Define $$\begin{aligned} b_1^{(1)} &= \# \{j \geq h+1 : c_j^{(1)} \geq 1 \}, & b_1^{(2)} &= \# \{j \leq h : c_j^{(2)} \geq 1 \},\\ b_2^{(1)} &= \# \{j \geq h+1 : c_j^{(1)} \geq 2 \}, & b_{k+1}^{(2)} &= \# \{j \leq h : c_j^{(2)} \geq 2 \},\\ &\vdots & &\vdots \\ b_k^{(1)} &= \# \{j \geq h+1 : c_j^{(1)} \geq k \}, & b_{m-1}^{(2)} &= \# \{j \leq h : c_j^{(2)} \geq m-k \},\\ b_m^{(1)} &= \# \{j \geq h+1 : c_j^{(1)} \geq k+1 \}, & b_m^{(2)} &= \# \{j \leq h : c_j^{(2)} \geq m-k+1 \}.\\\end{aligned}$$ For $1 \leq i \leq k$ we have $$b_i^{(1)} = \# \{j \geq h+1 : c_j^{(1)} \geq i \} = \# \{j \leq n : c_j \geq i\} - \# \{j \leq h : c_j \geq i\} = b_i - h.$$ Also, $b_m^{(1)} = 0 = b_m$. For $k+1 \leq i \leq m$ we have $$b_{i}^{(2)} = \# \{j \leq h : c_j^{(2)} \geq i-k+1 \} = \# \{j \leq h : c_j \geq i \}$$$$= \# \{j \leq n : c_j \geq i\} - \#\{j \geq h+1 : c_j \geq i\} = b_i - 0 = b_i.$$ Also, $b_1^{(2)} = h = b_1 - (n-h)$. Now define $d_i^{(1)} = b_i^{(1)} - r_i^{(1)}$ for $i \in \{1, 2, \ldots, k-1, k, m\}$ and $d_i^{(2)} = b_i^{(2)} - r_i^{(2)}$ for $i \in \{1, k+1, k+2, \ldots, m-1, m\}$. We find $$d_i^{(1)} = b_i - h - (r_i - h) = d_i, \quad \text{for $1 \leq i \leq k$,}$$ $$d_m^{(1)} = b_m - r_m = d_m,$$ $$d_i^{(2)} = b_i - r_i = d_i \quad \text{for $k+1 \leq i \leq m$}$$ $$d_1^{(2)} = b_1 - (n-h) - (r_1 - (n-h)) = d_1.$$ All in all we conclude $d_i^{(1)} = d_i$ and $d_i^{(2)} = d_i$ for all $i$. The total length of the horizontal boundary of $F$ in the columns $j$ with $c_j \leq k$ is exactly the same as the total length $L_h(F_1)$ of the horizontal boundary of $F_1$. The total length of the horizontal boundary of $F$ in the columns $j$ with $c_j > k$ is exactly the same as the total length $L_h(F_2)$ of the horizontal boundary of $F_2$. So $L_h(F) = L_h(F_1) + L_h(F_2)$. Note that $F_1$ has $n-b_{k+1}$ columns and $F_2$ has $b_{k+1}$ columns. By Theorem \[altgrens\] applied to $F_1$ we know that for any integer $t \geq 0$ and any set $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{1, 2, \ldots, k-1, k, m\}$ with $i_1 < i_2 < \ldots < i_{2t+1}$ we have $$L_h(F_1) \geq 2(n-b_{k+1}) + d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}}.$$ By the same theorem applied to $F_2$ we know that for any integer $t \geq 0$ and any set $\{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\} \subset \{1, k+1, k+2, \ldots, m-1, m\}$ with $\tilde i_1 < \tilde i_2 < \ldots < \tilde i_{2s+1}$ we have $$L_h(F_2) \geq 2b_{k+1} + d_{\tilde i_1} - d_{\tilde i_2} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}}.$$ Adding these two results yields . Now let $\sigma \geq 1$ and suppose that we have already proven the theorem for any image with $\sum_{i=1}^k d_i < \sigma$. Let $$\begin{aligned} A_1 &= \max\{ d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}}\}, \\ A_2 &= \max\{ d_{\tilde i_1} - d_{\tilde i_2} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}} \},\end{aligned}$$ where the first maximum is taken over all integers $t \geq 0$ and sets $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{1, 2, \ldots, k-1, k, m\}$ with $i_1 < i_2 < \ldots < i_{2t+1}$, and the second maximum over all integers $s \geq 0$ and sets $\{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\} \subset \{1, k+1, k+2, \ldots, m-1, m\}$ with $\tilde i_1 < \tilde i_2 < \ldots < \tilde i_{2s+1}$. Furthermore, fix $i_1, i_2, \ldots, i_{2t+1}$ and $\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}$ such that these maxima are attained. Since $d_k<0$ by definition of $k$, and since $d_m = 0$, we have $$d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{k} < d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{m}.$$ If $i_{2t+1} = k$, this would contradict the maximality of $A_1$, so we conclude $$\label{eqfact} i_{2t+1} \neq k.$$ We also know $d_{k+1} \geq 0$ by definition of $k$, and $d_1 = 0$. So if $s \geq 1$, then $$d_{1} - d_{k+1} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}} \leq d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}}.$$ This means that if $s \geq 1$, we may assume without loss of generality that $(\tilde i_1, \tilde i_2) \neq (1, k+1)$. Also, $$d_{1} - d_{\tilde i_2} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}} \leq d_{k+1} - d_{\tilde i_2} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}}.$$ This means that if $s \geq 1$ and $\tilde i_2 > k+1$, we may assume that $\tilde i_1 \neq 1$. Finally, $$2 d_1 \leq 2d_{k+1},$$ so if $s=1$ we may also assume that $\tilde i_1 \neq 1$. All in all we may assume in all cases that $$\label{eqassume} \tilde i_1 \neq 1.$$ It suffices to prove $$\label{tebewijzen} L_h(F) \geq 2n + A_1 + A_2 - \sigma.$$ Let $j$ with $1 \leq j \leq n$ be such that $\# \big( \{(1,j), (2,j), \ldots, (k,j) \} \cap F \big) < \min(c_j, k)$, i.e. in column $j$ there is at least one one in rows $k+1, k+2, \ldots, m$ and at least one zero in rows $1, 2, \ldots, k$. Such a column exists, because $$\sum_{i=1}^k r_i < \sum_{i=1}^k b_i = \sum_{i=1}^k \#\{j: c_j \geq i\} = \sum_{j \mid c_j \leq k} c_j + \sum_{j \mid c_j>k} k.$$ We will now consider various cases. *Case* 1. Suppose that there exist integers $l \geq 2$, $h \geq k+1$ and $u \geq 0$ such that $l+u \leq k$, $h+u \leq m-1$ and - $(l-1, j) \in F$, and - $(l, j), (l+1, j), \ldots, (l+u, j) \not\in F$, and - $(h,j), (h+1, j), \ldots, (h+u,j) \in F$, and - $(h+u+1,j) \not\in F$, and - $(l+u+1, j) \in F$ or $(h-1,j) \not\in F$. ![Two possibilities for column $j$ in Case 1. The grey cells have value 1, the other cells value 0.[]{data-label="figcase1before"}](plaatje.4) We define a new image $F'$ by moving the ones at $(h,j), (h+1,j), \ldots, (h+u,j)$ to $(l,j), (l+1,j), \ldots, (l+u,j)$; that is, $$F' = F \cup \{ (l,j), (l+1,j), \ldots, (l+u,j) \} \backslash \{ (h,j), (h+1,j), \ldots, (h+u,j) \}.$$ The column sums of $F'$ are identical to the column sums of $F$. The row sums $r_i'$ of $F'$ are given by $$r_i' = \begin{cases} r_i + 1 & \text{if $l \leq i \leq l+u$}, \\ r_i - 1 & \text{if $h \leq i \leq h+u$}, \\ r_i & \text{else.} \end{cases}$$ Define $d_i' = b_i - r_i'$ and $\sigma' = \sum_{i=1}^k d_i' = \sigma - (u+1)$. By the induction hypothesis, we have for the total length $L_h(F')$ of the horizontal boundary of $F'$ $$L_h(F') \geq 2n + A_1' + A_2' - \sigma',$$ where $$\begin{aligned} A_1' &= d_{i_1}' - d_{i_2}' + d_{i_3}' - \cdots - d_{i_{2t}}' + 2d_{i_{2t+1}}',\\ A_2' &= d_{\tilde i_1}' - d_{\tilde i_2}' + d_{\tilde i_3}' - \cdots - d_{\tilde i_{2s}}' + 2d_{\tilde i_{2s+1}}'.\end{aligned}$$ By moving the $u+1$ ones in column $j$, the piece of horizontal boundary between row $l-1$ and row $l$ has vanished, just like the piece of horizontal boundary between row $h+u$ and $h+u+1$. If $(l+u+1,j) \in F$, the piece of horizontal boundary between row $l+u$ and row $l+u+1$ has also vanished, but there may be a new piece of horizontal boundary between row $h-1$ and $h$. On the other hand, if $(h-1,j) \not\in F$, the piece of horizontal boundary between row $h-1$ and row $h$ has vanished, but there may be a new piece of horizontal boundary between row $l+u$ and $l+u+1$. At least one of both is the case. All in all, we have $L_h(F') \leq L_h(F) - 2$. ![Moving ones in Case 1, in both possible configurations. The grey cells have value 1, the other cells value 0.[]{data-label="figcase1after"}](plaatje.7) Furthermore, some of the $d_i'$ involved in $A_1'$ or $A_2'$ may be different from the corresponding $d_i$. Since $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{1, 2, \ldots, k-1, k, m\}$, we have $d_i' = d_i$ or $d_i' = d_i-1$ for $i \in \{i_1, i_2, \ldots, i_{2t+1} \}$. The values of $i$ for which $d_i' = d_i-1$, are all consecutive. Since the coefficients for $d_i$ in $A_1$ are alternatingly positive and negative, and there is only one positive coefficient that is $+2$ rather than $+1$, we have $$A_1' = d_{i_1}' - d_{i_2}' + d_{i_3}' - \cdots - d_{i_{2t}}' + 2d_{i_{2t+1}}' \geq d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}} - 2 = A_1 - 2.$$ Since $\{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\} \subset \{1, k+1, k+2, \ldots, m-1, m\}$, we have $d_i' = d_i$ or $d_i' = d_i + 1$ for $i \in \{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\}.$ By a similar argument as above and by the fact that all negative coefficients in $A_2$ are equal to $-1$, we have $$A_2' \geq A_2 - 1.$$ Finally, we have $\sigma' = \sigma - (u+1) \leq \sigma - 1$. We conclude $$\begin{aligned} L_h(F) &\geq L_h(F') + 2 \\ &\geq 2n + A_1' + A_2' - \sigma' + 2 \\ &\geq 2n + (A_1 -2) + (A_2 - 1) - (\sigma-1) +2 \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ This proves in Case 1. *Case* 2. Suppose that the conditions of Case 1 do not hold and furthermore that $(k,j) \in F$ and $(k+1,j) \in F$. Then there exist integers $l \geq 2$, $h \leq k$ and $u \geq 0$ such that $h \geq l+1$, $k+1 \leq h+u \leq m-1$ and - $(l-1, j) \in F$, and - $(l, j), (l+1, j), \ldots, (h-1, j) \not\in F$, and - $(h,j), (h+1, j), \ldots, (h+u,j) \in F$, and - $(h+u+1,j) \not\in F$. As Case 1 does not apply, we cannot change all zeroes in $(l, j)$, $(l+1,j)$, …, $(h-1,j)$ into ones by moving ones from $(k+1,j)$, $(k+2,j)$, …, $(h+u,j)$. This implies that $h-l > (h+u)-k \geq 1$, so $l < h-1$. We will now distinguish between several cases. *Case* 2a. Suppose that there does not exist an integer $r$ with $0 \leq r \leq t$ such that $l = i_{2r+1}$. We define a new image $F'$ by moving the one at $(h+u,j)$ to $(l,j)$; that is, $$F' = F \cup \{(l,j)\} \backslash \{(h+u,j)\}.$$ We define $r_i'$, $d_i'$, $\sigma'$, $A_1'$, $A_2'$ and $L_h(F')$ similarly as in Case 1. As in Case 1 we have $A_2' \geq A_2 -1$. However, of the $d_i$ with $i \in \{1, 2, \ldots, k-1, k, m\}$ only one has changed (namely $d_l' = d_l - 1$), and we know that $d_l$ does not have a positive coefficient in $A_1$. So $A_1' \geq A_1$. Furthermore, $L_h(F') = L_h(F)$ and $\sigma' = \sigma - 1$. By applying the induction hypothesis to $F'$, we find $$\begin{aligned} L_h(F) &= L_h(F') \\ &\geq 2n + A_1' + A_2' - \sigma' \\ &\geq 2n + A_1 + (A_2 - 1) - (\sigma-1) \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ This proves in Case 2a. *Case* 2b. Suppose that there does not exist an integer $r$ with $0 \leq r \leq t$ such that $h-1 = i_{2r+1}$. We define a new image $F'$ by moving the one at $(h+u,j)$ to $(h-1,j)$; the rest of the proof is the same as in Case 2a. *Case* 2c. Suppose neither Case 2a nor Case 2b applies. Then there are integers $r_1$ and $r_2$ with $0 \leq r_1 < r_2 \leq t$ such that $l=i_{2r_1+1}$ and $h-1=i_{2r_2+1}$. Note that $r_1 < t$, so $d_l$ has coefficient $+1$ in $A_1$. Now let $v = i_{2r_1+2} < h-1$. Again, we distinguish between two cases. *Case* 2c1. Suppose that $k+1 \leq h+u-v+l$. Then we define a new image $F'$ by moving the ones at $(h+u-v+l, j)$, $(h+u-v+l+1,j)$, …, $(h+u,j)$ to $(l,j)$, $(l+1,j)$, …, $(v,j)$; that is, $$F' = F \cup \{(l,j), (l+1,j), \ldots, (v,j)\} \backslash \{(h+u-v+l, j), (h+u-v+l+1,j), \ldots, (h+u,j) \}.$$ We define $r_i'$, $d_i'$, $\sigma'$, $A_1'$, $A_2'$ and $L_h(F')$ similarly as in Case 1. As in Case 2a we have $A_2' \geq A_2 -1$ and $L_h(F') = L_h(F)$. Also, $\sigma' \leq \sigma - 1$. Furthermore, of the $d_i$ with $i \in \{1, 2, \ldots, k-1, k, m\}$ exactly two have changed: $d_l' = d_l -1$ and $d_v' = d_v - 1$. As $d_l$ has coefficient $+1$ in $A_1$ and $d_v$ has coefficient $-1$ in $A_1$, we have $A_1' = A_1$. By applying the induction hypothesis to $F'$, we find $$\begin{aligned} L_h(F) &= L_h(F') \\ &\geq 2n + A_1' + A_2' - \sigma' \\ &\geq 2n + A_1 + (A_2 - 1) - (\sigma-1) \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ This proves in Case 2c1. *Case* 2c2. Suppose that $k+1 > h+u-v+l$. Then we define a new image $F'$ by moving the ones at $(k+1, j)$, $(k+2,j)$, …, $(h+u,j)$ to $(l,j)$, $(l+1,j)$, …, $(l+h+u-k-1,j)$; that is, $$F' = F \cup \{(l,j), (l+1,j), \ldots, (l+h+u-k-1,j)\} \backslash \{(k+1, j), (k+2,j), \ldots, (h+u,j) \}.$$ We define $r_i'$, $d_i'$, $\sigma'$, $A_1'$, $A_2'$ and $L_h(F')$ similarly as in Case 1. As in Case 2c1 we have $L_h(F') = L_h(F)$ and $\sigma' \leq \sigma - 1$. Since $l+h+u-k-1 < v$, of the $d_i$ with $i \in \{1, 2, \ldots, k-1, k, m\}$ exactly one has changed: $d_l' = d_l -1$. As $d_l$ has coefficient $+1$ in $A_1$, we have $A_1' = A_1-1$. Now we consider $A_2'$. Some of the $d_i$ with $i \in \{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\}$ may have increased by 1. If $\tilde i_1 > h+u$, none of the row indices $k+1$, $k+2$, …, $h+u$ occurs in $\{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\}$, and we have $A_2' = A_2$. If not, then $k+1 \leq \tilde i_1 \leq h+u$ (using ). The values of $i$ for which $d_i' = d_i+1$, are all consecutive. Since the coefficients for $d_i$ in $A_1$ are alternatingly positive and negative, and since $\tilde i_1$ (which has a positive coefficient in $A_1$) is included in $\{k+1, k+2, \ldots, h+u\}$, we have $A_2' \geq A_2$. By applying the induction hypothesis to $F'$, we find $$\begin{aligned} L_h(F) &= L_h(F') \\ &\geq 2n + A_1' + A_2' - \sigma' \\ &\geq 2n + (A_1-1) + A_2 - (\sigma-1) \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ This proves in Case 2c2, which completes the proof of Case 2. *Case* 3. Suppose that the conditions of Case 1 and Case 2 do not hold. By definition of $j$ we know that in column $j$ there is at least one one in rows $k+1$, $k+2$, …, $m$. As Case 2 does not apply, we have $(k,j) \notin F$ or $(k+1,j) \not\in F$. If $(k,j) \in F$ (so $(k+1, j) \not\in F$) we can apply Case 1: let $l$ be the smallest integer such that $(l,j)\not\in F$, let $h'$ be the greatest integer such that $(h',j) \in F$, and let $u$ be maximal such that $(i,j) \not\in F$ for $l \leq i \leq l+u$ and $(i,j) \in F$ for $h'-u \leq i \leq h'$. Define $h = h'-u$. Since $(k,j) \in F$ and $(k+1,j) \not\in F$, we have $l+u < k$ and $h > k+1$, so all conditions of Case 1 are satisfied. Hence we have $(k,j) \not\in F$. Now there exist integers $h \geq k+1$ and $u \geq 0$ such that $h+u \leq m-1$ and - $(h-1,j) \not\in F$, and - $(i,j) \in F$ for $h \leq i \leq h+u$, and - $(h+u+1,j) \not\in F$. Furthermore, let $l \leq k$ be such that $(l-1,j)\in F$ and $(l,j) \not\in F$. Since Case 1 does not apply, there does not exist an integer $u'$ such that $l+u' \leq k$, $(i,j) \not\in F$ for $l \leq i \leq l+u'$ and $(l+u'+1, j) \in F$. This means that $(i,j) \not\in F$ for all $i$ with $l \leq i \leq k+1$. Also, we could still apply Case 1 if there are at least as many zeroes in $(l,j)$, $(l+1,j)$, …$(k,j)$ as there are ones in $(h,j)$, $(h+1, j)$, …, $(h+u,j)$. Hence we must have $u+1 > k-l+1$. We will distinguish between various cases. *Case* 3a. Suppose that either $i_{2t+1} < l$ or $i_{2t+1} = m$. This means that none of the $d_i$ with $l \leq i \leq k$ has coefficient $+2$ in $A_1$. Since $u+1 > k-l+1$, we have $h+k-l < h+u$, so there are ones at $(h,j)$, $(h+1,j)$, …, $(h+k-l,j)$. We define a new image $F'$ by moving those ones to $(l,j)$, $(l+1,j)$, …, $(k,j)$; that is $$F' = F \cup \{(l,j), (l+1,j), \ldots, (k,j)\} \backslash \{(h,j), (h+1,j), \ldots, (h+k-l,j)\}.$$ We define $r_i'$, $d_i'$, $\sigma'$, $A_1'$, $A_2'$ and $L_h(F')$ similarly as in Case 1. As in Case 1 we have $A_2' \geq A_2-1$. Furthermore, $L_h(F') = L_h(F)$. Suppose $l=k$. Then only one $d_i$ with $i \in \{1, 2, \ldots, k-1, k, m\}$ has changed, namely $d_k' = d_k - 1$. We know that $d_k$ does not have a positive coefficient in $A_1$, since $k \neq i_{2t+1}$ (see ) and $i_{2t-1} \leq k-1$. So $A_1' \geq A_1$. Also, $\sigma' = \sigma - 1$, so by applying the induction hypothesis to $F'$, we find $$\begin{aligned} L_h(F) &= L_h(F') \\ &\geq 2n + A_1' + A_2' - \sigma' \\ &\geq 2n + A_1 + (A_2-1) - (\sigma -1) \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ Now suppose that $l<k$. Then we have $\sigma' \leq \sigma - 2$. Furthermore, none of the $d_i$ with $l \leq i \leq k$ has coefficient $+2$ in $A_1$, so $A_1' \geq A_1-1$. By applying the induction hypothesis to $F'$, we find $$\begin{aligned} L_h(F) &= L_h(F') \\ &\geq 2n + A_1' + A_2' - \sigma' \\ &\geq 2n + (A_1-1) + (A_2-1) - (\sigma -2) \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ This proves in Case 3a. *Case* 3b. Suppose that $i_{2t+1} \geq l$, $i_{2t+1} \neq m$ and $i_{2t+1} \neq k-1$. Using , we then have $l \leq i_{2t+1} \leq k-2$. Since $u+1 > k-l+1$, we find that $u \geq k-l+1 \geq (l+2) -l +1 \geq 3$. We define a new image $F'$ by moving the ones at $(h,j)$, $(h+1,j)$ and $(h+2,j)$ to $(l,j)$, $(l+1,j)$ and $(l+2,j)$; that is, $$F' = F \cup \{(l,j), (l+1,j), (l+2,j)\} \backslash \{(h,j), (h+1,j), (h+2,j)\}.$$ We define $r_i'$, $d_i'$, $\sigma'$, $A_1'$, $A_2'$ and $L_h(F')$ similarly as in Case 1. As in Case 1, we have $A_1' \geq A_1 -2$ and $A_2' \geq A_2 - 1$. Furthermore, $L_h(F') = L_h(F)$ and $\sigma' = \sigma-3$. By applying the induction hypothesis to $F'$, we find $$\begin{aligned} L_h(F) &= L_h(F') \\ &\geq 2n + A_1' + A_2' - \sigma' \\ &\geq 2n + (A_1-2) + (A_2-1) - (\sigma -3) \\ &= 2n + A_1 + A_2 - \sigma.\end{aligned}$$ This proves in Case 3b. *Case* 3c. Suppose that neither Case 3a nor Case 3b applies. Then we have $i_{2t+1} = k-1$. Using , this means that $\tilde i_1 \geq k+1 > k-1 = i_{2t+1}$. We now apply Theorem \[altgrens\] to the image $F$ and the row indices $\{i_1, i_2, \ldots, i_{2t}, k-1, k, \tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\}$: $$\begin{aligned} L_h(F) &\geq 2n + d_{i_1} - d_{i_2} + \cdots - d_{i_{2t}} + d_{k-1} - d_k + d_{\tilde i_1} - d_{\tilde i_2} + \cdots - d_{\tilde i_{2s}} + 2 d_{\tilde i_{2s+1}} \\ &= 2n + A_1 - d_{k-1} - d_k + A_2.\end{aligned}$$ By Ryser’s Theorem [@ryser] we have $\sum_{i=1}^{k-2} d_i \geq 0$, since the line sums are consistent, so $$\sigma = \sum_{i=1}^k d_i = \sum_{i=1}^{k-2} d_i + d_{k-1} + d_k \geq d_{k-1} + d_k.$$ Hence $$L_h(F) \geq 2n+A_1 - d_{k-1} - d_k + A_2 \geq 2n + A_1 + A_2 - \sigma,$$ which proves in Case 3c. This finishes the proof of the theorem. \[exsplitsing\] Let $m=n=12$ and let row sums $(12, 8, 9, 8, 8, 5, 5, 2, 3, 2, 1, 0)$ and column sums $(10, 8, 8, 8, 6, 6, 6, 3, 2, 2, 2, 2)$ be given. We compute $b_i$ and $d_i$, $i=1, 2, \ldots, 12$ as shown below. [c|\*[12]{}[c]{}]{} $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\ $b_i$ & 12 & 12 & 8 & 7 & 7 & 7 & 4 & 4 & 1 & 1 & 0 & 0\ $r_i$ & 12 & 8 & 9 & 8 & 8 & 5 & 5 & 2 & 3 & 2 & 1 & 0\ $d_i$ & $0$ & $+4$ & $-1$ & $-1$ & $-1$ & $+2$ & $-1$ & $+2$ & $-2$ & $-1$ & $-1$ & $0$ Here yields at most $$L_h \geq 24 + 4 - (-1) + 2 - (-1) + 2 \cdot 2 = 36,$$ and yields at most $$L_h \geq 24 - (-2) + 2 - (-1) + 2 - (-1) + 4 - 2 \cdot 0 = 36.$$ However, we can apply Theorem \[thmsplits\] with $k=5$ (note that $d_5 < 0$ and $d_6 \geq 0$). We have $\sigma = 1$. If we take $t=0$, $s=0$, $i_1 = 2$, $\tilde i_1 = 6$, $\tilde i_2 = 7$ and $\tilde i_3 = 8$, then we find $$L_h \geq 24 + 2 \cdot 4 + 2 - (-1) + 2\cdot 2 - 1 = 38.$$ So in this example, Theorem \[thmsplits\] gives a better bound than Theorem \[altgrens\]. In fact, the bound of Theorem \[thmsplits\] is sharp in this example: in Figure \[figexsplitsing\] a binary image $F$ with the given row and column sums is shown, for which $L_h = 38$. ![The binary image from Examples \[exsplitsing\]. The grey cells have value 1, the other cells value 0. The numbers indicate the row and column sums. The length of the horizontal boundary of this image is 38.[]{data-label="figexsplitsing"}](plaatje.14) \[corollsplits\] Let row sums $\mathcal{R} = (r_1, r_2, \ldots, r_m)$ and column sums $\mathcal{C} = (c_1, c_2, \ldots, c_n)$ be given. Suppose there exists an image $F$ with line sums $(\mathcal{R}, \mathcal{C})$ and let $L_h(F)$ be the total length of the horizontal boundary of this image. Define $b_i = \#\{j: c_j \geq i\}$ and $d_i = b_i - r_i$ for $i = 1, 2, \ldots, m$. Also set $d_0 = d_{m+1} = 0$. Let $k$ be an integer with $1 \leq k \leq m$ such that $d_k < 0$ and $d_{k+1} \geq 0$. Let $\sigma = \sum_{i=1}^k d_k$. For any integers $t, s \geq 0$ and any sets $\{i_1, i_2, \ldots, i_{2t+1} \} \subset \{0, 1, \ldots, k-1, k, m+1\}$ with $i_1 < i_2 < \ldots < i_{2t+1}$ and $\{\tilde i_1, \tilde i_2, \ldots, \tilde i_{2s+1}\} \subset \{0, k+1, k+2, \ldots, m, m+1\}$ with $\tilde i_1 < \tilde i_2 < \ldots < \tilde i_{2s+1}$ we have $$\begin{aligned} L_h(F) \geq 2r_1 &+ d_{i_1} - d_{i_2} + d_{i_3} - \cdots - d_{i_{2t}} + 2d_{i_{2t+1}} \notag \\ &+ d_{\tilde i_1} - d_{\tilde i_2} + d_{\tilde i_3} - \cdots - d_{\tilde i_{2s}} + 2d_{\tilde i_{2s+1}} - \sigma.\end{aligned}$$ Completely analogous to the proof of Corollary \[corollaltgrens\]. [00]{} E. Balogh, A. Kuba, C. Dévényi, A. Del Lungo, Comparison of algorithms for reconstructing hv-convex discrete sets, *Linear Algebra and its Applications* 339 (2001) 23-35. E. Barcucci, A. Del Lungo, M. Nivat, R. Pinzani, Reconstructing convex polyominoes from horizontal and vertical projections, *Theoretical Computer Science* 155 (1996) 321-347. M. Chrobak, C. Dürr, Reconstructing hv-convex polyominoes from orthogonal projections, *Information Processing Letters* 69 (1999) 283-289. G. Dahl, T. Flatberg, Optimization and reconstruction of hv-convex (0,1)-matrices, *Discrete Applied Mathematics* 151 (2005) 93-105. R.J. Gardner, P. Gritzmann, D. Prangenberg, On the computational complexity of reconstructing lattice sets from their X-rays, *Discrete Mathematics* 202 (1999) 45-71. S.B. Gray, Local properties of binary images in two dimensions, *IEEE Transactions on Computers* 20 (1971) 551-561. G.T. Herman, A. Kuba, editors, *Discrete Tomography: Foundations, Algorithms and Applications*, Birkhäuser, Boston (1999). G.T. Herman, A. Kuba, editors, *Advances in Discrete Tomography and Its Applications*, Birkhäuser, Boston (2007). A. Rosenfeld, Connectivity in digital pictures, *Journal of the Association for Computing Machinery* 17 (1970) 146-160. H.J. Ryser, Combinatorial properties of matrices of zeros and ones, *Canadian Journal of Mathematics* 9 (1957) 371-377. G.J. Woeginger, The reconstruction of polyominoes from their orthogonal projections, *Information Processing Letters* 77 (2001) 225-229.

--- abstract: 'In this paper we introduce a mathematical model that captures some of the salient features of recommender systems that are based on popularity and that try to exploit social ties among the users. We show that, under very general conditions, the market always converges to a steady state, for which we are able to give an explicit form. Thanks to this we can tell rather precisely how much a market is altered by a recommendation system, and determine the power of users to influence others. Our theoretical results are complemented by experiments with real world social networks showing that social graphs prevent large market distortions in spite of the presence of highly influential users.' author: - | Marco Bressan[^1][^2]\ Sapienza University of Rome\ Rome, Italy\ - | Stefano Leucci\ Sapienza University of Rome\ Rome, Italy\ - | Alessandro Panconesi\ Sapienza University of Rome\ Rome, Italy\ - | Prabhakar Raghavan\ Google\ Mountain View, CA - | Erisa Terolli\ Sapienza University of Rome\ Rome, Italy\ title: | The Limits of Popularity-Based Recommendations,\ and the Role of Social Ties --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10003120.10003130.10003131.10003270&lt;/concept\_id&gt; &lt;concept\_desc&gt;Human-centered computing Social recommendation&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003120.10003130&lt;/concept\_id&gt; &lt;concept\_desc&gt;Human-centered computing Collaborative and social computing&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003120.10003130.10003131.10003292&lt;/concept\_id&gt; &lt;concept\_desc&gt;Human-centered computing Social networks&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10002951.10003317.10003347.10003350&lt;/concept\_id&gt; &lt;concept\_desc&gt;Information systems Recommender systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003752.10010070.10010099.10010106&lt;/concept\_id&gt; &lt;concept\_desc&gt;Theory of computation Market equilibria&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10002950.10003648.10003700&lt;/concept\_id&gt; &lt;concept\_desc&gt;Mathematics of computing Stochastic processes&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003033.10003106.10003114.10011730&lt;/concept\_id&gt; &lt;concept\_desc&gt;Networks Online social networks&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; [**Acknowledgments.**]{} The authors are grateful to Tiziana Castrignanò at CINECA for her precious help with the technical setup of the experiments, and to Flavio Chierichetti for the stimulating and insightful discussions throughout the development of this work. [^1]: Supported in part by a Google Focused Research Award, by the Sapienza Grant C26M15ALKP, by the SIR Grant RBSI14Q743, and by the ERC Starting Grant DMAP 680153. [^2]: Corresponding author. Address: [bressan@di.uniroma1.it]{}

--- abstract: 'Europium nitride is semiconducting and contains non-magnetic Eu$^{3+}$, but sub-stoichiometric EuN has Eu in a mix of 2+ and 3+ charge states. We show that at Eu$^{2+}$ concentrations near 15-20% EuN is ferromagnetic with a Curie temperature as high as 120 K. The Eu$^{3+}$ polarization follows that of the Eu$^{2+}$, confirming that the ferromagnetism is intrinsic to the EuN which is thus a novel diluted magnetic semiconductor. Transport measurements shed light on the likely exchange mechanisms.' author: - Do Le Binh - 'B. J. Ruck' - 'F. Natali' - 'H. Warring' - 'H. J. Trodahl' - 'E.-M. Anton' - 'C. Meyer' - 'L. Ranno' - 'F. Wilhelm' - 'A. Rogalev' title: 'Europium nitride: A novel diluted magnetic semiconductor' --- Diluted magnetic semiconductors (DMSs), in which magnetic impurities are doped into a semiconducting host, offer important opportunities for use in spintronics technology as materials for spin injection or manipulation [@Awschalom_Flatte; @Zutic_DasSarma; @Dietl_Ohno]. Understanding the exchange interactions in these systems is challenging, with a range of theoretical models proposed to describe the various systems [@Coey_Venkatesan; @Dietl_Ohno; @Zunger_Raebiger; @Coey_Paul]. The understanding is further complicated by the possible existence of magnetic impurity phases distinct from the semiconducting host, as these can often be of small enough dimensions to escape conventional detection methods [@Ney; @Shinde_Venkatesan; @Chambers; @Dietl_Wu]. Nevertheless, numerous examples of DMS systems have been reported, with ferromagnetic transition temperatures ranging from a few kelvin to far above room temperature [@Sawicki_Bonanni; @Dhar_Ploog; @Cibert_Scalbert]. In the most well studied system, Mn-doped III-V semiconductors, the exchange mechanism is now reasonably well understood based on the modified Zener model of coupling mediated by carriers [@Dietl_Ohno]. By contrast, it is relatively rare to find intrinsically ferromagnetic semiconductors, where the ordered magnetic moments are provided directly by the host cations [@Nagaev; @Mauger_Godart; @Pappas_Fumagalli]. The most notable example is EuO [@Mauger_Godart], where the physics of the magnetic state in electron-doped samples remains controversial [@Monteiro_Langridge; @Liu_Tang]. The rare-earth nitride (REN) series, which are largely ionic with 3+ valence for the rare-earth and 3- for nitrogen, also contains such intrinsic ferromagnetic semiconductors, including GdN, DyN, and SmN [@Leuenberger_Hessler; @Granville_Trodahl; @Preston_Lambrecht; @Azeem_Kamba; @Meyer_Trodahl; @Larson_Schilfgaarde]. Europium nitride has also been demonstrated to be semiconducting [@Richter_Lambrecht], but EuN stands out amongst the RENs because the ground state of the Eu$^{3+}$ ion has configuration 4$f^6$ giving it a total angular momentum $J=0$, and thus it is non-magnetic [@Johannes_Pickett]. Accordingly there are no ferromagnetic compounds based on Eu$^{3+}$. However, trivalent Eu does possess a non-zero spin angular momentum quantum number $S=3$ which has led to the suggestion that it might support “hidden ferromagnetism” [@Johannes_Pickett]. Furthermore, the first excited state $J=1$ lies close in energy to the ground state, so Eu$^{3+}$ has a relatively strong van Vleck susceptibility [@VanVleck]. Thus, the magnetic properties of stoichiometric or doped EuN are of substantial interest. We have previously demonstrated that epitaxial EuN films display a dominant paramagnetic signal that is at odds with that expected for a collection of Eu$^{3+}$ ions [@Ruck_Meyer]. The origin of this signal was shown to be a concentration of a few percent of Eu$^{2+}$, most likely related to doping by nitrogen vacancies in the material [@Richter_Lambrecht]. Furthermore, x-ray magnetic circular dichroism (XMCD) at the Eu L-edges showed that there is a partial polarization of the Eu$^{3+}$ that follows the Eu$^{2+}$ polarization [@Ruck_Meyer]. It is of fundamental interest to investigate the evolution of the magnetism in EuN as the quantity of Eu$^{2+}$ increases, thereby increasing the possibility of interaction between the localized magnetic moments. Here we present such a study based on EuN films with Eu$^{2+}$ concentrations as high as 15-20%, and we show that such films are ferromagnetic at temperatures well above 100 K. XMCD results show that the Eu$^{2+}$ ions polarize the neighboring Eu$^{3+}$, showing that the ferromagnetism is not an artefact of an impurity phase and suggesting that Eu$^{3+}$ plays a role in the exchange mechanism. The 100-200 nm thick EuN films were grown onto substrates of either sapphire or GaN templates on sapphire by thermal evaporation of Eu in the presence of a flux of ionized nitrogen. In contrast to other rare-earth nitrides the use of an excited nitrogen source is essential for obtaining near-stoichiometric EuN films. The nitrogen partial pressure in the growth chamber was $3-5\times 10^{-4}$ mbar and the ions were accelerated through 125 V at a beam current of 0.37 mA. The films were grown at either room temperature or 680$^\circ$C, and were capped to prevent oxidation after growth using layers of either GaN or AlN. The films were characterized by reflection high-energy electron diffraction and x-ray diffraction (XRD), which showed that the 680$^\circ$C grown films are epitaxial with \[111\] orientation, while the room temperature grown films are polycrystalline with \[111\] texturing. There is no evidence in the XRD for any impurity phase, and all films show the expected lattice constant of 4.99 Å [@Richter_Lambrecht; @Klemm_Winkelman; @Brown_Clark]. As we will show below, the key difference between the 680$^\circ$C and the room temperature grown films is that the latter are more heavily doped and contain a substantially larger Eu$^{2+}$ concentration. We show representative data from films grown at the two temperatures, but the repeatability of the magnetization and transport results has been checked on additional samples grown under similar conditions. The magnetization of the films was measured using a Quantum Design MPMS SQUID magnetometer. Further investigation of the magnetic state of the films was made by XMCD carried out at beam line ID12 of the European Synchrotron Radiation Facility in Grenoble, France. Measurements were made at grazing incidence in the total fluorescence yield detection mode, with the magnetic field applied in the film plane. Electrical transport measurements were conducted in a Quantum Design Physical Properties Measurement System using a four terminal geometry with contacts made using pressed indium. ![(color online) Field cooled (FC) and zero field cooled (ZFC) temperature dependent magnetization of a room temperature grown EuN film (solid lines) measured in a field of 500 Oe. Also shown are the Eu$^{2+}$ and Eu$^{3+}$ XMCD amplitudes (solid and open circles), which both follow the measured magnetisation. Inset: Hysteresis loop measured at 5 K.[]{data-label="SQUID"}](FMEuN_magnetisation2.eps){width="9cm"} In Figure \[SQUID\] we show our main result, namely that the room temperature grown films are ferromagnetic with a Curie temperature near 120 K as evidenced by the sharp rise in the temperature dependent magnetization. A clear hysteresis is observed at low temperatures (Fig. \[SQUID\] inset) along with saturation of the magnetization at around 1.4 $\mu_B$ per Eu ion. Assuming the magnetic response is associated with the Eu$^{2+}$ component of the film we estimate a rather large divalent fraction corresponding to about 20% of the cations. A series of similar room temperature grown films all showed ferromagnetism, with Curie temperatures ranging from 100 to 120 K (estimated by extrapolating the steepest part of the magnetization curve back to zero). By contrast the 680$^\circ$C grown films display only a paramagnetic response whose magnitude is consistent with Eu$^{2+}$ concentrations of around 2-5%. ![(color online) (a) Eu L$_2$-edge x-ray absorption and XMCD at various temperatures from ferromagnetic EuN. The fit to the absorption spectrum (dashed colored lines) implies that about 15% of the Eu ions are in the 2+ charge state. Strong XMCD with similar temperature dependence is observed for both the Eu$^{2+}$ and Eu$^{3+}$ features. (b) Eu$^{3+}$ versus Eu$^{2+}$ polarization of the $5d$ electrons extracted from the XMCD spectra. The black symbols are from the ferromagnetic film in (a), the red symbols from the paramagnetic film reported in Ref. \[\]. The solid line is a guide to the eye.[]{data-label="XMCD"}](FMEuN_XMCD2.eps "fig:"){width="9cm"} ![(color online) (a) Eu L$_2$-edge x-ray absorption and XMCD at various temperatures from ferromagnetic EuN. The fit to the absorption spectrum (dashed colored lines) implies that about 15% of the Eu ions are in the 2+ charge state. Strong XMCD with similar temperature dependence is observed for both the Eu$^{2+}$ and Eu$^{3+}$ features. (b) Eu$^{3+}$ versus Eu$^{2+}$ polarization of the $5d$ electrons extracted from the XMCD spectra. The black symbols are from the ferromagnetic film in (a), the red symbols from the paramagnetic film reported in Ref. \[\]. The solid line is a guide to the eye.[]{data-label="XMCD"}](Eu2plusvs3plus_2.eps "fig:"){width="9cm"} To understand the origin of the ferromagnetism we have carried out XMCD on a ferromagnetic EuN film grown at room temperature. The L$_{2,3}$-edge XMCD involves the transition $2p^6 5d^0 \rightarrow 2p^5 5d^1$ so it interrogates the polarization of the $5d$ empty-state conduction band orbitals. The x-ray absorption spectrum at the L$_2$-edge shown in Figure \[XMCD\](a) shows a clear shoulder at 7615 eV superimposed on the usual Eu$^{3+}$ white line absorption centered at 7624 eV. Atomic multiplet calculations clearly identify the shoulder as originating from absorption by Eu$^{2+}$ ions [@Richter_Lambrecht; @Thole_Esteva]. This feature is substantially stronger than the corresponding shoulder seen in a paramagnetic epitaxial EuN film [@Ruck_Meyer], confirming the much larger Eu$^{2+}$ concentration in the room temperature grown films. The curve fitting of the 2+ and 3+ peaks shown in the figure implies a Eu$^{2+}$ concentration of around 15%, consistent within uncertainty with the value extracted above from the saturation magnetization. The corresponding XMCD spectra taken at various temperatures in a field of 3 T are also plotted in Fig. \[XMCD\](a). The strongest feature near 7615 eV is clearly associated with Eu$^{2+}$. The strength of this XMCD feature at the lowest temperature is roughly three times stronger than in the paramagnetic epitaxial films [@Ruck_Meyer], and its temperature dependence follows closely the measured magnetization as shown by the solid symbols in Fig. \[SQUID\] (the disagreement between the SQUID and XMCD amplitudes at 105 K is a result of the much higher measurement field used for XMCD). These observations confirm the origin of the ferromagnetism to be the Eu in the film. The remainder of the XMCD features are associated mostly with Eu$^{3+}$, and interestingly these also show a strong signature of the ferromagnetism. Similar to the Eu$^{2+}$ XMCD the amplitude of the 3+ signal is much larger than that seen in paramagnetic films at similar applied fields. Furthermore, rather than following a van Vleck temperature dependence, the Eu$^{3+}$ signal closely follows the Eu$^{2+}$ signal (open symbols in Fig. \[SQUID\]), implying that there is a strong exchange coupling between the Eu$^{2+}$ and Eu$^{3+}$ ions [@Matsuda_Wada]. This is further demonstrated in Figure \[XMCD\](b) which shows the Eu$^{3+}$ polarization plotted against the Eu$^{2+}$ polarization determined from the XMCD using the method described in Ref. \[\]. The red triangles represent data from the paramagnetic sample of Ref. \[\]. The black circles, representing the ferromagnetic sample, show much larger polarization for both species but they follow the same trend as the paramagnetic sample implying that the coupling between the Eu$^{2+}$ and Eu$^{3+}$ ions is of the same nature in each case, with the key difference simply being the concentration. We stress that the strong coupling between the Eu$^{2+}$ and Eu$^{3+}$ is compelling evidence that the ferromagnetic phase is not simply an impurity, such as electron-doped EuO [@Mauger_Godart; @Liu_Tang], but rather represents the response of the EuN matrix containing a large concentration of Eu$^{2+}$ ions. ![(color online) (a) Temperature dependent resistivity of ferromagnetic (black) and paramagnetic (red) EuN. (b) Hall resistance of ferromagnetic EuN, showing an anomalous Hall effect below the T$_C$ of 120 K. (c) Magnetoresistance of ferromagnetic EuN showing cusp-like behavior at low fields for temperatures below T$_C$.[]{data-label="Transport"}](EuNRvsT.eps "fig:"){width="8cm"} ![(color online) (a) Temperature dependent resistivity of ferromagnetic (black) and paramagnetic (red) EuN. (b) Hall resistance of ferromagnetic EuN, showing an anomalous Hall effect below the T$_C$ of 120 K. (c) Magnetoresistance of ferromagnetic EuN showing cusp-like behavior at low fields for temperatures below T$_C$.[]{data-label="Transport"}](FMEuN_Hall2.eps "fig:"){width="8cm"} ![(color online) (a) Temperature dependent resistivity of ferromagnetic (black) and paramagnetic (red) EuN. (b) Hall resistance of ferromagnetic EuN, showing an anomalous Hall effect below the T$_C$ of 120 K. (c) Magnetoresistance of ferromagnetic EuN showing cusp-like behavior at low fields for temperatures below T$_C$.[]{data-label="Transport"}](FMEuN_MR.eps "fig:"){width="8cm"} We have further investigated the source of the doping and the nature of the exchange mechanism that couples the Eu$^{2+}$ by measuring the transport properties of the films. Figure \[Transport\](a) shows the temperature dependent resistivity of a room temperature grown ferromagnetic film and a paramagnetic film grown at 680$^\circ$C. The paramagnetic film shows a metallic temperature dependence at high temperature, developing a negative temperature coefficient of resistance below about 60 K. The magnitude of the resistivity is rather high ($\approx11~\mathrm{m}\Omega$cm), consistent with the conclusion that these EuN films are semiconductors doped to degeneracy by a high concentration of nitrogen vacancies. This is further supported by Hall effect measurements that give a carrier concentration at room temperature of $-8\times10^{20}$ cm$^{-3}$ (i.e., the carriers are electrons). The origin of the upturn in the resistivity below 60 K is uncertain. Magnetic scattering from the Eu$^{2+}$ in the film could lead to a Kondo effect [@Kondo; @He_Wang], and indeed the resistivity does follow the expected logarithmic temperature dependence. The room temperature grown ferromagnetic film shows a qualitatively similar temperature dependent resistivity, although the magnitude is substantially smaller and the carrier concentration is larger ($4\times10^{21}$ cm$^{-3}$). The ratio of carrier concentration between the two films is similar to the ratio of Eu$^{2+}$ content, indicating a link between the two quantities. The low temperature resistivity upturn occurs at higher temperature in the ferromagnetic sample than the paramagnetic sample, and it constitutes a larger fractional change in resistivity in the film with more Eu$^{2+}$. However, the ferromagnetic film has a larger mobility (1.5 cm$^2$V$^{-1}$s$^{-1}$) than the paramagnetic film (0.7 cm$^2$V$^{-1}$s$^{-1}$), supporting the conclusion that the upturn is related to magnetic scattering rather than weak localization [@Altshuler_Larkin]. There is no sharp anomaly at the Curie temperature in the ferromagnetic sample, as is often observed in ferromagnets where it can be caused either by scattering from magnetic fluctuations [@deGennes_Friedel; @Fisher_Langer; @He_Wang] or by a change in carrier concentration as the sample enters the ferromagnetic state [@Mauger_Godart; @Leuenberger_Hessler; @Granville_Trodahl]. On the other hand evidence for the magnetic ordering is clearly seen in the form of an anomalous Hall effect that sets in below $T_C$ \[Fig. \[Transport\](b)\], the strength of which is enhanced by the relatively large resistivity in these films. Evidence for coupling between the magnetic order and the electrical transport is also present in the magnetoresistance presented in Fig. \[Transport\](c). It shows a negative parabolic behavior above $T_C$, characteristic of scattering from uncorrelated magnetic impurities [@BealMonod_Weiner], with an additional positive contribution evident at low field [@Dietl_Wu]. These features disappear below $T_C$ to be replaced by a sharp negative cusp at low fields followed by a near-linear high-field behavior, similar to the behavior observed in other ferromagnets [@Csontos_Mihaly]. By contrast, the magnetoresistance of the paramagnetic sample is parabolic down to low temperature with a small cusp observable only below 10 K. Similarly, the paramagnetic films show evidence for an anomalous Hall effect only below 10 K where the Eu$^{2+}$ becomes strongly polarized in the large measurement fields. Based on the evidence presented above and previous calculations and measurements of the electronic structure of stoichiometric EuN we propose a simple model for the formation of Eu$^{2+}$ in EuN. The underlying band structure is semiconducting, but with the Eu$^{2+}$ $4f^7$ ($^8S$) level lying very close to the bottom of the conduction band [@Richter_Lambrecht]. The presence of large quantities of nitrogen vacancies shifts the Fermi level into the conduction band, and at carrier concentrations above $\sim10^{20}$ cm$^{-3}$ it approaches the $^8S$ level that thus becomes populated. This is similar to a proposed model of the electronic structure of sub-stoichiometric YbN [@Degiorgi_Wachter], although there the Yb$^{2+}$ is nonmagnetic and there is no magnetic ordering. Given the above model it is interesting to seek evidence for an enhancement of the effective mass in the heavily doped samples where the Fermi level approaches the Eu $^8S$ level. To investigate this possibility we write the resistivity as the sum of a phonon contribution ($\rho_{\mathrm{ph}}$) and a contribution from disorder scattering involving both lattice defects and magnetic inhomogeneity ($\rho_{\mathrm{dis}}$): $$\label{resmod} \rho(T)=\rho_{\mathrm{dis}}+\rho_{\mathrm{ph}}=\frac{m^*}{ne^2\tau_{\mathrm{dis}}}+\frac{m^*}{ne^2\tau_{\mathrm{ph}}},$$ where $m^*$ is the carrier effective mass, $n$ is the carrier concentration, $e$ is the electron’s charge, $\tau_{\mathrm{ph}}$ is the phonon scattering time, and $\tau_{\mathrm{dis}}$ is the combined magnetic and quenched disorder scattering time. At high temperature $\tau_{\mathrm{dis}}$ is temperature independent and the phonon scattering rate $\tau_{\mathrm{ph}}^{-1}=cT$ with $c$ a constant, so we can express the effective mass as $$m^* = \frac{1}{ne^2c}\frac{d\rho}{dT}.$$ Assuming the phonon scattering rate, and hence the constant $c$, is the same in all samples, and using the measured resistivity slopes and carrier concentrations, we can seek variations in $m^*$ between samples. Doing so we find that the paramagnetic film in Fig. \[Transport\](a) has a larger $m^*$ than the ferromagnetic film by a factor of nearly three. This is the same within the uncertainty among all of the films, and we see no evidence for a systematic variation in $m^*$ with Eu$^{2+}$ concentration. Once again this is consistent with conclusions obtained from YbN [@Degiorgi_Wachter]. Next, we consider the possible exchange interactions present in the films. The carrier concentration in the ferromagnetic samples is larger than in the paramagnetic films, suggesting that carrier mediated mechanisms may play an important role. Indeed, the conduction band states are formed primarily from Eu 5$d$ orbitals, and the XMCD results show a very clear polarization of these states. This is similar to the polarization of Eu$^{3+}$ seen in the mixed valence compounds EuNi$_2$P$_2$ and EuNi$_2$(Si$_{0.18}$Ge0.82)$_2$, although there the polarization was induced by a very large applied field [@Matsuda_Wada]. At the large Eu$^{2+}$ concentrations where ferromagnetism occurs there will be many nearest-neighbor Eu$^{2+}$ ions on the cation sublattice, allowing for short-ranged exchange interactions. This will naturally lead to a percolating type of magnetic ordering nucleating at regions of high Eu$^{2+}$ density, and this percolating nature might explain the lack of a cusp at T$_C$ in the temperature dependent resistivity. Finally, we note that the underlying matrix of Eu$^{3+}$ ions is also polarizable due to the small energy gap to the $J=1$ excited state, which could lead to a Van Vleck type contribution to the exchange interaction as has been reported for Cr-doped Bi$_2$Sb$_3$ [@Yu_Fang]. XMCD measurements at the Eu M-edge would be of interest to probe the 4$f$ levels directly. In summary, we have shown that EuN with a large fraction of the Eu ions in the 2+ charge state is ferromagnetic at temperatures as high as 120 K. It thus represents a novel dilute magnetic semiconducting system, with the magnetism contributed largely by the Eu$^{2+}$, but where the host lattice based on Eu$^{3+}$ is also polarizable. The concentration of Eu$^{2+}$ ions is correlated with the charge carrier concentration, allowing us to propose a simple model for the formation of Eu$^{2+}$. The large concentration of Eu$^{2+}$ in the ferromagnetic samples requires that many are nearest neighbors on the cation lattice allowing for short-ranged exchange interactions which may be supported by interactions involving the charge carriers and also even the polarizable Eu$^{3+}$ background. The relatively simple physical structure of this system may make it an attractive testing ground for theories of exchange interactions in diluted magnetic systems. We acknowledge funding from the NZ Foundation for Research, Science, and Technology (VICX0808) and the Marsden Fund (08-VUW-030). The MacDiarmid Institute is supported by the New Zealand Centres of Research Excellence Fund. We are grateful to Walter Lambrecht for helpful discussions. [41]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , , , ****, (). , , , , , ****, (). , , , ****, (). , , , ****, (). , , , , , ****, (). , ****, (). , , , , , , , , , ****, (). , ****, (). , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , ****, (). , ** (, ), vol. of **, pp. . , ** (, ). , ****, (). , , , , , , , , , ****, (). , , , , , , , , ****, (). , ****, (). , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , ** (, ). , , , , , , , , , , ****, (). , ****, (). , ****, (). , , , , , , ****, (). , , , , , , , , , ****, (). , ****, (). , , , , , , ****, (). , , , , ** (, ), . , ****, (). , ****, (). , ****, (). , , , , , , , ****, (). , , , ****, (). , , , , , , ****, ().

--- author: - |     [*Lane A. Hemaspaandra*]{} [^1]\     Dept. of Computer Science\     University of Rochester\     Rochester, NY 14627, USA\ \ - |     [*Zhigen Jiang*]{} [^2]\     Institute of Software\     Chinese Academy of Sciences\     Beijing 100080, China\ \ - | [*Jörg Rothe*]{} [^3]\ Institut für Informatik\ Friedrich-Schiller-Universität Jena\ 07743 Jena, Germany\ - | [*Osamu Watanabe*]{} [^4]\ Dept. of Computer Science\ Tokyo Institute of Technology\ Tokyo 152, Japan\ bibliography: - 'main.bib' title: ' Boolean Operations, Joins, and the Extended Low Hierarchy ' --- =by 60 =by -60 by =3000=10000 = currsize = currsize [**Abstract**]{} > We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy [@bal-boo-sch:j:low] than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, $\mbox{EL}_2$, yet their join [*is*]{} in $\mbox{EL}_2$. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of $\mbox{EL}_2$ and prove that $\mbox{EL}_2$ is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) $\mbox{EL}_2$ lower bounds for certain notions generalizing Selman’s P-selectivity [@sel:j:pselective-tally], which may be regarded as an interesting result in its own right. Introduction ============ The low hierarchy [@sch:j:low] provides a yardstick to measure the complexity of sets that are known to be in NP but that are seemingly neither in P nor NP-complete. In order to extend this classification beyond NP, the extended low hierarchy [@bal-boo-sch:j:low] has been introduced (see the surveys [@koe:c:survey-low; @hem:j:yardstick]). An informal way of describing the intuitive nature of these hierarchies might be the following: A set $A$ that is placed in the $k$th level of the low or the extended low hierarchy contains no more information than the empty set relative to the computation of a $\Sigma_k^p$ machine (see [@mey-sto:c:reg-exp-needs-exp-space; @sto:j:poly] for the definition of the $\Sigma$ levels of the polynomial hierarchy), either because $A$ is so chaotically organized that a $\Sigma_k^p$ machine is not able to extract useful information from $A$, or because $A$ is so simple that it has no useful information to offer a $\Sigma_k^p$ machine.[^5] The low and extended low hierarchies have been very thoroughly investigated in many papers (see, e.g., [@sch:j:low; @ko-sch:j:circuit-low; @bal-boo-sch:j:low; @sch:j:gi; @sch:j:pr-low; @ko:j:separating-low-high; @all-hem:j:low; @ami-bei-gas:j-subm:uni; @koe:j:locating; @she-lon:j:extended-low; @hem-nai-ogi-sel:j:refinements]). In light of the informal intuition given above—that classifying the level in the extended low hierarchy of a problem or a class gives insight into the amount of polynomial-hierarchy computational power needed to make access to the problem or the class redundant—one main motivation for the study of the extended low hierarchy is to understand which natural complexity classes and problems easily extend the power of the polynomial hierarchy and which do not. Among the important natural classes and problems that have been carefully classified in these terms are the Graph Isomorphism Problem (which in fact is known to be low), bounded probabilistic polynomial time (BPP), approximate polynomial time (APT), the class of complements of sets having Arthur-Merlin games (coAM), the class of sparse and co-sparse sets, the P-selective sets, and the class of sets having polynomial-size circuits (P/poly). Another motivation for the study of the low and extended low hierarchies is to relate their properties to other complexity-theoretic concepts. For instance, Schöning showed that the existence of an NP-complete set (under any “reasonable” reducibility) in the low hierarchy implies a collapse of the polynomial hierarchy [@sch:j:low]. Among the most important recent results about extended lowness are Sheu and Long’s result that the extended low hierarchy is a strictly infinite hierarchy [@she-lon:j:extended-low] and Köbler’s optimal location of P/poly in the extended low hierarchy [@koe:j:locating]. In this note, we seek to further explore the structure of the extended low hierarchy by studying its interactions with such operations as the join. In particular, we prove properties of $\mbox{EL}_2$ with regard to its interaction with the join and with Boolean operations. Our results add to the body of evidence that extended lowness does not provide a natural, intuitive measure of complexity. In light of the many ways in which extended lowness captures certain concepts of low information content (such as all sparse sets and certain reduction closures of the sparse sets—e.g., the Turing closure of the class of sparse sets, which is known to be equal to P/poly) as well as certain concepts of “almost” feasible computation (such as BPP, APT, and P-selectivity, etc.), it might be tempting to assume that extended lowness would provide a reasonable measure of complexity in the sense that a problem’s property of being extended low indicates that this problem is of “low” complexity. However, in Section \[sec:join\], we will prove that [*the join operator can lower difficulty as measured in terms of extended lowness*]{}: There exist sets that are not in $\mbox{EL}_2$, yet their join is in $\mbox{EL}_2$. Since in a strong intuitive sense the join does not lower complexity, our result suggests that, if one’s intuition about complexity is—as is natural—based on reductions, then the extended low hierarchy is not a natural measure of complexity. Rather, it is a measure that is related to the difficulty of information extraction, and it is in flavor quite orthogonal to more traditional notions of complexity. That is, our result sheds light on the orthogonality of “complexity in terms of reductions” versus “difficulty in terms of non-extended-lowness.” In fact, our result is possible only since the second level of the extended low hierarchy is not closed under polynomial-time many-one reductions (this non-closure is known, see [@all-hem:j:low], and it also follows as a corollary of our result). In Section \[sec:boolean\], we apply the technique developed in the preceding section to prove that the second level of the extended low hierarchy is not closed under the Boolean operations intersection, union, exclusive-or, and equivalence. Our result will follow from the proof of another result, which establishes the first known (and optimal) EL$_2$ lower bounds for generalized selectivity-like classes (that generalize Selman’s class of P-selective sets [@sel:j:pselective-tally], denoted P-Sel) such as the polynomial-time membership-comparable sets introduced by Ogihara [@ogi:j:comparable] and the multi-selective sets introduced by Hemaspaandra et al. [@hem-jia-rot-wat:t:multiselectivity]. These results sharply contrast with the known result that all P-selective sets are in EL$_2$ and they are thus interesting in their own right. Extended Lowness and the Join Operator {#sec:join} ====================================== The low hierarchy and the extended low hierarchy are defined as follows. \[def:low\] 1. [[@sch:j:low]]{} For each $k\geq 1$, define . 2. [[@bal-boo-sch:j:low]]{}For each $k\geq 2$, define EL$_k {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ L \mid \Sigma_k^{p,\,L} = \Sigma_{k-1}^{p,\, L \oplus \mbox{\scriptsize SAT}}\}$, where SAT is the set of all satisfiable Boolean formulas. For sets $A$ and $B$, their join, $A\oplus B$, is $\{0x \mid x\in A\}\cup\{1x \mid x\in B\}$. Theorem \[thm:el2-join\] below establishes that the join operator can lower the difficulty measured in terms of extended lowness. At first glance, this might seem paradoxical. After all, every set that $\leq_{m}^{p}$-reduces[^6] to a set $A$ or $B$ also reduces to $A\oplus B$, and thus intuition strongly suggests that $A\oplus B$ must be at least as hard as $A$ and $B$, as most complexity lower bounds (e.g., NP-hardness) are defined in terms of reductions. However, extended lowness merely measures the opacity of a set’s internal organization, and thus Theorem \[thm:el2-join\] is not paradoxical. Rather, Theorem \[thm:el2-join\] highlights the orthogonality of “complexity in terms of reductions” and “difficulty in terms of non-extended-lowness.” Indeed, note Corollary \[cor:el2-join\], which was first observed by Allender and Hemaspaandra (then Hemachandra) [@all-hem:j:low]. We interpret Theorem \[thm:el2-join\] as evidence that extended lowness is not an appropriate, natural complexity measure with regard to even very simple operations such as the join. \[thm:el2-join\] There exist sets $A$ and $B$ such that $A \not\in \mbox{\rm EL}_{2}$ and $B \not\in \mbox{\rm EL}_{2}$, and yet . Lemma \[lem:el2-join\] below will be used in the upcoming proof of Theorem \[thm:el2-join\]. First, we fix some notations. Fix the alphabet $\Sigma = \{ 0,1 \}$. Let $\Sigma^*$ denote the set of all strings over $\Sigma$. For each set $L {\subseteq}\Sigma^*$, $L^{=n}$ ($L^{\leq n}$) is the set of all strings in $L$ having length $n$ (less than or equal to $n$), and $\|L\|$ denotes the cardinality of $L$. Let $\Sigma^{n}$ be a shorthand for $(\Sigma^*)^{=n}$. Let $\leq_{\mbox{\protect\scriptsize lex}}$ denote the standard quasi-lexicographical ordering on ${\mbox{$\Sigma^\ast$}}$. The census function of a set $L$ is defined by $\mbox{\em census\/}_{L}(0^n) = \| L^{\leq n} \|$. $L$ is said to be sparse if there is a polynomial $p$ such that for every $n$, $\mbox{\em census\/}_{L}(0^n) \leq p(n)$. Let SPARSE denote the class of all sparse sets. For each class ${\hspace*{1pt}}{\cal C}$ of sets over ${\rm \Sigma}$, define $\mbox{co}{\hspace*{1pt}}{\cal C} {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{L\,|\,\overline{L}\in {\cal C}\}$. Let ${{\rm I\!N}}$ denote the set of non-negative integers. To encode a pair of integers, we use a one-one, onto, polynomial-time computable pairing function, ${\langle}{\cdot},{\cdot}{\rangle}: {{\rm I\!N}}\times {{\rm I\!N}}\rightarrow {{\rm I\!N}}$, that has polynomial-time computable inverses. FP denotes the class of polynomial-time computable functions. We shall use the shorthand NPTM to refer to “nondeterministic polynomial-time Turing machine.” For an NPTM $M$ (an NPTM $M$ and a set $A$, respectively), $L(M)$ ($L(M^A)$) denotes the set of strings accepted by $M$ (relative to $A$). \[lem:el2-join\] If $F$ is a sparse set and $\mbox{\em census\/}_{F} \in {\mbox{\rm FP}}^{F\oplus \mbox{\scriptsize SAT}}$, then $F \in \mbox{\rm EL}_{2}$. [**Proof.**]{} Let $L \in {\mbox{\rm NP}}^{\mbox{\scriptsize NP}^{F}}$ via NPTMs $N_1$ and $N_2$, i.e., $L = L(N_{1}^{L(N_{2}^{F})})$. Let $q(n)$ be a polynomial bounding the length of all queries that can be asked in the run of $N_{1}^{L(N_{2}^{F})}$ on inputs of length $n$. Below we describe an NPTM $N$ with oracle $F\oplus \mbox{SAT}$: On input $x$, $|x| = n$, $N$ first computes $\mbox{\em census\/}_{F}(0^i)$ for each relevant length $i \leq q(n)$, and then guesses all sparse sets up to length $q(n)$. Knowing the exact census of $F$, $N$ can use the $F$ part of its oracle to verify whether the guess for $F^{\leq q(n)}$ is correct, and rejects on all incorrect paths. On the correct path, $N$ uses itself, the SAT part of its oracle, and the correctly guessed set $F^{\leq q(n)}$ to simulate the computation of $N_{1}^{L(N_{2}^{F})}$ on input $x$. Clearly, $L(N^{F\oplus \mbox{\scriptsize SAT}}) = L$. Thus, ${\mbox{\rm NP}}^{\mbox{\scriptsize NP}^{F}} {\subseteq}{\mbox{\rm NP}}^{F\oplus \mbox{\scriptsize SAT}}$, i.e., $F \in \mbox{EL}_{2}$. $\Box$ [**Proof of Theorem \[thm:el2-join\].**]{} $A {\stackrel{\mbox{\protect\scriptsize df}}{=}}\bigcup_{i\geq 0} A_i$ and $B {\stackrel{\mbox{\protect\scriptsize df}}{=}}\bigcup_{i\geq 0} B_i$ are constructed in stages. In order to show $A \not\in \mbox{EL}_{2}$ and $B \not\in \mbox{EL}_{2}$ it suffices to ensure in the construction that ${\mbox{\rm NP}}^A \not{\subseteq}{\mbox{\rm coNP}}^{A \oplus \mbox{\scriptsize SAT}}$ and ${\mbox{\rm NP}}^B \not{\subseteq}{\mbox{\rm coNP}}^{B \oplus \mbox{\protect\scriptsize SAT}}$ (and thus, and ). Define function $t$ inductively by $t(0) {\stackrel{\mbox{\protect\scriptsize df}}{=}}2$ and $t(i) {\stackrel{\mbox{\protect\scriptsize df}}{=}}2^{2^{2^{t(i-1)}}}$ for $i \geq 1$. Let $\{ N_i \}_{i\geq 1}$ be a fixed enumeration of all coNP oracle machines having the property that the runtime of each $N_i$ is independent of the oracle and each machine appears infinitely often in the enumeration. Define $$L_A {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ 0^{t(i)}\mid (\exists j \geq 1 )\, [ i = {\langle}0,j {\rangle}\,\wedge\, \| A \cap \Sigma^{t(i)} \| \geq 1 ] \},$$ $$L_B {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ 0^{t(i)}\mid (\exists j \geq 1 )\, [ i = {\langle}1,j {\rangle}\,\wedge\, \| B \cap \Sigma^{t(i)} \| \geq 1 ] \}.$$ Clearly, $L_A \in {\mbox{\rm NP}}^A$ and $L_B \in {\mbox{\rm NP}}^B$. In stage $i$ of the construction, at most one string of length $t(i)$ will be added to $A$ and at most one string of length $t(i)$ will be added to $B$ in order (1) : to ensure $L(N_j^{A_i \oplus \mbox{\scriptsize SAT}}) \neq L_A$ if $i = {\langle}0,j{\rangle}$ (or $L(N_j^{B_i \oplus \mbox{\scriptsize SAT}}) \neq L_B$, respectively, if ), and (2) : to encode an easy to find string into $A$ if $i = {\langle}1,j{\rangle}$ (or into $B$ if $i = {\langle}0,j{\rangle}$) indicating whether or not some string has been added to $B$ (or to $A$) in (1). Let $A_{i-1}$ and $B_{i-1}$ be the content of $A$ and $B$ prior to stage $i$. Initially, let $A_0 = B_0 = \emptyset$. Stage $i$ is as follows: First assume $i = {\langle}0,j {\rangle}$ for some $j\geq 1$. If it is the case that no path of $N_j^{A_{i-1} \oplus \mbox{\scriptsize SAT}}(0^{t(i)})$ can query all strings in $\Sigma^{t(i)} - \{0^{t(i)}\}$ and $N_j^{A_{i-1} \oplus \mbox{\scriptsize SAT}}(0^{t(i)})$ cannot query any string of length $t(i+1)$ (otherwise, just skip this stage—we will argue later that the diagonalization still works properly), then simulate $N_j^{A_{i-1} \oplus \mbox{\scriptsize SAT}}$ on input $0^{t(i)}$. If it rejects (in the sense of coNP, i.e., if it has one or more rejecting computation paths), then fix some rejecting path and let $w_i$ be the smallest string in $\Sigma^{t(i)} - \{0^{t(i)}\}$ that is not queried along this path, and set and . Otherwise (i.e., if $0^{t(i)} \in L(N_j^{A_{i-1} \oplus \mbox{\scriptsize SAT}})$), set and . The case of $i = {\langle}1,j {\rangle}$ is analogous: just exchange $A$ and $B$. This completes the construction of stage $i$. Since each machine $N_i$ appears infinitely often in our enumeration and as the $t(i)$ are strictly increasing, it is clear that for only a finite number of the $N_{i_1}, N_{i_2}, \ldots $ that are the same machine as $N_i$ can it happen that stage $i_k$ must be skipped (in order to ensure that $w_{i_k}$, if needed to diagonalize against $N_{i_k}$, indeed exists, or that the construction stages do not interfere with each other), and thus each machine $N_i$ is diagonalized against eventually. This proves that $A \not\in \mbox{EL}_{2}$ and $B \not\in \mbox{EL}_{2}$. Now observe that $A\oplus B$ is sparse and that $\mbox{\em census\/}_{A\oplus B} \in {\mbox{\rm FP}}^{A\oplus B}$. Indeed, $$\mbox{\em census\/}_{A\oplus B}(0^n) = 2( \| A \cap \{ 0, 00, \ldots , 0^{n-1} \} \| + \| B \cap \{ 0, 00, \ldots , 0^{n-1} \} \|).$$ Thus, by Lemma \[lem:el2-join\], $A\oplus B \in \mbox{EL}_{2}$. $\Box$ \[cor:el2-join\] [[@all-hem:j:low]]{} $\mbox{\rm EL}_{2}$ is not closed under $\leq_{m}^{p}$-reductions. In contrast to the extended low hierarchy, every level of the low hierarchy within NP is clearly closed under $\leq_{m}^{p}$-reductions. Thus, the low hierarchy analog of Theorem \[thm:el2-join\] cannot hold. $(\forall k \geq 0)\, (\forall A,B)\, [(A \not\in \mbox{\rm Low}_{k} \,\vee\, B \not\in \mbox{\rm Low}_{k}) {\ \Longrightarrow \ }A\oplus B \not\in \mbox{\rm Low}_{k} ]$. [**Proof.**]{} Assume $A\oplus B \in \mbox{Low}_{k}$. Since for all sets $A$ and $B$, $A \leq_{m}^{p} A\oplus B$ and $B \leq_{m}^{p} A\oplus B$, the closure of $\mbox{Low}_{k}$ under $\leq_{m}^{p}$-reductions implies that both $A$ and $B$ are in $\mbox{Low}_{k}$. $\Box$ One of the most interesting open questions related to the results presented in this note is whether the join operator also can [ *raise*]{} the difficulty measured in terms of extended lowness. That is, do there exist sets $A$ and $B$ such that and , and yet for, e.g., $k=2$? Or is the second level of the extended low hierarchy (and more generally, are [*all*]{} levels of this hierarchy) closed under join? Regarding potential generalizations of our result, we conjecture that Theorem \[thm:el2-join\] can be generalized to higher levels of the extended low hierarchy. Such a result, to be sure, would probably require some new technique such as a clever modification of the lower-bound technique for constant-depth Boolean circuits developed by Yao, H[å]{}stad, and Ko (see, e.g., [@has:j:circuits; @ko:j:separating-low-high]). EL[$_2$]{} is not Closed Under Certain Boolean Connectives {#sec:boolean} ========================================================== In this section, we will prove that the second level of the extended low hierarchy is [*not*]{} closed under the Boolean connectives union, intersection, exclusive-or, or equivalence. We will do so by combining the technique of the previous section with standard techniques of constructing P-selective sets. To this end, we first seek to improve the known EL$_2$ lower bounds of P/poly, the well-studied class of sets having polynomial-size circuits [@kar-lip:c:nonuniform]. To wit, we will show that certain generalizations of the class of P-selective sets, though still contained in P/poly [@ogi:j:comparable; @hem-jia-rot-wat:t:multiselectivity], are not contained in EL$_2$. As interesting as this result may be in its own right, its proof will even provide us with the means required to show the above-mentioned main result of this section: EL$_2$ is not closed under certain Boolean connectives (and indeed P-selective sets can be used to witness the non-closure). This extends the main result of Hemaspaandra and Jiang [@hem-jia:j:psel], namely that P-Sel is not closed under those Boolean connectives. Let us first recall the following generalizations of Selman’s P-selectivity. Ogihara introduced the P-membership comparable sets [@ogi:j:comparable] and the present paper’s authors ([@hem-jia-rot-wat:t:multiselectivity], see also [@rot:phd]) introduced the notion of multi-selectivity as defined in Definition \[def:multisel\]. \[def:p-mc\] [@ogi:j:comparable] Fix a positive integer $k$. A function $f$ is called a $k$-membership comparing function for a set $A$ if and only if for every $w_1,\ldots ,w_m$ with $m \geq k$, $$\begin{aligned} f(w_1,\ldots ,w_m)\in \{0,1\}^m & \mbox{and} & (\chi_A(w_1),\ldots ,\chi_A(w_m)) \neq f(w_1,\ldots ,w_m),\end{aligned}$$ where $\chi_A$ denotes the characteristic function of $A$. If in addition $f \in {\mbox{\rm FP}}$, $A$ is said to be polynomial-time $k$-membership comparable. Let P-mc$(k)$ denote the class of all polynomial-time $k$-membership comparable sets. We can equivalently (i.e., without changing the class) require in the definition that must hold only if the inputs happen to be [*distinct*]{}. This is true because if there are $r$ and $t$ with $r \neq t$ and $w_r = w_t$, then $f$ simply outputs a length $m$ string having a “0” at position $r$ and a “1” at position $t$. \[def:multisel\] Fix a positive integer $k$. Given a set $A$, a function $f \in {\mbox{\rm FP}}$ is said to be an $\mbox{\rm S}(k)$-selector for $A$ if and only if $f$ satisfies the following property: For each set of distinct input strings $y_1,\ldots ,y_n$, 1. $f(y_1,\ldots ,y_n) \in \{ y_1,\ldots ,y_n\}$, and 2. if $\| A \cap \{ y_1,\ldots ,y_n\} \| \geq k$, then $f(y_1,\ldots ,y_n) \in A$. The class of sets having an $\mbox{\rm S}(k)$-selector is denoted by $\mbox{\rm S}(k)$. It is easy to see that $\mbox{P-mc}(1) = {\mbox{\rm P}}$ and . Furthermore, though the hierarchies $\bigcup_{k}\mbox{P-mc}(k)$ and $\bigcup_{k}\mbox{S}(k)$ are properly infinite, they both are still contained in P/poly [@ogi:j:comparable; @hem-jia-rot-wat:t:multiselectivity]. Among a number of other results, all the relations between the classes $\mbox{P-mc}(j)$ and $\mbox{S}(k)$ are completely established in Hemaspaandra et al. [@hem-jia-rot-wat:t:multiselectivity]. These relations are stated in Lemma \[lem:relations\] below, as they’ll be referred to in the upcoming proof of Theorem \[thm:s2-el2\]. \[lem:relations\] [@hem-jia-rot-wat:t:multiselectivity] 1. $\mbox{P-mc}(2) \not{\subseteq}\bigcup_{k \geq 1}\mbox{S}(k)$. 2. For each $k\geq 1$, $\mbox{S}(k) \subset \mbox{P-mc}(k+1)$ and $\mbox{S}(k) \not{\subseteq}\mbox{P-mc}(k)$.[^7] The following result establishes a structural difference between Selman’s P-selectivity and the generalized selectivity introduced above: Though clearly  [@ami-bei-gas:j-subm:uni] and , we show that there are sets (indeed, sparse sets) in $\mbox{S}(2) \cap \mbox{P-mc}(2)$ that are not in $\mbox{EL}_{2}$. Previously, Allender and Hemaspaandra [@all-hem:j:low] have shown that  (and indeed SPARSE and coSPARSE) is not contained in $\mbox{EL}_{2}$. Theorem \[thm:s2-el2\] and Corollary \[cor:s2-el2\], however, extend this result and give the first known (and optimal) $\mbox{EL}_{2}$ lower bounds for generalized selectivity-like classes. \[thm:s2-el2\] $\mbox{\rm SPARSE} \cap \mbox{\rm S}(2) \cap \mbox{\rm P-mc}(2) \not{\subseteq}\mbox{\rm EL}_{2}$. [**Proof.**]{} Let $t$ be the function defined in the proof of Theorem \[thm:el2-join\] that gives triple-exponentially spaced gaps. Let $T_k {\stackrel{\mbox{\protect\scriptsize df}}{=}}\Sigma^{t(k)}$, for $k \geq 0$, and $T {\stackrel{\mbox{\protect\scriptsize df}}{=}}\bigcup_{k \geq 0} T_k$. Let EE\[ind:ee\] be defined as $\bigcup_{c\ge 0} \mbox{DTIME}[2^{c2^{n}}]$. We will construct a set $B$ such that = currsize (a) : $B {\subseteq}T$, (b) : $B\in \mbox{EE}$, (c) : $\| B \cap T_k \| \leq 1$ for each $k \geq 0$, and (d) : $B \not\in \mbox{EL}_{2}$. Note that it follows from (a), (b), and (c) that $B$ is a sparse set in $\mbox{S}(2)$. Indeed, any input to the $\mbox{S}(2)$-selector for $B$ that is not in $T$ (which can easily be checked) is not in $B$ by (a) and may thus be ignored. If all inputs that are in $T$ are in the same $T_k$ then, by (c), the $\mbox{S}(2)$-promise (that $B$ contains at least two of the inputs) is never satisfied, and the selector may thus output an arbitrary input. On the other hand, if the inputs that are in $T$ fall in more than one $T_k$, then for all inputs of length smaller than the maximum length, it can be decided by brute force whether or not they belong to $B$—this is possible, as $B \in \mbox{EE}$ and the $T_k$ are triple-exponentially spaced. From these comments, the action of the $\mbox{S}(2)$-selector is clear. By Lemma \[lem:relations\], $B$ is thus in $\mbox{P-mc}(k)$ for each $k \geq 3$. But since $\mbox{S}(2)$ and $\mbox{P-mc}(2)$ are incomparable (again by Lemma \[lem:relations\]), we still must argue that . Again, this follows from (a), (b), and (c), since for any fixed two inputs, $u$ and $v$, if they are of different lengths, then the smaller one can be solved by brute force; and if they have the same length, then it is impossible by (c) that . In each case, one out of the four possibilities for the membership of $u$ and $v$ in $B$ can be excluded in polynomial time. Hence, $B \in \mbox{P-mc}(2)$. For proving (d), we will construct $B$ such that ${\mbox{\rm NP}}^B \not{\subseteq}{\mbox{\rm coNP}}^{B \oplus \mbox{\protect\scriptsize SAT}}$ (which clearly implies that ${\mbox{\rm NP}}^{\mbox{\protect\scriptsize NP}^{B}} \not{\subseteq}{\mbox{\rm NP}}^{B \oplus \mbox{\protect\scriptsize SAT}}$). Define $$L_B {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ 0^n \,|\, (\exists x)\, [ |x| = n\ \wedge\ x \in B] \}.$$ Clearly, $L_B \in {\mbox{\rm NP}}^B$. As in the proof of Theorem \[thm:el2-join\], let $\{ N_i \}_{i\geq 1}$ be a standard enumeration of all  oracle machines satisfying the condition that the runtime of each $N_i$ is independent of the oracle and each machine is repeated infinitely often in the enumeration. Let $p_i$ be the polynomial bound on the runtime of $N_i$. The set $B {\stackrel{\mbox{\protect\scriptsize df}}{=}}\bigcup_{i\geq 0} B_i$ is constructed in stages. In stage $i$, at most one string of length $n_i$ will be added to $B$, and $B_{i-1}$ will have previously been set to the content of $B$ up to stage $i$. Initially, $B_0 = \emptyset$ and $n_0 = 0$. Stage $i > 0$ is as follows: Let $n_i$ be the smallest number such that (i) $n_i > n_{i-1}$, (ii) $n_i = t(k)$ for some $k$, and (iii) $2^{n_i} > p_i(n_i)$. Simulate $N_{i}^{B_{i-1} \oplus \mbox{\protect\scriptsize SAT}}(0^{n_i})$. Case 1: : If $N_{i}^{B_{i-1} \oplus \mbox{\protect\scriptsize SAT}}(0^{n_i})$ rejects (in the sense of coNP, i.e., if there are one or more rejecting computation paths), then fix some rejecting path and let $w_i$ be the smallest string of length $n_i$ that is not queried along this path. Note that, by our choice of $n_i$, such a string $w_i$, if needed, must always exist. Set . Case 2: : If $0^{n_i} \in L(N_{i}^{B_{i-1} \oplus \mbox{\protect\scriptsize SAT}})$, then set . Case 3: : If the simulation of $N_{i}^{B_{i-1} \oplus \mbox{\protect\scriptsize SAT}}$ on input $0^{n_i}$ fails to be completed in double exponential (say, $2^{100 \cdot 2^{n_i}}$ steps) time (for example, because $N_i$ is huge in size relative to $n_i$), then abort the simulation and set . This completes the construction of stage $i$. Since we have chosen an enumeration such that the same machine as $N_i$ appears infinitely often and as the $n_i$ are strictly increasing, it is clear that for only a finite number of the $N_{i_1}, N_{i_2}, \ldots $ that are the same machine as $N_i$ can Case 3 occur (and thus $N_i$, either directly or via one of its clones, is diagonalized against eventually). Note that the construction meets requirements (a), (b), and (c) and shows $L_B \neq L(N_{i}^{B \oplus \mbox{\protect\scriptsize SAT}})$ for each $i\geq 1$. $\Box$ Since $\mbox{EL}_2$ and $\mbox{\rm P-mc}(2)$ are both closed under complementation, we have the following corollary. \[cor:s2-el2\] $\mbox{\rm coSPARSE} \cap \mbox{\rm coS}(2) \cap \mbox{P-mc}(2) \not{\subseteq}\mbox{\rm EL}_{2}$. When suitably combined with standard techniques of constructing P-selective sets, the proof of the previous theorem even proves that the second level of the extended low hierarchy is not closed under a number of Boolean operations, as we have claimed in the beginning of this section. These results extend the main result of Hemaspaandra and Jiang [@hem-jia:j:psel] which says that P-Sel is not closed under those Boolean connectives. Let us first adopt and slightly generalize some of the formalism used in [@hem-jia:j:psel] so as to suffice for our objective. The intuition is that we want to show that certain [ *widely-spaced*]{} and [*complexity-bounded*]{} sets whose definition will be based on the set $B$ constructed in the previous proof are P-selective. Fix some complexity-bounding function $f$ and some wide-spacing function $\mu$ such that the spacing is at least as wide as given by the following inductive definition: $\mu(0)=2$ and for each $i\ge 0$. Now define for each $k \geq 0$, $$R_k {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ i \,|\, i \in {{\rm I\!N}}\, \wedge\, \mu(k)\le i < \mu(k+1)\},$$ and the following two classes of languages (where we will implicitly use the standard correspondence between ${\mbox{$\Sigma^\ast$}}$ and ${{\rm I\!N}}$): $${\cal C_1} {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ A {\subseteq}{{\rm I\!N}}\,|\, (\forall j\ge 0)\, [R_{2j}\cap A = \emptyset\,\wedge\, (\forall x,y \in R_{2j+1})\, [(x\le y \,\wedge\, x\in A ) \, \Longrightarrow \, y\in A]]\};$$ $${\cal C_2} {\stackrel{\mbox{\protect\scriptsize df}}{=}}\{ A {\subseteq}{{\rm I\!N}}\,|\, (\forall j\ge 0)\, [R_{2j}\cap A = \emptyset\,\wedge\, (\forall x,y \in R_{2j+1})\, [(x\le y \,\wedge\, y\in A) \, \Longrightarrow \, x\in A]]\}.$$ In [@hem-jia:j:psel], the following lemma is proven for the particular complexity-bounding function $f'(n) = 2^{{\cal O}(n)}$ and for the classes ${\cal C}^{'}_{1}$ and ${\cal C}^{'}_{2}$ having implicit in their definition the particular wide-spacing function that is given by $\mu'(0)=2$ and , $i \geq 0$. However, there is nothing special about these functions $f'$ and $\mu'$, i.e., for Lemma \[lem:hemjia\] to hold it suffices that $f$ and $\mu$ relate to each other as required above. In light of this, the proof of Lemma \[lem:hemjia\] is quite analogous to the proof given in [@hem-jia:j:psel]. \[lem:hemjia\] ${\cal C_1} \cap \mbox{DTIME}[f] {\subseteq}{\mbox{\rm P-Sel}}\ $ and $\ {\cal C_2} \cap \mbox{DTIME}[f] {\subseteq}{\mbox{\rm P-Sel}}$. Now we are ready to prove the main result of this section. \[thm:el2-not-closed\] $\mbox{\rm EL}_{2}$ is not closed under intersection, union, exclusive-or, or equivalence. [**Proof.**]{} Using the technique of [@hem-jia:j:psel], it is not hard to prove that the set $B$ constructed in the proof of Theorem \[thm:s2-el2\] can be represented as $B = A_1 \cap A_2$ for P-selective sets $A_1$ and $A_2$. More precisely, let $$\begin{aligned} A_1 & {\stackrel{\mbox{\protect\scriptsize df}}{=}}& \{x \,|\, (\exists w \in B)\, [ |x| = |w| \,\wedge\, x \leq_{\mbox{\protect\scriptsize lex}} w ] \},\\ A_2 & {\stackrel{\mbox{\protect\scriptsize df}}{=}}& \{x \,|\, (\exists w \in B)\, [ |x| = |w| \,\wedge\, w \leq_{\mbox{\protect\scriptsize lex}} x] \}.\end{aligned}$$ Since $B\in \mbox{EE}$ and is triple-exponentially spaced, we have from Lemma \[lem:hemjia\] that $A_1$ and $A_2$ are in P-Sel and thus in $\mbox{EL}_2$. On the other hand, we have seen in the previous proof that $B = A_1 \cap A_2$ is not in $\mbox{EL}_2$. Similarly, if we define $$\begin{aligned} C_1 & {\stackrel{\mbox{\protect\scriptsize df}}{=}}& \{x \,|\, (\exists w \in B)\, [ |x| = |w| \,\wedge\, x <_{\mbox{\protect\scriptsize lex}} w ] \},\\ C_2 & {\stackrel{\mbox{\protect\scriptsize df}}{=}}& \{x \,|\, (\exists w \in B)\, [ |x| = |w| \,\wedge\, x \leq_{\mbox{\protect\scriptsize lex}} w] \},\end{aligned}$$ we have $B = C_1 \ {\mbox{$\sym$}}\ C_2$, where ${\mbox{$\sym$}}$ denotes the exclusive-or operation. As before, $C_1$ and $C_2$ are in P-Sel and thus in $\mbox{EL}_2$. Hence, $\mbox{EL}_2$ is not closed under intersection or exclusive-or. Since $\mbox{EL}_2$ is closed under complementation, it must also fail to be closed under union and equivalence. $\Box$ The proof of the above result also gives the following corollary. \[cor:el2-not-closed\] [[@hem-jia:j:psel]]{}  is not closed under intersection, union, exclusive-or, or equivalence. [**Acknowledgments**]{} We thank an anonymous referee for stressing that our results should be interpreted as evidence regarding the unnaturalness of extended lowness as a complexity measure. = currsize [^1]: Supported in part by grants NSF-INT-9513368/DAAD-315-PRO-fo-ab, NSF-CCR-8957604, NSF-INT-9116781/JSPS-ENG-207, and NSF-CCR-9322513. Work done in part while visiting Friedrich-Schiller-Universität Jena and the Tokyo Institute of Technology. [^2]: Supported in part by a postdoctoral fellowship from the Chinese Academy of Sciences, and by grant NSF-CCR-8957604. Work done in part while visiting the University of Rochester and while at McMaster University. Current address: Suite 1100, 123 Front Street, Toronto, Ontario, M5J 2M3 Canada. [^3]: Supported in part by a DAAD research visit grant, and by grants NSF-INT-9513368/DAAD-315-PRO-fo-ab, NSF-CCR-9322513, and NSF-CCR-8957604. Work done in part while visiting the University of Rochester. Contact via email: [rothe@informatik.uni-jena.de]{}. [^4]: Supported in part by grant NSF-INT-9116781/JSPS-ENG-207. Work done in part while visiting the University of Rochester. [^5]: = currsize We stress that this is a very loose and informal description. In particular, for the case of the extended low hierarchy, it would be more accurate to say: A set $A$ that is placed in the $(k+1)$st level of the extended low hierarchy, $k>1$, is such that ${\mbox{\rm NP}}^A$ contains no more information than ${\rm SAT} \oplus L$ relative to the computation of a $\Sigma_{k}^p$ machine. [^6]: = currsize For sets $X$ and $Y$, $X \leq_{m}^{p} Y$ if and only if there is a polynomial-time computable function $f$ such that . [^7]: = currsize This generalizes to $k$ larger than 1 a result of Ogihara who proves that the P-selective sets are strictly contained in $\mbox{P-mc}(2)$ [@ogi:j:comparable] as well as the known fact that P-Sel is strictly larger than P [@sel:j:pselective-tally].

--- abstract: 'A Bose gas in an external potential is studied by means of the semi-classical approximation. Analytical results are derived for the energy of an interacting Bose gas in a generic power-law trapping potential. An expression for the chemical potential below the critical temperature is also obtained. The theoretical results are in qualitative agreement with a recent energy measurement.' address: 'Institute of Theoretical Physics, Academia Sinica, Beijing 100080,China' author: - 'Hualin Shi and Wei-Mou Zheng' title: The Energy of a Trapped Interacting Bose Gas --- The condensation of an ideal Bose-Einstein gas is one of the most striking consequences of quantum statistics [@huang87]. When Bose-Einstein condensation (BEC) occurs at a sufficiently low temperature, the zero momentum state can become macroscopically occupied. For many years, it was considered hopeless to experimentally observe BEC in an atomic gas with weak interactions. With the development of techniques to trap and cool atoms, BEC was recently observed directly in dilute atomic vapors[@and95; @bra95; @dav95]. The new experimental achievements have stimulated great interest in the theoretical study of inhomogeneous Bose gases. The thermodynamic properties of trapped atomic Bose gases undergoing BEC can be altered by the spatially varying trapping potential. The interaction between atoms may have a significant effect on the thermodynamic properties. There have been several investigations analyzing the dependence of the critical temperature on the trapping potential and weak interaction in the Bose gas[@bag87; @gio96; @shi96b]. The thermodynamic properties of Bose gases in an external potential have also been discussed in Refs. [@bag87; @shi96c]. In a recent experiment[@ens96], after turning off the trapping field, the kinetic energy of a sufficiently expanded atom cloud was measured. The experimental data is in a good agreement with the theory of a trapped ideal Bose gas when the temperature is above critical temperature. However, a discrepancy with the theory of ideal Bose gases is found below the critical temperature. To explain this discrepancy it is important to know how the mutual interaction affects the energy. Here we shall derive some analytical expressions for the energy of a trapped non-interacting and interacting Bose gas under the semi-classical approximation. First, let us calculate the kinetic and potential energies of an ideal Maxwell-Boltzmann gas and of an ideal Bose-Einstein gas in an external potential. For an ideal Maxwell-Boltzmann gas, according to the Maxwell-Boltzmann distribution, the local atom number density is given by $$\label{stat-mb} n_{\hbox{\tiny MB}}({\bf r,p},T)= \exp [-\beta (p^2/2m+V({\bf r})-\mu )] ,$$ where $\beta=kT$ and $\mu$ is the chemical potential which comes from the normalization. Integration of the number density (\[stat-mb\]) over the whole phase space gives the total number of atoms $$N = \frac 1{h^3}\int n_{\hbox{\tiny MB}}({\bf r,p},T)d{\bf p}d{\bf r} = \frac{z}{\lambda ^3}\int \exp [-\beta V({\bf r})]d{\bf r} , \label{n-mb}$$ where $$\lambda= \left(\frac{2 \pi \hbar^2}{m kT} \right)^{1/2}, \qquad z = e^{\beta \mu} .$$ For the generic power-law potential discussed in Ref. [@bag87] $$\label{pot}V({\bf r})=\epsilon _1\left| \frac x{L_1}\right| ^p+\epsilon _2\left| \frac y{L_2}\right| ^l+\epsilon _3\left| \frac z{L_3}\right| ^q,$$ where $L_1$, $L_2$ and $L_3$ are the linear sizes of the volume, and $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_3$, $p$, $l$ and $q$ the parameters, Eq. (\[n-mb\]) yields $$\label{nt} N=z\lambda ^{-3} \chi I(p,l,q)\beta^{-\eta +1/2} ,$$ with $$\chi = \frac{L_1 L_2 L_3}{\epsilon_1^{1/p}\epsilon_2^{1/l}\epsilon_3^{1/q}}, \qquad I(p,l,q) = \frac{8}{plq}\; \Gamma (1/p)\;\Gamma (1/l)\;\Gamma (1/q), \qquad \eta =\frac{1}{p}+\frac{1}{l}+\frac{1}{q} + \frac{1}{2} .$$ In the derivation we have used the formula for the gamma function $$\label{gamma} \Gamma (z)=\int_0^\infty t^{z-1}e^{-t}\;dt.$$ Relation (\[nt\]) determines the chemical potential $\mu$ from a given total number $N$ of atoms. The total kinetic energy can be obtained as $$K_{\hbox{\tiny MB}} =\int \frac{p^2}{2m}\;dN({\bf r,p},T) =\frac 32kT\frac{z}{\lambda ^3}\int \exp [-\beta V({\bf r})] \; d{\bf r} =\frac 3 2 NkT, \label{t-mb}$$ which is just a result of the energy equipartition principle. Similarly, the total potential energy is $$V_{\hbox{\tiny MB}} = \int V({\bf r})\;dN({\bf r,p},T) = \left( \frac 1p+\frac 1l+\frac 1q\right) \frac{z \chi I(p,l,q)}{\lambda^3 \beta^{\eta+1/2}} .$$ Thus, the total energy of the trapped ideal Maxwell-Boltzmann gas is $$E_{\hbox{\tiny MB}}^{t}=\left( \eta +1\right) NkT .$$ We next consider an ideal Bose-Einstein gas in the generic power law potential. Under the semi-classcial approximation[@chou96; @bag87; @gio96] the local number density is given by the Bose-Einstein distribution function $$\label{stat-be} n_{\hbox{\tiny BE}}({\bf r,p},T)= \{\exp [\beta (p^2/2m+V({\bf r})-\mu )]-1\}^{-1},$$ from which the reduced spatial distribution of atoms is [@bag87; @chou96] $$\rho({\bf r}) = \lambda^{-3} g_{3/2}[\exp(-\beta (V(\bf{r})-\mu))], \label{den-be}$$ where $g_\nu(x)=\sum_{j=1} x^j/j^{\nu}$. The critical temperature for an ideal Bose-Einstein gas in the generic power law potential to undergo BEC has been found in Refs. [@bag87; @shi96b] to be $$T_c= \frac 1 k\;\left(\frac{N\;(2\pi \hbar ^2)^{3/2}}{\;m^{3/2}\; \zeta (\eta +1)\; \chi\; I(p,l,q) }\right)^{1/(\eta+1)}, \label{i-t}$$ where $\zeta (\nu )=g_\nu (1)$ is the Riemann zeta function. The cases of the temperature below and above $T_c$ should be treated separately. For $T>T_c$, integrating the number density function (\[stat-be\]) over the phase space, we obtain the total number of atoms [@shi96c] $$\label{ben} N =h^{-3} \int n_{\hbox{\tiny BE}}({\bf r,p},T) d{\bf r}d{\bf p} =\frac{\chi I(p,l,q) }{\lambda ^3\beta ^{\eta -1/2}} g_{\eta +1}(z).$$ In the derivation we have used the definition of the function $g_{\nu}$. Similarly, we have the total kinetic energy $$K_{\hbox{\tiny BE}} =h^{-3}\int \frac{p^2}{2m} n_{\hbox{\tiny BE}}({\bf r,p},T) d{\bf p}d{\bf r} =\frac 32 \frac{\chi I(p,l,q)}{\lambda ^3\beta ^{\eta +1/2}}\;g_{\eta +2}(z), \label{t-be-i}$$ and the total potential energy $$V_{\hbox{\tiny BE}} = \int V({\bf r})\rho ({\bf r}) d{\bf r} = \frac{\chi I(p,l,q)} {\lambda ^3\beta ^{\eta +1/2}}\left( \frac 1p+\frac 1l+\frac 1q\right) g_{\eta +2}(z) . \label{v-be-i}$$ The total energy of the trapped ideal Bose gas is then[@pin96] $$E_{\hbox{\tiny BE}}^{t}=\frac{g_{\eta +2}(z)}{g_{\eta +1}(z)} \left( \eta +1\right) NkT \label{tot-be-i}$$ Comparing with the trapped ideal Maxwell-Boltzmann gas, we see that the only difference in the two total energy expressions is the factor $g_{\eta +2}(z)/g_{\eta +1}(z)$ for the Bose gas, which breaks the linear dependence of the total energy on the temperature. When the temperature is below $T_c$, the contribution to the energy of the system comes only from the normal component of the Bose gas. From Eqs. (\[ben\]), (\[t-be-i\]), (\[v-be-i\]) and (\[tot-be-i\]) the kinetic, potential and total energy of the system can be written as $$\begin{aligned} K_{\hbox{\tiny BE}} &=& \frac 32\frac{g_{\eta +2}(1)}{g_{\eta +1}(1)}\left( \frac T{T_c}\right) ^{\eta +1}NkT , \label{t-bl-i}\\ V_{\hbox{\tiny BE}} &=& \left( \frac 1p+\frac 1l+\frac 1q\right) \frac{g_{\eta +2}(1)}{g_{\eta +1}(1)}\left( \frac T{T_c}\right) ^{\eta +1} NkT, \label{v-bl-i}\\ E^{t}_{\hbox{\tiny BE}} &=& (\eta +1)\frac{g_{\eta +2}(1)} {g_{\eta +1}(1)}\left( \frac T{T_c}\right)^{\eta +1} NkT .\end{aligned}$$ The energy of the system will go to zero when the temperature approaches zero. So far we have not considered the role of the mutual interaction of Bose atoms. For a trapped interacting Bose gas, we divide the total energy into three parts: the kinetic energy $K_{int}$, potential energy $V_{int}$ and interaction energy $U_{int}$, i.e. $$\label{en-tot}E^{t}_{int}=K_{int}+V_{int}+U_{int} ,$$ In the local density approximation or semi-classical approximation, the local number density function of a trapped interacting Bose gas is given by [@gio96; @chou96] $$\label{stat-be-i} n({\bf r,p},T)=\{\exp [\beta (p^2/2m+V({\bf r})+ 4a\lambda ^2\rho ({\bf r})-\mu )]-1\}^{-1}.$$ In similarity to the non-interacting Bose gas, the cases of the temperature above and below $\tilde T_c$ need to be dealt with separately, where $\tilde T_c$ is the critical temperature of the trapped interacting Bose gas [@shi96b]. At a temperature above $\tilde T_c$, from (\[stat-be-i\]) the total number of atoms is [@shi96c] $$N =\frac 1{h^3}\int n({\bf r,p},T)d{\bf p}d{\bf r} =\frac{\chi I(p,l,q) }{\lambda ^3\beta ^{\eta -1/2}} \left [ g_{\eta +1}(z) -\frac{2 a}{\lambda} F_{3/2,3/2,\eta -1}(z) \right ] ,$$ where $$F_{\delta ,\nu ,\eta }(x) =\sum_{i,j}^\infty \frac{x^{i+j}} {i^\delta j^\nu (i+j)^{\eta -1/2}} .$$ In the derivation we have expanded the exponential density function with respect to the small parameter $a/\lambda$, and kept only terms up to the lowest order in $a/\lambda$ [@shi96b; @shi96c]. Up to this first order correction, Eq. (\[den-be\]) can be used. By the same procedure, the total kinetic and potential energy are obtained as $$\begin{aligned} K_{int} &=&\frac 1{h^3}\int \frac{p^2}{2m}n({\bf r,p},T)d{\bf p}d{\bf r} =\frac 32 \frac{\chi I(p,l,q) }{\lambda ^3\beta ^{\eta +1/2}} \left [ g_{\eta +2}(z) -\frac{4 a}{\lambda} F_{3/2,3/2,\eta}(z) \right ] ,\\ V_{int} &=& \frac{1}{h^3} \int V({\bf r})n({\bf r,p},T)d{\bf p}d{\bf r} =\left(\frac{1}{p}+\frac{1}{l}+\frac{1}{q} \right) \frac{\chi I(p,l,q) } {\lambda ^3\beta ^{\eta +1/2}}\left [ g_{\eta +2}(z) -\frac{4 a}{\lambda} F_{1/2,3/2,\eta+1}(z) \right ] .\end{aligned}$$ Similarly, with $V({\bf r})$ being replaced by $2 a\lambda ^2kT\rho ({\bf r})$, the interaction energy is calculated as $$U_{int} =\int 2 a\lambda ^2kT\rho ^2({\bf r})d{\bf r} =\frac{2 a \chi I(p,l,q) }{\lambda ^4\beta ^{\eta +1/2}} F_{3/2,3/2,\eta }(z) .$$ From the above three expressions for $K_{int}$, $V_{int}$ and $U_{int}$, we see that the correction to the energy of the trapped ideal Bose gas due to the mutual interaction of atoms is proportional to $a/\lambda$, hence is very small for a weak interaction. This explains why the ideal Bose gas result agrees well with the experimental data above $\tilde T_c$. When the temperature is below $\tilde T_c$, a region where a great number of atoms are in the BEC state forms. In the local density approximation, a relation between density and potential inside the region reads [@chou96] $$V({\bf r})+4\pi a\rho ({\bf r})\hbar ^2/m=V({\bf r}_0)+4\pi a\rho _0\hbar ^2/m, \qquad r<r_0 ,$$ where $r_0$ corresponds to the edge of the region. At $r = r_0^-$, we have $\rho({r_0^-}) =\rho_0=\lambda^{-3} g_{3/2}(1)$, which is the critical density of the gaseous phase for the untrapped homogeneous ideal Bose gas at temperature $T$. From the above relation, we find the density of the condensed atoms as the difference of $\rho({\bf r})$ and $\rho_0$ $$\rho _s({\bf r})=\rho ({\bf r})-\rho_0 =\frac{m}{4 \pi a \hbar^2} [V({\bf r_0})-V({\bf r})],\qquad r<r_0 . \label{den-bec}$$ In order to obtain $\rho_s$, the potential $V(r_0)$ at the edge of the condensed region needs to be determined. Integrating $\rho_s$ over the whole volume, we have the total number of atoms in the condensed state $$N_s = \frac{m \chi }{4 \pi a \hbar^2} \frac{I(p,l,q)}{\Gamma (\eta +3/2)} V^{\eta+1/2}(r_0) , \label{ns-c}$$ where the derivation involves the integral $$\int_{|x|^p+|y|^l+|z|^q\leq 1} (1-|x|^p-|y|^l-|z|^q)\;dxdydz =\frac{I(p,l,q)}{\Gamma (\eta +3/2)}.$$ At the weak interaction limit we may approximate $N_s$ and $\tilde T_c$ by those for the trapped ideal Bose gas [@bag87; @shi96b], i.e. $$N_s \approx N \left [ 1-\left(\frac{T}{T_c} \right)^{\eta+1} \right ] ,\qquad T<T_c, \label{ns-t}$$ where we have written $\tilde T_c$ as $T_c$. From Eqs. (\[ns-c\]) and (\[ns-t\]) we find $$\label{v0} V(r_0) = \left \{ \frac{4 \pi a \hbar^2\Gamma (\eta +3/2)}{m \chi I(p,l,q)} \left[1-\left(\frac{T}{T_c} \right)^{\eta+1} \right ]N\right\}^{2/(2\eta+1)}.$$ For the condensation region at $T<T_c$ there exists another relation [@chou96] $$kT \ln z - V({\bf r}) = \frac{4 \pi a \hbar^2}{m} [\rho({\bf r})+ \rho_0 ] , \qquad (r<r_0) .$$ Setting $r=r_0^-$ in this formula, from Eq. (\[v0\]) we find the chemical potential as a descending function of temperature to be $$\mu = \frac{8 \pi a \hbar^2}{m} \rho_0 +\left \{ \frac{4 \pi a \hbar^2\Gamma (\eta +3/2)}{m \chi I(p,l,q)} \left[1-\left(\frac{T}{T_c} \right)^{\eta+1} \right ] N \right \}^{2/(2\eta+1)},\qquad (T<T_c) , \label{ch-bl}$$ which, for a cylindrically symmetric harmonic trapping potential, reduces to $$\mu = \frac{8\pi a\rho _0\hbar ^2}m+\frac{\hbar \omega_0 }2\left\{ \frac{15aN}{a_0}\left[ 1-\left( \frac T{T_c}\right) ^3\right] \right\} ^{2/5} \buildrel {T\to 0}\over\longrightarrow \frac{\hbar \omega_0 }2\left\{ \frac{15aN}{a_0}\right\} ^{2/5},$$ where $a_0$ is the geometric mean characteristic length of the trap, i.e. $a_0^2=\hbar /(m\omega_0)$ with $\omega_0^3=\omega_{\perp}^2 \omega_z$. This is consistent with the Ginzburg-Pitaevskii-Gross mean field theory [@bay96]. According to Refs.[@chou96; @huang57], the density of the mutual interaction energy in the condensation region is $$\begin{aligned} u_{int} &=& 2 a \lambda^2 kT \rho^2({\bf r}) -a \lambda^2 kT \rho_s^2({\bf r}) \nonumber \\ &=& a \lambda^2 kT [2 \rho_0^2({\bf r}) +4 \rho_0 \rho_s({\bf r}) +\rho_s^2({\bf r}) ],\qquad (r<r_0),\end{aligned}$$ which, after integrating over the condensation region ($r<r_0$), gives $$U_{int} = \chi I(p,l,q)V^{\eta +1/2}(r_0)\; \left[ \frac{4 \pi a \hbar^2 \rho_0^2}{m\Gamma (\eta+1/2) V(r_0)} +\frac{2 \rho_0}{\Gamma (\eta+3/2)} +\frac{m V(r_0)}{4\pi a \hbar^2 \Gamma (\eta+5/2)}\right]. \label{en-int}$$ The first two terms depending on $\rho_0$ vanish when the temperature approaches zero. So, at low temperature the main contribution to $U_{int}$ comes from the third term corresponding to the interaction within the condensed atoms. We may then neglect the first and second terms of (\[en-int\]) to write $$U_{int} \approx \frac{1}{\Gamma (\eta+5/2)} \left(\frac{4 \pi a \hbar^2}{m\chi I(p,l,q)}\right)^{2/(2\eta +1)} \left\{\left[1-\left(\frac{T}{T_c} \right)^{\eta+1} \right ]\Gamma (\eta+3/2)N \right \}^{(2 \eta +3)/(2\eta+1)}. \label{int-bec}$$ To an approximation, we may neglect any correction to the energy from the normal component, and express the energy of the trapped interacting Bose gas as that of the trapped ideal Bose gas plus the only correction from the mutual interaction within the condensed component. In the experiment the trapping potential is [@ens96] $$V({\bf r})=\frac 12m\omega_{\perp}^2r_{\perp}^2 + \frac{1}{2} \omega_z^2 r_z^2,$$ which is harmonic. In the ideal Bose gas approximation, at a temperature $T$ above $T_c$, Eqs. (\[t-be-i\]) and (\[v-be-i\]) reduce to $$K = \frac{3}{2} \frac{g_4(z)}{g_3(z)}NkT, \qquad V = \frac{3}{2} \frac{g_4(z)}{g_3(z)}NkT,$$ where the fugacity $z$ is related to the total number of atoms as $$N = \frac{g_3(z)}{\hbar^3 \omega_{\perp}^2 \omega_z \beta^3} ,$$ which yields $g_3(z)=(T_c/T)^3 g_3(1)$. The energy of the atom interaction is negligible at $T>T_c$. When the temperature is below $T_c$, from Eqs. (\[t-bl-i\]) and (\[v-bl-i\]), the approximate kinetic and potential energy are $$K^-= \frac{3}{2} \frac{\zeta(4)}{\zeta(3)} \left(\frac{T}{T_c} \right)^3 NkT ,\qquad V^- = K^- .$$ The contribution to the energy from the interaction within the condensation component is now significant. From Eqs. (\[i-t\]) and (\[int-bec\]), this interaction energy is $$U^-=\frac{1}{7} \left( \frac{15a}{a_0}\right) ^{2/5}N^{1/15} (\zeta(3))^{1/3} \left[ 1-\left( \frac T{T_c}\right) ^3\right] ^{7/5} N k T_c .$$ In the experiment the trapped atoms are $^{87}$Rb [@ens96]. The $s$-wave scattering length of $^{87}$Rb is $a \approx 100$ in the unit of the Bohr radius $a_B$. The trapping harmonic potential is axially symmetric with the frequency ratio of axial to radial being $\sqrt{8}$. The final stage of the evaporative cooling is performed at $\nu_z = 373$ Hz. The trapping field is then turned off to initiate the expansion of the atom cloud non-adiabatically. The kinetic energy $E_{\hbox{\tiny exp}}$ of atoms is measured after a sufficient expansion which ensures the interaction energy to convert to purely kinetic energy. Therefore, we may estimate $E_{\hbox{\tiny exp}}$ as the sum of $K^-$ and $U^-$, i.e. $$\frac{E_{\hbox{\tiny exp}}}{N kT_c} = \frac{3}{2}\frac{g_4(z)} {g_3(z)} \left( \frac{T}{T_c} \right), \hbox{\quad for\quad} T>T_c,$$ and $$\frac{E}{N kT_c} = \frac{3}{2}\frac{g_4(1)} {g_3(1)} \left( \frac{T}{T_c} \right)^4 + 0.449 \left(\frac{a}{a_0} \right)^{2/5} N^{1/15} \left[ 1-\left(\frac{T}{T_c}\right)^3 \right]^{7/5}, \hbox{\quad for\quad} T<T_c .$$ Assume that the total number of atoms in the trap is 40000. The scaled energy per particle, $E/NkT_c$, is plotted versus the scaled temperature, $T/T_c$, in Fig. 1, where the solid, dashed and dotted lines correspond to the ideal Bose gas, ideal Maxwell-Boltzmann and interacting Bose gas, respectively. When the temperature is rather high, they are very close. However, at low temperature the differences among them are prominent. The theoretical result of the interacting Bose gas is in qualitative agreement with the experimental data. We have considered only the contribution from the mutual interaction of the condensed atoms. If the finite size effect[@gro95; @ket96] and all other interaction terms are included, a quantitative comparison with experimental data can be made. [99]{} Kerson Huang, [*Statistical Mechanics*]{}, ( Wiley, New York, 1987), 2nd ed . O. Penrose and L. Onsager, [*Phys. Rev.*]{} [**104**]{}, 576 (1956). M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science [**269**]{}, 198 (1995). C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, [*Phys. Rev. Lett.*]{} [**75**]{}, 1687 (1995). K.B. Davis, M.-O. Mewes, M.R. Andrew, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, [*Phys. Rev. Lett.*]{} [**75**]{}, 3969 (1995). Vanderlei Bagnato, David E. Pritchard, and Daniel Kleppner, [*Phys. Rev.*]{} [**A35**]{}, 4354 (1987). S. Giorgini, L. Pitaevskii, and S. Stringari, [*Phys. Rev.*]{} [**A54**]{}, 4633 (1996). Hualin Shi, Wei-mou Zheng, [cond-mat/9609241]{} . Hualin Shi, Wei-mou Zheng, [cond-mat/9611171]{} . J.R. Ensher, D.S. Jin, M.R. Matthews, C.E. Wieman, and E.A. Cornell, [*Phys. Rev. Lett.*]{} 4984 (1996) . T.T. Chou, C.N. Yang, and L.W. Yu, [*Phys. Rev.*]{} [**A53**]{}, 4257 (1996). P.W.H. Pinkse, A.Mosk, M. Weidemüller, M.W. Reynolds, T.W. Hijmans, and J.T.M. Walraven, [physics/9611026]{} . G. Baym, and C.J. Pethick, [*Phys. Rev. Lett.*]{} [**76**]{}, 6 (1996). Kerson Huang and C.N. Yang, [*Phys. Rev.*]{} [**105**]{}, 776 (1957). S. Grossmann and M. Holthaus, [*phys. Lett.*]{} [**A208**]{}, 188 (1995). W. Ketterle and N.J. van Druten, [*Phys. Rev.*]{} [**A54**]{}, 656 (1996). Fig. 1 Plot of the scaled energy per particle $E/NkT_c$ of trapped gas vs. scaled temperature $T/T_c$. The solid, dashed and dotted lines correspond to an ideal Bose gas, ideal Maxwell-Boltzmann gas and interacting Bose gas, respectively.

--- abstract: 'We prove the local and global in time existence of the classical solutions to two general classes of the stress-assisted diffusion systems. Our results are applicable in the context of the non-Euclidean elasticity and liquid crystal elastomers.' address: - 'Marta Lewicka, University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, USA ' - 'Piotr B. Mucha, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02097 Warszawa, Poland' author: - Marta Lewicka - 'Piotr B. Mucha' title: | A local and global well-posedness results for\ the general stress-assisted diffusion systems --- Introduction and the main results. ================================== There are a number of phenomena where inhomogeneous and incompatible pre-strain is observed in $3$-dimensional bodies. Growing leaves, gels subjected to differential swelling, electrodes in electrochemical cells, edges of torn plastic sheets are but a few examples [@12a; @23a; @24a; @28a]. It has also been recently suggested that such incompatible pre-strains may be exploited as means of actuation of micro-mechanical devices [@18; @19]. The mathematical foundations for these theories has lagged behind but has recently been the focus of much attention. While the static theory involving thin structures such as pre-strained plates and shells is now reasonably well understood [@13a; @3a; @lepa; @BLS; @lemapa2], leading to the variationally reduced models constrained to appropriate types of isometries [@lepa; @BLS; @4a], and requiring bringing together the differential geometry of surfaces with the theory of elasticity appropriately modified [@lemapa1; @LOP; @13a; @14a], the parallel evolutionary PDE model seems to not have been considered in this context. The model and the main results. ------------------------------- In this paper, we are concerned with two systems of coupled PDEs in the description of stress-assisted diffusion. The first system: $$\label{maineq} \left\{\begin{split} & u_{tt} - \mbox{div}\Big(\partial_F W(\phi, \nabla u)\Big) = 0 \\ & \phi_t = \Delta \Big(\partial_\phi W(\phi,\nabla u)\Big). \end{split}\right.$$ consists of a balance of linear momentum in the deformation field $u:\mathbb{R}^3\times\mathbb{R_+}\to \mathbb{R}^3$, and the diffusion law of the scalar field $\phi:\mathbb{R}^3\times\mathbb{R_+}\to \mathbb{R}$ representing the inhomogeneity factor in the elastic energy density $W$. The field $\phi$ may be interpreted as the local swelling/shrinkage rate in morphogenesis at polymerization, or the localized conformation in liquid crystal elastomers. The second system is a quasi-static approximation of (\[maineq\]), in which we neglect the material inertia $u_{tt}$, consistent with the assumption that the diffusion time scale is much larger than the time scale of elastic wave propagation: $$\label{maineq2} \left\{\begin{split} & - \mbox{div}\Big(\partial_F W(\phi, \nabla u)\Big) = 0 \\ & \phi_s = \Delta \Big(\partial_\phi W(\phi,\nabla u)\Big). \end{split}\right.$$ In both systems, the deformation $u$ induces the deformation gradient, and the velocity and velocity gradients, respectively denoted as: $$F = \nabla u \in\mathbb{R}^{3\times 3}, \quad v=\xi_t\in\mathbb{R}^3, \quad Q=\nabla\xi_t = \nabla v = F_t\in\mathbb{R}^{3\times 3}.$$ We will be concerned with the local in time well-posedness of the classical solutions to (\[maineq\]), and the global well-posedness of (\[maineq2\]), subject to the (subset of) initial data: $$\label{initial1} u(0,\cdot) = u_0, \quad u_t(0,\cdot)=u_1 \quad \mbox{ in } \mathbb{R}^3,$$ $$\label{initial2} \phi(0,\cdot) = \phi_0 \quad \mbox{ in } \mathbb{R}^3,$$ and the non-interpenetration ansatz: $$\label{nonin} \det \nabla u > 0 \quad \mbox{ in } \mathbb{R}^3.$$ The main results of this paper are the following: \[th1\] Let $u_0 - \mathrm{id} \in H^4({\mathbb{R}}^3)$, $u_1 \in H^3({\mathbb{R}}^3)$ and $\phi_0\in H^3({\mathbb{R}}^3)$. Assume that $W$ is as in subsection \[ener\]. Fix $T>0$, and assume that the following quantities: $$\label{small} \|u_1,\nabla u_0 - \mathrm{Id}_3,\phi_0\|_{H^3}^2 + \|u_0 - \mathrm{id}\|_{L^2}^2 + \int_{{\mathbb{R}}^3} W(\phi_0,\nabla u_0)~\mathrm{d}x$$ are sufficiently small in comparison with $T$, and with the constant $\gamma$ in (\[ass\]). Then there exists a unique solution $(u, \phi)$ of the problem (\[maineq\]) (\[initial1\] - \[nonin\]), defined on the time interval $[0,T]$, and such that: $$\begin{split} & u-\mathrm{id} \in L^\infty(0,T;H^4(\mathbb{R}^3)), \quad u_{tt}\in L^\infty(0,T;H^2({\mathbb{R}}^3)), \\ & \phi \in L^\infty(0,T;H^3(\mathbb{R}^3)) \mbox{ \ \ and \ \ } \phi_t \in L^2(0,T;H^2(\mathbb{R}^3)). \end{split}$$ \[th2\] Let $\phi_0 \in H^2(\mathbb{R}^3)$ and assume that $W$ is as in subsection \[ener\]. Assume that $\|\phi_0\|_{H^2}$ is sufficiently small. Then there exists a unique global in time solution $(u, \phi)$ to (\[maineq2\]) (\[initial2\]) (\[nonin\]) such that: $$\begin{split} & u-\mathrm{id} \in L^\infty(\mathbb{R}_+;L^6(\mathbb{R}^3)), \quad \nabla^2 u \in L^2(\mathbb{R}_+;H^2(\mathbb{R}^3)), \\ & \phi \in L^\infty(\mathbb{R}_+; H^2(\mathbb{R}^3)) \mbox{\ \ and \ \ } \nabla \phi\in L^2(\mathbb{R}_+; H^2(\mathbb{R}^3)). \end{split}$$ The proof of Theorem \[th1\] relies on controlling the energy: $$\int_{{\mathbb{R}}^3} \frac 12 |u_t|^2 + W(\phi,\nabla u) ~\mbox{d}x,$$ where the hyperbolic character of the first equation in (\[maineq\]) suggests to seek the a-priori bounds on higher norms of $u$ and $\phi$ by the standard energy techniques. A detailed analysis reveals that the special structure of coupling in the stress-assisted diffusion system indeed allows for cancellation of those terms that otherwise prevent closing the bounds in each of the two equations in (\[maineq\]) alone. These terms are displayed in formulas (\[dodici\]) and (\[dicianove\]) in the proof of Lemma \[lemapriori\]. Existence of solutions in Theorem \[th1\] is then shown via Galerkin’s method, where we check that solutions to all appropriate $\epsilon$-approximations of the original system (\[maineq\]) still enjoy the same a-priori bounds in Theorem \[thap\]. This is carried out in section \[sec3\], while uniqueness of solutions is proved in section \[sec4\]. The proof of Theorem \[th2\], given in section \[sec5\], is based on the $L^2$-approach as well. The system (\[maineq2\]) is of elliptic-parabolic type, thus there is no loss of regularity with respect to the initial data (in contrast to (\[maineq\])). The analysis here is simpler than for (\[maineq\]) and we are able to show the global in time existence of small solutions. The toolbox we use for the proofs of both results is universal for hyperbolic-parabolic and elliptic-parabolic systems. Similar methods have been applied in [@BaMa; @FMNP; @MPZ; @PZ; @ZO] to study models of elasticity and their couplings with flows of complex fluids. A key element in these methods is the basic conservation law of energy and entropy type. The energy density $W$. {#ener} ----------------------- We now introduce the assumptions on the inhomogeneous elastic energy density $W$ in (\[maineq\]). Namely, the nonnegative scalar field $W: \mathbb{R}_+\times \mathbb{R}^{3\times 3}\rightarrow \overline{\mathbb{R}}_+$ is assumed to be $\mathcal{C}^4$ in a neighborhood of $(0, \mbox{Id}_3)$ and to satisfy, with some constant $\gamma >0$: $$\label{ass} \begin{split} & W(0, \mbox{Id}) = 0, \quad DW(0, \mbox{Id}) = 0, ~~~ \mbox{ and: } \\ & D^2W(0,\mbox{Id}) : (\tilde\phi, \tilde F)^{\otimes 2} \geq \gamma (|\tilde\phi|^2 + |\mbox{sym~}\tilde F|^2) \quad \mbox{ \ \ for all \ } (\tilde \phi, \tilde F)\in \mathbb{R}\times \mathbb{R}^{3\times 3}. \end{split}$$ The two main examples of $W$ that we have in mind, concern non-Euclidean elasticity and liquid crystal elastomers, where respectively: $$\label{exa} \begin{split} W_1(\phi,F) = W_0(FB(\phi)) + \frac{1}{2}|\phi|^2,\\ W_2(\phi,F) = W_0(B(\phi)F) + \frac{1}{2}|\phi|^2. \end{split}$$ are given in terms of the homogeneous energy density $W_0:\mathbb{R}^{3\times 3}\rightarrow \overline{\mathbb{R}}_+$ and the smooth tensor field $B:\mathbb{R}\to\mathbb{R}^{3\times 3}$. In both cases, we assume that $B(\phi)$ is symmetric and positive definite, and that $B(0)=\mbox{Id}$. Further, the principles of material frame invariance, material consistency, normalisation, and non-degeneracy impose the following conditions on $W_0$, valid for all $F\in\mathbb{R}^{3\times 3}$ and all $R\in SO(3)$: $$\label{elastic_dens} \begin{minipage}{14cm} \begin{itemize} \item[(i)] $W_0(RF) = W_0(F).$ \item[(ii)] $W_0(F)\to +\infty \quad \mbox{ as } \det F\to 0$. \item[(iii)] $W_0(\mbox{Id}) = 0$. \item[(iv)] $W_0(F) \geq c~\mbox{dist}^2(F, SO(3))$. \end{itemize} \end{minipage}$$ Examples of $W_0$ satisfying the above conditions are: $$\begin{split} W_{0,1}(F) & = |(F^TF)^{1/2} - \mbox{Id}|^2 + |\log \det F|^q \\ W_{0,2}(F) & = |(F^TF)^{1/2} - \mbox{Id}|^2 + \left|\frac{1}{\det F} - 1\right|^q \mbox{ for } \det F>0, \end{split}$$ where $q>1$ and $W_{0,i}$ is intended to be $+\infty$ if $\det F\leq 0$ [@MS]. Another case-study example, satisfying (i), (iii) but not (iv) is: $W_0(F)=|F^TF-\mbox{Id}|^2$. We have the following observation, which we will prove in the Appendix: \[prop\] For $W_0$ which is $\mathcal{C}^2$ in a neighborhood of $SO(3)$ and $B$ which is $\mathcal{C}^2$ in a neighborhood of $0$, assume (\[elastic\_dens\]) and assume that $B(0) = \mathrm{Id}$. Then $W_1$ and $W_2$ in (\[exa\]) satisfy (\[ass\]). Background and relation to previous works. ------------------------------------------ To put our results in a broader context, consider a general referential domain $\Omega$ which is an open, smooth and simply connected subset of $\mathbb{R}^3$. Let $G:\bar{\Omega}\rightarrow \mathbb{R}^{3\times 3}$ be a given smooth Riemann metric on $\Omega$ and denote its unique positive definite symmetric square root by $B=\sqrt{G}$. The “incompatible elastic energy” of a deformation $u$ of $\Omega$ is then given by: $$\label{functiona} E(u, \Omega) = \int_{\Omega} W_0(\nabla u(x) B(x)^{-1})~\mbox{d}x \qquad \forall u\in W^{1,2}(\Omega,\mathbb{R}^3),$$ where the elastic energy density $W_0$ is as in (\[elastic\_dens\]). It has been proved in [@lepa] that: $$\inf_{u\in W^{1,2}(\Omega, \mathbb{R}^3)} E(u, \Omega) = 0$$ if and only if the Riemann curvature tensor of $G$ vanishes identically in $\Omega$ and when (equivalently) the infimum above is achieved through a smooth isometric immersion $u$ of $G$. It is worth mentioning that in the context of thin films when $\Omega = \Omega^h = U\times (-\frac{h}{2}, \frac{h}{2})$ with some $U\subset \mathbb{R}^2$, there is a large body of literature relating the magnitude of curvatures of $G$ to the scaling of $\inf E(\cdot, \Omega^h)$ in terms of the film’s thickness $h$, and subsequently deriving the residual $2$-dimensional energies using the variational techniques. Firstly, in the Euclidean case of $G=\mbox{Id}_3$, where the residual energies are driven by presence of applied forces $f^h\sim h^\alpha$, three distinct limiting theories have been obtained [@FJMhier] for $\frac{1}{h}E(\cdot, \Omega^h)\sim h^\beta$ with $\beta>2$ (equivalently $\alpha>2$). Namely: $\beta\in (2,4)$ corresponded to the linearized Kirchhoff model (nonlinear bending energy), $\beta=4$ to the classical von-Kármán model, and $\beta>4$ to the linear elasticity. For $\beta=0$ the membrane energy has been derived in the seminal papers [@LR1; @LR2], while the case $\beta=2$ was considered in [@FJMM]. Secondly, in [@LPhier] a higher order (infinite) hierarchy of scalings and of the resulting elastic theories of shells, where the reference configuration is a thin curved film, has been derived by an asymptotic calculus. Thirdly, in the context of the non-Euclidean energy (\[functiona\]), it has been shown in [@BLS] that the scaling: $\inf \frac{1}{h} E(\cdot,\Omega^h)\sim h^2$ only occurs when the metric $G_{2\times 2}$ on the mid-plate $U$ can be isometrically immersed in $\mathbb{R}^3$ with the regularity $W^{2,2}$ and when, at the same time, the three appropriate Riemann curvatures of $G$ do not vanish identically; the relevant residual theory, obtained through $\Gamma$-convergence, yielded then a Kirchhoff-like residual energy. Further, in [@LRR] the authors proved that the only outstanding nontrivial residual theory is a von Kàrmàn-like energy, valid when: $\inf \frac{1}{h}E(\cdot, \Omega^h) \sim h^4$. This scale separation, contrary to [@FJMhier; @LPhier], is due to the fact that while the magnitude of external forces is adjustable at will, it seems not to be the case for the interior mechanism of a given metric $G$ which does not depend on $h$. In fact, it is the curvature tensor of $G$ which induces the nontrivial stresses in the thin film and it has only six independent components, namely the six sectional curvatures created out of the three principal directions, which further fall into two categories: including or excluding the thin direction variable. The simultaneous vanishing of curvatures in each of these categories correspond to the two scenarios at hand in terms of the scaling of the residual energy. Other types of the residual energies, pertaining to different contexts and scalings, have been studied and derived by the authors in [@3a; @4a; @12a; @13a; @14a; @lemapa1; @lemapa2; @lemapa2new; @LOP; @23a]. Note that, at the formal level, the Euler-Lagrange equations of (\[functiona\]) are precisely the first equation in the system (\[maineq\]). The dynamical viscoelasticity has been the subject of vast studies in the last decades (see for example [@AM; @4; @1; @Demoulini; @BLZ; @11; @25] and references therein), where various results on existence, asymptotics and stability have been obtained for a large class of models. For the coupled systems of stress-assisted diffusion of the type (\[maineq\]), we found a substantial body of literature in the Applied Mechanics community [@v1; @v2; @v3; @v4; @v5], deriving these equations from basic principles of continuum mechanics and irreversible thermodynamics. For example, the system derived in [@v5] is quite close to (\[maineq2\]) from the view point of theory of PDEs; indeed the structure of nonlinearity in both systems is almost the same. However, derivation from the first principles aside, it seems that the analytical study of the Cauchy problem, particularly in long temporal ranges, has not been yet carried out. The closest investigation in this direction has been recently proposed in [@JiWa], concerning existence of solutions for models of nonlinear thermoelasticity, and in [@ZO] where the authors examine further models of thermoviscoelasticity from the viewpoint of mathematical well-posedness. We refer here to [@RaSh; @LeMucha] as well. Notation. --------- Throughout the paper we use the following notation. In (\[maineq\]) the operator ${div}$ stands for the spacial divergence of an appropriate field. We use the convention that the divergence of a matrix field is taken row-wise. We use the matrix norm $|F|=(\mbox{tr}(F^TF))^{1/2}$, which is induced by the inner product: $\langle F_1:F_2\rangle = \mbox{tr}(F_1^TF_2)$. The derivatives of $W$ are denoted by $DW$, $D^2W$ etc, while their action on the appropriate variations $(\tilde\phi, \tilde F)\in\mathbb{R}\times \mathbb{R}^{3\times 3}$ is denoted by: $DW(\phi,\nabla u) : (\tilde \phi, \tilde F)$, $D^2W(\phi,\nabla u) : (\tilde \phi, \tilde F)^{\otimes 2}$ etc, often abbreviating to $DW : (\tilde \phi, \tilde F)$ and $(D^2W) : (\tilde \phi, \tilde F)^{\otimes 2}$ when no confusion arises. The partial derivative of $W$ with respect to its second argument is denoted by $\partial_FW\in \mathbb{R}^{3\times 3}$. The derivative in the direction of the variation $\tilde F\in \mathbb{R}^{3\times 3}$ is then $\langle (\partial_FW):\tilde F\rangle\in\mathbb{R}$. By $(\partial^k_FW):(\tilde F_1\otimes \tilde F_2\ldots \otimes \tilde F_{k-1})\in\mathbb{R}^{3\times 3}$ we denote the linear map acting on $F\in\mathbb{R}^{3\times 3}$ as the $k$th derivative of $W$ in the direction of $\tilde F_1, \tilde F_2\ldots \tilde F_{k-1}, F$. Hence, differentiating in $F$ gives: $$(\partial^k_FW):(\tilde F_1\otimes \ldots \tilde F_k) = \Big\langle \big((\partial^k_FW):(\tilde F_1\otimes \ldots \tilde F_{k-1})\big) : \tilde F_k\Big\rangle \in\mathbb{R}.$$ Finally, $C, c>0$ stand for universal constants, independent of the variable quantities at hand. Acknowledgments. ---------------- M.L. was partially supported by the NSF grant DMS-0846996 and the NSF grant DMS-1406730. P.B.M. was partly supported by the NCN grant No. 2011/01/B/ST1/01197. The crucial a priori estimate. {#abd} ============================== \[gaga5\] Every solution $(u,\phi)$ to (\[maineq\]), with regularity prescribed in Theorem \[th1\], satisfies: $$\label{nove} \frac{\mathrm{d}}{\mathrm{d}t} \int_{\mathbb{R}^3} \frac{1}{2} |u_t|^2 + W(\phi, \nabla u)~\mathrm{d}x + \int|\nabla (- \Delta)^{-1}\phi_t|^2 ~\mathrm{d}x = 0.$$ Testing the first equation in (\[maineq\]) by $u_t$ and integrating by parts gives: $$\begin{split} \frac{\mathrm{d}}{\mathrm{d}t} \int_{\mathbb{R}^3} \frac{1}{2} |u_t|^2 + & W(\phi, \nabla u)~\mathrm{d}x = \int \langle u_t, u_{tt}\rangle + \partial_\phi W(\phi,\nabla u) \phi_t +\langle \partial_FW(\phi,\nabla u) : \nabla u_t\rangle ~\mathrm{d}x\\ & = \int \langle u_t, \mbox{div}\left(\partial_FW(\phi,\nabla u)\right)\rangle + \langle \partial_FW(\phi,\nabla u) : \nabla u_t\rangle + \partial_\phi W(\phi,\nabla u) \phi_t ~\mathrm{d}x \\ & = \int_{\mathbb{R}^3} \partial_\phi W(\phi,\nabla u) \phi_t ~\mathrm{d}x. \end{split}$$ Define $\psi=(-\Delta)^{-1}\phi$ and integrate the second equation in (\[maineq\]) against $\psi_t$: $$\int_{\mathbb{R}^3} |\nabla\psi_t|^2 ~\mathrm{d}x = \int \phi_t \psi_t ~\mathrm{d}x = \int \psi_t \Delta \big(\partial_\phi W(\phi, \nabla u)\big)~\mathrm{d}x = - \int_{\mathbb{R}^3} \partial_\phi W(\phi,\nabla u) \phi_t ~\mathrm{d}x.$$ Summing the above two equalities yields (\[nove\]) and achieves the proof. For every $i,j,k\in\{1,2,3\}$ we now define the correction terms: $$\label{R2} \begin{split} \mathcal{R}_{ijk} = & (\partial_\phi\partial_F^2W) : \big(\nabla u_{x_i, x_j}\otimes \nabla u_{x_k} + \nabla u_{x_i, x_k}\otimes \nabla u_{x_j} + \nabla u_{x_j, x_k}\otimes \nabla u_{x_i} \big) \\ & + (\partial_\phi^2\partial_F\partial_\phi W) : \big(\nabla u_{x_i, x_j}\phi_{x_k} + \nabla u_{x_i, x_k}\phi_{x_j} +\nabla u_{x_j, x_k}\phi_{x_i} \\ & \qquad\qquad \qquad\qquad \qquad\qquad + \nabla u_{x_i}\phi_{x_j, x_k} + \nabla u_{x_j}\phi_{x_i, x_k}+ \nabla u_{x_k}\phi_{x_i, x_j}\big) \\ & + (\partial_\phi\partial_F^3W) : \nabla u_{x_i}\otimes \nabla u_{x_j} \otimes \nabla u_{x_k} \\ & + (\partial^2_\phi\partial_F^2 W) : \big(\nabla u_{x_i} \otimes \nabla u_{x_j} \phi_{x_k} + \nabla u_{x_i} \otimes \nabla u_{x_k} \phi_{x_j} +\nabla u_{x_j} \otimes \nabla u_{x_k} \phi_{x_1} \big) \\ & + (\partial^3_\phi\partial_F W) : \big(\nabla u_{x_i} \phi_{x_j} \phi_{x_k} + \nabla u_{x_j} \phi_{x_i} \phi_{x_k} +\nabla u_{x_k} \phi_{x_i} \phi_{x_j}\big)\\ & + (\partial_\phi^4 W) \phi_{x_i} \phi_{x_j}\phi_{x_k} \\ & + (\partial^3_\phi W) \big(\phi_{x_i, x_j} \phi_{x_k} + \phi_{x_j, x_k} \phi_{x_i} +\phi_{x_i, x_k} \phi_{x_j}\big). \end{split}$$ \[lemapriori\] Let $(u,\phi)$ be a solution to (\[maineq\]), with regularity prescribed in Theorem \[th1\]. For $t>0$, define the two quantities: $$\begin{split} & \mathcal{E}(t) = \int_{\mathbb{R}^3} |u_t|^2 + |\nabla^3u_t|^2 + 2W(\phi,\nabla u) \\ & \qquad \qquad + \sum_{i,j,k=1..3} D^2W(\phi,\nabla u) : (\phi_{x_i, x_j, x_k}, \nabla u_{x_i, x_j, x_k})^{\otimes 2} + 2 \sum_{i,j,k=1..3} \mathcal{R}_{ijk}\phi_{x_i, x_j, x_k} ~\mathrm{d}x, \\ & \mathcal{Z}(t) = \|u_t\|^2_{H^{3}(\mathbb{R}^3)} + \|\nabla u - \mathrm{Id}\|^2_{H^{3}(\mathbb{R}^3)} + \|\phi\|^2_{H^{3}(\mathbb{R}^3)}. \end{split}$$ Then: $$\label{ventinove} \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{E}\leq C(\mathcal{Z}^{4} + \mathcal{Z}^{3/2}).$$ [**1.**]{} We differentiate the first equation in (\[maineq\]) in a spacial direction $x_i\in \{x_1, x_2, x_3\}$: $$u_{x_i,tt} - \mbox{div}\left(\partial_F^2W(\phi, \nabla u) : \nabla u_{x_i}\right) = \mbox{div}\left(\partial_F\partial_\phi W(\phi, \nabla u)\phi_{x_i}\right).$$ We now differentiate the above twice more in the directions $x_i, x_j\in \{x_1, x_2, x_3\}$: $$\label{undici} u_{x_i, x_j, x_k, tt} - \mbox{div}\left(\partial_F^2W(\phi, \nabla u) : \nabla u_{x_i, x_j, x_k}\right) = \mbox{div}\left(\partial_F\partial_\phi W(\phi, \nabla u)\phi_{x_i, x_j, x_k}\right) + \mathcal{R}_1.$$ The error term $\mathcal{R}_1$ above has the following form, where we suppress the distinction between different $x_i, x_j, x_k$, retaining hence only the structure of different terms: $$\label{R1} \begin{split} \mathcal{R}_1 = \mbox{div}\Big(&3(\partial_F^3W) : \nabla u_x\otimes \nabla u_{xx} + 3(\partial_F^2\partial_\phi W) : (\nabla u_{xx}\phi_x +\nabla u_x\phi_{xx}) \\ & + 3(\partial_F\partial^2_\phi W)\phi_x\phi_{xx} + (\partial_F^4W) : (\nabla u_x)^{\otimes 3} + 3(\partial_F^3\partial_\phi W) : (\nabla u_x)^{\otimes 2} \phi_x \\ & + 3(\partial_F^2\partial^2_\phi W) : \nabla u_x (\phi_x)^2 + (\partial_F\partial_\phi^3 W)(\phi_x)^3 \Big). \end{split}$$ Above and in what follows, we also write $(\partial_F^3W)$ instead of $\partial_F^3W(\phi, \nabla u)$, and $(\partial_F^2\partial_\phi W)$ instead of $\partial_F^2\partial_\phi W(\phi, \nabla u)$, etc. Integrating (\[undici\]) by parts against $u_{x_i, x_j, x_k, t}$ we get: $$\label{dodici} \begin{split} & \frac{\mbox{d}}{\mbox{d}t}\int_{\mathbb{R}^3} \frac{1}{2} (u_{x_i, x_j, x_k, t})^2 +\phi_{x_i, x_j, x_k}\langle(\partial_F\partial_\phi W):\nabla u_{x_i, x_j, x_k}\rangle \\ & \qquad\qquad\qquad \qquad \qquad + \frac{1}{2} (\partial^2_FW) : \nabla u_{x_i, x_j, x_k}\otimes \nabla u_{x_i, x_j, x_k} ~\mbox{d}x \\ & = \boxed{\int_{\mathbb{R}^3} \phi_{x_i, x_j, x_k, t}\langle(\partial_F\partial_\phi W):\nabla u_{x_i, x_j, x_k}\rangle ~\mathrm{d}x } \\ & \qquad + \int_{\mathbb{R}^3} \phi_{x_i, x_j, x_k}\langle(\partial_t\partial_F\partial_\phi W):\nabla u_{x_i, x_j, x_k}\rangle ~\mathrm{d}x \\ & \qquad + \frac{1}{2} \int_{\mathbb{R}^3} (\partial_t\partial^2_F\partial_\phi W) : \nabla u_{x_i, x_j, x_k} \otimes \nabla u_{x_i, x_j, x_k} ~\mathrm{d}x + \int_{\mathbb{R}^3} \mathcal{R}_1 u_{x_i, x_j, x_k, t} ~\mathrm{d}x. \end{split}$$ [**2.**]{} Differentiate now the second equation in (\[maineq\]) in $x_i\in \{x_1, x_2, x_3\}$: $$\phi_{x_i, t} = \Delta\big(\langle\partial_\phi\partial_FW (\phi, \nabla u) : \nabla u_{x_i}\rangle + \partial^2_\phi W(\phi, \nabla u)\phi_{x_i}\big).$$ As before, differentiate twice more in $x_i, x_j\in \{x_1, x_2, x_3\}$, to obtain: $$\label{diciotto} \phi_{x_i, x_j, x_k, t} = \Delta\left((\partial_\phi\partial_F W) : \nabla u_{x_i, x_j, x_k} + (\partial^2_\phi W) \phi_{x_i, x_j, x_k} +\mathcal{R} _{ijk}\right),$$ where $\mathcal{R}$ is given in (\[R2\]). Testing (\[diciotto\]) against $(-\Delta)^{-1}\phi_{x_i, x_j, x_k, t} = \psi_{x_i, x_j, x_k, t}$, we get: $$\label{dicianove} \begin{split} -\int_{\mathbb{R}^3} |\nabla\psi_{x_i, x_j, x_k, t}|^2&~\mbox{d}x = \boxed{\int_{\mathbb{R}^3} \phi_{x_i, x_j, x_k, t}\langle(\partial_F\partial_\phi W):\nabla u_{x_i, x_j, x_k}\rangle ~\mbox{d}x} \\ & +\frac{\mbox{d}}{\mbox{d}t}\int_{\mathbb{R}^3} \frac{1}{2} (\partial^2_\phi W) (\phi_{x_i, x_j, x_k})^2 ~\mbox{d}x \\ & - \frac{1}{2} \int_{\mathbb{R}^3} (\partial_t(\partial^2_\phi W)) (\phi_{x_i, x_j, x_k})^2~\mbox{d}x + \int_{\mathbb{R}^3} \mathcal{R}_{ijk} \phi_{x_i, x_j, x_k, t} ~\mbox{d}x . \end{split}$$ Note now that the first terms in the right hand side of both (\[dodici\]) and (\[dicianove\]), namely the terms displayed in boxes, are the same. Consequently, subtracting (\[dicianove\]) from (\[dodici\]), we get: $$\label{venti} \begin{split} \int_{\mathbb{R}^3} |\nabla&\psi_{x_i, x_j, x_k, t}|^2~\mbox{d}x \\ & + \frac{1}{2}\frac{\mbox{d}}{\mbox{d}t}\int_{\mathbb{R}^3} (u_{x_i, x_j, x_k, t})^2 + D^2W(\phi, \nabla u) : (\phi_{x_i, x_j, x_k}, \nabla u_{x_i, x_j, x_k})^{\otimes 2} ~\mbox{d}x \\ & + \int_{\mathbb{R}^3}\mathcal{R}_{ijk}\phi_{x_i, x_j, x_k, t} ~\mbox{d}x\\ & \qquad\qquad = \int_{\mathbb{R}^3} \phi_{x_i, x_j, x_k}\langle(\partial_t\partial_F\partial_\phi W):\nabla u_{x_i, x_j, x_k}\rangle ~\mbox{d}x \\ & \qquad \qquad\qquad + \frac{1}{2} \int_{\mathbb{R}^3} (\partial_t\partial^2_F\partial_\phi W) : \nabla u_{x_i, x_j, x_k} \otimes \nabla u_{x_i, x_j, x_k}~\mbox{d}x \\ & \qquad \qquad \qquad + \frac{1}{2} \int_{\mathbb{R}^3} (\partial_t(\partial^2_\phi W)) (\phi_{x_i, x_j, x_k})^2 ~\mbox{d}x + \int_{\mathbb{R}^3} \mathcal{R}_1 u_{x_i, x_j, x_k, t}~\mbox{d}x. \end{split}$$ [**3.**]{} We will now estimate terms in the right hand side of (\[venti\]) and prove that: $$\label{estim1} \begin{split} \int_{\mathbb{R}^3}& |\phi_{x_i, x_j, x_k}| |\partial_t\partial_F\partial_\phi W | |\nabla u_{x_i, x_j, x_k}|~\mbox{d}x + \int_{\mathbb{R}^3} |\partial_t\partial^2_F\partial_\phi W| |\nabla u_{x_i, x_j, x_k}|^2~\mbox{d}x \\ & \qquad\qquad + \int_{\mathbb{R}^3} |\partial_t(\partial^2_\phi W)| |\phi_{x_i, x_j, x_k}|^2 ~\mbox{d}x \leq C \left(\mathcal{Z}^{3/2} + \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \mathcal{Z}\right). \end{split}$$ and: $$\label{estim2} |\int_{\mathbb{R}^3} \mathcal{R}_1 u_{x_i, x_j, x_k, t}~\mbox{d}x |\leq C \left(\mathcal{Z}^{3/2} + \mathcal{Z}^2 + \mathcal{Z}^{5/2}\right) + C \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \left( \mathcal{Z}^{3/2} + \mathcal{Z}^2\right).$$ For the first term in (\[estim1\]), we note that by the Sobolev embedding $\mathcal{C}^{0, 1/2}(\mathbb{R}^3) \hookrightarrow H^{2}(\mathbb{R}^3)$ one easily gets: $$\begin{split} \int_{{\mathbb{R}}^3} |\phi_{x_i, x_j, x_k}| |\partial_t\partial_F\partial_\phi W | |\nabla u_{x_i, x_j, x_k}| & \leq \| (\partial^2_F\partial_\phi W) : \nabla u_t + (\partial_F\partial^2_\phi W)\phi_t\|_{L^\infty} \|\nabla^3\phi\|_{L^2} \|\nabla^4u\|_{L^2} \\ & \leq C\left( \|\nabla u_t\|_{L^\infty} + \|\phi_t\|_{L^\infty}\right) \|\nabla^3\phi\|_{L^2} \|\nabla^4u\|_{L^2} \\ & \leq C\left( \|\nabla u_t\|_{H^{2}} + \|\Delta\psi_t\|_{H^{2}}\right) \|\nabla^3\phi\|_{L^2} \|\nabla^4u\|_{L^2} \\ & \leq C \left(\mathcal{Z}^{3/2} + \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \mathcal{Z}\right). \end{split}$$ Similarly, the other two terms in (\[estim1\]) are bounded by: $$C\left( \|\nabla u_t\|_{L^\infty} + \|\phi_t\|_{L^\infty}\right) \left( \|\nabla^4u\|_{L^2}^2 + \|\nabla^3\phi\|_{L^2}^2\right),$$ which implies the same estimate as before. Regarding (\[estim2\]), the first term in $\int_{\mathbb{R}^3} \mathcal{R}_1u_{x_i, x_j, x_k, t} ~\mbox{d}x $, is bounded by: $$\begin{split} \int_{\mathbb{R}^3} |\mbox{div}\big((\partial_F^3 W) : \nabla u_x\otimes \nabla u_{xx}\big)| |\nabla^3u_t|~\mbox{d}x & \leq C \Big( (\|\nabla u_t\|_{L^\infty} + \|\phi_t\|_{L^\infty}) \|\nabla^2 u\|_{L^\infty} \|\nabla^3 u\|_{L^2} \\ & \qquad + \|\nabla^3 u\|_{L^4}^2 + \|\nabla^2 u\|_{L^\infty} \|\nabla^4 u\|_{L^\infty} \Big) \|\nabla^3 u_t\|_{L^2} \\ & \leq C \left(\mathcal{Z}^{3/2} + \mathcal{Z}^{2} + \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \mathcal{Z}^{3/2}\right), \end{split}$$ because of the Sobolev embedding $W^{1,2}(\mathbb{R}^3)\hookrightarrow L^p(\mathbb{R}^3)$ valid for any $p\in [2,6]$. Also: $$\begin{split} \int_{\mathbb{R}^3} |\mbox{div}\big((\partial_F^4 W) : (\nabla u_x)^{\otimes 3}\big)| |\nabla^3u_t|~\mbox{d}x & \leq C \Big( (\|\nabla u_t\|_{L^\infty} + \|\phi_t\|_{L^\infty}) \|\nabla^2 u\|^3_{L^6} \\ & \qquad\qquad + \|\nabla^2 u\|_{L^\infty}^2 \|\nabla^3 u\|_{L^2} \Big) \|\nabla^3 u_t\|_{L^2} \\ & \leq C \left(\mathcal{Z}^{5/2} + \mathcal{Z}^2 + \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \mathcal{Z}^2\right). \end{split}$$ Other terms in $\mathcal{R}_1$ induce the same estimate as above. This establishes (\[estim2\]). [**4.**]{} We now consider the last term in the right hand side of (\[venti\]): $$\label{ma} \int_{\mathbb{R}^3} \mathcal{R}_{ijk}\phi_{x_i, x_j, x_k, t} ~\mbox{d}x = \left(\frac{\mbox{d}}{\mbox{d}t} \int_{\mathbb{R}^3}\mathcal{R}_{ijk}\phi_{x_i, x_j, x_k}~\mbox{d}x \right) - \int_{\mathbb{R}^3}(\mathcal{R}_{ijk})_t\phi_{x_i, x_j, x_k}~\mbox{d}x.$$ We now prove that: $$\label{estim3} |\int_{\mathbb{R}^3} (\mathcal{R}_{ijk})_t \phi_{x_i, x_j, x_k}~\mbox{d}x |\leq C \left(\mathcal{Z}^{3/2} + \mathcal{Z}^{5/2}\right) + C \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \left( \mathcal{Z}^{3/2} + \mathcal{Z}^2\right).$$ First, using the notational convention as in (\[R1\]), $\mathcal{R}$ can be replaced by: $$\label{R3} \begin{split} \mathcal{R}_2 = & ~3(\partial_\phi\partial_F^2W) : \nabla u_x\otimes \nabla u_{xx} + 3(\partial_\phi^2\partial_F\partial_\phi W) : (\nabla u_{xx}\phi_x +\nabla u_x\phi_{xx}) \\ & + (\partial_\phi\partial_F^3W) : (\nabla u_x)^{\otimes 3} + 3(\partial^2_\phi\partial_F^2 W) : (\nabla u_x)^{\otimes 2} \phi_x \\ & + 3(\partial_F\partial^3_\phi W) : \nabla u_x (\phi_x)^2 + (\partial_\phi^4 W)(\phi_x)^3 + 3(\partial^3_\phi W)\phi_x\phi_{xx}. \end{split}$$ The first term in (\[R3\]) can be estimated as before, using embedding and interpolation theorems: $$\begin{split} \int_{\mathbb{R}^3} |\big((\partial_\phi\partial_F^2 &W) : \nabla u_x\otimes \nabla u_{xx}\big)_t| |\nabla^3\phi|~\mbox{d}x \\ & \leq C \Big( (\|\nabla u_t\|_{L^\infty} + \|\phi_t\|_{L^\infty}) \|\nabla^2 u\|_{L^\infty} \|\nabla^3 u\|_{L^2} \\ & \qquad \qquad + \|\nabla^2 u_t\|_{L^4}\|\nabla^3 u\|_{L^4} + \|\nabla^2 u\|_{L^\infty} \|\nabla^3 u_t\|_{L^2} \Big) \|\nabla^3 \phi\|_{L^2} \\ & \leq C \left(\mathcal{Z}^{3/2} + \mathcal{Z}^{2} + \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \mathcal{Z}^{3/2}\right), \end{split}$$ while the third term in $\mathcal{R}_2$ is estimated by: $$\begin{split} \int_{\mathbb{R}^3} |\big(\partial_\phi\partial_F^3 W) : & (\nabla u_x)^{\otimes 3}\big)_t| |\nabla^3\phi|~\mbox{d}x \\ & \leq C \Big( (\|\nabla u_t\|_{L^\infty} + \|\phi_t\|_{L^\infty}) \|\nabla^2 u\|^3_{L^6} + \|\nabla^2 u\|_{L^\infty}^2 \|\nabla^3 u_t \|_{L^2} \Big) \|\nabla^3 \phi\|_{L^2} \\ & \leq C \left(\mathcal{Z}^{5/2} + \mathcal{Z}^2 + \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)} \mathcal{Z}^2\right). \end{split}$$ Other terms in $\mathcal{R}_2$ induce the same estimate as above. This establishes (\[estim3\]). [**5.**]{} Summing now (\[venti\]) over all triples $x_i, x_j, x_k$, adding (\[nove\]), and taking into account (\[ma\]), (\[estim1\]), (\[estim2\]) and (\[estim3\]), we obtain: $$\begin{split} \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{E} + \left( 2\|\nabla\psi_t\|_{L^2}^2 + \|\nabla^4\psi_t\|^2_{L^2}\right) & \leq C \left(\mathcal{Z}^{5/2} + \mathcal{Z}^{3/2}\right) + C \|\nabla \psi_t\|_{W^{3}_{2}(\mathbb{R}^3)} \left(\mathcal{Z}^2 + \mathcal{Z}\right) \\ & \leq \epsilon \|\nabla \psi_t\|_{H^{3}(\mathbb{R}^3)}^2 + C \left(\mathcal{Z}^{4} + \mathcal{Z}^{3/2}\right). \end{split}$$ in view of Young’s inequality. Consequently, (\[ventinove\]) follows and the proof is complete. We now deduce the main a-priori estimate of this section: \[thap\] Under the assumptions of Theorem \[th1\], any solution on the time interval $[0, T]$ to (\[maineq\]) (\[initial1\] - \[initial2\]) satisfies: $$\label{a12A} \sup_{t\leq T} \mathcal{Z}(t) \leq C\Big(\mathcal{E}(0) + T^2\mathcal{E}_0(0) + \|u_0-\mathrm{id}\|_{L^2}^2\Big),$$ where: $\mathcal{E}_0(0)=\int_{\mathbb{R}^3} |u_1|^2 + 2W(\phi_0,\nabla u_0)~\mathrm{d}x$, and $C$ is a universal constant. Assume that the quantities in (\[small\]) are sufficiently small. In particular, we require that $\mathcal{Z}(0)\ll 1$ and that $\mathcal{Z}$ is sufficiently small on an interval $[0, t_0]$, where we also choose an appropriate $t_0\ll T$. Lemma \[lemapriori\] implies: $\mathcal{E}'(t) \leq C \mathcal{Z}^{3/2}(t)$, which is equivalent to: $$\label{a1} \mathcal{E}(t) \leq C\int_0^t \mathcal{Z}^{3/2}(s) ~\mbox{d}s + \mathcal{E}(0).$$ Further, by (\[nove\]) it follows that: $$\label{a2} \sup_t \|u_t\|_{L^2}^2 \leq \mathcal{E}_0(0).$$ Since: $$\label{a3} \begin{split} \forall t\leq t_0\qquad \|u(t)-\mathrm{id}\|_{L^2}^2 & = 2\int_0^t \int_{\mathbb{R}^3} \langle u -\mathrm{id}, u_t\rangle ~\mbox{d}x + \|u_0-\mathrm{id}\|_{L^2}^2 \\ & \leq 2T\big(\sup_{s\leq t} \|u_t\|_{L^2}\big) \Big(\sup_{s\leq t}\|u-\mathrm{id}|_{L^2}\Big) + \|u_0-\mathrm{id}\|_{L^2}^2, \end{split}$$ we easily obtain in view of (\[a2\]): $$\label{a4} \begin{split} \sup_{t\leq t_0} \|u(t)-\mathrm{id}\|_{L^2}^2& \leq 4t_0^2\sup_{t\leq t_0} \|u_t(t)\|_{L^2}^2 + 2\|u_0-\mathrm{id}\|_{L^2}^2 \\ & \leq 4 t_0^2 \mathcal{E}_0(0) + 2\|u_0-\mathrm{id}\|_{L^2}^2. \end{split}$$ Further, we observe that thanks to (\[ass\]), to Korn’s inequality and to Poincaré’s inequality, there exist constants $c, C > 0$ so that: $$\begin{split} c &\mathcal{Z}(t) \leq \\ & \int_{\mathbb{R}^3} |u_t|^2 + |\nabla^3u_t|^2 + 2W(\phi,\nabla u) + \sum_{i,j,k=1..3} D^2W(\phi,\nabla u) : (\phi_{x_i, x_j, x_k}, \nabla u_{x_i, x_j, x_k})^{\otimes 2} ~\mbox{d}x \\ & \qquad\qquad\qquad \qquad\qquad\qquad \qquad + \int_{\mathbb{R}^3} |u-\mathrm{id}|^2 ~\mbox{d}x \leq C \mathcal{Z}(t), \end{split}$$ as well as: $$\big |\int_{\mathbb{R}^3} \sum_{i,j,k=1..3} \mathcal{R}_{ijk}\phi_{x_i, x_j, x_k} ~\mathrm{d}x\big |\leq C \|\mathcal{R}\|_{L^2} \|\nabla^3\phi\|_{L^2} \leq C \mathcal{Z}^{3/2}(t) \mathcal{Z}^{1/2}(t) = C \mathcal{Z}^{2}(t),$$ where we estimated each term in (\[R2\]) by the Cauchy-Schwartz inequality and noted the appropriate Sobolev embedding. Consequently, we arrive at: $$\label{a7} \forall t\leq t_0\qquad \mathcal{E}(t)+\|u-\mathrm{id}\|_{L^2}^2(t) \geq c\mathcal{Z}(t) - C\mathcal{Z}^{2}(t) \geq c \mathcal{Z}(t),$$ provided that $\mathcal{Z}\ll 1$ is sufficiently small on the time interval we consider. In view of (\[a1\]), (\[a7\]) and (\[a4\]), we now get: $$\label{a8} \forall t\leq t_0\qquad \mathcal{Z}(t) \leq C\Big(\int_0^t \mathcal{Z}^{3/2}(s)~\mbox{d}s + \mathcal{E}(0) + t_0^2\mathcal{E}_0(0) + \|u_0-\mathrm{id}\|_{L^2}^2\Big).$$ Calling $\bar{\mathcal{Z}} =\sup_{t\in [0,t_0]} \mathcal{Z}(t)$, we have: $$\label{a10} \bar{\mathcal{Z}} \leq \big(C t_0 {\bar{\mathcal{Z}}}^{1/2}\Big) \bar{\mathcal{Z}} + C\big(\mathcal{E}(0) + t_0^2\mathcal{E}_0(0) + \|u_0-\mathrm{id}\|_{L^2}^2\big),$$ which combined with the requirement: $ C T {\bar{\mathcal{Z}}}^{1/2} \leq \frac12$ yields: $$\label{a12} \mathcal{Z}(t_0) \leq \bar{\mathcal{Z}}\leq C\big(\mathcal{E}(0) + t_0^2\mathcal{E}_0(0) + \|u_0-\mathrm{id}\|_{L^2}^2\big).$$ The above clearly implies the Theorem in view of the smallness of initial data in (\[small\]). Proof of Theorem \[th1\]: Existence of solutions to (\[maineq\]). {#sec3} ================================================================= In this section we construct approximate solutions to the Cauchy problem (\[maineq\]) (\[initial1\] - \[initial2\]), which satisfy the same a-priori bounds as in section \[abd\]. Given $\epsilon >0$, consider the regularized problem: $$\label{approx} \left\{\begin{split} & u_{tt} - \mbox{div}\Big(\partial_F W(\phi, \nabla u)\Big) - \epsilon \Delta u = 0 \\ & \phi_t = \Delta \Big(\partial_\phi W(\phi,\nabla u)\Big) \end{split}\right.$$ with the same initial data as in (\[initial1\] - \[initial2\]). \[th3.1\] Assume that all quantities in (\[small\]) are sufficiently small. Then, there exists $T_\epsilon >0$ and a solution $(u^\epsilon, \phi^\epsilon)$ of (\[approx\]) (\[initial1\] - \[initial2\]) on $\mathbb{R}^3\times [0, T_\epsilon)$, such that: $$\label{oldbounds} \begin{split} & u^\epsilon -\mathrm{id} \in L^\infty(0,T;H^4(\mathbb{R}^3)), \quad u^\epsilon_{tt}\in L^\infty(0,T;H^2({\mathbb{R}}^3)), \\ & \phi^\epsilon \in L^\infty(0,T;H^3(\mathbb{R}^3)) \mbox{ \ \ and \ \ } \phi^\epsilon_t \in L^2(0,T;H^2(\mathbb{R}^3)). \end{split}$$ [**1.**]{} Since $\epsilon>0$ is fixed, we drop the superscript $^\epsilon$ in order to lighten the notation in the next two steps. We proceed by the Galerkin method. Choose an orthonormal base $\{w^k\}_{k=1}^\infty$ in the space $H^4({\mathbb{R}}^3,{\mathbb{R}}^3)$ equipped with the scalar product: $$\label{e2} \langle w, \tilde w\rangle_{H^4}=\langle w, \tilde w\rangle_{L^2} + \langle \nabla^4 w : \nabla^4 \tilde w\rangle_{L^2}.$$ Similarly, let $\{v^k\}_{k=1}^\infty$ be an orthonormal basis in $H^3({\mathbb{R}}^3)$ equipped with: $$\label{e3} \langle v, \tilde v\rangle_{H^3} = \langle v,\tilde v\rangle_{L^2} + \langle \nabla^3 v : \nabla^3 \tilde v\rangle_{L^2}.$$ Denote: $W^N=\mbox{span} \{w^1,...,w^N\}$ and $V^N=\mbox{span} \{v^1,...,v^N\}$. We now introduce the auxiliary scalar products: $$\label{e2a} \begin{split} &\langle w, \tilde w\rangle_{W}=\langle w, \tilde w\rangle_{L^2} + \langle \nabla^3 w : \nabla^3 \tilde w\rangle_{L^2}\qquad \forall w,\tilde w \in H^4({\mathbb{R}}^3,{\mathbb{R}}^3),\\ & \langle v,\tilde v\rangle_{V} = \langle v,(-\Delta)^{-1} \tilde v\rangle_{L^2} + \langle \nabla^3 v : (-\Delta)^{-1} \nabla^3 \tilde v\rangle_{L^2} \qquad \forall v, \tilde v\in H^3({\mathbb{R}}^3,{\mathbb{R}}). \end{split}$$ Clearly, these products are not equivalent to (\[e2\]), (\[e3\]), however their properties will allow for using the energy estimates of the proof of Lemma \[lemapriori\] to prove the regularity of approximate solutions $(u^N,\phi^N)$ which we define below. Let $(u^N, \phi^N)\in W^N\times V^N$ be the solution to: $$\label{e5} \left\{ \begin{split} & \big \langle u^N_{tt} - {{\rm div}\,}\partial_FW(\phi^N,\nabla u^N) - \epsilon \Delta u^N, w^l\big\rangle_W=0, \\ & \big \langle \phi_t^N - \Delta \partial_\phi W(\phi^N,\nabla u^N), v^l\big\rangle_V=0, \qquad\qquad\qquad \forall l:1\ldots N \\ & u^N(0,\cdot) = \mathbb{P}_{W^N}(u_0), \quad (u^N)_t(0,\cdot) = \mathbb{P}_{W^N}(u_1), \quad \phi^N(0,\cdot) = \mathbb{P}_{V^N}(\phi_0). \end{split}\right.$$ By $\mathbb{P}$ we denote here the orthogonal projections on appropriate subspaces. The classical theory of systems of ODEs guarantees existence of solutions to (\[e5\]) on some time interval $[0,T_N)$. We now prove that these time intervals may be taken uniform for all sequences $\{u^N,\phi^N\}$. [**2.**]{} Since $u_t^N\in W^N$ and $\phi_t^N\in V^N$, (\[e5\]) implies: $$\label{e6} \begin{split} & \big\langle u^N_{tt} - {{\rm div}\,}\partial_F W(\phi^N,\nabla u^N) - \epsilon \Delta u^N, u^N_t\big\rangle_W=0, \\ & \big \langle \phi_t^N - \Delta \partial_\phi W(\phi^N,\nabla u^N), \phi^N_t\big\rangle_V=0. \end{split}$$ Note that the first equation in (\[e6\]) is equivalent to: $$\label{e6a} \begin{split} & \Big \langle u^N_{tt} - {{\rm div}\,}\partial_FW(\phi^N,\nabla u^N) - \epsilon \Delta u^N, u^N_t\Big \rangle_{L^2} \\ & \quad + \sum_{i,j,k = 1..3} \Big\langle u^N_{x_i, x_j, x_k, tt} - {\rm div}\big(\partial_F^2W(\phi^N, \nabla u^N) : \nabla u^N_{x_i, x_j, x_k}\big) -\epsilon \Delta u^N_{x_i, x_j, x_k}, u^N_{x_i,x_j,x_k,t} \Big \rangle_{L^2} \\ & = \sum_{i,j,k = 1..3} \Big\langle {\rm div}\big(\partial_F\partial_\phi W(\phi^N, \nabla u^N)\phi_{x_i, x_j, x_k}\big), u^N_{x_i,x_j,x_k,t} \Big \rangle_{L^2} + \big\langle \mathcal{R}_1^{N}: \nabla^3 u^N_t\big\rangle, \end{split}$$ where by $\mathcal{R}_1^N$ we denote the error terms induced by the functions $u^N, \phi^N$ as in (\[undici\]), (\[R1\]). Likewise, the second equation in (\[e6\]) becomes: $$\label{e6b} \begin{split} & \Big\langle \phi_t^N - \Delta \partial_\phi W(\phi^N,\nabla u^N), (-\Delta)^{-1} \phi^N_t\Big\rangle_{L^2}\\ & \quad + \sum_{i,j,k =1..3 } \Big\langle \phi^N_{x_i, x_j, x_k, t} - \Delta\Big((\partial_\phi\partial_F W^N) : \nabla u^N_{x_i, x_j, x_k} - (\partial^2_\phi W^N) \phi^N_{x_i, x_j, x_k}\Big), (-\Delta)^{-1} \phi^N_{x_i, x_j, x_k, t} \Big\rangle_{L^2}\\ & = \sum_{i,j,k =1..3 } \Big\langle \Delta \mathcal{R}^N_{ijk}, (-\Delta)^{-1} \phi^N_{x_i, x_j, x_k, t} \Big\rangle_{L^2}, \end{split}$$ where we used the identity (\[diciotto\]) and the notation (\[R2\]), with the superscript $^N$ indicating that they concern $u^N$ and $\phi^N$. Let $\mathcal{E}[u^N, \phi^N](t)$ be as in Lemma \[lemapriori\] with $(u,\phi)$ replaced by $(u^N, \phi^N)$, and define: $$\label{e7} \mathcal{E}_\epsilon[u^N, \phi^N](t) = \mathcal{E}[u^N,\phi^N](t) + \epsilon \big\langle (-\Delta)u^N,u^N\big\rangle_V,$$ so that: $$\begin{split} \mathcal{E}_\epsilon[\phi^N,u^N](t) = \int_{\mathbb{R}^3} & |u_t^N|^2 + |\nabla^3u^N_t|^2 + 2W(\phi^N,\nabla u^N) +\epsilon |\nabla u^N|^2 \\ & + \sum_{i,j,k=1..3} D^2W(\phi^N,\nabla u^n) : (\phi^N_{x_i, x_j, x_k}, \nabla u^N_{x_i, x_j, x_k})^{\otimes 2} + \epsilon |\nabla u^N_{x_i, x_j, x_k}|^2\\ & + 2 \sum_{i,j,k=1..3} \mathcal{R}^N_{ijk}\phi^N_{x_i, x_j, x_k} ~\mathrm{d}x. \end{split}$$ Following the proof of Lemmas \[gaga5\] and \[lemapriori\], we find the counterpart of the inequality (\[ventinove\]): $$\label{e8} \mathcal{E}_\epsilon[u^N, \phi^N](t) \leq C\int_0^t \mathcal{Z}^{3/2}[u^N, \phi^N](s)~\mbox{d}s + \mathcal{E}_\epsilon(0),$$ where the constant $C$ is independent from $\epsilon$, and where: $$\mathcal{Z}[u^N, \phi^N] (t) = \|u^N_t\|^2_{H^{3}(\mathbb{R}^3)} + \|\nabla u^N - \mathrm{Id}\|^2_{H^{3}(\mathbb{R}^3)} + \|\phi^N\|^2_{H^{3}(\mathbb{R}^3)}.$$ Note that in order to obtain (\[e8\]) we use only the equivalent formulations of (\[e6\]) above, hence indeed all the steps from the proof of Lemma \[lemapriori\] are valid with universal constants. Since the initial data in (\[e5\]) consists of projections of the original data, their norms are uniformly controlled as well. [**3.**]{} We now consider the equivalence of $\mathcal{E}_\epsilon$ with $\mathcal{Z}$. Since for small $\mathcal{Z}$ one has: $$\int \mathcal{R}^N_{ijk}\phi^N_{x_i, x_j, x_k} ~\mathrm{d}x \leq C \mathcal{Z}^{3/2}[u^N, \phi^N],$$ we easily see that: $$\label{e8a} \mathcal{E}_\epsilon[u^N, \phi^N] \leq C\mathcal{Z}[u^N, \phi^N].$$ On the other hand, in view of (\[e7\]): $$\label{e9} \mathcal{E}_\epsilon[u^N, \phi^N] \geq c_\epsilon (\mathcal{Z} - C_\epsilon \mathcal{Z}^{3/2}) \geq c_\epsilon \mathcal{Z}[u^N, \phi^N],$$ where by $c_\epsilon, C_\epsilon$ we denote positive constants independent of $N$ but depending on $\epsilon$. By (\[e8\]) we now arrive at: $$\mathcal{Z}[u^N, \phi^N](t)\leq C_\epsilon \int_0^t \mathcal{Z}^{3/2}[u^N, \phi^N](s)~\mbox{d}s + C_\epsilon \mathcal{E}_\epsilon(0).$$ Consequently, for $t_{0,\epsilon}$ sufficiently small, we have: $$\sup_{t\leq t_{0,\epsilon}} \mathcal{Z}[u^N, \phi^N](t) \leq C_\epsilon \mathcal{E}_\epsilon(0).$$ The above estimates, in particular (\[e8a\]) and (\[e9\]) imply the uniform in $N$ boundedness of the following quantities, on their common interval of existence $[0, T_\epsilon]$: $$\label{e10} \begin{split} & u^N - \mathrm{id} \in L^\infty(0,T_\epsilon;H^4(\mathbb{R}^3)), \qquad u^N_t \in L^\infty(0,T_\epsilon;H^3(\mathbb{R}^3)),\\ & \phi^N \in L^\infty(0,T_\epsilon;H^3(\mathbb{R}^3)),\qquad \phi_t^N \in L^2(0,T_\epsilon;H^2(\mathbb{R}^3)), \end{split}$$ yielding the weak-$*$ convergence in $L^\infty$ as $N\to\infty$ (up to a subsequence), of the quantities: $u^N - \mathrm{id}$, $u_t^N$, $\phi^N$, $\phi_t^N$ to the limiting quantities: $u^\epsilon - \mathrm{id}$, $u_t^\epsilon$, $\phi^\epsilon$, $\phi_t^\epsilon$. Additionally, passing if necessary to a further subsequence and invoking a diagonal argument, we may also assure that: $$\nabla u^N \to \nabla u^\epsilon \quad \mbox{ and } \quad \phi^N \to \phi^\epsilon \qquad \mbox{point-wise in } \mathbb{R}^3.$$ The Sobolev compact embedding: $ H^1(0,T_\epsilon; H^2(B(R))) \hookrightarrow \mathcal{C}^\alpha((0,T_\epsilon)\times B(R))$, valid on any ball $B(R)\subset\mathbb{R}^3$, justifies now that $\phi^\epsilon \in \mathcal{C}^\alpha((0,T_\epsilon)\times B(R))$. Thus, in particular: $$(\partial_\phi W,~ \partial_FW)(\phi^N,\nabla u^N) \to (\partial_\phi W,~ \partial_FW)(\phi^\epsilon,\nabla u^\epsilon) \quad \mbox{ as } N\to\infty.$$ It follows that $(\phi^\epsilon, u^\epsilon)$ is a distributional solution to (\[approx\]) (\[initial1\] - \[initial2\]). By (\[e10\]) we obtain the desired regularity, completing the proof of Lemma \[th3.1\]. [**Proof of Theorem \[th1\] (existence part).**]{} Let $\phi^\epsilon, u^\epsilon$ be as in Lemma \[th3.1\]. We first observe that a common interval of existence of $(\phi^\epsilon, u^\epsilon)$ can be taken as $[0,T]$ with $T$ prescribed by Theorem \[th1\]. This follows through repeating the estimates in section \[abd\], dealing with estimates of the first and the third order separately, and noting that the $\epsilon$-term appears exclusively in $\mathcal{E}_\epsilon$ with a “good” sign. Consequently: $$\label{e13} \sup_{t\leq t_0} \mathcal{Z}[u^\epsilon, \phi^\epsilon]\leq C(t_0, \mbox{initial data}),$$ and we see that indeed the solutions $\phi^\epsilon, u^\epsilon$ can be extended over appropriate $[0,T]$ with the quantities in (\[oldbounds\]) enjoying common bounds, independent of $\epsilon$. The same argument as in the last part of the proof of Lemma \[th3.1\] implies now that the weak-$*$ limit (up to a subsequence) of $(\phi^\epsilon, u^\epsilon)$ yield the desired regular solution $(\phi, u)$ to the original problem (\[maineq\]) (\[initial1\] - \[initial2\]). Condition (\[nonin\]) is automatically satisfied because of the smallness of initial data. Proof of Theorem \[th1\]: Uniqueness of solutions to (\[maineq\]). {#sec4} ================================================================== Let $(\phi, u)$ and $(\bar \phi, \bar u)$ be two solutions to (\[maineq\]) with the same initial data. Define: $$\label{u1} (\delta \phi) = \phi - \bar \phi, \qquad (\delta u) = u - \bar u,$$ and observe that: $$\label{u2} \begin{split} & {\partial}_FW(\phi,\nabla u) - {\partial}_FW(\bar \phi,\nabla \bar u) \\ & = ~ {\partial}^2_FW(\bar \phi,\nabla \bar u) : \nabla (\delta u) + {\partial}_\phi{\partial}_F W (\bar \phi,\nabla \bar \phi) (\delta \phi) + D^2{\partial}_F W(\tilde \phi, \nabla \tilde u) : \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2}, \\ & {\partial}_\phi W(\phi,\nabla u) - {\partial}_\phi W(\bar \phi,\nabla \bar u) \\ & = ~ {\partial}_\phi^2 W(\bar \phi,\nabla \bar u) (\delta \phi) + {\partial}_\phi{\partial}_F W(\bar \phi,\nabla \bar u) : \nabla (\delta u) + D^2 {\partial}_\phi W(\tilde \phi, \nabla \tilde u) : \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2}, \end{split}$$ where $\tilde \phi$ and $\tilde u$ are suitable linear combinations of $\phi,\bar \phi$ and $u,\bar u$, given by the application of the Taylor formula. Subtracting equations (\[maineq\]) for $(\phi,u)$ and $(\bar \phi,\bar u)$ and using (\[u2\]), it follows that: $$\label{u4} \begin{split} & (\delta u)_{tt} - {{\rm div}\,}\Big({\partial}^2_FW(\bar \phi,\nabla \bar u) : \nabla (\delta u) + {\partial}_\phi{\partial}_F W (\bar \phi,\nabla \bar u) (\delta \phi)\Big) \\ & \qquad\qquad\qquad\qquad \qquad = {{\rm div}\,}\Big( D^2{\partial}_F W(\tilde \phi, \nabla \tilde u) : \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2}\Big), \\ & (\delta \phi)_t - \Delta \Big( {\partial}_\phi^2 W(\bar \phi,\nabla \bar u) (\delta \phi) + {\partial}_\phi{\partial}_F W(\bar \phi,\nabla \bar u) : \nabla (\delta u) \Big) \\ & \qquad\qquad\qquad\qquad \qquad = \Delta \Big( D^2 {\partial}_\phi W(\tilde \phi, \nabla \tilde u) : \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2} \Big). \end{split}$$ We now test the first equation above by $(\delta u)_t$, while the second equation by $(-\Delta)^{-1} (\delta \phi)_t$, to obtain: $$\label{u6} \begin{split} & \frac{1}{2} \int_{\mathbb{R}^3} |\delta u_t|^2 + {\partial}_F^2 W(\bar \phi, \nabla \bar u) : \big (\nabla (\delta u)\big )^{\otimes 2} ~\mbox{d}x + \int_{\mathbb{R}^3} {\partial}_\phi{\partial}_F W(\bar \phi,\nabla \bar u) : (\delta \phi) \nabla (\delta u)_t ~\mbox{d}x \\ & \qquad\qquad\qquad = \int_{\mathbb{R}^3} D^2{\partial}_F W(\tilde \phi, \nabla \tilde u) : \Big( \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2}\otimes \nabla (\delta u)_t\Big) ~\mbox{d}x, \\ & \int_{\mathbb{R}^3} |\nabla (\delta \phi)_t|^2 ~\mbox{d}x + \int_{\mathbb{R}^3} \Big( {\partial}_\phi^2 W(\bar \phi,\nabla \bar u) (\delta \phi) + {\partial}_\phi{\partial}_F W(\bar \phi,\nabla \bar u) : \nabla (\delta u) \Big)(\delta \phi)_t ~\mbox{d}x \\ & \qquad\qquad\qquad = \int_{\mathbb{R}^3} D^2 {\partial}_\phi W(\tilde \phi, \nabla \tilde u) : \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2} (\delta \phi)_t ~\mbox{d}x. \end{split}$$ Consequently: $$\label{u8} \begin{split} & \frac 12 \frac{\mbox{d}}{\mbox{d}t} \int_{\mathbb{R}^3} {\partial}_\phi ^2 W(\bar \phi,\nabla \bar u) (\delta \phi)^{2}~\mbox{d}x + \int_{\mathbb{R}^3} {\partial}_\phi{\partial}_F W(\bar \phi,\nabla \bar u): (\delta \phi_t)\nabla (\delta u) ~\mbox{d}x \\ & \qquad\qquad\qquad\qquad\qquad\qquad \qquad ~~ ~~ + \int_{\mathbb{R}^3} |\nabla (\delta \phi)_t|^2 ~\mbox{d}x \\ & = \int_{\mathbb{R}^3} D^2 {\partial}_\phi W(\tilde \phi, \nabla \tilde u) : \big((\delta \phi),\nabla (\delta u)\big)^{\otimes 2} (\delta \phi)_t ~\mbox{d}x \\ & \qquad\qquad\qquad\qquad\qquad\qquad \qquad ~~ ~~ + \int_{\mathbb{R}^3} {\partial}_t \Big({\partial}_\phi^2 W(\bar \phi,\nabla \bar u)\Big) (\delta \phi)^{2} ~\mbox{d}x. \end{split}$$ Adding (\[u6\]) and (\[u8\]), we arrive at: $$\begin{split} & \frac{\mbox{d}}{\mbox{d}t}\frac{1}{2} \int_{\mathbb{R}^3} |(\delta u)_t|^2 + {\partial}_F^2 W(\bar \phi, \nabla \bar u) : \big(\nabla (\delta u)\big)^{\otimes 2} + {\partial}_\phi^2 W(\bar \phi,\nabla \bar u) (\delta \phi)^{2} \\ & \qquad\qquad\qquad\qquad \qquad \qquad\qquad\qquad\qquad \quad + 2{\partial}_\phi{\partial}_F W(\bar \phi,\nabla \bar u) : (\delta\phi)\nabla (\delta u) ~\mbox{d}x \\ & \leq C \|(\delta \phi)_t, \phi_t,\bar \phi_t\|_{L_\infty(\mathbb{R}^3)}(t) \cdot \sup_t \|(\delta \phi),\nabla (\delta u)\|_{L^2(\mathbb{R}^3)}^2(t), \end{split}$$ which implies that: $$\label{u10} \begin{split} & \sup_t \int_{\mathbb{R}^3} |\delta u_t|^2 + D^2 W(\bar \phi, \nabla \bar u) : \big((\delta \phi), \nabla (\delta u)\big)^{\otimes 2} ~\mbox{d}x \leq \\ & \qquad\qquad\qquad C \sup_t \|(\delta \phi),\nabla (\delta u)\|_{L^2}^2 \int_0^t \|(\delta \phi)_t,\phi_t,\bar \phi_t\|_{L_\infty}(s) ~\mbox{d}s. \end{split}$$ As before, assumptions on $W$ guarantee that the left hand side in (\[u10\]) bounds from above the quantity: $\sup_t \|(\delta \phi),\nabla (\delta u)\|_{L^2}^2$. Since the integral quantity above is small for $t\ll 1$, it follows by (\[u10\]) that $(\delta \phi)$ and $\nabla (\delta u)$ are zero. Proof of Theorem \[th2\]: The elliptic-parabolic problem (\[maineq2\]). {#sec5} ======================================================================= As in section \[sec3\], we first derive an a priori estimate for solutions of (\[maineq2\]), whose existence will follow then via Galerkin’s method, in the same manner as for the system (\[maineq\]). \[aprio\] Assume that $(\phi, u)$ is a sufficiently smooth solution to (\[maineq2\]) which remains in a vicinity of $(0, \mathrm{id})$ for all $t\geq 0$, in the sense that: $$\Xi[\phi,\nabla u - \mathrm{Id}] := \sup_{t\geq 0}( \|\phi\|_{H^2(\mathbb{R}^3)}^2 +\|\nabla u -Id\|_{H^2}) + c \int_0^\infty \|\nabla \phi,\nabla^2 u\|_{H^2(\mathbb{R}^3)}^2 ~\mathrm{d}t \ll 1$$ Then: $$\begin{split} \sup_{t\geq 0} \big(\|\nabla u(t) -\mathrm{Id}\|^2_{H^2} + \|u(t) - \mathrm{id}\|^2_{L^6}\big) + \Xi[\phi,\nabla u - \mathrm{Id}] \leq C\|\phi_0\|^2_{H^2(\mathrm{R}^3)}. \end{split}$$ [**1.**]{} Observe first the following elementary fact: $$\label{o12} \|\nabla^2 u(t)\|_{H^1(\mathbb{R}^3)} \leq C \|\nabla \phi(t)\|_{H^1(\mathbb{R}^3)}.$$ To see (\[o12\]), consider the first equation in $(\ref{maineq2})$: $$\label{o13} \partial^2_{F}W(\phi,\nabla u) : \nabla u_{x_i} = - \partial_{\phi}\partial_{F}W(\phi,\nabla u)\phi_{x_i} \qquad i=1\ldots 3.$$ Condition (\[ass\]) and Korn’s inequality imply that the system (\[o13\]) is elliptic, hence its solutions (normalised so that $\nabla u - \mbox{Id} \in L^6(\mathbb{R}^3)$) obey: $$\label{o14} \|\nabla^2 u\|_{L^p(\mathbb{R}^3)} \leq C\|\nabla \phi\|_{L^p(\mathbb{R}^3)} \qquad p=2,4.$$ Differentiating (\[o13\]) with respect to $x$ leads further to: $$\|\nabla^3 u\|_{L^2} \leq C(\|\nabla^2 \phi\|_{L^2} + \|\nabla \phi,\nabla^2 u\|_{L^4}^2)\leq C(\|\nabla^2 \phi\|_{L^2} + \|\nabla \phi\|_{L^4}^2) \leq C(\|\nabla^2 \phi\|_{L^2} + \|\nabla \phi\|_{H^1}^2),$$ proving (\[o12\]) in view of the assumption in the Lemma. We also observe the resulting control of pointwise smallness of $\phi$ and $(\nabla u - \mbox{Id})$, by the Sobolev embedding: $$\label{o23} \sup_{t\geq 0} \|\phi, \nabla u - \mbox{Id}\|_{L^\infty} \leq C \sup_{t\geq 0} \|\phi, \nabla u - \mbox{Id}\|_{H^2} \leq C\Xi^{1/2} \ll 1.$$ [**2.**]{} Testing the first equation in (\[maineq2\]) by $u_t$ and the second one by $\psi_t = (-\Delta)^{-1} \phi_t$, we obtain the energy estimate, as in Lemma \[gaga5\]: $$\frac{\mbox{d}}{~\mbox{d}t}\int_{\mathbb{R}^3} W(\phi,\nabla u) ~\mbox{d}x + \int_{\mathbb{R}^3} |\nabla (-\Delta)^{-1} \phi_t|^2~\mbox{d}x =0.$$ To derive the second energy estimate we proceed slightly differently. Differentiating (\[maineq2\]) in a spatial direction $x_i\in \{x_1,x_2,x_3$}, we get: $$\label{o3} \begin{split} & {{\rm div}\,}\big(\partial_{F}^2W(\phi,\nabla u):\nabla u_{x_i} +\partial_{F}\partial_\phi W(\phi,\nabla u)\phi_{x_i}\big)=0, \vspace{1mm}\\ &\phi_{x_i, t}=\Delta \big(\partial_{\phi}\partial_{F}W(\phi, \nabla u) : \nabla u_{x_i} + \partial^2_{\phi}W(\phi,\nabla u)\phi_{x_i}\big). \end{split}$$ Now, testing the first equation above by $u_{x_i}$, testing the second one by $\psi_{x_i} = (-\Delta)^{-1} \phi_{x_i}$, and summing up the results, yields: $$\label{o4} \frac{1}{2} \frac{\mbox{d}}{~\mbox{d}t} \int_{\mathbb{R}^3} |\nabla(-\Delta)^{-1} \phi_{x_i}|^2 ~\mbox{d}x + \int_{\mathbb{R}^3} D^2W(\phi,\nabla u) : (\phi_{x_i},\nabla u_{x_i})^{\otimes 2} ~\mbox{d}x=0 .$$ Consequently, thanks to (\[ass\]), the strict convexity of $W$ implies: $$\frac{1}{2}\frac{\mbox{d}}{\mbox{d}t} \int_{{\mathbb{R}}^3}|\nabla^2\psi|^2 + \frac{\gamma}{2}\int_{{\mathbb{R}}^3} \big(|\nabla\phi|^2 + \sum_{i=1}^3 | (\mbox{sym}\nabla u_{x_i})|^2\big)\leq 0.$$ Using Korn’s inequality and integrating in time we see that: $$\label{o4a} \sup_{t>0} \int_{{\mathbb{R}}^3} \phi^2 ~\mbox{d}x + c \int_0^\infty \int_{{\mathbb{R}}^3} (|\nabla \phi|^2 + |\nabla^2 u|^2) ~\mbox{d}x \mbox{d}t\leq C\|\phi_0\|^2_{L^2(\mathbb{R}^3)}.$$ [**3.**]{} We now differentiate (\[o3\]) in a spatial direction $x_j \in \{x_1,x_2,x_3\} $, getting: $$\label{o5} \begin{array}{l} {{\rm div}\,}\big({\partial}_{F}^2W(\phi,\nabla u) : \nabla u_{x_i,x_j} +{\partial}_{\phi}{\partial}_FW(\phi,\nabla u) \phi_{x_i,x_j}\big)={{\rm div}\,}\mathcal{R}_1, \\[5pt] \phi_{x_i,x_j,t} - \Delta (\partial^2_{\phi}W(\phi,\nabla u)\phi_{x_i,x_j} + \partial_{\phi}\partial_F W(\phi,\nabla u) : \nabla u_{x_i,x_j}\big)=\Delta \mathcal{R}_2, \end{array}$$ where the error terms $\mathcal{R}_1$ and $\mathcal{R}_1$ have the following structure (we suppress the distinction between different $x_i, x_j$): $$\label{o6} \mathcal{R}_1 , \mathcal{R}_2 \sim D^3W(\phi,\nabla u) : \big( (\nabla u_x)^{\otimes 2} + (\nabla u_x) \phi_x + (\phi_x)^2\big).$$ Integrating (\[o5\]) by parts against $u_{x_i, x_j}$ and $(-\Delta)^{-1}\phi_{x_i, x_j}$, respectively, it follows that: $$\label{o7} \begin{split} \frac{\mbox{d}}{~\mbox{d}t} \int_{{\mathbb{R}}^3} |\nabla \psi_{x_i,x_j}|^2 ~\mbox{d}x ~ + c\int_{{\mathbb{R}}^3} D^2 W(\phi,\nabla u)& : (\phi_{x_i,x_j},\nabla u_{x_i,x_j})^{\otimes 2} ~\mbox{d}x \\ & \qquad \leq C \int_{{\mathbb{R}}^3} |\nabla^2 u|^4 + |\nabla \phi|^4 ~\mbox{d}x, \end{split}$$ because of (\[o6\]) and: $$\begin{split} |\int_{{\mathbb{R}}^3} ({{\rm div}\,}\mathcal{R}_1) u_{x_i,x_j} ~\mbox{d}x| + & |\int_{{\mathbb{R}}^3} (\Delta \mathcal{R}_2) (-\Delta)^{-1} \phi_{x_i,x_j} ~\mbox{d}x| \\ & \qquad \leq \epsilon \| \nabla u_{x_i,x_j}, \phi_{x_i,x_j}\|_{L^2(\mathbb{R}^3)}^2 + C\|\mathcal{R}_1,\mathcal{R}_2\|^2_{L^2(\mathbb{R}^3)}. \end{split}$$ Differentiating (\[o5\]) further, we obtain: $$\begin{array}{l} {{\rm div}\,}\big(\partial_{F}^2W(\phi,\nabla u) : \nabla u_{x_i,x_j,x_k} + \partial_{\phi}\partial_FW(\phi,\nabla u)\phi_{x_i,x_j,x_k}\big) = {{\rm div}\,}\mathcal{R}_3, \\[5pt] \phi_{x_i,x_j,x_k,t} - \Delta \big(\partial^2_{\phi}W(\phi,\nabla u)\phi_{x_i,x_j,x_k} + \partial_{\phi}\partial_F W(\phi,\nabla u) : \nabla u_{x_i,x_j,x_k}\big)=\Delta \mathcal{R}_4, \end{array}$$ where, as before: $$\begin{gathered} \mathcal{R}_3 , \mathcal{R}_4 \sim D^3W(\phi,\nabla u) : \big( \nabla u_{xx} \otimes \nabla u_x + \nabla u_{xx}\phi_x + \nabla u_x\phi_{xx} +\phi_x\phi_{xx}\big)\\ + D^4W(\phi,\nabla u) : \big( (\nabla u_x)^{\otimes 3} + (\nabla u_x)^{\otimes 2}\phi_x + \nabla u_{x}(\phi_x)^2 + \nabla u_x(\phi_{x})^2 + (\phi_x)^2\big).\end{gathered}$$ Testing by $u_{x_i,x_j,x_k}$ and $\psi_{x_i,x_j,x_k}$, we find: $$\label{o10} \begin{split} \frac{\mbox{d}}{~\mbox{d}t} \int_{{\mathbb{R}}^3} |\nabla \psi_{x_i,x_j,x_k}|^2 &~\mbox{d}x + c\int_{{\mathbb{R}}^3} D^2 W(\phi,\nabla u) : (\phi_{x_i,x_j,x_k},\nabla u_{x_i,x_j,x_k})^{\otimes 2} ~\mbox{d}x \\ & \leq C \int_{{\mathbb{R}}^3} |\nabla^3 u|^4 + |\nabla^2 u|^4 + |\nabla\phi|^4 + |\nabla^2\phi|^4 + |\nabla^2 u|^6 + |\nabla\phi|^6 ~\mbox{d}x. \end{split}$$ Summing (\[o4\]), (\[o7\]), (\[o10\]), integrating the result in time in the same manner as in (\[o4a\]), and recalling (\[o12\]), we obtain: $$\label{o11} \begin{split} \Xi[\phi,\nabla u - \mbox{Id}] & \leq C \int_0^\infty \int_{\mathbb{R}^3} |\nabla^2 u|^4 + |\nabla \phi|^4 + |\nabla^2 u|^6 + |\nabla \phi|^6 \\ & \qquad \qquad \qquad\qquad + |\nabla^3 u|^4 + |\nabla^2 u|^4 + |\nabla^2\phi |^4 ~\mbox{d}x \mbox{d}t + C\|\phi_0\|^2_{H^2(\mathbb{R}^3)} \\ & \leq C \int_0^\infty \int_{\mathbb{R}^3} |\nabla \phi|^4 + |\nabla \phi|^6 + |\nabla^2 \phi|^4 ~\mbox{d}x \mbox{d}t + C\|\phi_0\|^2_{H^2(\mathbb{R}^3)}. \end{split}$$ We further have: $$\begin{split} \int_0^\infty \int_{\mathbb{R}^3} |\nabla &\phi|^4 + |\nabla \phi|^6 + |\nabla^2 \phi|^4 ~\mbox{d}x \mbox{d}t \\ & \leq C \sup_{t\geq 0}\big(\|\nabla \phi\|_{L^\infty}^2 + \|\nabla \phi\|_{L^\infty}^4 + \|\nabla^2 \phi\|_{L^\infty}^2\big) \int_0^\infty \|\nabla\phi\|_{H^1(\mathbb{R}^3)}^2~\mbox{d}t \leq C(\Xi^2 + \Xi^3) \end{split}$$ Consequently, (\[o11\]) becomes: $ \Xi \leq C(\Xi^2+\Xi^3) + C\|\phi_0\|_{H^2}^2$. By the assumed smallness of $\Xi[\phi,\nabla u - \mathrm{Id}]$, we see that: $$\label{o21} \Xi \leq 2C\|\phi_0\|_{H^2}^2.$$ [**4.**]{} We now conclude the proof of the a-priori bound. Test (\[o13\]) by $(u-\mbox{id})$ to get: $$\begin{split} \int_{{\mathbb{R}}^3} {\partial}_{F}^2 W(\phi,&\nabla u) : (\nabla u - \mbox{Id})^{\otimes 2} \mbox{d}x \\ & \leq C \int_{{\mathbb{R}}^3} |\phi||\nabla u - \mbox{Id}| + |u-\mbox{id}|\big(|\nabla u - \mbox{Id}| + |\phi|\big) \big(|\nabla \phi|+|\nabla^2 u|\big) ~\mbox{d}x \\ & \leq C\|\phi\|^2_{L^2} + (\epsilon + C\Xi^{1/2}) \|\nabla u - \mbox{Id}\|^2_{L^2} \end{split}$$ Indeed, by (\[o23\]) and (\[o12\]): $$\begin{split} \int_{{\mathbb{R}}^3} |u-\mbox{id}|\big(|\nabla u &- \mbox{Id}| + |\phi|\big) \big(|\nabla \phi|+|\nabla^2 u|\big) ~\mbox{d}x \\ & \leq C\|u-\mbox{id}\|_{L^6} \|\nabla u - \mbox{Id}\|_{L^2} \|\nabla \phi,\nabla^2\phi\|_{L^3} + C\|u-\mbox{id}\|_{L^6} \|\phi\|_{L^2} \|\nabla \phi,\nabla^2\phi\|_{L^3} \\ & \leq C\|\nabla u-\mbox{Id}\|^2_{L^2} \Xi^{1/2} + C\|\phi\|_{L^2}^2 + C\Xi \|\nabla u-\mbox{id}\|^2_{L^2} \end{split}$$ Thus, we obtain the bound on $\|\nabla u - \mbox{Id}\|_{L^2}$, and subsequently on $\|u-\mbox{id}\|_{L^6}$. [**A proof of Theorem \[th2\].**]{} Given $(\bar\phi, \bar u)$, consider the following problem which is the linearization of (\[maineq2\]) at $(0,\mbox{id})$: $$\label{ex1} \begin{split} & {{\rm div}\,}\Big(\partial^2_{F}W(0,\mbox{Id}) (\nabla u -\mbox{Id}) + \partial_{\phi}\partial_F W(0,\mbox{Id})\phi\Big) = {{\rm div}\,}A, \\ & \phi_t - \Delta \Big(\partial^2_{\phi} W(0,\mbox{Id})\phi + \partial_{\phi}\partial_F W(0,\mbox{Id})(\nabla u - \mbox{Id})\Big) = \Delta B, \end{split}$$ where: $$\label{ex2} \begin{split} & A = \partial^2_{F}W(0,\mbox{Id})(\nabla \bar u-\mbox{Id}) + \partial_{\phi}\partial_F W(0,\mbox{Id})\bar \phi- \partial_FW(\bar \phi,\nabla \bar u), \\ & B = \partial_\phi W(\bar \phi,\nabla \bar u)-\partial^2_{\phi}W(0,\mbox{Id})\bar \phi + \partial_{\phi}\partial_F W(0,\mbox{Id})(\nabla \bar u - \mbox{Id}). \end{split}$$ Let ${\mathcal T}$ be its solution operator, so that $\mathcal{T}[\bar \phi,\bar u] = (\phi, u)$. We will prove that $\mathcal{T}$ has a fixed point in the space $X$, where: $$\begin{split} X = \big\{ (\phi, u); ~~& \phi \in L^\infty({\mathbb{R}}_+;H^2(\mathbb{R}^3)), ~\nabla(\nabla u - \mbox{Id}) \in L^\infty({\mathbb{R}}_+;H^1(\mathbb{R}^3)), \\ & \nabla \phi \in L^2({\mathbb{R}}_+;H^2(\mathbb{R}^3))), ~\nabla(\nabla u - \mbox{Id}) \in L^2({\mathbb{R}}_+;H^2(\mathbb{R}^3))) \big\}. \end{split}$$ Note first that the well-posedness of the system (\[ex1\]) follows by the Galerkin method in exactly the same manner as in section \[sec3\], under the regularity of the right hand side: $$A \in L^\infty({\mathbb{R}}_+;H^2(\mathbb{R}^3)) \cap L^2({\mathbb{R}}_+;H^3(\mathbb{R}^3)), \quad B \in L^2({\mathbb{R}}_+;H^3(\mathbb{R}^3))$$ Approximative spaces are constructed for $\phi\in H^3({\mathbb{R}}^3)$ and for $u-\mbox{id}$ such that $\nabla u - \mbox{Id} \in H^3(\mathbb{R}^3)$. We leave this construction to the reader and note that it is simpler than the one for the system (\[maineq\]). As in the proof of Lemma \[aprio\], solutions to (\[ex1\]) then satisfy: $$\begin{split} \sup_{t\geq 0} \int_{{\mathbb{R}}^3} \phi^2 ~\mbox{d}x + \sum_{i} \int_0^\infty \int_{{\mathbb{R}}^3} D^2W(0,\mbox{Id}) &: (\phi_{x_i},\nabla u_{x_i})^{\otimes 2} ~\mbox{d}x \mbox{d}t \\ & \leq C\|\nabla A,\nabla B\|^2_{L^2({\mathbb{R}}_+;L^2({\mathbb{R}}^3))} + C\|\phi_0\|^2_{L^2}, \end{split}$$ which is obtained by testing with $u_{x_i}$ and $(-\Delta)^{-1}\phi_{x_i}$. Similarly, the second and third derivatives bounds eventually yield: $$\label{ex3} \begin{split} \sup_{t\geq 0} \|\phi\|^2_{H^2(\mathbb{R}^3)} + \|\nabla \phi, \nabla(&\nabla u - \mbox{Id})\|_{L^2({\mathbb{R}}_+;H^2(\mathbb{R}^3))}^2 \\ & \leq C\|\nabla A,\nabla B\|_{L^2({\mathbb{R}}_+;H^2(\mathbb{R}^3))}^2 + C\|\phi_0\|^2_{H^2(\mathbb{R}^3))}, \end{split}$$ while: $$\sup_{t\geq 0} \|\nabla u - \mbox{Id}\|_{H^2(\mathbb{R}^3)}^2\leq C\|A\|_{L^\infty({\mathbb{R}}_+;H^2(\mathbb{R}^3))}^2.$$ Directly from (\[ex2\]) we observe that: $$\begin{split} & \|A\|_{L^\infty({\mathbb{R}}_+;H^2(\mathbb{R}^3)) \cap L^2({\mathbb{R}}_+;H^3(\mathbb{R}^3))} \leq C~\Xi[\bar \phi,\nabla \bar u - \mbox{Id}]\\ & \|B\|_{ L^2({\mathbb{R}}_+;H^3(\mathbb{R}^3))} \leq C~\Xi[\bar \phi,\nabla \bar u - \mbox{Id}] \end{split}$$ provided the quantity $\Xi$ is small. Then, by (\[ex3\]): $$\Xi[\phi,\nabla u -\mbox{Id}] \leq C \Xi[\bar \phi,\nabla \bar u - \mbox{Id}]^2 +C_0\|\phi_0\|_{H^2(\mathbb{R}^3)}^2.$$ Based on the considerations from the part about the a priori bound we observe that: $$\Xi[\phi,\nabla u - \mbox{Id}] \leq 2 C_0\|\phi_0\|^2_{H^2({\mathbb{R}}^3)},$$ provided that $\Xi$ is sufficiently small. Hence the operator $\mathcal{T}$ maps a ball $\mathcal{B}\subset X$ with a sufficiently small radius, into itself. Observe further that $\mathcal{T}$ is a contraction over $\mathcal{B}$, whose fixed point yields the unique solution to the system (\[maineq2\]). Theorem \[th2\] is proved. Appendix: Proof of Proposition \[prop\]. ======================================== The first condition in (\[ass\]) is obvious. A direct calculation shows that: $$DW_1(\phi, F) : (\tilde\phi,\tilde F) = \phi\tilde\phi + \langle DW_0(FB(\phi)) : \tilde \phi F B'(\phi)\rangle + \langle DW_0(FB(\phi)) : \tilde F B(\phi)\rangle,$$ which implies the second condition in (\[ass\]). Further: $$\begin{split} D^2W_1(0,\mbox{Id}) : (\tilde\phi,\tilde F)^{\otimes 2} & = |\tilde \phi|^2 + D^2W_0(\mbox{Id}) : (\tilde\phi B'(0))^{\otimes 2} \\ & \qquad + 2 D^2W_0(\mbox{Id}) : (\tilde B'(0)\otimes \tilde F) + D^2W_0(\mbox{Id}) : {\tilde F}^{\otimes 2} \\ & = |\tilde \phi|^2 + D^2W_0(\mbox{Id}) : (\tilde F +\tilde \phi B'(0))^{\otimes 2} \\ & \geq |\tilde\phi|^2 + c\left|\mbox{sym } \tilde F + \tilde \phi B'(0)\right|^2, \end{split}$$ where we concluded from (\[elastic\_dens\]) that $DW_0(\mbox{Id}) = 0$ and that $D^2W_0(\mbox{Id})$ is positive definite on symmetric matrices. We also note that: $ D^2W_2(0,\mbox{Id}) = D^2W_1(0,\mbox{Id}).$ To conclude the proof, it is hence enough to show that: $$\label{want} |\tilde\phi|^2 + |\mbox{sym } \tilde F + \tilde\phi B'(0)|^2 \geq c (|\tilde\phi|^2 + |\mbox{sym } \tilde F|^2 ),$$ for all $\tilde\phi$ and $\tilde F$. Expanding the square in the left hand side, dividing by $|\tilde \phi|$ and collecting terms, this is equivalent to: $$(1-c + |B'(0)|^2) + (1-c) |\mbox{sym} (\frac{1}{\tilde\phi}\tilde F)|^2 + 2\langle \mbox{sym} (\frac{1}{\tilde\phi}\tilde F) : B'(0)\rangle \geq 0,$$ which becomes: $$1-c -\frac{c}{1-c} |B'(0)|^2 + |\sqrt{1-c} \mbox{ sym} \big(\frac{1}{\tilde\phi}\tilde F\big) + \frac{1}{\sqrt{1-c}} B'(0)|^2 \geq 0.$$ The above inequality follows from: $1-c - \frac{c}{1-c} |B'(0)|^2 >0$, which is true whenever $c>0$ is sufficiently small. [999999]{} G. Andrews, *On the existence of solutions to the equation $u_{tt} = u_{xxt} + σ(u_x)_x$*, J. Diff. Eqs. [**35**]{}, 200–231, 1980. S. Antmann and R. Malek-Madani, *Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media*, Quart. Appl. Math. [**46**]{}, 77–-93, 1988. S. Antman and T. Seidman, *Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity*, J. Diff. Eqs. [**124**]{}, 132–-184, 1996. B. Barker, M. Lewicka, and K. Zumbrun, *Existence and stability of viscoelastic shock profiles*, Arch. Rational Mech. Anal. [**200**]{}, Number 2, (2011) 491–532. H. Barucq, M. Madaune-Tort, P. Saint-Macary, *Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models.* Nonlinear Anal. 61 (2005), no. 4, 591–612. O. Besov, V. Ilin, and S. Nikolski, *Integral representations of functions and imbedding theorems. Vol. I.*, Translated from the Russian. Scripta Series in Mathematics, Washington, D.C., Halsted Press 1978. K. Bhattacharya, M. Lewicka, M. Schaffner, *Plates with incompatible prestrain*, to appear. C. Dafermos, *The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity*, J. Diff. Eqs. [**6**]{}, 71–-86, 1969. S. Demoulini, *Weak solutions for a class of nonlinear systems of viscoelasticity*, Arch. Rat. Mech. Anal. [**155**]{} (4), 299-334, 2000. J. Dervaux, P. Ciarletta, and M. Ben Amar, *Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Foppl-von Karman limit*, Journal of the Mechanics and Physics of Solids, 57 (3), (2009), 458–471. E. Efrati, E. Sharon and R. Kupferman, *Elastic theory of unconstrained non-Euclidean plates*, J. Mech. Phys. Solids, 57 (2009), 762–775. E. Feireisl, P.B. Mucha, A. Novotny, M. Pokorny, *Time-periodic solutions to the full Navier-Stokes-Fourier system.* Arch. Ration. Mech. Anal. [**204**]{} (2012), no. 3, 745–786. G. Friesecke, R. James and S. Müller, *A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence*, Arch. Ration. Mech. Anal., [**180**]{} (2006), no. 2, 183–236. G. Friesecke, R. James, M.G. Mora and S. Müller, *Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence*, C. R. Math. Acad. Sci. Paris, [**336**]{} (2003), no. 8, 697–702. S. Govindjee, J. Simo, *Coupled stress-diffusion: Case II*, Journal of the Mechanics and Physics of Solids [**41**]{}, Issue 5, May 1993, 863–887. K. Garikipati, L. Bassman, M. Deal, *A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids*, Journal of the Mechanics and Physics of Solids [**49**]{}, Issue 6, June 2001, 1209–1237. J. Hill, *Plane steady solutions for stress-assisted diffusion*, Mechanics Research Communications [**6(3)**]{}, (1979), 147–150. S. Jiang, Y.-G. Wang, *Global Existence and Exponential Stability in Nonlinear Thermoelasticity*, Encyclopedia of Thermal Stresses [**2014**]{}, pp 1998-2006 Y. Klein, E. Efrati and E. Sharon, *Shaping of elastic sheets by prescription of non-Euclidean metrics*, Science [**315**]{} (2007), 1116–1120. A. Korn, *Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen*, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat., (1909) 705–724. R. Kupferman and Y. Shamai, *Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space*, Israel J. Math. 190 (2012) 135–156. R. Kupferman and C. Maor, *A Riemannian approach to the membrane limit in non-Euclidean elasticity*, to appear in Comm. Contemp. Math. O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, *Linear and quasilinear eqs of parabolic type* Translation of Mathematical Monographs [**23**]{}, AMS 1968. H. LeDret and A. Raoult, *The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity*, J. Math. Pures Appl. [**73**]{} (1995), 549–578. H. Le Dret and A. Raoult, *The membrane shell model in nonlinear elasticity: a variational asymptotic derivation*, J. Nonlinear Sci., [**6**]{} (1996), 59–84. M. Lewicka, L. Mahadevan, M.R. Pakzad, *The Foppl-von Karman equations for plates with incompatible strains*, Proceedings of the Royal Society A, [**467**]{} (2011), 402–426 . M. Lewicka, L. Mahadevan, M.R. Pakzad, *Models for elastic shells with incompatible strains*, Proceedings of the Royal Society A, [**470**]{} (2014). M. Lewicka, L. Mahadevan, M.R. Pakzad, *The Monge-Ampère constrained elastic theories of shallow shells*, to appear. M. Lewicka and P.B. Mucha, *A local existence result for a system of viscoelasticity with physical viscosity*, Evolution Equations and Control Theory, Vol 2, Issue 2, (2013) 337–353. M. Lewicka, P. Ochoa, M.R. Pakzad, *Variational models for prestrained plates with Monge-Ampere constraint*, to appear. M. Lewicka, M.R. Pakzad, *Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics*, ESAIM: Control, Optimisation and Calculus of Variations, [**17**]{}, no 4 (2011), 1158–1173. M. Lewicka and M. Pakzad, *The infinite hierarchy of elastic shell models; some recent results and a conjecture*, Infinite Dimensional Dynamical Systems, Fields Institute Communications [**64**]{}, 407–420 (2013). M. Lewicka, A. Raoult, D. Ricciotti, *Plates with incompatible prestrain of higher order*, to appear. H. Liang and L. Mahadevan, *Growth, geometry and mechanics of the blooming lily*, Proc. Nat. Acad. Sci., 108, 5516–21, (2011). M. Marder, (2003) The shape of the edge of a leaf. Found. of Physics [**33**]{}, 1743–1768. C.D. Modes, K. Bhattacharya and M. Warner, *Disclination-mediated thermo-optical response in nematic glass sheets*, Phys. Rev. E [**81**]{} (2010). C.D. Modes, K. Bhattacharya and M. Warner, *Gaussian curvature from flat elastica sheets*, Proc. Roy. Soc. A 467 1121–1140 (2011). M.G. Mora and L. Scardia, *Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density*, J. Differential Equations [**252**]{} (2012), 35–55. P.B. Mucha, M. Pokorny, E. Zatorska, *Chemically reacting mixtures in terms of degenerated parabolic setting.* J. Math. Phys. [**54**]{} (2013), no. 7, 071501, 17 pp. I. Pawlow, W.M. Zajaczkowski, *Unique global solvability in two-dimensional non-linear thermoelasticity.* Math. Methods Appl. Sci. [**28**]{} (2005), no. 5, 551–592. R. Pego, *Phase transitions in one-dimensional nonlinear viscoelasticity*, Arch. Rational Mech. Anal. [**97**]{}, 353–394, 1987. E.K. Rodriguez, A. Hoger and A. McCulloch, J. Biomechanics 27, 455 (1994). R. Racke, Y. Shibata, *Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity*. Arch. Rational Mech. Anal. [**116**]{} (1991), no. 1, 1–34. Y. Weitsman, *Stress assisted diffusion in elastic and viscoelastic materials*, Journal of the Mechanics and Physics of Solids [**35**]{}, Issue 1, 1987, 73–94. C.H. Wu, *The role of Eshelby stress in composition-generated and stress-assisted diffusion*, Journal of the Mechanics and Physics of Solids [**49**]{}, Issue 8, August 2001, 1771–1794. V. G. Zvyagin, V. P. Orlov, *Existence and uniqueness results for a coupled problem in continuum thermomechanics*, Vestnik: Fizika, Matematika [**2**]{}, 120–141, 2014.

--- abstract: 'We revisit Besicovitch’s 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper’s influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch’s methods.' author: - 'P. Pollack and J. Vandehey' title: 'Besicovitch, bisection, and the normality of $0.(1)(4)(9)(16)(25)\dots$' --- Introduction {#section:introduction} ============ A real number $x$ is said to be normal (to base $10$) if every string of decimal digits appears in the decimal expansion of $x$ as frequently as every other string of the same length, so the digit $4$ should appear as often as $9$, and $299$ should appear as often as $058$. More concretely, given a fixed integer base $g \ge 2$, let $\nu(x,N,{\mathbf{s}})$ denote the number of times the string ${\mathbf{s}}$, consisting of $k$ base $g$ digits, appears in the first $N$ digits of the base $g$ expansion for $x$: then $x$ is normal to base $g$ if for every non-empty string ${\mathbf{s}}$, we have $$\label{eq:mainlimit} \lim_{N\to \infty} \frac{\nu(x,N,{\mathbf{s}})}{N} = \frac{1}{g^k}.$$ Borel showed that almost all real numbers are normal to a given base $g$, in the sense that the set of numbers that are not normal has Lebesgue measure $0$. Despite this, to this day, no well-known mathematical constant, such as $e$, $\pi$, or $\ln 2$, is known to be normal to any integer base. All known examples of normal numbers were numbers constructed to be normal. The first such explicit construction was given by Champernowne [@champernowne33]: he showed that if we concatenate all the integers in succession—like so, $ 0.123456789101112\cdots$—then the resulting number is normal to base $10$. Champernowne’s result inspired many mathematicians to look at sequences of positive integers $\{a_n\}_{n=1}^\infty$ which make the number $0.\widebar{a_1}\widebar{a_2}\widebar{a_3}\widebar{a_4}\cdots$ normal in a given base. Here, given an integer $a$, we shall let $\widebar{a}$ denote the string composed of its base $g$ digits. Shortly after Champernowne, Besicovitch studied the sequence $a_n=n^2$. Although it is commonly stated that Besicovitch proved that the number $x_B=0.14916253649\cdots$ is normal to base $10$, in fact he showed a different result from which the normality of $x_B$ can be derived relatively quickly. Nonetheless, Besicovitch’s work was important for two main reasons. First, Besicovitch’s result inspired Davenport and Erdős [@DE52] to develop a method of proving normality through exponential sum estimates, and they used this method to show that if $f(n)$ is a positive, non-constant, integer-valued polynomial, then $0.\widebar{f(1)}\widebar{f(2)}\widebar{f(3)}\dots$ is a normal number to the given base. (Taking $f(n)=n^2$, this gives a second proof that $x_B$ is normal to base $10$.) This has given rise to what we call the *analytic method of normality proofs*, which has resulted in a number of different normality results by applying different results on exponential sums (see, for example, [@MTT08; @NS97; @vandehey13]). Second, Besicovitch’s result inspired the definition of an integer being $(\epsilon,k)$-normal. Let $\nu(a,{\mathbf{s}})$ denote the number of times the finite string ${\mathbf{s}}$ appears in $\widebar{a}$, and let $L(a)$ denote the number of digits in the string $\widebar{a}$. Then an integer $n$ is said to be $(\epsilon,k)$-normal if $$\left| \frac{\nu(a,{\mathbf{s}})}{L(a)} - \frac{1}{g^k}\right| \le \epsilon$$for every string ${\mathbf{s}}$ with $k$ digits. This definition inspired the following result, which may be called *the combinatorial method of normality proofs*. \[thm:combinatorial\]Consider a sequence $\{a_n\}_{n=1}^\infty$. Suppose that the lengths of the strings $\widebar{a_n}$ are growing on average, but that no one length dominates; more precisely, suppose that as $m$ tends to infinity, we have $$\label{eq:lengths} m = o\left( \sum_{n=1}^m L(a_n) \right) \qquad \text{ and } \qquad m\cdot \max_{1\le n \le m} L(a_n) =O\left( \sum_{n=1}^m L(a_n)\right) .$$ In addition, suppose that for any fixed $\epsilon>0$ and $k\in \mathbb{N}$, almost all $a_n$ are $(\epsilon,k)$-normal, in the sense that the number of $n\le m$ for which $a_n$ is *not* $(\epsilon,k)$-normal is $o(m)$ as $m$ tends to infinity again. Then $x=0.\widebar{a}_1\widebar{a}_2\widebar{a}_3\widebar{a}_4\dots$ is normal. Here, we use the notation $f(x)=O(g(x))$ to mean that $|f(x)/g(x)|\le C $ for some constant $C$ (called an *implicit constant*), and the notation $f(x)=o(g(x))$ to mean that $f(x)/g(x)$ approaches $0$ as $x$ approaches infinity. We shall also use the notation $f(x) \sim g(x)$ to mean $f(x) = g(x)(1+o(1))$ or, equivalently, $\lim_{x\to \infty} f(x)/g(x)=1$. Theorem \[thm:combinatorial\] says, in essence, that if almost all $a_n$ exhibit small-scale normality results, then we expect the full number $x$ to exhibit large-scale normality results. Although Theorem \[thm:combinatorial\] is implicitly used in almost every combinatorial normality proof, we are unaware of it ever being given explicitly in the literature, and so provide a proof in Section \[sec:combproof\]. Copeland and Erdős [@CE46] gave a fairly powerful counting result on the number of integers that are *not* $(\epsilon,k)$-normal. The first half of the following proposition is due to them; the second half is derived from Lemma $4.7$ in [@Bugeaud]. \[prop:CE\] Let $\epsilon>0$ and $k\in \mathbb{N}$ be fixed. There exists a $\delta=\delta(\epsilon,k)>0$ such that the number of integers in the interval $[1,m]$ that are not $(\epsilon,k)$-normal is at most $m^{1-\delta}$ for all sufficiently large $m$. There also exists a $\delta'=\delta'(\epsilon,k)>0$ such that the number of base-$g$ strings of length $\ell$ (including those that start with $0$) that are not $(\epsilon,k)$-normal is at most $g^{\ell(1-\delta)}$ for all sufficiently large $\ell$. Combining Theorem \[thm:combinatorial\], the first half of Proposition \[prop:CE\], and the Prime Number Theorem, it is immediate that the Copeland-Erdős number $0.2357111317\dots$, formed by taking $a_n$ to be the $n$th prime, is normal to base $10$. Many early normality results focused on sequences $\{a_n\}_{n=1}^\infty$ that were increasing. More recent variants have allowed more chaotic and oscillating functions $a_n=f(n)$ to be considered. Recently, the authors of this note looked at functions $f(n)$ that are *almost bijective*, which we shall define in the following way. First, we say a set $S\subset \mathbb{N}$ is *meager* if $\#\{n\in S: n \le m \} \le m^{1-\delta}$ for some fixed $\delta>0$ and all sufficiently large $m$. We say a set $S\subset \mathbb{N}$ has *asymptotic density $0$* if $\#\{n\in S: n \le m \} =o(m)$. We say a function $f\colon\mathbb{N}\to \mathbb{N}$ is *almost bijective* if the pre-image of any meager set has asymptotic density $0$. By Proposition \[prop:CE\], the set of integers which are not $(\epsilon,k)$-normal is a meager set. Thus, if $f\colon\mathbb{N}\to \mathbb{N}$ is almost bijective, then $f(n)$ will be $(\epsilon,k)$-normal for almost all $n$. This gives the following variant on the combinatorial method, which appears explicitly in [@PV]: Suppose the function $f\colon\mathbb{N}\to\mathbb{N}$ is almost bijective. If, in addition, $$m = o\left( \sum_{n=1}^m L(a_n) \right) \qquad \text{ and } \qquad m\cdot \max_{1\le n \le m} L(a_n) =O\left( \sum_{n=1}^m L(a_n)\right)$$ then $x=0.\widebar{f(1)}\widebar{f(2)}\widebar{f(3)}\dots$ is normal. This result covers a fairly wide variety of functions. De Koninck and Kátai [@DKK11; @DKK13] implicitly applied this result with certain variants of the largest prime divisor function. Pollack and Vandehey [@PV] showed that one could take $f(n)$ to be various classical number-theoretic functions, including the Euler totient function and the sum-of-divisors function. Szüsz and Volkmann [@SV] gave fairly general analytic conditions guaranteeing that the values $f(n)$ are $(\epsilon,k)$-normal for almost all $n$. (See the following table for some explicit examples.) Function Discoverer Resulting normal number ---------------------------------------------------------------- ---------------------- --------------------------- $P(n)$, the largest prime divisor of $n$ De Koninck and Kátai $0.123253723511213\dots$ $\phi(n)$, the Euler totient function Pollack and Vandehey $0.112242646410412\dots$ $\sigma(n)$, the sum of the divisors of $n$ Pollack and Vandehey $0.1347612815131812\dots$ $\lfloor n^{1/2} \rfloor$, the floor of the square root of $n$ Szüsz and Volkmann $0.1112222233333334\dots$ Despite all these results inspired by Besicovitch’s work, none of them are strong enough to prove the normality of $x_B=0.14916253649\dots$.[^1] In particular, for the function $f(n)=n^2$, the pre-image of the meager set $S=\{n^2:n \in \mathbb{N}\}$ is the whole domain $\mathbb{N}$. Our goal in the rest of this paper is to update and simplify the proof of the normality of $x_B=0.\widebar{f(1)}\widebar{f(2)}\dots$ with $f(n)=n^2$ and some integer base $g\ge 2$, to reflect modern work on normal numbers, in the hope that it can inspire further combinatorial results. In Section \[sec:combproof\], we prove Theorem \[thm:combinatorial\]. In Sections \[sec:half\]–\[sec:secondhalf\], we present our proof of the normality of $x_B$. In Section \[sec:final\], we briefly discuss how our proof differs from Besicovitch’s and the difficulties in extending this method combinatorially. Proof of Theorem \[thm:combinatorial\] {#sec:combproof} ====================================== Consider a sequence $\{a_n\}_{n=1}^\infty$ satisfying the conditions of Theorem \[thm:combinatorial\]. Let $x=0.\widebar{a_1}\widebar{a_2}\widebar{a_3}\dots$ in the appropriate base $g$. To show that $x$ is normal to base $g$, we must show for any given string ${\mathbf{s}}$ of length $k$ that $$\lim_{N\to \infty} \frac{\nu(x,N,{\mathbf{s}})}{N} = \frac{1}{g^k}.$$ We fix an $\epsilon>0$, which will be allowed to tend towards zero at the end of the proof. For a given integer $N$, let $m=m(N)$ be such that the $N$th digit of $x$ lies in the string given by $\widebar{a_m}$. Then $$\sum_{n=1}^{m-1} L(a_n) < N \le \sum_{n=1}^m L(a_n).$$ The second part of implies that $L(a_m) = o(\sum_{n=1}^m L(a_n))$, so we have that $N\sim \sum_{n=1}^m L(a_n)$, $m=o(N)$, and $L(a_m)=o(N)$. Therefore, $$\nu(x,N,{\mathbf{s}}) = \nu(\widebar{a_1}\widebar{a_2}\dots \widebar{a_m},{\mathbf{s}}) + O(L(f(m)) )= \nu(\widebar{a_1}\widebar{a_2}\dots \widebar{a_m},{\mathbf{s}}) + o(N).$$ The number of times a string of length $k$ can appear in $\widebar{a_1}\widebar{a_2}\dots \widebar{a_m}$ starting in some $\widebar{a_n}$ and ending in some $\widebar{a_{n'}}$ with $n< n'$ is at most $km=o(N)$. Therefore, $$\nu(x,N,{\mathbf{s}}) = \nu(\widebar{a_1}\widebar{a_2}\dots \widebar{a_m},{\mathbf{s}}) + o(N) = \sum_{n\le m} \nu(\widebar{a_n},{\mathbf{s}}) +o(N).$$ Let $T\subset\mathbb{N}$ be the set of integers $n$ such that $a_n$ is *not* $(\epsilon,k)$-normal. Note that by the assumptions of the theorem, we have $\#\{n\le m: n \in T\} = o(m)$. We always have that $\nu(\widebar{a_n},{\mathbf{s}})=O(L(a_n))$, and therefore $$\begin{aligned} \sum_{\substack{n\le m \\ n \in T}} \nu(\widebar{a_n},{\mathbf{s}}) &= O\left( \sum_{\substack{n\le m \\ n \in T}} L(a_n) \right) = O\left( \max_{n\le m} L(a_n) \cdot \sum_{\substack{n\le m \\ n \in T}} 1 \right)\\ &= o \left( m \cdot \max_{n\le m} L(a_n) \right) = o\left( N\right).\end{aligned}$$ Now we let $S=\mathbb{N}\setminus T$ be the set of integers $n$ such that $a_n$ is $(\epsilon,k)$-normal. If $n\in S$ then $\nu(\widebar{a_n},{\mathbf{s}}) = L(a_n)g^{-k}+O(\epsilon L(a_n))$, and thus $$\begin{aligned} \sum_{\substack{n\le m\\ n\in S }} \nu(\widebar{a_n},{\mathbf{s}}) &= \sum_{\substack{n\le m \\ n\in S}} \left( L(a_n) \left(g^{-k} +O(\epsilon)\right)\right)\\ &= g^{-k}\left( \sum_{n \le m} L(a_n) - \sum_{\substack{n \le m \\n \in T}} L(a_n) \right) +O\left( \epsilon \cdot \sum_{\substack{n\le m \\ n\in S}} L(a_n)\right) \\ &= g^{-k} \left( N(1+o(1)) + o(N)\right) +O\left( \epsilon \cdot \sum_{n\le m } L(a_n)\right)\\ &= g^{-k}N +o(N)+O(\epsilon N)\end{aligned}$$ Since the sum over $n\le m$ is equal to the sum over $n\le m$ with $n\in S$ plus the sum over $n\le m$ with $n \in T$, we have shown that $$\frac{\nu(x,N,{\mathbf{s}})}{N} = g^{-k}+o(1)+O(\epsilon).$$ Since $\epsilon>0$ was arbitrary, $\nu(x,N,{\mathbf{s}})/N\to g^{-k}$ as $N\to \infty$. Cutting the squares in half {#sec:half} =========================== From here on, we shall be interested in the specific case when $a_n=f(n)=n^2$ with a fixed integer base $g\ge 2$. Implicit constants may depend on $g$. In this case we have $L(f(n))=\lfloor \log_{g} f(n) \rfloor+1=2\log_{g} n +O(1)$, and thus $$\label{eq:mlengths} \sum_{n=1}^m L(f(m)) = \frac{2}{\log g} m \log m (1+o(1)).$$ Here the sum of $\log_g(n)$ has been estimated using the proof of the integral test. It is clear from that $a_n=n^2$ satisfies the restrictions on $L(a_n)$ from the statement of Theorem \[thm:combinatorial\]. To prove the normality of $x_B=0.\widebar{f(1)}\widebar{f(2)}\widebar{f(3)}\dots$ in this case, it suffices by Theorem \[thm:combinatorial\] to show that for a fixed $\epsilon>0$ and $k\in \mathbb{N}$, that the number of $n \in [1,m]$ for which $f(n)$ is not $(2\epsilon,k)$-normal is $o(m)$ as $m$ tends to infinity. (Note that using $2\epsilon$ is intentional here.) Let $\delta=\delta(\epsilon,k)$ be the constant from Proposition \[prop:CE\], and let $m'=\lfloor m^{1-\frac{\delta}{2}} \rfloor$. We may ignore values of $n\in [1,m'-1]$, since there are only $o(m)$ of these. Let $\ell := \lfloor L(f(m))/2 \rfloor$. For sufficiently large $m$, we have $L(f(n))> \ell$ for all $n \in [m',m]$. We now consider two new auxiliary functions $b(n,m)$ and $c(n,m)$ for a fixed $m$. We let $b(n,m)=\lfloor n^2/g^{\ell} \rfloor$, and we let $c(n,m)$ be the least nonnegative residue of $n^2$ modulo $g^{\ell}$. While we shall define $\widebar{b(n,m)}$ in the usual way as the string of base $g$ digits of $b(n,m)$, we shall modify our definition slightly for $\widebar{c(n,m)}$. If $c(n,m)$ has fewer than $\ell$ base $g$ digits, then append enough $0$’s to the beginning of the string $\widebar{c(n,m)}$ so that it has length $\ell$. With these definitions, the string $\widebar{f(n)}$ is the concatenation of $\widebar{b(n,m)}$ and $\widebar{c(n,m)}$, for all $n\in [m',m]$. As a quick example, consider $m=500$ and $n=179$. Then $f(m)=250000$, so that $l=3$, and $f(n)=32041$. In this case, $b(n,m)=32$ and $c(n,m)=41$, so that $\widebar{b(n,m)}=32$ and $\widebar{c(n,m)} =041$. Since $\widebar{f(n)}$ has close to $2\ell$ digits, and both $\widebar{b(n,m)}$ and $\widebar{c(n,m)}$ have approximately $\ell$ digits, we may think of this as bisecting $f(n)$ into halves. Now we make two claims which we will prove in subsequent sections: The number of $n\in [m',m]$ for which $\widebar{b(n,m)}$ is not $(\epsilon,k)$-normal is $o(m)$ as $m$ tends to infinity. The number of $n \in [m',m]$ for which $\widebar{c(n,m)}$ is not $(\epsilon,k)$-normal is $o(m)$ as $m$ tends to infinity. We now finish proving the normality of $x_B$ assuming these two claims. Suppose that both $\widebar{b(n,m)}$ and $\widebar{c(n,m)}$ are $(\epsilon,k)$-normal. Then $$\begin{aligned} \nu(f(n),{\mathbf{s}}) &= \nu(\widebar{b(n,m)},{\mathbf{s}})+\nu(\widebar{c(n,m)},{\mathbf{s}})+O(k)\\ &= L(\widebar{b(n,m)})(g^{-k} + O(\epsilon )) + L(\widebar{c(n,m)})(g^{-k}+O(\epsilon ))+O(k)\\ &= L(f(n))(g^{-k}+O(\epsilon))+O(k).\end{aligned}$$ (Here we are using the big-O notation with implicit constant $1$ in all cases.) Since $k = O(\epsilon L(f(n)))$ for all $n\in [m',m]$ provided $m$ is sufficiently large, we have that $f(n)$ is $(2\epsilon,k)$-normal in this case. So $f(n)$ is not $(2\epsilon,k)$-normal only if $\widebar{b(n,m)}$ or $\widebar{c(n,m)}$ is not $(\epsilon,k)$-normal, and there are only $o(m)$ such $n$ in the interval $[m',m]$ by our two claims. This completes the proof. How often is the first half of $n^2$ normal? {#sec:firsthalf} ============================================ Here we will prove that the number of $n\in [m',m]$ for which $\widebar{b(n,m)}$ is not $(\epsilon,k)$-normal is $o(m)$ as $m$ tends to infinity. This is comparatively simple. As the first half of the digits of $f(n)$ grow fairly regularly, they cannot take any given non-$(\epsilon,k)$-normal value too frequently. Recall that $b(n,m)=\lfloor n^2/g^{\ell} \rfloor$. For the remainder of this section we will often suppress the dependence on $m$ and just write $b(n)$. Since $\ell \ge \log_{g} m - 1$, we have $g^\ell \ge m/g$, and thus $b(n)$ is always in the interval $[1,gm]$. By the first half of Proposition \[prop:CE\], there are at most $(gm)^{1-\delta}$ integers in the interval $[1,gm]$ that are not $(\epsilon,k)$-normal. Now suppose that $m'\le n_1< n_2 \le m$. Then, since we have $\ell \le \log_g m+1$ as well, $$\begin{aligned} b(n_2)-b(n_1) &= \frac{n_2^2-n_1^2}{g^{\ell}} +O(1) = g^{-\ell} (n_2-n_1)(n_2+n_1) +O(1)\\ & \ge (gm)^{-1} (n_2-n_1)(2m') +O(1) \end{aligned}$$ Thus, if $n_2-n _1 \ge m^{3\delta/4}$, then $$b(n_2)-b(n_1)\ge (gm)^{-1} \cdot m^{3\delta/4} \cdot 2m' +O(1) \ge 2g^{-1}m^{\delta/4}+O(1).$$ Thus, for sufficiently large $m$, we have $b(n_1)\neq b(n_2)$ for $n_2-n_1 \ge m^{3\delta/4}$. Since $b(n)$ is non-decreasing, this means that $b(n)$ can take a given value in the interval $[1,gm]$ at most $m^{3\delta/4}$ times. Since there are at most $(gm)^{1-\delta}$ integers in the interval $[1,gm]$ that are not $(\epsilon,k)$-normal, we have that at most $O(m^{1-\delta/4})=o(m)$ of the integers $n\in [m',m]$ have $\widebar{b(n,m)}$ not $(\epsilon,k)$-normal. How often is the second half of $n^2$ normal? {#sec:secondhalf} ============================================= Here we will prove that the number of $n\in [m',m]$ for which $\widebar{c(n,m)}$ is not $(\epsilon,k)$-normal is $o(m)$ as $m$ tends to infinity. As before, we will write $c(n)$ in place of $c(n,m)$. Let $B$ be the set of integers $b\in [0,g^{\ell}-1]$ such that $\widebar{b}$ is not $(\epsilon,k)$-normal. Here again, we assume $\widebar{b}$ to be padded with initial zeros so as to have length $\ell$. By the second half of Proposition \[prop:CE\], the cardinality of $B$ is at most $g^{\ell(1-\delta')}$ for a certain $\delta'=\delta'(\epsilon,k)>0$. How often is $c(n)\in B$? Since $[m',m] \subset [1,g^{\ell+1}]$, the number of $n\in [m',m]$ with $c(n) \in B$ is at most the $g$ times the count of such $n$ in $[1,g^{\ell}]$. By Cauchy–Schwarz, $$\begin{aligned} \sum_{\substack{1 \le n \le g^{\ell}\\c(n)=b}} 1 &= \sum_{b\in B} \#\{1\le n \le g^{\ell}\mid n^2 \equiv b \pmod{g^{\ell}} \} \\ &\le\left( \sum_{b\in B} 1\right)^{1/2} \left( \sum_{b\in B} \#\{1\le n_1,n_2 \le g^{\ell}\mid n_1^2 \equiv n_2^2 \equiv b \pmod{g^{\ell}} \}\right)^{1/2}.\end{aligned}$$ We are now left with the problem of counting how many pairs of integers $(x,y)$ there are with $1\le x,y\le g^l$ and $x^2\equiv y^2 \pmod{g^\ell}$. When $g$ is a prime power, a satisfactory answer is contained in the next lemma. \[lem:secondhalf\] Let $p$ be a prime and $e$ be a positive integer. The number of solutions to the congruence $x^2 \equiv y^2 \pmod{p^e}$ is at most $$\begin{cases} 2e \cdot p^e, & \text{if }p\text{ is odd,}\\ 4e \cdot p^e, & \text{if }p = 2. \end{cases}$$ Certainly, if $p^{\lceil e/2\rceil}$ divides each component of the pair $(x,y)$, then $x^2\equiv y^2\pmod{p^e}$. There are $(p^{e-\lceil e/2\rceil})^2$ solutions of this kind. For all other solutions, $p^{\lceil e/2\rceil}$ divides neither component. Group the remaining solutions according to the largest exponent $r$ for which $p^r \mid x$. Then $0 \leq r < e/2$. Since $p^{2r}\mid p^e \mid x^2-y^2$ and $p^{2r}$ divides $x^2$, we see that $p^r$ divides $y$. Write $x=p^r x'$ and $y=p^r y'$, and notice that determining the pair $(x,y)$ modulo $p^e$ amounts to determining the pair $(x', y')$ modulo $p^{e-r}$. Now $x^2 \equiv y^2\pmod{p^e}$ precisely when $$\label{eq:congsq} x'^2 \equiv y'^2\pmod{p^{e-2r}}.$$ This final congruence looks very similar to the one we started with, and one might wonder if we have gained anything. Indeed we have: by the maximality of $r$, we know that $x'$ must be coprime to $p$, which forces $y'$ to be coprime to $p$ as well. In other words, $x'$ and $y'$ represent elements of the unit group modulo $p^{e-2r}$. The key word here is “group." As was known to Gauss, if $p$ is an odd prime, the units group modulo $p^{e-2r}$ is cyclic of order $p^{e-2r}(1-1/p)$. In any cyclic group of even order, each element has precisely two square roots. So if $p$ is odd, the congruence has $2p^{e-2r}(1-1/p)$ solutions modulo $p^{e-2r}$, and so has $p^{2r} \cdot 2p^{e-2r}(1-1/p) = 2p^e (1-1/p)$ solutions modulo $p^{e-r}$. Putting everything together, we see that the number of solutions to $x^2\equiv y^2\pmod{p^e}$ is exactly $$2p^e (1-1/p) \lceil e/2\rceil + (p^{e-\lceil e/2\rceil})^2.$$ Here we used that the number of integers in the range $0 \leq r < e/2$ is precisely $\lceil e/2\rceil$. What if $p=2$? In this case, the group of units modulo $p^{e-2r}$ is either cyclic or the direct sum of two cyclic groups, and so each element can have at most four square roots. Modifying the above argument accordingly, we find that the number of solutions in this case is at most $4p^e (1-1/p) \lceil e/2\rceil + (p^{e-\lceil e/2\rceil})^2$. Finally, it is straightforward to check that these bounds do not exceed the upper bounds specified in the statement of the lemma. Now suppose that $g$ has the prime factorization $p_1^{e_1}p_2^{e_2}\dots p_j^{e_j}$. We use the above lemma combined with the Chinese Remainder Theorem to see that the number solutions to $x^2\equiv y^2 \pmod{g^\ell}$ is at most $2(\prod_{i=1}^{j} 2e_i \ell) g^{\ell}$, where the initial $2$ comes from the possibility that $2$ is a factor of $g$. Since $g^\ell \le gm$ and $\ell \le \log_g m+1$, the total number of pairs is $O((\log m)^j m)$. Thus, the number of $n \in [m',m]$ for which $c(n)$ belongs to $B$ is is $O(\sqrt{\#B} \cdot (\log{m})^{j/2} m^{1/2})$. Recalling that $\#B \le g^{\ell(1-\delta')}$ while $g^{\ell} \leq gm$, we see that $\#B = O(m^{1-\delta'})$. Since $(\log{m})^{j/2}$ is smaller than $m^{\delta'/4}$ for large $m$, our final count of $n$ is $O(m^{1-\delta'/4})$, which is $o(m)$. This completes the proof. Revisiting and extending Besicovitch {#sec:final} ==================================== As mentioned earlier, the main result of Besicovitch’s paper was not a proof that $x_B=0.1491625\cdots$ is normal base $10$. What Besicovitch showed was that, almost all of the integers $n^2$, for $n\in \mathbb{N}$, are $(\epsilon,1)$-normal to a given base $g\ge 2$. Besicovitch, like we did above, split the string $\widebar{n^2}$ into approximate halves. His method for showing that the first half has good normality properties is very similar to the method we used. His method for the second half is a long direct counting argument quite different from ours. Besicovitch relies heavily on results from Diophantine approximation about how well real numbers can be approximated by rational numbers with small denominators. It’s natural to ask if we could show combinatorially that $n^3$ is $(\epsilon,k)$-normal for almost all $n$. We could divide the string $\widebar{n^3}$ into $3$ roughly equal length pieces as we did with $\widebar{n^2}$. The methods of Section \[sec:firsthalf\] (and those of Besicovitch) would show that the first third of $\widebar{n^3}$ would almost always be $(\epsilon,k)$-normal. The methods of Section \[sec:secondhalf\], with minor modifications[^2], would show the same for the last third of $\widebar{n^3}$. (Besicovitch’s methods would not work here due to the limits of Diophantine approximation of real numbers.) However, neither our methods nor Besicovitch’s would be sufficient to show the middle third of $\widebar{n^3}$ is almost always $(\epsilon,k)$-normal. New techniques are required for that. [10]{} A. S. Besicovitch, *The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers*, Math. Zeit. **39** (1935), 146–156. E. Borel, *Les probabilités dénombrables et leurs applications arithmétiques*, Supplemento di rend. circ. Mat. Palermo **27** (1909), 247–271. Y. Bugeaud, *Distribution modulo one and diophantine approximation*, Cambridge Tracts in Mathematics vol. 193 (2012). D. G. Champernowne, *The construction of decimals normal in the scale of ten*, J. London Math. Soc. **3** (1933), 254–260. A. H. Copeland and P. Erdős, *Note on normal numbers*, Bull. Amer. Math. Soc. **52** (1946), 857–860. H. Davenport and P. Erdős, *Note on normal decimals*, Canadian J. Math. **4** (1952), 58–63. J.-M. De Koninck and I. K[á]{}tai, *On a problem on normal numbers raised by [I]{}gor [S]{}hparlinski*, Bull. Aust. Math. Soc. **84** (2011), 337–349. , *Using large prime divisors to construct normal numbers*, Ann. Univ. Sci. Budapest. Sect. Comput. **39** (2013), 45–62. M. G. Madritsch, J. M. Thuswaldner, and R. F. Tichy, *Normality of numbers generated by the values of entire functions*, J. Number Theory **128** (2008), 1127–1145. Y. Nakai and I. Shiokawa, *Normality of numbers generated by the values of polynomials at primes*, Acta Arith. **81** (1997), 345–356. P. Pollack and J. Vandehey, *Some normal numbers generated by arithmetic functions*, preprint available at <http://arxiv.org/abs/1309.7386>. P Szüsz and B. Volkmann, *A combinatorial method for construction normal numbers*, Forum Mathematicum **6**:4 (1994), 399–414. J. Vandehey, *The normality of digits in almost constant additive functions*, Monatsh. Math. **171** (2013), 481–497. [^1]: Szusz and Volkmann mistakenly claim that their result is strong enough to prove a result like this. In Theorem 2 of their paper, they need an additional condition that $\beta \le 1$, because if $\beta>1$ then the bound in line (3.11) would be $M_k=O(1)$, which would cause their condition (v) to fail. [^2]: We could count the $n$ for which $p_i^{\lceil \log \ell \rceil} \mid c(n)$ for some $i$ first, and this will be $o(m)$. Then we apply our Cauchy-Schwarz estimate to the remaining $n$, so that in Lemma \[lem:secondhalf\] we could assume that $p^{\lceil \log \ell \rceil}$ does not divide either $x$ and $y$. The rest of the proof would be mostly unchanged.

--- abstract: 'We study the physics potential of the 8TeV LHC (LHC-8) to discover signals of extended gauge models or extra dimensional models whose low energy behavior is well represented by an $SU(2)^2 \otimes U(1)$ electroweak gauge structure. We find that with a combined integrated luminosity of $40$fb$^{-1}$, the first new Kaluza-Klein mode of the $W$ gauge boson can be discovered up to a mass of about $400$GeV, when produced in association with a $Z$ boson.' address: - | Department of Physics and Astronomy, Michigan State University\ East Lansing, Michigan 48824, USA\ $^*$ Speaker at Conference - | Center for High Energy Physics, Tsinghua University\ Beijing 100084, China - | Pittsburgh Particle Physics, Astrophysics and Cosmology Center\ Department of Physics and Astronomy\ University of Pittsburgh, Pittsburgh, PA 15260, USA author: - 'R. Sekhar Chivukula$^*$, Elizabeth H. Simmons' - 'Chun Du, Hong-Jian He, Yu-Ping Kuang, Bin Zhang' - 'Neil D. Christensen' title: | Vector Bosons Signals of\ Electroweak Symmetry Breaking[^1] --- Introduction ============ The ATLAS and CMS experiments at the LHC have now each collected over 20$\ifb$ of data at an 8TeV collision energy. These data will enable the LHC to make incisive tests of the predictions of many competing models of the origin of electroweak symmetry breaking (EWSB), from the Standard Model (SM) with a single Higgs boson, to models with multiple Higgs bosons, and to so-called Higgsless models of the EWSB. The Higgsless models [@Csaki:2003dt] contain new spin-1 gauge bosons which play a key role in EWSB by delaying unitarity violation of longitudinal weak boson scattering up to a higher ultraviolet (UV) scale [@SekharChivukula:2001hz]. Very recently, the effective UV completion of the minimal three-site Higgsless model [@3site] was presented and studied in [@Abe:2012fb] which showed that the latest LHC signals of a Higgs-like state with mass around $125-126$GeV [@July4] can be readily explained, in addition to the signals of new spin-1 gauge bosons studied in the present paper. In this talk, we explore the physics potential of the LHC-8 to discover a relatively light fermiophobic electroweak gauge boson $W_1$ with mass $250-400$GeV, as predicted by the minimal three-site moose model[@3site] and its UV completion[@Abe:2012fb]. Being fermiophobic or nearly so, the $W_1$ state is allowed to be fairly light. More specifically, the 5d models that incorporate ideally [@SekharChivukula:2005xm] delocalized fermions [@Cacciapaglia:2004rb; @Foadi:2004ps], in which the ordinary fermions propagate appropriately in the compactified extra dimension (or in deconstructed language, derive their weak properties from more than one $SU(2)$ group in the extended electroweak sector [@Chivukula:2005bn; @Casalbuoni:2005rs]), yield phenomenologically acceptable values for all $Z$-pole observables [@3site]. In this case, the leading deviations from the SM appear in multi-gauge-boson couplings, rather than the oblique parameters $S$ and $T$. Ref.[@tri-3site] demonstrates that the LEP-II constraints on the strength of the coupling of the $Z_0^{}$-$W_0^{}$-$W_0^{}$ vertex allow a $W_1$ mass as light as 250GeV, where $W_0$ and $Z_0$ refer to the usual electroweak gauge bosons. The Model ========= ![Moose diagram of the minimal linear moose model (MLMM) with the gauge structure $SU(2)_0 \times SU(2)_1 \times U(1)_2$ as well as two independent link fields $\Phi_1^{}$ and $\Phi_2^{}$ for spontaneous symmetry breaking. The relevant parameter space of phenomenological interest is where the gauge couplings obey $\,g,g' \ll \tilde{g}$.[]{data-label="fig:models"}](fig1.eps) We study the minimal deconstructed moose model at LHC-8 in a limit where its gauge sector is equivalent to the “three site model" [@3site] or its UV completed “minimal linear moose model" (MLMM)[@Abe:2012fb]. Both the three site model and the MLMM are based on the gauge group $\,SU(2)_0\otimes SU(2)_1\otimes U(1)_2$, as depicted by Fig.\[fig:models\] and its gauge sector is the same as that of the BESS models [@Casalbuoni:1985kq; @Casalbuoni:1996qt] or the hidden local symmetry model [@Bando:1985ej; @Bando:1985rf; @Bando:1988ym; @Bando:1988br; @Harada:2003jx]. The extended electroweak symmetry spontaneously breaks to electromagnetism when the distinct Higgs link-fields $\Phi_1$ connecting $SU(2)_0$ to $SU(2)_1$ and $\Phi_2$ connecting $SU(2)_1$ to $U(1)_2$ acquire vacuum expectation values (VEVs) $\,f_{1}^{}$ and $\,f_{2}^{}$. The weak scale $\,v \simeq 246$GeV is related to those VEVs via $\,v^{-2} = f_1^{-2} + f_2^{-2}$ and, for illustration, we take $\,f_1^{}=f_2^{}=\sqrt{2}v$. Below the symmetry breaking scale, the gauge boson spectrum includes an extra set of weak bosons $(W_1^{},\,Z_1^{})$, in addition to the standard-model-like weak bosons $(W_0^{},\,Z_0^{})$ and the photon. Furthermore, the scalar sector of the MLMM[@Abe:2012fb] contains two neutral physical Higgs bosons $(h^0,\,H^0)$, as well as the six would-be Goldstones eaten by the corresponding gauge bosons $(W_0^{},\,Z_0^{})$ and $(W_1^{},\,Z_1^{})$. One distinctive feature of the MLMM is that the unitarity of high-energy longitudinal weak boson scattering is maintained jointly by the exchange of both the new spin-1 weak bosons and the spin-0 Higgs bosons [@Abe:2012fb]. This differs from either the SM (in which unitarity of longitudinal weak boson scattering is ensured by the exchange of the Higgs boson alone)[@SMuni] or the conventional Higgsless models (in which unitarity of longitudinal weak boson scattering is ensured by the exchange of spin-1 new gauge bosons alone)[@SekharChivukula:2001hz]. It has been shown [@tri-3site] that the scattering amplitudes in such highly deconstructed models with only three sites can accurately reproduce many aspects of the low-energy behavior of 5d continuum theories. The unitarity of the generic longitudinal scattering amplitude of $\,W_{0}^{L}W_{0}^{L}\to W_{0}^{L}W_{0}^{L}$, in the presence of any numbers of spin-1 new gauge bosons $V_k^{}\,(=W_k^{},Z_k^{})$ and spin-0 Higgs bosons $h_k$, was recently studied in Ref.[@Abe:2012fb]. For the MLMM, tree-level unitarity implies sum rule[@Abe:2012fb], \[eq:SR-MLMM\] G\_[4W\_0]{}\^ - G\_[W\_0W\_0Z\_0]{}\^2 = G\_[W\_0W\_0Z\_1]{}\^2 + , where the symbols $(h,\,H)$ denote the two mass-eigenstate Higgs bosons. The sum rule illustrates how exchanging both the new spin-1 weak bosons $W_1/Z_1$ and the spin-0 Higgs bosons $h/H$ is required to ensure the unitarity of longitudinal weak boson scattering in the MLMM[@Abe:2012fb].[^2] Analysis of $\,{\bf W}_{\bf 1}^{\mathbf{\pm}}$ Detection at the LHC-8 ===================================================================== Extrapolating from our previous work [@PRD2008] at a 14 TeV LHC, we have found that the best process for detecting $W_1^{}$ at LHC-8 is associated production, $\,pp\rightarrow W_1^{}Z_0^{}\rightarrow W_0^{}Z_0^{}Z_0^{} \rightarrow jj \ell^+ \ell^-\ell^+ \ell^- $, where we select the $W_0^{}$ decays into dijets and the $Z_0^{}$ decays into electron or muon pairs. We have systematically computed all the major SM backgrounds for the $jj4\ell$ final state, including the irreducible backgrounds $\,pp\to W_0Z_0Z_0\to jj4\ell\,$ ($jj=qq'$) without the contribution of $W_1$, as well as the reducible backgrounds $\,pp\to ggZ_0Z_0\to jj4\ell$, $\,pp\to Z_0Z_0Z_0\to jj4\ell\,$, and the SM $\,pp\to jj4\ell\,$ other than the above reducible backgrounds. We performed parton level calculations at tree-level using two different methods and two different gauges to check the consistency. In one calculation, we used the helicity amplitude approach [@helicity] to generate the signal and backgrounds. We also calculated both the signal and background using CalcHEP [@Pukhov:1999gg; @Pukhov:2004ca]. For the signal calculation in CalcHEP, we used FeynRules [@Christensen:2008py] to implement the minimal Higgsless model [@Christensen:2009jx]. We found satisfactory agreement between these two approaches and between both unitary and ’tHooft-Feynman gauge. We used a scale of $\,\sqrt{\hat s}$ for the strong coupling in the backgrounds and $\,\sqrt{\hat s}/2$ for the CTEQ6L [@cteq6] parton distribution functions. We included both the first and second generation quarks in the protons and jets, and both electrons and muons in the final-state leptons. In our calculations, we impose basic acceptance cuts, $$\begin{aligned} p_{T \ell}^{} > 10\,{\rm GeV}, ~~~~~ |\eta_\ell^{}|<2.5 \,, \nn\\[0.9mm] p_{T j}^{} > 15\,{\rm GeV}, ~~~~~ |\eta_j^{}|<4.5 \,,\end{aligned}$$ and also a reconstruction cut for identifying $W_0$ bosons that decay to dijets, $$\begin{aligned} M_{jj}^{} = 80 \pm 15\,{\rm GeV} \,.\end{aligned}$$ We further analyzed the distributions of the dijet opening-angle $\,\Delta R(jj)$ in the decays of $\,W_0\to jj$ for both the signal and SM background events. We find that the signal events are peaked in the small opening-angle region around $\,\Delta R(jj)\sim 0.6$, while the SM backgrounds tend to populate the range of larger opening angles, with a broad bump around $\,\Delta R(jj)=1.5-3.3$. In order to sufficiently suppress the SM backgrounds, we find the following opening-angle cut[^3] to be very effective [@j-separation], \[eq:DR-cut1\] R(jj)  &lt;  1.6. At the LHC-8, we note that the above cut reduces the signal events by only $10-15\%$, but removes about $72-80\%$ of the SM backgrounds. ![Predicted signal cross section for $\,pp\to W_1Z_0\to W_0Z_0Z_0\to jj4\ell\,$ as a function of the $W_1^{}$ mass in the MLMM after all cuts at the LHC-8.[]{data-label="Fig:3"}](fig4.eps){width="8cm"} In Fig.\[Fig:3\], we display the predicted total signal cross section for the process $\,pp\to W_0Z_0Z_0\to jj4\ell\,$ after all cuts at the LHC-8 have been imposed; this is shown as a function of the $W_1^{}$ mass for the range $\,250-400$GeV.[^4] In Fig.\[Fig:5\], we display the required integrated luminosities for detecting the $W_1^\pm$ signal at the $3\sigma$ and $5\sigma$ levels as a function of the $W_1^\pm$ mass $M_{W_1}^{}$. We see the LHC-8 should be able to observe the $W_1^\pm$ gauge bosons of the minimal linear moose model studied up to masses of order 400 GeV. We look forward to seeing the results. ![Integrated luminosities required for detection of new $W_1^\pm$ gauge bosons at the $3\sigma$ level in the MLMM (lower blue curve), and at the $5\sigma$ level (upper red curve) as a function of the $W_1^{}$ mass, at the LHC-8.[]{data-label="Fig:5"}](fig5.eps){width="8cm"} [**Acknowledgments**]{}\ This research was supported by the NSF of China (grants 11275101, 10625522, 10635030, 11135003, 11075086) and the National Basic Research Program of China (grant 2010CB833000); by the U.S. NSF under Grants PHY-0854889 and PHY-0705682; and by the University of Pittsburgh Particle Physics, Astrophysics, and Cosmology Center. HJH thanks CERN Theory Division for hospitality. [99]{} C. Du, H. -J. He, Y. -P. Kuang, B. Zhang, N. D. Christensen, R. S. Chivukula, E. H. Simmons and , Phys. Rev. D [**86**]{}, 095011 (2012) \[arXiv:1206.6022 \[hep-ph\]\]. C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D [**69**]{}, 055006 (2004) \[hep-ph/0305237\]; C. Csaki, C. Grojean, L. Pilo and J. Terning, Phys. Rev. Lett.  [**92**]{}, 101802 (2004) \[hep-ph/0308038\]. R. S. Chivukula, D. A. Dicus, H. J. He, Phys. Lett. B [**525**]{}, 175 (2002) \[hep-ph/0111016\]; R. S. Chivukula and H. J. He, Phys. Lett. B [**532**]{}, 121 (2002) \[hep-ph/0201164\]; R. S. Chivukula, D. A. Dicus, H. J. He, S. Nandi, Phys. Lett. B [**562**]{}, 109 (2003) \[hep-ph/0302263\]; H. J. He, Int. J. Mod. Phys. A [**20**]{}, 3362 (2005) \[hep-ph/0412113\] R. S. Chivukula, H. J. He, M. Kurachi, E. H. Simmons, M. Tanabashi, Phys. Rev. D [**78**]{}, 095003 (2008) \[arXiv:0808.1682\] R. S. Chivukula, B. Coleppa, S. DiChiara, E. H. Simmons, H. J. He, M. Kurachi, M. Tanabashi, Phys. Rev. D [**74**]{}, 075011 (2006) \[arXiv:hep-ph/0607124\]. T. Abe, N. Chen, H. J. He, JHEP (2012), \[arXiv:1207.4103 \[hep-ph\]\]. G. Aad [*et al.,*]{} \[ATLAS Collaboration\], Phys. Lett. B[**716**]{}, 1 (2012) \[arXiv:1207.7214 \[hep-ex\]\];\ S. Chatrchyan [*et al.,*]{} \[CMS Collaboration\], Phys. Lett. B[**716**]{}, 30 (2012) \[arXiv:1207.7235 \[hep-ex\]\]. R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev.  D [**72**]{}, 015008 (2005) \[arXiv:hep-ph/0504114\]. G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, Phys. Rev.  D [**71**]{}, 035015 (2005) \[arXiv:hep-ph/0409126\]. R. Foadi, S. Gopalakrishna and C. Schmidt, Phys. Lett.  B [**606**]{}, 157 (2005) \[arXiv:hep-ph/0409266\]. R. S. Chivukula, E. H. Simmons, H. J. He, M. Kurachi and M. Tanabashi, Phys. Rev. D [**71**]{}, 115001 (2005) \[arXiv:hep-ph/0502162\]. R. Casalbuoni, S. De Curtis, D. Dolce and D. Dominici, Phys. Rev. D [**71**]{}, 075015 (2005) \[arXiv:hep-ph/0502209\]. A. Belyaev, R. S. Chivukula, N. D. Christensen, H. J. He, M. Kurachi, E. H. Simmons, M. Tanabashi, Phys. Rev. D [**80**]{}, 055022 (2009) \[arXiv:0907.2662\]; and presentation in the Proceedings of International Workshop on “Strong Coupling Gauge Theories in LHC Era" (SCGT-2009), arXiv:1003.1786 \[hep-ph\]. H. J. He, Y. P. Kuang, Y. Qi, B. Zhang, A. Belyaev, R. S. Chivukula, N. D. Christensen, A. Pukhov, E. H. Simmons, Phys. Rev. D [**78**]{}, 031701 (2008) \[arXiv:0708.2588\]. R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett.  B [**155**]{}, 95 (1985). R. Casalbuoni [*et al.*]{}, [Phys. Rev.]{} [**D53**]{}, 5201 (1996) \[hep-ph/9510431\]. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett.  [**54**]{}, 1215 (1985). M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys.  B [**259**]{}, 493 (1985). M. Bando, T. Fujiwara, and K. Yamawaki, Prog. Theor. Phys.  [**79**]{}, 1140 (1988). M. Bando, T. Kugo, and K. Yamawaki, [Phys. Rept.]{} [**164**]{}, 217 (1988). M. Harada and K. Yamawaki, Phys. Rept. [**381**]{}, 1 (2003) \[hep-ph/0302103\]. J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, Phys. Rev. Lett. [**30**]{} (1973) 1268; Phys. Rev. D[**10**]{} (1974) 1145; C. H. Llewellyn Smith, Phys. Lett. [**46**]{}B (1973) 233. D. A. Dicus and V. S. Mathur, Phys. Rev. D[**7**]{} (1973) 3111; B. W. Lee, C. Quigg, and H. B. Thacker, Phys. Rev. Lett. [**38**]{} (1977) 883; Phys. Rev. D[**16**]{} (1977) 1519; M. S. Chanowitz and M. K. Gaillard, Nucl. Phys. B [**261**]{} (1985) 379. N. D. Christensen and C. Duhr, Comput. Phys. Commun.  [**180**]{}, 1614 (2009) \[arXiv:0806.4194 \[hep-ph\]\]. N. D. Christensen, P. de Aquino, C. Degrande, C. Duhr, B. Fuks, M. Herquet, F. Maltoni and S. Schumann, Eur. Phys. J. C [**71**]{}, 1541 (2011) \[arXiv:0906.2474 \[hep-ph\]\]. K. Hagiwara and D. Zeppenfeld, DESY-85-133; K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, DESY-86-058 (1986); K. Hagiwara and D. Zeppenfeld, KEK Preprint 87-158 (1988). J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. Nadolsky and W. K. Tung, JHEP [**07**]{}, 012 (2002). This kind of cut should be applied to separated jets. Experimentally, two jets are separable if $\,\Delta R_{jj}>0.5$ \[see for instance, S. Ask (ATLAS Collaboration), arXiv:1106.2061\], and two jet-cones do not overlap at all if $\,\Delta R_{jj}>1$. Hence the cut (\[eq:DR-cut1\]) can be realized. A. Pukhov, E. Boos, M. Dubinin, V. Edneral, V. Ilyin, D. Kovalenko, A. Kryukov and V. Savrin, [*et al.*]{}, arXiv:hep-ph/9908288. A. Pukhov, arXiv:hep-ph/0412191. [^1]: This contribution is an abbrievated version of ref. [@Du:2012vh]. [^2]: We also note that the $hWW$ and $hZZ$ couplings are generally suppressed[@Abe:2012fb] relative to the SM values because of the VEV ratio $\,f_2^{}/f_1^{}=O(1)\,$ and the $h-H$ mixing. [^3]: These are somewhat weaker than the cut of $\,\Delta R(jj)< 1.5\,$ imposed in [@PRD2008]. [^4]: Here, we define the signal region to include all events satisfying the condition, $ M({Z_0jj}) ~= ~M_{W_1}^{} \pm 20 \,\text{GeV}$ .

--- abstract: 'Let $f$ be a newform of weight at least $2$ and squarefree level with Fourier coefficients in a number field $K$. We give explicit bounds, depending on congruences of $f$ with other newforms, on the set of primes $\lambda$ of $K$ for which the deformation problem associated to the mod $\lambda$ Galois representation of $f$ is obstructed. We include some explicit examples.' address: 'Department of Mathematics, University of California, Berkeley' author: - Tom Weston title: Explicit unobstructed primes for modular deformation problems of squarefree level --- [^1] Introduction ============ Let $f$ be a newform of weight $k \geq 2$, level $N$, and character $\omega$. Let $K$ be the number field generated by the Fourier coefficients of $f$. For any prime $\lambda$ of $K$ Deligne has constructed a semisimple Galois representation $${{\bar{\rho}}_{f,\lambda}}: {G_{{\mathbf{Q}},S\cup\{\ell\}}}\to \operatorname{GL}_{2} {k_{\lambda}}$$ over the residue field ${k_{\lambda}}$ of $K$ at $\lambda$; here ${G_{{\mathbf{Q}},S\cup\{\ell\}}}$ is the Galois group of the maximal extension of ${\mathbf{Q}}$ unramified outside the set $S$ of places dividing $N\infty$ and the characteristic $\ell$ of ${k_{\lambda}}$. The representation ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible for almost all primes $\lambda$; we write $\operatorname{Red}(f)$ for the set of $\lambda$ such that ${{\bar{\rho}}_{f,\lambda}}$ is not absolutely irreducible. Following Mazur, we say that a prime $\lambda \notin \operatorname{Red}(f)$ is an [*obstructed prime*]{} for $f$ if the cohomology group $H^{2}({G_{{\mathbf{Q}},S\cup\{\ell\}}},\operatorname{ad}{{\bar{\rho}}_{f,\lambda}})$ of the adjoint representation of ${{\bar{\rho}}_{f,\lambda}}$ is non-zero. We write $\operatorname{Obs}(f)$ for the set of such primes. The importance of this notion rests on the fact that for $\lambda \notin \operatorname{Obs}(f) \cup \operatorname{Red}(f)$, the universal deformation ring associated to ${{\bar{\rho}}_{f,\lambda}}$ is isomorphic to a power series ring in three variables over the Witt vectors of ${k_{\lambda}}$; see Section \[s2\] for details. It was shown in [@Weston2] that $\operatorname{Obs}(f)$ is finite for $f$ of weight $k \geq 3$. In this paper we obtain an explicit bound on $\operatorname{Obs}(f)$ in the case that the level $N$ of $f$ is squarefree. We state our result here only for $N >1$; see Section \[s42\] for the general statement (where we also allow $k=2$ and $S$ non-minimal) and a partial converse. Assume that $k \geq 3$ and that $N > 1$ is squarefree. Let $M$ denote the conductor of the Dirichlet character $\omega$. Then $$\operatorname{Obs}(f) \subseteq \bigl\{ \lambda \mid \ell \,;\, \ell \leq k+1 \text{~or~} \ell \mid N\varphi(N){{\textstyle}\underset{p \mid \frac{N}{M}}{\prod}(p+1)} \bigr\} \cup \operatorname{Cong}(f)$$ with $\operatorname{Cong}(f)$ the set of congruence primes for $f$ (as defined in Section \[s41\]) and $\varphi$ the Euler totient function. We note that the set $\operatorname{Cong}(f)$ is computable using the results of [@Sturm] and a tool such as [@Stein]. It is not immediately clear to the author what form to expect the analogue of this result for to take for $N$ not squarefree. In Section \[s2\] we give a brief review of deformation theory and use standard duality arguments to reduce the vanishing of $H^{2}({G_{{\mathbf{Q}},S}},\operatorname{ad}{{\bar{\rho}}_{f,\lambda}})$ to the vanishing of certain local and global cohomology groups. The local groups are the subject of Section \[s3\]; the computations rest on some simple cases of the local Langlands correspondence. In Section \[s41\] we use results of Hida (as refined in [@Ghate]) to relate the global cohomology group to a certain Selmer group studied by Diamond, Flach, and Guo. The main results of the paper are proved in Section \[s42\]. We give several explicit examples in Section \[s5\]. It is a pleasure to thank Matthias Flach, Elena Mantovan, Robert Pollack, and Ken Ribet for helpful conversations related to this paper. Notation {#notation .unnumbered} -------- If $\rho : G \to \operatorname{GL}_{2}\!R$ is a representation of a group $G$ over a ring $R$, we write $\operatorname{ad}\rho : G \to \operatorname{GL}_{4}\!R$ for the adjoint representation of $G$ on $\operatorname{End}(\rho)$ and ${\operatorname{ad}^{0}\!}\rho : G \to \operatorname{GL}_{3}\!R$ for the kernel of the trace map from $\operatorname{ad}\rho$ to the trivial representation. If $\rho : G \to \operatorname{GL}_{n}\!R$ is any representation, we write $H^{i}(G,\rho)$ for the cohomology group $H^{i}(G,V_{\rho})$ with $V_{\rho}$ a free $R$-module of rank $n$ with $G$-action via $\rho$. We write ${G_{{\mathbf{Q}}}}$ for the absolute Galois group of ${\mathbf{Q}}$. We fix now and forever embeddings ${\bar{{\mathbf{Q}}}}{\hookrightarrow}{{\bar{{\mathbf{Q}}}}_{p}}$ for each $p$, yielding injections $G_{p} {\hookrightarrow}{G_{{\mathbf{Q}}}}$ with $G_{p}$ the absolute Galois group of ${{\mathbf{Q}}_{p}}$. We write $I_{p}$ for the inertia subgroup of $G_{p}$. Let ${\varepsilon_{\ell}}: {G_{{\mathbf{Q}}}}\to {{\mathbf{Z}}_{\ell}}^{\times}$ be the $\ell$-adic cyclotomic character and let ${\bar{\varepsilon}_{\ell}}: {G_{{\mathbf{Q}}}}\to {{\mathbf{F}}_{\ell}}^{\times}$ be its reduction, the mod $\ell$ Teichmüller character. If $M$ is a ${{\mathbf{Z}}_{\ell}}[{G_{{\mathbf{Q}}}}]$-module, we write $M(1)$ for its first Tate twist $M \otimes_{{{\mathbf{Z}}_{\ell}}} {\varepsilon_{\ell}}$. If $S$ is a set of places of ${\mathbf{Q}}$ containing the infinite place, the expression “$p \in S$” is to be interpreted as “$p \in S - \{\infty\}$”. Obstructions {#s2} ============ Deformation theory {#s21} ------------------ In this section we review the fundamentals of the deformation theory of representations of profinite groups as in [@Mazur]. Let $k$ be a finite field and let ${\mathcal{C}}$ denote the category of local rings which are inverse limits of artinian local rings with residue field $k$; a morphism $A \to B$ in ${\mathcal{C}}$ is a continuous local homomorphism inducing the identity map on residue fields. Note that any ring $A$ in ${\mathcal{C}}$ is canonically an algebra for the Witt vectors $W(k)$ of $k$. Let $G$ be a profinite group and fix an absolutely irreducible continuous representation $${\bar{\rho}}: G \to \operatorname{GL}_{n}\!k$$ for some $n \geq 1$. A [*lifting*]{} of ${\bar{\rho}}$ to a ring $A$ in ${\mathcal{C}}$ is a continuous representation $\rho : G \to \operatorname{GL}_{n}\!A$ such that the composition $$G \overset{\rho}{{\longrightarrow}} \operatorname{GL}_{n}\!A {\longrightarrow}\operatorname{GL}_{n}\!k$$ is equal to $\rho$. Two liftings $\rho_{1},\rho_{2}$ of ${\bar{\rho}}$ are said to be [*strictly equivalent*]{} if there is a matrix $M$ in the kernel of $\operatorname{GL}_{n}\!A \to \operatorname{GL}_{n}\!k$ such that $\rho_{1} = M \cdot \rho_{2} \cdot M^{-1}$. A [*deformation*]{} of ${\bar{\rho}}$ to $A$ is a strict equivalence class of liftings. Let $$D_{{\bar{\rho}}} : {\mathcal{C}}\to \operatorname{Sets}$$ be the functor sending a ring $A$ to the set of deformations of ${\bar{\rho}}$ to $A$. The deformation functor $D_{{\bar{\rho}}}$ is representable by [@Mazur Section 1.2]; that is, there is a ring $R_{{\bar{\rho}}}$ in ${\mathcal{C}}$ (called the [*universal deformation ring*]{} of ${\bar{\rho}}$) and an isomorphism of functors $$\label{eq:func} D_{{\bar{\rho}}}(-) \cong \operatorname{Hom}_{{\mathcal{C}}}(R_{{\bar{\rho}}},-).$$ Note that via (\[eq:func\]) the identity map on $R_{{\bar{\rho}}}$ corresponds to a deformation $$\rho^{\operatorname{univ}} : G \to \operatorname{GL}_{n}\!R_{{\bar{\rho}}}$$ of ${\bar{\rho}}$ to $R_{{\bar{\rho}}}$; this is the [*universal deformation*]{} of ${\bar{\rho}}$, and the isomorphism (\[eq:func\]) sends $f : R_{{\bar{\rho}}} \to A$ to the deformation $f \circ \rho^{\operatorname{univ}}$ of ${\bar{\rho}}$ to $A$. The next proposition gives the fundamental connection between the deformation problem $D_{{\bar{\rho}}}$ and the cohomology groups $H^{i}(G,\operatorname{ad}{\bar{\rho}})$. We say that $D_{{\bar{\rho}}}$ is [*unobstructed*]{} if $H^{2}(G,\operatorname{ad}{\bar{\rho}}) = 0$. \[prop:defthy\] Assume that $H^{i}(G,\operatorname{ad}{\bar{\rho}})$ is finite-dimensional over $k$ for each $i$; set $d =dim_{k} H^{1}(G,\operatorname{ad}{\bar{\rho}})$. Then there exists a (non-canonical) surjection $$\label{eq:surj} W(k)[[T_{1},\ldots,T_{d}]] {\twoheadrightarrow}R_{{\bar{\rho}}}$$ with kernel generated by at most $\dim_{k} H^{2}(G,\operatorname{ad}{\bar{\rho}})$ elements. In particular, if $D_{{\bar{\rho}}}$ is unobstructed, then (\[eq:surj\]) is an isomorphism. This is proved in [@Mazur Section 1.6]. The existence of the surjection (\[eq:surj\]) follows from an isomorphism $$D_{{\bar{\rho}}}\bigl(k[\epsilon]/\epsilon^{2}\bigr) {\overset{\simeq}{\longrightarrow}}H^{1}(G,\operatorname{ad}{\bar{\rho}})$$ (sending a deformation $\rho$ to the cocycle $c_{\rho}$ such that $\rho(g) = {\bar{\rho}}(g)(1+\epsilon \cdot c_{\rho}(g))$ for all $g \in G$) and the interpretation of these groups as the tangent space of $R_{{\bar{\rho}}}$ via (\[eq:func\]). The statement about the kernel $J$ of (\[eq:surj\]) follows from an injection $$\operatorname{Hom}(J,k) {\hookrightarrow}H^{2}(G,\operatorname{ad}{\bar{\rho}})$$ constructed using an obstruction two-cocycle measuring the failure of $\rho^{\operatorname{univ}}$ to lift via (\[eq:surj\]). The next lemma will be useful later in the paper. \[lemma:adjvan\] Let ${\bar{\rho}}: G \to \operatorname{GL}_{2}\!k$ be continuous and absolutely irreducible and let $\chi : G \to k^{\times}$ be a character of order at least $3$. Then $H^{0}(G,\chi \otimes \operatorname{ad}{\bar{\rho}})=0$. If the image of ${\bar{\rho}}$ is dihedral, then the $G$-representation $\operatorname{ad}{\bar{\rho}}$ is the sum of the trivial character, a quadratic character, and an irreducible two-dimensional representation of $G$. If the image of ${\bar{\rho}}$ is not dihedral, then $\operatorname{ad}{\bar{\rho}}$ is the sum of the trivial character and an irreducible three-dimensional representation of $G$. In either case the lemma follows since $\chi$ is neither trivial nor quadratic. Galois cohomology {#s22} ----------------- Let $k$ be a finite field of odd characteristic $\ell$. We now apply the discussion of the previous section to the case of a two-dimensional Galois representation over $k$. Fix a finite set $S$ of places of ${\mathbf{Q}}$ including $\ell$ and the infinite place. Let ${\mathbf{Q}}_{S}$ denote the maximal extension of ${\mathbf{Q}}$ unramified outside $S$; set ${G_{{\mathbf{Q}},S}}:= \operatorname{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}})$. Let $${\bar{\rho}}: {G_{{\mathbf{Q}},S}}\to \operatorname{GL}_{2}\!k$$ be continuous and absolutely irreducible. We assume further that ${\bar{\rho}}$ is [*odd*]{} in the sense that the image of complex conjugation has distinct eigenvalues. In this section we study the cohomology groups $H^{i}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}})$. \[lemma:delta\] Each cohomology group $H^{i}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}})$ is finite-dimensional over $k$ and $$\dim_{k} H^{1}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}}) - \dim_{k} H^{2}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}}) = 3.$$ The first statement is [@Milne Corollary 4.15], while the second is a straightforward calculation using Tate’s global Euler characteristic formula as in [@Mazur Section 1.10]. \[cor:unob\] If $H^{2}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}})=0$, then the universal deformation ring $R_{{\bar{\rho}}}$ is (non-canonically) isomorphic to $W(k)[[T_{1},T_{2},T_{3}]]$. We will use global duality theorems of Poitou and Tate to study $H^{2}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}})$. For a $k[{G_{{\mathbf{Q}},S}}]$-module $M$, define $${\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},M) := \ker\bigl(H^{1}({G_{{\mathbf{Q}},S}},M) \to \underset{p \in S}{\oplus} H^{1}(G_{p},M)\bigr).$$ \[lemma:d2\] One has $$\dim_{k} H^{2}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}}) \leq \dim_{k} {\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}}) + \underset{p \in S}{{\textstyle}\sum} \dim_{k} H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}})$$ with equality if $\ell \neq 3$. The trace pairing $\operatorname{ad}{\bar{\rho}}\otimes \operatorname{ad}{\bar{\rho}}\to k$ identifies ${\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}}$ with the Cartier dual of $\operatorname{ad}{\bar{\rho}}$. Thus by [@Milne Theorem 4.10] there is an exact sequence $$\begin{gathered} 0 \to H^{0}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}}) \to \underset{p \in S}{\oplus} H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}}) \to \\ \operatorname{Hom}\bigl(H^{2}({G_{{\mathbf{Q}},S}},\operatorname{ad}{\bar{\rho}}),k\bigr) \to {\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}}) \to 0.\end{gathered}$$ Since ${\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}}= {\bar{\varepsilon}_{\ell}}\oplus ({\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}})$ and ${\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}})$ vanishes by [@Weston Lemma 10.6], the lemma follows from the exact sequence and Lemma \[lemma:adjvan\]. We will study the local terms $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{\bar{\rho}})$ in Section \[s3\]. The global term ${\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}})$ is difficult to control directly; instead we now relate it to a certain Selmer group, which in turn is often computable using the results of Section \[s41\]. Fix a totally ramified extension $K$ of the field of fractions of $W(k)$. The ring of integers ${\mathcal{O}}$ of $K$ lies in ${\mathcal{C}}$; we write ${\mathfrak{m}}$ for its maximal ideal. Let $\rho : {G_{{\mathbf{Q}},S}}\to \operatorname{GL}_{2}\!{\mathcal{O}}$ be a lifting of ${\bar{\rho}}$ to ${\mathcal{O}}$. Let $V_{\rho}$ (resp. $A_{\rho}$) denote a three-dimensional $K$-vector space (resp. $(K/{\mathcal{O}})^{3}$) endowed with a ${G_{{\mathbf{Q}},S}}$-action via ${\operatorname{ad}^{0}\!}\rho : {G_{{\mathbf{Q}},S}}\to \operatorname{GL}_{3}\!{\mathcal{O}}$. Let $V$ (resp. $A$) denote either $V_{\rho}$ (resp. $A_{\rho}$) or else its Tate twist. For a prime $p$, define $${H^{1}_{f}}(G_{p},V) := \begin{cases} H^{1}(G_{p}/I_{p},V^{I_{p}}) & p \neq \ell; \\ \ker\bigl(H^{1}(G_{p},V) \to H^{1}(G_{p},V \otimes B_{\operatorname{crys}})\bigr) & p = \ell; \end{cases}$$ regarded as a $K$-subspace of $H^{1}(G_{p},V)$; here $B_{\operatorname{crys}}$ is the crystalline period ring of Fontaine. Let ${H^{1}_{f}}(G_{p},A)$ denote the image of ${H^{1}_{f}}(G_{p},V)$ under the pushforward from $H^{1}(G_{p},V)$ to $H^{1}(G_{p},A)$. For $M$ denoting either of $V$ or $A$, the [*Selmer group*]{} of $M$ is defined by $${H^{1}_{f}}({G_{{\mathbf{Q}}}},M) := \bigl\{ c \in H^{1}({G_{{\mathbf{Q}}}},M) \,;\, c|_{G_{p}} \in {H^{1}_{f}}(G_{p},M) \text{~for all~}p \bigr\}.$$ Following [@DFG Section 7], we will also need a slight variant of this construction. Define $$\begin{gathered} {H^{1}_{w}}(G_{p},A) := \begin{cases} H^{1}(G_{p}/I_{p},A^{I_{p}}) & p \neq \ell; \\ {H^{1}_{f}}(G_{\ell},A) & p = \ell; \end{cases} \\ H^{1}_{\emptyset}({G_{{\mathbf{Q}}}},A) := \bigl\{ c \in H^{1}({G_{{\mathbf{Q}}}},A) \,;\, c|_{G_{p}} \in {H^{1}_{w}}(G_{p},A) \text{~for all~}p \bigr\}.\end{gathered}$$ Clearly one has ${H^{1}_{f}}(G_{p},A) \subseteq {H^{1}_{w}}(G_{p},A)$ for all $p$, so that $$\label{eq:comp} {H^{1}_{f}}({G_{{\mathbf{Q}}}},A) \subseteq H^{1}_{\emptyset}({G_{{\mathbf{Q}}}},A).$$ In fact, this inclusion is an equality if $A^{I_{p}}$ is divisible for all $p \neq \ell$. \[lemma:ineq\] Assume that $\ell > 3$ and ${H^{1}_{f}}({G_{{\mathbf{Q}}}},V_{\rho}) = {H^{1}_{f}}({G_{{\mathbf{Q}}}},V_{\rho}(1))=0$. Then $$\dim_{k} {\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}}) \leq \dim_{k} H^{1}_{\emptyset}({G_{{\mathbf{Q}},S}},A_{\rho})[{\mathfrak{m}}].$$ Since $\rho$ is a lifting of ${\bar{\rho}}$, the $k[{G_{{\mathbf{Q}},S}}]$-module $A_{\rho}(1)[{\mathfrak{m}}]$ is a realization of ${\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}}$. We thus obtain a natural map $$\label{eq:inj} H^{1}({G_{{\mathbf{Q}}}},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}}) = H^{1}\bigl({G_{{\mathbf{Q}}}},A_{\rho}(1)[{\mathfrak{m}}]\bigr) \to H^{1}\bigl({G_{{\mathbf{Q}}}},A_{\rho}(1)\bigr)$$ which is injective by Lemma \[lemma:adjvan\]. The image of ${\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}})$ under (\[eq:inj\]) is easily seen to lie in ${H^{1}_{f}}({G_{{\mathbf{Q}}}},A_{\rho}(1))$, so that we obtain an injection $$\label{eq:inj2} {\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{\bar{\rho}}) {\hookrightarrow}{H^{1}_{f}}\bigl({G_{{\mathbf{Q}}}},A_{\rho}(1)\bigr).$$ By [@Flach1 Theorem 1], the latter group is (non-canonically) isomorphic to ${H^{1}_{f}}({G_{{\mathbf{Q}}}},A_{\rho})$ (see [@Weston2 Proposition 2.2]; this also uses the assumption on the vanishing of the Selmer group of $V_{\rho}$ and $V_{\rho}(1)$). The lemma thus follows from (\[eq:inj2\]) and (\[eq:comp\]). The only difficulty in analyzing the failure of (\[eq:inj2\]) to be an isomorphism on ${\mathfrak{m}}$-torsion is the determination of the image of the restriction map $${H^{1}_{f}}\bigl({G_{{\mathbf{Q}}}},A_{\rho}(1)\bigr)[{\mathfrak{m}}] \to {H^{1}_{f}}\bigl(G_{\ell},A_{\rho}(1) \bigr)[{\mathfrak{m}}].$$ Unfortunately, this question appears to be quite difficult in general. Local invariants {#s3} ================ Let $f = \sum a_{n}q^{n}$ be a newform of weight $k \geq 2$, squarefree level $N$, and character $\omega$. Let $K$ be the number field generated by the Fourier coefficients $a_{n}$ of $f$. For any prime $\lambda$ of $K$, Deligne has constructed a continuous $\lambda$-adic Galois representation $${\rho_{f,\lambda}}: {G_{{\mathbf{Q}}}}\to \operatorname{GL}_{2}\! {K_{\lambda}}.$$ This representation is unramified at $p \nmid N\ell$ (with $\ell$ the characteristic of the residue field ${k_{\lambda}}$ of ${K_{\lambda}}$) and for such $p$ the trace (resp. the determinant) of the image of an arithmetic Frobenius element $\operatorname{Frob}_{p}$ under ${\rho_{f,\lambda}}$ is equal to $a_{p}$ (resp. $p^{k-1}\omega(p)$). As usual we identify $\omega : ({\mathbf{Z}}/N{\mathbf{Z}})^{\times} \to \mu_{\varphi(N)}$ with a Galois character via the canonical isomorphism $\operatorname{Gal}({\mathbf{Q}}(\mu_{N})/{\mathbf{Q}}) \cong ({\mathbf{Z}}/N{\mathbf{Z}})^{\times}$; the determinant of ${\rho_{f,\lambda}}$ is then ${\varepsilon_{\ell}}^{k-1}\omega$. Let $M$ denote the conductor of $\omega$ and let $\omega_{0} : ({\mathbf{Z}}/M{\mathbf{Z}})^{\times} \to \mu_{\varphi(M)}$ be the associated primitive Dirichlet character. Then $\omega$ is ramified at $p$ if and only if $p$ divides $M$, in which case the restriction of $\omega$ to the inertia group $I_{p}$ is a non-trivial character taking values in $\mu_{p-1}$. For the remainder of this section we fix a prime $\lambda$ of $K$ dividing a rational prime $\ell$. Let $${{\bar{\rho}}_{f,\lambda}}: {G_{{\mathbf{Q}}}}\to \operatorname{GL}_{2}\!{k_{\lambda}}$$ be the semisimple reduction of ${\rho_{f,\lambda}}$; this is well-defined independent of any choice of integral model of ${\rho_{f,\lambda}}$. We are interested in the local invariants $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}})$ for all primes $p$. As $${\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}\cong {\bar{\varepsilon}_{\ell}}\oplus ({\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}})$$ and $$H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}) \neq 0 \,\, \Leftrightarrow \,\, p \equiv 1 \pmod{\ell},$$ we will restrict our attention below to the case that $\ell$ does not divide $p-1$ and to the study of $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}})$. In the analysis below we make use of the local Langlands correspondence and the compatibility results completed in [@Carayol]. Rather than review these results in detail, we will only recall the consequences we need; see [@Weston2] for more details and references. $\boldsymbol{p \nmid N\ell}$ {#s31} ---------------------------- Fix $\alpha_{p},\beta_{p} \in {\bar{K}}$ with $\alpha_{p}+\beta_{p} = a_{p}$ and $\alpha_{p}\beta_{p} = p^{k-1}\omega(p)$. In this case, we have $${\rho_{f,\lambda}}|_{G_{p}} \otimes {{\bar{K}}_{\lambda}}\cong \chi_{1} \oplus \chi_{2}$$ where the $\chi_{i} : G_{p} \to {{\bar{K}}_{\lambda}}^{\times}$ are unramified characters with $$\label{eq:char} \chi_{1}(\operatorname{Frob}_{p}) = \alpha_{p}; \qquad \chi_{2}(\operatorname{Frob}_{p}) = \beta_{p}.$$ We write ${\bar{\chi}}_{i} : G_{p} \to {\bar{k}_{\lambda}}^{\times}$ for the reduction of $\chi_{i}$. \[lemma:unramps\] Assume $p \nmid N\ell$ and $p \not\equiv 1 \pmod{\ell}$. Then $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$ if and only if $a_{p}^{2} \equiv (p+1)^{2}p^{k-2}\omega(p) \pmod{\lambda}$. Since the existence of eigenvectors with ${k_{\lambda}}$-rational eigenvalues is invariant under base extension, the existence of $G_{p}$-invariants in ${\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}}$ is equivalent to the existence of $G_{p}$-invariants in $$\bigl({\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}}|_{G_{p}} \bigr) \otimes {\bar{k}_{\lambda}}\cong {\bar{\varepsilon}_{\ell}}\oplus {\bar{\varepsilon}_{\ell}}{\bar{\chi}}_{1}^{\vphantom{-1}}{\bar{\chi}}_{2}^{-1} \oplus {\bar{\varepsilon}_{\ell}}{\bar{\chi}}_{1}^{-1}{\bar{\chi}}_{2}^{\vphantom{-1}}.$$ As $p \not\equiv 1 \pmod{\ell}$, this has non-trivial $G_{p}$-invariants if and only if one of the characters ${\bar{\varepsilon}_{\ell}}{\bar{\chi}}_{1}^{\vphantom{-1}}{\bar{\chi}}_{2}^{-1}$, ${\bar{\varepsilon}_{\ell}}{\bar{\chi}}_{1}^{-1}{\bar{\chi}}_{2}^{\vphantom{-1}}$ is trivial. By (\[eq:char\]) this occurs if and only if $$\begin{aligned} {2} \frac{\alpha_{p}}{\beta_{p}} &\equiv p^{\pm 1} & &\pmod{\lambda}. \\ \intertext{This in turn is equivalent to} \frac{\alpha_{p}}{\beta_{p}} + \frac{\beta_{p}}{\alpha_{p}} &\equiv p + \frac{1}{p} & &\pmod{\lambda}\\ \frac{(\alpha_{p}+\beta_{p})^{2}}{\alpha_{p}\beta_{p}} &\equiv \frac{(p+1)^{2}}{p} & &\pmod{\lambda}\\ a_{p}^{2} &\equiv (p+1)^{2}p^{k-2}\omega(p) & &\pmod{\lambda}\end{aligned}$$ as claimed. $\boldsymbol{p \mid M}$, $\boldsymbol{p \neq \ell}$ {#s32} --------------------------------------------------- In this case the $p$-component $\pi_{p}$ of the automorphic representation associated to $f$ has conductor $1$ and ramified central character. It follows that $\pi_{p}$ is a principal series representation associated to one ramified character and one unramified character. On the Galois side, this translates to $${\rho_{f,\lambda}}|_{G_{p}} \otimes {{\bar{K}}_{\lambda}}\cong \chi_{1} \oplus \chi_{2}$$ for continuous characters $\chi_{i} : G_{p} \to {{\bar{K}}_{\lambda}}^{\times}$ with $\chi_{1}$ ramified and $\chi_{2}$ unramified. Since ${\rho_{f,\lambda}}|_{G_{p}}$ has determinant ${\varepsilon_{\ell}}^{k-1}\omega|_{G_{p}}$, we have $\chi_{1}\chi_{2} = {\varepsilon_{\ell}}^{k-1}\omega|_{G_{p}}$. In particular $\chi_{1}|_{I_{p}} = \omega|_{I_{p}}$ is a non-trivial character taking values in $\mu_{p-1}$. If $p \not \equiv 1 \pmod{\ell}$, then $\mu_{p-1}$ injects into ${k_{\lambda}}^{\times}$ and consequently the reduction $\bar{\chi}_{1} : G_{p} \to {\bar{k}_{\lambda}}^{\times}$ is still ramified at $p$. \[lemma:ramps\] Assume $p \mid M$, $p \neq \ell$, and $p \not\equiv 1 \pmod{\ell}$. Then $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) = 0$. As in Lemma \[lemma:unramps\], it suffices to show that the two characters ${\bar{\varepsilon}_{\ell}}{\bar{\chi}}_{1}^{\vphantom{-1}}{\bar{\chi}}_{2}^{-1}$, ${\bar{\varepsilon}_{\ell}}{\bar{\chi}}_{1}^{-1}{\bar{\chi}}_{2}^{\vphantom{-1}}$ are non-trivial. Since ${\bar{\varepsilon}_{\ell}}$ and ${\bar{\chi}}_{2}$ are unramified at $p$ while ${\bar{\chi}}_{1}$ is ramified at $p$, this is clear. $\boldsymbol{p \mid \frac{N}{M}}$, $\boldsymbol{p \neq \ell}$ {#s33} ------------------------------------------------------------- In this case $\pi_{p}$ has conductor $1$ and unramified central character. It follows that $\pi_{p}$ is the special representation associated to an unramified character. This means that there exists an unramified character $\chi : G_{p} \to {{\bar{K}}_{\lambda}}^{\times}$ such that $$\label{eq:spec} {\rho_{f,\lambda}}|_{G_{p}} \otimes {{\bar{K}}_{\lambda}}\cong \left(\begin{array}{cc} {\varepsilon_{\ell}}\chi & * \\ 0 & \chi \end{array}\right)$$ with the upper right corner ramified. \[lemma:specss\] Assume $p \mid \frac{N}{M}$, $p \neq \ell$, and $p^{2} \not\equiv 1 \pmod{\ell}$. Then $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$ if and only if ${{\bar{\rho}}_{f,\lambda}}$ is unramified at $p$. Since $p^{2} \not\equiv 1 \pmod{\ell}$, by [@Weston2 Lemma 5.1] we have $${{\bar{\rho}}_{f,\lambda}}|_{G_{p}} \otimes {\bar{k}_{\lambda}}\cong \left(\begin{array}{cc} {\bar{\varepsilon}_{\ell}}{\bar{\chi}}& \nu \\ 0 & {\bar{\chi}}\end{array}\right)$$ for some $\nu : G_{p} \to {\bar{k}_{\lambda}}$; in fact, one checks directly that ${\bar{\chi}}^{-1}\nu$ is naturally an element of $H^{1}(G_{p},{\bar{k}_{\lambda}}(1))$. Since ${\bar{\varepsilon}_{\ell}}$ and ${\bar{\chi}}$ are unramified, ${{\bar{\rho}}_{f,\lambda}}|_{G_{p}}$ is unramified if and only if ${\bar{\chi}}^{-1}\nu$ is unramified. However, since $p \not\equiv 1 \pmod{\ell}$ every non-zero element of $H^{1}(G_{p},{\bar{k}_{\lambda}}(1))$ is ramified. We conclude that ${{\bar{\rho}}_{f,\lambda}}|_{G_{p}}$ is unramified if and only if it is semisimple. [@Weston2 Lemma 5.2] now completes the proof. \[lemma:spec\] Assume $p \mid \frac{N}{M}$, $p \neq \ell$, $p^{2} \not\equiv 1 \pmod{\ell}$, and ${{\bar{\rho}}_{f,\lambda}}$ absolutely irreducible. Then $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$ if and only if there exists a newform $f'$, of weight $k$ and level dividing $\frac{N}{p}$, such that ${\bar{\rho}}_{f,{\bar{\lambda}}} \cong {\bar{\rho}}_{f',{\bar{\lambda}}}$ for some prime ${\bar{\lambda}}$ of ${\bar{{\mathbf{Q}}}}$ over $\lambda$. Here by ${\bar{\rho}}_{f,{\bar{\lambda}}}$ (resp. ${\bar{\rho}}_{f',{\bar{\lambda}}}$) we mean ${\bar{\rho}}_{f,\lambda} \otimes {\bar{k}_{\lambda}}$ (resp.  ${\bar{\rho}}_{f',\lambda'} \otimes \bar{k}'_{\lambda'}$ with $\lambda'$ the intersection of ${\bar{\lambda}}$ with the field $K'$ of Fourier coefficients of $f'$ and with $k'_{\lambda'}$ the residue field of $K'$ at $\lambda'$.) By [@Edixhoven (B) of p. 221], the existence of such an $f'$ is equivalent to ${{\bar{\rho}}_{f,\lambda}}$ being unramified at $p$. Thus the lemma follows from Lemma \[lemma:specss\]. \[rmk:spec\] If one further assumes that $p' \not\equiv 1 \pmod{\ell}$ for all $p'$ dividing $N$, then the newform $f'$ of Lemma \[lemma:spec\] must have level a multiple of $M$ and character lifting $\omega_{0}$, so that $\lambda$ is a congruence prime for $f$ of level dividing $\frac{N}{p}$ in the terminology of Section \[s41\]. Indeed, ${\bar{\rho}}_{f',\lambda}$ is isomorphic to ${\bar{\rho}}_{f,\lambda}$ and thus has determinant ${\bar{\varepsilon}_{\ell}}^{k-1}\bar{\omega}$; therefore the character $\omega'$ of $f'$ must have reduction equal to $\bar{\omega}$. However, since $p \not\equiv 1 \pmod{\ell}$ for all $p$ dividing $N$, the only such characters of conductor dividing $N$ are those which lift $\omega_{0}$. Thus $f'$ must have level divisible by $M$ and character lifting $\omega_{0}$, as claimed. $\boldsymbol{p = \ell}$, $\boldsymbol{\ell \nmid N}$ {#s34} ---------------------------------------------------- We now give some mild improvements on the results of [@Weston2 Section 4] on the vanishing of $H^{0}(G_{\ell},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}})$. Recall that $f = \sum a_{n}q^{n}$ is said to be [*ordinary*]{} (resp. [*supersingular*]{}) at $\lambda$ if $v_{\lambda}(a_{\ell}) = 0$ (resp. $v_{\lambda}(a_{\ell}) > 0$), with $v_{\lambda}$ the $\lambda$-adic valuation. If $f$ is ordinary at $\lambda$, then the semisimplification of ${\rho_{f,\lambda}}|_{I_{\ell}} \otimes {{\bar{K}}_{\lambda}}$ is isomorphic to ${\varepsilon_{\ell}}^{k-1} \oplus 1$, while if $f$ is supersingular at $\lambda$, then ${{\bar{\rho}}_{f,\lambda}}|_{G_{\ell}}$ is absolutely irreducible. (This all follows from the discussion of [@Edixhoven pp. 214–215], for example.) \[lemma:ord\] Assume $\ell \nmid N$. If $f$ is ordinary at $\lambda$ and $H^{0}(G_{\ell},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$, then $k \equiv 0,2 \pmod{\ell - 1}$. It suffices to prove the corresponding result for the $I_{\ell}$-invariants of the semisimplification of $({\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \otimes {\bar{k}_{\lambda}}$. By the above discussion this semisimplification is isomorphic to $${\bar{\varepsilon}_{\ell}}\oplus {\bar{\varepsilon}_{\ell}}\oplus {\bar{\varepsilon}_{\ell}}^{k} \oplus {\bar{\varepsilon}_{\ell}}^{2-k}.$$ Since ${\bar{\varepsilon}_{\ell}}$ has order $\ell-1$, the lemma follows. Note that the above lemma is vacuous in the case of weight $2$. \[lemma:ss\] Assume $\ell \nmid N$. If $f$ is supersingular at $\lambda$ and $\ell > 3$, then $H^{0}(G_{\ell},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}})=0$. As ${{\bar{\rho}}_{f,\lambda}}|_{G_{\ell}}$ is absolutely irreducible, this is immediate from Lemma \[lemma:adjvan\]. Global results {#s4} ============== We continue with a newform $f=\sum a_{n}q^{n}$ of weight $k$, squarefree level $N$, and character $\omega$ of conductor $M$ as in Section \[s3\]. Let ${\mathcal{O}}$ be the ring of integers of the field $K$ of Fourier coefficients of $f$. For each prime $\lambda$ of $K$, let $V_{\rho,\lambda}$ be a three-dimensional ${K_{\lambda}}$-vector space with ${G_{{\mathbf{Q}}}}$-action via ${\operatorname{ad}^{0}\!}{\rho_{f,\lambda}}$. Fix a ${G_{{\mathbf{Q}}}}$-stable ${{\mathcal{O}}_{\lambda}}$-lattice $T_{\rho,\lambda}$ in $V_{\rho,\lambda}$ and set $A_{\rho,\lambda} := V_{\rho,\lambda}/ T_{\rho,\lambda}$. In general the ${k_{\lambda}}[{G_{{\mathbf{Q}}}}]$-module $A_{\rho,\lambda}[\lambda]$ need not agree with the semisimple reduction ${{\bar{\rho}}_{f,\lambda}}$; however, these two representations must be isomorphic when ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible, which is the only case we will consider below. Congruences and Selmer groups {#s41} ----------------------------- The purpose of this section is to explain how the results of [@Hida] (as refined in [@Ghate]) and [@DFG] relate adjoint Selmer groups with congruences of modular forms. Let $d$ be a divisor of $N$ which is divisible by $M$. We say that a prime $\lambda$ of $K$ is a [*congruence prime of level $d$*]{} for $f$ if there exists a newform $f'$ of weight $k$ and level $d$ such that: 1. $f'$ has character lifting $\omega_{0}$; 2. $f'$ is not a Galois conjugate of $f$; 3. ${\bar{\rho}}_{f,{\bar{\lambda}}} \cong {\bar{\rho}}_{f',{\bar{\lambda}}}$ for some prime ${\bar{\lambda}}$ of ${\bar{{\mathbf{Q}}}}$ above $\lambda$. (Of course, by the Cebatorev density theorem the last condition is equivalent to a congruence $a_{p}(f) \equiv a_{p}(f') \pmod{{\bar{\lambda}}}$ for all primes $p$ not dividing $N$.) We say that a congruence prime $\lambda$ of level $d$ for $f$ is [*proper*]{} (resp.  [*strict*]{}) if $d<N$ (resp. $d=N$). Let $\operatorname{Cong}(f)$ (resp. $\operatorname{Cong}_{<N}(f)$, resp. $\operatorname{Cong}_{N}(f)$) denote the set of congruence primes (resp. proper congruence primes, resp. strict congruence primes) for $f$. We need a simple lemma. \[lemma:dfg\] Let $\lambda$ be a prime of $K$ dividing a rational prime $\ell$ not dividing $N$. Assume that $N > 1$ and that ${{\bar{\rho}}_{f,\lambda}}$ is ramified at some $p$ dividing $N$. If ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible, then ${{\bar{\rho}}_{f,\lambda}}|_{G_{F}}$ is absolutely irreducible as well, where $F = {\mathbf{Q}}(\sqrt{(-1)^{(\ell-1)/2}\ell})$. As in [@DFG Lemma 7.14], if ${{\bar{\rho}}_{f,\lambda}}|_{G_{F}}$ is absolutely reducible, then ${{\bar{\rho}}_{f,\lambda}}$ is induced from a character of $G_{F}$. In particular, it follows that the conductor $N'$ of ${{\bar{\rho}}_{f,\lambda}}$ (in the sense of [@Edixhoven]) is a square. However, $N'$ must also divide the level $N$ of $f$; since $N'$ is non-trivial by hypothesis and $N$ is squarefree, this is impossible. \[prop:hida\] Let $\lambda$ be a prime of $K$ dividing the rational prime $\ell$. Assume that: 1. \[a1\] ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible; 2. \[a2\] $\ell > k$; 3. \[a3\] Either $N > 1$ or $\ell \nmid (2k-3)(2k-1)$; 4. \[a4\] $\ell \nmid N$; 5. \[a5\] $\ell \nmid \varphi(N)$ (that is, $p \not\equiv 1 \pmod{\ell}$ for all $p \mid N$); 6. \[a6\] ${{\bar{\rho}}_{f,\lambda}}$ is ramified at $p$ for all $p \mid \frac{N}{M}$; Then $H^{1}_{\emptyset}({G_{{\mathbf{Q}}}},A_{\rho,\lambda}) \neq 0$ if and only if $\lambda \in \operatorname{Cong}_{N}(f)$. Conditions (\[a4\])–(\[a6\]) guarantee that $A_{\rho,\lambda}$ is minimally ramified in the sense of [@Diamond2 Section 3]. Using (\[a1\]), (\[a2\]), (\[a4\]), and Lemma \[lemma:dfg\] (or (\[a3\]) and [@DFG Lemma 7.14] for $N=1$), we may apply [@DFG Theorem 7.15] to conclude that $$\label{eq:len} \text{length}_{{{\mathcal{O}}_{\lambda}}} H^{1}_{\emptyset}({G_{{\mathbf{Q}}}},A_{\rho,\lambda}) = v_{\lambda}(\eta_{f}^{\emptyset});$$ here $\eta_{f}^{\emptyset}$ is the fractional ideal of $K$ defined in [@DFG Section 6.4] and $v_{\lambda}$ is the $\lambda$-adic valuation. By definition, the ideal $\eta_{f}^{\emptyset}$ is generated by the discriminant $d(L_{f}({\mathcal{O}}_{K}))$ of [@Ghate proof of Theorem 5], which in turn equals the square of the algebraic special value of the adjoint $L$-function of $f$: $$\label{eq:disc} d\bigl(L_{f}({\mathcal{O}}_{K})\bigr) = \left(\frac{W(f)\Gamma(1,\operatorname{ad}f)L(1,\operatorname{ad}f)} {\Omega(f,+)\Omega(f,-)}\right)^{2}.$$ (All of this is only true up to factors of primes violating (\[a1\])–(\[a4\]).) In particular, (\[eq:len\]) implies that $H^{1}_{\emptyset}({G_{{\mathbf{Q}}}},A_{\rho,\lambda})$ is non-zero if and only if (\[eq:disc\]) has positive $\lambda$-adic valuation. By [@Ghate Theorems 1 and 2], the latter condition is equivalent to the existence of a newform $f'$ of weight $k$ and level dividing $N$, not Galois conjugate to $f$, such that ${\bar{\rho}}_{f,{\bar{\lambda}}} \cong {\bar{\rho}}_{f',{\bar{\lambda}}}$ for some prime ${\bar{\lambda}}$ above $\lambda$. It remains to show that $f'$ has level $N$ and character $\omega$. Since ${\bar{\rho}}_{f',{\bar{\lambda}}}$ has determinant ${\bar{\varepsilon}_{\ell}}^{k-1}\bar{\omega}$ and $\mu_{\varphi(N)}$ injects into ${k_{\lambda}}^{\times}$ (by (\[a5\])), $f'$ has level divisible by $M$ and character lifting $\omega_{0}$. Hypothesis (\[a6\]) guarantees that ${\bar{\rho}}_{f',{\bar{\lambda}}}$ is ramified at all $p \mid \frac{N}{M}$ as well, so that $f'$ must in fact have level $N$. Vanishing of cohomology {#s42} ----------------------- Let $S$ be a finite set of places of ${\mathbf{Q}}$ containing all places dividing $N\infty$; let $N_{S}$ denote the product of all primes in $S$. Fix a prime $\lambda$ of $K$ dividing a rational prime $\ell$. We are now in a position to compute $H^{2}({G_{{\mathbf{Q}},S\cup\{\ell\}}},\operatorname{ad}{{\bar{\rho}}_{f,\lambda}})$. \[thm:van\] Assume that ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible and $\ell > 3$. If $$\label{eq:h2} H^{2}({G_{{\mathbf{Q}},S\cup\{\ell\}}},\operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0,$$ then one of the following holds: 1. \[b1\] $\ell \leq k$; 2. \[b2\] $\ell \mid N$; 3. \[b3\] $\ell \mid \varphi(N_{S})$; 4. \[b3.5\] $\ell \mid p+1$ for some $p \mid \frac{N}{M}$; 5. \[b4\] $a_{p}^{2} \equiv (p+1)^{2}p^{k-2}\omega(p) \pmod{\lambda}$ for some $p \mid \frac{N_{S}}{N}$, $p \neq \ell$; 6. \[b5\] $\ell = k + 1$ and $f$ is ordinary at $\lambda$; 7. \[b6\] $k=2$ and $a_{\ell}^{2} \equiv \omega(\ell) \pmod{\lambda}$; 8. \[b7\] $N=1$ and $\ell \mid (2k-3)(2k-1)$; 9. \[b8\] $\lambda \in \operatorname{Cong}(f)$. Using Lemma \[lemma:d2\] and the results of Sections \[s3\] and \[s41\], the reader should have little difficulty in detecting the source of each of the conditions above. We shall nevertheless endeavor to give a complete proof. If (\[eq:h2\]) holds, then Lemma \[lemma:d2\] implies that either $$\label{eq:pos1} H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$$ for some $p \in S \cup \{\ell\}$ or $$\label{eq:pos2} {\mbox{\cyrr Sh}}^{1}({G_{{\mathbf{Q}},S}},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0.$$ Suppose first that (\[eq:pos1\]) holds for a prime $p \in S \cup \{\ell\}$; we may assume $\ell \nmid N$ by (\[b2\]). If $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}) \neq 0$, then $\ell$ divides $p-1$ which in turn divides $\varphi(N_{S})$, so that (\[b3\]) holds. We may thus assume that $p \not\equiv 1 \pmod{\ell}$ and $$H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}}) \neq 0.$$ By Lemma \[lemma:ramps\] we know that $p$ does not divide $M$. If $p$ divides $\frac{N}{M}$, then by Lemma \[lemma:spec\] and Remark \[rmk:spec\] one of (\[b3.5\]) or (\[b8\]) holds, while if $p$ does not divide $N\ell$, then Lemma \[lemma:unramps\] implies that (\[b4\]) must hold. Finally, if $p = \ell$ and $k > 2$, then Lemmas \[lemma:ord\] and \[lemma:ss\] force (\[b1\]) or (\[b5\]) to hold; if $k = 2$, then [@Weston2 Proposition 4.4] forces (\[b6\]) to hold. It remains to consider the case that (\[eq:pos2\]) holds, (\[eq:pos1\]) does not hold for any $p \in S \cup \{\ell\}$, and none of (\[b1\])–(\[b7\]) hold. Then by [@DFG Theorem 8.2] $${H^{1}_{f}}({G_{{\mathbf{Q}}}},V_{f,\lambda}) = {H^{1}_{f}}({G_{{\mathbf{Q}}}},V_{f,\lambda}(1)) = 0.$$ Lemma \[lemma:ineq\] and (\[eq:pos2\]) thus imply that $$H^{1}_{\emptyset}({G_{{\mathbf{Q}}}},A_{\rho,\lambda}) \neq 0.$$ Since $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) = 0$ for all $p$ dividing $\frac{N}{M}$, Lemma \[lemma:specss\] implies that ${{\bar{\rho}}_{f,\lambda}}$ is ramified at all such $p$. Proposition \[prop:hida\] now applies to show that $\lambda \in \operatorname{Cong}_{N}(f)$. Thus (\[b8\]) holds, completing the proof. \[cor:def\] If ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible, $\ell > 3$, and $\lambda$ does not satisfy (\[b1\])–(\[b8\]), then $$R_{{{\bar{\rho}}_{f,\lambda}}} \cong W({k_{\lambda}})[[T_{1},T_{2},T_{3}]].$$ This follows immediately from Theorem \[thm:van\] and Corollary \[cor:unob\]. We also obtain the following partial converse to Theorem \[thm:van\]. \[thm:vanc\] Assume that ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible. Suppose that $\ell > 3$ and one of the following holds: 1. \[m1\] $\ell \mid \varphi(N_{S})$; 2. \[m2\] $a_{p}^{2} \equiv (p+1)^{2}p^{k-2}\omega(p) \pmod{\lambda}$ for some $p \mid \frac{N_{S}}{N}$, $p \neq \ell$; 3. \[m3\] $\lambda$ is a congruence prime for $f$ of level dividing $\frac{N}{p}$ for some $p \mid \frac{N}{M}$, $\ell \nmid p(p+1)$; 4. \[m4\] $k=2$, $\ell \nmid N$, and $a_{\ell}^{2} \equiv \omega(\ell) \pmod{\lambda}$. Then $H^{2}({G_{{\mathbf{Q}},S\cup\{\ell\}}},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$. By Lemma \[lemma:d2\] it suffices to show that these conditions guarantee that $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$ for some $p \in S$. If (\[m1\]) holds, then $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}) \neq 0$ for some $p \in S$, so that this is clear. If (\[m2\]) holds, then by Lemma \[lemma:unramps\] we have $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}}) \neq 0$. If (\[m1\]) does not hold but (\[m3\]) does hold, then Lemma \[lemma:spec\] implies that $H^{0}(G_{p},{\bar{\varepsilon}_{\ell}}\otimes {\operatorname{ad}^{0}\!}{{\bar{\rho}}_{f,\lambda}}) \neq 0$. Finally, if (\[m4\]) holds, then the proof of [@Weston2 Proposition 4.4] shows that $H^{0}(G_{\ell},{\bar{\varepsilon}_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0$. Examples {#s5} ======== In this section we use the data of [@Stein] to bound the obstructed primes for the deformation problems associated to a few specific modular forms. Of course, the most interesting aspect of these computations are the determination of congruences between newforms. Using [@Stein] we can check such congruences on the $p^{\text{th}}$ Fourier coefficients for all $p < 1000$; by the results of [@Sturm] these checks are more than sufficient to prove that these congruences actually exist in our examples. We will not comment further on this issue. For a modular form $f$, we let $\operatorname{Red}(f)$ denote the set of primes $\lambda$ of $K$ such that ${{\bar{\rho}}_{f,\lambda}}$ is absolutely reducible. We recall the following well-known facts regarding $\operatorname{Red}(f)$; see [@DFG Lemma 7.13] for example. \[lemma:red\] Let $f = \sum a_{n}q^{n}$ be a newform of weight $k$ and level $N$ with coefficient field $K$. Let $\lambda$ be a prime of $K$ dividing a rational prime $\ell$. Suppose that $\lambda \in \operatorname{Red}(f)$, so that $${{\bar{\rho}}_{f,\lambda}}\otimes {\bar{k}_{\lambda}}\cong \chi_{1} \oplus \chi_{2}$$ for characters $\chi_{1},\chi_{2} : {G_{{\mathbf{Q}}}}\to {\bar{k}_{\lambda}}^{\times}$. If $\ell$ does not divide $N$, then each $\chi_{i}$ has conductor dividing $N\ell$. If also $\ell > k$, then one of the $\chi_{i}$ has conductor dividing $N$, so that $$a_{p} \equiv p^{k-1} + 1 \pmod{\lambda}$$ for all $p \equiv 1 \pmod{N}$. In practice one uses the second condition to bound the set $\operatorname{Red}(f)$ and the first condition to check each remaining $\lambda$ not dividing $N$. For a prime $\lambda$ dividing $N$, one can still check that ${{\bar{\rho}}_{f,\lambda}}$ is absolutely reducible, but it is much more difficult to show that ${{\bar{\rho}}_{f,\lambda}}$ is absolutely irreducible; we will make no attempt to deal with this case below. For a finite set of places $S$ containing all places dividing $N\infty$, we let $\operatorname{Obs}_{S}(f)$ denote the set of $\lambda \notin \operatorname{Red}(f)$ such that $$H^{2}({G_{{\mathbf{Q}},S\cup\{\ell\}}},{\varepsilon_{\ell}}\otimes \operatorname{ad}{{\bar{\rho}}_{f,\lambda}}) \neq 0,$$ or equivalently such that the deformation problem associated to $${{\bar{\rho}}_{f,\lambda}}: {G_{{\mathbf{Q}},S\cup\{\ell\}}}\to \operatorname{GL}_{2}\! {k_{\lambda}}$$ is obstructed. We simply write $\operatorname{Obs}(f)$ for $\operatorname{Obs}_{\{p\mid N\infty\}}(f)$. In the interests of space, we make the following notational conventions. Fix a quadratic extension $K$ of ${\mathbf{Q}}$ and let $p$ be a rational prime. If $p$ ramifies in $K$, then we simply write ${\mathfrak{p}}_{p}$ for the prime of $K$ above $p$. If $p$ splits, then we will write ${\mathfrak{p}}_{p}$ and $\bar{{\mathfrak{p}}}_{p}$ for the two primes of $K$ above $p$, at least when it is not important to distinguish between them. Weight $\boldsymbol{12}$, level $\boldsymbol{5}$, trivial character ------------------------------------------------------------------- There are three newforms of weight $12$, level $5$, and trivial character. The first has rational Fourier coefficients and $q$-expansion $$f_{1} = q + 34q^{2} - 792q^{3} - 892q^{4} + 3125q^{5} - 26928q^{6} -17556q^{7} + \cdots$$ while the other two, $$\begin{gathered} f_{2} = q + (-10+6\sqrt{151})q^{2} + (-110+32\sqrt{151})q^{3} + (3448-120\sqrt{151})q^{4} \\ - 3125q^{5} +(30092-980\sqrt{151})q^{6} + (28950+1056\sqrt{151})q^{7} + \cdots\end{gathered}$$ and its Galois conjugate, have field of Fourier coefficients ${\mathbf{Q}}(\sqrt{151})$. Note that $\operatorname{Obs}(\bar{f}_{2})$ is simply the set of conjugates of elements of $\operatorname{Obs}(f_{2})$, so that it suffices to study $f_{1}$ and $f_{2}$. Using Lemma \[lemma:red\], one computes that: $$\begin{gathered} \operatorname{Red}(f_{1}) = \{2,5,31\}; \\ \operatorname{Red}(f_{2}) = \bigl\{{\mathfrak{p}}_{2}, {\mathfrak{p}}_{5},\bar{{\mathfrak{p}}}_{5},(601,358+\sqrt{151})\bigr\}.\end{gathered}$$ We now consider congruences. By comparing Fourier coefficients, one sees that $f_{1}$ and $f_{2}$ are congruent modulo primes above $2$ and $5$: $$\begin{gathered} \operatorname{Cong}_{5}(f_{1}) = \{2,5\}; \\ \operatorname{Cong}_{5}(f_{2}) = \bigl\{{\mathfrak{p}}_{2},(5,1+\sqrt{151})\bigr\}.\end{gathered}$$ The only possible proper congruences is with the unique newform $$\Delta = q -24q^{2} +252q^{3}-1472q^{4}+4830q^{5}-6048q^{6} -16744q^{7} + \cdots$$ of weight $12$ and level $1$; one computes that: $$\begin{gathered} \operatorname{Cong}_{<5}(f_{1}) = \{2,29\}; \\ \operatorname{Cong}_{<5}(f_{2}) = \bigl\{{\mathfrak{p}}_{2},(5,4+\sqrt{151}), (131,46+\sqrt{151})\bigr\}.\end{gathered}$$ Both $f_{1}$ and $f_{2}$ are ordinary at $13$, so that by Theorems \[thm:van\] and \[thm:vanc\] we conclude that: $$\begin{gathered} \{29\} \subseteq \operatorname{Obs}(f_{1}) \subseteq \{3,7,11,13,29\}; \\ \bigl\{(131,46+\sqrt{151})\bigr\} \subseteq \operatorname{Obs}(f_{2}) \subseteq \bigl\{{\mathfrak{p}}_{3},\bar{{\mathfrak{p}}}_{3},{\mathfrak{p}}_{7},\bar{{\mathfrak{p}}}_{7}, (11),(13),(131,46+\sqrt{151})\bigr\}.\end{gathered}$$ Weight $\boldsymbol{6}$, level $\boldsymbol{30}$, trivial character ------------------------------------------------------------------- There are two newforms of weight $6$, level $30$, and trivial character, both with rational Fourier coefficients: $$\begin{gathered} f_{1} = q+4q^{2}+9q^{3}-16q^{4}+25q^{5}+36q^{6}+32q^{7} -192q^{8}\\-162q^{9}+100q^{10}+12q^{11}-144q^{12}-154q^{13} + \cdots\end{gathered}$$ $$\begin{gathered} f_{2} = q-4q^{2}+9q^{3}-16q^{4}-25q^{5}-36q^{6}+164q^{7}+ 192q^{8}\\-162q^{9}+100q^{10}+720q^{11}-144q^{12}+698q^{13} + \cdots\end{gathered}$$ Using Lemma \[lemma:red\], one computes that: $$\begin{gathered} \operatorname{Red}(f_{1}) = \{2,3,5\}; \\ \{2,3\} \subseteq \operatorname{Red}(f_{2}) \subseteq \{2,3,5\}.\end{gathered}$$ The newforms $f_{1}$ and $f_{2}$ have a congruence modulo $12$ (remember that one only checks the Fourier coefficients with exponent prime to $30$), so that: $$\operatorname{Cong}_{30}(f_{1}) = \operatorname{Cong}_{30}(f_{2}) = \{2,3\}.$$ There are ten newforms of level dividing $30$ and trivial character to consider for proper congruences. The most interesting occur for $f_{2}$: it has a congruence modulo $19$ with the newform $$\begin{gathered} q+7q^{2}+9q^{3}+17q^{4}-25q^{5}+63q^{6}+12q^{7}-105q^{8} \\ -162q^{9} -175q^{10} + 112q^{11} + 153q^{12} - 974q^{13} + \cdots\end{gathered}$$ of level $15$, and modulo $31$ with the newform $$\begin{gathered} q-4q^{2}-26q^{3}-16q^{4}-25q^{5}+104q^{6}-22q^{7}+192q^{8}\\ +433q^{9} +100q^{10}-768q^{11}+416q^{12}-46q^{13}+\cdots\end{gathered}$$ of level $10$. In any event, one computes: $$\operatorname{Cong}_{<30}(f_{1}) = \{2,3,5\}; \quad \operatorname{Cong}_{<30}(f_{2}) = \{2,3,19,31\}.$$ Both $f_{1}$ and $f_{2}$ are ordinary at $7$, so that we conclude that: $$\begin{gathered} \operatorname{Obs}(f_{1}) \subseteq \{7\}; \\ \{19,31\} \subseteq \operatorname{Obs}(f_{2}) \subseteq \{5,7,19,31\}.\end{gathered}$$ To give an explicit example of an obstructed set, using (\[m3\]) of Theorem \[thm:vanc\] one finds that $$\operatorname{Obs}_{\{2,3,5,17,\infty\}}(f_{1}) = \{7\}.$$ Weight $\boldsymbol{3}$, level $\boldsymbol{35}$, character of conductor $\boldsymbol{7}$ ----------------------------------------------------------------------------------------- There are four newforms of weight $3$, level $35$, and with quadratic character $$\omega : ({\mathbf{Z}}/35{\mathbf{Z}})^{\times} \to \{\pm 1\}$$ the Legendre symbol $\left(\frac{\cdot}{7}\right)$. All four are defined over ${\mathbf{Q}}(\sqrt{-5})$: two are $$f_{1} = q - q^{2} - 2\sqrt{-5}q^{3} -3q^{4} -\sqrt{-5}q^{5} + 2\sqrt{-5}q^{6} +7q^{7}+ \cdots$$ and its Galois conjugate while the other two are $$f_{2} = q +2q^{2} +\sqrt{-5}q^{3}-\sqrt{-5}q^{5}+2\sqrt{-5}q^{6} -(2+3\sqrt{-5})q^{7} + \cdots$$ and its Galois conjugate. As before, it suffices to study $f_{1}$ and $f_{2}$. Using Lemma \[lemma:red\], one finds that: $$\begin{gathered} \bigl\{{\mathfrak{p}}_{2},{\mathfrak{p}}_{3},\bar{{\mathfrak{p}}}_{3}, {\mathfrak{p}}_{5}\bigr\} \subseteq \operatorname{Red}(f_{1}) \subseteq \bigl\{{\mathfrak{p}}_{2},{\mathfrak{p}}_{3},\bar{{\mathfrak{p}}}_{3},{\mathfrak{p}}_{5},{\mathfrak{p}}_{7},\bar{{\mathfrak{p}}}_{7} \bigr\} \\ \bigr\{{\mathfrak{p}}_{3},\bar{{\mathfrak{p}}}_{3}\bigr\} \subseteq \operatorname{Red}(f_{2}) \subseteq \bigr\{{\mathfrak{p}}_{3},\bar{{\mathfrak{p}}}_{3},{\mathfrak{p}}_{5},{\mathfrak{p}}_{7},\bar{{\mathfrak{p}}}_{7}\bigr\}.\end{gathered}$$ The newforms $f_{1}$ and $f_{2}$ have a congruence modulo $3$, while $f_{1}$ and $\bar{f}_{2}$ have no congruences; thus: $$\operatorname{Cong}_{35}(f_{1}) = \operatorname{Cong}_{35}(f_{2}) = \bigl\{{\mathfrak{p}}_{3},\bar{{\mathfrak{p}}}_{3}\}.$$ Since $\omega$ has conductor $7$, the only proper congruences we need to check are with the unique newform $$q-3q^{2}+5q^{4}-7q^{7}-3q^{8}-9q^{9} -6q^{11} + \cdots$$ of weight $3$, level $7$, and character $\left(\frac{\cdot}{7}\right)$. One finds that: $$\operatorname{Cong}_{<35}(f_{1}) = \bigl\{{\mathfrak{p}}_{2}\bigr\};$$ $$\operatorname{Cong}_{<35}(f_{2}) = \bigl\{{\mathfrak{p}}_{5}\bigr\}.$$ Theorem \[thm:van\] allows us to conclude that: $$\operatorname{Obs}(f_{1}) \subseteq \bigl\{{\mathfrak{p}}_{7},\bar{{\mathfrak{p}}}_{7}\bigr\}$$ $$\operatorname{Obs}(f_{2}) \subseteq \bigl\{{\mathfrak{p}}_{2},{\mathfrak{p}}_{5},{\mathfrak{p}}_{7},\bar{{\mathfrak{p}}}_{7} \bigr\}.$$ [10]{} Henri Carayol, *Sur les représentations [$l$]{}-adiques associées aux formes modulaires de [H]{}ilbert*, Ann. Sci. École Norm. Sup. (4) **19** (1986), no. 3, 409–468. Fred Diamond, *An extension of [W]{}iles’ results*, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 475–489. Fred Diamond, Matthias Flach, and Li Guo, *Adjoint motives of modular forms and the [T]{}amagawa number conjecture*, pre-print. Bas Edixhoven, *Serre’s conjecture*, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 209–242. Matthias Flach, *A generalisation of the [C]{}assels-[T]{}ate pairing*, J. Reine Angew. Math. **412** (1990), 113–127. Eknath Ghate, *Adjoint [$L$]{}-values and primes of congruence for [H]{}ilbert modular forms*, Compositio Math. **132** (2002), no. 3, 243–281. Haruzo Hida, *Congruence of cusp forms and special values of their zeta functions*, Invent. Math. **63** (1981), no. 2, 225–261. [to3em]{}, *On congruence divisors of cusp forms as factors of the special values of their zeta functions*, Invent. Math. **64** (1981), no. 2, 221–262. Barry Mazur, *Deforming [G]{}alois representations*, Galois groups over ${\bf Q}$ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. James Milne, *Arithmetic duality theorems*, Perspectives in Mathematics, vol. 1, Academic Press Inc., Boston, MA, 1986. Kenneth Ribet, *Mod [$p$]{} [H]{}ecke operators and congruences between modular forms*, Invent. Math. **71** (1983), no. 1, 193–205. William Stein, *The modular forms explorer*, available at `http://modular.fas.harvard.edu/ mfd/mfe`. Jacob Sturm, *On the congruence of modular forms*, Number theory (New York, 1984–1985), Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275–280. Tom Weston, *An overview of a theorem of [F]{}lach*, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 345–375. [to3em]{}, *Unobstructed modular deformation problems*, preprint. [^1]: Supported by an NSF postdoctoral fellowship

--- abstract: 'The lamplighter group over ${\mathbb Z}$ is the wreath product ${\mathbb Z}_q \wr {\mathbb Z}$. With respect to a natural generating set, its Cayley graph is the Diestel-Leader graph ${\mbox{\sl DL}}(q,q)$. We study harmonic functions for the “simple” Laplacian on this graph, and more generally, for a class of random walks on ${\mbox{\sl DL}}(q,r)$, where $q,r \ge 2$. The ${\mbox{\sl DL}}$-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive harmonic functions.' author: - '**Wolfgang WOESS**' date: 'Version of November 27, 2003, to appear in Combinatorics, Probability & Computing' title: | Lamplighters, Diestel-Leader graphs,\ random walks, and harmonic functions --- [^1] Introduction ============ Think of a (typically infinite) connected graph $X$ where in each vertex there is a lamp that may be switched off (state $0$), or switched on with $q-1$ different intensities (states $1, \dots, q-1$). Initially, all lamps are turned off, and a lamplighter starts at some vertex of $X$ and walks around. When he visits a vertex, he may switch the lamp sitting there into one of its $q$ different states (including “off”). Our information consists of the position $x\in X$ of the lamplighter and of the finitely supported configuration $\eta: X \to {\mathbb Z}_q = \{0, \dots q-1 \}$ of the lamps that are switched on, including their respective intensities. The set ${\mathbb Z}_q \wr X$ of all such pairs $(\eta,x)$ can be equipped in several ways with a natural connected graph structure, giving rise to a *lamplighter graph*. When $X$ is a Cayley graph of a group ${\Gamma}$ then underlying this construction, there is the *wreath product* ${\mathbb Z}_q \wr {\Gamma}$, which is the semidirect product of ${\Gamma}$ with the group of all finitely supported functions $\eta: {\Gamma}\to {\mathbb Z}_q$ (i.e., a direct sum), on which ${\Gamma}$ acts by $g\eta(h) = \eta(g^{-1}h)$. Instead of ${\mathbb Z}_q = {\mathbb Z}/(q{\mathbb Z})$, one may of course take any other group $L$ of “lamps”, leading to the wreath product $L \wr {\Gamma}$. Various aspects of random walks on lamplighter groups have received considerable attention recently: Poisson boundary ([Kaimanovich and Vershik]{} [@KaiVer] and [Kaimanovich]{} [@Kai]), rate of escape ([Lyons, Pemantle and Peres]{} [@LyoPemPer], [Erschler]{} [@Ers], [Revelle]{} [@Rev1]), spectral theory ([Grigorchuk and Żuk]{} [@GriZuk]), and the asymptotic behaviour of transition probabilites ([Saloff-Coste and Pittet]{} [@PitSal1], [@PitSal2], [Revelle]{} [@Rev2]). Here, we shall consider *harmonic functions.* A harmonic function on a locally finite graph is a real-valued function whose value at each vertex coincides with the arithmetic average of its values in the neighbour vertices. More generally, we can consider the transition matrix $P$ of a random walk on the graph, suitably adapted to the graph’s geometry; a harmonic function $h$ is then one that satisfies $Ph=h$. In the present paper, we shall determine all positive harmonic functions on certain Cayley graphs of the simplest lamplighter group, ${\Gamma}= {\mathbb Z}_q \wr {\mathbb Z}$. Namely, we first explain that the Diestel-Leader graph ${\mbox{\sl DL}}(q,q)$ is a Cayley graph of ${\Gamma}$. More generally, if $q, r \ge 2$ then ${\mbox{\sl DL}}(q,r)$ is obtained as a “horocyclic product” of two homogeneous trees ${\mathbb T}_q$ and ${\mathbb T}_r$ with degrees $q+1$ and $r+1$, respectively. We remark that this does not mean that ${\mbox{\sl DL}}(q,r)$ is “almost” a tree in any sense; indeed, it is a one-ended, vertex-transitive graph which is a Cayley graph only when $r=q$. When $r \ne q$, it is believed to be an example of a transitive graph that is not quasi-isometric with any Cayley graph of some finitely generated group – see [Diestel and Leader]{} [@DieLea]. Nevertheless, we can use the boundary of each of the two trees that compose ${\mbox{\sl DL}}(q,r)$ for giving an integral representation of all positive harmonic functions: in that representation, we start with the projections of the random walk on ${\mbox{\sl DL}}(q,r)$ to each of the two trees and the corresponding Martin kernels. Our main result is that every positive harmonic function on ${\mbox{\sl DL}}(q,r)$ is of the form $h = h_1 + h_2$, where $h_1$ is obtained by lifting a harmonic function from ${\mathbb T}_q$ to ${\mbox{\sl DL}}(q,r)$, and $h_2$ is obtained analogously from ${\mathbb T}_r$. Thereby, we also determine all minimal positive harmonic functions. We now give an outline of the contents of this paper. Section \[geometry\], although it does not contain *proofs,* is crucial, since it *explains* the geometry of the structures that we are working with, and in particular, the correspondence between lamplighter groups and Diestel-Leader graphs. As a matter of fact, it is precisely this geometric realization that allows us to determine all positive harmonic functions on ${\mathbb Z}_q \wr {\mathbb Z}$. At the end of §\[geometry\], we state the first main result, regarding the decomposition of positive harmonic functions over the two trees (Theorem \[split-theorem\]). In Section \[basics\], we recall basic results on positive harmonic functions for irreducible Markov chains. In particular, we consider finite sets with boundaries, the Martin boundary at infinity and its minimal part, and the Martin compactification for nearest neighbour random walks on trees. In Section \[principal\], we use all the preceding ingredients to prove the Decomposition Theorem \[split-theorem\]. It is then quite simple to determine all minimal positive harmonic functions (Theorem \[minimal-theorem\]); they are the Martin kernels of the two projected random walks, up to one, resp. two exceptions. We then retranslate these results to the lamplighter group ${\mathbb Z}_q \wr {\mathbb Z}$ (Example \[SRW-example\]). In Section \[extension\], we adapt the preceding results to the “switch-walk-switch” random walk, which is in some sense more natural from the point of view of the lamplighter than the simple random walk on ${\mbox{\sl DL}}(q,q)$. Section \[final\] is devoted to some additional remarks and speculations. Diestel-Leader graphs and lamplighters {#geometry} ====================================== Let ${\mathbb T}= {\mathbb T}_q$ be the homogeneous tree with degree $q+1$, $q \ge 2$. A *geodesic path*, resp. *geodesic ray*, resp. *infinite geodesic* in ${\mathbb T}$ is a finite, resp. one-sided infinite, resp. doubly infinite sequence $(x_n)$ of vertices of ${\mathbb T}$ such that $d(x_i,x_j) = |i-j|$ for all $i, j$, where $d(\cdot,\cdot)$ denotes the graph distance. Two rays are *equivalent* if their symmetric difference is finite. An *end* of ${\mathbb T}$ is an equivalence class of rays. The space of ends is denoted ${\partial}{\mathbb T}$, and we write ${\widehat}{\mathbb T}= {\mathbb T}\cup {\partial}{\mathbb T}$. For all $w, z \in {\widehat}{\mathbb T}$ there is a unique geodesic ${\overline{w\,z}}$ that connects the two. In particular, if $x \in {\mathbb T}$ and $\xi \in {\partial}{\mathbb T}$ then ${\overline{x\,\xi}}$ is the ray that starts at $x$ and represents $\xi$. Furthermore, if $\xi, \zeta \in {\partial}{\mathbb T}$ ($\xi \ne \zeta$) then ${\overline{\zeta\,\xi}}$ is the infinite geodesic whose two halves (split at any vertex) are rays that respresent $\zeta$ and $\xi$, respectively. For $x,y \in {\mathbb T}$, $x \ne y$, we define the cone ${\widehat}{\mathbb T}(x,y) = \{ w \in {\widehat}{\mathbb T}: y \in {\overline{x\,w}} \}$. The collection of all cones is the basis of a topology wich makes ${\widehat}{\mathbb T}$ a compact, totally disconnected Hausdorff space with ${\mathbb T}$ as a dense, discrete subset. We denote ${\mathbb T}(x,y) = {\mathbb T}\cap{\widehat}{\mathbb T}(x,y)$ and ${\partial}{\mathbb T}(x,y) = {\partial}{\mathbb T}\cap {\mathbb T}(x,y)$. We fix a root $o \in {\mathbb T}$. If $w, z \in {\widehat}{\mathbb T}$, then their *confluent* $c=w \wedge z$ with respect to the root vertex $o$ is defined by ${\overline{o\,w}} \cap {\overline{o\,z}} = {\overline{o\,c}}$. Similarly, we choose and fix a reference end ${\omega}\in {\partial}{\mathbb T}$. For $z, v \in {\widehat}{\mathbb T}\setminus \{ {\omega}\}$, their confluent $b = v {\curlywedge}z$ with respect to ${\omega}$ is defined by ${\overline{v\,{\omega}}} \cap {\overline{z\,{\omega}}} = {\overline{b\,{\omega}}}$. The *Busemann function* ${\mathfrak{h}}: {\mathbb T}\to {\mathbb Z}$ and the *horocycles* $H_k$ with respect to ${\omega}$ are defined as $${\mathfrak{h}}(x) = d(x,x {\curlywedge}o) - d(o,x {\curlywedge}o) {\quad\mbox{and}\quad}H_k = \{ x \in {\mathbb T}: {\mathfrak{h}}(x) = k \}\,.$$ Every horocycle is infinite. Every vertex $x$ in $H_k$ has one neighbour $x^-$ (its predecessor) in $H_{k-1}$ and $q$ neighbours (its successors) in $H_{k+1}$. We set ${\partial}^* {\mathbb T}= {\partial}{\mathbb T}\setminus \{{\omega}\}$. $$\beginpicture \setcoordinatesystem units <.7mm,1.04mm> \setplotarea x from -10 to 104, y from -84 to -4 \arrow <6pt> [.2,.67] from 2 -2 to 80 -80 \plot 32 -32 62 -2 / \plot 16 -16 30 -2 / \plot 48 -16 34 -2 / \plot 8 -8 14 -2 / \plot 24 -8 18 -2 / \plot 40 -8 46 -2 / \plot 56 -8 50 -2 / \plot 4 -4 6 -2 / \plot 12 -4 10 -2 / \plot 20 -4 22 -2 / \plot 28 -4 26 -2 / \plot 36 -4 38 -2 / \plot 44 -4 42 -2 / \plot 52 -4 54 -2 / \plot 60 -4 58 -2 / \plot 99 -29 64 -64 / \plot 66 -2 96 -32 / \plot 70 -2 68 -4 / \plot 74 -2 76 -4 / \plot 78 -2 72 -8 / \plot 82 -2 88 -8 / \plot 86 -2 84 -4 / \plot 90 -2 92 -4 / \plot 94 -2 80 -16 / \setdots <3pt> \putrule from -4.8 -4 to 102 -4 \putrule from -4.5 -8 to 102 -8 \putrule from -2 -16 to 102 -16 \putrule from -1.7 -32 to 102 -32 \putrule from -1.7 -64 to 102 -64 \put {$\vdots$} at 32 3 \put {$\vdots$} at 64 3 \put {$\dots$} [l] at 103 -6 \put {$\dots$} [l] at 103 -48 \put {$H_{-3}$} [l] at -12 -64 \put {$H_{-2}$} [l] at -12 -32 \put {$H_{-1}$} [l] at -12 -16 \put {$H_0$} [l] at -12 -8 \put {$H_1$} [l] at -12 -4 \put {$\vdots$} at -10 3 \put {$\vdots$} [B] at -10 -70 \put {$\omega$} at 82 -82 \put {\scriptsize $\bullet$} at 44 -4 \put {\scriptsize $1$} at 44 -6.5 \put {\scriptsize $0$} at 41 -12 \put {\scriptsize $1$} at 43 -24 \put {\scriptsize $0$} at 44 -48 \put {\scriptsize $0$} at 71.5 -75 \endpicture$$ *Figure 1* \[tree-seq\] [**Tree and sequences.**]{} Now consider the set $\Sigma_q$ of all sequences $\bigl( \sigma(n) \bigr)_{n \le 0}$ over ${\mathbb Z}_q$ with finite support $\{ n : \sigma(n) \ne 0 \}$. We denote by $\tau$ the (negative) shift, $\tau\sigma(n) = \sigma(n-1)$. Then the set $\Sigma_q \times {\mathbb Z}$ carries the structure of ${\mathbb T}_q$ in horocylic layers as above: the $k$-th horocycle is $H_k = \Sigma_q \times \{k\}$, and the predecessor of vertex $x = (\sigma, k)$ is $x^- = (\tau\sigma,k-1)$. This corresponds to labelling the edges of ${\mathbb T}_q$ by elemets of ${\mathbb Z}_q$ such that all edges on the ray from ${\omega}$ to $o$ have label $0$, and for every vertex $x$ and every $\ell \in {\mathbb Z}_q$ there is a successor $y$ among the $q$ successors of $x$ such that the edge $[x,y]$ carries label $\ell$. See Figure 1: the origin is the leftmost point on the horocycle $H_0$, we have indicated the labels on the edges that lead to the point marked with a “[$\bullet$]{}”, and that point has coordinates $(\sigma,k)$ with $\sigma = (\dots,0,\dots,0,1,0,1)$ and $k=-1$. Now consider two trees ${\mathbb T}_q$ and ${\mathbb T}_r$ with roots $o_1$ and $o_2$ and reference ends ${\omega}_1$ and ${\omega}_2$, respectively. \[DLdef\] The Diestel-Leader graph ${\mbox{\sl DL}}(q,r)$ is $${\mbox{\sl DL}}(q,r) = \{ x_1x_2 \in {\mathbb T}_q \times {\mathbb T}_r : {\mathfrak{h}}(x_1)+{\mathfrak{h}}(x_2) = 0 \}\,,$$ and neighbourhood is given by $$x_1x_2 \sim y_1y_2 \iff x_1 \sim y_1 {\quad\mbox{and}\quad}x_2 \sim y_2\,.$$ To visualize ${\mbox{\sl DL}}(q,r)$, draw ${\mathbb T}_q$ in horocyclic layers with ${\omega}_1$ at the top and ${\partial}^*{\mathbb T}_q$ at the bottom, and right to it ${\mathbb T}_r$ in the same way, but upside down, with the respective horocycles $H_k({\mathbb T}_q)$ and $H_{-k}({\mathbb T}_r)$ on the same level. Connect the two origins $o_1$, $o_2$ by an elastic spring. It is allowed to move along each of the two trees, may expand infinitely, but must always remain in horizontal position. The vertex set of ${\mbox{\sl DL}}_{q,r}$ consists of all admissible positions of the spring. From a position $x_1x_2$ with ${\mathfrak{h}}(x_1) + {\mathfrak{h}}(x_2) =0$ the spring may move downwards to one of the $r$ successors of $x_2$ in ${\mathbb T}_r$, and at the same time to the predecessor of $x_1$ in ${\mathbb T}_q$, or it may move upwards in the analogous way. Such a move corresponds to going to a neighbour of $x_1x_2$. Figure 2 depicts ${\mbox{\sl DL}}(2,2)$. $$\beginpicture \setcoordinatesystem units <3mm,3.5mm> \setplotarea x from -4 to 30, y from -3.8 to 6.4 \arrow <5pt> [.2,.67] from 4 4 to 1 7 \put{$\omega_1$} [rb] at 1.2 7.2 \put{$o_1$} [lb] at 8.15 0.2 \plot -4 -4 4 4 / \plot 4 4 12 -4 / \plot -2 -2 -2.95 -4 / \plot -.5 -2 -1.9 -4 / \plot -.5 -2 -.85 -4 / \plot 1 -2 .2 -4 / \plot 1 -2 1.25 -4 / \plot 2.5 -2 2.3 -4 / \plot 2.5 -2 3.35 -4 / \plot 5.5 -2 4.65 -4 / \plot 5.5 -2 5.7 -4 / \plot 7 -2 6.75 -4 / \plot 7 -2 7.8 -4 / \plot 8.5 -2 8.85 -4 / \plot 8.5 -2 9.9 -4 / \plot 10 -2 10.95 -4 / \plot 0 0 -.5 -2 / \plot 2 0 1 -2 / \plot 2 0 2.5 -2 / \plot 6 0 5.5 -2 / \plot 6 0 7 -2 / \plot 8 0 8.5 -2 / \plot 2 2 2 0 / \plot 6 2 6 0 / \arrow <5pt> [.2,.67] from 22 -4 to 25 -7 \put{$\omega_2$} [lt] at 25.2 -7.2 \put{$o_2$} [rt] at 17.95 -.2 \plot 14 4 22 -4 / \plot 22 -4 30 4 / \plot 16 2 15.05 4 / \plot 17.5 2 16.1 4 / \plot 17.5 2 17.15 4 / \plot 19 2 18.2 4 / \plot 19 2 19.25 4 / \plot 20.5 2 20.3 4 / \plot 20.5 2 21.35 4 / \plot 23.5 2 22.65 4 / \plot 23.5 2 23.7 4 / \plot 25 2 24.75 4 / \plot 25 2 25.8 4 / \plot 26.5 2 26.85 4 / \plot 26.5 2 27.9 4 / \plot 28 2 28.95 4 / \plot 18 0 17.5 2 / \plot 20 0 19 2 / \plot 20 0 20.5 2 / \plot 24 0 23.5 2 / \plot 24 0 25 2 / \plot 26 0 26.5 2 / \plot 20 -2 20 0 / \plot 24 -2 24 0 / \put {$\circ$} at 8 0 \put {$\circ$} at 18 0 \plot 8.25 0 12.1 0 / \plot 13.9 0 17.78 0 / \plot 12.1 0 12.25 .4 12.25 -.4 12.55 .4 12.55 -.4 12.85 .4 12.85 -.4 13.15 .4 13.15 -.4 13.45 .4 13.45 -.4 13.75 .4 13.75 -.4 13.9 0 13.9 0 / \setdashes <2pt> \putrule from -4.5 -7 to 12.5 -7 \putrule from 13.5 7 to 30.5 7 \put {${\partial}^*{\mathbb T}_q$} [r] at -5 -7 \put {${\partial}^*{\mathbb T}_r$} [l] at 31 7 \put {$\vdots$} at 4 -5.2 \put {$\vdots$} at 22 5.5 \endpicture$$ [*Figure 2*]{} As the reference point in ${\mbox{\sl DL}}(q,r)$, we choose $o=o_1o_2$. We shall keep in mind that ${\mathbb T}_q$ is the first and ${\mathbb T}_r$ the second tree; when $r=q$, it will be sometimes convenient to write ${\mathbb T}^1$ and ${\mathbb T}^2$ for the first and second trees, both copies of ${\mathbb T}_q$. Next, we explain what the lamplighter group ${\mathbb Z}_q \wr {\mathbb Z}$ has to do with ${\mbox{\sl DL}}(q,q)$. Let $(\eta,k) \in {\mathbb Z}_q \wr {\mathbb Z}$, and recall that $\eta: {\mathbb Z}\to {\mathbb Z}_q$ is a finitely supported configuration. We identify $(\eta,k)$ with the vertex $x_1x_2 \in {\mbox{\sl DL}}(q,q)$, where according to (\[tree-seq\]), the vertices $x_i$ are given by $$\label{identif} \begin{gathered} x_1 = (\eta_k^-,k) {\quad\mbox{and}\quad}x_2=(\eta_k^+,-k)\,,\quad\mbox{where}\\ \eta_k^- = \bigl(\eta(k+n)\bigr)_{n \le 0} {\quad\mbox{and}\quad}\eta_k^+ = \bigl(\eta(k+1-n)\bigr)_{n \le 0}\,, \end{gathered}$$ that is, we split $\eta$ at $k$, with $\eta_k^- = \eta|_{(-\infty\,,\,k]}$ and $\eta_k^+=\eta|_{[k+1\,,\,\infty)}$, both written as sequences over the non-positive integers. This is clearly a one-to-one correspondence between ${\mbox{\sl DL}}(q,q)$ and ${\mathbb Z}_q \wr {\mathbb Z}$, and it is also straighforward that this group acts transitively and fixed-point-freely on the graph: the action of $m \in {\mathbb Z}$ is given by $x_1x_2 = (\sigma_1,k)(\sigma_2,-k) \mapsto y_1y_2 = (\sigma_1,k+m)(\sigma_2,-k+m)$, and the action of the group of configurations is pointwise addition modulo $q$ in the obvious way; the reader is invited to work out the simple details. We have to determine the symmetric set of generators of our group with respect to which ${\mbox{\sl DL}}(q,q)$ is its Cayley graph. (Here, we mean the *right* Cayley graph, where an edge corresponds to multiplying with a generator on the right.) Stepping from a vertex $x_1x_2$ to $y_1x_2^-$, where $y_1$ is one of the successors of $x_1 \in H_k({\mathbb T}^1)$ (horocycle in the first tree) means that the lamplighter walks from position $k$ to $k+1$ and then switches the lamp at the new position to some state in ${\mathbb Z}_q$. Thus, the “downward” edges of this type correspond to multiplying on the right with the group elements $({\delta}_1^{\ell},1)$, $\ell \in {\mathbb Z}_q$, where ${\delta}_k^{\ell}$ is the configuration with value $\ell$ at $k$ and $0$ elsewhere. On the other hand, we have the “upward” edges from $x_1x_2$ to $x_1^-y_2$, where $y_2$ is one of the successors of $x_2 \in H_{-k}({\mathbb T}^2)$ (horocycle in the second tree). They correspond to multiplying on the right with the inverses of the above generators, i.e., the elements $({\delta}_0^{\ell},-1)$, where $\ell \in {\mathbb Z}_q$. Thus the simple random walk on ${\mbox{\sl DL}}(q,q)$ is the following lamplighter walk: its *law,* the probability measure on ${\mathbb Z}_q \wr {\mathbb Z}$ that describes the one step transition probabilites, is equidistribution on $$\label{walk-switch} \{ ({\delta}_1^{\ell},1)\,,\;({\delta}_0^{\ell},-1) : \ell \in {\mathbb Z}_q \}\,.$$ If at some step, the lamplighter stands at $k \in {\mathbb Z}$, (s)he chooses with equal probability either to step to $k+1$ and then to switch the lamp at $k+1$ to a random state, or (s)he chooses to switch the lamp at $k$ to a random state (before leaving $k$) and then to step to $k-1$. While this is a symmetric random walk on ${\mathbb Z}_q \wr {\mathbb Z}$, resp. ${\mbox{\sl DL}}(q,q)$, this type of action does not appear “symmetric” from the point of view of the lamplighter. For this reason, other types of “simple” random walks have been considered in the past: the one whose law is equidistribution on $$\label{walk-or-switch} \{ ({\mathbf 0},\pm 1)\,,\;({\delta}_0^{\ell},0) : 0 \ne \ell \in {\mathbb Z}_q \}$$ (“walk or switch”), and the one where the lamplighter standing at $k$ first switches the lamp where he stands to a random state, then walks to $k \pm 1$, and then switches the lamp at the arrival point to a random state (“switch-walk-switch”). The corresponding law is equidistribution on $$\label{switch-walk-switch} \{ ({\delta}_0^{\ell}+{\delta}_{\pm 1}^{m},\pm 1) : \ell,m \in {\mathbb Z}_q \}$$ Harmonic functions for the “walk or switch” model cannot be determined by the methods that we elaborate here, since it is not very well adapted to the structure of ${\mbox{\sl DL}}(q,q)$, see the comments at the end. On the other hand, the “switch-walk-switch” model (\[switch-walk-switch\]) corresponds to simple random walk on the following modification of ${\mbox{\sl DL}}(q,q)\,$: in the first of the two trees, we add edges between every vertex and the siblings of its predecessor (i.e., its “uncles”), and the resulting neighbourhood relation on the horocyclic product is as in Definition \[DLdef\]. It will be easy to adapt our results to this random walk. In the first place, we shall study the following slight generalization $P = P_{{\alpha}}$ of simple random walk on ${\mbox{\sl DL}}(q,r)$, where $0 < {\alpha}< 1$. For $x_1x_2 \in {\mbox{\sl DL}}(q,r)$ $$\label{random-walk} p(x_1x_2,y_1y_2) = \begin{cases} {\alpha}/q & \text{if}\; y_1^- = x_1 \;\text{and}\;y_2=x_2^-\\ (1-{\alpha})/r & \text{if}\; y_1 = x_1^- \;\text{and}\;y_2^-=x_2\\ 0 & \text{otherwise.} \end{cases}$$ $P$ acts on functions $h: {\mbox{\sl DL}}(q,r) \to {\mathbb R}$ by $$Ph(x_1x_2) = \sum_{y_1y_2} p(x_1x_2,y_1y_2)h(x_1x_2)$$ A *harmonic,* or more precisely, *$P_{\alpha}$-harmonic function,* is one that satisfies $Ph = h$. We can consider the projections $P_1 = P_{1,{\alpha}}$ and $P_2 = P_{2,1-{\alpha}}$ of $P_{{\alpha}}$ on ${\mathbb T}_q$ and ${\mathbb T}_r$, respectively: $$\label{projections} p_1(x_1,y_1) = \begin{cases} {\alpha}/q & \text{if}\; y_1^- = x_1 \\ (1-{\alpha}) & \text{if}\; y_1 = x_1^- \\ 0 & \text{otherwise,} \end{cases} \qquad p_2(x_1,y_2) = \begin{cases} {\alpha}& \text{if}\; y_2 = x_2^- \\ (1-{\alpha})/r & \text{if}\; y_2^- = x_2 \\ 0 & \text{otherwise.} \end{cases} $$ The following is straightforward. \[lift-harmonic\] [(a)]{} If $h_1$ is a $P_1$-harmonic function on ${\mathbb T}_q$, then $h(x_1x_2) = h_1(x_1)$, $x_1x_2 \in {\mbox{\sl DL}}(q,r)$, defines a $P$-harmonic function on ${\mbox{\sl DL}}(q,r)$.\ [(b)]{} If $h_2$ is a $P_2$-harmonic function on ${\mathbb T}_q$, then $h(x_1x_2) = h_2(x_2)$, $x_1x_2 \in {\mbox{\sl DL}}(q,r)$, defines a $P$-harmonic function on ${\mbox{\sl DL}}(q,r)$.\ Our first main result is the following. \[split-theorem\] If $h$ is a non-negative $P$-harmonic function on ${\mbox{\sl DL}}(q,r)$, then there are non-negative $P_i$-harmonic functions $h_i\,$, $i=1,2$, on ${\mathbb T}_q$ and ${\mathbb T}_r$, respectively, such that $$h(x_1x_2) = h_1(x_1) + h_2(x_2) \quad \text{for all}\; x_1x_2 \in {\mbox{\sl DL}}(q,r)$$ Conversely, it is of course clear that every sum of the latter form defines a $P$-harmonic function. Recall that in this type of decomposition, $x_2$ cannot vary independently of $x_1$, since one must have ${\mathfrak{h}}(x_1) + {\mathfrak{h}}(x_2) = 0$. The next short section contains some basic preparatory material for the proof of Theorem \[split-theorem\]. Basic results about harmonic functions {#basics} ====================================== Let $X$ be a denumerable set and $P = \bigl(p(x,y)\bigr)_{x,y \in X}$ the stochastic transition matrix of a Markov chain $(Z_n)_{n \ge 0}$ on $X$. We write $\Pr_x$ for probability conditioned to the starting point $Z_0=x$. The $n$-step transition probability $p^{(n)}(x,y) = \Pr_x[Z_n=y]$ is the $(x,y)$-entry of the matrix power $P^n$. We assume that $P$ is *irreducible:* $\forall\ x,y\ \exists\ n: p^{(n)}(x,y) > 0$. As above, a function $h$ on $X$ is called *$P$-harmonic* or just *harmonic at* $x$, if $Ph(x) = h(x)$, where $Ph(x) = \sum_y p(x,y)h(y)$. It is called harmonic when it is harmonic at each $x$. For a subset $A \subset X$, we define the stopping time $$s^A = \inf \{ n \ge 0 : Z_n \in A \}\,.$$ For $y \in X$, we write $s^y = s^{\{y\}}$. Given $x,y \in X$, let $$\label{hit-def} F(x,y) = {\Pr}_x[s^y < \infty] {\quad\mbox{and}\quad}F^A(x,y) = {\Pr}_x[s^y \le s^A\,,\;s^y < \infty]$$ Thus, $F(x,y)$ is the probability to ever reach $y$, starting from $x$. The function $F(\cdot,y)$ is harmonic in $X \setminus \{y\}$. Furthermore, if $y \in A$, then $F^A(\cdot,y)$ is harmonic in $X \setminus A$. \[finite\] [**Harmonic functions on finite sets.**]{} Let $S$ be a finite subset of $X$. Define its *boundary* and *interior* by $${\partial}S = \{ y \in S : p(y, X \setminus S) > 0\} {\quad\mbox{and}\quad}S^o = S \setminus {\partial}S\,.$$ For the sake of simplicity, we assume that the restriction of $P$ to $S^o$ is irreducible. We define $${\mathcal H}(P,S) = \{ h: S \to {\mathbb R}\mid h \;\text{is harmonic in}\; S^o \}$$ The following is very well known. \[dirichlet\] Under the above assumptions, the functions $F^{{\partial}S}(\cdot, y)\,$, $y \in {\partial}S$, constitute a basis of the linear space ${\mathcal H}(P,S)$. Every $h \in {\mathcal H}(P,S)$ is uniquely respresented as $$h = \sum_{y \in {\partial}S} F^{{\partial}S}(\cdot,y)\,h(y).$$ The functions $F^{{\partial}S}(\cdot, y)\,$, $y \in {\partial}S$, are linearly independent, since $F^{{\partial}S}(x, y)={\delta}_x(y)$ for $x, y \in {\partial}S$. Given $h \in {\mathcal H}(P,S)$, let $g = \sum_{y \in {\partial}S} F^{{\partial}S}(\cdot,y)\,h(y)$. Then $g \in {\mathcal H}(P,S)$, and $(g-h)|_{{\partial}S} \equiv 0$. By the *Minimum Principle,* every function in ${\mathcal H}(P,S)$ attains its minimum (and its maximum) on the boundary. Therefore $g = h$ on $S$. \[positive-harmonic\] [**Positive and minimal harmonic functions.**]{} Regarding the following material, see [Woess]{} [@Wbook], §24 for a more detailed outline and many references. We return to the infinite set $X$ with irreducible transition matrix $P$. For the sake of simplicity, we assume that $P$ has *finite range,* i.e., $\{ y: p(x,y) > 0 \}$ is finite for all $x \in X$. The set ${\mathcal H}^+ = {\mathcal H}^+(P,X)$ of non-negative $P$-harmonic functions constitutes a convex cone that is closed in the topology of pointwise convergence. We choose a reference point $o \in X$. Then the set ${\mathcal B}= \{ h \in {\mathcal H}^+ : h(o)=1 \}$ is a compact, convex base of the cone ${\mathcal H}^+$. Its extremal elements are called *minimal harmonic functions.* Thus, $h \in {\mathcal H}^+$ is minimal if $$h(o) = 1 {\quad\mbox{and}\quad}h \ge h_1 \in {\mathcal H}^+ \Longrightarrow h_1/h \equiv \text{constant.}$$ The set ${\mathcal B}_{\min}$ of minimal harmonic functions is a Borel subset of ${\mathcal B}$, and every $h \in {\mathcal H}^+$ is an integral of minimal ones with respect to a Borel measure on ${\mathcal B}_{\min}$. This can be made more precise by the following construction. Define the *Martin kernel* $$K(x,y) = F(x,y)/F(o,y)\,.$$ The *Martin compactification* is the smallest metrizable compactification of $X$ containing $X$ as a discrete, dense subset, and to which all functions $K(x,\cdot)$, $x \in X$, extend continuously. The *Martin boundary* ${\mathcal M}= {\mathcal M}(P)$ is the ideal boundary added to $X$ in this compactification. Then every minimal harmonic function is of the form $K(\cdot,\xi)$ for some $\xi \in {\mathcal M}$, and the set $${\mathcal M}_{\min} = \{ \xi \in {\mathcal M}: K(\cdot,\xi) \;\text{is minimal harmonic} \}$$ is a Borel set. The *Poisson-Martin Representation Theorem* says that for every $h \in {\mathcal H}^+$ there is a unique Borel measure $\nu^h$ on ${\mathcal M}$ with $\nu^h({\mathcal M}\setminus {\mathcal M}_{\min})=0$ such that $$h(x) = \int_{{\mathcal M}} K(x,\cdot)\,d\nu^h \quad \forall\ x \in X\,.$$ Furthermore, considering the constant harmonic function $\mathbf 1$, we set $\nu = \nu^{\mathbf 1}$. Then every *bounded* harmonic function $h$ has a unique representation as above, where $d\nu^h(\xi) = \varphi(\xi)\,d\nu(\xi)$ with $\varphi \in L^{\infty}({\mathcal M},\nu)$. The probability space $({\mathcal M},\nu)$ is a model of the *Poisson boundary* of the random walk. While the Martin boundary is a topological object, the Poisson boundary is a measure theoretical one, and finding it means to determine it up to isomorphisms between measure spaces. See [Kaimanovich and Vershik]{} [@KaiVer] for a profound introduction and impressive results regarding Poisson boundaries of random walks on groups, [Kaimanovich]{} [@Kai] for lamplighter groups over ${\mathbb Z}^d$ and other semidirect products, and [Kaimanovich and Woess]{} [@KaiWoe] for Poisson boundaries of random walks on homogenenous graphs, including the Diestel-Leader graphs. \[trees\] [**Harmonic functions on trees.**]{} Next, let us suppose that $X=T$ carries the structure of an infinite, locally finite tree. We assume that $P$ is of *nearest neighbour* type, i.e., $$p(x,y) > 0 \iff x \sim y \; \text{in}\; T\,.$$ ($\sim$ denotes neighbourhood.) We also assume that the random walk (Markov chain) with transition matrix $P$ is *transient,* that is, $\sum_n p^{(n)}(x,y) < \infty$ for some ($\!\!\iff\!$ all) $x,y \in T$. Geodesics and boundary of $T$ are defined as in §\[geometry\], with the general tree $T$ in the place of ${\mathbb T}_q$. The results regarding the Martin compactification in this setting are contained in the seminal paper by [Cartier]{} [@Car]. The basic link between tree structure and random walk is the following well-known lemma, see e.g. [Cartier]{} [@Car], or [Woess]{} [@Wbook], Lemmas 1.23 and 1.13(d). \[tree-lemma\] For a nearest neighbour random walk on a tree $T$, $$F(x,y) = F(x,w)F(w,z) \quad \text{for all}\;x,y \in T\;\text{and}\; w \in {\overline{x\,y}}\,.$$ Furthermore, if $x \sim y$, then $$F(y,x) = p(y,x) + \sum_{w \ne x} p(y,w)F(w,y)F(y,x)\,.$$ For $x, y \in T$, let $c= x \wedge y$ be their confluent with respect to $o$. Then the lemma implies that $K(x,y) = K(x,c)$. From here, the following is almost immediate. \[tree-martin\] Suppose that $P$ defines a transient nearest neighbour random walk on the locally finite tree $T$ with root $o$. Then the Martin compactification is the end compactification ${\widehat}T$, and for $\xi \in {\partial}T$, the Martin kernel is given by $$K(x,\xi) = K(x,c) = \frac{F(x,c)}{F(o,c)}\,,\quad \text{where}\quad c = x \wedge \xi\,.$$ Furthermore, each function $K(\cdot,\xi)$, $\xi \in {\partial}T$, is minimal harmonic. For various different proofs, see [Cartier]{} [@Car], [Picardello, Taibleson and Woess]{} [@PicTaiWoe], or [Woess]{} [@Wbook], §26, or also the one which is implicit in the proof of Theorem \[split-theorem\] below. \[alpha-walk\] Consider the random walk on ${\mathbb T}_q$ with transition matrix $P_1=P_{1,{\alpha}}$, defined in \[projections\]. It is clear that for this random walk, the probabilities $$F_1^- = F_1(x,x^-) {\quad\mbox{and}\quad}F_1^+ = F_1(x^-,x)$$ are independent of $x \in {\mathbb T}_q$ ($x^-$ is the predecessor with respect to ${\omega}$). Using Lemma \[tree-lemma\], we find the two quadratic equations $$F_1^- = (1-{\alpha}) + {\alpha}(F_1^-)^2 {\quad\mbox{and}\quad}F_1^+ = \frac{{\alpha}}{q} + (q-1)\frac{{\alpha}}{q}F_1^-F_1^+ + (1-{\alpha})(F_1^+)^2\,.$$ Among the two solutions of each equation, the smaller one is the right one (compare e.g. with the generating functions argument in the proof of Lemma 1.24 in [Woess]{} [@Wbook]). Thus $$\label{F-computation} F_1^- = \begin{cases} {\displaystyle}\frac{1-{\alpha}}{{\alpha}} &{\displaystyle}\text{if}\; {\alpha}\ge \frac12\,, \\[7pt] {\displaystyle}1& {\displaystyle}\text{if}\; {\alpha}\le \frac12\,, \end{cases} \qquad F_1^+ = \begin{cases} {\displaystyle}\frac{1}{q} & {\displaystyle}\text{if}\; {\alpha}\ge \frac12\,, \\[7pt] {\displaystyle}\frac{{\alpha}}{(1-{\alpha})q} & {\displaystyle}\text{if}\; {\alpha}\le \frac12\,. \end{cases}$$ We can now compute the associated Martin kernels $K_1(\cdot,\xi)$, $\xi \in {\partial}{\mathbb T}_q$. First, since $x \wedge {\omega}= x {\curlywedge}o$ (where $\wedge$ and ${\curlywedge}$ denote confluents with respect to $o$ and ${\omega}$), it is immediate that $$\label{martin-omega} K_1(x,{\omega}) = (F_1^-)^{{\mathfrak{h}}(x)} = \begin{cases} {\displaystyle}\left(\frac{1-{\alpha}}{{\alpha}}\right)^{\mbox{\footnotesize ${\mathfrak{h}}(x)$}} & {\displaystyle}\text{if}\; {\alpha}\ge \frac12\,, \\[7pt] {\displaystyle}1 & {\displaystyle}\text{if}\; {\alpha}\le \frac12\,. \end{cases}$$ Next, if $\xi \in {\partial}^*{\mathbb T}_q$, we set $c = x \wedge \xi$ and write $k = d(o,o {\curlywedge}x)$, $l = d(x,o {\curlywedge}x)$, so that ${\mathfrak{h}}(x) = l-k$. We distinguish two cases, see Figure 3. $$\beginpicture \setcoordinatesystem units <1mm,1.6mm> \setplotarea x from -2 to 100, y from -10 to 10 \arrow <6pt> [.2,.67] from 0 0 to 17 -17 \arrow <6pt> [.2,.67] from 6 -6 to 16.5 4.5 \plot 12 -12 28 4 / \put {$\omega$} at 18 -18 \put {$\xi$} at 18 6 \put {$o$} at -1.5 1.5 \put {$x$} at 29.5 5.5 \put {$c$} at 4 -7.5 \put {$o {\curlywedge}x$}[r] at 10.5 -13 \multiput {\scriptsize $\bullet$} at 0 0 6 -6 12 -12 28 4 / \arrow <6pt> [.2,.67] from 70 0 to 87 -17 \arrow <6pt> [.2,.67] from 88 -6 to 77.5 5.5 \plot 82 -12 98 4 / \put {$\omega$} at 88 -18 \put {$\xi$} at 76 6 \put {$o$} at 68.5 1.5 \put {$x$} at 99.5 5.5 \put {$c$} at 89.5 -7.5 \put {$o {\curlywedge}x$}[r] at 80.5 -13 \multiput {\scriptsize $\bullet$} at 70 0 82 -12 88 -6 98 4 / \put{\it Figure 3}[c] at 49 -22 \endpicture$$ *Case 1.* $c$ lies between $o$ and $o {\curlywedge}x$. Let $s = d(o,c)$. Then $$K_1(x,\xi) = \frac{(F_1^-)^l(F_1^+)^{k-s}}{(F_1^-)^s} = K_1(x,{\omega})(F_1^-F_1^+)^{k-s} = K_1(x,{\omega})(F_1^-F_1^+)^{\mbox{\footnotesize ${\mathfrak{h}}(o{\curlywedge}\xi)-{\mathfrak{h}}(x{\curlywedge}\xi)$}}$$ *Case 2.* $c$ lies between $o {\curlywedge}x$ and $x$. Let $r = d(x,c)$. Then $$K_1(x,\xi) = \frac{(F_1^-)^r}{(F_1^-)^k(F_1^+)^{l-r}} = K_1(x,{\omega})(F_1^-F_1^+)^{r-l} = K_1(x,{\omega})(F_1^-F_1^+)^{\mbox{\footnotesize ${\mathfrak{h}}(o{\curlywedge}\xi)-{\mathfrak{h}}(x{\curlywedge}\xi)$}}$$ We write ${\mathfrak{h}}(x,\xi) = d(x,c) - d(o,c)$, the horocycle number with respect to $\xi$, while ${\mathfrak{h}}(x) = {\mathfrak{h}}(x,{\omega})$. Also, we set $\rho = (F_1^-F_1^+)^{1/2}$. Then we find in both cases $$\label{martin-xi} K_1(x,\xi) = K_1(x,{\omega})\, \rho^{\mbox{\footnotesize ${\mathfrak{h}}(x,\xi) - {\mathfrak{h}}(x)$}} \,,\quad\text{where}\quad \rho = \min \left\{ \frac{1-{\alpha}}{{\alpha}q}\,, \frac{{\alpha}}{(1-{\alpha})q} \right\}^{1/2}\,.$$ In particular, if ${\alpha}=1/2$ then $$\label{martin-1} K_1(x,{\omega}) = 1 {\quad\mbox{and}\quad}K_1(x,\xi) = q^{\mbox{\footnotesize $({\mathfrak{h}}(x) - {\mathfrak{h}}(x,\xi))/2$}}\quad \text{for}\quad \xi \in {\partial}^*{\mathbb T}_q\,.$$ Minimal harmonic functions on ${\mbox{\sl DL}}(q,r)$ {#principal} ==================================================== After all these preliminaries, the proof of Theorem \[split-theorem\] depends in the first place on the way how we look at the underlying structure. In ${\mbox{\sl DL}}(q,r)$, consider the subgraph spanned by all vertices $x_1x_2$ with $-n \le {\mathfrak{h}}(x_1) \le n$. It is not connected. We denote by $S = S^{(n)}$ the connected component of the root $o_1o_2$. Let $a_1 = a_1^{(n)} \in {\mathbb T}_q$ be the vertex on ${\overline{o_1\,{\omega}_1}}$ at distance $d(a_1,o_1) = n$. Then $a_1$ can be viewed as the root of the $q$-ary rooted tree $S_1 = S_1^{(n)} = \{ x_1 \in {\mathbb T}_q : -n \le {\mathfrak{h}}(x_1) \le n\,,\; a_1 \in {\overline{x_1\,{\omega}_1}}\,\}$ of height $2n$, whose set of leaves (elements with ${\mathfrak{h}}(x_1)=n$) is denoted ${\partial}^* S_1$. Analogously, we define $a_2 = a_2^{(n)}$, the $r$-ary rooted tree $S_2 = S_2^{(n)}$, and it set of leaves ${\partial}^*S_2$. Then $$S = \{ x_1x_2 \in S_1 \times S_2 : {\mathfrak{h}}(x_1)+{\mathfrak{h}}(x_2)=0\}$$ is the horocyclic product of $S_1$ and $S_2$, see Figure 4. $$\beginpicture \setcoordinatesystem units <3mm,3.5mm> \setplotarea x from -4 to 30, y from -2.3 to 2.3 \put{$o_1$} [lb] at 8.15 0.2 \plot -4 -4 4 4 / \plot 4 4 12 -4 / \plot -2 -2 -2.95 -4 / \plot -.5 -2 -1.9 -4 / \plot -.5 -2 -.85 -4 / \plot 1 -2 .2 -4 / \plot 1 -2 1.25 -4 / \plot 2.5 -2 2.3 -4 / \plot 2.5 -2 3.35 -4 / \plot 5.5 -2 4.65 -4 / \plot 5.5 -2 5.7 -4 / \plot 7 -2 6.75 -4 / \plot 7 -2 7.8 -4 / \plot 8.5 -2 8.85 -4 / \plot 8.5 -2 9.9 -4 / \plot 10 -2 10.95 -4 / \plot 0 0 -.5 -2 / \plot 2 0 1 -2 / \plot 2 0 2.5 -2 / \plot 6 0 5.5 -2 / \plot 6 0 7 -2 / \plot 8 0 8.5 -2 / \plot 2 2 2 0 / \plot 6 2 6 0 / \multiput {\scriptsize $\bullet$} at 4 4 -4 -4 -2.95 -4 -1.9 -4 -.85 -4 .2 -4 1.25 -4 2.3 -4 3.35 -4 4.65 -4 5.7 -4 6.75 -4 7.8 -4 8.85 -4 9.9 -4 10.95 -4 12 -4 / \put{$a_1$}[b] at 4 4.5 \put{${\partial}^*S_1$}[t] at 4 -5 \put{$o_2$} [rt] at 17.95 -.2 \plot 14 4 22 -4 / \plot 22 -4 30 4 / \plot 16 2 15.05 4 / \plot 17.5 2 16.1 4 / \plot 17.5 2 17.15 4 / \plot 19 2 18.2 4 / \plot 19 2 19.25 4 / \plot 20.5 2 20.3 4 / \plot 20.5 2 21.35 4 / \plot 23.5 2 22.65 4 / \plot 23.5 2 23.7 4 / \plot 25 2 24.75 4 / \plot 25 2 25.8 4 / \plot 26.5 2 26.85 4 / \plot 26.5 2 27.9 4 / \plot 28 2 28.95 4 / \plot 18 0 17.5 2 / \plot 20 0 19 2 / \plot 20 0 20.5 2 / \plot 24 0 23.5 2 / \plot 24 0 25 2 / \plot 26 0 26.5 2 / \plot 20 -2 20 0 / \plot 24 -2 24 0 / \multiput {\scriptsize $\bullet$} at 22 -4 14 4 15.05 4 16.1 4 17.15 4 18.2 4 19.25 4 20.3 4 21.35 4 22.65 4 23.7 4 24.75 4 25.8 4 26.85 4 27.9 4 28.95 4 30 4 / \put{$a_2$}[t] at 22 -4.5 \put{${\partial}^*S_2$}[b] at 22 5 \put {$\circ$} at 8 0 \put {$\circ$} at 18 0 \plot 8.25 0 12.1 0 / \plot 13.9 0 17.78 0 / \plot 12.1 0 12.25 .4 12.25 -.4 12.55 .4 12.55 -.4 12.85 .4 12.85 -.4 13.15 .4 13.15 -.4 13.45 .4 13.45 -.4 13.75 .4 13.75 -.4 13.9 0 13.9 0 / \endpicture$$ [*Figure 4*]{} One may imagine $S$ as a tetrahedron. Two of its faces are copies of $S_1$ that meet at the common bottom edge ${\partial}^*S_1 \times \{a_2\}$, and the other two faces are copies of $S_2$ that meet at the common top edge $\{a_1\}\times {\partial}^*S_2$. The boundary of $S_i$ is $\{a_i\} \cup {\partial}^* S_i$. We now restrict $P$ to $S$, and also the projections $P_1$ to $S_1$ and $P_2$ to $S_2$. Then the boundary of $S$ in the sense of (\[finite\]) is $${\partial}S = \bigl({\partial}^*S_1 \times \{a_2\}\bigr) \cup \bigl(\{a_1\}\times {\partial}^*S_2\bigr)\,.$$ As in Lemma \[lift-harmonic\], if $h_i \in {\mathcal H}(P_i,S_i)$, then it lifts to a function in ${\mathcal H}(P,S)$. In particular, if $y_1 \in {\partial}^*S_1$ then $h(x_1x_2)= F_1^{{\partial}S_1}(x_1,y_1)$ defines a function in ${\mathcal H}(P,S)$ with value one at $y_1a_2$ and value $0$ in ${\partial}S \setminus \{ y_1a_2 \}$. But, by Proposition \[dirichlet\], these properties characterize the function $x_1x_2 \mapsto F^{{\partial}S}(x_1x_2,y_1a_2)$ on $S$. Therefore, for all $x_1x_2 \in S$, $$\label{hitting} \begin{aligned} F^{{\partial}S}(x_1x_2,y_1a_2) &= F_1^{{\partial}S_1}(x_1,y_1) \quad \forall\ y_1 \in {\partial}^*S_1 \quad\text{and}\\ F^{{\partial}S}(x_1x_2,a_1y_2) &= F_2^{{\partial}S_2}(x_2,y_2) \quad \forall\ y_2 \in {\partial}^*S_2 \end{aligned}$$ Applying Proposition \[dirichlet\] once more, we see that every $h \in {\mathcal H}(P,S)$ can be written uniquely as $$\label{finite-split} \begin{aligned} h(x_1x_2) &= h_1(x_1) + h_2(x_2) \quad \forall\ x_1x_2 \in S\,,\quad\text{where}\\[10pt] h_1(x_1) &= \sum_{y_1 \in {\partial}^*S_1} F_1^{{\partial}S_1}(x_1,y_1) h(y_1a_2) {\quad\mbox{and}\quad}\\ h_2(x_2) &= \sum_{y_2 \in {\partial}^*S_2} F_2^{{\partial}S_2}(x_2,y_2) h(a_1y_2)\,. \end{aligned}$$ This is true, in particular, if $h$ is $P$-harmonic on the whole of ${\mbox{\sl DL}}(q,r)$, since its restriction to $S = S^{(n)}$ is in ${\mathcal H}(P,S)$. Furthermore, if $h$ is non-negative then so are $h_1$ and $h_2$. Note, however, that $h_i = h_i^{(n)}$ ($i=1,2$) *depend on $n$,* and it is by no means true that the restriction of $h_i^{(n+1)}$ to $S_i^{(n)}$ might coincide with $h_i^{(n)}$. We have to study the behaviour of $h_i^{(n)}$ when $n \to \infty$, and this is the point where the assumption of non-negativity of $h$ will be used. We define $$K_i^{(n)}(x_i,y_i) = \frac{F_i^{{\partial}S_i}(x_i,y_i)}{F_i^{{\partial}S_i}(o_i,y_i)}\,,\; x_i \in S_i\,,\; y_i \in {\partial}S_i\,,\; S_i=S_i^{(n)}\,,\; i=1,2\,.$$ Then we can rewrite the functions $h_i$ of (\[finite-split\]) as $$\begin{aligned} &&h_i(x_i) = \sum_{y_i \in {\partial}S_i} K_i^{(n)}(x_i,y_i) {\lambda}_i(y_i)\,, \quad\text{where}\\ {\lambda}_1(a_1) \!\! &=&\!\! 0\,,\quad {\lambda}_1(y_1) = h(y_1a_2)/F_1^{{\partial}S_1}(o_1,y_1)\,,\;y_1 \in {\partial}^*S_1\,,\\ {\lambda}_2(a_2) \!\! &=&\!\! 0\,, \quad {\lambda}_2(y_2) = h(a_1y_2)/F_2^{{\partial}S_2}(o_2,y_2)\,,\;y_2 \in {\partial}^*S_2\,.\end{aligned}$$ Of course, also ${\lambda}_i(y_i) = {\lambda}_i^{(n)}(y_i)$ depends on $n$. For $\xi_1 \in {\partial}{\mathbb T}_q$, we define $$K_1^{(n)}(x_1,\xi_1) = K_1^{(n)}(x_1,y_1)\,,\quad\text{if}\; \xi_1\in {\mathbb T}_q(o_1,y_1) \;\text{with}\;y_1 \in {\partial}S_1\,.$$ Then $K_1^{(n)}(x_1,\cdot)$ is locally constant (whence continuous) on ${\partial}{\mathbb T}_q$. (Recall that ${\partial}{\mathbb T}_q$ is compact and totally disconnected.) We define a non-negative Borel-measure $\nu_1^{(n)}$ on ${\partial}{\mathbb T}_q$ by $$\begin{aligned} \nu_1^{(n)}\bigl({\partial}{\mathbb T}_q(o_1,a_1^-)\bigr) &=& 0 {\quad\mbox{and}\quad}\\ \nu_1^{(n)}\bigl({\partial}{\mathbb T}_q(o_1,w_1)\bigr) &=& {\lambda}_1^{(n)}(y_1)\, q^{{\mathfrak{h}}(w_1)-n} \quad\text{if}\; w_1 \in {\mathbb T}_q(o_1,y_1)\,,\;y_1 \in {\partial}^*S_1^{(n)}\,.\end{aligned}$$ This defines a finitely additive, non-negative measure on the semiring of all sets ${\partial}{\mathbb T}_q(o_1,x_1)$, $x_1\in {\mathbb T}_q \setminus \{o_1\}$. Since all these sets are open and compact, the measure is sigma-additive on that semiring and extends to a unique non-negative Borel measure on ${\partial}{\mathbb T}_q$. We proceed in precisely the same way on the second tree, and also get a non-negative Borel measure $\nu_2^{(n)}$ on ${\partial}{\mathbb T}_r$, such that for all $x_1x_2 \in S^{(n)}\,$, $$\label{finite-integral} h_1^{(n)}(x_1) = \int_{{\partial}{\mathbb T}_q} K_1^{(n)}(x_1,\cdot)\,d\nu_1^{(n)} {\quad\mbox{and}\quad}h_2^{(n)}(x_2) = \int_{{\partial}{\mathbb T}_r} K_2^{(n)}(x_2,\cdot)\,d\nu_2^{(n)} \,.$$ Since $K_i^{(n)}(o_i,\cdot) \equiv 1$, we have $\nu_1^{(n)}({\partial}{\mathbb T}_q) + \nu_2^{(n)}({\partial}{\mathbb T}_r) = h(o_1o_2)$ for all $n$. Thus, by compactness (Helly’s theorem), there are a subsequence $(n')$ and non-negative measures $\nu_1$ on ${\partial}{\mathbb T}_q$ and $\nu_2$ on ${\partial}{\mathbb T}_r$ such that $\nu_i^{(n')} \to \nu_i$ weakly for $i=1,2$. If $x_1x_2 \in {\mbox{\sl DL}}(q,r)$, then we choose $n_0=n_0(x_1x_2)$ large enough such that for all $n \ge n_0$, the geodesics ${\overline{o_i\,x_i}}$ are contained in the interior of $S_i^{(n)}$, $i=1,2$. For every $\xi_i \in {\partial}{\mathbb T}_q$, resp. $\in {\partial}{\mathbb T}_r$, the confluent $c_i = x_i \wedge \xi_i$ is one of the finitely many points on ${\overline{o_i\,x_i}}$. It is clear that $F_i^{{\partial}S_i^{(n)}}(x_i,c_i) \to F_i(x_i,c_i)$ as $n \to \infty$, and the same is true with $o_i$ in the place of $x_i$. Therefore, using Lemma \[tree-lemma\], we find $$K_i^{(n)}(x_i,\xi_i) = K_i^{(n)}(x_i,c_i) \to K_i(x_i,c_i) = K_i(x_i,\xi_i)\,,$$ as $n \to \infty$, where $K_i(\cdot,\cdot)$ is the Martin kernel of $P_i$ on ${\mathbb T}_q$, resp. ${\mathbb T}_r$, $i=1,2$. Thus, $K_i^{(n)}(x_i,\cdot) \to K_i(x_i,\cdot)$ uniformly, and using (\[finite-integral\]), we get $$h_1^{(n')}(x_1) \to \int_{{\partial}{\mathbb T}_q} K_1(x_1,\cdot)\,d\nu_1 =: h_1(x_1){\quad\mbox{and}\quad}h_2^{(n')}(x_2) \to \int_{{\partial}{\mathbb T}_r} K_1(x_2,\cdot)\,d\nu_2 =: h_2(x_2)\,.$$ Then $h_1$ is $P_1$-harmonic on ${\mathbb T}_q$ and $h_2$ is $P_2$-harmonic on ${\mathbb T}_r$, and $h(x_1x_2) = h_1(x_1)+h_2(x_2)$ for all $x_1x_2 \in {\mbox{\sl DL}}(q,r)$. We remark that the last part is is the key point of the argument, namely a way of recovering a Poisson integration formula from the finite approximations $K_i^{(n)}(\cdot,\cdot)$ of the Martin kernel. \[minimal-theorem\] [(a)]{} Each of the functions $$x_1x_2 \mapsto K_1(x_1,\xi_1)\,,\; \xi_1 \in {\partial}^*{\mathbb T}_q\,, {\quad\mbox{and}\quad}x_1x_2 \mapsto K_2(x_2,\xi_2)\,,\; \xi_2 \in {\partial}^*{\mathbb T}_r\,,$$ is minimal $P_{{\alpha}}$-harmonic on ${\mbox{\sl DL}}(p,q).$ [(b)]{} If ${\alpha}\ne 1/2$ then these are all minimal harmonic functions. [(c)]{} If ${\alpha}=1/2$, then these together with the constant function $\mathbf 1$ are all minimal harmonic functions. \(a) Let $\xi_1 \in {\partial}^* {\mathbb T}_q$ and suppose that $K_1(x_1,\xi_1) \ge h(x_1x_2)$ for all $x_1x_2 \in {\mbox{\sl DL}}(q,r)$, where $h \ge 0$ is $P_{{\alpha}}$-harmonic. By Theorem \[split-theorem\], $h(x_1x_2) = h_1(x_1) + h_2(x_2)$, where $h_i \ge 0$ is $P_i$-harmonic, $i=1,2$. Then $K_1(\cdot,\xi_1) \ge h_1$. By Proposition \[tree-martin\], $h_1 = c\cdot K_1(\cdot,\xi_1)$, where $0 \le c \le 1$. If $c=1$ then we are done. Otherwise, $K_1(x_1,\xi_1) \ge c\cdot K_1(\cdot,\xi_1)+ h_2(x_2)$, that is, $$K_1(x_1,\xi_1) \ge \frac{1}{1-c}h_2(x_2) = \int_{{\partial}T_r} K_2(x_2,\cdot)\, d\nu_2 \quad \forall\ x_1x_2 \in {\mbox{\sl DL}}(q,r)\,,$$ where $\nu_2$ is a non-negative Borel measure on ${\partial}{\mathbb T}^r$. Setting $x_2 = o_2$, we obtain $$K_1(x_1,\xi_1) \ge \nu_2({\partial}{\mathbb T}_r) \quad \forall\ x_1 \in H_0({\mathbb T}_q)\,.$$ If $x_1 \to {\omega}_1$ then ${\mathfrak{h}}(x_1,\xi_1) \to \infty$. Therefore, (\[martin-xi\]) yields $$\nu_2({\partial}{\mathbb T}_r) \le \lim_{x_1 \to {\omega}_1,\, {\mathfrak{h}}(x_1)=0} K_1(x_1,\xi_1) = 0\,,$$ and $\nu_2({\partial}{\mathbb T}_r) = 0$. Therefore $h_2 \equiv 0$. This proves minimality of $K_1(\cdot,\xi_1)$ for all $\xi_1 \in {\partial}^*{\mathbb T}_q$. Exchanging the roles of the two trees, we get the other “half” of statement (a).\ (b,c) Conversely, let $h$ be a minimal $P_{{\alpha}}$-harmonic function on ${\mbox{\sl DL}}(q,r)$. By Theorem \[split-theorem\], $h(x_1x_2)=h_1(x_1) + h_2(x_2)$ with $h_i$ non-negative and $P_i$-harmonic. One of the $h_i\,$, say $h_1\,$, must be positive. Minimality yields $h_1(x_1) = c_1\cdot h(x_1x_2)$ for all $x_1x_2 \in {\mbox{\sl DL}}(q,r)$, where $c_1 > 0$. Thus $h(x_1x_2)$ depends only on $x_1$. Without loss of generality, $c_1 = 1$, and $h(x_1x_2) = h_1(x_1)$ for all $x_1x_2$. Minimality of $h$ with respect to $P_{{\alpha}}$ yields minimality of $h_1$ with respect to $P_1$. Thus, by Proposition \[tree-martin\], $h(x_1x_2) = K_1(x_1,\xi_1)$ for some $\xi_1 \in {\partial}{\mathbb T}_q$. The case when $h(x_1x_2)$ depends only on $x_2$ is analogous.\ To complete the proof of statments (b) and (c), we have to study minimality of the functions $x_1x_2 \mapsto K_i(\cdot,{\omega}_i)$, $i=1,2$ with respect to $P_{{\alpha}}$.\ If ${\alpha}= 1/2$ then by (\[martin-omega\]), $K_i(\cdot,{\omega}_i) \equiv 1$. In this case it is known from [Kaimanovich and Woess]{} [@KaiWoe], §6.2, that the Poisson boundary is trivial, i.e., all bounded harmonic functions are constant, which is the same as minimality of the constant function $\mathbf 1$. This proves (c).\ Suppose ${\alpha}\ne 1/2$. Then – again by [@KaiWoe] – the Poisson boundary is nontrivial, and the constant function $\mathbf 1$ is non-minimal. If ${\alpha}> 1/2$, then this yields that $x_1x_2 \mapsto K_2(x_2,{\omega}_2) = 1$ is non-minimal. On the other hand, we know from (\[martin-omega\]) that $$g(x_1x_2):=K_1(x_1,{\omega}_1) = \left(\frac{1-{\alpha}}{{\alpha}}\right)^{\mbox{\footnotesize ${\mathfrak{h}}(x_1)$}}.$$ We can conjugate $P_{{\alpha}}$ by $g$, that is, we set $$\check p(x_1x_2,y_1y_2) = \frac{p(x_1x_2,y_1y_2)g(y_1y_2)}{g(x_1x_2)}\,.$$ Then $g$ is minimal $P_{{\alpha}}$-harmonic if and only if $\mathbf 1$ is minimal $\check P_{{\alpha}}$-harmonic. However, $\check P_{{\alpha}} = P_{1-{\alpha}}$ by a straightforward computation, and the constant function $\bf 1$ is not minimal $P_{1-{\alpha}}$-harmonic by non-triviality of the Poisson boundary. Thus, also $g$ is non-minimal for $P_{{\alpha}}$. Again, the case ${\alpha}< 1/2$ follows by exchanging the roles of the two trees. We remark that for our nearest neighbour case, minimality, resp. non-minimalty of $\mathbf 1$ can be proved in a more elementary (somewhat longer) way than by appealing to the results of [Kaimanovich and Woess]{} [@KaiWoe].\ \[SRW-example\] We conclude this section by retranslating the results for simple random walk on ${\mbox{\sl DL}}(q,q)$ to the setting and notation of the random walk (\[walk-switch\]) on ${\mbox{\sl DL}}(q,q)$ (“walk forward and switch or switch and walk backward”). We write ${\mathbb T}^1$ and ${\mathbb T}^2$ for the first and the second tree, respectively. (Both are copies of ${\mathbb T}_q$.) We have ${\alpha}= 1/2$, and the constant harmonic function $\mathbf 1$ is minimal. In terms of configurations, each $\xi_1 \in {\partial}^*{\mathbb T}^1$ corresponds to an infinite configuration $$\xi_1: {\mathbb Z}\to {\mathbb Z}_q \quad\text{with}\quad \xi_1(n) = 0 \; \forall\ n \le n_0(\xi_1) \in {\mathbb Z}\,.$$ If we label the edges of ${\mathbb T}^1$ by elements of ${\mathbb Z}_q$, as described in (\[tree-seq\]), then $\xi_1(n)$ is the label of the edge between the horocycles $H_{n-1}({\mathbb T}^1)$ and $H_n({\mathbb T}^1)$ on the infinite geodesic ${\overline{{\omega}_1\,\xi_1}}$. Now let $(\eta,k) \in {\mathbb Z}_q \wr {\mathbb Z}$, and consider the associated pair $x_1x_2 \in {\mbox{\sl DL}}(q,q)$. From the computations in Example \[alpha-walk\], we know that $$K_1(x_1,\xi_1) = q^{\mbox{\footnotesize ${\mathfrak{h}}(x_1 {\curlywedge}\xi_1) - {\mathfrak{h}}(o_1 {\curlywedge}\xi_1)$}}$$ It is easy to compute in terms of $(\eta,k)$ $$\label{positive-defect} \begin{aligned} {\mathfrak{h}}(x_1{\curlywedge}\xi_1) &= \begin{cases} \min \{ n \le k : \xi_1(n+1) \ne \eta(n+1) \} & \text{if such $n$ exists,} \\ k &\text{otherwise,} \end{cases} \\[4pt] {\mathfrak{h}}(o_1{\curlywedge}\xi_1) &= \begin{cases} \min \{ m \le 0 : \xi_1(m+1) \ne 0 \} & \text{if such $m$ exists,} \\ 0 &\text{otherwise.} \end{cases} \end{aligned}$$ We shall write ${\operatorname{\sf def}}^+\bigl((\eta,k),\xi_1) = {\mathfrak{h}}(x_1 {\curlywedge}\xi_1) - {\mathfrak{h}}(o_1 {\curlywedge}\xi_1)$, the *(positive) defect* of $(\eta,k)$ with respect to $\xi_1$. Analogously, each $\xi_2 \in {\partial}^*{\mathbb T}^2$ corresponds to an infinite configuration $$\xi_2: {\mathbb Z}\to {\mathbb Z}_q \quad\text{with}\quad \xi_2(n) = 0 \; \forall\ n \ge n_0(\xi_2) \in {\mathbb Z}\,.$$ Here, we label the edges of ${\mathbb T}^2$ by elements of ${\mathbb Z}_q$, and $\xi_2(n)$ is the label of the edge between the horocycles $H_{-n}({\mathbb T}^2)$ and $H_{-n+1}({\mathbb T}^2)$ on the infinite geodesic ${\overline{{\omega}_2\,\xi_2}}$. If $(\eta,k) \in {\mathbb Z}_q \wr {\mathbb Z}\leftrightarrow x_1x_2 \in {\mbox{\sl DL}}(q,q)$, then $$K_1(x_2,\xi_2) = q^{\mbox{\footnotesize ${\mathfrak{h}}(x_2 {\curlywedge}\xi_2) - {\mathfrak{h}}(o_2 {\curlywedge}\xi_2)$}}\,,$$ and we compute $$\label{negative-defect} \begin{aligned} -{\mathfrak{h}}(x_2{\curlywedge}\xi_2) &= \begin{cases} \max \{ n > k : \xi_2(n) \ne \eta(n) \} & \text{if such $n$ exists,} \\ k &\text{otherwise,} \end{cases} \\[4pt] -{\mathfrak{h}}(o_2{\curlywedge}\xi_2) &= \begin{cases} \max \{ m > 0 : \xi_2(m) \ne 0 \} & \text{if such $m$ exists,} \\ 0 &\text{otherwise.} \end{cases} \end{aligned}$$ We shall write ${\operatorname{\sf def}}^-\bigl((\eta,k),\xi_2) = {\mathfrak{h}}(x_2 {\curlywedge}\xi_2) - {\mathfrak{h}}(o_2 {\curlywedge}\xi_2)$, the *(negative) defect* of $(\eta,k)$ with respect to $\xi_2$. We conclude: write ${\partial}^+ ({\mathbb Z}_q \wr {\mathbb Z})$ and ${\partial}^-({\mathbb Z}_q \wr {\mathbb Z})$ for all infinite configurations $\xi_1$, resp. $\xi_2$ as above. (Every *finitely supported* configuration appears once in each of the two parts of the boundaries !) Then all non-constant minimal $P$-harmonic functions are given by $$\begin{aligned} K_1(\cdot, \xi_1) &= q^{\mbox{\footnotesize ${\operatorname{\sf def}}^+(\cdot,\xi_1)$}}\,,\;\xi_1 \in {\partial}^+({\mathbb Z}_q\wr{\mathbb Z})\,, {\quad\mbox{and}\quad}\\ K_2(\cdot,\xi_2) &= q^{\mbox{\footnotesize ${\operatorname{\sf def}}^-(\cdot,\xi_2)$}}\,,\;\xi_2 \in {\partial}^-({\mathbb Z}_q\wr{\mathbb Z})\,. \end{aligned}$$ The constant function $\mathbf 1$ is also minimal harmonic. Switch-walk-switch {#extension} ================== We now turn our attention to the random walk (\[switch-walk-switch\]), where at each step, the lamplighter first switches the lamp at his actual position to a random state, then walks, and then switches the lamp at the arrival point to a random state. As explained in Section \[geometry\], this corresponds to simple random walk on the modification of ${\mbox{\sl DL}}(q,q)$ where in the first tree, additional edges are drawn between every vertex and the siblings of its predecessor, while the second tree remains as it is. More generally, consider ${\mbox{\sl DL}}(q,r)$. For $x_1,u_1 \in {\mathbb T}_q$, we introduce the *sibling relation* $x_1 {\,{\buildrel s \over \sim}\,}u_1 \!\!\iff\!\! x_1^-=u_1^-$. We extend this relation to ${\mbox{\sl DL}}(q,r)$ by setting $x_1x_2 {\,{\buildrel s \over \sim}\,}u_1x_2 \!\!\iff\!\! x_1 {\,{\buildrel s \over \sim}\,}u_1$. The new ege set on the same vertex set $\{ x_1x_2 \in {\mathbb T}_q \times {\mathbb T}_r : {\mathfrak{h}}(x_1)+{\mathfrak{h}}(x_2)=0\}$ is now given by $$\{ [x_1x_2, y_1y_2] : y_1^- {\,{\buildrel s \over \sim}\,}x_1 \;\text{and}\; y_2=x_2^- \}\,.$$ We write ${\mbox{\sl DL}}^s(q,r)$ for the resulting graph. Every vertex $x_1x_2$ with ${\mathfrak{h}}(x_1) = k$ has $q^2$ neighbours $y_1y_2$ with ${\mathfrak{h}}(y_1)=k+1$ and $qr$ neighbours with ${\mathfrak{h}}(y_1)=k-1$. Adapted to this structure, we choose $0 < {\alpha}< 1$ and consider the random walk on ${\mbox{\sl DL}}^s(q,r)$ with transition matrix $Q =Q_{{\alpha}}$ given by $$\label{sws-walk} q(x_1x_2,y_1y_2) = \begin{cases} {\alpha}/q^2 & \text{if}\; y_1^- {\,{\buildrel s \over \sim}\,}x_1 \;\text{and}\;y_2=x_2^-\\ (1-{\alpha})/(qr) & \text{if}\; y_1 {\,{\buildrel s \over \sim}\,}x_1^- \;\text{and}\;y_2^-=x_2\\ 0 & \text{otherwise,} \end{cases}$$ Now note that when $x_1 {\,{\buildrel s \over \sim}\,}u_1$ then transitions from $x_1x_2$ and $u_1x_2$ go to the same neighbours with the same probabilities. Thus, $Qh(x_1x_2) = Qh(u_1x_2)$ whenever $x_1 {\,{\buildrel s \over \sim}\,}u_1$, and we have the following. \[sibling-harmonic\] Every $Q_{{\alpha}}$-harmonic function is constant on each equivalence class of the sibling relation. We can construct the *factor graph* of ${\mbox{\sl DL}}^s(q,r)$ with respect to the sibling relation. We write $[x_1]x_2$ for the equivalence class of $x_1x_2$, since all its elements have the same second “coordinate”, and $[x_1]$ is the sibling class in the first tree. Then the vertex set of the factor graph is $\{ [x_1]x_2 : x_1x_2 \in {\mbox{\sl DL}}^s(q,r) \}$, and two classes $[x_1]x_2$ and $[y_1]y_2$ are connected by an edge of the factor graph if and only if there is an edge bewteen a pair of representatives in ${\mbox{\sl DL}}^s(q,r)$. Thus, if ${\mathfrak{h}}(y_2) = {\mathfrak{h}}(x_2) - 1$, there is an edge from $[x_1]x_2$ to $[y_1]y_2$ precisely when $[y_1^-] = [x_1]$ and $x_2^- = y_2$. We write $\pi$ for the natural projection. The next lemma is now immediate. \[factor-graph\] The factor graph of ${\mbox{\sl DL}}^s(q,r)$ with respect to the sibling relation is ${\mbox{\sl DL}}(q,r)$. The transition matrix $Q_{{\alpha}}$ is compatible with the factorization, and its image under the projection $\pi: {\mbox{\sl DL}}^s(q,r) \to {\mbox{\sl DL}}(q,r)$ is $P_{{\alpha}}$, as defined in (\[random-walk\]). By “compatible” we mean that $ q_{{\alpha}}(v_1x_2,[y_1]y_2) = \sum_{w_1 {\,{\buildrel s \over \sim}\,}y_1} q_{{\alpha}}(x_1x_2,w_1y_2) $ is the same for each representative $v_1 \in [x_1]$, and this common value is the transition probability from $[x_1]x_2$ to $[y_1]y_2$ of the projection of $Q_{{\alpha}}$. \[factor-harmonic\] Every $Q_{{\alpha}}$-harmonic function is of the form $h \circ \pi$, where $h$ is a $P_{{\alpha}}$-harmonic function on ${\mbox{\sl DL}}(q,r)$ and $\pi: {\mbox{\sl DL}}^s(q,r) \to {\mbox{\sl DL}}(q,r)$ is the factor map with respect to the sibling relation. \[sws-example\] Our final task is to retranslate once more to the lamplighter group, by giving a direct description of the minimal harmonic functions for the switch-walk-switch model that does not involve the above factor map. We have $Q = Q_{1/2}$ on ${\mbox{\sl DL}}^s(q,q)$. Now, it is clear what the factor map does to a pair $(\eta,k) \in {\mathbb Z}_q \wr {\mathbb Z}\,$: it “forgets” (cancels) the value $\eta(k)$, and what remains is the pair $(\eta_{\not \,k}, k)$, where $$\eta_{\not \,k}(n) = \begin{cases} \eta(n-1) & \text{if}\; n \le k\,,\\ \eta(n) & \text{if}\;n > k\,. \end{cases}$$ Thus, with respect to the computations of Example \[SRW-example\], (\[negative-defect\]) remains unchanged, while instead of the positive defect we need $$\label{new-positive-defect} \begin{aligned} {\operatorname{\sf def}}^{\oplus}\bigl((\eta,k),\xi_1) &= {\mathfrak{h}}([x_1] {\curlywedge}\xi_1) - {\mathfrak{h}}([o_1] {\curlywedge}\xi_1)\,, \quad\text{where}\\[6pt] {\mathfrak{h}}([x_1]{\curlywedge}\xi_1) &= \begin{cases} \min \{ n \le k : \xi_1(n) \ne \eta(n) \} & \text{if such $n$ exists,} \\ k &\text{otherwise,} \end{cases} \\[4pt] {\mathfrak{h}}([o_1]{\curlywedge}\xi_1) &= \begin{cases} \min \{ m \le 0 : \xi_1(m) \ne 0 \} & \text{if such $m$ exists,} \\ 0 &\text{otherwise.} \end{cases} \end{aligned}$$ Again, the constant function $\mathbf 1$ is minimal $Q$-harmonic. All non-constant minimal $Q$-harmonic functions are given by $$\begin{aligned} K_1(\cdot, \xi_1) &= q^{\mbox{\footnotesize ${\operatorname{\sf def}}^{\oplus}(\cdot,\xi_1)$}}\,,\;\xi_1 \in {\partial}^+({\mathbb Z}_q\wr{\mathbb Z})\,, {\quad\mbox{and}\quad}\\ K_2(\cdot,\xi_2) &= q^{\mbox{\footnotesize ${\operatorname{\sf def}}^-(\cdot,\xi_2)$}}\,,\;\xi_2 \in {\partial}^-({\mathbb Z}_q\wr{\mathbb Z})\,. \end{aligned}$$ Final observations and speculations {#final} =================================== Even when $r \ne q$, one can interpret ${\mbox{\sl DL}}(q,r)$ as a “lamplighter graph” over ${\mathbb Z}\,$: at each point of ${\mathbb Z}$, there are green lamps with $q$ different states, including “off”, and red lamps with $r$ different states, again including “off”. The lamplighter walking along ${\mathbb Z}$ has to make sure that when his actual position is $k \in {\mathbb Z}$, then the lamps in $(-\infty\,,\,k]$ have to be in one of the green states, and those in $[k+1\,,\,+\infty)$ in one of the red states. Several basic properties of random walks on ${\mbox{\sl DL}}(q,r)$ that are not necessarily of nearest neighbour type, but invariant under a transitive group of automorphisms of ${\mbox{\sl DL}}(q,r)$, were studied by [Bertacchi]{} [@Ber]. For a large class of random walks of this type, the Poisson boundary was determined by [Kaimanovich and Woess]{} [@KaiWoe], as mentioned above. As a general principle, the three problems of (i) determining the Poisson boundary ($\equiv$ all bounded harmonic functions), (ii) determining all minimal harmonic functions ($\equiv$ all positive harmonic functions), and (iii) determining the full Martin compactification ($\equiv$ finding the directions of convergence of the Martin kernels) have an increasing degree of difficulty. (As a matter of fact, these three problems get sometimes mixed up even by advanced non-experts.) Thus, the reader should not be a priori astonished by the fact that in this paper, we were able to solve problem (ii) for a much smaller class of random walks than those for which (i) was solved in [@KaiWoe]. In particular, to the author’s knowledge, the present results provide the first example of a complete solution of problem (ii) on a finitely generated, solvable group. In addition, our group is non-polycyclic. On the other hand, the situation is much better understood for connected solvable Lie groups, because more structure theory is at hand. For the basic example, namely the *affine group* over ${\mathbb R}$, random walks and harmonic functions were studied in much detail, see [Molchanov]{} [@Mol], [Elie]{} [@Eli], or [Bougerol and Elie]{} [@BouEli]. We recall here that the main result of [@BouEli] implies existence of non-constant positive harmonic functions for finite range random walks on finitely generated polycyclic groups with exponential growth, but a complete solution of (ii) is not available for those groups. As for the affine group over ${\mathbb R}$, the Poisson boundary and the Martin compactification are well understood for random walks on the affine group over the $p$-adic numbers, see [Cartwright, Kaimanovich and Woess]{} [@CarKaiWoe] and [Brofferio]{} [@Bro]. As pointed out by [Bertacchi]{} [@Ber], there is a natural geometric compactification of ${\mbox{\sl DL}}(q,r)$. Namely, this graph is a subgraph of the direct product ${\mathbb T}_q \times {\mathbb T}_r$, for which a natural compactification is ${\widehat}{\mathbb T}_q \times {\widehat}{\mathbb T}_r$. Thus, we define ${\widehat}{\mbox{\sl DL}}(q,r)$ as the closure of ${\mbox{\sl DL}}(q,r)$ in ${\widehat}{\mathbb T}_q \times {\widehat}{\mathbb T}_r$, and ${\partial}{\mbox{\sl DL}}(q,r) = {\widehat}{\mbox{\sl DL}}(q,r) \setminus {\mbox{\sl DL}}(q,r)$. Almost sure convergence of random walks to a boundary-valued random variable is studied in [@Ber]. However, at present, we are far from proving that this compactification is in some sense comparable or identical with the Martin compactification even in the case of simple random walk. The following seems noteworthy regarding the two classes of examples that we have studied: in the case of drift (${\alpha}\ne 1/2$), the minimal harmonic functions are parametrized (continuously on each part) by ${\partial}^*{\mathbb T}_q \cup {\partial}^*{\mathbb T}_r$. In the drift-free case (${\alpha}= 1/2$), the *additional* minimal harmonic function $\mathbf 1$ enters the stage. Thus, in some sense, the cone of positive harmonic functions is *bigger* in the driftfree case than in presence of drift. This contrasts with all examples known so far. (Of course, the constant function $\mathbf 1$ is a Martin kernel even when ${\alpha}\ne 1/2$, but then it does not belong to ${\mathcal M}_{\min}$.) Finally, the reason why our method does not adapt to the “walk or switch” model (\[walk-or-switch\]) appears to be that in contrast with the cases that we have solved here, this random walk is *not* invariant under a group of automorphisms of ${\mbox{\sl DL}}(q,r)$ that acts *transitively* both on ${\partial}^*{\mathbb T}^1$ and ${\partial}^*{\mathbb T}^2$. A next step could be to try to prove the Decomposition Theorem \[split-theorem\] for all irreducible random walks with the latter transitivity property. [**Acknowledgement.**]{} The author is grateful to Rögnvaldur I. Möller for pointing out the relationship between the Diestel-Leader graphs ${\mbox{\sl DL}}(q,q)$ and the lamplighter groups. [22]{} Bertacchi, D.: *Random walks on Diestel-Leader graphs,* Abh. Math. Sem. Univ. Hamburg [**71**]{} (2001) 205–224. Bougerol, Ph., and Elie, L.: *Existence of non-negative harmonic functions on groups and on covering manifolds,* Ann. Inst. Poincaré (Probab. Stat.) [**31**]{} (1995) 59–80. Brofferio, S.: *Marches aléatories sur les groupes affines de l’arbre et de la droite réelle et processus localment contractifs,* Thèse, Univ. Paris 6 (2002). Cartier, P.: *Fonctions harmoniques sur un arbre,* Symposia Math. [**9**]{} (1972) 203–270. Cartwright, D. I., Kaimanovich, V. A., and Woess, W.: *Random walks on the affine group of local fields and of homogeneous trees,* Ann. Inst Fourier (Grenoble) [**44**]{} (1994) 1243–1288. Diestel, R., and Leader, I.: *A conjecture concerning a limit of non-Cayley graphs,* J. Algebraic Combin. [**14**]{} (2001) 17–25. Elie, L.: *Fonctions harmoniques positives sur le groupe affine,* in: *Probability Measures on Groups,* (ed. H. Heyer.) pp. 96–110, Lect. Notes in Math. [**706**]{}, Springer, Berlin, 1978. Erschler, A. G.: *On the asymptotics of the rate of departure to infinity* (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) [**283**]{} (2001) 251–257, 263. Grigorchuk, R. I., and Żuk, A.: *The lamplighter group as a group generated by a 2-state automaton, and its spectrum,* Geom. Dedicata [**87**]{} (2001) 209–244. Kaimanovich, V. A.: *Poisson boundaries of random walks on discrete solvable groups,* in: *Probability Measures on Groups X* (ed. H. Heyer), pp. 205–238, Plenum, New York, 1991. Kaimanovich, V. A., and Vershik, A. M.: *Random walks on discrete groups: boundary and entropy,* Ann. Probab. [**11**]{} (1983) 457–490. Kaimanovich, V. A., and Woess, W.: *Boundary and entropy of space homogeneous Markov chains,* Ann. Probab. [**30**]{} (2002) 323-363. Lyons, R., Pemantle, R., and Peres, Y.: *Random walks on the lamplighter group,* Ann. Probab. [**24**]{} (1996) 1993–2006. Molchanov, S. A.: *Martin boundaries for invariant Markov processes on a solvable group,* Theory of Probability and Appl. [**12**]{} (1967) 310–314. Picardello, M. A., Taibleson, M. H., and Woess, W.: *Harmonic functions on Cartesian products of trees with finite graphs,* J. Functional Analysis [**102**]{} (1991) 379–400. Pittet, C., and Saloff-Coste, L.: *Amenable groups, isoperimetric profiles and random walks,* in: *Geometric Group Theory Down Under* (Canberra, 1996), pp. 293–316, de Gruyter, Berlin, 1999. Pittet, C., and Saloff-Coste, L.: *On random walks on wreath products,* Ann. Probab. [**30**]{} (2002) 948–977. Revelle, D.: *Rate of escape of random walks on wreath products,* Ann. Probab. [**31**]{} (2003), to appear. Revelle, D.: *Heat kernel asymptotics on the lamplighter group,* preprint, Cornell Univ. (2002). Woess, W.: *Random Walks on Infinite Graphs and Groups*, Cambridge Tracts in Mathematics [**138**]{}, Cambridge University Press, Cambridge, 2000. [^1]: Partially supported by FWF (Austrian Science Fund) project P15577

--- abstract: 'For a gauge theory which includes a light massive vector field interacting with the familiar photon $U(1)_{QED}$ via a Chern-Simons- like coupling, we study the static quantum potential. Our analysis is based on the gauge-invariant, but path-dependent, variables formalism. The result is that the theory describes an exactly screening phase. Interestingly enough, this result displays a marked departure of a qualitative nature from the axionic elctrodynamics result. However, the present result is analogous to that encountered in the coupling between the familiar photon $U(1)_{QED}$ and a second massive gauge field living in the so-called hidden-sector $U(1)_h$, inside a superconducting box.' author: - Patricio Gaete$^1$ - 'José A. Helayel-Neto$^2$' - Euro Spallucci$^3$ title: 'Some aspects of a Chern-Simons-like coupling in an external magnetic field' --- Introduction ============ Nowadays, one of the most actively pursued areas of research in physics consists of the investigation of extensions of the Standard Model (SM) such as axion-like particles and light extra hidden $U(1)$ gauge bosons, in order to explain cosmological and astrophysical results. These hidden $U(1)$ gauge bosons are frequently encountered in string theories physics. Also, this subject has had a revival after recent results of the PVLAS collaboration [@Zavattini; @Cameron; @Chen; @Zavattini2; @Robilliard; @Chou; @Ahlers; @Ahlers2; @Popov; @Popov2; @Chelouche; @Gninenko; @Jaeckel]. Nevertheless, although none of these searches ultimately has yielded a positive signal, the arguments in favor of the existence of axion-like particles or light extra hidden $U(1)$ gauge bosons, remain as cogent as ever. In this perspective, it is useful to recall that the axion-like scenario can be qualitatively understood by the existence of light pseudoscalar bosons $\phi$ (“axions”), with a coupling to two photons. In other terms, the interaction term in the effective Lagrangian has the form $\mathcal{L}_I = - \frac{1}{4}F_{\mu \nu } \tilde F^{\mu \nu } \phi$, where $\tilde F^{\mu \nu } = \frac{1}{2}\varepsilon _{\mu \nu \lambda \rho} F^{\lambda \rho }$. However, the crucial feature of axionic electrodynamics is the mass generation due to the breaking of rotational invariance induced by a classical background configuration of the gauge field strength [@Spallucci], which leads to confining potentials in the presence of nontrivial constant expectation values for the gauge field strength $F_{\mu \nu}$ [@GaeteGuen]. In fact, in the case of a constant electric field strength expectation value the static potential remains Coulombic, while in the case of a constant magnetic field strength expectation value the potential energy is the sum of a Yukawa and a linear potential, leading to the confinement of static charges. Also it is important to point out that the magnetic character of the field strength expectation value needed to obtain confinement is in agreement with the current chromo-magnetic picture of the $QCD$ vacuum [@Savvidy]. In addition, similar results have been obtained in the context of the dual Ginzburg-Landau theory [@Suganuma], as well as for a theory of antisymmetric tensor fields that results from the condensation of topological defects as a consequence of the Julia-Toulouse mechanism [@GaeteW]. In a general perspective, we draw attention to the fact that much of this work has been inspired by studies coming from the realms of string theory [@Jaeckel2; @Batell] and quantum field theory [@Holdom; @Nath; @Nath2; @Castelo; @Masso; @Foot; @Singleton; @Langacker]. Indeed, as was observed in [@Singleton2], the introduction of a second gauge field in addition to the usual photon was pioneered in Ref. [@Cabibbo], in the context of electrodynamics in the presence of magnetic monopoles [@Dirac]. Whereas the quantization for a system with two photons was later carried out in [@Hagen]. The possible existence of massive vector fields was also proposed in [@Okun]. A kinetic term between the familiar photon $U(1)_{QED}$ and a second gauge field has also been considered in order to explain recent unexpected observations in high energy astrophysics [@Nima]. We further note that recently another possible candidate for extensions of the SM has been studied [@Antoniadis]. It is the so-called Chern-Simons-like coupling scenario, which includes a light massive vector field interacting with the familiar photon $U(1)_{QED}$ via a Chern-Simons-like coupling. As a result, it was argued that this new model reproduces the effects of rotation of the polarization plane. Given the relevance of these studies, it is of interest to improve our understanding of the physical consequences presented by this new scenario (Chern-Simons-like coupling scenario). Of special interest will be to study the connection or equivalence with the axion-like particles and light extra hidden $U(1)$ gauge bosons scenarios. Thus, our purpose here is to further explore the impact of a light massive vector field in the Chern-Simons-like coupling scenario on physical observables. To this end, we will study the screening and confinement issue. This issue is generally not discussed. Our calculation is accomplished by making use of the gauge-invariant but path-dependent variables formalism along the lines of Ref. [@GaeteS; @GaeteJPA; @GaeteSpa09; @GaeteHel09], which is a physically-based alternative to the usual Wilson loop approach. As we shall see, in the case of a constant magnetic field the theory describes an exactly screening phase. This then implies that the static potential profile obtained from both the Chern-Simons-like coupling and axionic electrodynamics models are quite different. This means that the two theories are not equivalent. As it was shown in [@GaeteGuen], axionic electrodynamics has a different structure which is reflected in a confining piece, which is not present in the Chern-Simons-like coupling scenario. Incidentally, the above static potential profile (Chern-Simons-like coupling scenario) is similar to that encountered in the coupling between the familiar massless electromagnetism $U(1)_{QED}$ and a hidden-sector $U(1)_h$ inside a superconducting box [@Gaete0410]. In this way one obtains a new connection among different models describing the same physical phenomena. In this connection, the benefit of considering the present approach is to provide unifications among different models, as well as exploiting the equivalence in explicit calculations, as we shall see in the course of the discussion. Interaction energy {#s2} ================== We now discuss the interaction energy between static point-like sources for the model under consideration. To carry out such study, we will compute the expectation value of the energy operator $H$ in the physical state $|\Phi\rangle$ describing the sources, which we will denote by $ {\langle H\rangle}_\Phi$. As anticipated above, the gauge theory we are considering describes the interaction between the familiar massive photon $U(1)_{QED}$ with a light massive vector field via a Chern-Simons- like coupling. In this case the corresponding theory is governed by the Lagrangian density [@Antoniadis]: $$\begin{aligned} \mathcal{L} &=& - \frac{1}{4}F_{\mu \nu }^2 (A) - \frac{1}{4}F_{\mu \nu }^2 (B) + \frac{{m_\gamma ^2 }}{2}A_\mu ^2 + \frac{{m_B^2 }}{2}B_\mu ^2 \nonumber\\ &-& \frac{{\kappa }}{2}\varepsilon ^{\mu \nu \lambda \rho } A_\mu B_\nu F_{\lambda \rho } (A), \label{Csmag05}\end{aligned}$$ where $m_\gamma$ is the mass of the photon, and $m_{B}$ represents the mass for the gauge boson $B$. In particular, this alternative theory exhibits an effective mass for the component of the photon along the direction of the external magnetic field, exactly as it happens with axionic electrodynamics. If we consider the model in the limit of a very heavy $B$-field ($m_{B} \gg m_\gamma$) and we are bound to energies much below $m_{B}$, we are allowed to integrate over $B_\mu$ and to speak about an effective model for the $A_\mu$-field. Then, the first crucial point is that by eliminating the gauge field $B_\mu$ in terms of $A_\mu$ in the original Lagrangian (\[Csmag05\]) one gets an effective Lagrangian $\mathcal{L}_{eff}$. This is accomplished by making use of the following shifting for the $B_\mu$-field: $$B_\mu \equiv \tilde B_\mu + \frac{\kappa }{2}\frac{1} {{\left( {\Delta + m_B^2 } \right)}}\left[ {\eta _{\mu \nu } + \frac{{\partial _\mu \partial _\nu }}{{m_B^2 }}} \right]\varepsilon ^{\mu \lambda \rho \nu } A_\mu F_{\lambda \rho } \left( A \right). \label{Csmag10}$$ Once this is done, and dropping the constant factor which emerges by integrating out the $\tilde B_\mu$-field, we arrive at the following effective Lagrangian density $$\begin{aligned} \mathcal{L}_{eff} &=& - \frac{1}{4}F_{\mu \nu }^2 + \frac{{m_\gamma ^2 }} {2}A_\mu A^\mu + \frac{{\kappa ^2 }}{4}A_\alpha F_{\beta \gamma } \frac{1} {{\left( {\Delta + m_B^2 } \right)}}A^\alpha F^{\beta \gamma } \nonumber\\ &+& \frac{{\kappa ^2 }}{2}A_\alpha F_{\beta \gamma } \frac{1}{{\left( {\Delta + m_B^2 } \right)}}A^\gamma F^{\alpha \beta } \nonumber\\ &+& \frac{{\kappa ^2 }}{8}\tilde F^{\alpha \beta } F_{\alpha \beta } \frac{1} {{m_B^2 \left( {\Delta + m_B^2 } \right)}}\tilde F^{\gamma \delta } F_{\gamma \delta }, \label{Csmag15}\end{aligned}$$ where ${\widetilde F}_{\mu \nu } \equiv {\raise0.7ex\hbox{$1$} \!\mathord{\left/{\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}\varepsilon _{\mu \nu \lambda \rho } F^{\lambda \rho }$. The same result can be obtained by integrating out the $B$ field in a path integral formulation of the model. The integral is gaussian in $B$ and can be exactly computed leading to the effective Lagrangian (\[Csmag15\]). Before going ahead, we would like to note that from Eq. (\[Csmag15\]) the gauge invariance is broken and one could argue about the possibility of getting a gauge invariant result for the static potential between test charges from (\[Csmag15\]). There are at least two available options to solve this apparent inconsistency. One way is to restore gauge invariance by inserting Stuckelberg compensating fields either into (\[Csmag05\]) or into (\[Csmag15\]). Once the compensators are integrated out the resulting model is explicitly gauge invariant. Unfortunately, this procedure introduces non-local effective interaction terms which are difficult to handle. In alternative we shall follow the Hamiltonian formulation discussed below. As a second point, if we wish to study quantum properties of the electromagnetic field in the presence of external electric and magnetic fields, we should split the $A_\mu$-field as the sum of a classical background, $\langle A_\mu \rangle$, and a small quantum fluctuation, $a_\mu$, $$A_\mu = \langle A_\mu \rangle + a_\mu. \label{Csmag20}$$ Therefore the previous Lagrangian density, up to quadratic terms in the fluctuations, is also expressed as $$\begin{aligned} {\cal L}_{eff} &=& - \frac{1}{4}f_{\mu \nu } \Omega f^{\mu \nu} + \frac{1}{2}a_\mu M^2 a^\mu -\frac{{\kappa ^2 }}{2}f_{\gamma \beta }\left\langle {A^\gamma } \right\rangle \frac{{1}}{{\left( {\Delta + m_B^2 } \right)}} \left\langle {A_\alpha } \right\rangle f^{\alpha \beta } + \frac{{\kappa ^2 }}{8}f_{\mu \nu } v^{\mu \nu } \frac{1} {{m_B^2 \left( {\Delta + m_B^2 } \right)}}v^{\lambda \rho } f_{\lambda \rho} \nonumber\\ &-& \kappa ^2 \left( {\varepsilon ^{jk0i} v_{0i} \left\langle {A_j } \right\rangle a^m \frac{1}{{\left( {\Delta + m_B^2 } \right)}}f_{km} } \right) + \kappa ^2 \left( {\varepsilon ^{jk0i} v_{0i} \left\langle {A^m } \right\rangle a_k \frac{1}{{\left( {\Delta + m_B^2 } \right)}}f_{jm} } \right) \nonumber\\ &-& \frac{{\kappa ^2 }}{2}\left( {\varepsilon ^{jk0i} v_{0i} \left\langle {A^l } \right\rangle a_l \frac{1}{{\left( {\Delta + m_B^2 } \right)}}f_{jk} } \right) , \label{Csmag25}\end{aligned}$$ where $f_{\mu \nu } = \partial _\mu a_\nu - \partial _\nu a_\mu$, and $\Delta \equiv \partial_\mu\partial^\mu$. $\Omega \equiv 1 - \kappa ^2 \frac{{\left\langle {A^i } \right\rangle \left\langle {A_i } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}$, and $M^2 \equiv m_\gamma ^2 + \frac{{\kappa ^2 }}{2}\frac{{v_{0i} v^{0i} }}{{\left( {\Delta + m_B^2 } \right)}}$. In the above Lagrangian we have considered the $v^{0i}\neq0$ and $v^{ij}=0$ case (referred to as the magnetic one in what follows), and simplified our notation by setting $\varepsilon ^{\mu \nu \alpha \beta } \left\langle{F_{\mu \nu } } \right\rangle \equiv v^{\alpha \beta }$. As a consequence, the Lagrangian (\[Csmag25\]) becomes a Maxwell-Proca-like theory with a manifestly Lorentz violating term. This effective theory provide us with a suitable starting point to study the interaction energy. However, before proceeding with the determination of the energy, it is necessary to restore the gauge invariance in (\[Csmag25\]). For this end we now carry out a Hamiltonian analysis. The canonically conjugate momenta are found to be $\Pi ^\mu = - \left[ {1 - \kappa ^2 \frac{{\left\langle {A^k } \right\rangle \left\langle {A_k } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}} \right]f^{0\mu } + \kappa ^2 \frac{{\left\langle {A^\mu } \right\rangle \left\langle {A_i } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}f^{i0} - \frac{{\kappa ^2 }}{{4m_B^2 }}v^{\mu 0} \frac{1}{{\left( {\Delta + m_B^2 } \right)}}v^{0i} f_{0i}$. This yields the usual primary constraint $\Pi^{0}=0$, while the momenta are $\Pi^i = - \left[ {1 - \kappa ^2 \frac{{\left\langle {A^k } \right\rangle \left\langle {A_k } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}} \right]f^{0i} + \kappa ^2 \frac{{\left\langle {A^i} \right\rangle \left\langle {A_k } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}f^{k0} - \frac{{\kappa ^2 }}{{4m_B^2 }}v^{i0} \frac{1}{{\left( {\Delta + m_B^2 } \right)}}v^{0k} f_{0k}$. This leads us to the canonical Hamiltonian, $$\begin{aligned} H_C &=& - \int {d^3 x} a_0 \left( {\partial _i \Pi ^i + \frac{1}{2}\left\{ {m_\gamma ^2 - \frac{{{\raise0.7ex\hbox{${\kappa ^2 {\bf v}^2 }$} \!\mathord{\left/ {\vphantom {{\kappa ^2 v^2 } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}{{\left( {\Delta + m_B^2 } \right)}}} \right\}a^0 } \right) + \int {d^3 x} \left( { - \frac{1}{2}a_i \left\{ {m_\gamma ^2 - \frac{{{\raise0.7ex\hbox{${\kappa ^2 {\bf v}^2 }$} \!\mathord{\left/ {\vphantom {{\kappa ^2 v^2 } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}{{\left( {\Delta + m_B^2 } \right)}}} \right\}a^i } \right) \nonumber\\ &+& \int {d^3 x} \left( {\frac{1}{2}\Pi _i \left\{ {1 - \frac{{\kappa ^2 \left\langle {\bf A} \right\rangle ^2 }}{{\left( {\Delta + m_B^2 } \right)}}} \right\}\Pi ^i } \right) + \int {d^3 x} \left( {\frac{{\kappa ^2 }}{2}\frac{1}{{\left( {\Delta + m_B^2 } \right)}}\left( {\left\langle {\bf A} \right\rangle \cdot {\bf \Pi }} \right)^2 } \right) \nonumber\\ &+& \int {d^3 x} \left( {\frac{1}{2}B_i \left\{ {1 + \frac{{\kappa ^2 \left\langle {\bf A} \right\rangle ^2 }}{{\left( {\Delta + m_B^2 } \right)}}} \right\}B^i } \right) + \int {d^3 x} \left\{ { - \frac{1}{2}\frac{{\kappa ^2 }}{{\left( {\Delta + m_B^2 } \right)}}\left( {\varepsilon _{ikj} \left\langle {A^k } \right\rangle B^j } \right)^2 } \right\} \nonumber\\ &+& \int {d^3 x} \left( {\kappa ^2 \varepsilon ^{jk0i} v_{0i} \left\langle {A_j } \right\rangle a^m \frac{1}{{\left( {\Delta + m_B^2 } \right)}}f_{km} } \right) - \int {d^3 x} \left( {\kappa ^2 \varepsilon ^{jk0i} v_{0i} \left\langle {A^m } \right\rangle a_k \frac{1}{{\left( {\Delta + m_B^2 } \right)}}f_{jm} } \right) \nonumber\\ &+& \int {d^3 x} \left( {\frac{{\kappa ^2 }}{2}\varepsilon ^{jk0i} v_{0i} \left\langle {A^l } \right\rangle a_l \frac{1}{{\left( {\Delta + m_B^2 } \right)}}f_{jk} } \right) , \label{Csmag30}\end{aligned}$$ where ${\bf v}$ stands for the external magnetic field ($v^{0i}$) and $B^i$ is now the magnetic field associated to the fluctuation, namely, $B^i\equiv\epsilon^{ijk} f_{jk}$. Time conservation of the primary constraint $ \Pi_0=0$ yields the following secondary constraint $\Gamma \left( x \right) \equiv \partial _i \Pi ^i + \left( {m_\gamma ^2 - \frac{{\kappa ^2 {\bf v}^2 }} {2}\frac{1}{{\left( {\Delta + m_B^2 } \right)}}} \right)a^0 = 0$. Notice that the nonvanishing bracket $\left\{ {\Pi ^0 ,\partial _i \Pi ^i + \left( {m_\gamma ^2 - \frac{{\kappa ^2 {\bf v}^2 }}{2}\frac{1}{{\left( {\Delta + m_B^2 } \right)}}} \right)a^0 } \right\}$ shows that the above pair of constraints are second class constraints, as expected for a theory with an explicit mass term which breaks the gauge invariance. To convert the second class system into first class we enlarge the original phase space by introducing a canonical pair of fields $\theta$ and $\Pi _\theta$ [@GaeteSpa09]. It follows, therefore, that a new set of first class constraints can be defined in this extended space: $$\Lambda _1 = \Pi _0 + \left( {m_\gamma ^2 - \frac{{\kappa ^2 {\bf v}^2 }}{2}\frac{1}{{\left( {\Delta + m_B^2 } \right)}}} \right)a^0, \label{Csmag35a}$$ and $$\Lambda _2 \equiv \Gamma + \Pi _\theta. \label{Csmag35b}$$ In this way the gauge symmetry of the theory under consideration has been restored. Then, the new effective Lagrangian, after integrating out the $\theta$-field, becomes $$\begin{aligned} \mathcal{L}_{eff} &=& - \frac{1}{4}f_{\mu \nu } \left[ {\frac{{\left( {\Delta ^2 + a^2 \Delta + b^2 } \right)}}{{\Delta \left( {\Delta + m_B^2 } \right)}}} \right]f^{\mu \nu } -\left\langle {A^i } \right\rangle f_{i0} \frac{{1}} {{\left( {\Delta + m_B^2 } \right)}} \left\langle {A_k } \right\rangle f^{k0} - \frac{{\kappa ^2 }}{2}f_{ki} \left\langle {A^k } \right\rangle \frac{{1}} {{\left( {\Delta + m_B^2 } \right)}} \left\langle {A_l } \right\rangle f^{li} \nonumber\\ &+& \frac{{\kappa ^2 }}{8}v^{0i} f_{0i} \frac{1}{{m_B^2 \left( {\Delta + m_B^2 } \right)}}v^{0k} f_{0k}, \label{Csmag40}\end{aligned}$$ where $a^2 \equiv m_B^2 + m_\gamma ^2 \left( {1 - \kappa ^2 \frac{{\left\langle {A_k } \right\rangle \left\langle {A^k } \right\rangle }} {{m_\gamma ^2 }}} \right)$, and $ b^2 = m_\gamma ^2 \left( {m_B^2 - \frac{{\kappa ^2 {\bf v}^2 }}{{2m_\gamma ^2 }}} \right)$. We observe that to get the above theory we have ignored the last three terms in (\[Csmag30\]) because it add nothing to the static potential calculation, as we will show it below. In other words, the new effective action (\[Csmag30\]) provide us with a suitable starting point to study the interaction energy without loss of physical content. We now turn our attention to the calculation of the interaction energy. In order to obtain the corresponding Hamiltonian, the canonical quantization of this theory from the Hamiltonian analysis point of view is straightforward and follows closely that of our previous work [@GaeteSpa09; @GaeteHel09]. The canonical momenta read $\Pi ^\mu = - \left( {\frac{{\Delta ^2 + a^2 \Delta + b^2 }} {{\Delta \left( {\Delta + m_B^2 } \right)}}} \right)f^{0\mu } + \kappa ^2 \frac{{\left\langle {A^\mu } \right\rangle \left\langle {A_k } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}f^{k0} \\ + \frac{{\kappa ^2 }}{4}\frac{{v^{0\mu } }}{{m_B^2 }}\frac{1} {{\left( {\Delta + m_B^2 } \right)}}v^{0k} f_{0k}$, and one immediately identifies the usual primary constraint $\Pi ^i = - \left( {\frac{{\Delta ^2 + a^2 \Delta + b^2 }} {{\Delta \left( {\Delta + m_B^2 } \right)}}} \right)f^{0i} + \kappa ^2 \frac{{\left\langle {A^i } \right\rangle \left\langle {A_k } \right\rangle }}{{\left( {\Delta + m_B^2 } \right)}}f^{k0} + \frac{{\kappa ^2 }}{4}\frac{{v^{0i} }}{{m_B^2 }}\frac{1} {{\left( {\Delta + m_B^2 } \right)}}v^{0k} f_{0k}$. The canonical Hamiltonian is thus given by $$\begin{aligned} H_C &=& \int {d^3 x} \left\{ { - a_0 \partial _i \Pi ^i + \frac{1}{2}B^i \frac{{\left( {\Delta ^2 + a^2 \Delta + b^2 } \right)}}{{\Delta \left( {\Delta + m_B^2 } \right)}}B^i } \right\} - \frac{1}{2}\int {d^3 } x\Pi _i \frac{{\left( {\Delta + m_B^2 } \right)}} {{\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)}} \Pi^i \nonumber\\ &+& \frac{{\kappa ^2 }}{2}\int {d^3 x} \Pi _i \left\langle {A^i } \right\rangle \frac{{\left( {\Delta + m_B^2 } \right)}}{{\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)^2}}\left\langle {A^k } \right\rangle \Pi _k + \frac{{\kappa ^2 }}{2}\int {d^3 x} f_{ki} \left\langle {A^k } \right\rangle \frac{1}{{\left( {\Delta + m_B^2 } \right)}}\left\langle {A_l } \right\rangle f^{li} \nonumber\\ &+& \frac{{\kappa ^2 {\bf v}^2 }}{{8m_B^2 }}\int {d^3 x} \Pi _i \frac{{\left( {\Delta + m_B^2 } \right)}}{{\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)^2 }}\Pi ^i ,\label{Csmag45}\end{aligned}$$ where $a^2 = m_B^2 + m_\gamma ^2 + \kappa ^2 \left\langle {\bf A} \right\rangle ^2$ and $b^2 = m_\gamma ^2 m_B^2 + \frac{{\kappa ^2 }}{2}{\bf v}^2$. Since our energies are all much below $m_B$, it is consistent with our considerations to neglect $\kappa ^2 \left\langle {\bf A} \right\rangle ^2$ with respect to $m_B^2$. This implies that $a^2$ and $b^2$ should be taken as: $a^2 = m_B^2$ and $b^2 = m_\gamma ^2 m_B^2 + \frac{{\kappa ^2 }}{2}{\bf v}^2$. The consistency condition $\dot \Pi _0 = 0$ leads to the usual Gauss constraint $\Gamma_1 \left( x \right) \equiv \partial _i \Pi ^i=0$. It is also possible to verify that no further constraints are generated by this theory. Consequently, the extended Hamiltonian that generates translations in time then reads $H = H_C + \int {d^2}x\left( {c_0 \left( x \right)\Pi _0 \left( x \right) + c_1 \left(x\right)\Gamma _1 \left( x \right)} \right)$, where $c_0 \left( x\right)$ and $c_1 \left( x \right)$ are arbitrary Lagrange multipliers. Moreover, it follows from this Hamiltonian that $\dot{a}_0 \left( x \right)= \left[ {a_0 \left( x \right),H} \right] = c_0 \left( x \right)$, which is completely arbitrary. Since $ \Pi^0 = 0$ always, neither $a^0$ and $\Pi^0$ are of interest in describing the system and may be discarded from the theory. If a new arbitrary coefficient $c(x) = c_1 (x) - A_0 (x)$ is introduced the Hamiltonian may be rewritten as $$\begin{aligned} H &=& \int {d^3 x} \left\{ {c(x)\partial _i \Pi ^i + \frac{1}{2}B^i \frac{{\left( {\Delta ^2 + a^2 \Delta + b^2 } \right)}}{{\Delta \left( {\Delta + m_B^2 } \right)}}B^i } \right\} - \frac{1}{2}\int {d^3 } x\Pi _i \frac{{\left( {\Delta + m_B^2 } \right)}}{{\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)}}\Pi^i \nonumber\\ &+& \frac{{\kappa ^2 }}{2}\int {d^3 x} \Pi _i \left\langle {A^i } \right\rangle \frac{{\left( {\Delta + m_B^2 } \right)}}{{\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)^2}}\left\langle {A^k } \right\rangle \Pi _k + \frac{{\kappa ^2 }}{2}\int {d^3 x} f_{ki} \left\langle {A^k } \right\rangle \frac{1}{{\left( {\Delta + m_B^2 } \right)}}\left\langle {A_l } \right\rangle f^{li} \nonumber\\ &+& \frac{{\kappa ^2 {\bf v}^2 }}{{8m_B^2 }}\int {d^3 x} \Pi _i \frac{{\left( {\Delta + m_B^2 } \right)}}{{\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)^2 }}\Pi ^i. \label{Csmag50}\end{aligned}$$ We can at this stage impose a subsidiary on the vector potential such that the full set of constraints become second class. A particularly convenient choice is found to be $$\Gamma _2 \left( x \right) \equiv \int\limits_{C_{\xi x} } {dz^\nu } a_\nu \left( z \right) \equiv \int\limits_0^1 {d\lambda x^i } a_i \left( {\lambda x} \right) = 0, \label{Csmag55}$$ where $\lambda$ $(0\leq \lambda\leq1)$ is the parameter describing the spacelike straight path $ x^i = \xi ^i + \lambda \left( {x - \xi } \right)^i $, and $ \xi $ is a fixed point (reference point). There is no essential loss of generality if we restrict our considerations to $ \xi ^i=0 $. The choice (\[Csmag55\]) leads to the Poincaré gauge [@Gaete99]. As a consequence, we can now write down the only nonvanishing Dirac bracket for the canonical variables $$\begin{aligned} \left\{ {a_i \left( x \right),\Pi ^j \left( y \right)} \right\}^ * &=&\delta{ _i^j} \delta ^{\left( 2 \right)} \left( {x - y} \right) \nonumber\\ &-& \partial _i^x \int\limits_0^1 {d\lambda x^j } \delta ^{\left( 2 \right)} \left( {\lambda x - y} \right). \label{Csmag60}\end{aligned}$$ At this point, we have all the elements necessary to find the interaction energy between point-like sources for the model under consideration. As we have already indicated, we will calculate the expectation value of the energy operator $H$ in the physical state $|\Phi\rangle$. In this context, we recall that the physical state $|\Phi\rangle$ can be written as $$\begin{aligned} \left| \Phi \right\rangle & \equiv & \left| {\overline \Psi \left( \bf y \right)\Psi \left( {\bf 0} \right)} \right\rangle \nonumber\\ &=& \overline \psi \left( \bf y \right)\exp \left( {iq\int\limits_{{\bf 0}}^{\bf y} {dz^i } a_i \left( z \right)} \right)\psi \left({\bf 0} \right)\left| 0 \right\rangle, \label{Csmag65}\end{aligned}$$ where $\left| 0 \right\rangle$ is the physical vacuum state. The line integral is along a spacelike path starting at $\bf 0$ and ending at $\bf y$, on a fixed time slice. Notice that the charged matter field together with the electromagnetic cloud (dressing) which surrounds it, is given by $\Psi \left( {\bf y} \right) = \exp \left( { - iq\int_{C_{{\bf \xi} {\bf y}} } {dz^\mu A_\mu (z)} } \right)\psi ({\bf y})$. Thanks to our path choice, this physical fermion then becomes $\Psi \left( {\bf y} \right) = \exp \left( { - iq\int_{\bf 0}^{\bf y} {dz^i } A_{i} (z)} \right)\times \psi ({\bf y})$. In other terms, each of the states ($\left| \Phi \right\rangle$) represents a fermion-antifermion pair surrounded by a cloud of gauge fields to maintain gauge invariance. Further, by taking into account the structure of the Hamiltonian above, we observe that $$\begin{aligned} \Pi _i \left( x \right)\left| {\overline \Psi \left( \bf y \right)\Psi \left( {{\bf y}^ \prime } \right)} \right\rangle &=& \overline \Psi \left( \bf y \right)\Psi \left( {{\bf y}^ \prime } \right)\Pi _i \left( x \right)\left| 0 \right\rangle \nonumber\\ &+& q\int_ {\bf y}^{{\bf y}^ \prime } {dz_i \delta ^{\left( 3 \right)} \left( {\bf z - \bf x} \right)} \left| \Phi \right\rangle.\nonumber\\ \label{Csmag65b}\end{aligned}$$ Having made this observation and since the fermions are taken to be infinitely massive (static sources), we can substitute $\Delta$ by $-\nabla^{2}$ in Eq. (\[Csmag50\]). Therefore, the expectation value $\left\langle H \right\rangle _\Phi$ is expressed as $$\left\langle H \right\rangle _\Phi = \left\langle H \right\rangle _0 + \left\langle H \right\rangle _\Phi ^{\left( 1 \right)} + \left\langle H \right\rangle _\Phi ^{\left( 2 \right)}, \label{Csmag70}$$ where $\left\langle H \right\rangle _0 = \left\langle 0 \right|H\left| 0 \right\rangle$. The $\left\langle H \right\rangle _\Phi ^{\left( 1 \right)}$ and $\left\langle H \right\rangle _\Phi ^{\left( 2 \right)}$ terms are given by $$\begin{aligned} \left\langle H \right\rangle _\Phi ^{\left( 1 \right)} &=& - \frac{{b^2 B}}{2} \int {d^3 x} \left\langle \Phi \right|\Pi _i \frac{{\nabla ^2 }} {{\left( {\nabla ^2 - M_1^2 } \right)}}\Pi ^i \left| \Phi \right\rangle +\frac{{b^2 B}}{2}\int {d^3 x} \left\langle \Phi \right|\Pi _i \frac{{M_2^2 }} {{M_1^2 }}\frac{{\nabla ^2 }}{{\left( {\nabla ^2 - M_2^2 } \right)}} \Pi ^i \left| \Phi \right\rangle, \nonumber\\ \label{Csmag75a}\end{aligned}$$ and $$\begin{aligned} \left\langle H \right\rangle _\Phi ^{\left( 2 \right)} &=& m_B^2 b^2 B\int {d^3 x} \left\langle \Phi \right|\Pi _i \frac{1}{{\left( {\nabla ^2 - M_1^2 } \right)}} \Pi ^i \left| \Phi \right\rangle - m_B^2 b^2 B\int {d^3 x} \left\langle \Phi \right|\Pi _i \frac{{M_2^2 }} {{M_1^2 }}\frac{1}{{\left( {\nabla ^2 - M_2^2 } \right)}}\Pi ^i \left| \Phi \right\rangle, \nonumber\\ \label{Csmag75b}\end{aligned}$$ where $B = \frac{1}{{M_2^2 \left( {M_2^2 - M_1^2 } \right)}}$, $M_1^2 \equiv \frac{{a^2 }}{2}\left[ {1 + \sqrt {1 - \frac{{4b^2 }} {{a^4 }}} } \right]$, and $M_2^2 \equiv \frac{{a^2 }} {2}\left[ {1 - \sqrt {1 - \frac{{4b^2 }}{{a^4 }}} } \right]$. We have neglected the terms in (\[Csmag50\]) where $\left( {\Delta + a^2 + \frac{{b^2 }}{\Delta }} \right)^2$ appears in the denominator, the reason being that we wish to compute an interparticle potential, which expresses the effects of photons exchange in the low-energy (or low-frequency) limit. Therefore, these terms we are mentioning are suppressed in view of the presence of higher power of the frequency in the denominator. Another important point to be highlighted in our discussion comes from the expressions for $M_1^2$ and $M_2^2$. Our treatment is only valid under the assumption that $a^4 > 4b^2$. However, this condition is equivalent to taking $\kappa ^2 {\bf v}^2 < \frac{{m_B^4 }}{2}$, which is perfectly compatible with our approximation. So, we restrict ourselves to an external magnetic field such that $|{\bf v}| < \frac{{m_B^2 }}{{2\kappa ^2 }}$. Using Eq. (\[Csmag65b\]), the $\left\langle H \right\rangle _\Phi ^ {\left( 1 \right)}$ and $\left\langle H \right\rangle _\Phi ^ {\left( 2 \right)}$ terms can be rewritten as $$\begin{aligned} \left\langle H \right\rangle _\Phi ^{\left( 1 \right)} &=& - \frac{{b^2 Bq^2 }}{2}\int {d^3 x} \int_{\bf y}^{{\bf y}^ \prime } {dz_i^ \prime \delta ^{(3)} \left( {{\bf x} - {\bf z}^ \prime } \right)} \frac{{\nabla ^2 }}{{\left( {\nabla ^2 - M_1^2 } \right)}} \times \int_{\bf y}^{{\bf y}^ \prime } {dz^i } \delta ^{(3)} \left( {{\bf x} - {\bf z}} \right) \nonumber\\ &+& \frac{{b^2 Bq^2 }}{2}\frac{{M_2^2 }}{{M_1^2 }}\int {d^3 x} \int_{\bf y}^{{\bf y}^ \prime } {dz_i^ \prime \delta ^{(3)} \left( {{\bf x} - {\bf z}^ \prime } \right)} \frac{{\nabla ^2 }} {{\left( {\nabla ^2 - M_2^2 } \right)}} \int_{\bf y}^{{\bf y}^ \prime } {dz^i } \delta ^{(3)} \left( {{\bf x} - {\bf z}} \right), \label{Csmag80a}\end{aligned}$$ and $$\begin{aligned} \left\langle H \right\rangle _\Phi ^{\left( 2 \right)} &=& m_B^2 b^2 Bq^2 \int {d^3 x} \int_{\bf y}^{{\bf y}^ \prime } {dz_i^ \prime \delta ^3 \left( {{\bf x} - {\bf z}^ \prime } \right)} \frac{1}{{\left( {\nabla ^2 - M_1^2 } \right)}} \int_{\bf y}^{{\bf y}^ \prime } {dz^i } \delta ^{(3)} \left( {{\bf x} - {\bf z}} \right) \nonumber\\ &-& \frac{{m_B^2 b^2 Bq^2M_2^2 }}{{M_1^2 }}\int {d^3 x} \int_{\bf y}^{{\bf y}^ \prime } {dz_i^ \prime \delta ^{(3)} \left( {{\bf x} - {\bf z}^ \prime } \right)} \frac{1} {{\left( {\nabla ^2 - M_2^2} \right)}} \int_{\bf y}^{{\bf y}^ \prime } {dz^i } \delta ^{(3)} \left( {{\bf x} - {\bf z}} \right). \nonumber\\ \label{Csmag80b}\end{aligned}$$ Following our earlier procedure [@GaeteS; @GaeteHel09], we see that the potential for two opposite charges located at ${\bf y}$ and ${\bf y^{\prime}}$ takes the form $$\begin{aligned} V &=& - \frac{{q^2 }}{{4\pi }}\frac{1}{{a^2 \sqrt {1 - {\raise0.7ex\hbox{${4b^2 }$} \!\mathord{\left/ {\vphantom {{4b^2 } {a^4 }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${a^4 }$}}} }} \nonumber\\ &\times& \left[ {\left( {M_1^2 - m_B^2 } \right)\frac{{e^{ - M_1 L} }}{L} - \left( {M_2^2 - m_B^2 } \right)\frac{{e^{ - M_2 L} }}{L}} \right] . \nonumber\\ \label{Csmag85}\end{aligned}$$ Consequently, our analysis reveals that the theory under consideration describes an exactly screening phase. It is important to realize that expression (\[Csmag85\]) displays a marked departure of a qualitative nature from the result from axionic elctrodynamics. As already mentioned, axionic electrodynamics has a different structure which is reflected in a confining piece, which is not present in the Chern-Simons-like coupling scenario. It is to be noted that the choice of the gauge is in this development really arbitrary. Put another way, being the formalism completely gauge invariant, we would obtain exactly the same result in any gauge. We also note here that by considering the limit $b \to 0$, we obtain a theory of two independent uncoupled $U(1)$ gauge bosons, one of which is massless. In such a case, one can easily verify that the static potential is a Yukawa-like correction to the usual static Coulomb potential. Finally, the following remark is pertinent at this point. It should be noted that by substituting $B_\mu$ by $\partial_\mu \phi$ in (\[Csmag05\]), the theory under consideration assumes the form [@Gaete06] $${\cal L} = - \frac{1}{4}F_{\mu \nu }^2 + \frac{{m_\gamma ^2 }}{2} + \frac{1}{2}\partial _\mu \phi \partial ^\mu \phi - \frac{\kappa }{{2m_B }}\varepsilon ^{\mu \nu \lambda \rho } F_{\mu \nu } F_{\lambda \rho } \phi, \label{Csmag90}$$ which is similar to axionic electrodynamics. In fact, it is worth recalling here that axionic electrodynamics is described by [@GaeteGuen] $${\cal L} = - \frac{1}{4}F_{\mu \nu } F^{\mu \nu } - \frac{g}{8}\phi \varepsilon ^{\mu \nu \rho \sigma } F_{\mu \nu } F_{\rho \sigma } + \frac{1}{2}\partial _\mu \phi \partial ^\mu \phi - \frac{{m_A^2 }}{2}\phi ^2, \label{Csmag90b}$$ hence we see that both theories are quite different. Thus, after performing the integration over $\phi$ in (\[Csmag90\]), the effective Lagrangian density reads $$\begin{aligned} {\cal L} &=& - \frac{1}{4}F_{\mu \nu }^2 + \frac{{m_\gamma ^2 }}{2}A_\mu ^2 - \frac{{\kappa ^2 }}{{8m_B^2 }}\varepsilon ^{\mu \nu \lambda \rho } F_{\mu \nu } F_{\lambda \rho } \frac{1}{{\nabla ^2 }} \nonumber\\ &\times&\varepsilon ^{\alpha \beta \gamma \delta } F_{\alpha \beta } F_{\gamma \delta }. \label{Csmag95}\end{aligned}$$ This expression can now be rewritten as $$\begin{aligned} {\cal L} &=& - \frac{1}{4}f_{\mu \nu }^2 + \frac{{m_\gamma ^2 }}{2}a_\mu ^2 - \frac{{\kappa ^2 }}{{2m_B^2 }}\varepsilon ^{\mu \nu \alpha \beta } \left\langle {F_{\mu \nu } } \right\rangle \varepsilon ^{\lambda \rho \gamma \delta } \left\langle {F_{\lambda \rho } } \right\rangle \nonumber\\ &\times& f_{\alpha \beta } \frac{1}{{\nabla ^2 }}f_{\gamma \delta } , \label{Csmag100}\end{aligned}$$ where $\left\langle {F_{\mu \nu } } \right\rangle$ represents the constant classical background. Here $f_{\mu \nu } =\partial _\mu a_\nu -\partial _\nu a_\mu$ describes fluctuations around the background. The above Lagrangian arose after using $ \varepsilon ^{\mu \nu \alpha \beta } \left\langle {F_{\mu \nu } } \right\rangle \left\langle {F_{\alpha \beta } } \right\rangle=0$ (which holds for a pure electric or a pure magnetic background). By introducing the notation $\varepsilon ^{\mu \nu \alpha \beta } \left\langle{F_{\mu \nu } } \right\rangle \equiv v^{\alpha \beta } $ and $\varepsilon ^{\rho \sigma \gamma \delta } \left\langle {F_{\rho \sigma } } \right\rangle \equiv v^{\gamma \delta }$, expression (\[Csmag100\]) then becomes $${\cal L} = - \frac{1}{4}f_{\mu \nu }^2 + \frac{{m_\gamma ^2 }}{2}a_\mu ^2 - \frac{{\kappa ^2 }}{{2m_B^2 }}v^{\alpha \beta } f_{\alpha \beta } \frac{1}{{\nabla ^2 }}v^{\gamma \delta } f_{\gamma \delta} , \label{Csmag105}$$ where the tensor $v^{\alpha \beta }$ is not arbitrary, but satisfies $\varepsilon ^{\mu \nu \alpha \beta } v_{\mu \nu } v_{\alpha \beta }=0$. Following the same steps employed for obtaining (\[Csmag85\]), the static potential is expressed as $$V= - \frac{{q^2 }}{{4\pi }}\frac{{e^{ - ML} }}{L}, \label{Csmag130}$$ with $ M^2 \equiv m_\gamma ^2 + \frac{{\kappa ^2 }}{{m_B^2 }}{\bf v}^2$ . Again, the theory describes a screening phase, as we have just seen above. Final Remarks ============= Let us summarize our work. Once again we have advocared a key point for understanding the physical content of gauge theories, that is, the identification of field degrees of freedom with observable quantities. We have showed that the static potential profile obtained from both a gauge theory which includes a light massive vector field interacting with the familiar photon $U(1)_{QED}$ via a Chern-Simons-like coupling and axionic electrodynamics models are quite different. This means that the two theories are not equivalent. As it was shown in [@GaeteGuen], axionic electrodynamics has a different structure which is reflected in a confining piece, which is not present in the gauge theory which includes a light massive vector field interacting with the familiar photon $U(1)_{QED}$ via a Chern-Simons-like coupling. However, our result is analogous to that encountered in the coupling between the familiar photon $U(1)_{QED}$ and a second massive gauge field living in the so-called hidden-sector $U(1)_h$, inside a superconducting box.\ Acknowledgments =============== One of us (PG) wants to thank the Field Theory Group of the CBPF and the Physics Department of the Università di Trieste for hospitality. This work was supported in part by Fondecyt (Chile) grant 1080260 (PG). (J.H-N) expresses his gratitude to CNPq. E. Zavattini et al. \[PVLAS collaboration\], Phys. Rev. Lett. [**96**]{}, 110406 (2006). R. Cameron et al. \[BFRT collaboration\], Phys. Rev. [**D47**]{}, 3707 (1993). S. J. Chen et al. \[Q and A collaboration\], Mod. Phys. Lett. [**A22**]{}, 2815 (2007). E. Zavattini et al. \[PVLAS collaboration\], Phys. Rev. [**D77**]{}, 032006 (2008). C. Robilliard et al. \[BMV collaboration\], Phys. Rev. Lett. [**99**]{}, 190403 (2007). A. S. Chou et al. \[Gamme V (T-969) collaboration\], Phys. Rev. Lett. [**100**]{}, 080402 (2008). M. Ahlers, H. Gies, J. Jaeckel, J. Redondo and R. Ringwald, Phys. Rev. [**D77**]{}, 095001 (2008). M. Ahlers, H. Gies, J. Jaeckel, J. Redondo and R. Ringwald, Phys. Rev. [**D76**]{}, 115005 (2007). V. Popov, Tr. J. of Physics, [**23**]{}, 943 (1999). V. Popov and O. Vasil’ev, Europhys. Lett. [**15**]{}, 7 (1991). D. Chelouche et al., Astrophys. J. Suppl. [**180**]{}, 1 (2009). S. N. Gninenko, arXiv: 0802.1315 \[hep-ph\]. J. Jaeckel and J. Redondo, Europhys. Lett. [**84**]{}, 31002 (2008). S. Ansoldi, E. I. Guendelman and E. Spallucci, JHEP [**09**]{}, 044 (2003). P. Gaete and E. I. Guendelman, Mod. Phys. Lett. [**A20**]{}, 319 (2005). G. K. Savvidy, Phys. Lett. [**B71**]{}, 133 (1977); H. B. Nielsen and P. Olesen, Nucl. Phys. [**B160**]{}, 380 (1979); H. B. Nielsen and M. Ninomiya, Nucl. Phys. [**B156**]{}, 1 (1979). H. Suganuma, S. Sasaki and H. Toki, Nucl. Phys. [**B435**]{}, 207 (1995). P. Gaete and C. Wotzasek, Phys. Lett. [**B601**]{}, 108 (2004). S. A. Abel, M. D. Goodsell, J. Jaeckel, V. V. Khoze and R. Ringwald, JHEP [**0807**]{}, 124 (2008). B. Batell and T. Gherghetta, Phys. Rev. [**D73**]{}, 045016 (2006). B. Holdom, Phys. Lett. [**B166**]{}, 196 (1986). D. Feldman, Z. Liu and P. Nath, Phys. Rev. [**D75**]{}, 115001 (2007). B. Körs and P. Nath, Phys. Lett. [**B586**]{}, 366 (2004). P. Castelo Ferreira, Phys. Lett. [**B651**]{}, 74 (2007). E. Massó and J. Redondo, JCAP [**0509**]{}, 015 (2005). R. Foot, A. Yu. Ignatiev and R. R. Volkas, Phys. Lett. [**B503**]{}, 355 (2001). D. Singleton, Int. J. Theor. Phys. [**34**]{}, 37 (1995). P. Langacker, arXiv: 0801.1345 \[hep-ph\]. D. Singleton, Am. J. Phys. [**64**]{}, 452 (1996). N. Cabibbo and E. Ferrari, Nuovo Cimento [**23**]{}, 1147 (1962). P. A. M. Dirac, Proc. R. Soc. [**A133**]{}, 60 (1931); Phys. Rev. [**74**]{}, 817 (1948). C. R. Hagen, Phys. Rev. [**140**]{}, B804 (1965). L. B. Okun, Sov. Phys. JETP [**56**]{}, 502 (1982). N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer and N. Weiner, Phys. Rev. [**D79**]{}, 015014 (2009). I. Antoniadis, A. Boyarsky and O. Ruchayskiy, Nucl. Phys. [**B793**]{}, 246 (2008). P. Gaete and I. Schmidt, Phys. Rev. [**D76**]{}, 027702 (2007). P. Gaete and E. Spallucci, J. Phys. A: Math. Gen. [**39**]{}, 6021 (2006). P. Gaete and E. Spallucci, Phys.Lett. [**B675**]{}, 145 (2009). P. Gaete and J. A. Helayel-Neto, Phys. Lett. [**B683**]{}, 211 (2010). P. Gaete and I. Schmidt, “Coulomb’s law corrections from a gauge-kinetic mixing”, arXiv: 1004.1591 \[hep-th\]. P. Gaete, Phys.Rev. [**D59**]{}, 127702, (1999); Z. Phys. [**C76**]{}, 355 (1997). P. Gaete and I. Schmidt, “Remarks on Axion-like models”, hep-th/0612303.

--- abstract: 'EC paper authors form a complex network of co-authorship which is, by itself, an example of an evolving system with its own rules, concept of fitness, and patterns of attachment. In this paper we explore the network of authors of evolutionary computation papers found in a major bibliographic database. We examine its macroscopic properties, and compare it with other co-authorship networks; the EC co-authorship network yields results in the same ballpark as other networks, but exhibits some distinctive patterns in terms of internal cohesion. We also try to find some hints on what makes an author a sociometric star. Finally, the role of proceeding editorship as the origin of long-range links in the co-authorship network is studied as well. [**Keywords**]{}: Evolutionary computation, sociometric studies, complex networks, scale-free networks, power laws, co-authorship networks.' author: - | [**Carlos Cotta**]{}\ [ccottap@lcc.uma.es]{}\ Dept. Lenguajes y Ciencias de la Computación,\ University of Málaga, Spain,\ \ [**Juan-Julián Merelo**]{}\ [jj@merelo.net]{}\ Dept. Arquitectura y Tecnología de Computadores,\ University of Granada, Spain bibliography: - 'coauthorship.bib' title: '**The Complex Network of Evolutionary Computation Authors: an Initial Study**' --- Introduction ============ The study of all kind of networks has undergone an accelerated expansion in the last few years, after the introduction of models for power-law [@emerg:barabasi1999] and scale-free networks [@watt98], which, in turn, has induced the study of many different phenomena under this new light. One of them have been co-authorship networks: nodes in these networks are paper authors, joined by edges if they have written at least a paper in common. Even as most papers are written by a few authors staying at the same institution, science is a global business nowadays, and lots of papers are co-authored by scientists continents apart from each other. There are several interesting facts that can be computed on these co-authorship networks: first, what kind of macroscopic values they yield, and second, which are the most outstanding *actors* (authors) and edges (co-authorships) within this network. A better understanding of the structure of the network and what makes some nodes stand out goes beyond mere curiosity to give us some insight on the deep workings of science, what makes an author popular, or some co-authors preferred over others. Co-authorship networks are studied within the field of sociometry, and, in the case at hand, scientometry. First studies date back to the second half of the nineties: Kretschmer [@kretschmer97] studied the [*invisible colleges*]{} of physics, finding that their behavior was not much different to other collaboration networks, such as co-starring networks in movies. However, it was at the beginning of this century when Newman [@newman01a; @newman01b] studied co-authorship networks as complex networks, giving the first estimations of their overall shape and macroscopic properties. In general, these kind of networks are both small worlds [@watt98], that is, there is, on average, a short distance between any two scientists taken at random, and scale free, which means they follow a power law [@emerg:barabasi1999] in several node properties (e.g., the in-degree, or number of nodes linking a particular one) . Newman made measurements on networks from several disciplines: physics, medicine and computer science, showing results for clustering coefficients (related to transitivity in co-authorship networks), and mean and maximum distances (which gives an idea of the shape of the network). Barabási and collaborators [@barabasi02] later proved that the scale free structure of these co-authorship networks can be attributed to preferential attachment: authors that have been more time in business publish more papers on average, and thus get more new links than new authors. However, even as this model satisfactorily explains the overall structure of the network, there must be much more in the author positions in the network than just having been there for more time. In addition to these general works, several studies have also focused in particular scientific communities: computer support of cooperative work [@horn2004], psychology and philosophy [@psy2003], chemistry [@chem2004], SIGMOD authors [@SIGMOD2003] and sociology [@moody04], to name a few. In this work, we analyze the co-authorship network of evolutionary computation researchers. Studying this network gives us a better understanding of its cohesiveness as a discipline, and sheds some light on the collaboration patterns of the community. It also provides interesting hints about who are the central actors in the network, and what determines their prominency in the area. Materials and Methods {#sec:mm} ===================== The bibliographical data used for the construction of the scientific-collaboration network in EC has been gathered from the DBLP[^1] –*Digital Bibliography & Library Project*– computer Science bibliography server, maintained by Michael Ley at the University of Trier. This database provides bibliographic information on major computer science journals and proceedings, comprising more than 610,000 articles and several thousand computer scientists (as of March 2005). The database provides bibliographical data indexed by author and by conference/journal. This turns out to be one of its advantages since, for example, the URL of the page containing the information for a certain author can be used as identifying key for that author. To some extent this alleviates one of the problems typically found in this kind of studies, namely the fact that a single author may report his/her name differently on different papers (e.g., using the first name or just initials, including a middle name or not, etc.)[^2]. Of course, this kind of situation is still possible in this database, and indeed we have found some instances of it. However, it seems that the maintainers of the database have put some care in avoiding this issue. Besides this indexing issue, the DBLP exhibits two additional advantages. Firstly, it is a “moderated” database, meaning that it is not updated via authors’ submitting their references. On the contrary, the maintainers add themselves new entries by inspecting published volumes, or incorporate full BibTeX collections provided by publishers or editors. This eliminates a potential source of bias in the sample of publications, i.e., some authors being very active in submitting their bibliographical entries while other being less proactive in this sense. Finally, the second additional advantage is the fact that DBLP pages are highly structured and regular. Hence they are very amenable for automated parsing by a scraping program. In particular, hyperlinks are provided for every co-author of a paper, making navigation through the database very easy. The process to obtain the raw data is the following: our scraping robot is firstly fed with a collection of DBLP author keys, stored in a stack. Subsequently, while this stack is not empty, a key is extracted from it, and the corresponding HTML page is downloaded. Then, it is parsed to extract the textual name of the author, and the papers he/she has authored. For each of these papers, the hyperlinks of co-authors are identified, and added to the stack (cycles are avoided by keeping track of processed authors using an ordered binary tree). An important issue to be taken into account is the fact that we are interested in obtaining a network for the EC community. However, an EC author may also publish articles in other fields; hence, we cannot blindly parse all entries in a certain page since non-EC papers (and later on, non-EC authors) would be included in the network. To avoid this, we have used a double check: firstly we look for certain patterns in the publication reference. These include the acronyms of EC-specific conferences –such as GECCO, PPSN, EuroGP, etc.– or keywords –such as “Evolutionary Computation”, “Genetic Programming”, etc.– that account for the relevant journals and/or additional conferences. Papers with any of these strings in its publication reference are directly classified as EC papers and parsed as described above. If this criterion is not fulfilled, then the title is scanned in order to detect another set of relevant keywords such as “evolutionary algorithm”, “genetic algorithm”, etc., or acronyms such as “EA”, “GA” or “GP”. Again, if a paper triggers this criterion, it is classified as an EC paper and processed accordingly. It must be noted that this system has turned out to be rather accurate in detecting EC papers. Actually, the visual inspection of the resulting network indicated that only a small fraction of false positives (well below 1% of the total number of papers) passed the filters. These were mostly computational biology papers, and were readily removed from the network. As a final consideration, we have chosen a large representative sample of authors as the seed of our search robot. To be precise, we have used a collection composed of all authors that have published at least one paper in the last five years in any of the following large EC conferences: GECCO, PPSN, EuroGP, EvoCOP, and EvoWorkshops (unfortunately, CEC is not indexed in the DBLP; however, this does not alter the macroscopic properties of the network, as it will be shown below). This way, the immense majority of active EC researchers is guaranteed to be included in the sample. Actually, active authors not publishing in these fora are in practice linked –directly or indirectly– with all likelihood with authors who do publish in them. Just as an indication, the number of authors used as seed is 2,536 whereas the final number of authors in the network is 5,492, that is, more than twice as many. Macroscopic Network Properties {#sec:macro} ============================== The overall characteristics of the EC co-authorship network are shown in Table \[tab:todo\] alongside with results obtained by Newman [@newman01a]. The latter correspond to co-authorship networks in Medline (biomedical research), the Physics E-print Archive and SPIRES (several areas of physics and high-energy physics respectively), and NCSTRL (several areas of computer science). ----------------------------- ------- --------- --------- -------- -------- -- EC Medline Physics SPIRES NCSTRL total papers 6199 2163923 98502 66652 13169 total authors 5492 1520251 52909 56627 11994 mean papers per author 2.9 6.4 5.1 11.6 2.6 mean authors per paper 2.56 3.75 2.53 8.96 2.22 collaborators per author 4.2 18.1 9.7 173.0 3.6 size of the giant component 3686 1395693 44337 49002 6396      as a percentage 67.1% 92.6% 85.4% 88.7% 57.2% 2nd largest component 36 49 18 69 42 clustering coefficient 0.798 0.066 0.43 0.726 0.496 mean distance 6.1 4.6 5.9 4.0 9.7 diameter (maximum distance) 18 24 20 19 31 ----------------------------- ------- --------- --------- -------- -------- -- : Summary of results of the analysis of five scientific collaboration networks.\[tab:todo\] 6.5cm 6.5cm First of all, the number of EC papers and authors is much smaller than those for the communities studied by Newman; however, it must be taken into account that these communities are much more general and comprise different subareas. Notice also that in most aspects, EC data seems closer to the NCSTRL database than to any other. This indicates that despite the interdisciplinary nature of EC, the publication practices of this area are in general those of computer science. This way, average scientific productivity per author (2.9) is not so high as in physics (5.1, 11.6) and biomedicine (6.4). It nevertheless follows quite well Lotka’s Law of Scientific Productivity [@lotka26], as shown by the power law distribution illustrated in Fig. \[fig:fig1\] (left). The most interesting feature is the [*long tail*]{}: while most authors appear only once in the database, there are quite a few that have authored dozens of papers. The average size of collaborations (2.56) is also smaller than in biomedical research (3.75) or high-energy physics (8.96), although similar to those of average physicists (2.53), and slightly superior to average computer scientists (2.22). It also follows a power law (up from 3 authors) as shown in Fig. \[fig:fig1\] (right). Notice the peak in the tail of the distribution, caused by the large collaborations implied by proceedings. Their role will be examined in Sect. \[sec:sociometric\_stars\] Relevant considerations can be also done regarding the total number of collaborators per author (4.2); physics and biomedicine are areas in which new collaborations seem more likely than in EC (9.7, 173.0, and 18.1). However, the figure for NCSTRL (3.6) is lower than for EC, thus suggesting that the EC author is indeed open to new collaborations, as regarded from a computer science perspective. The histogram of number of collaborators per authors (not shown) also fits quite well to a power law with exponent -2.58. In this case, this power law can be attributed to a model of preferential attachment such as the one proposed by Barabási [@barabasi02]: *new* authors tend to link (be co-authors) of those that have published extensively before. However, as we pointed out before, that cannot be the whole story. For starters, information on who is the most prolific author is not usually available (although educated guesses can go a long way), and, besides, there are strong constraints that avoid free linking: a person can only tutor so many PhD students at the same time, for instance, and not everybody is ready, or able, to move to the university of the professor she wants to work with. However, let us point out that actors with many links do not necessarily coincide with the most prolific; they are rather persons that have diverse interests, reflected in their choice or co-authors, participate in transnational projects, or have a certain wanderlust, being visiting professors in many different institutions, which leads them to co-author papers with their sponsors or hosts in those institutions. The fact that the clustering coefficient (that is, the average fraction of an actor’s collaborators that are collaborators themselves) in the EC co-authorship networks is so high, and the mean degree of separation is so close to the proverbial six degrees, means that in general all authors in this field are no more than 6 degrees of separation of those *sociometric stars* with a wide variety of interests, projects or visits. These sociometric stars will be analyzed more in depth in next section. Another interesting aspect refers to the so-called *giant component*. This is a connected subset of vertices whose size encompass most of the network. The remaining vertices group in components of much smaller size (actually, independent of the total size of the network). As pointed out in [@newman01a], the existence of this giant component is a healthy sign, for it shows that most of the community is connected via collaboration, and hence by person-to-person contact ultimately. In the case of the EC network, the giant component comprises more than 2/3 of the network (see Fig. \[fig:fig2\]), again superior to the computer science network, but significantly smaller than for physics or biomedicine. This fact is nevertheless counteracted by the high clustering coefficient (actually the highest of the set). This indicates a much closer contact among actors, since one’s collaborators are very likely to collaborate among themselves too. It is also significant that the mean distance among actors is halfway between the medical/physics communities (around 4) and the computer science community (around 9), while diameter is the second-smallest. This shows that the EC community is halfway between computer science and more theoretical fields, such as physics. Evolutionary Computation Sociometric Stars {#sec:sociometric_stars} ========================================== In the previous section we have considered global collaboration patterns that can be inferred from macroscopic properties of the network. Let us know take a closer look at the fine detail of the network structure. More precisely, we are going to identify which actors play a more prominent role in the network, and analyze why they are important. The term *centrality* is used to denote this prominency status for a certain node. Centrality can be measured in multiple ways. We are going to focus on metrics based on geodesics, i.e., the shortest paths between actors in the network. These geodesics constitute a very interesting source of information: the shortest path between two actors defines a “referral chain” of intermediate scientists through whom contact may be established – cf. [@newman01b]. It also provides a sequence of research topics (recall that common interests exist between adjacent links of this chain, as defined by the co-authored papers) that may suggest future joint works. The first geodesic-based centrality measure that we are going to analyze is *betweenness* [@freeman77], i.e., the total number of geodesics between any two actors $i,j$ that passes through a third actor $k$. The rationale behind this measure lies in the information flow between actors: when a joint paper is written, the authors exchange lots of information (research ideas, unpublished results, etc.) which can in turn be transmitted (at least to some extent) to their colleagues in other papers, and so on. Hence, actors with high betweenness are in some sense “hubs” that control this information flow; they are recipients –and emitters– of huge amounts of cutting-edge knowledge; furthermore, their removal from the network would result in the increase of geodesic distances among a large number of actors [@wassermanFaust94]. The second centrality measure we are going to consider is precisely based on this geodesic distance. Intuitively, the length of the shortest path indicates the number of steps that research ideas (and in general, all kind of memes) require to jump from one actor to another. Hence, scientists whose average distance to other scientists is small are likely to be the first to learn new information, and information originating with them will reach others quicker than information originating with other sources. Average distance (i.e., *closeness*) is thus a measure of centrality of an actor in terms of their access to information. ----- --------------- ---- -- --------------- ------- -- --------------- -------       1. K. Deb 98 K. Deb 19.06 K. Deb 28.60 2. D.E. Goldberg 75 D.E. Goldberg 14.24 W. Banzhaf 27.28 3. R. Poli 67 D. Corne 10.23 D.E. Golberg 26.87 4. M. Schoenauer 62 X. Yao 7.90 R. Poli 26.86 5. W. Banzhaf 58 W. Banzhaf 7.70 H.-G. Beyer 26.55 6. D. Corne 56 H. de Garis 6.92 P.L. Lanzi 26.50 7. X. Yao 56 R. Poli 6.86 D. Corne 25.93 8. J.A. Foster 54 J.J. Merelo 6.50 M. Schoenauer 25.73 9. J.J. Merelo 53 H. Iba 6.48 E.K. Burke 25.62 10. J.F. Miller 51 M. Schoenauer 6.33 D.B. Fogel 25.54 ----- --------------- ---- -- --------------- ------- -- --------------- ------- : Most central actors in the EC network. D. E. Goldberg, author of one of the most famous books on EC, figures prominently in all rankings, as well as Kalyanmoy Deb, who is a well known author in theoretical EC and multi-objective optimization. The rest of the authors are well known as conference organizers, or as leaders of some subfields within EC. The three columns show rankings for three quantities: number of co-authors, and two centrality measures: betweenness and closeness. \[tab:centrality\_1\] 6.5cm 6.5cm The result of our centrality analysis of the EC network is shown in Table \[tab:centrality\_1\]. The numbers provided for each actor indicate the normalized values of betweenness and closeness (that is, their actual values divided by the maximum possible value, expressed as a percentage). Regarding betweenness, the analysis provides clear winners, with large numerical differences among the top actors. These differences are not so marked for closeness values with all top actors clustered in a short interval. Notice that there are some actors that appear in both top-lists. Using Milgram’s terminology [@milgram67], these constitute the *sociometric superstars* of the EC field. Several factors are responsible for the prominent status of these actors. Obviously, scientific excellence is one of them. This excellence is difficult to measure in absolute, objective terms, but the number of collaborators provides some hints on it[^3]. This quantity is shown for the top ten actors in the network in Table \[tab:centrality\_1\]. Certainly, some correlation between degree and centrality is evident. This is further illustrated in Fig. \[fig:distance\] (left). As it can be seen, there is a trend of decreasing average distance to other actors as the actor degree increases. By crossing this information with the percentile distribution of distances shown in Fig. \[fig:distance\] (right) we can obtain some interesting facts about the collaborative strength of elite scientists. For example, consider the top 5% percentile; it is composed of actors whose average distance to the remaining actors is at most 4.61. According to Fig. \[fig:distance\] (left), 23 collaborators are required at least to have an average distance below this value. A more sensitive analysis indicates that 33 collaborators are required to have an statistically significant (using a standard t-test) result. 7.2cm Another important factor influencing the particular ranking shown above is the presence of conference proceedings among authors’ publications. These play a central role in the creation and structure of the network, to the point that its features change dramatically if links arising from proceedings co-authorship are removed. To begin with, the visual aspect of the network is different, as is shown in the left hand side of Fig. \[fig:noproc\] (compare it to the network with proceedings included, shown in Fig. \[fig:fig2\]). The reader should notice that the core is much more diffuse (actually, it looks like there are several micro-cores, plausibly corresponding to different EC subareas). This change is also reflected in the right hand side of Fig. \[fig:noproc\], which plots the histogram of average distances from each node to the rest of the network: without proceedings, the average distance and maximum distance increase by 2 units, and the modal distance increases by 3 units. The resulting distribution is also much more symmetric than the original distribution, which was notably skewed towards low values. This can be explained by the very distinctive *authoring* (in property, *editing*) patterns of proceedings: they are usually edited by a larger number of researchers, typically corresponding to the different thematic areas included in the conference or symposium. These are often senior researchers, with a prominent position in their subareas (thus, centrality and proceeding editorship reinforce each other). Furthermore, the fact that editors come from different areas contribute to the creation of long-distance links, resulting in a dramatic overall decrease of inter-actor distances. ----- ---------------- ---- -- ---------------- ------- -- ---------------- -------       1. D.E. Goldberg 63 D.E. Goldberg 22.68 Z. Michalewicz 20.21 2. K. Deb 55 K. Deb 20.04 K. Deb 20.05 3. M. Schoenauer 52 M. Schoenauer 12.68 M. Schoenauer 19.89 4. X. Yao 42 H. de Garis 12.62 A.E. Eiben 19.77 5. H. de Garis 41 Z. Michalewicz 12.58 B. Paechter 19.70 6. T. Higuchi 40 T. Bäck 10.31 D.E. Goldberg 19.64 7. Z. Michalewicz 40 R.E. Smith 9.46 T. Bäck 18.70 8. L.D. Whitley 39 X. Yao 9.07 D.B. Fogel 18.59 9. M. Dorigo 38 A.E. Eiben 8.61 J.J. Merelo 18.52 10. J.J. Merelo 38 B. Paechter 8.05 T.C. Fogarty 18.50 ----- ---------------- ---- -- ---------------- ------- -- ---------------- ------- : Most central actors in the EC network after removing proceedings. \[tab:noprocs\] Although proceeding editorship is certainly a scientific activity, and constitutes a valuable contribution to the community, putting them at the same level of research papers is arguable at the very least. It thus seems reasonable to exclude proceedings from the network to obtain a more unbiased figure of centrality. We have done this, obtaining the results shown in Table \[tab:noprocs\]. As it can be seen, there is now a higher agreement between the two centrality measures (7/10 are the same, vs. 6/10 before). Furthermore, researchers of unquestionable scientific excellence who were not in the previous ranking do appear now. For example, Z. Michalewicz, author of several excellent EC books, is now the author with the highest closeness, the 5th-highest betweeness, and the 7th-highest number of collaborators. Overall, this may provide a more objective view on the central actors of our field. Discussion and Conclusion ========================= In this paper, we have made a preliminary study of the co-authorship network in the field of evolutionary computation, paving the way to study the impact of certain measures, such as grants, the establishment of scientific societies or new conferences, has on the subject. The general features of the network suggest that it is quite similar to the field it can be better placed, computer science, but, at the same time, authors are much more closely related with each other. We have also taken into account the impact co-editorship of proceedings have on the overall aspect of the network and most centrality measures. To the best of our knowledge, this issue had not been considered in previous related works, and we believe it plays an important role in distorting some network properties. We suggest to not consider them in the future in this kind of studies. In connection to this latter issue, we believe that co-authorship networks created by different kind of papers (technical reports, conference papers, journal papers) might be different owing to the different kind of collaboration they imply. Consider that while technical reports may be written in a hurry and present very preliminary results, conference papers are usually somewhat more long term, and journal papers really indicate a committed scientific relationship (due to the long time they take to be published and the several iterations of the revision process). The authors suggest to approach them separately and analyze the features of the networks they yield. In addition to this, our future lines of work along this topic will include the analysis of the network evolution through time, as well as the impact funded scientific networks and transnational grants (such as EU grants) have had on it. We also plan to study the existence of *invisible colleges* or communities within the EC field, and analyze which their axes of development are, e.g., topical or regional. [^1]: `http://www.informatik.uni-trier.de/\simley/db/` [^2]: A second kind of problem is possible: having two different authors with exactly the same name. We are not aware of any glaring instance of this duplicity in the EC community. [^3]: This quantity is strongly correlated with the number of papers ($\rho=.82$), and thus provides information on the efficiency in knowledge transmission, which is the ultimate goal of scientific publishing. Involvement in PhD supervision and research projects, and wide research interests will typically result in a higher number of collaborators as well.

--- abstract: 'Quasar feedback has most likely a substantial but only partially understood impact on the formation of structure in the universe. A potential direct probe of this feedback mechanism is the Sunyaev-Zeldovich effect: energy emitted from quasar heats the surrounding intergalactic medium and induce a distortion in the microwave background radiation passing through the region. Here we examine the formation of such hot quasar bubbles using a cosmological hydrodynamic simulation which includes a self-consistent treatment of black hole growth and associated feedback, along with radiative gas cooling and star formation. From this simulation, we construct microwave maps of the resulting Sunyaev-Zeldovich effect around black holes with a range of masses and redshifts. The size of the temperature distortion scales approximately with black hole mass and accretion rate, with a typical amplitude up to a few micro-Kelvin on angular scales around 10 arcseconds. We discuss prospects for the direct detection of this signal with current and future single-dish and interferometric observations, including ALMA and CCAT. These measurements will be challenging, but will allow us to characterize the evolution and growth of supermassive black holes and the role of their energy feedback on galaxy formation.' author: - | Suchetana Chatterjee$^{1}$ Tiziana Di Matteo$^{2}$, Arthur Kosowsky$^{1}$ & Inti Pelupessy$^{2}$\ $^{1}$Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260 USA\ $^{2}$McWilliam’s Center for Cosmology, Carnegie Mellon University, Pittsburgh, PA 15213 USA title: 'Simulations of the Sunyaev-Zeldovich Effect from Quasars' --- \[firstpage\] cosmic microwave background – intergalactic medium – galaxies: active. Introduction ============ The temperature fluctuations in the cosmic microwave background, as measured by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite (Bennett et al.2003) and numerous other microwave experiments (e.g., Dawson et al. 2002; Rajguru et al. 2005; Reichardt et al. 2008) have proven to be the single most powerful tool in constraining cosmology (Spergel et al.2007). The temperature anisotropy has been mapped with large statistical significance on angular scales down to around a quarter degree, where the dominant physical mechanisms contributing to the fluctuations arise from density perturbations at the epoch of recombination. Attention is now turning to arcminute angular scales, where temperature fluctuations arise due to interaction of the microwave photons with matter in the low-redshift universe (for a brief review, see Kosowsky 2003). These low-redshift and small-angle anisotropies are collectively known as “secondary anisotropies” in the microwave background. The most prominent among them is the Sunyaev-Zeldovich (SZ) effect (Sunyaev & Zeldovich 1972) from the inverse Compton scattering of the microwave photons due to hot electrons. The SZ effect provides a powerful method for detecting accumulations of hot gas in the universe (Carlstrom Holder & Reese 2002). Galaxy clusters, which contain the majority of the thermal energy in the universe, provide the largest SZ signal; clusters were first detected this way through pioneering measurements over the past decade (e.g, Marshall et al. 2001) and thousands of them will be detected by the upcoming SZ surveys like the Atacama Cosmology Telescope (ACT) (Kosowsky et al. 2006) and the South Pole Telescope (SPT) (Ruhl et al. 2004). However, a number of other astrophysical processes will also create SZ distortions. This includes SZ distortion from peculiar velocities during reionization (McQuinn et al. 2005, Illiev et al. 2006), supernova-driven galactic winds (Majumdar, Nath, & Chiba 2001), black hole seeded proto-galaxies (Aghanim, Ballad & Silk 2000), kinetic SZ from Lyman Break Galaxy outflow (Babich & Loeb 2007), effervescent heating in groups and clusters of galaxies (Roychowdhury, Ruszkowski & Nath 2005) and supernova from first generation of stars (Oh, Cooray, & Kamionkowski 2003). The SZ distortion in galactic scales (hot proto galactic gas) have been studied by different authors (e.g, de Zotti et al. 2004, Rosa-Gonz’alez et al. 2004, Massardi et al. 2008 ). Here we investigate one generic class of SZ signals: the hot bubble surrounding a quasar powered by a supermassive black hole. Probing black hole energy feedback via SZ distortions is one direct observational route to understanding the growth and evolution of supermassive black holes and their role in structure formation. Analytic studies of this signal have been done by several authors (e.g., Natarajan & Sigurdsson 1999; Yamada, Sugiyama & Silk 1999; Lapi, Cavaliere & De Zotti 2003; Platania et al. 2002; Chatterjee & Kosowsky 2007); the current numerical work complements a similar study by Scannapieco Thacker and Couchman 2008. -------------- ------- ------------------ -------------------- -------------------- ------ ------ D4 33.75 $2\times216^{3}$ $2.75\times10^{8}$ $4.24\times10^{7}$ 6.25 0.00 D6 (BHCosmo) 33.75 $2\times486^{3}$ $2.75\times10^{7}$ $4.24\times10^{6}$ 2.73 1.00 -------------- ------- ------------------ -------------------- -------------------- ------ ------ Analytic models and numerical simulations of galaxy cluster formation indicate that the temperature and the X-ray luminosity relation should be related as $L_{x} \simeq T^{2}$ in the absence of gas cooling and heating (Peterson & Fabian 2006 and references therein). Observations show instead that $L_{x} \simeq T^{3}$ over the temperature range 2 to 8 kev with a wide dispersion at lower temperature, and a possible flattening above (Markevitch 1998; Arnaud & Evrard 1999; Peterson & Fabian 2006). The simplest explanation for this result is that the gas had an additional heating of 2 to 3 keV per particle (Wu, Fabian & Nulsen 2000; Voit et al. 2003). Several nongravitational heating sources have been discussed in this context (Peterson & Fabian 2006; Morandi Ettori & Moscardini 2007); quasar feedback (e.g., Binney & Tabor 1995; Silk & Rees 1998; Ciotti & Ostriker 2001; Nath & Roychowdhury 2002; Kaiser & Binney 2003; Nulsen et al. 2004) is perhaps the most realistic possibility. The effect of this feedback mechanism on different scales of structure formation have been addressed by several authors (e.g., Mo & Mao 2002; Oh & Benson 2003; Granato et al. 2004). The mechanism of quasar heating in cluster cores has been observationally motivated by studies from McNamara et al. 2005, Voit & Donahue 2005 and Sanderson, Ponman & O’Sullivan 2006 (see McNamara & Nulsen 2007 for a recent review). The impact of this nongravitational heating in galaxy groups, which have shallower potential wells and thus smaller intrinsic thermal energy than galaxy clusters, can also be substantial (Arnaud & Evrard 1999; Helsdon & Ponman 2000; Lapi, Cavaliere & Menci 2005). Observational efforts to detect the impact of this additional heating source in the context of quasar feedback have been carried out using galaxy groups in the Sloan Digital Sky Survey (SDSS) by Weinmann et al. 2006, and with a Chandra group sample by Sanderson, Ponman & O’Sullivan 2006. Detailed theoretical studies of galaxy groups using simulations which include quasar feedback have been undertaken by, e.g., Zanni et al. 2005, Sijacki et al. 2007, and Bhattacharya, Di Matteo & Kosowsky 2007. At smaller scales, the impact of quasar feedback has been investigated by Schawinski et al. 2007 with early-type galaxies in SDSS, and has also been studied in several theoretical models of galaxy evolution (e.g, Kawata & Gibson 2005; Bower et al. 2006; Cattaneo et al. 2007). Growing observational evidence points to a close connection between the formation and evolution of galaxies, their central supermassive black holes (e.g., Magorrian et al. 1998, Ferrarese & Merritt 2000, Tremaine et al.2002) and their host dark matter halos (Merritt & Ferrarese 2001; Tremaine et al. 2002). Several different groups have now investigated black hole growth and the effects of quasar feedback in the cosmological context (e.g., Scannapieco & Oh 2004; Di Matteo, Springel & Hernquist 2005; Lapi et al. 2006; Croton et al. 2006; Thacker, Scannapieco & Couchman 2006, Sijacki et al. 2007). Recently Di Matteo et al. (2008) carried out a hydrodynamic cosmological simulation following in detail the growth of supermassive black holes, using a simple but realistic model of gas accretion and associated feedback. Using this simulation, we construct maps of the SZ distortion around black holes of different masses at various redshifts. We demonstrate that the SZ signal around quasars scales with black hole mass and accretion rate. A similar approach has been taken by Scannapieco, Thacker & Couchman (2008) from a cosmological simulation with a different implementation of quasar feedback. Predictions for the SZ distortion due to a phenomenological treatment of galactic winds from simulations have been obtained previously by White, Hernquist & Springel (2002). [cc]{} \ \ \ \ Typically, the SZ distortions from quasar feedback result in effective temperature distortions at the micro-Kelvin level on arcminute angular scales. Recent advances in millimeter-wave detector technology and the construction of several single-dish and interferometric experiments (see Birkinshaw & Lancaster 2007 for a recent review on SZ observations), including the Sub Millimeter Array (SMA), the combined array for research in Millimeter wave Astronomy (CARMA), the Cornell Caltech Atacama Telescope (CCAT), the Atacama Large Millimeter Array (ALMA), and the Large Millimeter Telescope (LMT), along with SZ surveys like ACT and SPT, have brought detection of this signal into the realm of possibility. Although the direct detection of this signal from current SZ surveys seems unlikely, since the amplitude of fluctuation observed is at or below the noise threshold of ACT or SPT (Chatterjee & Kosowsky 2007), proposed submillimeter facilities offer some possibility for direct detection of this signal. An additional route for detection is cross-correlation of optically-selected quasar with microwave maps (Chatterjee & Kosowsky 2007; Scannapieco Thacker & Couchman 2008). However, this approach likely requires multifrequency observations to discriminate the SZ effect from intrinsic quasar emission or infrared sources. The paper is organized as follows. Section 2 describes the simulation that has been used in this work. In Section 3 we give a brief review of the SZ distortion and display the SZ maps derived from the simulation. Astrophysical results from the maps are presented in Section 4, including radial profiles around individual black holes and the correlation between black hole mass and SZ distortion. The concluding Section estimates detectability of these signals and summarizes future prospects. Throughout we use units with $c=k_B=1$. ----- --- ------- ----- --- ------- 3.0 0 0.034 3.0 0 0.240 2.0 3 0.003 2.0 2 0.013 1.0 4 0.013 1.0 1 0.005 ----- --- ------- ----- --- ------- [cc]{} \ \ \ \ [cc]{}\ Simulation ========== The numerical code uses a standard $\Lambda$CDM cosmological model with cosmological parameters from the first year WMAP results (Spergel et al.2003). The cosmological parameters are $\Omega_{m} = 0.3$, $\Omega_{\Lambda} = 0.7$, $H_{0} = 70$ km/s Mpc$^{-1}$ and Gaussian initial adiabatic density perturbations with a spectral index $n_{s}=1$ and normalization $\sigma_{8}=0.9$. (While the current lower value of $\sigma_8$ will affect the total number of black holes in a given volume, it should have little impact on the results for individual black holes presented here.) The simulation uses an extended version of the parallel cosmological Tree Particle Mesh-Smoothed Particle Hydrodynamics code GADGET2 (Springel 2005). Gas dynamics are modeled with Lagrangian smoothed particle hydrodynamics (SPH) (Monaghan 1992); radiative cooling and heating processes are computed from the prescription given by Katz, Weinberg, & Hernquist (1996). The relevant physics of star formation and the associated supernova feedback has been approximated based on a sub-resolution multiphase model for the interstellar medium developed by Springel & Hernquist (2003a). A detailed description of the implementation of black hole accretion and the associated feedback model is given in Di Matteo et al. 2008. Black holes are represented as collisionless “sink” particles that can grow in mass by accreting gas or by merger events. The Bondi-Hoyle relation (Bondi 1952; Bondi & Hoyle 1944; Hoyle & Lyttleton 1939) is used to model the accretion rate of gas onto a black hole. The accretion rate is given by $\dot{M}_{BH} = 4\pi[G^{2}M_{BH}^{2}\rho]/(c_{s}^{2} + v^{2})^{3/2}$, where $\rho$ and $c_{s}$ are density and speed of sound of the local gas, $v$ is the velocity of the black hole with respect to the gas, and $G$ is the gravitational constant. The radiated luminosity is taken to be $L_{r} = \eta (\dot{M}_{BH}c^{2})$ where $\eta=0.1$ is the canonical efficiency for thin disk accretion. It is assumed that a small fraction of the radiated luminosity couples to the surrounding gas as feedback energy $E_{f}$, such that $\dot{E_{f}} = \epsilon_{f} L_{T}$ with the feedback efficiency $\epsilon_f$ taken to be $5\%$. This feedback energy is put directly into the gas smoothing kernel at the position of the black hole (Di Matteo et. al 2008). The efficiency $\epsilon_{f}$ is the only free parameter in our quasar feedback model, and is chosen to reproduce the observed normalization of the $ M_{BH} -\sigma$ relation (Di Matteo, Springel & Hernquist 2005). This number is also consistent with the preheating in groups and clusters that is required to explain their X-ray properties (Scannapieco & Oh 2004). The feedback energy is assumed to be distributed isotropically for the sake of simplicity; however the response of the gas can be anisotropic. This model of quasar feedback as isotropic thermal coupling to the surrounding gas is likely a good approximation to any physical feedback mechanism which leads to a shock front which isotropizes and becomes well mixed over physical scales smaller than those relevant to our simulations and on timescales smaller than the dynamical time of the galaxies (see Di Matteo et al. 2008 and Hopkins & Hernquist 2006 for more detailed discussions). In actual active galaxies, the accretion energy is often released anisotropically through jets. As radio galaxy lobes can have substantial separations, it is conceivable that actual hot gas bubble morphology could differ somewhat from that in the simulations. This difference needs to be investigated with further simulations, but the overall detectability of the signal depends primarily on its amplitude and characteristic angular scale, which are determined mainly by the total energy injection as a function of time. The results for the signals and detectability presented here are unlikely to differ significantly due to more detailed modeling of the energy injection morphology. [cc]{}\ The formation mechanism for the seed black holes which evolve into the observed supermassive black holes today is not known. The simulation creates seed black holes in haloes which cross a specified mass threshold. At a given redshift, haloes are defined by a friends-of-friends group finder algorithm run on the fly. For any halo with mass $M>10^{10}h^{-1}M_\odot$ which does not contain a black hole, the densest gas particle is converted to a black hole of mass $M_{BH}=10^5h^{-1}M_\odot$; the black hole then grows via the accretion prescription given above and by efficient mergers with other black holes (Di Matteo et al. 2008). The simulations used in this paper have a box size of $33.75h^{-1}$ Mpc with periodic boundary conditions. The characteristics of the simulation are listed in Table 1, where $N_{p}$ is the total number of dark matter plus gas particles in the simulation, $m_{DM}$ and $m_{\rm gas}$ are their respective masses, $\epsilon$ gives the comoving softening length, and $z_{\rm end}$ is the final redshift of the run. For redshifts lower than 1, the fundamental mode in the box becomes nonlinear, so large-scale properties of the simulation are unreliable after $z=1$. The current results are derived for the D4 run with $2\times 216^{3}$ particles; we will present brief comparisons with the higher-resolution D6 (BHCosmo) run to demonstrate that our results are reasonably independent of resolution. A different simulation and feedback model have recently been used by Scannapieco, Thacker & Couchman (2008) to study the same issues. They associate the remnant circular velocity within a post merger event with black hole mass. The time scale on which the black hole shines at its Eddington luminosity is assumed to be a fixed fraction of the dynamical time scale of the system; the time scale and black hole mass scale are used to estimate the energy output from a black hole. Their feedback energy efficiency into the intergalactic medium is 5%, consistent with the assumption in our simulation. In contrast, our simulation tracks the time-varying feedback from a given black hole due to changing local gas density as the surrounding cosmological structure evolves. This simulation offers the possibility of tracking the accretion history and duty cycle of black hole emission for individual black holes, which we plan to address in future work. [c]{}\ \ ----- ------ ----- -------------------------------------------- ----------------------------------------------------- 3.0 2378 127 $\log y = 0.56 \log(M_{BH}/M_{\odot})-9.8$ $\log(dM_{BH}/dt) = 0.74 log(M_{BH}/M_{\odot})-8.1$ 2.0 3110 336 $\log y = 1.00 \log(M_{BH}/M_{\odot})-14$ $\log(dM_{BH}/dt) = 0.65 logM_{BH}/M_{\odot})-8.4$ 1.0 3404 404 $\log y = 1.90 \log(M_{BH}/M_{\odot})-22$ $\log(dM_{BH}/dt) = 1.4 log(M_{BH}/M_{\odot})-15$ ----- ------ ----- -------------------------------------------- ----------------------------------------------------- Results from the simulations ============================ The Sunyaev-Zeldovich Distortion and Maps ----------------------------------------- The Compton $y$-parameter characterizing the non-relativistic thermal SZ spectral distortion is proportional to the line-of-sight integral of the electron pressure: $$y = 2\int dl \,\sigma_{T}n_{e}\frac {T_{e}}{m_{e}} \label{ydef}$$ where $\sigma_{T}$ is the Thompson cross section, $n_{e}$ and $T_{e}$ are electron number density and temperature, and the integral is along the line of sight. The effective temperature distortion at a frequency $\nu$ is given by (Sunyaev & Zeldovich 1972) $$\frac{\Delta T}{T_{0}} = \left[x\coth(x/2)-4\right]y,$$ where $x=h\nu/T_0$ and $T_{0}$ is the CMB temperature equal to 2.73 K. Figure 1 shows $y$-distortion maps centered around two representative black holes in the simulation at redshifts 3, 2 and 1 (from left to right respectively). The two black holes are the most massive and the second most massive black hole at redshift 3.0 in the simulation. We have chosen the two most massive black holes in the simulation since the amplitude of the SZ distortion from the most massive black holes is relevent within the realm of current and future experiments. These maps were made by evaluating the line-of-sight integral in Eq. \[ydef\] through the appropriate portion of the simulation box. In order to characterize the large scale structure and associated $y$-distortions surrounding the black holes, we show a large region of the simulation within a comoving radius of 2.5 Mpc of the black hole in question, displayed with a comoving box size of 5 Mpc (top and third row for the most massive black hole and for another black hole in the simulation respectively) as well as a zoom into the central 200 kpc box (second and forth rows). The smaller region (200 Kpc) is the relevent scale of interest when looking at the direct impact of the central black hole to its surrounding gas; in the larger box multiple black holes are present. The mass of the central black hole is $7.35\times 10^{8}M_{\odot}$ at $z=3$, $2.76\times 10^{9}M_{\odot}$ at $z=2$, and $4.32\times 10^{9}M_{\odot}$ at $z=1$ (top two rows) and $7.11\times 10^{8}M_{\odot}$ at $z=3$, $8.2\times 10^{8}M_{\odot}$ at $z=2$, and $2.11\times 10^{9}M_{\odot}$ at $z=1$ (third and forth row). The feedback energy associated with black hole accretion creates a hot bubble of gas surrounding the black hole, which, as shown in the figures, grows significantly in size as redshift decreases. The growing hot bubble is roughly spherical by $z=1$, in agreement with the assumption of the analytic spherical blast wave model in Chatterjee and Kosowsky (2007). In order to further characterize this expanding hot bubble, Fig. 2 displays maps of the difference between the two simulation with black hole modeling and without, in the same 200 kpc regions of Fig. 1. The top and second rows show the most massive black hole at $z=3$ and $1$ respectively while the third and forth row show the second black hole at the same redshifts. In this figure, the left column shows the logarithm of the $y$ distortion, the [**central**]{} column is the logarithm of the mass-weighted temperature in units of Kelvin and the right is the logarithm of projected electron number density in units of cm$^{-2}$,. At $z=3$ a residual $y$ distortion is evident and concentrated around the black hole, with little effect further out; the peak $y$ distortion due to the black hole is on the order of $10^{-7}$, corresponding to an effective temperature shift of the order of 1 $\mu$K. By $z=1$, the energy injected into the center has propagated outwards, forming a hot halo around the black hole. Table 2 shows the respective black hole accretion rates at different redshift for the two black holes in Figure 1 and 2. It is evident that the highest amplitude of $y$ distortion is associated with the most active, high-redshift epochs of accretion, when large amounts of energy are coupled to the surrounding gas via the feedback process. At $z=1$ the black hole accretion rate has dropped so the $y$ distortion has a smaller amplitude but has spread over a larger region (Fig. 2). Angular Profiles ---------------- For the two black holes shown in Figure 1 we see an overall enhancement in the SZ signal due to quasar feedback. This agrees with the simulations in Scannapieco, Thacker & Couchman 2008. To further quantify the effects of quasar feedback we average the SZ signal in annuli around the black hole and examine the angular profile of the resulting $y$ from the hot bubble in Figure 3 and 4. Figure 3 shows the average angular profiles of the total $y$ distortion around the two objects in the maps in Figure 1. The black dashed, blue dot-dashed and red solid lines are for $z=1$, $z=2$, and $z=3$ respectively. In both cases the $y$ increases with time between $\sim$ 10 to 25 arcsecond separation from the black hole. $y$ gets steadily larger as the feedback energy spreads over this volume (see also Fig.4). At $z=3$ the $y$ profile is steeper in the central regions with a significant peak (in particular for the second quasar) at scales below 5 arcseconds. The bumps in the profiles are due to concentrations of hot gas or occasional other black holes which are included in the total average signal. $y$ typically reaches its highest central peaks at time when the quasar is most active (the black hole accretion rate is high - see Table 2), and hence large amounts of energy are coupled to the surrounding gas according to our feedback prescription. For example, the $z=3$ curve in the right panel shows the black hole at a particularly active phase; the central $y$ distortion corresponds to a temperature difference of over 4 $\mu$K. At $z=2$ this central distortion is smaller by a factor of 20, while it is larger by a factor of 10 at an angular separation of 10 arcseconds. Figure 3 shows the total SZ effect in the direction of a quasar resulting from the superposition of the SZ signature from quasar feedback plus the SZ distortion from the rest of the line of sight due to the surrounding adiabatic gas compression, which is expected to form an average background level in the immediate vicinity of the back hole. In order to clearly disentangle the contribution due to quasar feedback, in Figure 4, we plot the fractional change in $y$ distortion between the simulation with and without black hole modeling, at two different redshifts. These are the profiles corresponding to the maps shown in Figure 2. It is clear that the local SZ signature is largely dominated by the energy output from the black hole, giving a factor between 300 to over 3000 (for the second black hole at $z=3$ in right panel) increase in $y$ near the black hole. Our results are also consistent with the expected $y$ distortion from the thermalized gas in the host halos containing these black holes (which are on the order $10^{12}M_{\odot}$ to $10^{13}M_{\odot}$) and is the range $10^{-9}$ to $10^{-7}$ (see also Komatsu & Seljak 2002). The largest peak in $y$ distortion enhancement due to quasar feedback generally lies within 5 arcseconds of the black hole. Black Hole Mass Scaling Relations --------------------------------- Since the SZ effect from the region around the black holes we analyzed in the previous section is dominated by the quasar feedback, we investigate whether a correlation between black hole mass and $y$ distortion exists for the population as a whole (see also Colberg & Di Matteo 2008 for other scaling relations between $M_{BH}$ and host properties). The top row of Figure 5 plots the mean $y$ distortion, computed over a sphere of radius 200 kpc/$h$ (i.e. the same as in the maps, corresponding to 20 arcseconds) versus black hole mass for all black holes in the simulations with $M_{BH} > 10^{7}M_{\odot}$ at $z=1,2$ and $3$ (from right to left respectively). The size of the region is chosen to sample the entire region of distortion due to the quasar feedback, while minimizing bias from the local environment (Fig. 3 and 4). The mass cut-off is chosen to (a) minimize effects due to lack of appropriate resolution in the simulations as well as (b) produce SZ distortions that may be detectable by current or upcoming experiments. Simple power law fits to the $y$ distortion as a function of black hole mass show a redshift evolution with the scaling becoming steeper with decreasing redshift. Table 3 summarizes our results from the fits. The trends show a close correspondence between the mean $y$ parameter and the total feedback energy as measured from $y$. In order to further investigate the reason for $y-M_{BH}$ relations, in the bottom row of Figure 5 we plot the accretion rate versus black hole mass at redshifts 3, 2, and 1 for the same sample as in the top panel and perform similar power-law fits (see Table 3). The trends in accretion rate versus $M_{BH}$ are qualitatively similar to the top panel, demonstrating the connection of the $y$ distortion due to quasar feedback with the black hole accretion rate and black hole mass. In particular, at $z=1$ the relations get steeper as expected if the largest fraction of black holes are accreting according to the Bondi scaling (e.g., $\dot{m} \propto M_{BH}^2$) and shallower with increasing redshift when most black holes are accreting close to the critical Eddington value (e.g., $\dot{m} \propto M_{BH}$). Of course, the accretion rate depends not only on black hole mass but also on the properties of the local gas and is also regulated by the large scale gas infall driven by major mergers, which peak at higher redshifts (Di Matteo et al. 2008). The ratio of the slopes (accretion rate to y distortion) for the fits shown in table 3 are 1.32, 0.65 and 0.73 at redshifts 3.0, 2.0 and 1.0 respectively. This shows the agreement of the top and the bottom panels in Figure 5, and the close connection between accretion history and SZ distortion: the SZ effect tracks closely quasar feedback and is promising probe of black hole accretion. The largest amplitudes of SZ signal from quasar is expected from $z\sim 2-3$ at a time close to the peak of the quasar phase in galaxies. [c]{}\ \ Resolution Test --------------- In the previous section we have made use of the D4 (Table 1) simulations from our analysis. At this resolution we have two identical realizations, with and without black hole modeling, allowing us to carry out detailed comparisons of the effects of the quasar feedback. We now wish to assess possible effects due to numerical resolution by making use of the D6 (BHCosmo) run (see also Di Matteo et al. 2008, Croft et al. 2008 and Bhattacharya DiMatteo & Kosowsky 2008 for additional resolution studies). Figure 6 shows the $y$ distortion maps for the most massive black hole at redshifts 3, 2, and 1. The top row is for the higher-resolution BHCosmo run and the bottom row is for the lower-resolution run (D4). Our results at the lower resolution appear reasonably well converged, though with some differences. The central black hole masses in the two runs differ somewhat. At $z=1$, $2$, and $3$, the black hole masses in the D4 and BHCosmo run are ($4.29 \times 10^{9}M_{\odot}$, $2.96 \times 10^{9} M_{\odot}$), ($2.76 \times 10^{9} M_{\odot}$, $1.85 \times 10^{9} M_{\odot}$) and ($7.35 \times 10^{8}M_{\odot}$, $8.56 \times 10^{8}M_{\odot}$) respectively. It clear that the difference in resolution is affecting the black hole mass as expected from modest changes in mass accretion rate (which is sensitive to the gas properties close to the black hole). Also, more small scale structure in the gas distribution is evident at higher resolution, as expected. This affects the amplitude of the total SZ flux which is enhanced by about 6% at $z=2$ and by about 22% at $z=3$ (when it is most peaked around the black hole) in the higher resolution run. Detectability and Discussion ============================ Observationally, quasar feedback is directly detectable by resolving Sunyaev-Zeldovich peaks on small angular scales of tens of arcseconds with amplitudes of up to a few $\mu$K above the immediately surrounding region. The combination of angular scale and small amplitude make detecting this effect very challenging, at the margins of currently planned experiments. The necessary sensitivity requires large collecting areas, while the angular resolution needed points to an interferometer in a compact configuration, or a large single-dish experiment. Since the SZ signal is manifested as a peak over the surrounding background level, a region substantially larger than the SZ peak must be imaged. This requires a telescope having sufficient resolution to resolve the central peak in the SZ distortion in an SZ image and enough field of view so that the peak could be identified. An example is the compact ALMA subarray known as the Atacama Compact Array (ACA), composed of 12 7-meter dishes. The ALMA sensitivity calculator gives that the synthesized beam for this array is about 14 arcseconds, and the integration time required to attain 1 $\mu$K sensitivity per beam at a frequency of 145 GHz and a maximum band width of 16 GHz is on the order of 1000 hours (ALMA sensitivity calculator). A very deep survey with this instrument could detect the SZ effect from individual black holes. The 50-meter Large Millimeter-Wave Telescope instrumented with the AzTEC bolometer array detector will have a somewhat similar sensitivity but detectibility would require a very deep (thousands of hours) integration time. The Cornell-Caltech Atacama Telescope (CCAT), a 25-meter telescope, estimates a possible pixel sensitivity for SZ detection at 150 GHz of 310 $\mu{\rm K}\,{\rm s}^{1/2}$ for 26 arcsecond pixels, so a 30 hour observation could give 1$\mu$K pixel noise. These pixels would not be small enough to resolve the hot halo around a black hole, but might be able to detect the difference in a single pixel due to black hole emission compared to the surrounding pixels. Aside from raw sensitivity and angular resolution, a serious difficulty with direct detection is the confusion limit from infrared point source emission; these sources are generally high-redshift star forming galaxies with a high dust emission. CCAT estimates show that their one-source-per-beam confusion limit will be around 6 $\mu$K at 150 GHz (Golwala 2006). This will present substantial difficulties for detecting a 1 $\mu$K temperature distortion if accurate. It is noted that the observations in the sub-millimeter band is limited by confusion noise and so another possibility of direct detection of the signal through radio frequency telescopes could be considered. Massardi et al. 2008 shows that the confusion due to dusty galaxies is lower at 10 GHz then at 100 GHz. The authors show that for galactic scale SZ effect the optimal frequency range for detection is between 10 to 35 GHz. However substantial confusion from radio galaxies at these low frequency observations would still be a challenging issue in the direct detection of the signal. Given these substantial difficulties associated with direct detection, an alternate route may be necessary. Cross-correlation of arcminute-resolution microwave maps with optically selected quasar or massive galaxies is a second possible detection strategy (Chatterjee and Kosowsky 2007, Scannapieco, Thacker, and Couchman 2008). By averaging over large numbers of objects, we can have an estimate of a small mean black hole distortion signal from the noise in the maps. The primary challenge with this technique is the direct emission from quasar in the microwave band. It may be possible to select a sample of quasar which is sufficiently radio-quiet that the cross-correlation is not dominated by the intrinsic emission. Another possibility is to select massive field galaxies under the assumption that they harbor a central massive black hole which at one time was active; the hot bubble produced has a cooling time comparable to the Hubble time, so formerly active galaxies should still have an SZ signature. Finally, the SZ effect from black holes is in addition to the SZ emission from any hot gas in which the black hole’s host galaxy is embedded. Massive galaxies trace large-scale structure, and any cross-correlation will also detect this signal. Although the observational requirements for the cross-correlation method are plausible the scopes for detectibility with this method is still limited by confusion noise. Stacking microwave (SZ) maps in the direction of known quasars would also serve as an independent route in detecting the signal (Chatterjee & Kosowsky 2007). This can improve the signal to noise by a substantial amount although this method would still be limited by the uncertainties described above. Quantifying in detail the observable signal (which will need to be disentangled from other confusions such as dusty galaxies, radio galaxies etc.) for the possible direct detections methods or from cross-correlation analysis that we have discussed is beyond the scope of this paper and we defer it to a future work. The simulations and maps presented here provide a basis for further modeling of all these effects. The main conclusions drawn from this work are summarized as follows. We have used the first cosmological simulations to incorporate realistic black hole growth and feedback to produce simulated maps of the Sunyaev-Zeldovich distortion of the microwave background due to the feedback energy from accretion onto supermassive black holes. These simulations address the rapid accretion phases of black holes: periods of strong emission are typically short-lived and require galaxy mergers to produce strong gravitational tidal forcing necessary for sufficient nuclear gas inflow rates (Hopkins, Narayan & Hernquist 2006; Di Matteo et al. 2008). The result is heating of the gas surrounding the black hole, so that the largest black holes produce a surrounding hot region which induces a $y$-distortion (related to a temperature distortion) with a characteristic amplitude of a few $\mu$K. We have obtained a scaling relation between the black hole mass and their SZ temperature decrement, which in turn is a measure of the amount of feedback energy output. The correspondence between the y distortion and the accretion rates is not exact but there is a close association which shows the correlation between feedback output and black hole activity. From our results we have shown that with the turn on of AGN feedback the signal gets enhanced largely and the enhancement is predominant at angular scales of 5 arcseconds. Finally we have shown that there is a fair probability of detecting this signal even from the planned sub millimeter missions. The role of energy feedback from quasars and from star formation is known to have substantial impact on the process of galaxy formation and evolution of the intergalactic medium, but the details of this process are not well understood. Probes based on Sunyaev-Zeldovich distortions are challenging, but an eventual detection can be used to put useful constraints and checks on models of AGN feedback. acknowledgments =============== SC would like to thank Bruce Partridge and James Moran for helpful discussions on experimental capabilities of various telescopes. Special thanks to Mark Gurwell for helping with the sensitivity calculation for SMA. SC and AK would also like to thank Christoph Pfrommer for some initial discussions on the project. Thanks to Jonathan Las Fargeas who helped with the analysis, supported by NSF grant 0649184 to the University of Pittsburgh REU program. We would also like to thank the referee for valuable suggestions on improvement of the paper.This work was supported at the University of Pittsburgh by the National Science Foundation through grant AST-0408698 to the ACT project, and by grant AST-0546035. At CMU this work has been supported in part through NSF AST 06-07819 and NSF OCI 0749212. SC was also partly funded by the Zaccheus Daniel Fellowship at the University of Pittsburgh. Aghanim N., Balland, C., & Silk, J. 2000, A&A, 357, 1 Arnaud M., & Evrard A. E., 1999, [MNRAS]{}, 305, 631 Babich, D. & Loeb, A. 2007, [MNRAS]{}, 374, L24 Bennett et al., 2003, ApJs, 148, 1 Bhattacharya S., Di Matteo T., & Kosowsky A., 2007, ArXiv e-prints, 710, arxiv:07105574 Binney J., & Tabor G., 1995, [MNRAS]{}, 276, 663 Birkinshaw M., & Lancaster K., 2007, NewAR, 51, 346 Bondi H., 1952, [MNRAS]{}, 112, 195 Bondi H., & Hoyle F., 1944, [MNRAS]{}, 104, 273 Bower R. G., Benson A. J., Mallbon R., Helly J.C., Frenk C. S., Baugh C. M., Cole S., & Lacey C. G., 2006, [MNRAS]{}, 370, 645 Carlstrom J., E., Holder, G., P., & Reese, E. D., 2002, ARA&A, 40, 643 Cattaneo A. et al., 2007, [MNRAS]{}, 377, 63 Chatterjee S., & Kosowsky A., 2007, ApJL, 661, L113 Ciotti L., & Ostriker J. P., 2001, 551, 131 Colberg J. & Di Matteo T., 2008, [MNRAS]{}, in press Croft R.A.C., Di Matteo T., Springel, V., Hernquist, L., 2008, [MNRAS]{}, submitted Croton D. et al., 2006, [MNRAS]{}, 365, 11 Dawson K. S. et al., 2002 [ApJ]{}, 581, 86 De Zotti G., Burigana C., Cavaliere A., Danese L., Granato G. L., Lapi, A., Platania, P., & Silva, L., 2004, AIPC, 703, 375D Di Matteo T., Springel V., & Hernquist L., 2005, Nature, 433, 604 Di Matteo T., Colberg J., Springel V., Hernquist L., & Sijacki D., 2008, [ApJ]{}, 676, 33 Ferrarese L., & Merritt D., 2000, [ApJ]{}, 539, L9 Golwala S., 2006, http://www.astro.caltech.edu/$\sim$golwala/talks /CCATFeasibilityStudySZScience.pdf Granato G. L., De Zotti G., Silva, L., Bressan A., & Danese L., 2004, [ApJ]{}, 600, 580 Helsdon S. F., & Ponman T. J., 2000, [MNRAS]{}, 315, 356 Hopkins P F., Hernquist L., 2006, ApJS, 163, 50 Hopkins P F., Narayan R., & Hernquist L., 2006, [ApJ]{}, 643, 641 Hoyle F., & Lyttleton R. A., 1939, in proceedings of the Cambridge Philosophical Society, 405 Iliev, I. T., Pen, U.-L., Bond, J. R., Mellema, G., & Shapiro, P. R. 2006, New Astron. Rev. 50, 909. Kaiser C. R., & Binney J. J., 2003, [MNRAS]{}, 338, 837 Katz N., Weinberg D. H., & Hernquist L., 1996, ApJS, 105, 19 Kawata D., & Gibson B. K., 2005, [MNRAS]{}, 358, L16 Komatsu E., & Seljak U., 2002, [MNRAS]{}, 336, 1256 Kosowsky A., 2003, AIP Conf. Proc. 666, 325. Kosowsky A. et al., 2006, New Astron. Rev. 50, 969. Lapi A., Cavaliere A., & De Zotti G., 2003, ApJl, 597, L93 Lapi A., Cavaliere A., & Menci N., 2005, ApJ, 619, 60 Lapi A., Shankar F., Mao J., Granato G. L., Silva L., De Zotti G., & Danese L., 2006, ApJ, 650, 42 Magorrian, J., et al., 1998, [AJ]{}, 115, 2285 Majumdar, S. Nath, B., & Chiba, M. 2001, [MNRAS]{}, 324, 537 Markevitch M., 1998, [ApJ]{}, 504, 27 Marshall et al. 2001, [ApJ]{}, 551, L1-L4 Massardi M., Lapi A., de Zotti G., Ekers R. D., & Danese L., 2008, [MNRAS]{}, 384, 701 McNamara B. R., Nulsen P. E. J., Wise M. W., Rafferty D. A., Carilli C., Sarazin C. l., & Blanton E. L., 2005, Nature, 433, 45 McNamara B. R., & Nulsen P. E. J., 2007, ARA&A, 45, 117 McQuinn, M., Furlanetto, S. R., Hernquist, L., Zahn, O., & Zaldarriaga, M. 2005, [ApJ]{}, 630, 643 Merritt D., & Ferrarese L., 2001, [ApJ]{}, 547, 140 Mo H. J., & Mao S., 2002, [MNRAS]{}, 333, 768 Morandi A., Ettori S., & Moscardini L., 2007, [MNRAS]{}, 379, 518 Monaghan J. J., 1992, ARA&A, 30, 543 Natarajan P., & Sigurdsson S., 1999, [MNRAS]{}, 302, 288 Nath B. B., & Roychowdhury S., 2002, [MNRAS]{}, 333, 145 Nulsen P., Mcnamara B., David L., & Wise M., 2004, cosp, 35, 3235 Oh S. P., & Benson A., 2003, [MNRAS]{}, 342, 664 Oh, S. P., Cooray, A., & Kamionkowski, M. 2003, [MNRAS]{}, 342, 20 Peterson R., & Fabian A., 2006, Phys. Rep., 427, 1 Platania P., Burigana C., De Zotti G., Lazzaro E., & Bersanelli M., 2002, [MNRAS]{}, 337, 242 Reichardt et al., 2008, Arxiv e-prints, 801, arxiv:08011491 Rajguru et al., 2005, [MNRAS]{}, 363, 1125 Rosa-Gonzalez D., Terlvich R., Terlvich E., Friaca A., & Gaztanaga E., 2004, [MNRAS]{}, 348, 669 Roychowdhury S., Ruszkowski, M., & Nath, B. B. 2005, [ApJ]{}, 634, 90 Ruhl J. E. et al., 2004, Proc. SPIE, 5498, 11 Sanderson A. J. R., Ponman T. J., & O’Sullivan E., 2006, [MNRAS]{}, 372, 1496 Scannapieco E., & Oh S. P., 2004, [ApJ]{}, 608, 62 Scannapieco E., Thacker R. J., Couchman H. M. P., 2008, [ApJ]{}, 678, 674 Schawinski K., Tomas D., Sarzi M., Maraston C., Kaviraj S., Joo S., Yi S., Silk J., 2007, ArXiv e-prints, 709, arXiv:0709.3015 Sijacki D., Springel V., DiMatteo T., & Hernquist L., 2007, [MNRAS]{}, 380, 877 Silk J., & Rees M. J., 1998, A&A, 331, L1 Spergel D. N. et al., 2003, ApJS, 148, 175 Spergel D. N. et al., 2007, ApJS, 170, 377 Springel V., 2005, [MNRAS]{}, 364, 1105 Springel V., & Hernquist L., 2003a, [MNRAS]{}, 339, 312 Sunyaev R. A., & Zel’dovich Ya. B., 1972, Comments Astrophys. Space Phys., 4, 173 Thacker R. J., Scannapieco E., & Couchman H. M. P., 2006, [ApJ]{}, 653, 86 Tremaine, S. et.al. 2002, [ApJ]{}, 574, 740 Voit G. M., Balogh M. L., Bower R. G., Lacey C. G., & Bryan G. L., 2003, [ApJ]{}, 593, 272 Voit G. M., & Donahue M., 2005, [ApJ]{}, 634, 955 Weinmann S. M., vandenBosch F. C., Yang X., Mo H. J., Croton D. J., & Moore B., 2006, [MNRAS]{}, 372, 1161 White M., Hernquist L., Springel V., 2002, [ApJ]{}, 579, 16 Wu K. K. S., Fabian A. C., & Nulsen P. E. J., 2000, [MNRAS]{}, 318, 889 Yamada M., Sugiyama N., & Silk J. 1999, [ApJ]{}, 522, 66 Zanni C., Murante G., Bodo G., Massaglia S., Rossi P., & Ferrari A., 2005, A&A, 429, 399

--- abstract: 'Feature engineering is one of the most important but most tedious tasks in data science. This work studies automation of feature learning from relational database. We first prove theoretically that finding the optimal features from relational data for predictive tasks is NP-hard. We propose an efficient rule-based approach based on heuristics and a deep neural network to automatically learn appropriate features from relational data. We benchmark our approaches in ensembles in past Kaggle competitions. Our new approach wins late medals and beats the state-of-the-art solutions with significant margins. To the best of our knowledge, this is the first time an automated data science system could win medals in Kaggle competitions with complex relational database.' author: - | Hoang Thanh Lam[^1]\ Dublin Research Laboratory\ IBM Research\ Dublin, Ireland\ `t.l.hoang@ie.ibm.com`\ Tran Ngoc Minh\ Dublin Research Laboratory\ IBM Research\ Dublin, Ireland\ `m.n.tran@ibm.com`\ Mathieu Sinn\ Dublin Research Laboratory\ IBM Research\ Dublin, Ireland\ `mathsinn@ie.ibm.com`\ Beat Buesser\ Dublin Research Laboratory\ IBM Research\ Dublin, Ireland\ `beat.buesser@ie.ibm.com`\ Martin Wistuba\ Dublin Research Laboratory\ IBM Research\ Dublin, Ireland\ `martin.wistuba@ie.ibm.com`\ bibliography: - 'sigproc.bib' title: Neural Feature Learning From Relational Database --- Introduction ============ Often data science problems require machine learning models to be trained on table with one label and multiple feature columns. Data scientists must hand-craft these features from raw data. This process is called *feature engineering* and is one of the most tedious tasks in data science. Data scientists report that up to 95% of the total project time must be allocated to carefully hand-crafting features to achieve competitive models, for example see [^2]. Here, we study methods to automate this step of feature engineering specifically for relational database because of four reasons. First, as mentioned data science projects involve many tedious trial-and-error steps and automation of these steps can significantly improve the productivity of data scientists. Second, before investing in data science projects with uncertain outcome quick estimates of the results can be achieved using automation. Third, automation democratizes data science by facilitating access for people with limited data science skills. Fourth, relational data is reported as the most popular type of data in industry reported by a recent survey of 14000 data scientists by [@survey] with at least 65% working daily with relational data. The full automation of feature engineering for general purposes is very challenging, especially in applications where specific domain knowledge is an advantage. However, recent work by [@DFS] indicates that for relational data impressive performances like top 24-36% of all participants on Kaggle competitions can be achieved fully automated. A disadvantage of the cited work is its limitation to numerical data and neglection of temporal information. Moreover, the set of features contains redundant information because it is extracted using a set of predefined rules irrespective of the domain and targeted problems. In this work, we extend this prior art with a rule-based approach to deal with non-numerical and temporally ordered data and further improve this approach via feature learning using a deep neural network to learn relevant transformations. We present experiments with different completed Kaggle competitions where our methods outperform the state-of-the-art baselines and achieve the top 3-8% of all participants. These results are achieved with minimal effort on data preparation and manual feature engineering within one week, while the competitions lasted for at least two months. Backgrounds =========== Let $D = \{T_0,T_1,\cdots,T_n\}$ be a database of tables. Consider $T_0$ as the main table which has a target column, several foreign key columns and optional attribute columns. Each entry in the main table corresponds to a training example. ![A toy database and its relational graph.[]{data-label="fig:database"}](./toys.pdf){width="1.0\columnwidth"} Figure \[fig:database\] shows an example database with 3 tables. User table (main) contains a prediction target column indicating whether a user is a loyal customer. User shopping transactions are kept in the order table and the product table includes product price and names. A *relational graph* is a graph where nodes are tables and edges are links between tables via foreign-key relationships. Figure \[fig:database\] shows the relational graph of the database in the same figure. \[Joining path\] A joining path is a sequence $p = T_0 \xrightarrow{c_1} T_1 \xrightarrow{c_2} T_2 \cdots \xrightarrow{c_k} T_k \mapsto c$, where $T_0$ is the main table, each $T_i$ is a table in the database, $c_i$ is a foreign-key column connecting tables $T_{i-1}$ and $T_i$, and $c$ is a column (or a list of columns) in the last table $T_k$ on the path. Joining the tables following the path $p = main \xrightarrow {UserID} order \xrightarrow {ProductID} product \mapsto Price$, we can obtain the price of all products that have been purchased by a user. The joined result can be represented as a relational tree defined in Definition \[def:relational tree\] below. \[Relational tree\] Given a training example with identifier $e$ and a joining path $p = T_0 \xrightarrow{c_1} T_1 \xrightarrow{c_2} T_2 \cdots \xrightarrow{c_k} T_k \mapsto c$, a relational tree, denoted as $t^p_e$, is a tree representation of the joined result for the entity $e$ following the joining path $p$. The tree $t^p_e$ has maximum depth $d = k$. The root of the tree corresponds to the training example $e$. Intermediate nodes at depth $0 < j < k$ represent the rows in the table $T_j$. A node at depth $j-1$ connects to a node at depth $j$ if the corresponding rows in tables $T_{j-1}$ and table $T_j$ share the same value of the foreign-key column $c_j$. Each leaf node of the tree represents the value of the data column $c$ in the last table $T_k$. \[def:relational tree\] \[exp:relational tree\] Figure \[fig:relational tree transformation prior art\].a shows a relational tree for $UserID=1$ following the joining path $p = main \xrightarrow {UserID} order \xrightarrow {ProductID} product \mapsto Price$. As can be seen, the that user made two orders represented by two intermediate nodes at depth $d=1$. Besides, order 1 includes two products with $ProductID = 1$ and $ProductID = 2$, while order 4 consists of products with $ProductID = 1$ and $ProductID = 3$. The leaves of the tree carry the price of the purchased products. \[Tree transformation\] A transformation function $f$ is a map from a relational tree $t^p_e$ to a fixed size vector $x \in R^l$, i.e. $f(t^p_e) = x$. Vector $x$ is called a feature vector. In general, feature engineering looks for relevant tree transformations to convert a tree into input feature vectors for machine learning models. For example, if we sum up the prices of all products carried at the leaves of the tree in Figure \[fig:relational tree transformation prior art\].a we obtain the purchased product price sum which can be a good predictor for the loyalty customer target. Problem definition and complexity analysis ========================================== Given a relational database $DB$ with $m$ training examples $E=\{(e_1, y_1), (e_2, y_2), \cdots, (e_m, y_m)\}$, where $Y=\{y_1, \cdots, y_m\}$ is a set of labels. Let denote $P =\{p_1,p_2,\cdots,p_q\}$ as a set of joining paths extracted from the relational graph of $DB$. Recall that for each entity $e_j$ following a joining path $p_i$ we can obtain a relational tree $t^{p_i}_{e_j}$. Let $f_{p_i} \in F$ (the set of candidate transformations) be a tree transformation function associated with the path $p_i$, denote $f_{p_i}(t^{p_i}_{e_j}) = x^i_j$ as the feature vector extracted for $e_j$ by following the path $p_i$. Let $g(x^1_j \oplus x^2_j \oplus \cdots \oplus x^q_j) = \hat{y}_j$, be a machine learning model that estimates $y_j$ from a concatenation of the feature vectors obtained from $q$ joining paths. Denote $L_{P, F, g}(Y,\hat{Y})$ as the loss function defined over the set of ground-truth labels and the set of estimated labels $\hat{Y}=\{\hat{y}_1, \cdots, \hat{y}_m\}$ \[prob:Feature learning\] Given a relational database, find the set of joining paths, transformations and models such that $P^*, F^*, g^* = argmin L_{P, F, g}(Y,\hat{Y})$. The following theorem shows that Problem \[prob:Feature learning\] as an optimization problem is NP-hard even when $F$ and $g$ are given (see the proof in supplementary material). Given a relational graph, the candidate set of transformations $F$ and model $g$ , searching for the optimal path for predicting the correct label is an NP-Hard problem. Since the problem is hard, in sub-section \[subsec:path generation\], we explain efficient heuristic approaches for joining path generation. On the other hand, finding the best model $g$ when the features are given is a model selection problem. That problem has been intensively studied in the machine learning literature, therefore, we limit the scope of this work to finding the relevant tree transformations. A rule based approach for tree transformation ============================================= Given relational trees, there are different ways to transform the trees into features. In this section, we discuss rule-based approaches predefining tree transformations based on heuristics. The method discussed in this section is an extension of the Data Science Machine (DFS) by [@DFS] so we first briefly recall the DFS algorithm. ![Tree transformations: DFS uses augmented aggregation functions at each level of the tree. OneBM flattens out the trees and transforms the flattened results using transformations that support unstructured and with temporally ordered data. R2N uses supervised learning to transform and embed any type of data using relational recurrent neural nets.[]{data-label="fig:relational tree transformation prior art"}](./tree_transform.pdf){width="1.0\columnwidth"} In DFS, transformation function $f$ is a composition of basic aggregation functions such as AVG, SUM, MIN and MAX augmented at each depth level of a tree. For instance, for the relational tree $t^p_1$ in Figure \[fig:relational tree transformation prior art\].a, a feature can be collected for $UserID=1$ by applying MEAN and MAX at the root and first depth level respectively. The aggregation function at each node takes input from the children and outputs a value which is in turn served as an input to its parent. The example in Figure \[fig:relational tree transformation prior art\].a produces a feature $MEAN(MAX(10,20),MAX(10,5)) = 15$ corresponding to the average of the maximum price of purchased products by a user, which could be a good predictor for the user loyalty target. DFS works well for numerical data, however, it does not support non-numerical data. For instance, if the product name instead of price is considered, the given set of basic transformations become irrelevant. Moreover, when the nodes of the trees have temporal order the basic aggregations ignore temporal patterns in the data. Therefore, we extend DFS to One Button Machine (OneBM) to deal with such situation (see the supplementary material). In OneBM, the tree is first flattened out by a GroupBy operation at the root node to produce a set or a sequence of values as can be seen in Figure \[fig:relational tree transformation prior art\].b . Depending on the type of the values carried at the leaves, different transformations are applied. For instance, the tree $t^p_1$ in Figure \[fig:relational tree transformation prior art\] can be flattened out into a multi-set: $s = \{10, 20, 10, 5\}$. When there is an order associated with these values the multi-sets turn into timeseries. On the other hand, if the data column is the product name instead of price, we obtain an itemset or a sequence of purchased products. For each type of such data, popular transformations such as correlated itemsets, correlated subsequences, autocorrelation coefficients can be extracted. A full list of the most popular transformations for each special type of data is described in Table \[tab:rules\]. This simple approach allows us to incorporate additional user-defined transformations and being able to handle complex transformations beyond simple aggregations. As can be seen in experiments, this approach produces very good results in most Kaggle competitions. Once features are generated by the given set of rules, feature selection is needed to remove irrelevant features. Feature selection is a well-studied topic, please refer to the supplementary material for the discussion of our choice in feature selection. The rule based approaches like OneBM and DFS specify the transformation functions based on heuristics regardless of the domain. In practice, predefined transformations can’t be universally relevant for any use-case. In the next section we introduce an approach to go around this issue. **Data type** **transformation functions** ------------------------------- -------------------------------------------------------- numerical as is categorical order by frequency and use order as the transformation timestamp calendar features, gap to cut-off time if exists timestamp series series of gaps to cut-off time if exists number multi-set avg, variance, max, min, sum, count multi-set of items count, distinct count, high correlated items timeseries avg, max, min, sum, count, variance, recent($k$), normalized count and sum to the max gap to cut-off sequence, texts, set of texts count, distinct count, high correlated symbols : Transformation rules in OneBM[]{data-label="tab:rules"} Neural feature learning ======================= In this section, we discuss an approach that learns transformation from labelled data rather than being specified *a-priori* by the user. Relational recurrent neural network {#sec:r2n} ----------------------------------- To simplify the discussion, we make some assumptions as follows (an extension to the general case is discussed in the next section): - the last column $c$ in the joining path $p$ is a fixed-size numerical vector. - all nodes at the same depth of the relational tree are ordered according to a predefined order. With the given simplification, transformation function $f$ and prediction function $g$ can be learned from data by training a deep neural network structure that includes a set of recurrent neural networks (RNNs). We call the given network structure *relational recurrent neural network* (r2n) as it transforms relational data using recurrent neural networks. There are many variants of RNN, in this work we assume that an RNN takes as input a sequence of vectors and outputs a vector. An RNN is denoted as $rnn(s, \theta)$, where $s$ is a variable size sequence of vectors and $\theta$ is the network parameter. Although the discussion focuses on RNN cells, our framework also works for Long Short Term Memory (LSTM) or Gated Recent Unit (GRU) cells. \[Relational Recurrent Neural Network\] For a given relational tree $t^p_e$, a relational recurrent neural network is a function denoted as $r2n (t^p_e, \theta)$ that maps the relational tree to a target value $y_e$. An $r2n$ is a tree of $RNNs$, in which at every intermediate node, there is an $RNN$ that takes as input a sequence of output vectors of the $RNNs$ resident at its children nodes. In an $r2n$, all RNNs, resident at the same depth $d$, share the same parameter set $\theta_d$. \[exp:r2n\] Figure \[fig:relational tree transformation prior art\].c shows an r2n of the tree depicted in Figure \[fig:relational tree transformation prior art\].a . As it is observed, an $r2n$ summarizes the data under every node at depth $d$ in the relation tree via a function parametrized by an RNN with parameters $\theta_d$ (shared for all RNNs at the same depth). Compared to the DFS method in Figure \[fig:relational tree transformation prior art\].a, the transformations are learned from the data rather than be specified a-priori by the users. A universal r2n {#sec:r2n universal} --------------- In this section, we discuss a neural network structure that works for the general case even without the two assumptions made in Section \[sec:r2n\]. ### Dealing with unstructured data When input data is unstructured, we add at each leaf node an embedding layer that embeds the input into a vector of numerical values. The embedded layers can be learned jointly with the $r2n$ network as shown in Figure \[fig:relational tree transformation prior art\].c. For example, if the input is a categorical value, a direct look-up table is used, that maps each categorical value to a fixed size vector. If the input is a sequence, an RNN is used to embed a sequence to a vector. In general, the given list can be extended to handle more complicated data types such as graphs, images and GPS trajectories. ### Dealing with unordered data When data is not associated with an order, the input is a set instead of a sequence. In that case, the transformation function $f(s)$ takes input as a set, we call such function as set transformation. It is important to notice $f(s)$ is invariant in any random permutation of $s$. An interesting question to ask is if recurrent neural network can be used to approximate any set function $f(s)$. Unfortunately, the following theorem shows that, there is no recurrent neural network that can approximate any set function except the constant or the sum function. \[theo:set function\] If a recurrent neural network $rnn(s, W, H, U)$ with linear activation is a set function, it is either a constant function or can be represented as: $$\begin{aligned} rnn(s, W, H, U) &=& c + h_0U + |s|*bU + UW*sum(s) \label{eq:rnn set}\end{aligned}$$ From equation \[eq:rnn set\] we can imply that an RNN cannot approximate the Max set and Min set functions unless we define an order on the input data. Therefore, we sort the input vectors according to vector mean value to ensure an order for input data., Joining path generation {#subsec:path generation} ----------------------- So far, we have discussed feature learning from relational trees extracted from a database when a joining path is given. In this section, we discuss different strategies to generate relevant joining paths. Because finding the optimal paths is hard, we limit the maximum depth of the joining paths and propose simple heuristic traversing strategies: simple: only allows simple paths (no repeated nodes); forward only: nodes are assigned depth numbers based on a breadth-first traversal starting from the main table. Path generation only considers the paths such that latter nodes must be deeper than the former ones; all: all paths are considered In our experiments, forward only is the most efficient which is our first choice. The other strategies are supported for the completeness. For any strategy, the joined tables can be very large, especially when the maximum depth is set high. Therefore, we apply sampling strategies and caching intermediate tables to save memory and speed up the join operations (see the supplementary material). Networks for multiple joining paths ----------------------------------- Recall that for each joining path $p_i$, we create an $r2n_i$ network that learns features from the data generated by the joining path. In order to jointly learn features from multiple joining paths ${p_1,p_2,\cdots,p_m}$, we use a fully connected layer that transform the output of the $r2n_i$ to a fixed size output vector before concatenating these vectors and use a feed-forward network to transform them into a desired final output size. The entire network structure is illustrated in Figure \[fig:bigr2n\]. For classification problems, additional softmax function is applied on the final output vector to obtain the class prediction distribution for classification problem. ![Network for multiple paths: data from each joining path is transformed using an embedding layer, an r2n before being combined by a fully connected layers and a feedforward layer.[]{data-label="fig:bigr2n"}](./bigr2n.pdf){width="1.0\columnwidth"} Experiments =========== In this section we discuss the experimental results. DFS is considered as a baseline to compare to in addition to manual feature engineering approaches provided by Kaggle participants. Due to space limit, more details about data preparation, hyper-parameter automatic tuning, experimental settings can be found in Appendices (see supplementary material). **Data** **$\sharp$ tables** **$\sharp$ columns** **size** **OneBM** **r2n** **DFS** ----------------- --------------------- ---------------------- ---------- ----------- --------- ---------------- KDD Cup 2014 4 51 0.9 GB 12.9 h 1 week NA Coupon purchase 7 50 2.2 GB 2.8 h 1 week 84.04 hours Grupo Bimbo 5 19 7.2 GB 56 min. 1 week 2 weeks (not finished) : Datasets and running time of OneBM (60 GPUs), DFS (1 GPU) and r2n (4 CPUs + 1 GPU)[]{data-label="tab:data size"} Datasets -------- Three Kaggle competitions with complex relational graphs and different problem types (classification, regression and ranking) were chosen for validation (see Figure \[fig:egraph\]). The data characteristics are described in Table \[tab:data size\] and the graphs are illustrated in Figure \[fig:egraph\] where Coupon data has the most complex graph. {width="100.00000%"} Experimental settings and running time -------------------------------------- A Spark cluster with 60 cores, 300 GB of memory and 2 TB of disk space was used to run the rule-based OneBM algorithm. Features from OneBM were fit into an XGBOOST model known as the most popular model in the Kaggle community. Hyper-parameters of XGBOOST were auto-tuned via the Bayesian optimization for 50 iterations (see the settings in supplementary material). To train the r2n networks, we used a machine with one GPU with 12 GB of memory and 4 CPU cores with 100 GB of memory. Training one model until convergence needed 7 days. Auto-tuning the r2n hyper-parameters was not considered because of limited time budget. The network structure hyperparameters are fixed based on computational feasibility in our available computing resource (see supplementary material). All results are reported based on the Kaggle private leaderboard ranking information. The running time of the algorithms is reported in Table \[tab:data size\]. Kaggle competition results and discussion ----------------------------------------- Competition Task Metric DFS OneBM R2N Ensemble ----------------- ---------------- -------- ---------- -------------- ---------- ----------- KDD Cup 2014 Classification AUC 0.586 0.622 0.619 **0.626** Groupo Bimbo Regression LRMSE NA 0.475 0.485 **0.472** Coupon Purchase Ranking MAP@10 0.005635 **0.007416** 0.004228 0.006065 : Data science competition results[]{data-label="tab:results "} Competition Rank Top(%) Medal ----------------- ---------- -------- -------- KDD Cup 2014 31/472 6.5 Silver Groupo Bimbo 152/1969 7.7 Bronze Coupon Purchase 30/1076 2.7 Silver : Ranking of the best among r2n, OneBM and their ensemble[]{data-label="tab:ranking "} Table \[tab:results \] reports the results of DFS, OneBM, r2n and a linear ensemble (with equal weights) of r2n and OneBM on three Kaggle competitions where results are available (the result of DFS on Bimbo was not available as its run didn’t finish within 2-weeks time budget. In the KDD Cup 2014 competition, both OneBM and r2n outperformed DFS with a significant margin. OneBM always achieves better results than r2n because of the robustness of the XGBOOST model which was auto-tuned by Bayesian optimization. However, r2n provides additional benefits to the rule-based approach as the linear ensemble of these methods shows better results than each individual model in both KDD Cup 2014 and Grupo Bimbo competitions. The result of r2n in the coupon purchase dataset is worse than the other methods. As shown in Figure \[fig:egraph\] coupon purchase’s graph is more complex which might need careful network structure search and hyper-parameter tuning to achieve better performance. {width="100.00000%"} Figure \[fig:rank\] shows a comparison between the best of OneBM, r2n and their linear ensemble (marked with a dash-line) and all Kaggle participants. As it is observed, in terms of prediction accuracy, our method outperformed most participants and achieved results that were very close to the best teams. The ranking and achieved medals are reported in Table \[tab:ranking \]. To the best of our knowledge, before our solution, no other automatic system could win medals in Kaggle competitions. Related Work ============ The data science work-flow includes five basic steps: problem formulation, data acquisition, data curation, feature engineering, model selection and hyper-parameter tuning. Most related works focus on automating the last two steps which will be reviewed in the following subsections. Automatic model selection and tuning ------------------------------------ Auto-Weka by [@kotthoff2016auto; @ThoHutHooLey13-AutoWEKA] and Auto-SkLearn by [@feurer2015efficient] are two popular tools trying to find the best combination of data pre-processing, hyper-parameter tuning and model selection. Both works are based on Bayesian optimization [@brochu2010tutorial] to avoid exhaustive grid-search. Cognitive Automation of Data Science (CADS) [@cads] is another system built on top of Weka, SPSS and R to automate model selection and hyper-parameter tuning processes. Besides these works, TPOT by [@tpot] is another system that uses genetic programming to find the best model configuration and pre-processing work-flow. In summary, automation of hyper-parameter tuning and model selection is a very attractive research topic with very rich literature. The key difference between our work and these works is that, while the state-of-the-art focuses on optimization of models given a ready set of features stored in a single table, our work focuses on preparing features as an input to these systems from relational databases with multiple tables. Therefore, these works are orthogonal to each other. Automatic feature engineering ----------------------------- Different from automation of model selection and tuning where the literature is very rich, only a few works have been proposed to completely automate feature engineering for general problems. The main reason is that feature engineering is both domain and data specific. Therefore, we discuss related work for relational database. DFS by [@DFS] is the first system that automates feature engineering from relational data with multiple tables. DFS has been shown to achieve good results on public data science competitions. OneBM is closely related to work in inductive logic programming, e.g. see [@luc] where relational data is unfolded via propositionalizing, e.g. see [@propositionalization] or Wordification by [@wordification] discretises the data into words from which the joined results can be considered as a bag of words. Each word in the bag is a feature for further predictive modelling. Wordification is a rule-based approach, which does not support unstructured and temporally ordered data. In [@arno], the authors proposed an approach to learn multi-relational decision tree induction for relational data. This work does not support temporally ordered data and is limited to decision tree models. Our work extended DFS to deal with non-numerical and temporally ordered data. Moreover, we resolved the redundancy issues of rule-based approaches via learning features rather than relying on predefined rules. Besides, works in statistical relational learning (StarAI) presented in [@lisa] are also related to our work. Recently, a deep relational learning approach was proposed by [@deeprl] to learn to predict object’s properties using object’s neighbourhood information. However, the given prior art does not support temporally ordered data and unstructured properties of objects. Besides, an important additional contribution of our work is the study of the theoretical complexity of the feature learning problem for relational data as well as the universal expressiveness of the network structures used for feature learning. Cognito by [@cognito] automates feature engineering for one table. It applies recursively a set of predefined mathematical transformations on the table’s columns to obtain new features from the original data. Since it does not support relational databases with multiple tables and is orthogonal to our approach. Conclusion and future work ========================== We have shown that feature engineering for relational data can be automated using predefined sets of heuristic transformations or by the r2n network structure. This opens many interesting research directions for the future. For example, the r2n network structure in this work is not auto-tuned due to the efficiency issue. Future work could focus on efficient methods for network structure search to boost the current results even more. Second, the rule based approach seems to be the best choice so far because of its effectiveness and efficiency, yet there are chances to improve the results further if a smarter graph traversal policy is considered. Although we have proved that finding the best joining path is NP-hard, the theoretical analysis assumes that there is no domain knowledge about the data. We believe that exploitation of semantic relation between tables and columns can lead to better search algorithm and better features. NP-hardness proof ================= ![A reduction from Hamiltonian cycle problem to the problem finding the optimal joining path for engineering features from relational data for a given predictive analytics problem.[]{data-label="fig:np-hardness"}](./Hamiltonian.pdf){width="1.0\columnwidth"} Problem \[prob:Feature learning\] is an optimization problem, the decision version asks whether there exists a solution such that $ L_{P, F, g}(Y,\hat{Y}) = 0$. We prove the NP-hardness by reducing the given decision problem to the *Hamiltonian cycle problem* which was well-known as an NP-Complete problem as discussed in [@karp1972reducibility]. Given a graph $G(E,V)$, where $E$ is a set of undirected edges and $V = \{v_1,v_2,\cdots,v_n\}$ is a set of $n$ nodes, the Hamiltonian cycle problem asks if there exists a cycle on the graph such that all nodes are visited on the cycle and every node is visited exactly twice. Given an instance of the Hamiltonian cycle problem, we create an instance of the feature learning problem with a relational graph as demonstrated in Figure \[fig:np-hardness\]. We assume that we have a database $D = \{T_1, T_2, \cdots, T_{n}\}$ with $n$ tables, each table $T_i$ corresponds to exactly one node $v_i \in V$. Assume that each table has only one row. For each pair of tables $T_i$ and $T_j$ (where $i < j$), there is a foreign key $k_{ij}$ presenting in both tables such that the values of $k_{ij}$ in $T_i$ and $T_j$ are the same if and only if there is an edge $(v_i,v_j) \in E$. Assume that $T_1$ is the main table which has an additional label column with value equal to $n$. We also assume that all the keys $k_{ij}$ have unique value in each table it presents which means that for each entry in $T_i$ there is at most one entry in $T_j$ with the same $k_{ij}$ value and vice versa, we call such relations between tables one-one. Recall that the relational graph $G_D$ constructed for the database $D$, where nodes are tables and edges are relational links, is a fully connected graph. Let $p$ is a path defined on $G_D$ and starts from the main table $T_1$. Because all the relations between tables are $one-one$, following the joining path $p$ we can either obtain an empty or a set containing at most one element, denoted as $J_p$. A cycle in a graph is simple if all nodes are visited exactly twice. Let assume $F$ as the set of functions such that: if $J_p$ is empty then $f(J_p) = 0$ and if $J_p$ is not empty then $f(J_p) = k$ where $k$ is the length of the longest simple cycle which is a sub-graph of $p$. Let $g$ be the identity function. The decision problem asks whether there exists a path $p$ such that $ L_{p, F, g}(Y,\hat{Y}) = 0$ is equivalent to asking whether $g(f(J_p)) = n$ or $f(J_p) = n$ assuming $g$ is an identity function. Assume that $g(f(J_p)) = n$, we can imply that $J_p$ is not empty and $p$ is a Hamiltonian cycle in $G_D$. Since $J_p$ is not empty, $p$ is a sub-graph of $G$. Hence $G$ also possess at least one Hamiltonian cycle. On the other hands, if $p$ is a sub-graph of $G$ and it is a Hamiltonian cycle, then since $G$ is a sub-graph of the fully connected graph $G_D$ we must have $g(f(J_p)) = n$ as well. The given reduction from the Hamiltonian cycle problem is a polynomial time reduction because the time for construction of the database $D$ is linear in the size of the graph $G$. Therefore, the NP-hardness follows. Proof of expressiveness theorem ================================ First, we need to prove the following lemma: A recurrent neural network $rnn(s, W, H, U)$ with linear activation is a set function if and only if $H=1$ or $rnn(s, W, H, U)$ is a constant. \[lemma:set function rnn\] Denote $s$ as a set of numbers and $p(s)$ is any random permutation of $s$. A set function $f(s)$ is a map from any set $s$ to a real-value. Function $f$ is invariant with respect to set-permutation operation, i.e. $f(s) = f(p(s))$. For simplicity, we prove the lemma when the input is a set of scalar numbers. The general case for a set of vectors is proved in a similar way. Consider the special case when $s = \{x_0, x_1\}$ and $p(s) = \{x_1, x_0\}$. According to definition of recurrent neural net we have: $$\begin{aligned} h_t &=& b + H*h_{t-1} + W*x_t \\ o_t &=& c + U*h_t\end{aligned}$$ from which we have $rnn(s) = o_2$, where: $$\begin{aligned} h_1 &=& b + H*h_0 + W*x_0 \\ o_1 &=& c + U*h_1 \\ h_2 &=& b + H*h_1 + W*x_1 \\ o_2 &=& c + U*h_2 \end{aligned}$$ In a similar way we can obtain the value of $rnn(p(s)) = o^*_2$, where: $$\begin{aligned} h^*_1 &=& b + H*h^*_0 + W*x_1 \\ o^*_1 &=& c + U*h^*_1 \\ h^*_2 &=& b + H*h^*_1 + W*x_0 \\ o^*_2 &=& c + U*h^*_2 \end{aligned}$$ Since $rnn(p(s)) = rnn(s)$, we infer that: $$\begin{aligned} U*(H-1)*W *(x_0-x_1) = 0\end{aligned}$$ The last equation holds for all value of $x_0, x_1$, therefore, either $H=1$, $W=0$ or $U=0$. The lemma is proved. According to Lemma \[lemma:set function rnn\], $rnn(s, W, H, U)$ is either a constant function or $H=1$. Replace $H = 1$ to the formula of an RNN we can easily obtain equation \[eq:rnn set\]. Efficient implementation on GPU =============================== Deep learning techniques takes the advantage of fast matrix computation capabilities of GPU to speed up its training time. The speed-up is highly dependent upon if the computation can be packed into a fixed size tensor before sending it to GPUs for massive parallel matrix computation. A problem with the network is that the structures of relational trees are different even for a given joining path. For instances, the relational trees in Figure \[exp:relational tree\] have different structures depending on input data. This issue makes it difficult to normalize the computation across different relational trees in the same mini-batch to take the advantage of GPU computation. In this section, we discuss a computation normalization approach that allows speeding up the implementation 5x-10x using GPU computation under the assumption that the input to an r2n network are relational trees we set $\{D_{p_1}, D_{p_2}, \cdots, D_{p_q} \}$, where $D_{p_i} = \{t^{p_i}_1, t^{p_i}_2, \cdots, t^{p_i}_m\}$. It is important to notice that $t^{p_k}_i$ and $t^{p_l}_i$ have different structure when $p_l$ and $p_k$ are different. Therefore, normalization across joining paths is not a reasonable approach. For a given joining path $p_i$, the trees $t^{p_i}_k$ and $t^{p_i}_l$ in the set $D_{p_i}$ may have different structures as well. Fortunately, those trees share commons properties: - they have the same maximum depth equal to the length of the path $p_i$ - transformation based on RNN at each depth of the trees are shared Thanks to the common properties between the trees in $D_{p_i}$ the computation across the trees can be normalized. The input data at each depth of all the trees in $D_{p_i}$ (or a mini-batch) are transformed at once using the shared transformation network at the given depth. The output of the transformation is a list, for which we just need to identify which output corresponds to which tree for further transformation at the parent nodes of the trees. Model hyper-parameter tuning with Bayesian optimization ======================================================== The quality of the prediction highly depends on various hyper-parameters present in the machine learning method. Since their optimal choice highly depends on the data, fixing them arbitrarily will provide suboptimal results. Therefore, we make use of Bayesian optimization [@Mockus1975], the state-of-the-art approach for efficient and automated hyperparameter optimization [@Snoek2012]. For Bayesian optimization, the problem of hyperparameter optimization is formulated as a black-box function minimization problem. We define this black-box function $f$ by $$f\ :\ \Lambda\rightarrow\mathbb{R}\enspace,$$ where $f$ maps a hyperparameter configuration $\boldsymbol{\lambda}\in\Lambda$ to its loss on the validation data set. The evaluation of $f$ is a time-consuming step because it involves training our neural network on the training data set and evaluating it on the validation data set. However, minimizing $f$ will provide us the optimal hyperparameter configuration. $$\boldsymbol{\lambda}^{\ast}=\underset{\boldsymbol{\lambda}\in\Lambda}{\arg\,\min}f\left(\boldsymbol{\lambda}\right)\enspace.$$ Bayesian optimization efficiently minimizes this expensive black-box function sequentially. In each optimization iteration the function $f$ is approximated by a Bayesian machine learning model, the surrogate model. We selected a Gaussian process with Matérn $\sfrac{5}{2}$ kernel as our surrogate model. This Bayesian model provides point and uncertainty estimates for the whole hyperparameter search space. According to a heuristic considering both mean and uncertainty prediction, the most promising hyperparameter configuration is evaluated next. We select expected improvement as our acquisition function [@Mockus1975]. \[sec:tuning\] Data preparation {#sec:preparation} ================ The following steps are needed to turn the raw data into the format that our system requires: 1. For every dataset we need to create a main table with training instances. The training data must reflect exactly how the test data was created. This ensures the consistency between training and test settings. 2. Users need to explicitly declare the database schema. 3. Each interested entity is identified by a key column. We added additional key columns to represent those entities if the keys are missing in the original data. It is important to notice that, the first step is an obligation for all Kaggle participants. The second step is trivial as it only requires declaring the table column’s special types and primary/foreign key columns. Basic column types such as numerical, Boolean, timestamps, categorical etc., are automatically determined by our system. The last step requires knowledge about the data but time spent on creating additional key columns is negligible compared to creating hand-crafted features. #### Grupo Bimbo participants were asked to predict weekly sales of fresh bakery products on the shelves of over 1 million stores across Mexico. The database contains 4 different tables: - *sale series*: the sale log with weekly sale in units of fresh bakery products. Since the evaluation is based on Root Mean Squared Logarithmic Error (RMSLE), we take the logarithm of the demand. - *town state*: geographical location of the stores - *product*: additional information, e.g. product names - *client*: information about the clients The historical sale data spans from week 1-9 while the test data spans from weeks 10-11. We created the main table from the sale series table with data of the weeks 8-9. Data of prior weeks was not considered because there was a shortage of historical sales for the starting weeks. The main table has a target column which is the demand of the products and several foreign key columns and some static attributes of the products. #### Coupon Purchase participants were asked to predict the top ten coupons which were purchased by the users in the test weeks. The dataset includes over one year of historical logs about coupon purchases and user activities: - *coupon list*: coupon’s info: location, discount price and the shop - *coupon detail*: more detailed information about the coupons - *coupon area*: categorical information about the coupon types and its display category on the website - *coupon visit*: historical log about user activities on the coupon websites. User and coupon keys are concatenated to create a user-coupon key that represents the user-coupon pair which is the target entity of our prediction problem. - *user*: demographic information about the users - *prefecture*: user and coupon geographical information We cast the recommendation problem into a classification problem by creating a main table with 40 weeks of data before the testing week. To ensure that the training data is consistent with the test data, for each week, we find coupons with released date falling into the following week and create an entry in the main table for each user-coupon pair. We label the entry as positive if the coupon was purchased by that user in the following week and negative otherwise. The main table has three foreign keys to represent the coupons, the users and the user-coupon pairs. #### KDD Cup 2014 participants were asked to predict which project proposals are successful based on their data about: - *projects*: project descriptions, school and teacher profiles and locations. The project table is considered as the main table in our experiment as it contains the target column. - *essays*: written by teachers who proposed the proposal as a project goal statement. - *resources*: information about the requested resources - *donation*: ignored as no data for test set. - *outcome*: historical outcome of the past projects. We add three missing key columns (school ID, teacher ID, school NCES ID) to the outcome table to connect it to the main table. This allows our system to explore the historical outcome for each school, teacher and school NCES ID. The relational graphs of the datasets are shown in Figure \[fig:egraph\]. In all datasets, we experimented with the forward only graph traversal policy. In the given policy, the maximum search depth is always set to the maximum depth of the breadth-first search of the relational graph starting from the main table. Parameter settings ================== Table \[tab:parameters\] reports the hyper-parameters used in our experiments. Since running full automatic hyper-parameter tuning and network structure search is computationally expensive we chose the size of the network based on our available computing resource. While the parameters related to optimization like the learning rate was chosen based on popular choice in the literature. **parameter** **value** ------------------------------------------------------- ------------------- The number of recent values 10 Maximum joined table size $10^9$ The number of highest correlated items 10 Min correlation $10^{-16}$ Min info-gain $10^{-16}$ Optimization algorithm for backprop ADAM Learning rate of ADAM 0.01 Initial weights for FC and feed-forwards Xavier Output size of FCs 10 The number of hidden layers in feedforward layers 1 The number of hidden layer size in feedforward layers 1024 RNN cell LSTM LSTM cell size 18 Max input sequence size 50 Early termination after no improvement on 25% training data Validation ratio 10% training data : Parameter settings for OneBM and the r2n networks[]{data-label="tab:parameters"} {width="100.00000%"} Baseline method settings ======================== DFS is currently considered as the state of the art for automation of feature engineering for relational data. Recently, DFS was open-sourced[^3]. We compared OneBM and r2n to DFS (version 0.1.14). It is important to notice that the open-source version of DFS has been improved a lot since its first publication [@DFS]. For example, in the first version described in the publication there is no concept of temporal index which is very important to avoid mining leakages. To use DFS properly, it requires knowledge about the data to create additional tables for interesting entities and to avoid creating diamond relational graphs because DFS doesn’t support diamond loops in the graph and does not allow many-many relations. The results of Grupo Bimbo and Coupon purchase competitions were reported using the open-source DFS after consulting with the authors on how to use DFS properly on these datasets. For the Bimbo and Coupon purchased data, the relational graphs shown in Figure \[fig:egraph\] are not supported by DFS as they contain many-many relations and diamond subgraphs. Therefore, we tweaked these graphs to let it run under the DFS framework. Particularly, for Bimbo data the relation between main and series tables is many-many. To go around this problem, we created an additional table called product-client from the sale series table. Each entry in the new table encodes the product, client pairs. The product-client is the main table correspond to product-client pair. Since the competition asked for predicting sales of every pair of product-client at different time points, we created a cut-off time-stamp table, where each entry corresponds to exactly one cut-off timestamp. The new relational graph is presented in Figure \[tweak-database\]. We run DFS with maximum depth set to 1 and 2 and 3. For Coupon datasets, more efforts are needed to prepare a proper input for DFS because the original relational graph contains both diamond loops and many-many relations. The latter issue can be resolved by changing the connections as demonstrated in Figure \[tweak-database\]. To avoid diamond loops, we need to delete some relations. We decided to delete the relations (marked with an X) in Figure \[tweak-database\]. Alternatively, we also tried to delete the relation between the main and coupon-visit table but that led to much worse prediction than the given choice. In general, the results are highly dependent on how we use the tool so for the KDD cup 2014, to have a fair comparison, we used the results in the original publication [@DFS]. For KDD Cup 2015 and IJCAI 2015, the competitions are closed for submissions so we couldn’t report the results. Acknowledgements ================ We would like to thank Johann Thiebaut, Dr. Tiep Mai, Dr. Bei Chen, Dr. Oznur Alkan, Dr. Olivier Verscheure, Dr. Eric Bouillet, Dr. Horst C. Samulowitz, Dr. Udayan Khurana and Tejaswina Pedapat for useful discussion and support during the development of the project. We would like to thank Max Kanter for explaining us how to use DFS (featuretools) for the experiments. [^1]: Use footnote for providing further information about author (webpage, alternative address)—*not* for acknowledging funding agencies. [^2]: http://blog.kaggle.com/2016/09/27/grupo-bimbo-inventory-demand-winners-interviewclustifier-alex-andrey/ [^3]: https://www.featuretools.com/

--- abstract: 'This paper provides a new instance of quantum deletion error-correcting codes. This code can correct any single quantum deletion error, while our code is only of length 4. This paper also provides an example of an encoding quantum circuit and decoding quantum circuits. It is also proven that the length of any single deletion error-correcting codes is greater than or equal to 4. In other words, our code is optimal for the code length.' author: - 'Manabu HAGIWARA [^1]' - 'Ayumu NAKAYAMA [^2]' bibliography: - 'reference.bib' title: ' A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length' --- Introduction ============ Similar to classical error-correcting codes, quantum error-correcting codes are an essential factor for implementing practical quantum communication and quantum computation. In recent classical coding theory, deletion codes have attracted researcher’s attention. For example, while only three papers on deletion codes were presented at the symposium ISIT 2015, more than ten papers were presented at ISIT 2017, 2018 and 2019. Furthermore, two technical sessions for deletion codes were organized at the last ISIT. Classical deletion error-correcting codes have applications to reliable communication for synchronization error [@sala2017exact; @helberg1993coding], error-correction for DNA storage [@buschmann2013levenshtein], error-correction for racetrack memory [@chee2017coding], etc. On the other hand, only a few studies on quantum deletion correcting codes have been published. In 2019, Leahy et.al. introduced quantum insertion/deletion channel [@leahy2019quantum] and a technique to reduce quantum deletion error to quantum erasure error under the specific assumption. The first quantum binary deletion codes under the general scenario was constructed in [@nakayama2019first]. The code length was $8$. Deletion error-correction is a problem that to determine the original information from a partial information. The difference from erasure error-correction is that we are not given the information on the error positions. For example, a bit-sequence $00111000$ is changed to $001?1000$ by a single erasure error but is changed to $0011000$ by a single deletion error. The symbol “$?$” tells the position where error occurred. An erasure bit-sequence $001?1000$ can be transformed to $0011000$ by deleting the symbol “$?$”. Hence deletion error-correction is more difficult than erasure error-correction. In quantum information theory, quantum deletion error-correction is a problem to determine the quantum state in the entire quantum system from a quantum state in a partial system. Therefore it is related to various topics, e.g., quantum erasure error-correcting codes [@grassl1997codes], partial trace, quantum secret sharing [@cleve1999share; @hillery1999quantum], purification of quantum state [@hughston1993complete], quantum cloud computing [@biamonte2017quantum] and etc. This paper provides a new and a shorter length single deletion error-correcting code. The code space is not an instance of previously known quantum error-correcting codes, e.g., CSS codes [@calderbank1996good; @steane1996multiple], stabilizer codes [@gottesman1997stabilizer], surface codes [@fowler2012surface], and etc. Remark that our code length is only $4$. This is the same length to the optimal shortest code length of quantum erasure error-correcting codes [@grassl1997codes]. In fact, we show that $4$ is also the optimal length of quantum deletion error-correcting codes. This paper also provides examples of one encoding circuit and two different decoding circuits. The depth of the circuits are small. The readers are assumed to be familiar with quantum information theory and coding theory, in particular, quantum error-correcting codes and classical deletion error-correcting codes. Deletion Errors and Code Construction ===================================== Deletion Error and Deletion Error-Correcting Codes -------------------------------------------------- Set $\ket{0}, \ket{1} \in \mathbb{C}^2$ as $$\ket{0}:= \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \ket{1}:= \left( \begin{array}{c} 0 \\ 1 \end{array} \right)$$ respectively. For a binary sequence $ \bm{x} = x_1 x_2 \dots x_n \in \{0,1\}^n$, $ \ket{ \bm{x} } $ denotes $$\ket{ x_1 } \otimes \ket{ x_2 } \otimes \dots \ket{ x_n } \in \mathbb{C}^{2 \otimes n} .$$ We denote the set of all density matrices of order $N$ by $S(\mathbb{C}^N)$. An element of $S(\mathbb{C}^N)$ is called a quantum state in this paper. We also use a complex vector for representing a quantum state if the state is pure. For an integer $1 \le i \le n$ and a square matrix $$A = \sum_{\bm{x},\bm{y} \in \{0,1\}^n } a_{\bm{x},\bm{y}} \cdot \ket{x_1} \bra{y_1} \otimes \cdots \otimes \ket{x_n} \bra{y_n}$$ with $a_{\bm{x},\bm{y}} \in \mathbb{C}$, define the map $\mathrm{Tr}_i : S( \mathbb{C}^{2 \otimes n} ) \rightarrow S( \mathbb{C}^{2 \otimes (n-1)} )$ as $$\begin{aligned} \mathrm{Tr}_i(A) := &\sum_{\bm{x},\bm{y} \in \{0,1\}^n} a_{\bm{x},\bm{y}} \cdot \mathrm{Tr}(\ket{x_i} \bra{y_i}) \cdot \ket{x_1}\bra{y_1} \otimes \\ & \cdots \otimes \ket{x_{i-1}} \bra{y_{i-1}} \otimes \ket{x_{i+1}} \bra{y_{i+1}} \otimes \\ & \cdots \otimes \ket{x_n} \bra{y_n}.\end{aligned}$$ The map $\mathrm{Tr}_i$ is called a partial trace. Recall that in the classical coding theory, a single deletion error is defined as an operator that maps a sequence $x_1 x_2 \dots x_i \dots x_n$ to a short sequence $x_1 x_2 \dots x_{i-1} x_{i+1} \dots x_n$ for some $i$. For an integer $1 \le i \le n$, we call $\mathrm{Tr}_i$ a single deletion error $D_i$, i.e., $$D_i(\rho):= \mathrm{Tr}_i(\rho),$$ where $\rho \in S(\mathbb{C}^{2 \otimes n})$ is a quantum state. If a quantum state $\rho$ is corresponding to $n$-photons $p_1, p_2, \dots, p_n$, the state $D_i( \rho )$ is corresponding to $n-1$-photons $p_1, p_2, \dots, p_{i-1}, p_{i+1}, \dots, p_n$. We call a vector space $Q \subset \mathbb{C}^{2 \otimes n}$ an $[n , k]$ single deletion error-correcting code if - there exists a complex linear bijection $\mathrm{Enc} : \mathbb{C}^{2 \otimes k} \rightarrow Q$, - there exists a map $\mathrm{Dec} : S( \mathbb{C}^{2 \otimes (n-1)} ) \rightarrow \mathbb{C}^{2 \otimes k}$ such that for any $ \ket{\phi} \in \mathbb{C}^{2 \otimes k}$ and for any $1 \le i \le n$, $$\mathrm{Dec} \circ D_i \circ \mathrm{Enc} ( \ket{ \phi }) = \ket{ \phi },$$ where $\circ$ is the composite for operations. In other words, there exist an encoder $\mathrm{Enc}$ and a decoder $\mathrm{Dec}$ that correct any single deletion errors. Comparing to erasure errors, deletion errors do not tell the position where the information is deleted. Hence to correct deletion errors is more difficult than to correct erasure errors. Code Construction ----------------- Most famous classical error-correcting codes for single deletion errors are “non-linear” codes that are called VT codes [@tenengolts1965correction] discovered by Levenshtein [@levenshtein1966binary]. One of the reasons why “non-linear” codes are preferable for deletion error-correction is unveiled in [@abdel2010correcting]. It is stated that a code rate of any single deletion error-correcting code cannot exceed $1/2$ if a code is linear. This implies that if a CSS code is constructed from two classical linear deletion error-correcting codes then the code rate of the CSS code is $0$. Let us define a linear map $\mathrm{En}_4 : \mathbb{C}^{2} \rightarrow \mathbb{C}^{2 \otimes 4}$. For a quantum state $| \phi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle \in \mathbb{C}^2$, $\mathrm{En}_4$ maps the state $\ket{ \phi }$ to the following state $\ket{ \Phi }$, $$\begin{aligned} | \Phi \rangle :=& \frac{\alpha}{\sqrt{2}} (| 0000 \rangle + | 1111 \rangle )\\ & + \frac{\beta}{\sqrt{6}} (| 0011 \rangle + |0101 \rangle + |0110 \rangle \\ & + |1001 \rangle + |1010 \rangle + |1100 \rangle).\end{aligned}$$ Set $Q_4$ as the image of $\mathrm{En}_4$ for quantum messages, i.e., $$Q_4 := \{ \mathrm{En}_4 ( \ket{ \phi } ) \mid \ket{\phi} \in \mathbb{C}^2, \ket{\phi} \bra{\phi} \in S( \mathbb{C}^2 ) \}.$$ The QEC code with $4$ qubits [@grassl1997codes] is closely related to our code $Q_4$. We can say that more symmetry is introduced to our code for correcting error at the unknown position. In fact, we will show that $Q_4$ is a $[4,1]$ single deletion error-correcting code with the encoder $\mathrm{En}_4$. We can characterize this encoding in the following manner. Let $A$ be the set of four bit sequences with Hamming weight $0$ or $4$ and $B$ the set of four bit sequences with Hamming weight $2$, i.e., $$\begin{aligned} A &= \{0000, 1111\},\\ B &= \{0011, 0101, 0110, 1001, 1010, 1100\}.\end{aligned}$$ Then the codeword $| \Phi \rangle$ is $$\frac{\alpha}{\sqrt{2}} \sum_{a \in A} | a \rangle + \frac{\beta}{\sqrt{6}} \sum_{b \in B} | b \rangle.$$ By this encoding, $\ket{0}$ and $\ket{1}$ are encoded to quantum states which are superpositions of two and six orthonormal states. Hence this encoding is neither an instance of CSS codes nor stabilizer codes. Let us define a map $\mathrm{De}_4$ from $S( \mathbb{C}^{2 \otimes 3} )$ to $\mathbb{C}^{2}$. The map $\mathrm{De}_4$ consists of the following steps: - (Step 1) Perform the measurement $\{ P_0, P_1 \}$ to a quantum state in $S( \mathbb{C}^{2 \otimes 3} )$ and obtain the outcome $i \in \{0, 1\}$, where $P_0$ (resp. $P_1$) is the projection from $\mathbb{C}^{2 \otimes 3}$ to the linear space $V_0$ (resp. $V_1$) spanned by $\ket{000}, \ket{011}, \ket{101}$ and $\ket{110}$ (resp. $\ket{111}, \ket{100}, \ket{010}$ and $\ket{001}$). Hence, the outcome is $i$ if the quantum state in $S( \mathbb{C}^{2 \otimes 3})$ is changed into a state in $S( V_{i} )$. - (Step 2) Only if the outcome is $1$, act the quantum operation $F : V_1 \rightarrow V_0$ to $S$, where $F$ is defined as $$\begin{aligned} F( a \ket{111} + b \ket{100} + c \ket{010} + d \ket{001} ) \\ := a \ket{000} + b \ket{011} + c \ket{101} + d \ket{110}.\end{aligned}$$ - (Step 3) Act the quantum operation $G : V_0 \rightarrow \mathbb{C}^{2 \otimes 3}$ to the state, where $G$ is defined as $$\begin{aligned} G( a \ket{000} + b \ket{\overline{100}} + c \ket{\overline{010}} + d \ket{\overline{001}}\\ := a \ket{000} + b \ket{100} + c \ket{010} + d \ket{011},\end{aligned}$$ and where $$\begin{aligned} \ket{\overline{100}} &= \frac{1}{\sqrt{3}} \left( \ket{011} + \ket{101} + \ket{110} \right)\\ \ket{\overline{010}} &= \frac{1}{\sqrt{3}} \left( \ket{011} + \omega \ket{101} + \omega^2 \ket{110} \right)\\ \ket{\overline{001}} &= \frac{1}{\sqrt{3}} \left( \ket{011} + \omega^2 \ket{101} + \omega \ket{110} \right).\end{aligned}$$ - (Step 4) Finally, delete the 3rd and the 2nd qubits by the partial traces $\mathrm{Tr}_3$ and $\mathrm{Tr}_2$. Proof for Error-Correcting Property ----------------------------------- In this subsection, we prove our code $Q_4$ corrects any single deletion errors $D_1, D_2, D_3,$ and $D_4$. Let us set $$\begin{aligned} A_0 &:= \{000\},\\ A_1 &:= \{111\},\\ B_0 &:= \{011, 101, 110\},\\ B_1 &:= \{001, 010, 100\}.\end{aligned}$$ Let $\ket{\Phi} := \alpha \sum_{ \bm{a} \in A } | \bm{a} \rangle + \beta \sum_{ \bm{b} \in B} | \bm{b} \rangle$ and $\rho := | \Phi \rangle \langle \Phi |$. For any $1 \le i \le 4$, $$D_i (\rho) = \frac{1}{2}| \Phi_0 \rangle \langle \Phi_0 | + \frac{1}{2} | \Phi_1 \rangle \langle \Phi_1 |,$$ where $| \Phi_0 \rangle = \alpha \sum_{ \bm{a} \in A_0 } | \bm{a} \rangle + \frac{\beta}{\sqrt{3}} \sum_{\bm{b} \in B_0} | \bm{b} \rangle$ and $| \Phi_1 \rangle = \alpha \sum_{ \bm{a} \in A_1 } | \bm{a} \rangle + \frac{\beta}{\sqrt{3}} \sum_{\bm{b} \in B_1} | \bm{b} \rangle.$ At the first, we show a case where $i=1$. By using $A_0, A_1, B_0$ and $B_1$, we can rewrite $\ket{ \Phi }$ as $$\begin{aligned} \ket{ \Phi } = &\ket{ 0 } \left( \frac{ \alpha }{\sqrt{2}} \sum_{\bm{a} \in A_0 } \ket{ \bm{a} } + \frac{ \beta }{\sqrt{6}} \sum_{\bm{b} \in B_0 } \ket{ \bm{b} } \right)\\ &+\ket{ 1 } \left( \frac{ \alpha }{\sqrt{2}} \sum_{\bm{a} \in A_1 } \ket{ \bm{a} } + \frac{ \beta }{\sqrt{6}} \sum_{\bm{b} \in B_1 } \ket{ \bm{b} } \right).\end{aligned}$$ Hence, $$\begin{aligned} \rho &= \ket{ \Phi} \bra{ \Phi }\\ &= \ket{ 0} \bra{ 0 } \otimes \\ & \biggl( \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{a }, \bm{ a'} \in A_0 } \ket{ \bm{a} } \bra{ \bm{a '} } + \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{a } \in A_0, \bm{b' } \in B_0 } \ket{ \bm{a} } \bra{ \bm{b'} }\\ &+ \frac{ \beta }{\sqrt{6}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{b } \in B_0, \bm{ a'} \in A_0 } \ket{ \bm{b} } \bra{ \bm{a '} } + \frac{ \beta }{\sqrt{6}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{b }, \bm{b' } \in B_0 } \ket{ \bm{b} } \bra{ \bm{b'} } \biggr)\\ &+ \ket{ 1} \bra{ 1 } \otimes \\ & \biggl( \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{a }, \bm{ a'} \in A_1 } \ket{ \bm{a} } \bra{ \bm{a '} } + \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{a } \in A_1, \bm{b' } \in B_1 } \ket{ \bm{a} } \bra{ \bm{b'} }\\ &+ \frac{ \beta }{\sqrt{6}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{b } \in B_1, \bm{ a'} \in A_1 } \ket{ \bm{b} } \bra{ \bm{a '} } + \frac{ \beta }{\sqrt{6}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{b }, \bm{b' } \in B_1 } \ket{ \bm{b} } \bra{ \bm{b'} } \biggr)\\ &+ \ket{ 0} \bra{ 1 } \otimes \rho' + \ket{1} \bra{0} \otimes \rho'',\end{aligned}$$ for some matrices $\rho'$ and $\rho''$. Note that $\mathrm{Tr}( \ket{0} \bra{0} ) = \mathrm{Tr}( \ket{1} \bra{1} ) = 1$ and $\mathrm{Tr}( \ket{0} \bra{1} ) = \mathrm{Tr}( \ket{1} \bra{0} ) = 0$. By the definition of the partial trace, $$\begin{aligned} &\mathrm{Tr_1} ( \rho )\\ &= \biggl( \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{a }, \bm{ a'} \in A_0 } \ket{ \bm{a} } \bra{ \bm{a '} } + \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{a } \in A_0, \bm{b' } \in B_0 } \ket{ \bm{a} } \bra{ \bm{b'} }\\ &+ \frac{ \beta }{\sqrt{6}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{b } \in B_0, \bm{ a'} \in A_0 } \ket{ \bm{b} } \bra{ \bm{a '} } + \frac{ \beta }{\sqrt{6}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{b }, \bm{b' } \in B_0 } \ket{ \bm{b} } \bra{ \bm{b'} } \biggr)\\ &+ \biggl( \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{a }, \bm{ a'} \in A_1 } \ket{ \bm{a} } \bra{ \bm{a '} } + \frac{ \alpha }{\sqrt{2}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{a } \in A_1, \bm{b' } \in B_1 } \ket{ \bm{a} } \bra{ \bm{b'} }\\ &+ \frac{ \beta }{\sqrt{6}} \frac{ \overline{\alpha} }{\sqrt{2}} \sum_{\bm{b } \in B_1, \bm{ a'} \in A_1 } \ket{ \bm{b} } \bra{ \bm{a '} } + \frac{ \beta }{\sqrt{6}} \frac{ \overline{\beta} }{\sqrt{6}} \sum_{\bm{b }, \bm{b' } \in B_1 } \ket{ \bm{b} } \bra{ \bm{b'} } \biggr)\\ &= \frac{1}{2}\biggl( \alpha \overline{\alpha} \sum_{\bm{a }, \bm{ a'} \in A_0 } \ket{ \bm{a} } \bra{ \bm{a '} } + \alpha \frac{ \overline{\beta} }{\sqrt{3}} \sum_{\bm{a } \in A_0, \bm{b' } \in B_0 } \ket{ \bm{a} } \bra{ \bm{b'} }\\ &+ \frac{ \beta }{\sqrt{3}} \overline{\alpha} \sum_{\bm{b } \in B_0, \bm{ a'} \in A_0 } \ket{ \bm{b} } \bra{ \bm{a '} } + \frac{ \beta }{\sqrt{3}} \frac{ \overline{\beta} }{\sqrt{3}} \sum_{\bm{b }, \bm{b' } \in B_0 } \ket{ \bm{b} } \bra{ \bm{b'} } \biggr)\\ &+ \frac{1}{2} \biggl( \alpha \overline{\alpha} \sum_{\bm{a }, \bm{ a'} \in A_1 } \ket{ \bm{a} } \bra{ \bm{a '} } + \alpha \frac{ \overline{\beta} }{\sqrt{3}} \sum_{\bm{a } \in A_1, \bm{b' } \in B_1 } \ket{ \bm{a} } \bra{ \bm{b'} }\\ &+ \frac{ \beta }{\sqrt{3}} \overline{\alpha} \sum_{\bm{b } \in B_1, \bm{ a'} \in A_1 } \ket{ \bm{b} } \bra{ \bm{a '} } + \frac{ \beta }{\sqrt{3}} \frac{ \overline{\beta} }{\sqrt{3}} \sum_{\bm{b }, \bm{b' } \in B_1 } \ket{ \bm{b} } \bra{ \bm{b'} } \biggr)\\ &= \frac{1}{2}| \Phi_0 \rangle \langle \Phi_0 | + \frac{1}{2} | \Phi_1 \rangle \langle \Phi_1 |.\end{aligned}$$ By the symmetry of $\ket{\Phi}$, $$D_1 (\rho) = D_2( \rho) = D_3 (\rho) = D_4 (\rho)$$ holds. The code $Q_4$ is a $[4,1]$ single deletion error-correcting code with the encoder $\mathrm{En}_4$ and the decoder $\mathrm{De}_4$. Since $\ket{\Phi_0} \in V_0$ and $\ket{\Phi_1} \in V_1$, by the Step 1, The obtained quantum state $s$ is $\ket{\Phi_0} \bra{\Phi_0}$ or $\ket{\Phi_1} \bra{\Phi_1}$. At the Step 2, if the outcome is $1$, the state is changed to $\ket{\Phi_0} \bra{\Phi_0}$ by the operation $F$. Hence after the Step 2, obtained quantum state is $\ket{\Phi_0} \bra{\Phi_0}$. Since $G( \Phi_0 ) = \alpha \ket{000} + \beta \ket{100} = \ket{\phi} \otimes \ket{00}$, by the Step 3, the quantum state is changed to $ \ket{ \phi } \bra{ \phi} \otimes \ket{00} \bra{00}. $ Hence at the Step 4, the obtained state is $\ket{\phi} \bra{ \phi }$, i.e. the original quantum state. Encoding Circuit and Decoding Circuit ===================================== Encoding Circuit ---------------- ![Encoder []{data-label="figure:encoder"}](encoder.jpg){width="5.5cm"} Figure \[figure:encoder\] shows an example of an encoder for our four qubits code $Q_4$. The gate $H$ is the Hadamard matrix, i.e., $$H = \left( \begin{array}{cc} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{array} \right),$$ and the gate $X$ is the bit-flip matrix, i.e., $$X := \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right).$$ The black dot is the control gate for the connected gate. For example, the final gate of Figure \[figure:encoder\] is the C-NOT gate with the 3rd qubit as the control qubit and the 1st qubit as the target bit. The gates $U$ and $V$ are defined as $$U := \left( \begin{array}{cc} 1/\sqrt{3} & -\sqrt{2}/\sqrt{3} \\ \sqrt{2}/\sqrt{3} & 1/\sqrt{3} \end{array} \right), V := \left( \begin{array}{cc} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{array} \right)$$ respectively. The input of the circuit is a single qubit $\ket{ \phi } \in \mathbb{C}^2$. The input position is settled to the left-most position. For the other three positions, initialized three qubits $| 000 \rangle$ are input. By the first two depth operations, i.e., the controlled $U$ gate, the controlled $V$ gate and the Hadamard gate, a quantum state $\ket{ 0000 }$ is changed to $$\frac{1}{ \sqrt{2}} \left( \ket{ 0000 } + \ket{ 0010 } \right).$$ and a quantum state $\ket{ 1000 }$ is changed to $$\frac{1}{ \sqrt{6}} \left( \ket{ 1000 } + \ket{ 1010 } + \ket{ 0100 } + \ket{ 0110 } + \ket{ 1100 } + \ket{ 1110 } \right).$$ By the remaining four controlled $X$ gates, i.e. C-NOTs, a quantum state $\ket{ x_1 x_2 x_3 0 }$ is changed to $\ket{(x_1 + x_3) (x_2 + x_3) (x_3) (x_1 + x_2 + x_3)}$, e.g., $$\begin{aligned} \ket{0000} &\rightarrow \ket{0000},\\ \ket{0010} &\rightarrow \ket{1111},\\ \ket{1000} &\rightarrow \ket{1001},\\ \ket{1010} &\rightarrow \ket{0110},\\ \ket{0100} &\rightarrow \ket{0101},\\ \ket{0110} &\rightarrow \ket{1010},\\ \ket{1100} &\rightarrow \ket{1100},\\ \ket{1110} &\rightarrow \ket{0011}.\end{aligned}$$ Hence $(\alpha \ket{ 0 } + \beta \ket{ 1} ) \otimes \ket{ 000 }$ is encoded to $$\alpha \left( \frac{1}{\sqrt{2}} \sum_{ \bm{a} \in A} \ket{ \bm{a} } \right) + \beta \left( \frac{1}{\sqrt{6}} \sum_{ \bm{b} \in B} \ket{ \bm{b} } \right).$$ Decoding Circuits ----------------- ![Decoding Circuit after Step 1[]{data-label="figure:decoder_3"}](decoder_3.jpg){width="4cm"} Figure \[figure:decoder\_3\] shows a quantum circuit that changes a pure state $\left( \ket{011} + \ket{101} + \ket{110} \right)/\sqrt{3}$ to a pure state $\ket{100}$ and keeps a pure state $\ket{000}$. Hence the circuit changes a pure state $\ket{ \Psi_{0} } = \alpha \sum_{ \bm{a} \in A_0 } \ket{ \bm{a} } + \frac{\beta}{3} \sum_{ \bm{b} \in B_0 } \ket{ \bm{b} }$ to a pure state $\ket{ \phi } = \alpha \ket{0} + \beta \ket{1}$. Remember that the step 1 of the decoding algorithm changes a single deleted quantum state $D_i ( \rho )$ to $\ket{\Psi_0}$. Hence the circuit is available as a decoding circuit after step 1. Let us provide another quantum circuit that is depicted as Figure \[figure:decoder\_4\]. The quantum circuit consists of two parts. The first part has six C-NOT and the last part is the same as the circuit defined by Figure \[figure:decoder\_3\]. The first part changes a quantum pure state $\ket{x_1 x_2 x_3}$ to $\ket{x_1 x_2 x_3 0}$ if the Hamming weight of $x_1 x_2 x_3$ is even and to $\ket{(x_1 + 1) (x_2 +1) (x_3 + 1) 1}$ if the weight is odd. This implies that the first part changes a single deleted quantum state $D_i (\rho)$ to $$\ket{\Psi_0} \bra{\Psi_0} \otimes \left( \begin{array}{cc} 1/2 & 0 \\ 0 & 1/2 \end{array} \right).$$ Since the last part is the same as the circuit corresponding to Figure \[figure:decoder\_3\], the state is changed to $$\ket{ \phi} \bra{\phi} \otimes \ket{0} \bra{0} \otimes \ket{0} \bra{0} \otimes \left( \begin{array}{cc} 1/2 & 0 \\ 0 & 1/2 \end{array} \right).$$ In other words, the state at the first position of output is a pure state $\ket{ \phi }$. ![Decoding Circuit without Measurement[]{data-label="figure:decoder_4"}](decoder_4.jpg){width="5.5cm"} Generalization ============== Number of Information Qubits ---------------------------- We generalize our $[4, 1]$ single deletion error-correcting code $Q_4$ to a $[2^{k+2} - 4, k]$ single quantum deletion error-correcting code for any positive integer $k$. Let $l$ be a positive integer. For $0 \le i \le l-1$, let us define $$\mathcal{A}_i := \{ \bm{x} \in \{ 0, 1 \}^{4(l-1)} \mid \mathrm{wt}( \bm{x} ) = 2i \text{ or } 4(l-1) - 2i\}$$ and $$\mathrm{En}_{4 (l-1)} \left( \ket{ \phi_l } \right) := \sum_{0 \le i \le l-1} c_i \left( \sum_{\bm{x} \in \mathcal{A}_i } \ket{ \bm{x} } \right),$$ where $\ket{ \phi_l } := \sum_{0 \le i \le l-1} c_i \ket{i}$ is a quantum pure state of level $l$, i.e. $\ket{ \phi_l } \in \mathbb{C}^{l}$, and $\ket{0}, \ket{1}, \dots, \ket{l-1}$ is the standard orthonormal basis of $\mathbb{C}^l$. We claim that the image of $\mathrm{En}_{4(l-1)}$ is a single quantum deletion error-correcting code for any $l \ge 2$. A decoding algorithm $\mathrm{De}_{4(l-1)}$ can be defined similar to $\mathrm{De}_{4}$. Due to the limit of page numbers, we only explain how to define its measurement $\{ \mathcal{P}_0, \mathcal{P}_1 \}$. $\mathcal{P}_0$ (resp. $\mathcal{P}_1$) is the projection from $\mathbb{C}^{2 \otimes 4(l-1)}$ to the linear space $\mathcal{V}_0$ (resp. $\mathcal{V}_1$) spanned by $$\begin{aligned} \{ \ket{ \bm{x}} \mid & \; \bm{x} \in \{0, 1\}^{4(l-1) - 1}, \\ & \text{ the Hamming weight of $\bm{x}$ is even (resp. odd) } \}.\end{aligned}$$ The outcome is $i \in \{0, 1 \}$ if the quantum state in $S( \mathbb{C}^{2 \otimes 4(l-1)})$ is changed into a state in $S( \mathcal{V}_{i} )$. For the case $l= 2^k$, we obtain a $[2^{k+2} - 4, k]$ single quantum deletion error-correcting code. Permutation of The Received Qubits ---------------------------------- Our code word has symmetry for position permutations. In other words, any permutation for four qubits does not change the codeword. Additionally, any received word after any single deletion has also symmetry for position permutation. This means that even if we permute the three input to the quantum circuits corresponding to Figure \[figure:decoder\_3\], we can obtain the original quantum information $\ket{ \phi }$ at the left most position of output. There is no Quantum Deletion Codes with less than $4$ Qubits ============================================================ \[thm:noLeng2\] There is no single deletion error-correcting code of length $2$. Let us assume that there exist a code of length $2$. Then we have an encoder $\mathrm{En}_2 : \mathbb{C}^2 \rightarrow \mathbb{C}^{2 \otimes 2}$ and a decoder $\mathrm{De}_2 : S(\mathbb{C}^2) \rightarrow \mathbb{C}^2$. Let $\rho \in \mathbb{C}^{2}$ and encode it to $\mathrm{En}_2 ( \rho )$. Assume that the encoded state is corresponding to two photons $p_1$ and $p_2$. The quantum states of these photons are $D_1 \circ \mathrm{En}_2 (\rho)$ and $D_2 \circ \mathrm{En}_2 ( \rho )$ respectively. Perform the decoder $\mathrm{De}_2$ to the photons $p_1$ and $p_2$ simultaneously. Then the states for $p_1$ and $p_2$ are changed to $\mathrm{De}_2 \circ D_1 \circ \mathrm{En}_2 (\rho) = \rho$ and $\mathrm{De}_2 \circ D_2 \circ \mathrm{En}_2 (\rho) = \rho$ respectively. It contradicts to non-cloning theorem. The following is the remarkable result by Grassl et.al. \[fact:noLeng3\] There is no quantum error-correcting code of length three that can correct one erasure and encodes one qubit. A similar result to deletion error-correcting codes holds. \[thm:noLeng3\] There is no single deletion error-correcting code of length $3$. Assume that there exists a single deletion error-correcting code of length $3$. Let us explain that this code can be regarded as a quantum erasure error-correcting code of length $3$. If we have a received word with a single quantum erasure error, we can find the position where the error occurs. Then delete the state at the position of the received word. In other words, we can convert an erasure error to a deletion error. By the assumption, we can correct the error by deletion error-correcting decoder. This contradicts to Fact \[fact:noLeng3\]. The shortest length of single quantum deletion error-correcting codes is $4$. Conclusion ========== This paper provided a single quantum deletion error-correcting code of optimal length, i.e. $4$. The construction of this code is far from known quantum error-correcting codes, e.g., CSS codes [@calderbank1996good; @steane1996multiple], stabilizer codes [@gottesman1997stabilizer], surface codes [@fowler2012surface], and etc. Acknowledgment {#acknowledgment .unnumbered} ============== This paper is partially supported by KAKENHI 18H01435. The authors thank to Professor Mikio Nakahara, Professor Akinori Kawachi, and Professor Akihisa Tomita for valuable discussion. [^1]: Department of Mathematics and Informatics, Graduate School of Science, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba City, Chiba Pref., JAPAN, 263-0022 [^2]: Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba City, Chiba Pref., JAPAN, 263-0022

--- abstract: 'In this paper, we answer a set of research questions that are required to develop service identification approach based on the analysis of object-oriented software. Such research questions are: (1) what is a service, (2) how are services different from software components, (3) what are types of services, (4) what are existing service identification approaches that consider service types into account, and (5) how to identify services based on the object-oriented source code with respect to their types. Our methodology is based on performing a literature review to identify the answers of these research questions. Also, we propose a taxonomy of service types.' bibliography: - 'main.bib' --- {width="50.00000%"}\ **Laboratoire de Recherches sur les Technologies du Commerce Électronique**\ {width="50.00000%"}\ **Université du Québec à Montréal**\ **What Should You Know Before Developing a Service Identification Approach**\ \ *LATECE Technical Report 2017-2, LATECE Laboratoire, Université du Québec à Montréal, Canada* **What Should You Know Before Developing a Service Identification Approach**\ [Anas Shatnawi[^1], Hafedh Mili[^2], Manel Abdellatif, Ghizlane El Boussaidi,\ Yann-Gaël Guéhéneuc, Naouel Moha, Jean Privat]{}\ *LATECE Technical Report 2017-2, LATECE Laboratoire, Université du Québec à Montréal, Canada* Introduction {#Introduction} ============ Legacy software are software that have been developed based on *outdated* technologies, however they still give *significant values* to the enterprises [@sneed2006integrating]. Besides their well-known advantages, legacy software still suffer from several drawbacks: maintenance cost, scalability, portability, etc. Due to the knowledge embedded in these legacy software, enterprises cannot easily *replaced* such software. Therefore, enterprises need to migrate their legacy software to *more loosely coupled architectures* such as *Service-Oriented Architectures* (SOA). The migration of such software to SOA does not only allow enterprises to *invest the values* of legacy software, but it also enables *the integration with new advanced technologies*. i.e., services can be shared and (re)used as *stand-alone functionalities* through their provided and required interfaces, and can be successfully *deployed in a cloud environment*. Legacy software migration to SOA involves two processes: (i) service identification and (ii) service packaging. Service identification aims to reverse engineer clusters of useful functionalities that can be good candidates for services. It was the goal of several approaches like [@nakamura2009extracting] [@alahmari2010service] [@fuhr2011using] [@grieger2014architectural] [@adjoyanservice] [@zhang2005service]. Service packaging aims to repackage these clusters in the “format” of a SOA model. This has been supported by many approaches such as [@alshara2016materializing]. After analyzing the literature of service identification process, we identify that existing approaches do not have a clear idea about several important issues. Such issues are, but not limited to what is a service and how does it differ from a software component, taxonomies of service types. Therefore, in this paper, we identify-and solve- a set of five research questions that are related to these issues. These research questions are: 1. What is a service? 2. How are services different from components? 3. [What are the different service types?]{} 4. [What are the existing service identification approaches that consider service types into account?]{} 5. [How to identify services based on the analysis of legacy source code with respect to service types?]{} Our methodology to answer these research question is based on analyzing existing definitions of services, analyzing existing service taxonomies that define the different types of services, building a service taxonomy of service types as a result of the analysis of the existing ones, analyzing existing service identification approaches that considered different service types into account during the identification process, and providing criteria that can be used to identify services within legacy applications based on service types. The rest of this report is organized as follows. We discuss service definitions in Section \[sec:what-services\]. Then, Section \[sec:service-vs-component\] differentiates between services and components. Next, we identify the different types of services in Section \[sec:service-taxonomy\]. The analysis of existing service identification approaches that consider service types into account is performed in Section \[sec:existing-identification-approaches\]. In Section \[sec:propose-approach\] and Section \[sec:conclusion\], we provide proposition to develop a service identification approach, and discuss the conclusion of this report, respectively. What is a Service? {#sec:what-services} ================== In the literature, many definitions have been proposed for defining services [@barry2003web] [@nakamura2009extracting] [@erradi2006soaf] [@brown2002using] [@openGroup]. Each definition describes a service based on different details (e.g., granularity, communication mechanism, composition, etc.). In this section, we present some of these definitions. Defining a simple service: {#defining-a-simple-service .unnumbered} -------------------------- Barry [@barry2003web] defined a service as *a well-defined, self-contained function that does not depend on the context or state of other services*[@barry2003web]. However, this definition focuses on the functional attributes of the service, while it does not define service interfaces. Considering service interfaces: {#considering-service-interfaces .unnumbered} ------------------------------- Nakamura et al[^3] [@nakamura2009extracting] and Erradi et al[^4] [@erradi2006soaf] added to these definitions the specification of service interfaces such that services should communicate through open and discoverable interfaces. Additional issues: {#additional-issues .unnumbered} ------------------ Some definitions, such as [@barry2003web] [@openGroup], added more constraints to the service. For example, a service should be a black-box functionality based on the Open Group definition[^5] [@openGroup]. Some others allow a service to be composed of multiple other services to implement a larger functionality [@barry2003web] [@openGroup]. Definition of service interfaces: {#definition-of-service-interfaces .unnumbered} --------------------------------- Brown et al[^6] [@brown2002using] clarified the communication model for service interfaces. This is based on a loosely coupled, (sometimes) asynchronous, message-based. Meaning of Service Characteristics {#Terminologies .unnumbered} ---------------------------------- **Open Interface:** “A service has an open interface, by which external entities can access to the service independently of the implementation of the service. For the access, a service cannot require platform specific operations, nor implementation-specific data that are only used within the system.” [@nakamura2009extracting] **Self-contained:** “A service can be executed by itself without any other services. Thus, a process cannot be a service if the process requires execution and/or data of any other processes. Such mutually-dependent processes should be aggregated within the same service.” [@nakamura2009extracting] **Coarse-Grained:** “A service is a coarse-grained process that can be a business construct by itself. Also, multiple services can be integrated to achieve a more sophisticated and coarser-grained service.” [@nakamura2009extracting] **Stateless** “means there is no record of previous interactions and each interaction request has to be handled based entirely on information that comes with it.” \[Internet dictionary\] **Discoverable** “means that services can be found at both design time and run time, not only by unique identity but also by interface identity and by service” [@adjoyanservice]. How are services different from components? {#sec:service-vs-component} =========================================== Following the definitions of services [@barry2003web] [@nakamura2009extracting] [@erradi2006soaf] [@brown2002using] [@openGroup] and software components [@Baster2001BCC][^7], [@szyperski2002component][^8] [@luer2002composition][^9], we find that services are very similar to software components in terms of their characteristics (*loose coupling*, *reusability*, *autonomy*, *composability*, etc.). However, we can distinguish between services and components based on two aspects: The Granularity Range --------------------- Services start at higher level of abstractions compared to components. A service can be a part of a business process (at the requirement level), an architectural element (at the design level) and a function (at the implementation level). However, components appear only at the design level and the implementation level in terms of architectural elements and functions (Figure \[fig:service-vs-component\]. ). The Deployment Technologies and Models -------------------------------------- Services and components are different in terms of deployment technologies and models that are used to technically implement them. For examples, services can be *web-services* [@alonso2004web], *micro-services* [@namiot2014micro], *REST services* [@riva2009designing], etc., while components can be *OSGi* [@alliance2003osgi], *Fractal* [@bruneton2006fractal], *SOFA* [@plasil1998sofa], etc. These have variations in their specification that make the implementation of services and components varied and respectively their provided and required interfaces. ![The granularity range of services and components[]{data-label="fig:service-vs-component"}](images/service-component-granularity.png){width="50.00000%"} What are the Different Types of Services? {#sec:service-taxonomy} ========================================= Several taxonomies [@alahmari2010service] [@fuhr2011using] [@gu2010service] [@grieger2014architectural] [@cohen2007ontology] [@dikmans2012soa] were presented to categorize different service types. In this section, we first present these six taxonomies. Then, we discuss our service taxonomy of service types as a result of the analysis of the existing ones. Existing Taxonomies of Service Types ------------------------------------ ### Alahmari’s Service Taxonomy Alahmari et al. [@alahmari2010service] classified services into seven types based on their granularities (coarse-grained or fine-grained), purposes (CRUD) and the data they manipulate. These types are as follows: - **Process service:** is the largest coarse-grained service that performs a sequence of tasks corresponding to a business process. It can be composed of other types of services. - **Business service:** implements a business logic/value. - **Transactional-data service:** performs a CRUD functions (Create, Retrieve, Update, Delate) on transactional data. - **Master-data service:** performs a CRUD functions on master data. - **Utility service:** offers a domain functionality that is required by other services to perform their tasks. - **Infrastructure service:** offers a technical functional service that is needed by other services. - **Composite service:** is the aggregation and orchestration of different atomic services. ### Fuhr’s Service Taxonomy Fuhr et al. [@fuhr2011using] distinguished three types of services based on their relationships with the business process activities. These are: - **Business service:** provides the implementation of a specific business functionality corresponds only to one business activity. - **Utility service:** implements a functionality required by some other services. This type corresponds to services participating in different business activities. - **Helper service:** implements a general functionality that is required by most of the other services. Normally, it is used by all of the business activities. ### Qu’s Service Taxonomy Qu and Lago [@gu2010service] presented six service types. This is based on studying 30 service identification approaches. These are: - **Business process service:** implements a business logic/value. - **Data service:** concerns an entity object and/or a data one. - **Composite service:** is the aggregation of other services. - **IT service:** offers functionalities related to technology. This can be infrastructure or utility services. - **Partner service:** is provided to an external partner. - **Web service:** is implemented using Web-based technology. ### Grieger’s Service Taxonomy Grieger et al. [@grieger2014architectural] defined three types of services are identified, These are as follows: - **Initial design service:** is a fine-grained functionality that is used to implement one single business logic/value. - **Composite service:** denotes to the building of a larger service based on aggregating other initial design service. - **Utility or technical service:** offers crosscutting domain and technical functionalities requires by other serviced. ### Cohen’s Service Taxonomy Cohen [@cohen2007ontology] recognized six types of services that can be classified into two main categories based on whether a service is a part of the application implementation (i.e., application service) or a part of the platform (i.e., infrastructure service). Application services can be of four kinds: (1) entity, (2) activity, (3) capability, and (4) process services. Infrastructure services can be of two kinds: (5) communication and (6) utility services. These services are defined as follows: - **Process service:** is the implementation of a business process by aggregating and orchestrating other types of services. - **Capability service:** is the implementation of an action-centric functionality corresponds to a generic business logic/value. Its scope is organizational resource. - **Activity service:** is the implementation of an action-centric functionality corresponds to a generic business value. But, its scope is smaller than the capability service. It can be for a single application or a family composed of some applications. - **Entity service:** is a data-centric service corresponding to a business entity like employee or customer. - **Communication service:** is a server that has message transportation capabilities, regardless the message content. - **Utility service:** offers infrastructural functionality that is not tied to any specific business services. ### Dikmans’s Service Taxonomy Dikmans [@dikmans2012soa] provided a comprehensive service categorization schema that is built based on six axes. These are related to the service granularity (elementary, composite and process), the actor who executes the service (human or IT-system), the channel used to offer the service (telephone, web, email, etc.), security level (public or confidential), organizational boundaries (department, enterprise or external), and the architecture layer (business, application or technical). {width="\textwidth"} Synthesis of Our Service Taxonomy --------------------------------- In the previous subsection, we presented six service taxonomies that distinguish different service types. These taxonomies are similar in terms of some service’s types and differ in terms of some others. In this section, we want to provide a comprehensive view of these taxonomies and build our taxonomy that covers all of them. Figure \[fig:servicetypes\] shows our taxonomy. We classify all service mainly into two types of services based on application and IT perspectives. In the following subsections, we will discuss them in details. ### Applications services Application services implement functionalities related to business values in the application implementation space. We classify them based on their granularities into three categories: 1. **Process services** implement largest coarse-grained functionalities correspond to business processes. This type of services is only considered in the taxonomies of Alahmari et al. [@alahmari2010service] Cohen [@cohen2007ontology] and Dikmans [@dikmans2012soa]. 2. **Business services** implement specific coarse-grained business functionalities correspond to business activities or tasks. This type is defined by all of the mentioned taxonomies. Some researchers distinguished different business services. Cohen [@cohen2007ontology] and Qu [@gu2010service] classified business services into two kinds. The first one refers to data services that implement entities and data related services. The second kind denotes to business services that implement functionalities related to business logics/values. The latter is also classified by Cohen [@cohen2007ontology] into two types based on the service scope. A service is called capability service when it has a large scope such as organization scope, while it is called activity service when it has a small scope such as a single application or a single family composed of few applications (i.e., a service in a software product line). 3. **Helper services** that implements fine-grained general functionalities related to small business values shared by different business logics. This means that these services are used by other application services to perform their activities. Helper services are referred by Alahmari et al. [@alahmari2010service], Grieger et al. [@grieger2014architectural] and Fuhr et al. [@fuhr2011using]. For example, a helper service provides CRUD functions for another business service [@alahmari2010service]. Cohen [@cohen2007ontology] and Qu et al. [@gu2010service] do not consider this type of services. ### IT services IT services provide technical functionalities offered by the infrastructure platforms and operating systems. These services are provided through programming languages API (e.g., *Java SDK*, *Android SDK*, *.Net Framework*, etc.). Application services require them to perform their tasks. This type is used by Alahmari et al. [@alahmari2010service], Qu et al. [@gu2010service], and Grieger et al. [@grieger2014architectural] and Cohen [@cohen2007ontology]. Cohen [@cohen2007ontology] considered additional sub-type of IT services. This refers to communication services that only provide message transportation capabilities, regardless the message content. IT services are not considered by Fuhr et al. [@fuhr2011using] taxonomy. ### Orthogonal Service Types Other orthogonal service types are distinguished in the service taxonomies. We consider these types of services as attributes that can be applied to any of the previously mentioned services regardless their types. Qu et al. [@gu2010service] considered a service as a partner service if it offers interfaces to external software, and a service is considered as a web service if it is implemented using a Web technology. We consider composite service type as a methodology to build a service/application by orchestrating different services, in order to produce a larger service that implements a larger business value. For example, we orchestrate several business services and infrastructure services to form a single process service related to a business process. Composite services are presented in Alahmari et al. [@alahmari2010service], Qu et al [@gu2010service], Grieger et al. [@grieger2014architectural], and Cohen [@cohen2007ontology] taxonomies. Many other orthogonal service types are presented by Dikmans [@dikmans2012soa]. These types are based on the actor (human or IT-system), the channel used (telephone, web, email, etc.), security level (public or confidential), organizational boundaries (department, enterprise or external), and the architecture layer (business, application or technical). What are the Existing Service Identification Approaches that Consider Service Types in to Account? {#sec:existing-identification-approaches} ================================================================================================== In the literature, several approaches were presented to identify services from legacy software like [@nakamura2009extracting] [@alahmari2010service] [@fuhr2011using] [@grieger2014architectural] [@adjoyanservice] [@zhang2005service]. However, we identify only three approaches [@alahmari2010service] [@fuhr2011using] [@grieger2014architectural] that consider different service types into account during the service identification process. In the following subsections, we summarize these approaches. Alahmari’s Service Identification Approach ------------------------------------------ Alahmari et al. [@alahmari2010service] identified services based on analyzing business process models. These business process models are derived from questionnaires, interviews and available documentations that provide atomic business processes and entities on the one hand, and activity diagrams that provide primitive functionalities on the other hand. The activity diagrams are manually identified from UML class diagrams extracted from the legacy code using IBM Rational Rose. Different service granularity are distinguished in relation to atomic business processes and entities. Dependent atomic processes as well as the related entities are grouped together at the same service to maximize the cohesion and minimize the coupling. There is no details about how to identify the different service types. Fuhr’s Service Identification Approach -------------------------------------- In [@fuhr2011using], Fuhr et al. identified three types of services. These are business, utility and helper services. The services are identified from legacy codes based on a dynamic analysis technique. The authors relied on a business process model to identify correlation among classes. Each activity in the business process model is executed. Classes that have got called during the execution are considered as related. The identification of services is based on a clustering technique where the similarity measurement is based on how many classes are used together in the activity executions. The identified clusters are manually interpreted and mapped into the different service types. Classes used only for the implementation of one activity are grouped into a business service corresponding to this activity. Utility services are composed of clusters of classes that contribute to implement multiple activities but not all of them. A Cluster of classes that are used by all of the activities represent the implementation of helper services. A strong assumption regarding this approach is that business process model should be available to identify execution scenarios. Grieger’s Service Identification Approach ----------------------------------------- Grieger et al. [@grieger2014architectural] presented an approach that identified three service types based on analyzing existing legacy modules. The first one refers to initial design services that implement business values. These are identified based on refining the existing legacy modules related to business values. The second one denotes to coarse-grained services, e.g., related to business processes. These are identified based on orchestrating other services related to the same underlying business process (i.e., structural dependent services). To this end, a hierarchical clustering algorithm is utilized to identify a dendrogram tree that represents the aggregation of initial design services based on their dependencies. The last service type is related to services that implement crosscutting concerns and technical functionalities used across different services (i.e., utility or technical services). The identification of these services is based on partitioning the functionalities of multiple services to recover individual and common parts. To this end, the authors relied on a clone detection algorithm to extract cloned functionalities shared among different services. Then, the identified cloned functionalities are given to software architects to decide if these cloned functionalities need to be moved into an existing service or to create a new one. How to Identify Services Based on the Object-Oriented Source Code with Respect to their Types? {#sec:propose-approach} ============================================================================================== In this section, we provide an overview analysis of how to propose a service identification approach that identify different service types from legacy object-oriented software based on the analysis source code. We identify three main elements that have to be defined for any service identification approach. These are: 1. **Service Compared to Object:** we want to to explain what do services structurally mean compared to objects. Thus, we map service-oriented concepts to object-oriented concepts. 2. **Target Service Types:** service types mentioned previously by service taxonomies can not be always identified only based on the source code analysis. Some types require additional software artifacts to be analyzed. Thus, we discuss what types of services that we are interested (based on the possibilities to identify them) to identify based on the source code analysis. 3. **Identification Algorithms:** the identification approach can be applied based on different algorithms to identify different service types. This is based on where service types are differentiated compared to the service identification step. Thus, we provide our proposition about how these algorithms work. Service Compared to Object -------------------------- In object-oriented software, functionalities are implemented in source codes in terms of a set of object-oriented classes organized in a set of object-oriented packages. Each class encapsulates a set of methods (operations) and attributes (data) related to one object. We structurally define a service as a collection of object-oriented classes that participate to implement a specific set of related functionalities. The service’s functionalities are accessible by other services through the public methods that are encapsulated in the object-oriented implementation of the service. Thus, the service interfaces are structured based on the set of all public methods existed in the object-oriented classes composing the implementation of a service. Target Service Types -------------------- In the context of service reverse engineering, distinguishing between different service types cannot be always identified based on the analysis of source codes. Some service types require further investigation based on additional software artifacts. The identification step of some service types applied the same process, while the difference between their types is based on the software architect knowledge. For example, the difference between capability services and activity services requires to identify the service scope, which is not available in the source codes. This requires to recover where the service can be deployed, in a single system or in the organization[^10]. These are identified as business services with different attributes. We focus on the following three service types during the design of service identification approaches: **Process service type** is used to implement a large-grained service related to business process. This business process is composed of a set of business activities that are implemented based on a set of business services. These business services depend on other helper services as well as IT ones. Thus, we invest the service composition type to define process service type. Therefore, a process service can be identified as a composition of a collection of dependent business, helper and IT services that are participate to implement the corresponding business process. **Business service type** is used to realize the implementation of coarse-grained services corresponding to business activities. Thus, we define a business service as a specific coarse-grained functionality that provides the implementation of one business activity. **Helper service type** refers to services that implement functionalities that do not have business values. However, they are used by business services to perform their tasks. For example, non-functional functionalities. **IT service type** is used to implement services related to technical infrastructural services that are related to the platforms. These are used by other service types to access the platform capabilities through programming languages. Identification Algorithms {#section:IdentificationAlgorithm} ------------------------- We distinguish three service identification algorithms based on where service types are taken into account in relation to the service identification method. These are pre-identification, in-identification and post-identification algorithms. The pre-identification algorithm concerns the idea of partitioning object-oriented source code into different parts based on the target service types. Then, a service identification method is applied on each part to recover the corresponding services (see Fig. \[fig:processPreIdenti\]). The in-identification one is related to design several identification algorithms where each algorithm is interested at identifying a specific service type based on the analysis of source codes (see Fig. \[fig:processInIdenti\]). The post-identification algorithm is related to one service identification process followed by refinement and classification processes to differentiate service types (see Fig. \[fig:processPostIdenti\]). {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} Conclusion {#sec:conclusion} ========== In this report, we discussed a set of issues that help developing a service identification approach based on the analysis of object-oriented source of legacy software. Such issues answer a set of research questions that are: 1. **What is a service?** We analyzed existing service definitions. We organized these definitions based on the details that they provide about services. 2. **How are services different from components?** We clarified how services are different from software components based on the granularity range and the deployment technologies and models. 3. **What are the different types of services?** We analyze six service taxonomies that were presented to classify different service types. Also, we discussed a new service taxonomy of service types based on the analysis of the existing ones. 4. **What are the existing service identification approaches that consider service types into account?** We presented three service identification approaches that considered different service types into account during the service identification process. 5. **How to identify services based on the object-oriented source code with respect to their types?** We showed the mapping of service-oriented concepts to object-oriented concepts. Then, we identify the set of service types that can be identified based on the source code analysis. Next, we discussed the algorithm that can be used to identify services and distinguishing their types. [^1]: anasshatnawi@gmail.com [^2]: mili.hafedh@uqam.ca [^3]: A service is a set of processes that: (1) has an open interface. (2) self-contained (3) coarse-grained [@nakamura2009extracting]. [^4]: A service is a self-contained business functionality that has well-defined and discoverable interfaces [@erradi2006soaf]. [^5]: A service is a black-box logical representation of a self-contained functionality that has a specified outcome and can be composed of other services [@openGroup]. [^6]: A service is: (1) composed of a loosely-coupled, coarse-grained, self-contained, discoverable, and composable functionality, (2) composed of multiple services that can be depends on each other, (3) communicates with other services based on clear interface via asynchronous messages [@barry2003web]. [^7]: A component is abstract, self-contained packages of functionality performing a specific business function within a technology framework. These business components are reusable with well-defined interfaces[@Baster2001BCC]. [^8]: A component is a unit of composition with contractually specified interfaces and explicit context dependencies only. A software component can be deployed independently and is subject to composition by third parties[@szyperski2002component]. [^9]: A component is a software element that (a) encapsulates a reusable implementation of functionality, (b) can be composed without modification, and (c) adheres to a component model[@luer2002composition]. [^10]: Can be related to software product lines!

--- abstract: 'We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of non-point classical and conditional symmetry. Then we find solution to nonlinear heat equation which can not be obtained in the framework of the classical Lie approach. By using operators of Lie–Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too. We show that the method can be applied to nonevolutionary partial differential equations.' author: - 'Ivan Tsyfra$^{1}$' - 'Tomasz Czyżycki$^{2}$' title: 'Non-point symmetry reduction method of partial differential equations ' --- $^1$ AGH University of Science and Technology, Faculty of Applied Mathematics, 30 Mickiewicza Avenue, 30-059 Krakow, Poland\ $^2$ Institute of Mathematics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland\ Email: tsyfra@agh.edu.pl, tomczyz@math.uwb.edu.pl\ MSC classification numbers: 35A20, 35Q55, 35Q72, 82D75\ Keywords: non-point symmetry, symmetry group, invariants, reduction, nonlinear wave type equation. Introduction ============ It is well known that the classical Lie symmetry method of point transformations is often used for reducing the number of independent variables in partial differential equation to obtain ordinary differential equations. After integration of reduced differential equations one can obtain partial solutions of the equation under study [@BK; @O; @Ovs]. The main problem is that the maximal invariance group of point transformations of differential equations used in applications is not sufficiently wide and thus the group approach can not be successfully applied to these equations. The concept of generalized conditional symmetry has been introduced in [@Fok; @Zhd] to extend the applicability of the symmetry method to the construction of solutions of evolution equations. The relationship of generalized conditional symmetries of evolution equations to compatibility of system of differential equations is studied in [@Kun]. The method for construction nonlocally related partial differential equation systems for a given partial differential equation has been proposed in [@BluKyt]. The starting point for the method is the existence of operator of point symmetry of the equation under study. Through a nonlocally related systems one can construct operators of nonlocal symmetry and nonlocal conservation laws of initial equation. We use operators of non-point classical and conditional symmetries to extend the class of differential equations to which the symmetry method is applicable. In this paper we study the symmetry reduction of partial differential equations with two independent variables by using the operators of non–point symmetry because the prolongated operators of classical point symmetry lead to the classical invariant solutions. The method can be naturally generalized to the multidimensional case. We construct the ansatz for dependent variable $u$ or its derivatives which reduces the scalar partial differential equation to a system of ordinary differential equations. We use the operators of the classical point symmetry [@BK; @O] of the corresponding system which are not the prolongated operators of point symmetries admitted by the original equation to construct the ansatz for derivatives. The ansatz for $u$ we construct by using ordinary differential equation admitting the operators of Lie–Bäcklund symmetry (in the classical sense [@O; @Fok]). We consider nonlinear evolution and wave type equations and present the operator of conditional symmetry for the corresponding system which generates the Bäcklund transformations for nonlinear wave equation. Recall, that the well-known integrable nonlinear differential equations such as Korteweg-de-Vries, sine-Gordon, cubic Schrödinger equations admit an infinite number of Lie–Bäcklund symmetry operators [@BK; @O]. Another goal of this paper is to show that such important properties of nonlinear partial differential equations as existence of Bäcklund transformations, linearization, existence of the class of solutions depending on arbitrary function can be related to their invariance under the finite number of non-point symmetry operators. Non-point symmetry and reduction of nonlinear wave type and evolution equations with two independent variables =============================================================================================================== The concept of differential invariant solutions based on infinite Lie group $G$ is introduced in [@Ovs]. This group is a classical symmetry group of point transformations of dependent and independent variables for the equation under study. Generally speaking, analysis similar to that in constructing differential invariant solutions enables us to obtain the ansätze for derivatives $u_{x_1}$, $u_{x_2}$ by virtue of operators of non-point symmetry [@Ovs; @T2]. Let us consider nonlinear differential equation $$\label{2} u_{x_2x_2}=\frac{1}{1-u_{x_1}^r}, \quad r \not =0,\pm 1.$$ We search for the ansatz for the derivatives of such form $$\label{1} \frac{\partial u}{\partial x_1}=R_1(x_1, x_2, u,\varphi_1(\omega) ,\varphi_2(\omega)), \quad \frac{\partial u}{\partial x_2}=R_2(x_1, x_2, u, \varphi_1(\omega), \varphi_2(\omega)) ,$$ where $\omega=\omega(x_1, x_2, u)$. Operators of classical and conditional symmetry of the corresponding system can be used to find $R_1$, $R_2$. The corresponding system has the form $$\label{3} v^1_2=v^2_1, \quad v^2_2=\frac{1}{1-(v^1)^r},$$ where $v^1=u_{x_1}$,$v^2=u_{x_2}$, $v^i_k=v^i_{x_k}$, $i, k=1,2$. To construct ansatz of type (\[1\]) we use the symmetry operator $$Q=(r+1)x_1\partial_{x_1}+rv^2\partial_{x_2}-v^1\partial_{v^1}+rv^2\partial_{v^2}.$$ of system (\[3\]). It is obvious that operator $Q$ generates non-point group transformations for variables $x_1, x_2, u$. It is easy to find the invariants of one-parameter Lie group with generator $Q$ $$\omega=x_2-v^2, \quad \omega_1=v^1(x_1)^{\frac{1}{r+1}}, \quad \omega_2=v^2(x_1)^{\frac{-r}{r+1}}.$$ By using these invariants one can construct the ansatz for $v^1$, $v^2$ $$\label{4d} v^1=(x_1)^{\frac{-1}{r+1}}\varphi_1(\omega), \quad v^2=(x_1)^{\frac{r}{r+1}}\varphi_2(\omega).$$ From (\[4d\]) we have $$\label{5d} v^2_2=\frac{(x_1)^{\frac{r}{r+1}}\varphi'_2}{1+(x_1)^{\frac{r}{r+1}}\varphi'_2}$$ where $\varphi'_2=\frac{d\varphi_2}{d\omega}$. Substituting (\[4d\]) and (\[5d\]) into the equation $$v^2_2=\frac{1}{1-(v^1)^r}$$ yields $$(x_1)^{\frac{r}{r+1}}\varphi'_2-\varphi'_2\varphi^r_1=1+(x_1)^{\frac{r}{r+1}}\varphi'_2.$$ Thus we get the first reduced ordinary differential equation $$\label{6d} \varphi'_2\varphi^r_1=-1.$$ The second one we obtain from the compatibility condition $v^1_2=v^2_1$. It has the form $$\label{7d} \frac{r}{r+1}\varphi_2=\varphi'_1.$$ We take the particular solution of reduced system of ordinary differential equations (\[6d\]), (\[7d\]) in the form $$\varphi_1=\left (\sqrt{\frac{r(r+1)}{2(r-1)}}\omega+C_1 \right )^{\frac{2}{r+1}},$$ $$\varphi_2=\sqrt{\frac{2(r+1)}{r(r-1)}}\left (\sqrt{\frac{r(r+1)}{2(r-1)}}\omega+C_1 \right )^{\frac{1-r}{r+1}}$$ where $C_1=const$. Thus one has to integrate overdetermined compatible system of differential equations $$u_{x_1}=(x_1)^{\frac{-1}{r+1}}\left (\sqrt{\frac{r(r+1)}{2(r-1)}}(x_2-u_{x_2})+C_1 \right )^{\frac{2}{r+1}}$$ $$u_{x_2}=(x_1)^{\frac{r}{r+1}}\sqrt{\frac{2(r+1)}{r(r-1)}}\left (\sqrt{\frac{r(r+1)}{2(r-1)}}(x_2-u_{x_2})+C_1 \right )^{\frac{1-r}{r+1}}$$ to construct the solution of equation (\[2\]). Nevertheless it is easy to get the solution of the equation $$\label{8d} w_t- \frac{1}{r}\left (\frac{(w-1)^{\frac{1-r}{r}}w_x}{w^{\frac{1+r}{r}}}\right )_x=0$$ in such form $$\left (1-\frac{1}{w} \right )^{\frac{1}{r}}=(t)^{\frac{-1}{r+1}}\left (\sqrt{\frac{r(r+1)}{2(r-1)}}(x-\theta )+C_1 \right )^{\frac{2}{r+1}},$$ $$\theta=(t)^{\frac{r}{r+1}}\sqrt{\frac{2(r+1)}{r(r-1)}}\left (\sqrt{\frac{r(r+1)}{2(r-1)}}(x-\theta)+C_1 \right )^{\frac{1-r}{r+1}}.$$ The tangent transformations groups are also used in the framework of this approach. Let us consider the nonlinear evolution equation $$\label{9d} u_t=e^{\frac{1}{u_{xx}}}.$$ One can construct operator of tangent transformations of the form $$K= -t\partial_{t}+u_x\partial_{x}+\frac{u_x^2}{2}\partial_u+ u_t\partial_{u_t}$$ admitted by (\[9d\]). The first order functionally independent differential invariants of the corresponding one-parameter Lie group of tangent transformations can be chosen in the form $$\omega=xu_x-2u, \quad \omega_1=\ln u_t-\frac{x}{u_x}, \quad \omega_2=u_x, \quad \omega_3=tu_t.$$ In order to construct ansatz of type (\[1\]) reducing equation (\[9d\]) to system of ordinary differential equations we consider two-dimensional Lie algebra with basic operators $\{K, P_t=\frac{\partial}{\partial t} \}$. The operators satisfy the commutation relation $[K,P_t]=P_t$. The invariants of two-parameter Lie group with generators $K, P_t$ are $\omega, \omega_1, \omega_2$. Then we construct the ansatz by using these invariants in the form $$\label{11d} u_x=f(\omega), \quad u_t=\exp \left (\varphi(\omega)+\frac{x}{f(\omega)}\right ).$$ From (\[11d\]) and (\[9d\]) we have $$u_{xx}=-\frac{f'f}{1-xf'}$$ and first ordinary differential equation $$\label{12d} f'f\varphi =-1.$$ From the condition $u_{xt}=u_{tx}$ it follows that $f, \varphi$ satisfy the second ordinary differential equation $$\label{13d} \frac{f-\varphi' f^3}{f^2}=-2f'.$$ Thus the reduced system consists of equations (\[12d\]) and (\[13d\]). From (\[12d\]), (\[13d\]) and (\[11d\]) it follows that the solutions of equation (\[9d\]) can be constructed by integrating overdetermined compatible system $$u_t=\exp{\left [\frac{2(\sqrt{C_1-4(xu_x-2u)}+x)}{2C_2-\sqrt{C_1-4(xu_x-2u)}}\right ]},$$ $$u_x=\frac{2C_2-\sqrt{C_1-4(xu_x-2u)}}{2}$$ where $C_1$, $C_2$ are arbitrary real constants. Next we emphasize that the operators of conditional symmetry of corresponding system can be used for construction the Bäcklund transformations for nonlinear wave equation $$\label{14d} u_{x_1x_2}=[1-k^2u_{x_2}^2]^{1/2}\sin u.$$ Indeed we showed that $$\label{14d1} Q=\partial_{x_3}+k\cos x_3\partial_{v^1}+k^{-1}\sqrt{1-k^2(v^2)^2}\partial_{v^2}$$ is the operator of conditional symmetry of the corresponding system $$\label{8} v^1_2+v^1_3v^2=v^2_1+v^2_3v^1,$$ $$\label{9} v^2_1+v^2_3v^1=\sqrt{1-k^2(v_2)^2}\sin x_3,$$ where $u\equiv x_3$. Using operator $Q$ we can write the ansatz in the following form $$\label{15d} u_{x_1}=\varphi_2+k\sin u, \quad u_{x_2}=k^{-1}\sin (u-\varphi_1),$$ where $\varphi_1$, $\varphi_2$ are unknown functions on $x_1$, $x_2$ and hence the Bäcklund transforms $$\label{16d} u_{x_2}=k^{-1}\sin (u-w), \quad u_{x_1}=w_{x_1}+k\sin u$$ relating equation (\[14d\]) and sine-Gordon equation $w_{x_1x_2}=\sin w$. These Bäcklund transforms (\[16d\]) have been obtained for the first time in [@BD] by another technique. Note that this approach is also applicable for linearization nonlinear partial differential equations with two independent variables. Indeed, consider the second order differential equation $$\label{17d} u_{x_0x_0}=F(u_{x_0x_1,} u_{x_1x_1}),$$ where $F$ is a smooth function. Using the invariance of (\[17d\]) under Lie group of transformations with corresponding five-dimensional Lie algebra given by basic elements $\partial_{x_0}$, $\partial_{x_1}$ $\partial_u$, $x_0\partial_u$, $x_1\partial_u$ we write the corresponding system in the form $$\label{18d} \frac{\partial F}{\partial v^2} \frac{\partial v^2}{\partial x_1}+\frac{\partial F}{\partial v^3} \frac{\partial v^3}{\partial x_1} =\frac{\partial v^2}{\partial x_0}, \quad \frac{\partial v^3}{\partial x_0}=\frac{\partial v^2}{\partial x_1},\quad v^1=F(v^2, v^3),$$ where $u_{x_0x_0}\equiv v^1(x_0,x_1)$, $u_{x_0x_1}\equiv v^2(x_0,x_1)$, $u_{x_1x_1}\equiv v^3(x_0,x_1)$. One can prove that (\[18d\]) possesses infinite Lie classical symmetry and can be linearized by hodograph transformations. Thus we obtained the method of linearization of the second order partial differential equation of the form (\[17d\]) for arbitrary function $F$. Let us note that the symmetry group of corresponding system written in the general form contains the symmetry group of point transformations of initial equation as a subgroup and generators of point transformations can be used to construct ansatz (\[1\]). However these operators lead to invariant solutions in the classical Lie sense. We shall illustrate this property by the following example. Let us consider the wave equation $$\label{19d} u_{x_1x_2}=F(u),$$ where $F$ is a smooth function. It is invariant with respect to the three-parameter Lie group. The basis of Lie algebra is given by $\{\partial_{x_1}, \partial_{x_2}, x_1\partial_{x_1}-x_2\partial_{x_2} \}$. Consider two-dimensional subalgebra with basic elements $\{ \partial_{x_2}, x_1\partial_{x_1}-x_2\partial_{x_2} \}$. By using the differential invariants $u, x_1u_{x_1}, \frac{u_{x_2}}{x_1}$ of the corresponding two-parameter Lie group we construct ansatz of the form $$\label{20d} u_{x_1}=\frac{f(u)}{x_1}, \quad u_{x_2}=x_1\varphi (u)$$ which reduces (\[19d\]) to the system $$f'\varphi=\varphi +\varphi' f=F(u).$$ Let $F(u)=0$. Then we obtain two cases\ $$1. f'=0, \quad \varphi +\varphi' f=0$$ and solution of reduced system has the form $$f=C_1=const, \quad \varphi= C_2\exp{\left (-\frac{u}{C_1}\right )}, \quad C_2=const.$$ By integrating system $$\frac{\partial u}{\partial x_1}=\frac{C_1}{x_1}, \quad \frac{\partial u}{\partial x_2}=C_2x_1\exp{\left (-\frac{u}{C_1} \right )}$$ one obtains the solution $$\label{21d} u=C_1\ln \left (\frac{C_2}{C_1}x_1x_2+C_3x_1\right )$$ where $C_3$ is arbitrary real constant and $C_1\not =0$, of equation (\[19d\]) with $F=0$. In the second case we have $$2. \hspace{5mm} \varphi =0,$$ $$u_{x_1}=\frac{1}{x_1}f(u), \quad u_{x_2}=0$$ and solution has the form $$\label{22d} u=h(x_1),$$ where $h(x_1)$ is arbitrary differentiable function. Let consider the operator $$\label{23d} Q=\alpha \partial_{x_2}+\beta ( x_1\partial_{x_1}-x_2\partial_{x_2})$$ where $\alpha$, $\beta$ are arbitrary real constants. One can verify that $$Q\left (u-C_1\ln \left (\frac{C_2}{C_1}x_1x_2+C_3x_1\right )\right )=0$$ iff $$\label{24d} \alpha \frac{C_2}{C_1}+\beta C_3=0.$$ It means that solution (\[21d\]) is invariant with respect to one-dimensional subgroup of symmetry group of equation (\[19d\]) with generator $Q$ where $\alpha$, $\beta$ satisfy the condition (\[24d\]). It is obvious that the solution (\[22d\]) is invariant with respect to one-parameter group with generator $Q=\alpha \partial_{x_2}$ ($\beta=0$). Thus we conclude that any solution of equation (\[19d\]) when $F=0$ constructed by this method with the help of two-dimensional Lie algebra with basic elements $\{ \partial_{x_2}, x_1\partial_{x_1}-x_2\partial_{x_2} \}$ is an invariant one in the classical Lie sense. Further we show how the operators of Lie–Bäcklund symmetry [@O; @Fok] are used for reducing partial differential equations. Let us consider equation $$\label{25d} U ( x, u,{\mathop u\limits_1},{\mathop u\limits_2},\ldots , {\mathop u\limits_k} ) =0,$$ where $x=(x_1,x_2,\ldots ,x_n)$, $u=u(x)\in C^k(\mathbb{R}^n,{\mathbb R}^1)$, and ${\mathop u\limits_k}$ denotes all partial derivatives of $k$-th order and the $m$-th order ordinary differential equation of the form $$\label{26d} H\left (x_1, x_2, \ldots , x_n, u, \frac{\partial u}{\partial x_1}, \ldots, \frac{\partial^m u}{\partial x_1^m} \right )=0.$$ Let $$u=F(x, C_1,\ldots,C_{m}),$$ where $F$ is a smooth function on variables $x, C_1,\ldots ,C_{m}$, and $C_1,\ldots, C_{m}$ are arbitrary functions on variables $x_2, x_3,\ldots, x_n$, be a general solution of equation (\[26d\]). We use the Theorem 1 from [@T8] which implies that if equation (\[26d\]) is invariant with respect to the Lie–Bäcklund operator $X=U ( x, u,{\mathop u\limits_1},{\mathop u\limits_2},\ldots ,{\mathop u\limits_k} )\partial_u$ then the ansatz $$\label{28d} u=F(x, \varphi_1, \varphi_2,\ldots,\varphi_{m}),$$ where $\varphi_1, \varphi_2,\ldots,\varphi_{m}$ depend on $n-1$ variables $x_2, x_3,\ldots, x_n$ reduces partial differential equation (\[25d\]) to the system of $k_1$ equations for unknown functions $\varphi_1, \varphi_2,\ldots ,\varphi_m$ with $n-1$ independent variables and $k_1 \le m$. We show the application of the theorem to nonlinear partial differential equation. Consider linear ordinary differential equation $$\label{29d} u_{x_1x_1}+\alpha^2u_{x_1}=0$$ where $\alpha=const$. Recall, that the concepts of local theory of differential equations such as symmetry, conditional symmetry, conservation laws, Lax representations are defined by differential equalities which must be satisfied only for solutions of the equations under study. One can prove that equation (\[29d\]) admits the following Lie–Bäcklund operator $$X=(u_{x_1x_2}-u_{x_1}F(u_{x_1}+\alpha^2u))\partial_u,$$ where $F\in C^2(\mathbb{R}^1,{\mathbb R}^1)$. It means that the following criterium of invariance $$X_{(2)}(u_{x_1x_1}+\alpha^2u_{x_1})=0 \quad \textrm{whenever} \quad u_{x_1x_1}+\alpha^2u_{x_1}=0,$$ where $X_{(2)}$ is the prolongated operator of the second order [@O], is fulfilled. We have proved that (\[29d\]) admits operator of non-point (tangent) symmetry $$X_1=f(u, u_x)\partial u$$ if $f(u, u_x)$ satisfies the following equation $$f''_{uu} -2\alpha^2f''_{uu_x}+ \alpha^4f''_{u_xu_x}=0.$$ The general solution of this equation has the form $$\label{1m} f=A(u_{x_1}+\alpha^2u)u+B(u_{x_1}+\alpha^2u)$$ where $A$, $B$ are arbitrary smooth functions of one variable. One can verify that equation (\[29d\]) also admits operator $$X_2=e^{-\alpha^2x_1}h(u_{x_1}+\alpha^2u)\partial u$$ where $h$ is arbitrary function on variable $u_{x_1}+\alpha^2u$. Then the ansatz $$\label{30d} u=\varphi_1(x_2)+e^{-\alpha^2x_1}\varphi_2(x_2)$$ obtained from the general solution of equation (\[29d\]) reduces wave type partial differential equations $$\label{31d} u_{x_1x_2}=u_{x_1}F(u_{x_1}+\alpha^2u)+A(u_{x_1}+\alpha^2u)u+B(u_{x_1}+\alpha^2u)+ ku_{x_2}+e^{-\alpha^2x_1}h(u_{x_1}+\alpha^2u)$$ where $k$ is a real constant. In general, the $x_1$ dependent coefficients in partial differential equations enable us to study the effects of field gradients. Substituting (\[30d\]) into (\[31d\]) we obtain the reduced system of two ordinary differential equations $$\label{32d} -\alpha^2\varphi'_2=-\alpha^2\varphi_2F(\alpha^2\varphi_1)+A(\alpha^2\varphi_1)\varphi_2+k\varphi'_2+h(\alpha^2\varphi_1),$$ $$\label{32m} A(\alpha^2\varphi_1)\varphi_1+ B(\alpha^2\varphi_1)+k\varphi'_1=0$$ for unknown functions $\varphi_1(x_2)$, $\varphi_2(x_2)$. One can obtain partial solutions of (\[31d\]) from solutions of system (\[32d\]), (\[32m\]). In particular, if $A=B=0$ and $k=0$ then system (\[32d\]), (\[32m\]) is reduced to one ordinary differential equation of the form $$\varphi'_2=\varphi_2F(\alpha^2\varphi_1)-\displaystyle \frac{1}{\alpha^2}h(\alpha^2\varphi_1)$$ where $\varphi_1(x_2)$, $\varphi_2(x_2)$ are unknown functions. This equation is integrable by quadratures for arbitrary $\varphi_1(x_2)$. Its general solution has the form $$\label{35m} \varphi_2=\left (C_1-\displaystyle \frac{1}{\alpha^2}\int h(\alpha^2\varphi_1(x_2))H(x_2)dx_2 \right ) \exp{\left (\int F(\alpha^2\varphi_1(x_2))dx_2 \right )}$$ where $C_1=const$, $$\label{35v} H(x_2)=\exp{\left (-\int F(\alpha^2\varphi_1(x_2))dx_2 \right )}.$$ Using (\[30d\]) one can construct the solution of nonlinear wave equation $$\label{31w} u_{x_1x_2}=u_{x_1}F(u_{x_1}+\alpha^2u)+e^{-\alpha^2x_1}h(u_{x_1}+\alpha^2u)$$ in the following form $$\label{33d} u=\varphi_1(x_2)+\left (C_1-\displaystyle \frac{1}{\alpha^2}\int h(\alpha^2\varphi_1(x_2))H(x_2)dx_2 \right ) \exp{\left (\int F(\alpha^2\varphi_1(x_2))dx_2 -\alpha^2 x_1\right )},$$ where $\varphi_1(x_2)$ is arbitrary smooth function. So, in the framework of this approach we have constructed solution with arbitrary function $\varphi_1(x_2)$ to nonlinear wave type partial differential equation (\[31w\]) for arbitrary functions $F$ and $h$. Conclusions =========== We have constructed ansätze (\[4d\]) and (\[11d\]) which reduce nonlinear evolution equations (\[2\]) and (\[9d\]) to ordinary differential equations and can not be obtained by using classical Lie method. We have found the solution of nonlinear heat equation (\[8d\]). It turns out that some of these ansätze result in the classical invariant solutions. Obviously, one can construct such ansätze by prolongated operators of point symmetry admitted by the initial equation but they lead to the invariant solutions too. It is necessary that operators of non-point and conditional symmetry should be applied to obtain new results. As was noted above the linearization of class of nonlinear partial differential equations (\[17d\]) is possible in the framework of this approach. Finally we show that the existence even at least one operator of Lie-Bäcklund symmetry to ordinary differential equations (\[29d\]) gives the possibility of constructing the solutions (\[33d\]) defined by arbitrary functions to equation (\[31w\]). To our knowledge the inverse scattering tranformation method is not applicable in this case. We show that this approach is applicable to nonevolutionary differential equations. [99]{} Bluman G.W. and Kumei S., Symmetries and Differential Equations, (1989), Appl. Math. Sci. 81, Springer–Verlag, Berlin Olver P.J., Applications of Lie Groups to Differential Equations, Springer–Verlag, New York, 1986 Ovsiannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982 Fokas A.S. and Q.M.Liu, Nonlinear interaction of travelling waves of nonintegrable equations, [*Phys. Rev. Lett.*]{} [**72, No.21**]{} (1994), 3293–3296. Zhdanov R.Z., Conditional Lie-Bäcklund symmetry and reduction of evolution equations, [*J. Phys. A: Math. Gen.* ]{} [**28**]{} (1995), 3841–3850. Kunzinger M. and Popovych R., Generalized conditional symmetries of evolution equations, [*J. Math. Anal. Appl*]{} [**379**]{} (2011), 444-460. Bluman G.W. and Zhengzheng Yang, A symmetry-based method for constructing nonlocally related partial differential equation systems, [*J. Math. Phys*]{} [**54**]{}, 093504 (2013). Tsyfra I., Napoli A., Messina A., Tretynyk V., On new ways of group methods for reduction of evolution-type equations , [*J. Math. Anal. Appl.*]{} [**307**]{}, (2005), 724–735. Dodd R.K. and Bullogh R.K. Bäcklund transformations for the sine–Gordon equations, [*Proc. R. Soc. A*]{} [**351**]{}, (1976), 499–523. Tsyfra I.M. Symmetry reduction of nonlinear differential equations, Proceedings of Institute of Mathematics, Kiev, 2004, [**50**]{}, pp. 266–270.

--- abstract: 'General relativity predicts that the Kerr black hole develops qualitatively new and surprising features in the limit of maximal spin. Most strikingly, the region of spacetime near the event horizon stretches into an infinitely long throat and displays an emergent conformal symmetry. Understanding dynamics in this NHEK (Near-Horizon Extreme Kerr) geometry is necessary for connecting theory to upcoming astronomical observations of high-spin black holes. We review essential properties of NHEK and its relationship to the rapidly rotating Kerr black hole. We then completely solve the geodesic equation in the NHEK region and describe how the resulting trajectories transform under the action of its enhanced symmetries. In the process, we derive explicit expressions for the angular integrals appearing in the Kerr geodesic equation and obtain a useful formula, valid at arbitrary spin, for a particle’s polar angle in terms of its radial motion. These results will aid in the analytic computation of astrophysical observables relevant to ongoing and future experiments.' author: - Daniel Kapec - Alexandru Lupsasca bibliography: - 'NHEKgeo.bib' --- Introduction ============ Extremal black holes have served as a rich source of novel ideas and techniques in quantum gravity and field theory for several decades [@Strominger1996; @Maldacena1997; @Guica2009; @Maldacena2016a; @Maldacena2016b]. These fundamental advances have led to a mathematical description of numerous interesting quantum-mechanical and gravitational systems, but have yet to connect directly with experiment. However, with the advent of a new generation of powerful astronomical detectors such as LIGO and the Event Horizon Telescope [@Abbott2016; @EHT2019], a subclass of astrophysically realistic near-extremal black holes stands poised to bridge this gap between formal theoretical investigation and successful experimental verification. The near-extremal Kerr black hole exhibits a number of striking phenomena showcasing strong-field general relativity, and a confirmation of even the most basic, qualitative prediction derived from the emergent symmetries of its near-horizon region would mark a huge success for both theory and experiment. If high-spin black holes do exist and come within observational reach, they will provide a window into a region of our Universe that is qualitatively similar to the extensively studied Anti-de Sitter (AdS) spacetime, which plays an outsize role in the modern holographic perspective on quantum gravity. The traditional approach to the modeling of astrophysical black holes is based on extensive numerical simulation across large swaths of parameter space. While this method of analysis is perfectly adequate in the general setting, it must confront new complications that arise in the specific regime of high spin. As a black hole rotates faster, it develops an increasingly deep throat that nonetheless remains confined within a small coordinate distance from the event horizon. As a result, resolving near-horizon physics requires a spacetime mesh of increasingly fine resolution as the spin grows. Meanwhile, the overall size of the grid must remain large in order to accurately capture the asymptotically flat region far from the black hole. Eventually, this large separation of scales can incur a prohibitive computational cost. Fortunately, the same phenomenon that renders the problem numerically intractable also enables the application of a complementary analytic method. The analytic approach proceeds from the key observation that near extremality, the Kerr spacetime separates into two distinct regions. While the extreme Kerr metric resolves physics away from the horizon, the throat region is instead described by the dramatically simpler Near-Horizon Extreme Kerr (NHEK) geometry, which possesses additional symmetries and can be viewed as a spacetime in its own right. The enhanced symmetry of the NHEK metric fixes the behavior of many near-horizon processes and turns complex dynamical questions into considerably simpler kinematic ones [@Lupsasca2014; @Zhang2014; @Lupsasca2015; @Compere2015; @Gralla2016a; @Chen2017; @Gralla2017b; @Gralla2018; @Hadar2019]. Once a problem has been solved in the NHEK geometry, it can be matched onto the far region in order to produce astrophysical predictions. Indeed, there is a growing body of work that seeks to exploit the symmetries of the NHEK region to constrain physical observables targeted by ongoing and upcoming experiments [@Porfyriadis2014; @Hadar2014; @Hadar2015; @Hadar2017; @Compere2018; @Gralla2015; @Gralla2016b; @Burko2016; @Gralla2016c; @Compere2017; @Gralla2017a; @Porfyriadis2017; @Lupsasca2018; @Gates2018]. Many, if not all, of these analyses require approximate or exact solutions to the geodesic equation in NHEK and Kerr. While Kerr geodesic motion is an old and well-studied subject, most treatments reduce the problem to first-order form and then simply seek numerical integration of the equations. On the other hand, NHEK geodesic motion has received far less attention. Previous work has focused primarily on equatorial geodesics and those geodesics obtainable from the equatorial case through a symmetry transformation [@Porfyriadis2014; @Hadar2014; @Hadar2015; @Hadar2017; @Compere2018]. In this paper, we present drastically simplified analytic expressions for the angular geodesic integrals in Kerr, and solve outright the full geodesic equation in NHEK. Our results will directly aid in the calculation of gravitational wave and electromagnetic signals from non-equatorial sources in the NHEK region, which are expected to be relevant for both current and future experiments. The outline of this paper is as follows. In Sec. \[sec:Kerr\], we revisit the problem of geodesic motion in the Kerr spacetime for arbitrary values of the black hole spin, and derive new, improved expressions for the motion in the poloidal $(r,\theta)$ plane. In Sec. \[sec:NHEK\], we present a pedagogical review of the NHEK geometry and its origin as the near-horizon scaling limit of a (near-)extreme Kerr black hole. We then completely solve the NHEK geodesic equation in Sec. \[sec:GeodesicsInNHEK\], working first in the global strip, then in the Poincaré patch, and finally in the near-NHEK patch. We classify all categories of geodesic motion in each coordinate system and obtain explicit formulas for the motion as a function of coordinate time in each case. We conclude with a description of how NHEK geodesics transform under the action of the $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ isometry group. Appendix \[app:EllipticIntegrals\] gathers mathematical definitions needed in the main body of the text. Appendix \[app:AdS2\] describes the dimensional reduction of the NHEK geometry to AdS$_2$ with a constant electric field, and the projection of NHEK geodesics to the trajectories of charged particles in AdS$_2$ subject to the Lorentz force exerted by the background electromagnetic field. Geodesics in Kerr {#sec:Kerr} ================= In this section, we review the standard treatment of geodesic motion in the background of a rotating black hole [@Carter1968; @Bardeen1973; @Chandrasekhar1983; @ONeill1995]. We begin in Sec. \[subsec:KerrGeodesics\] by re-deriving the Kerr geodesic equation in its first-order formulation. Then, we classify the different possible qualitative behaviors of the polar motion in Sec. \[subsec:QualitativeDescription\], before explicitly evaluating all the angular path integrals appearing in the geodesic equation in Sec. \[subsec:Computation\]. We take great care to unpack these integrals into sums of manifestly real and positive elliptic integrals, each of which is represented in Legendre canonical form. This results in compact expressions that are appreciably simpler than those previously given in the literature [@Rauch1994; @Vazquez2004; @Kraniotis2005; @Dexter2009; @Fujita2009; @Kraniotis2011; @Hackmann2010; @Hackmann2014], which either did not explicitly unpack the path integrals or did not reduce them to manifestly real and positive expressions. Our formulas then allow us to obtain in Sec. \[subsec:PolarInversion\] a simple expression for the polar angle as a function of the radial motion. Readers solely interested in the end results may skip directly to Sec. \[subsec:KerrSummary\] for a succinct summary. The Kerr geodesic equation in first-order form {#subsec:KerrGeodesics} ---------------------------------------------- The Kerr metric describes astrophysically realistic rotating black holes of mass $M$ and angular momentum $J=aM$. In Boyer-Lindquist coordinates ${{\mathopen{}\mathclose\bgroup\originalleft}(t,r,\theta,\phi{\aftergroup\egroup\originalright})}$ and natural units where $G=c=1$, the Kerr line element is \[eq:Kerr\] $$\begin{gathered} ds^2=-\frac{\Delta}{\Sigma}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}t-a\sin^2{\theta}{\mathop{}\!\mathrm{d}}\phi{\aftergroup\egroup\originalright})}^2+\frac{\Sigma}{\Delta}{\mathop{}\!\mathrm{d}}r^2+\Sigma{\mathop{}\!\mathrm{d}}\theta^2+\frac{\sin^2{\theta}}{\Sigma}{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}\phi-a{\mathop{}\!\mathrm{d}}t{\aftergroup\egroup\originalright}]}^2,\\ \Delta(r)=r^2-2Mr+a^2,\qquad \Sigma(r,\theta)=r^2+a^2\cos^2{\theta}.\end{gathered}$$ This metric admits two Killing vectors ${\mathop{}\!\partial}_t$ and ${\mathop{}\!\partial}_\phi$ generating time-translation symmetry and axisymmetry, respectively. In addition to these isometries, the Kerr metric also admits an irreducible symmetric Killing tensor[^1] $$\begin{aligned} \label{eq:KerrKilling} K_{\mu\nu}=-{J_\mu}^\lambda J_{\lambda\nu},\qquad J=a\cos{\theta}{\mathop{}\!\mathrm{d}}r\wedge{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}t-a\sin^2{\theta}{\mathop{}\!\mathrm{d}}\phi{\aftergroup\egroup\originalright})}+r\sin{\theta}{\mathop{}\!\mathrm{d}}\theta\wedge{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}\phi-a{\mathop{}\!\mathrm{d}}t{\aftergroup\egroup\originalright}]}.\end{aligned}$$ The motion of a free particle of mass $\mu$ and four-momentum $p^\mu$ is described by the geodesic equation, $$\begin{aligned} p^\mu\nabla_\mu p^\nu=0,\qquad g^{\mu\nu}p_\mu p_\nu=-\mu^2.\end{aligned}$$ In the Kerr geometry , geodesic motion is completely characterized by three conserved quantities, $$\begin{aligned} \omega=p_\mu{\mathop{}\!\partial}_t^\mu =-p_t,\qquad \ell=p_\mu{\mathop{}\!\partial}_\phi^\mu =p_\phi,\qquad k=K^{\mu\nu}p_\mu p_\nu =p_\theta^2+a^2\mu^2\cos^2{\theta}+{{\mathopen{}\mathclose\bgroup\originalleft}(p_\phi\csc{\theta}+p_ta\sin{\theta}{\aftergroup\egroup\originalright})}^2,\end{aligned}$$ denoting the total energy, angular momentum parallel to the axis of symmetry, and Carter constant, respectively. The first two quantities are the conserved charges associated with the Killing vectors ${\mathop{}\!\partial}_t$ and ${\mathop{}\!\partial}_\phi$, respectively, whereas the conservation of the third quantity follows from the existence of the Killing tensor . While the Carter constant has the advantage of being manifestly positive, it is often useful to work instead with the so-called Carter integral $$\begin{aligned} Q=k-{{\mathopen{}\mathclose\bgroup\originalleft}(\ell-a\omega{\aftergroup\egroup\originalright})}^2 =p_\theta^2+a^2{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2-p_t^2{\aftergroup\egroup\originalright})}\cos^2{\theta}+p_\phi^2\cot^2{\theta}.\end{aligned}$$ By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,\omega,\ell,k{\aftergroup\egroup\originalright})}$, we find that a particle following a geodesic in the Kerr geometry has an instantaneous four-momentum $p=p_\mu{\mathop{}\!\mathrm{d}}x^\mu$ of the form $$\begin{aligned} p(x^\mu,\omega,\ell,k)=-\omega{\mathop{}\!\mathrm{d}}t\pm_r\frac{\sqrt{\mathcal{R}(r)}}{\Delta}{\mathop{}\!\mathrm{d}}r\pm_\theta\sqrt{\Theta(\theta)}{\mathop{}\!\mathrm{d}}\theta+\ell{\mathop{}\!\mathrm{d}}\phi,\end{aligned}$$ where the two choices of sign $\pm_r$ and $\pm_\theta$ depend on the radial and polar directions of travel, respectively. Here, we also introduced radial and polar potentials $$\begin{aligned} \mathcal{R}(r)&={{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}^2-\Delta{{\mathopen{}\mathclose\bgroup\originalleft}(k+\mu^2r^2{\aftergroup\egroup\originalright})},\\ \Theta(\theta)&=k-a^2\mu^2\cos^2{\theta}-{{\mathopen{}\mathclose\bgroup\originalleft}(\ell\csc{\theta}-a\omega\sin{\theta}{\aftergroup\egroup\originalright})}^2.\end{aligned}$$ One can then raise $p_\mu$ to obtain the equations for the geodesic trajectory, $$\begin{aligned} \label{eq:RadialGeodesicEquation} \Sigma\frac{dr}{d\sigma}&=\pm_r\sqrt{\mathcal{R}(r)},\\ \label{eq:AngularGeodesicEquation} \Sigma\frac{d\theta}{d\sigma}&=\pm_\theta\sqrt{\Theta(\theta)},\\ \Sigma\frac{d\phi}{d\sigma}&=\frac{a}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+\ell\csc^2{\theta}-a\omega,\\ \label{eq:TimeGeodesicEquation} \Sigma\frac{dt}{d\sigma}&=\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+a{{\mathopen{}\mathclose\bgroup\originalleft}(\ell-a\omega\sin^2{\theta}{\aftergroup\egroup\originalright})}.\end{aligned}$$ The parameter $\sigma$ is the affine parameter for massless particles ($\mu=0$), and is related to the proper time $\delta$ by $\delta=\mu\sigma$ for massive particles. This system is completely integrable because it admits as many constants of motion as momentum variables, and can be integrated by quadratures. To do so, first note that $$\begin{aligned} \frac{1}{\pm_r\sqrt{\mathcal{R}(r)}}\frac{dr}{d\sigma}=\frac{1}{\Sigma}=\frac{1}{\pm_\theta\sqrt{\Theta(\theta)}}\frac{d\theta}{d\sigma}.\end{aligned}$$ Integration along the geodesic from $\sigma=\sigma_s$ to $\sigma=\sigma_o$ yields $$\begin{aligned} \fint_{\sigma_s}^{\sigma_o}\frac{1}{\pm_r\sqrt{\mathcal{R}(r)}}\frac{dr}{d\sigma}{\mathop{}\!\mathrm{d}}\sigma=\fint_{\sigma_s}^{\sigma_o}\frac{1}{\pm_\theta\sqrt{\Theta(\theta)}}\frac{d\theta}{d\sigma}{\mathop{}\!\mathrm{d}}\sigma,\end{aligned}$$ where the slash notation $\fint$ indicates that these integrals are to be evaluated along the geodesic, with turning points in the radial or polar motion occurring whenever the corresponding potential $\mathcal{R}(r)$ or $\Theta(\theta)$ vanishes. By definition, the signs $\pm_r$ and $\pm_\theta$ in front of $\sqrt{\mathcal{R}(r)}$ and $\sqrt{\Theta(\theta)}$ are always the same as that of ${\mathop{}\!\mathrm{d}}r$ and ${\mathop{}\!\mathrm{d}}\theta$, respectively, so these integrals grow secularly (rather than canceling out) over multiple oscillations. If the particle is located at ${{\mathopen{}\mathclose\bgroup\originalleft}(t_s,r_s,\theta_s,\phi_s{\aftergroup\egroup\originalright})}$ when $\sigma=\sigma_s$ and at ${{\mathopen{}\mathclose\bgroup\originalleft}(t_o,r_o,\theta_o,\phi_o{\aftergroup\egroup\originalright})}$ when $\sigma=\sigma_o$, then this simplifies to $$\begin{aligned} \fint_{r_s}^{r_o}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta(\theta)}}.\end{aligned}$$ Likewise, $$\begin{aligned} \phi_o-\phi_s&=\fint_{\phi_s}^{\phi_o}{\mathop{}\!\mathrm{d}}\phi =\fint_{\sigma_s}^{\sigma_o}\frac{d\phi}{d\sigma}{\mathop{}\!\mathrm{d}}\sigma =\fint_{\sigma_s}^{\sigma_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{a}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+\ell\csc^2{\theta}-a\omega{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}\sigma}{\Sigma}\\ &=\fint_{\sigma_s}^{\sigma_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{a}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}-a\omega{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}\sigma}{\Sigma} +\fint_{\sigma_s}^{\sigma_o}\ell\csc^2{\theta}\frac{{\mathop{}\!\mathrm{d}}\sigma}{\Sigma}.\end{aligned}$$ Aside from $\Sigma$, the first integrand only contains $r$-dependent terms and the second integrand only contains $\theta$-dependent terms. Thus, we naturally replace $\Sigma$ using Eq.  in the first integral and using Eq.  in the second, resulting in $$\begin{aligned} \phi_o-\phi_s=\fint_{r_s}^{r_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{a}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}-a\omega{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}+\fint_{\theta_s}^{\theta_o}\frac{\ell\csc^2{\theta}}{\pm_\theta\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta.\end{aligned}$$ After repeating the same procedure for $t$ and shuffling constant pieces from one integral into the other, we find that $$\begin{aligned} t_o-t_s&=\fint_{t_s}^{t_o}{\mathop{}\!\mathrm{d}}t =\fint_{\sigma_s}^{\sigma_o}\frac{dt}{d\sigma}{\mathop{}\!\mathrm{d}}\sigma =\fint_{\sigma_s}^{\sigma_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+a{{\mathopen{}\mathclose\bgroup\originalleft}(\ell-a\omega\sin^2{\theta}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}\sigma}{\Sigma}\\ &=\fint_{\sigma_s}^{\sigma_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+a{{\mathopen{}\mathclose\bgroup\originalleft}(\ell-a\omega{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}\sigma}{\Sigma}+\fint_{\sigma_s}^{\sigma_o}a^2\omega\cos^2{\theta}\frac{{\mathop{}\!\mathrm{d}}\sigma}{\Sigma}\\ &=\fint_{r_s}^{r_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+a{{\mathopen{}\mathclose\bgroup\originalleft}(\ell-a\omega{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}+\fint_{\theta_s}^{\theta_o}\frac{a^2\omega\cos^2{\theta}}{\pm_\theta\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta.\end{aligned}$$ To summarize, a geodesic labeled by $(\omega,\ell,k)$ connects spacetime points $x_s^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(t_s,r_s,\theta_s,\phi_s{\aftergroup\egroup\originalright})}$ and $x_o^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(t_o,r_o,\theta_o,\phi_o{\aftergroup\egroup\originalright})}$ if \[eq:KerrGeodesicEquation\] $$\begin{aligned} &\fint_{r_s}^{r_o}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta(\theta)}},\\ \phi_o-\phi_s&=\fint_{r_s}^{r_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{a}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}-a\omega{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}+\fint_{\theta_s}^{\theta_o}\frac{\ell\csc^2{\theta}}{\pm_\theta\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta,\\ t_o-t_s&=\fint_{r_s}^{r_o}{{\mathopen{}\mathclose\bgroup\originalleft}\{\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}}{\Delta}{{\mathopen{}\mathclose\bgroup\originalleft}[\omega{{\mathopen{}\mathclose\bgroup\originalleft}(r^2+a^2{\aftergroup\egroup\originalright})}-a\ell{\aftergroup\egroup\originalright}]}+a{{\mathopen{}\mathclose\bgroup\originalleft}(\ell-a\omega{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}\}}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}+\fint_{\theta_s}^{\theta_o}\frac{a^2\omega\cos^2{\theta}}{\pm_\theta\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta.\end{aligned}$$ Generically, Kerr geodesics undergo multiple librations (polar oscillations) and rotations about the axis of symmetry. They may also undergo radial oscillations when they are bound (${{\mathopen{}\mathclose\bgroup\originalleft}|\omega{\aftergroup\egroup\originalright}|}<\mu$) [@Wilkins1972]. Kerr geodesics are therefore characterized by integers $(w,m,n)$ denoting the number of turning points in the radial motion, the number of turning points in the polar motion, and the winding number about the axis of symmetry, respectively. Qualitative description of the polar motion {#subsec:QualitativeDescription} ------------------------------------------- We now wish to compute the angular integrals that appear in the Kerr geodesic equation , $$\begin{aligned} G_\theta=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta(\theta)}},\qquad G_\phi=\fint_{\theta_s}^{\theta_o}\frac{\csc^2{\theta}}{\pm_\theta\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta,\qquad G_t=\fint_{\theta_s}^{\theta_o}\frac{\cos^2{\theta}}{\pm_\theta\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta,\end{aligned}$$ and then solve for the final angle $\theta_o$ in the $(r,\theta)$ part of the equation, which is of the form $I_r=G_\theta$ with $$\begin{aligned} I_r=\fint_{r_s}^{r_o}\frac{{\mathop{}\!\mathrm{d}}r}{\pm_r\sqrt{\mathcal{R}(r)}}.\end{aligned}$$ To do so, it is convenient to rewrite the angular potential as $$\begin{aligned} \Theta(\theta)=Q+P\cos^2{\theta}-\ell^2\cot^2{\theta},\qquad P=a^2{{\mathopen{}\mathclose\bgroup\originalleft}(\omega^2-\mu^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ There are three qualitatively different cases that we will consider in turn: 1. corresponds to null geodesics with $\mu=0$, or unbound timelike geodesics with $0<\mu^2<\omega^2$. 2. corresponds to marginally bound timelike geodesics with $0<\mu^2=\omega^2$. 3. corresponds to bound timelike geodesics with $0\le\omega^2<\mu^2$. Here, the bound/unbound nomenclature refers to the radial motion—the polar motion is of course always bounded. The positivity condition $\Theta(\theta)\ge0$ implies that a geodesic can reach a pole at $\theta_N=0$ or $\theta_S=\pi$ if and only if $\ell=0$. We will assume the genericity condition $\ell\neq0$, in which case the polar motion is strictly restricted to oscillations bounded by turning points $\theta_\pm\in{{\mathopen{}\mathclose\bgroup\originalleft}(0,\pi{\aftergroup\egroup\originalright})}$.[^2] This oscillatory motion can be of two qualitatively different types: - Oscillatory motion about the equatorial plane between $\theta_-\in{{\mathopen{}\mathclose\bgroup\originalleft}(0,\pi/2{\aftergroup\egroup\originalright})}$ and $\theta_+\in{{\mathopen{}\mathclose\bgroup\originalleft}(\pi/2,\pi{\aftergroup\egroup\originalright})}$ with $\theta_+=\pi-\theta_-$. - “Vortical" motion between turning points $0<\theta_-<\theta_+<\pi/2$ or $\pi/2<\theta_+<\theta_-<\pi$, corresponding to geodesics that never cross the equatorial plane and are instead confined to a cone lying either entirely above or entirely below the equatorial plane [@deFelice1972]. The Kerr geometry also admits planar geodesics at any fixed polar angle $\theta_0$. These arise in the special limit where the turning points of the angular motion coalesce at $\theta_0=\theta_\pm$. In that case, the geodesic equation degenerates and a separate treatment is necessary. In the null case, the planar geodesics are the well-known principal null congruences, which endow the Kerr geometry with many of its special properties.[^3] We presently exclude this fine-tuned situation, which has been extensively studied in the literature [@Chandrasekhar1983]. To study the two generic types, it is useful to define signs $$\begin{aligned} {2} \eta_o&=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_o^\theta{\aftergroup\egroup\originalright})}\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta}_o{\aftergroup\egroup\originalright})}\hspace{10em} \eta_s&&=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta}_s{\aftergroup\egroup\originalright})}\\ &=(-1)^m\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta}_o{\aftergroup\egroup\originalright})}, &&=(-1)^m\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_o^\theta{\aftergroup\egroup\originalright})}\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta}_s{\aftergroup\egroup\originalright})}, \end{aligned}$$ where $p_s^\theta$ and $p_o^\theta$ denote the polar momentum evaluated at the endpoints $x_s^\mu$ and $x_o^\mu$ of the geodesic, respectively. By working through all the possible configurations, one can check that the angular path integral unpacks as follows: $$\begin{aligned} \text{Type A:}\qquad& \fint_{\theta_s}^{\theta_o}=2m{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_\pm}{\aftergroup\egroup\originalright}|}+\eta_s{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_s}{\aftergroup\egroup\originalright}|}-\eta_o{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_o}{\aftergroup\egroup\originalright}|},\\ \text{Type B:}\qquad& \fint_{\theta_s}^{\theta_o}={{\mathopen{}\mathclose\bgroup\originalleft}(m\pm\eta_o\frac{1-(-1)^m}{2}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_-}^{\theta_+}{\aftergroup\egroup\originalright}|}\pm\eta_s{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_\pm}^{\theta_s}{\aftergroup\egroup\originalright}|}\mp\eta_o{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_\pm}^{\theta_o}{\aftergroup\egroup\originalright}|},\end{aligned}$$ where for both types, we presented two equivalent representations that differ only in the choice of turning point taken as a reference for the integrals. It will turn out that the type of oscillation is picked out by the sign of $Q$: 1. corresponds to Type A oscillations. These are allowed for all signs of $P$. 2. corresponds to Type B (vortical) oscillations. These are only allowed for $P>0$.[^4] 3. corresponds to a singular limit of Type B motion in which the cone of oscillation touches the equatorial plane, where the integrals develop a nonintegrable singularity. Such geodesics are also only allowed for $P>0$. From now on, we will work with the variable $u=\cos^2{\theta}$, in terms of which $$\begin{aligned} \Theta(u)= \begin{cases} \displaystyle\frac{P}{1-u}{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}&P\neq0,\vspace{2pt}\\ \displaystyle\frac{Q+\ell^2}{1-u}{{\mathopen{}\mathclose\bgroup\originalleft}(u_0-u{\aftergroup\egroup\originalright})}&P=0. \end{cases}\end{aligned}$$ Here we defined $$\begin{aligned} \Delta_\theta=\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{Q+\ell^2}{P}{\aftergroup\egroup\originalright})},\qquad u_\pm=\Delta_\theta\pm\sqrt{\Delta_\theta^2+\frac{Q}{P}},\qquad u_0=\frac{Q}{Q+\ell^2}.\end{aligned}$$ For future convenience, we also introduce the quantities $$\begin{aligned} \Psi_j^\pm=\arcsin\sqrt{\frac{\cos^2{\theta_j}}{u_\pm}},\qquad \Upsilon_j^\pm=\pm\arcsin\sqrt{\pm\frac{\cos^2{\theta_j}-u_\mp}{u_+-u_-}}.\end{aligned}$$ Computation of the angular integrals {#subsec:Computation} ------------------------------------ ### Case 1: P=0 In this case, we must necessarily have $Q>0$ and $0\le\cos^2{\theta}\le u_0<1$. Hence, the oscillation is of Type A with turning points at $\theta_\mp=\arccos{\pm\sqrt{u_0}}$. The only integrals we need are $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{\Theta(\theta)}}{\aftergroup\egroup\originalright}|}&=\frac{1}{2}\sqrt{\frac{u_0}{Q}}\int_0^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_0-u{\aftergroup\egroup\originalright})}}} =\sqrt{\frac{u_0}{Q}}\arcsin\sqrt{\frac{u_j}{u_0}},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{\csc^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2}\sqrt{\frac{u_0}{Q}}\int_0^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-u{\aftergroup\egroup\originalright})}\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_0-u{\aftergroup\egroup\originalright})}}} =\frac{1}{\sqrt{Q}}\sqrt{\frac{u_0}{1-u_0}}\arcsin\sqrt{\frac{u_j}{u_0}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1-u_0}{1-u_j}{\aftergroup\egroup\originalright})}},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{\cos^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2}\sqrt{\frac{u_0}{Q}}\int_0^{u_j}\frac{u{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_0-u{\aftergroup\egroup\originalright})}}} =\frac{1}{2}\sqrt{\frac{u_0}{Q}}{{\mathopen{}\mathclose\bgroup\originalleft}[u_0\arcsin\sqrt{\frac{u_j}{u_0}}-\sqrt{u_j{{\mathopen{}\mathclose\bgroup\originalleft}(u_0-u_j{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}]},\end{aligned}$$ where, in order to ensure that each integral is real and positive, we used the substitution $$\begin{aligned} u=u_0t^2.\end{aligned}$$ Thus, in the $P=0$ case (where necessarily $Q>0$), we obtain [align]{} \[eq:P=0,Q&gt;0\] G\_&=[[\[m+\_s-\_o]{}\]]{},\ G\_&=[[\[m+\_s-\_o]{}\]]{},\ G\_t&=[[{u\_0G\_-]{}}]{}. ### Case 2: P&lt;0 In this case, we must necessarily have $Q>0$ and $0\le\cos^2{\theta}\le u_-<1$. Hence, the oscillation is of Type A with turning points at $\theta_\mp=\arccos{\pm\sqrt{u_-}}$. The only integrals we need are $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{\Theta(\theta)}}{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{-P}}\int_0^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u_--u{\aftergroup\egroup\originalright})}}} =\frac{1}{\sqrt{-u_+P}}F{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_j^-{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_-}{u_+}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{\csc^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{-P}}\int_0^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-u{\aftergroup\egroup\originalright})}\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u_--u{\aftergroup\egroup\originalright})}}} =\frac{1}{\sqrt{-u_+P}}\Pi{{\mathopen{}\mathclose\bgroup\originalleft}(u_-;\Psi_j^-{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_-}{u_+}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{\cos^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{-P}}\int_0^{u_j}\frac{u{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u_--u{\aftergroup\egroup\originalright})}}} =-\frac{2u_-}{\sqrt{-u_+P}}E'{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_j^-{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_-}{u_+}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ where we defined $E'(x|k)\equiv{\mathop{}\!\partial}_kE(x|k)={{\mathopen{}\mathclose\bgroup\originalleft}[E(x|k)-F(x|k){\aftergroup\egroup\originalright}]}/(2k)$ and, in order to ensure that each integral is real and positive, we used the substitution $$\begin{aligned} u=u_-t^2.\end{aligned}$$ Thus, in the $P<0$ case (where necessarily $Q>0$), we obtain [align]{} \[eq:P&lt;0,Q&gt;0\] G\_&=[[\[2mK[[(]{})]{}+\_sF[[(\_s\^-[|]{}.]{})]{}-\_oF[[(\_o\^-[|]{}.]{})]{}]{}\]]{},\ G\_&=[[\[2m+\_s-\_o]{}\]]{},\ G\_t&=-[[\[2mE’[[(]{})]{}+\_sE’[[(\_s\^-[|]{}.]{})]{}-\_oE’[[(\_o\^-[|]{}.]{})]{}]{}\]]{}. ### Case 3: P&gt;0 If $Q>0$, then $u_-<0\le\cos^2{\theta}\le u_+<1$ and the oscillation is of Type A with turning points at $\theta_\mp=\arccos{\pm\sqrt{u_+}}$. The only integrals we need are $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{\Theta(\theta)}}{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{P}}\int_0^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}}} =\frac{1}{\sqrt{-u_-P}}F{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_j^+{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_+}{u_-}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{\csc^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{P}}\int_0^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-u{\aftergroup\egroup\originalright})}\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}}} =\frac{1}{\sqrt{-u_-P}}\Pi{{\mathopen{}\mathclose\bgroup\originalleft}(u_+;\Psi_j^+{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_+}{u_-}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\pi/2}^{\theta_j}\frac{\cos^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{P}}\int_0^{u_j}\frac{u{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}}} =-\frac{2u_+}{\sqrt{-u_-P}}E'{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_j^+{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_+}{u_-}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ where, in order to ensure that each integral is real and positive, we used the substitution $$\begin{aligned} u=u_+t^2.\end{aligned}$$ Thus, in the $P>0$, $Q>0$ case, we obtain [align]{} \[eq:P&gt;0,Q&gt;0\] G\_&=[[\[2mK[[(]{})]{}+\_sF[[(\_s\^+[|]{}.]{})]{}-\_oF[[(\_o\^+[|]{}.]{})]{}]{}\]]{},\ G\_&=[[\[2m+\_s-\_o]{}\]]{},\ G\_t&=-[[\[2mE’[[(]{})]{}+\_sE’[[(\_s\^+[|]{}.]{})]{}-\_oE’[[(\_o\^+[|]{}.]{})]{}]{}\]]{}. If $Q<0$, then $0<u_-\le\cos^2{\theta}\le u_+<1$ and the oscillation is of Type B with turning points at $\theta_-=\arccos{\pm\sqrt{u_+}}$ and $\theta_+=\arccos{\pm\sqrt{u_-}}$, where the upper/lower sign corresponds to vortical oscillation within a cone lying entirely above/below the equatorial plane. If we use $\theta_\pm$ as the reference turning point ($i.e.$, in both hemispheres, we integrate from the turning point closest to/farthest from the equator), then the only integrals we need are $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_\pm}^{\theta_j}\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{\Theta(\theta)}}{\aftergroup\egroup\originalright}|}&=\pm\frac{1}{2\sqrt{P}}\int_{u_\mp}^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}}} =\pm\frac{1}{\sqrt{u_\mp P}}F{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_j^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_\pm}^{\theta_j}\frac{\csc^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\pm\frac{1}{2\sqrt{P}}\int_{u_\mp}^{u_j}\frac{{\mathop{}\!\mathrm{d}}u}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-u{\aftergroup\egroup\originalright})}\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}}} =\pm\frac{1}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-u_\mp{\aftergroup\egroup\originalright})}\sqrt{u_\mp P}}\Pi{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\pm-u_\mp}{1-u_\mp};\Upsilon_j^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_\pm}^{\theta_j}\frac{\cos^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\pm\frac{1}{2\sqrt{P}}\int_{u_\mp}^{u_j}\frac{u{\mathop{}\!\mathrm{d}}u}{\sqrt{u{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(u-u_-{\aftergroup\egroup\originalright})}}} =\pm\sqrt{\frac{u_\mp}{P}}E{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_j^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ where we used the substitution $$\begin{aligned} u=u_\mp\pm{{\mathopen{}\mathclose\bgroup\originalleft}(u_+-u_-{\aftergroup\egroup\originalright})}t^2\end{aligned}$$ in order to ensure that each integral is real and positive. Thus, in the $P>0$, $Q<0$ case, we obtain [align]{} \[eq:P&gt;0,Q&lt;0\] G\_&=[[\[[[(m\_o]{})]{}K[[(1-]{})]{}+\_sF[[(\_s\^.]{})]{}-\_oF[[(\_o\^.]{})]{}]{}\]]{},\ G\_&=,\ G\_t&=[[\[[[(m\_o]{})]{}E[[(1-]{})]{}+\_sE[[(\_s\^.]{})]{}-\_oE[[(\_o\^.]{})]{}]{}\]]{}. To be clear, in these equations, the choice of upper/lower sign leads to equivalent representations of the integrals corresponding to different choices of reference turning point. If $Q=0$, then $0=u_-\le\cos^2{\theta}\le u_+<1$ and the oscillation appears to be of Type B with turning points at the equator and $\theta_-=\arccos{\pm\sqrt{u_+}}$, where the upper/lower sign corresponds to vortical oscillation within a cone lying entirely above/below the equatorial plane. However, in practice, the geodesic motion can only turn at $\theta_-$, as it is barred from reaching the equator at $u=0$, which corresponds to a nonintegrable singularity of the angular integrals. Hence, the complete motion can undergo at most one libration, $i.e.$, it can only have $m=0$ or $m=1$. Therefore, in this special situation, $$\begin{aligned} \fint_{\theta_s}^{\theta_o}=\eta_o{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_-}^{\theta_o}{\aftergroup\egroup\originalright}|}-\eta_s{{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_-}^{\theta_s}{\aftergroup\egroup\originalright}|}.\end{aligned}$$ Thus, using $\theta_-$ as the reference turning point (which we must, in order to avoid the singularity at the equator), the only integrals we need are $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_-}^{\theta_j}\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{\Theta(\theta)}}{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{P}}\int_{u_j}^{u_+}\frac{{\mathop{}\!\mathrm{d}}u}{u\sqrt{u_+-u}} =\frac{1}{\sqrt{u_+P}}\operatorname{arctanh}\sqrt{1-\frac{u_j}{u_+}},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_-}^{\theta_j}\frac{\csc^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{P}}\int_{u_j}^{u_+}\frac{{\mathop{}\!\mathrm{d}}u}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-u{\aftergroup\egroup\originalright})}u\sqrt{u_+-u}} =\frac{1}{\sqrt{u_+P}}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}\sqrt{1-\frac{u_j}{u_+}}+\sqrt{\frac{u_+}{1-u_+}}\arctan\sqrt{\frac{u_+-u_j}{1-u_+}}{\aftergroup\egroup\originalright}]},\\ {{\mathopen{}\mathclose\bgroup\originalleft}|\int_{\theta_-}^{\theta_j}\frac{\cos^2{\theta}}{\sqrt{\Theta(\theta)}}{\mathop{}\!\mathrm{d}}\theta{\aftergroup\egroup\originalright}|}&=\frac{1}{2\sqrt{P}}\int_{u_j}^{u_+}\frac{{\mathop{}\!\mathrm{d}}u}{\sqrt{u_+-u}} =\sqrt{\frac{u_+-u_j}{P}},\end{aligned}$$ where we did not need any substitution to obtain simpler trigonometric representations of the integrals, which are all real and positive. In conclusion, in the $P>0$, $Q=0$ case, we obtain [align]{} \[eq:P&gt;0,Q=0\] G\_&=[[\[\_o-\_s]{}\]]{},\ G\_&=G\_+[[\[\_o-\_s]{}\]]{},\ G\_t&=[[\[\_o-\_s]{}\]]{}. Solution to the (r,theta) equation {#subsec:PolarInversion} ---------------------------------- The $(r,\theta)$ part of the Kerr geodesic equation is of the form $I_r=G_\theta$, with $$\begin{aligned} I_r=\fint_{r_s}^{r_o}\frac{{\mathop{}\!\mathrm{d}}r}{\pm\sqrt{\mathcal{R}(r)}}.\end{aligned}$$ We want to solve this equation for $\theta_o$. We will proceed by considering each case in turn, starting with the simplest. In the $P=0$ case (where necessarily $Q>0$), Eq.  tells us that $$\begin{aligned} \sqrt{\frac{Q}{u_0}}I_r=\pi m+\eta_s\arcsin\sqrt{\frac{u_s}{u_0}}-\eta_o\arcsin\sqrt{\frac{u_o}{u_0}}.\end{aligned}$$ Using the fact that $\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(-x{\aftergroup\egroup\originalright})}=-\arcsin{x}$ is an odd function, this can be rewritten $$\begin{aligned} \arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_o}}{\sqrt{u_0}}{\aftergroup\egroup\originalright})}=(-1)^m{{\mathopen{}\mathclose\bgroup\originalleft}[\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_0}}{\aftergroup\egroup\originalright})}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\pi m-\sqrt{\frac{Q}{u_0}}I_r{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]},\end{aligned}$$ from which it follows that $$\begin{aligned} \cos{\theta_o}=\sqrt{u_0}\sin{{\mathopen{}\mathclose\bgroup\originalleft}[(-1)^mW_m{\aftergroup\egroup\originalright}]},\qquad W_m=\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_0}}{\aftergroup\egroup\originalright})}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\pi m-\sqrt{\frac{Q}{u_0}}I_r{\aftergroup\egroup\originalright})}.\end{aligned}$$ This expression can be further simplified by noting that $$\begin{aligned} W_m=W_{m-1}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\pi.\end{aligned}$$ Since the function $\sin{x}$ satisfies the periodicity condition $\sin{{\mathopen{}\mathclose\bgroup\originalleft}(x\pm\pi{\aftergroup\egroup\originalright})}=-\sin{x}$, it follows that $$\begin{aligned} \sin{{\mathopen{}\mathclose\bgroup\originalleft}[(-1)^mW_m{\aftergroup\egroup\originalright}]}=\sin{{\mathopen{}\mathclose\bgroup\originalleft}[(-1)^{m-1}W_{m-1}{\aftergroup\egroup\originalright}]} =\sin{W_0},\end{aligned}$$ from which we conclude that $\cos{\theta_o}$ is in fact independent of the number of turning points along the trajectory: [align]{} =, W\_0=[[(]{})]{}-[[(p\_s\^]{})]{}I\_r. We now turn to the remaining cases (with $P\neq0$), which are slightly more complicated but can nonetheless be treated using a similar approach. As a preliminary, note that because $\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(-x{\aftergroup\egroup\originalright})}=-\arcsin{x}$ is an odd function, $$\begin{aligned} \label{eq:SignAbsorption} \operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_j}{\aftergroup\egroup\originalright})}\Psi_j^\pm=\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_j}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}.\end{aligned}$$ In the $P\neq0$ case with $Q>0$, Eqs.  and tell us that $$\begin{aligned} \sqrt{-u_\mp P}I_r=2mK{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}+\eta_sF{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_s^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}-\eta_oF{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_o^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ where $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P{\aftergroup\egroup\originalright})}$. Using Eq. , this can be rewritten $$\begin{aligned} \label{eq:IntermediateStep+} F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_o}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}=(-1)^mX_m^\pm,\end{aligned}$$ where we defined $$\begin{aligned} X_m^\pm=F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}[2mK{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}-\sqrt{-u_\mp P}I_r{\aftergroup\egroup\originalright}]},\end{aligned}$$ and used the fact that $F(-x|k)=-F(x|k)$ is odd in its first argument. The inverse function of the elliptic integral of the first kind is the Jacobi elliptic function $\operatorname{sn}(x|k)$, which satisfies $\operatorname{sn}(F(\arcsin{x}|k)|k)=x$. Using this identity, it immediately follows from Eq.  that $$\begin{aligned} \frac{\cos{\theta_o}}{\sqrt{u_\pm}}=\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}((-1)^mX_m^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}.\end{aligned}$$ This expression can be further simplified by noting that $$\begin{aligned} X_m^\pm=X_{m-1}^\pm+2\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}.\end{aligned}$$ Since the function $\operatorname{sn}(x|k)$ satisfies the periodicity condition $\operatorname{sn}(x\pm2K(k)|k)=-\operatorname{sn}(x|k)$, it follows that $$\begin{aligned} \operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}((-1)^mX_m^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}=\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}((-1)^{m-1}X_{m-1}^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})} =\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}(X_0^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ from which we again conclude that $\cos{\theta_o}$ is independent of the number of turning points along the trajectory: [align]{} \[eq:ObserverAngleQ&gt;0\] =[[(X\_0\^.]{})]{}, X\_0\^=F[[(.]{})]{}-[[(p\_s\^]{})]{}I\_r. In the $Q<0$ case (where necessarily $P>0$), Eq.  tells us that $$\begin{aligned} \sqrt{u_\mp P}I_r={{\mathopen{}\mathclose\bgroup\originalleft}(m\pm\eta_o\frac{1-(-1)^m}{2}{\aftergroup\egroup\originalright})}K{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}+\eta_sF{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_s^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}-\eta_oF{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_o^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ where either choice of sign $\pm$ is equally valid. This can be rewritten as $$\begin{aligned} \label{eq:IntermediateStep-} F{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_o^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}=(-1)^m\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}Y_m^\pm,\end{aligned}$$ where we introduced $$\begin{aligned} Y_m^\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}F{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_s^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(m\pm\eta_o\frac{1-(-1)^m}{2}{\aftergroup\egroup\originalright})}K{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}-\sqrt{u_\mp P}I_r{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Next, we use the fact that the Jacobi elliptic function $\operatorname{dn}(x,k)$ satisfies $\operatorname{dn}(F(\arcsin{x}|k)|k)=\sqrt{1-kx^2}$, and therefore $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|y{\aftergroup\egroup\originalright}|}=\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.F{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.\arcsin{\sqrt{\frac{1-y^2}{k}}}{\aftergroup\egroup\originalright}|k{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}|k{\aftergroup\egroup\originalright})}.\end{aligned}$$ Applying this identity to $y=\sin{\Psi_o^\mp}$ yields $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}|\sin{\Psi_o^\mp}{\aftergroup\egroup\originalright}|}=\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.F{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.\arcsin{\sqrt{\frac{1-\sin^2{\Psi_o^\mp}}{1-\frac{u_\pm}{u_\mp}}}}{\aftergroup\egroup\originalright}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})} =\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(\pm(-1)^m\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}Y_m^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}.\end{aligned}$$ The last step follows from Eq. , together with the observation that $$\begin{aligned} \label{eq:UpsilonPsiRelation} \arcsin\sqrt{\frac{1-\sin^2{\Psi_j^\mp}}{1-\frac{u_\pm}{u_\mp}}}=\pm\Upsilon_j^\pm,\end{aligned}$$ which follows from $$\begin{aligned} 1-\sin^2{\Psi_j^\mp}={{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}\sin^2{\Upsilon_j^\pm}.\end{aligned}$$ Using Eq.  and the fact that $\operatorname{dn}(-x|k)=\operatorname{dn}(x|k)$ is even in its first argument, we find that $$\begin{aligned} \frac{\cos{\theta_o}}{\sqrt{u_\mp}}=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}|\sin{\Psi_o^\mp}{\aftergroup\egroup\originalright}|} =\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(Y_m^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})} =\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(Y_m^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ where in the last step, we used the fact that $\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}$ for Type B vortical geodesics. This expression can be further simplified by noting that $$\begin{aligned} Y_m^\pm=Y_{m-1}^\pm+{{\mathopen{}\mathclose\bgroup\originalleft}[1\pm\eta_o(-1)^{m-1}{\aftergroup\egroup\originalright}]}\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}K{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright})}.\end{aligned}$$ Since the function $\operatorname{dn}(x|k)$ satisfies the periodicity condition $\operatorname{dn}(x\pm2K(k)|k)=\operatorname{dn}(x|k)$, it follows that $$\begin{aligned} \operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(Y_m^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}=\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(Y_{m-1}^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})} =\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(Y_0^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\end{aligned}$$ from which we again conclude that $\cos{\theta_o}$ is independent of the number of turning points along the trajectory: [align]{} \[eq:ObserverAngleQ&lt;0\] =[[(]{})]{}[[(Y\_0\^.]{})]{}, Y\_0\^=[[(]{})]{}F[[(\_s\^.]{})]{}-[[(p\_s\^]{})]{}I\_r. While this expression for $\cos{\theta_o}$ (valid for $Q<0$) superficially differs from that given in Eq.  (valid for $Q>0$), the two expressions can be brought into the same form. For $k\not\in \mathbb{R}$, the reciprocal-modulus theorem [@Fettis1970] $$\begin{aligned} \frac{1}{\sqrt{k}}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{k}{\aftergroup\egroup\originalright})}=K(k)\mp iK(1-k),\qquad \pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\operatorname{Im}{k}{\aftergroup\egroup\originalright})},\end{aligned}$$ determines the real and imaginary parts of $K(1/k)$ in terms of $K(k)$ and $K(1-k).$ The difference in sign results from the branch cut in $K(k)$ extending from $k=1$ to $+\infty$ along the positive real axis. For real $k$, choosing the primary branch for $k<1$ fixes $$\begin{aligned} \label{eq:EllipticIdentity} \frac{1}{\sqrt{k}}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{k}{\aftergroup\egroup\originalright})}=K(k)\mp iK(1-k),\qquad \pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(1-k{\aftergroup\egroup\originalright})}.\end{aligned}$$ With this choice, identical manipulations demonstrate that [alignat=2]{} \[eq:EllipticIdentity+\] &F[[([.]{}|k]{})]{}+iF[[([.]{}|1-k]{})]{}, 1&lt;y\^2&lt;k,\ \[eq:EllipticIdentity-\] &F[[([.]{}|k]{})]{}-iF[[([.]{}|1-k]{})]{}, 0&lt;k&lt;y\^2&lt;1. We now apply these identities with $y=\sin{\Psi_s^\mp}=\sqrt{u_s/u_\mp}$ and $k=u_\pm/u_\mp$. Since $0<u_-<u_s<u_+<1$, we apply for the upper choice of sign and for the lower choice of sign. Combined with Eq. , we find that $$\begin{aligned} \sqrt{\frac{u_\mp}{u_\pm}}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\mp}{u_\pm}{\aftergroup\egroup\originalright})}=F{{\mathopen{}\mathclose\bgroup\originalleft}(\Psi_s^\pm{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}+iF{{\mathopen{}\mathclose\bgroup\originalleft}(\Upsilon_s^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}.\end{aligned}$$ After multiplying by $\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}$, this becomes $$\begin{aligned} \label{eq:IntermediateStep} \operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}\sqrt{\frac{u_\mp}{u_\pm}}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\mp}{u_\pm}{\aftergroup\egroup\originalright})}=F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}+iY_0^\pm+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\sqrt{-u_\mp P}I_r.\end{aligned}$$ In order to proceed, we also need the identity[^5] $$\begin{aligned} \operatorname{dn}(x|1-k)=\pm\sqrt{k}\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.ix\pm\frac{1}{\sqrt{k}}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{k}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}|k{\aftergroup\egroup\originalright})}.\end{aligned}$$ This relation holds for either choice of sign, which we take to be $\pm=-\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}$. Substituting $k=u_\pm/u_\mp$ and combining with Eq.  yields $$\begin{aligned} \cos{\theta_o}=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}\sqrt{u_\mp}\operatorname{dn}{{\mathopen{}\mathclose\bgroup\originalleft}(Y_0^\pm{\mathopen{}\mathclose\bgroup\originalleft}|1-\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})} =\sqrt{u_\pm}\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}(-iY_0^\pm+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}\sqrt{\frac{u_\mp}{u_\pm}}K{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{u_\mp}{u_\pm}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}.\end{aligned}$$ Finally, using Eq. , this reduces to $$\begin{aligned} \cos{\theta_o}=\sqrt{u_\pm}\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}(F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\sqrt{-u_\mp P}I_r{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}.\end{aligned}$$ Therefore Eqs.  and can be combined into the single expression [align]{} =[[(X\_0\^.]{})]{}, X\_0\^=F[[(.]{})]{}-[[(p\_s\^Q]{})]{}I\_r. In the $Q=0$ case (where necessarily $P>0$), Eq.  tells us that $$\begin{aligned} \sqrt{u_+P}I_r&=\eta_o\operatorname{arctanh}\sqrt{1-\frac{u_o}{u_+}}-\eta_s\operatorname{arctanh}\sqrt{1-\frac{u_s}{u_+}},\end{aligned}$$ which can be rewritten $$\begin{aligned} \operatorname{arctanh}\sqrt{1-\frac{u_o}{u_+}}=(-1)^m\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}Z,\qquad Z=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}\operatorname{arctanh}\sqrt{1-\frac{u_s}{u_+}}+\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\sqrt{u_+P}I_r.\end{aligned}$$ Since $\tanh{{\mathopen{}\mathclose\bgroup\originalleft}(-x{\aftergroup\egroup\originalright})}=-\tanh{x}$ is an odd function, it follows that $$\begin{aligned} \sqrt{1-\frac{u_o}{u_+}}=(-1)^m\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}\tanh{Z}.\end{aligned}$$ Therefore, the overall signs disappear upon squaring, leaving $$\begin{aligned} u_o=u_+\operatorname{sech}^2{Z}.\end{aligned}$$ Using the fact that $\operatorname{sech}{{\mathopen{}\mathclose\bgroup\originalleft}(-x{\aftergroup\egroup\originalright})}=\operatorname{sech}{x}$ is an even function, and that $\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_s}{\aftergroup\egroup\originalright})}=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\cos{\theta_o}{\aftergroup\egroup\originalright})}$ for vortical geodesics, we finally conclude: [align]{} =[[(]{})]{}, Z=+\_sI\_r. Summary of results {#subsec:KerrSummary} ------------------ Here, we collect the simplified expressions for the angular integrals $G_\theta$, $G_\phi$ and $G_t$, as well as expressions for the final angle $\theta_o$ obtained by solving the $(r,\theta)$ part of the geodesic equation . Our conventions are such that all of the terms appearing in these expressions are real and positive. \[eq:Q&gt;0,P=0\] [align]{} G\_&=[[\[m+\_s-\_o]{}\]]{},\ G\_&=[[\[m+\_s-\_o]{}\]]{},\ G\_t&=[[{u\_0G\_-]{}}]{},\ &=, W\_0=[[(]{})]{}-[[(p\_s\^]{})]{}I\_r. \[eq:Q&gt;0,P!=0\] [align]{} G\_&=[[\[2mK[[(]{})]{}+\_sF[[(\_s\^.]{})]{}-\_oF[[(\_o\^.]{})]{}]{}\]]{},\ G\_&=[[\[2m+\_s-\_o]{}\]]{},\ G\_t&=-[[\[2mE’[[(]{})]{}+\_sE’[[(\_s\^.]{})]{}-\_oE’[[(\_o\^.]{})]{}]{}\]]{},\ \[eq:ObserverAngleP!=0,Q&gt;0\] &=[[(X\_0\^.]{})]{}, X\_0\^=F[[(.]{})]{}-[[(p\_s\^]{})]{}I\_r. [align]{} G\_&=[[\[[[(m\_o]{})]{}K[[(1-]{})]{}+\_sF[[(\_s\^.]{})]{}-\_oF[[(\_o\^.]{})]{}]{}\]]{},\ G\_&=,\ G\_t&=[[\[[[(m\_o]{})]{}E[[(1-]{})]{}+\_sE[[(\_s\^.]{})]{}-\_oE[[(\_o\^.]{})]{}]{}\]]{},\ \[eq:ObserverAngleP&gt;0,Q&lt;0\] &=[[(]{})]{}[[(Y\_0\^.]{})]{}, Y\_0\^=[[(]{})]{}F[[(\_s\^.]{})]{}-[[(p\_s\^]{})]{}I\_r. [align]{} G\_&=[[\[\_o-\_s]{}\]]{},\ G\_&=G\_+[[\[\_o-\_s]{}\]]{},\ G\_t&=[[\[\_o-\_s]{}\]]{},\ &=[[(]{})]{}, Z=+\_sI\_r. Finally, we note that Eqs.  and can be conveniently combined into a single expression: [align]{} \[eq:ObserverAngle\] &=[[(X\_0\^.]{})]{}, X\_0\^=F[[(.]{})]{}-[[(p\_s\^Q]{})]{}I\_r. Near-horizon geometry of (near-)extreme Kerr {#sec:NHEK} ============================================ The Kerr family of metrics has two adjustable parameters corresponding to the mass $M$ and angular momentum $J$ of the black hole. Geometries satisfying the Kerr bound ${{\mathopen{}\mathclose\bgroup\originalleft}|J{\aftergroup\egroup\originalright}|}\le M^2$ have smooth event horizons concealing a ring singularity, while solutions that violate this bound exhibit naked singularities visible from infinity. Black holes that (nearly) saturate the Kerr bound are termed (near-)extremal, and there is strong evidence to suggest that no physical process can drive a (sub-)extremal black hole over the Kerr bound [@Sorce2017] (such super-extremal black holes would behave very differently—see, $e.g.$, Refs. [@Stuchlik2010; @Stuchlik2011]). However, one would expect accretion of matter onto an astrophysical black hole to push it towards extremality, and indeed the vast majority of measured supermassive black holes spins are close to maximal [@Brenneman2013; @Reynolds2019]. The limiting behavior of the Kerr metric in the extremal limit ${{\mathopen{}\mathclose\bgroup\originalleft}|J{\aftergroup\egroup\originalright}|}\to M^2$ is therefore of both theoretical and astronomical interest. This section is dedicated to a pedagogical review of the qualitatively new and surprising features that the Kerr black hole develops in the high-spin regime. We begin in Sec. \[subsec:Puzzle\] by presenting a timelike equatorial orbit that seemingly lies on a null hypersurface. This apparent paradox is resolved in Sec. \[subsec:EmergentThroat\] by the presence of an infinitely deep throat-like region bunched up near the event horizon: the Near-Horizon Extreme Kerr (NHEK) geometry. This motivates the more systematic investigation of near-horizon scaling limits that we conduct in Sec. \[subsec:ScalingLimits\]. These different limits are then related by an emergent conformal symmetry of the throat in Sec. \[subsec:ConformalSymmetryInTheSky\]. Finally, in Sec. \[subsec:GlobalNHEK\], we introduce global coordinates and describe the causal structure of the throat geometry using its Carter-Penrose diagram. Peculiar features of the extremal limit {#subsec:Puzzle} --------------------------------------- The ${{\mathopen{}\mathclose\bgroup\originalleft}|J{\aftergroup\egroup\originalright}|}\to M^2$ limit of the Kerr geometry poses a series of puzzles whose resolution requires a careful analysis of the near-horizon geometry of the extreme black hole. A particle orbiting a Kerr black hole on a prograde, circular, equatorial geodesic at radius $r=r_s$ has four-velocity [@Bardeen1972] $$\begin{aligned} \label{eq:EquatorialGeodesics} u_s=u_s^t{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\partial}_t+\Omega_s{\mathop{}\!\partial}_\phi{\aftergroup\egroup\originalright})},\qquad u_s^t=\frac{r_s^{3/2}+aM^{1/2}}{\sqrt{r_s^3-3Mr_s^2+2aM^{1/2}r_s^{3/2}}},\qquad \Omega_s=\frac{M^{1/2}}{r_s^{3/2}+aM^{1/2}}.\end{aligned}$$ The energy and angular momentum of this geodesic is given by $$\begin{aligned} \label{eq:EquatorialGeodesicsParameters} \frac{\omega_s}{\mu}=\frac{r_s^{3/2}-2Mr_s^{1/2}+aM^{1/2}}{r_s^{3/4}\sqrt{r_s^{3/2}-3Mr_s^{1/2}+2aM^{1/2}}},\qquad \frac{\ell_s}{\mu}=\frac{M^{1/2}\big(r_s^2-2aM^{1/2}r_s^{1/2}+a^2\big)}{r_s^{3/4}\sqrt{r_s^{3/2}-3Mr_s^{1/2}+2aM^{1/2}}}.\end{aligned}$$ This orbit is stable provided that the orbital radius $r_s$ exceeds the marginally stable radius $r_\mathrm{ms}$ of the Innermost Stable Circular Orbit (ISCO), given by \[eq:ISCO\] $$\begin{gathered} r_\mathrm{ms}=M{{\mathopen{}\mathclose\bgroup\originalleft}(3+Z_2-\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(3-Z_1{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(3+Z_1+2Z_2{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright})},\\ Z_1=1+{{\mathopen{}\mathclose\bgroup\originalleft}(1-a_\star^2{\aftergroup\egroup\originalright})}^{1/3}{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(1+a_\star{\aftergroup\egroup\originalright})}^{1/3}+{{\mathopen{}\mathclose\bgroup\originalleft}(1-a_\star{\aftergroup\egroup\originalright})}^{1/3}{\aftergroup\egroup\originalright}]},\qquad Z_2={{\mathopen{}\mathclose\bgroup\originalleft}(3a_\star^2+Z_1^2{\aftergroup\egroup\originalright})}^{1/2},\qquad a_\star=\frac{a}{M}.\end{gathered}$$ In the context of black hole astrophysics, these orbits provide a simple model for accretion onto a black hole: to a very good approximation, a thin disk of slowly accreting matter consists of particles following the geodesics [@Novikov1973; @Page1974]. In reality, their trajectories also have a small inward radial component, but it can be neglected down to the ISCO radius, which delineates the innermost edge of the disk where accretion terminates. Beyond this edge, the particles quickly plunge into the black hole and their radial momentum can no longer be ignored. Instead, their motion is described by infalling geodesics with the conserved quantities of the marginally stable orbit [@Cunningham1975; @Penna2012]. Consider a Kerr black hole with spin parameter $a=M\sqrt{1-\kappa^2}$. When the deviation from extremality is small, $0<\kappa\ll1$, the black hole has a small Hawking temperature of order $\kappa$, $$\begin{aligned} T_H=\frac{1}{4\pi r_+}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{r_+-r_-}{r_++r_-}{\aftergroup\egroup\originalright})} =\frac{\kappa}{4\pi M}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\kappa^2{\aftergroup\egroup\originalright})}},\end{aligned}$$ where $r_\pm$ denotes the radius of the (outer/inner) event horizon, $$\begin{aligned} \label{eq:EventHorizon} r_\pm=M\pm\sqrt{M^2-a^2} =M{{\mathopen{}\mathclose\bgroup\originalleft}(1\pm\kappa{\aftergroup\egroup\originalright})}.\end{aligned}$$ Thus, the extremal limit ${{\mathopen{}\mathclose\bgroup\originalleft}|J{\aftergroup\egroup\originalright}|}\to M^2$ is equivalent to a low-temperature limit $\kappa\to0$. A detailed investigation of this limit raises several puzzles: 1. The first puzzle pertains to the fate of the ISCO in the extremal limit. According to Eq. , $$\begin{aligned} r_\mathrm{ms}\stackrel{\kappa\to0}{=}M{{\mathopen{}\mathclose\bgroup\originalleft}[1+2^{1/3}\kappa^{2/3}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\kappa^{4/3}{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Comparing with Eq. , it would appear that the ISCO moves onto the event horizon in the extremal limit: $$\begin{aligned} \lim_{a\to M}r_\mathrm{ms}=M =\lim_{a\to M}r_+.\end{aligned}$$ However, for any sub-extremal black hole, the ISCO is a timelike geodesic, while the event horizon is ruled by null geodesics. Clearly, the extreme Kerr metric fails to accurately portray the spacetime geometry in the ISCO region correctly. 2. Indeed, although the ISCO and extremal horizon appear to coincide, the proper radial distance (as measured on a Boyer-Lindquist time-slice) between the two actually diverges logarithmically in this limit: $$\begin{aligned} \int_{r_+}^{r_\mathrm{ms}}ds\stackrel{\kappa\to0}{\sim}M{{\mathopen{}\mathclose\bgroup\originalleft}|\log{\kappa}{\aftergroup\egroup\originalright}|}.\end{aligned}$$ Thus, even though generic timelike (null) geodesics fall into the horizon in finite proper (affine) time, the near-horizon region acquires an infinite proper three-volume in the extremal limit. These observations were noted early on by Bardeen, Press and Teukolsky [@Bardeen1972] and later revisited in Refs. [@Jacobson2011; @Gralla2016a]. Taken together, these peculiarities indicate that the extremal Kerr metric grossly misrepresents the spacetime geometry near the event horizon of the extremal black hole. While it is true that the Boyer-Lindquist coordinates become singular at the horizon, we stress that these problems are not a coordinate artifact: they still arise even in coordinates that are smooth across the horizon. The existence of the infinite throat region is a coordinate-invariant statement, and describing it requires a careful resolution of the near-horizon geometry. This was accomplished by Bardeen and Horowitz [@Bardeen1999] by introducing a horizon-scaling limit tailored to this task, to which we now turn. The extreme Kerr throat {#subsec:EmergentThroat} ----------------------- In Sec. \[subsec:Puzzle\], we saw that as a black hole spins up and approaches the limiting extremal geometry with ${{\mathopen{}\mathclose\bgroup\originalleft}|J{\aftergroup\egroup\originalright}|}\to M^2$, a deep throat of divergent proper depth develops outside of its event horizon. Moreover, from the perspective of a distant observer, particles on the ISCO co-rotate with the black hole horizon in this limit. This motivates a coordinate transformation from Boyer-Lindquist coordinates $(t,r,\phi)$ to Bardeen-Horowitz coordinates $(T,R,\Phi)$ given by $$\begin{aligned} \label{eq:BardeenHorowitzCoordinates} t=\frac{T}{\Omega_H},\qquad r=r_+{{\mathopen{}\mathclose\bgroup\originalleft}(1+R{\aftergroup\egroup\originalright})},\qquad \phi=\Phi+T,\end{aligned}$$ where $\Omega_H$ denotes the angular velocity of the extremal black hole horizon, $$\begin{aligned} \Omega_H=\frac{a}{2Mr_+} \stackrel{a\to M}{=}\frac{1}{2M}.\end{aligned}$$ These coordinates are adapted to a local near-horizon observer co-rotating with the black hole, since $$\begin{aligned} \Omega_H{\mathop{}\!\partial}_T={\mathop{}\!\partial}_t+\Omega_H{\mathop{}\!\partial}_\phi.\end{aligned}$$ Local finite-energy excitations near the horizon of a black hole have large gravitational redshift relative to an observer at infinity. For black holes far from extremality, this region of spacetime is small and contains no stable orbits. However, for extremal black holes, the stable orbits extend down the throat, and the high-redshift emissions from sources in this region are phenomenologically interesting. In order to resolve the degeneracy arising from the infinite redshift while zooming into the horizon, we perform an infinite dilation onto the horizon, implemented by the rescaling $$\begin{aligned} \label{eq:HorizonScaling} {{\mathopen{}\mathclose\bgroup\originalleft}(T,R{\aftergroup\egroup\originalright})}\to{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{T}{\epsilon},\epsilon R{\aftergroup\egroup\originalright})},\qquad \epsilon\to0.\end{aligned}$$ If the black hole is precisely extremal ($a=M$), this scaling procedure has a finite limit and yields the NHEK geometry, with non-degenerate line element [@Bardeen1999] \[eq:NHEK\] $$\begin{gathered} d\hat{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-R^2{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\Phi+R{\mathop{}\!\mathrm{d}}T{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\\ \Gamma(\theta)=\frac{1+\cos^2{\theta}}{2},\qquad \Lambda(\theta)=\frac{2\sin{\theta}}{1+\cos^2{\theta}}.\end{gathered}$$ Since the NHEK geometry arises as a non-singular scaling limit of the extreme Kerr solution, it manifestly solves the vacuum Einstein equations and can be studied as a spacetime in its own right. Moreover, since in the limit $\epsilon\to0$, the resulting metric is $\epsilon$-independent, further coordinate rescalings $(T,R)\to(T/\epsilon,\epsilon R)$ leave the NHEK line element invariant: physically, the throat-like region is sufficiently deep that it becomes self-similar in the extremal limit. Therefore, the region of spacetime in the throat displays an emergent scaling symmetry, which is generated at the infinitesimal level by the dilation Killing vector $H_0=T{\mathop{}\!\partial}_T-R{\mathop{}\!\partial}_R$. Surprisingly, yet another, no-less constraining symmetry—invariance under special conformal transformations generated by $H_-$—also emerges in this limit. Together, these symmetries generate the global conformal group $\mathsf{SL}(2,\mathbb{R})$, with commutation relations $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}[H_0,H_\pm{\aftergroup\egroup\originalright}]}=\mp H_\pm,\qquad {{\mathopen{}\mathclose\bgroup\originalleft}[H_+,H_-{\aftergroup\egroup\originalright}]}=2H_0.\end{aligned}$$ Hence, in the high-spin regime, the $\mathbb{R}\times\mathsf{U}(1)$ symmetry of the Kerr metric generated by the Killing vectors ${\mathop{}\!\partial}_t$ and ${\mathop{}\!\partial}_\phi$ (associated with stationarity and axisymmetry, respectively) is enlarged within the near-horizon region to an $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ isometry group generated by $$\begin{aligned} \label{eq:KillingFieldsNHEK} H_0=T{\mathop{}\!\partial}_T-R{\mathop{}\!\partial}_R,\qquad H_+={\mathop{}\!\partial}_T,\qquad H_-={{\mathopen{}\mathclose\bgroup\originalleft}(T^2+\frac{1}{R^2}{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_T-2TR{\mathop{}\!\partial}_R-\frac{2}{R}{\mathop{}\!\partial}_\Phi,\qquad W_0={\mathop{}\!\partial}_\Phi.\end{aligned}$$ In fact, although the Killing tensor in extreme Kerr is associated to a non-geometrically-realized symmetry, in the near-horizon limit, this irreducible Killing tensor descends to a reducible Killing tensor in NHEK [@Galajinsky2010; @AlZahrani2011]. More precisely, it is given (up to a mass term) by the Casimir of $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$: $$\begin{aligned} \label{eq:KillingTensor} \hat{K}^{\mu\nu}=M^2\hat{g}^{\mu\nu}-H_0^\mu H_0^\nu+\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(H_+^\mu H_-^\nu+H_-^\mu H_+^\nu{\aftergroup\egroup\originalright})}+W_0^\mu W_0^\nu.\end{aligned}$$ Thus, the Kerr metric’s hidden symmetries become explicit in the emergent throat region, where it decomposes into $$\begin{aligned} \hat{g}^{\mu\nu}=-\frac{1}{2M^2\Gamma}{{\mathopen{}\mathclose\bgroup\originalleft}[-H_0^\mu H_0^\nu+\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(H_+^\mu H_-^\nu+H_-^\mu H_+^\nu{\aftergroup\egroup\originalright})}-{\mathop{}\!\partial}_\theta^\mu{\mathop{}\!\partial}_\theta^\nu+{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}W_0^\mu W_0^\nu{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Finally, we note that the induced metric on a hypersurface of fixed $\theta$ is that of warped three-dimensional Anti de-Sitter space (WAdS$_3$) with warp factor $\Lambda(\theta)$ [@Song2009]. This warp factor goes to unity, $\Lambda(\theta_c)=1$, at the critical angle $\theta_c=\arctan{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{2}/3^{1/4}{\aftergroup\egroup\originalright})}$, which corresponds to the so-called “light surface" of the extreme Kerr black hole [@Aman2012]. The NHEK metric becomes precisely that of AdS$_3$ on this surface, which seems to play a priviledged role in the propagation of light out of the throat [@Gralla2017a]. Near-horizon scaling limits for near-extreme Kerr {#subsec:ScalingLimits} ------------------------------------------------- In Sec. \[subsec:Puzzle\], we saw that the extreme Kerr metric fails to resolve near-horizon physics. Then, we argued in Sec. \[subsec:EmergentThroat\] that this failure is caused by the emergence, in a certain scaling limit, of a near-horizon region of the Kerr spacetime that resembles an infinite gravitational potential well. This throat-like region is sufficiently deep that in the extremal limit, it becomes self-similiar and enjoys an enhanced isometry group: the global conformal group $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$. The scaling limit to the NHEK region is unique for a precisely extremal black hole. But, according to the classical laws of black hole thermodynamics, such a black hole is unphysical since it has zero temperature, and no adiabatic process can turn a non-extremal black hole extremal [@Israel1986]. Thus, it is more realistic to consider black holes with a small deviation from extremality and a correspondingly small temperature. However, for such a near-extremal black hole, there exist infinitely many bands of near-horizon radii that become infinitely separated from each other in the extremal limit [@Gralla2015]. More precisely, if one considers two Boyer-Lindquist radii $r_1$ and $r_2$ that scale to the horizon at different rates as extremality is approached, so that $$\begin{aligned} a=M\sqrt{1-{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon\kappa{\aftergroup\egroup\originalright})}^2},\qquad r_1=M{{\mathopen{}\mathclose\bgroup\originalleft}[1+\epsilon^qR_1{\aftergroup\egroup\originalright}]},\qquad r_2=M{{\mathopen{}\mathclose\bgroup\originalleft}[1+\epsilon^sR_2{\aftergroup\egroup\originalright}]},\qquad 0<s\leq q\leq1,\end{aligned}$$ then the proper radial separations along a Boyer-Lindquist time-slice have the limiting form $$\begin{aligned} \label{eq:GeodesicDistance} \lim_{\epsilon\to0}\int_{r_1}^{r_2}ds= \begin{cases} \displaystyle M\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{R_2}{R_1}{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(q-s{\aftergroup\egroup\originalright})}M{{\mathopen{}\mathclose\bgroup\originalleft}|\log{\epsilon}{\aftergroup\egroup\originalright}|}, &q<1,\ s<1,\vspace{2pt}\\ \displaystyle M\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2R_2}{R_1+\sqrt{R_1^2-\kappa^2}}{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(1-s{\aftergroup\egroup\originalright})}M{{\mathopen{}\mathclose\bgroup\originalleft}|\log{\epsilon}{\aftergroup\egroup\originalright}|}, &q=1,\ s<1,\vspace{2pt}\\ \displaystyle M\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{R_2+\sqrt{R_2^2-\kappa^2}}{R_1+\sqrt{R_1^2-\kappa^2}}{\aftergroup\egroup\originalright})}, &q=1,\ s=1. \end{cases}\end{aligned}$$ Only radii that scale to the horizon at the same rate have finite radial separation in the extremal limit. This indicates that there are in fact infinitely many physically distinct near-horizon limits, each of which resolves the throat physics at different scales. The relevant scaling limits straightforwardly generalize Eqs.  and to $$\begin{aligned} \label{eq:Parameters} a=M\sqrt{1-{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon\kappa{\aftergroup\egroup\originalright})}^2},\qquad T=\epsilon^p\frac{t}{2M},\qquad R=\frac{r-M}{\epsilon^pM},\qquad \Phi=\phi-\frac{t}{2M},\qquad 0<p\le1,\qquad \epsilon\to0.\end{aligned}$$ The $\epsilon\to0$ limit with $p=1$ (which physically amounts to zooming into the near-horizon region at the same rate that the black hole is dialed into extremality) and $(T,R,\Phi,\kappa)$ held fixed yields the so-called near-NHEK geometry [@Bredberg2010] $$\begin{aligned} \label{eq:NearNHEK} d\bar{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2-\kappa^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\Phi+R{\mathop{}\!\mathrm{d}}T{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ which is the finite-temperature analogue of the NHEK metric . It also has an $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ isometry group generated by $$\begin{aligned} \label{eq:KillingFieldsNearNHEK} H_0=\frac{1}{\kappa}{\mathop{}\!\partial}_T,\qquad H_\pm=\frac{e^{\mp\kappa T}}{\sqrt{R^2-\kappa^2}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{R}{\kappa}{\mathop{}\!\partial}_T\pm{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_R-\kappa{\mathop{}\!\partial}_\Phi{\aftergroup\egroup\originalright}]},\qquad W_0={\mathop{}\!\partial}_\Phi.\end{aligned}$$ This region lies deepest in the throat and resolves the horizon at $r=M{{\mathopen{}\mathclose\bgroup\originalleft}(1+\epsilon\kappa{\aftergroup\egroup\originalright})}$, along with all other radii that also scale like $r\sim M{{\mathopen{}\mathclose\bgroup\originalleft}(1+\epsilon R{\aftergroup\egroup\originalright})}$ in the $\epsilon\to0$ limit. Examples of physically interesting radii that scale into near-NHEK include the photon orbit at $r_\mathrm{ph}$ and the (prograde) marginally bound orbit at $r_\mathrm{mb}$, also known as the Innermost Bound Circular Orbit (IBCO) radius [@Bardeen1972], $$\begin{aligned} r_\mathrm{ph}&=4M\cos^2{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{1}{3}\arccos{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{a}{M}{\aftergroup\egroup\originalright})}-\frac{\pi}{3}{\aftergroup\egroup\originalright}]} =M{{\mathopen{}\mathclose\bgroup\originalleft}[1+\frac{2}{\sqrt{3}}\epsilon\kappa+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^2{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}]},\\ r_\mathrm{mb}&=2M-a+2\sqrt{M{{\mathopen{}\mathclose\bgroup\originalleft}(M-a{\aftergroup\egroup\originalright})}} =M{{\mathopen{}\mathclose\bgroup\originalleft}[1+\sqrt{2}\epsilon\kappa+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^2{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ The proper radial separation between two prograde, equatorial, circular geodesics in near-NHEK is given by $$\begin{aligned} d\bar{s}{{\mathopen{}\mathclose\bgroup\originalleft}(R_1,R_2{\aftergroup\egroup\originalright})}=M\int^{ R_2}_{ R_1}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{R^2-\kappa^2}} =M\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{R_2+\sqrt{R_2^2-\kappa^2}}{R_1+\sqrt{R_1^2-\kappa^2}}{\aftergroup\egroup\originalright})}.\end{aligned}$$ This expression matches the limiting radial separation of equatorial geodesics calculated in the Kerr geometry , provided that one identifies the near-NHEK radius $R$ with the radius in Kerr scaling as $r=M{{\mathopen{}\mathclose\bgroup\originalleft}(1+\epsilon R{\aftergroup\egroup\originalright})}$. Note that this distance is not scale-invariant due to the presence of the horizon: physically, the presence of a small temperature $\kappa$ breaks the scaling symmetry exhibited by the NHEK distance. Mathematically, this can also be seen from Eq. , where the presence of the temperature $\kappa>0$ precludes the scaling limit $\epsilon\to0$ from being a coordinate limit: unlike the NHEK scaling , the dilation into near-NHEK also acts on the parameter $a$, which is why it is not forced to become an isometry in the limit. The $\epsilon\to0$ limit with any $0<p<1$ physically corresponds to spinning up the black hole faster than one zooms into the horizon, and always produces the same NHEK metric . However, each geometry thus obtained corresponds to a physically distinct region of the throat: a given choice of $p$ resolves a band of Boyer-Lindquist radii that scale like $r=M{{\mathopen{}\mathclose\bgroup\originalleft}(1+\epsilon^pR{\aftergroup\egroup\originalright})}$.[^6] The proper radial separation in NHEK, $$\begin{aligned} d\hat{s}{{\mathopen{}\mathclose\bgroup\originalleft}(R_1,R_2{\aftergroup\egroup\originalright})}=M\int^{R_2}_{R_1}\frac{{\mathop{}\!\mathrm{d}}R}{R} =M\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{R_2}{R_1}{\aftergroup\egroup\originalright})},\end{aligned}$$ matches the corresponding limiting radial separation of equatorial geodesics calculated in the Kerr geometry . However, because the NHEK expression is scale-invariant, the identification of NHEK radii with Kerr radii is ambiguous: one identifies the NHEK radius $R$ with the radius in Kerr scaling as $r=M{{\mathopen{}\mathclose\bgroup\originalleft}(1+\epsilon^pR{\aftergroup\egroup\originalright})}$, up to an overall $R$-independent factor. It is only after the throat is reattached to the asymptotically flat region, and the dilation symmetry is broken, that NHEK radii can be unambiguously identified with Kerr radii. A physically interesting Kerr radius that scales into NHEK is the ISCO radius, which according to Eq.  has a near-horizon limit $r_\mathrm{ms}=M{{\mathopen{}\mathclose\bgroup\originalleft}[1+2^{1/3}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon\kappa{\aftergroup\egroup\originalright})}^{2/3}{\aftergroup\egroup\originalright}]}$. The band of radii in Kerr with a finite radial separation from the ISCO in the extremal limit all scale like $r=M{{\mathopen{}\mathclose\bgroup\originalleft}(1+\epsilon^{2/3}R{\aftergroup\egroup\originalright})}$, and the limit with $p=2/3$ produces precisely the NHEK metric . In this limit, the four-velocity of the ISCO becomes [@Gralla2016a] $$\begin{aligned} \label{eq:TangentISCO} U=\frac{1}{2M}\frac{4}{\sqrt{3}R}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\partial}_T-\frac{3}{4}R{\mathop{}\!\partial}_\Phi{\aftergroup\egroup\originalright})},\end{aligned}$$ which is both timelike and finite.[^7] Therefore, the “$p=2/3$ NHEK" resolves the part of the throat in which ISCO-scale physics occurs. Again, note that while the near-horizon limit of Eq.  appears to identify the ISCO radius with the NHEK radius $R=2^{1/3}\kappa^{2/3}$, the dilation invariance of the NHEK metric indicates that there is in fact no meaningful way to assign a definite radius to the ISCO within NHEK. In fact, in contrast to near-NHEK, all circular geodesics in NHEK are marginally stable: from the near-horizon viewpoint, the ISCO is in some sense *everywhere* within the $p=2/3$ NHEK [@Gralla2015]. -- ------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------- -- Far region $\Big\}\ ds\sim M{{\mathopen{}\mathclose\bgroup\originalleft}|\log{\epsilon}{\aftergroup\egroup\originalright}|}$ NHEK ${{\mathopen{}\mathclose\bgroup\originalleft}(0<p<1{\aftergroup\egroup\originalright})}$ $\Big\}\ ds\sim M{{\mathopen{}\mathclose\bgroup\originalleft}|\log{\epsilon}{\aftergroup\egroup\originalright}|}$ near-NHEK ${{\mathopen{}\mathclose\bgroup\originalleft}(p=1{\aftergroup\egroup\originalright})}$ -- ------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------- -- To summarize, any two radii that scale to the horizon at the same rate (“lie in the same band”) end up in the same near-horizon geometry at a finite proper radial distance from each other. For instance, the photon orbit and IBCO radii both lie in the horizon band, and hence scale to the same near-NHEK region. Accordingly, the proper radial distance between these scales tends to a finite limit: $$\begin{aligned} ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_+,r_\mathrm{ph}{\aftergroup\egroup\originalright})}&=M\log\sqrt{3}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon{\aftergroup\egroup\originalright})}},\\ ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_+,r_\mathrm{mb}{\aftergroup\egroup\originalright})}&=M\log{{\mathopen{}\mathclose\bgroup\originalleft}(1+\sqrt{2}{\aftergroup\egroup\originalright})}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon{\aftergroup\egroup\originalright})}},\\ ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_\mathrm{ph},r_\mathrm{mb}{\aftergroup\egroup\originalright})}& =M\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+\sqrt{2}}{\sqrt{3}}{\aftergroup\egroup\originalright})}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon{\aftergroup\egroup\originalright})}}.\end{aligned}$$ On the other hand, radii that lie in different bands, $i.e.$, that scale to the horizon at different rates, end up in different NHEKs that are infinitely far apart (separated by a divergent proper radial distance). For instance, the ISCO band is infinitely far from both the horizon band, as well as from the mouth of the throat, which we may for instance define as the spin-independent equatorial radius of the ergosphere, $r_0=2M$: $$\begin{aligned} ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_+,r_\mathrm{ms}{\aftergroup\egroup\originalright})}&=\frac{M}{3}\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2^4}{\epsilon\kappa}{\aftergroup\egroup\originalright})}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^{2/3}{\aftergroup\egroup\originalright})}},\\ ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_\mathrm{ms},r_0{\aftergroup\egroup\originalright})}&=M{{\mathopen{}\mathclose\bgroup\originalleft}[1+\frac{2}{3}\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{\sqrt{2}\epsilon\kappa}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^{2/3}{\aftergroup\egroup\originalright})}},\\ ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_+,r_0{\aftergroup\egroup\originalright})}& =M{{\mathopen{}\mathclose\bgroup\originalleft}[1+\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2}{\epsilon\kappa}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^2{\aftergroup\egroup\originalright})}}.\end{aligned}$$ These facts are summarized in Fig. \[fig:Throat\], where the different bands appear stacked on top of one another, with cracks denoting the logarithmically divergent proper radial distance separating them. From this point of view, the precisely extremal, zero-temperature black hole is a degenerate limit in which all the throat geometries merge: near-NHEK disappears and the different NHEKs coalesce into one. Finally, note that the expansion about extremality defined by Eq.  can be viewed both as a small-temperature expansion and an expansion in the divergent proper depth $D=ds{{\mathopen{}\mathclose\bgroup\originalleft}(r_+,r_0{\aftergroup\egroup\originalright})}=M{{\mathopen{}\mathclose\bgroup\originalleft}|\log{\epsilon}{\aftergroup\egroup\originalright}|}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^0{\aftergroup\egroup\originalright})}}$ of the throat: indeed, at leading order, $$\begin{aligned} \epsilon=e^{-D/M}.\end{aligned}$$ Thus, subleading corrections due to deviations from extremality are exponentially suppressed in the characteristic length scale of the system, which diverges in the extremal limit. Similar behavior is of course observed near critical points in condensed matter systems—this analogy was further developed in Ref. [@Gralla2016a]. When studying the extremal Kerr black hole, it is important to note that neither the far metric (extreme Kerr) nor the near metric (NHEK) is more fundamental than the other: away from extremality, the Kerr metric resolves physics in the entire spacetime, but near extremality, the spacetime decouples into two regions. Each of these two regions is described by its own metric, which fails in the other region: while NHEK resolves the near-horizon region, it fails to resolve the far region (for instance, it is not asymptotically flat), and the far metric does not resolve the throat region. As is usual for smooth extremal solutions in general relativity, the extreme Kerr geometry serves to interpolate between two separate vacuum solutions: flat space in the far region and NHEK in the near region. The two regions of spacetime are on equal footing. Emergent conformal symmetry {#subsec:ConformalSymmetryInTheSky} --------------------------- In many situations (including those of astrophysical interest), it is appropriate to treat the Kerr geometry as a fixed background while neglecting gravitational backreaction of the matter system (as well as gravitational excitations). When this approximation is valid, it suffices to work strictly with the NHEK metric and its exact isometries. In other applications, one is interested not only in the vacuum NHEK geometry, but in all spacetimes that approach NHEK asymptotically in some appropriate sense.[^8] Although these geometries all possess a long throat and approximate scale-invariance, generic members of this class of spacetimes have no exact isometries. It is the symmetries of the class of spacetimes, rather than the symmetries of a specific spacetime, that control gravitational dynamics in the throat. In attempting to compute this generalized symmetry group, one often finds an enhancement of the global conformal isometry group $\mathsf{SL}(2,\mathbb{R})$ to an infinite-dimensional local conformal symmetry [@Compere2012]. The details of calculations of this type depend delicately on the choice of boundary conditions. We will focus on a particular class of symmetry transformations that have been repeatedly utilized [@Porfyriadis2014; @Hadar2014; @Hadar2015; @Hadar2017; @Compere2018] in calculating geodesics in NHEK and near-NHEK, and defer a complete asymptotic symmetry group analysis to future work. These large diffeomorphisms are the NHEK analogue of boundary reparameterizations of the AdS$_2$ throat discussed in Ref. [@Maldacena2016b] and should be related to inequivalent ways of reattaching the Kerr throat region to the exterior geometry. Starting with the NHEK line element , $$\begin{aligned} d\hat{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-R^2{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\Phi+R{\mathop{}\!\mathrm{d}}T{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ we consider a coordinate transformation of the form[^9] $$\begin{aligned} \label{eq:ConformalTransformation} T=f(t)+\frac{2f''(t){{\mathopen{}\mathclose\bgroup\originalleft}[f'(t){\aftergroup\egroup\originalright}]}^2}{4r^2{{\mathopen{}\mathclose\bgroup\originalleft}[f'(t){\aftergroup\egroup\originalright}]}^2-{{\mathopen{}\mathclose\bgroup\originalleft}[f''(t){\aftergroup\egroup\originalright}]}^2},\qquad R=\frac{4r^2{{\mathopen{}\mathclose\bgroup\originalleft}[f'(t){\aftergroup\egroup\originalright}]}^2-{{\mathopen{}\mathclose\bgroup\originalleft}[f''(t){\aftergroup\egroup\originalright}]}^2}{4r{{\mathopen{}\mathclose\bgroup\originalleft}[f'(t){\aftergroup\egroup\originalright}]}^3},\qquad \Phi=\phi+\log{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{2rf'(t)-f''(t)}{2rf'(t)+f''(t)}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ The resulting line element is given by $$\begin{aligned} \label{eq:VirasoroNHEK} d\hat{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}\{-r^2{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}\{f(t);t{\aftergroup\egroup\originalright}\}}}{2r^2}{\aftergroup\egroup\originalright})}^2{\mathop{}\!\mathrm{d}}t^2+\frac{{\mathop{}\!\mathrm{d}}r^2}{r^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}[{\mathop{}\!\mathrm{d}}\phi+r{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}\{f(t);t{\aftergroup\egroup\originalright}\}}}{2r^2}{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}t{\aftergroup\egroup\originalright}]}^2{\aftergroup\egroup\originalright}\}},\end{aligned}$$ where we introduced the Schwarzian derivative $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}\{f(\cdot);\cdot{\aftergroup\egroup\originalright}\}}={{\mathopen{}\mathclose\bgroup\originalleft}(\frac{f''}{f'}{\aftergroup\egroup\originalright})}'-\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{f''}{f'}{\aftergroup\egroup\originalright})}^2.\end{aligned}$$ These metrics are the NHEK analogues of the AdS$_3$ Bañados metrics [@Banados1999; @Compere2016]. Note that at the boundary $r\to\infty$, this coordinate change implements a time reparameterization $T\to f(t)$, and that as a result, subleading components of the metric transform like the expectation value of a stress-tensor component in CFT$_2$. Infinitesimally, the conformal transformation is implemented by the action of the vector field $$\begin{aligned} \xi{{\mathopen{}\mathclose\bgroup\originalleft}[\epsilon(t){\aftergroup\egroup\originalright}]}={{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon(t)+\frac{\epsilon''(t)}{2r^2}{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_t-r\epsilon'(t){\mathop{}\!\partial}_r-\frac{\epsilon''(t)}{r}{\mathop{}\!\partial}_\phi,\qquad f(t)=t+\epsilon(t)+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\epsilon^2{\aftergroup\egroup\originalright})}}.\end{aligned}$$ This can be decomposed into modes $$\begin{aligned} \xi_n=\xi{{\mathopen{}\mathclose\bgroup\originalleft}[t^{1-n}{\aftergroup\egroup\originalright}]},\qquad n\in\mathbb{Z},\end{aligned}$$ which obey the Witt algebra at the boundary, $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}[\xi_m,\xi_n{\aftergroup\egroup\originalright}]}=(m-n)\xi_{m+n}+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{r^3}{\aftergroup\egroup\originalright})}}.\end{aligned}$$ The $\mathsf{SL}(2,\mathbb{R})$ isometry group of NHEK is generated by the vector fields $$\begin{aligned} \xi_0=H_0,\qquad \xi_{\pm1}=H_\pm,\end{aligned}$$ whose corresponding finite diffeomorphisms are given by the Möbius transformations with vanishing Schwarzian:[^10] $$\begin{aligned} f(t)=\frac{at+b}{ct+d},\qquad ad-bc=1.\end{aligned}$$ The rest of the symmetry transformations with nonvanishing ${{\mathopen{}\mathclose\bgroup\originalleft}\{f(t);t{\aftergroup\egroup\originalright}\}}$ are spontaneously broken. Of particular interest here is the exponential map $$\begin{aligned} f(t)=e^{\kappa t} \qquad\Longrightarrow\qquad {{\mathopen{}\mathclose\bgroup\originalleft}\{f(t);t{\aftergroup\egroup\originalright}\}}=-\frac{\kappa^2}{2},\end{aligned}$$ for which the metric becomes $$\begin{aligned} \label{eq:ExponentialNHEK} d\hat{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}\{-r^2{{\mathopen{}\mathclose\bgroup\originalleft}(1-{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\kappa}{2r}{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright})}^2{\mathop{}\!\mathrm{d}}t^2+\frac{{\mathop{}\!\mathrm{d}}r^2}{r^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}[{\mathop{}\!\mathrm{d}}\phi+r{{\mathopen{}\mathclose\bgroup\originalleft}(1+{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\kappa}{2r}{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}t{\aftergroup\egroup\originalright}]}^2{\aftergroup\egroup\originalright}\}}.\end{aligned}$$ Near the boundary, this diffeomorphism acts as the exponential map on the boundary time. It is the analogue of the usual conformal transformation from the plane to the cylinder, which puts a CFT$_2$ at finite temperature. In fact, the metric is actually near-NHEK, as can be seen by performing a further (small) diffeomorphism $$\begin{aligned} r=\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(\bar{r}+\sqrt{\bar{r}^2-\kappa^2}{\aftergroup\egroup\originalright})},\end{aligned}$$ which puts it in the form of Eq. . By composing these transformations, one can directly map near-NHEK, $$\begin{aligned} d\bar{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-{{\mathopen{}\mathclose\bgroup\originalleft}(r^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}t^2+\frac{{\mathop{}\!\mathrm{d}}r^2}{r^2-\kappa^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\phi+r{\mathop{}\!\mathrm{d}}t{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ into NHEK via the coordinate change $$\begin{aligned} \label{eq:NHEK2NearNHEK} T=e^{\kappa t}\frac{r}{\sqrt{r^2-\kappa^2}},\qquad R=e^{-\kappa t}\frac{\sqrt{r^2-\kappa^2}}{\kappa},\qquad \Phi=\phi+\frac{1}{2}\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{r-\kappa}{r+\kappa}{\aftergroup\egroup\originalright})}.\end{aligned}$$ This transformation also maps the NHEK Killing vectors and near-NHEK Killing vectors into each other. It is important to note that this map is not surjective: its range covers only a subset of the NHEK Poincaré patch. Within that image, the inverse transformation is $$\begin{aligned} t=\frac{1}{2\kappa}\log{{\mathopen{}\mathclose\bgroup\originalleft}(T^2-\frac{1}{R^2}{\aftergroup\egroup\originalright})},\qquad r=\kappa TR,\qquad \phi=\Phi-\frac{1}{2}\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{TR-1}{TR+1}{\aftergroup\egroup\originalright})}.\end{aligned}$$ Since the map is a diffeomorphism between near-NHEK and a subset of the Poincaré patch in NHEK (rather than its entirety), the near-NHEK and NHEK patches are locally (but not globally) diffeomorphic. Of course, since both near-NHEK and NHEK have horizons, they are not geodesically complete spacetimes. As we will discuss in the next section, they have the same maximal extension: global NHEK. The global strip and the Poincaré patch {#subsec:GlobalNHEK} --------------------------------------- To obtain the maximal extension of the NHEK spacetime, we pass from the Poincaré coordinates $(T,R,\Phi)$ with a coordinate singularity at $TR=1$ to global coordinates $(\tau,y,\varphi)$ that can be smoothly continued past this surface. The transformation from Poincaré NHEK to a patch $\tau\in(-\pi,\pi)$ of global NHEK is given by $$\begin{aligned} \label{eq:PoincareToGlobal} \tau=\arctan{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{2TR^2}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-T^2{\aftergroup\egroup\originalright})}R^2+1}{\aftergroup\egroup\originalright}]},\qquad y=\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(1+T^2{\aftergroup\egroup\originalright})}R^2-1}{2R},\qquad \varphi=\Phi-\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(1-TR{\aftergroup\egroup\originalright})}^2+R^2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(1+T^2{\aftergroup\egroup\originalright})}R^2-1{\aftergroup\egroup\originalright}]}^2+4R^2}}{\aftergroup\egroup\originalright})}.\end{aligned}$$ Different branches of the arctangent map Poincaré NHEK to diffeomorphic patches of global NHEK which differ by $\tau\to\tau+2\pi n$ translations in global time. For this reason, all geodesic motion in global NHEK is oscillatory with period $2\pi$. The inverse map from the patch $\tau\in(-\pi,\pi)$ of global NHEK to Poincaré NHEK is given by $$\begin{aligned} \label{eq:GlobalToPoincare} T=\frac{\sin{\tau}\sqrt{1+y^2}}{\cos{\tau}\sqrt{1+y^2}+y},\qquad R=\cos{\tau}\sqrt{1+y^2}+y,\qquad \Phi=\varphi+\log{{\mathopen{}\mathclose\bgroup\originalleft}|\frac{\cos{\tau}+y\sin{\tau}}{1+\sin{\tau}\sqrt{1+y^2}}{\aftergroup\egroup\originalright}|}.\end{aligned}$$ Under this coordinate transformation, the NHEK metric becomes $$\begin{aligned} \label{eq:GlobalNHEK} d\tilde{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}\tau^2+\frac{{\mathop{}\!\mathrm{d}}y^2}{1+y^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\varphi+y{\mathop{}\!\mathrm{d}}\tau{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ and the NHEK Killing vector fields are mapped into \[eq:KillingFieldsGlobalNHEK\] $$\begin{aligned} H_\pm&={{\mathopen{}\mathclose\bgroup\originalleft}(1\pm\frac{y\cos{\tau}}{\sqrt{1+y^2}}{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_\tau\pm\sin{\tau}\sqrt{1+y^2}{\mathop{}\!\partial}_y\pm\frac{\cos{\tau}}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\varphi,\\ H_0&=\frac{y\sin{\tau}}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\tau-\cos{\tau}\sqrt{1+y^2}{\mathop{}\!\partial}_y+\frac{\sin{\tau}}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\varphi,\\ W_0&={\mathop{}\!\partial}_\varphi.\end{aligned}$$ It will be convenient for us to complexify this algebra by introducing new (complex) generators $$\begin{aligned} L_0=i{\mathop{}\!\partial}_\tau,\qquad L_\pm=e^{\pm i\tau}{{\mathopen{}\mathclose\bgroup\originalleft}[\pm\frac{y}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\tau-i\sqrt{1+y^2}{\mathop{}\!\partial}_y\pm\frac{1}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\varphi{\aftergroup\egroup\originalright}]},\end{aligned}$$ which obey the same commutation relations: $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}[L_0,L_\pm{\aftergroup\egroup\originalright}]}=\mp L_\pm,\qquad {{\mathopen{}\mathclose\bgroup\originalleft}[L_+,L_-{\aftergroup\egroup\originalright}]}=2L_0,\qquad {{\mathopen{}\mathclose\bgroup\originalleft}[W_0,L_\pm{\aftergroup\egroup\originalright}]}={{\mathopen{}\mathclose\bgroup\originalleft}[W_0,L_0{\aftergroup\egroup\originalright}]}=0.\end{aligned}$$ These two sets of generators are related by $$\begin{aligned} \label{eq:Automorphism} H_\pm=-iL_0\pm\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(L_+-L_-{\aftergroup\egroup\originalright})},\qquad H_0=-\frac{i}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(L_++L_-{\aftergroup\egroup\originalright})}.\end{aligned}$$ The generator $L_0$ of global time translations is the analogue of the Hamiltonian of a CFT$_2$ defined on the cylinder. The causal structure of the global NHEK geometry is best understood by introducing a compactified radius[^11] $$\begin{aligned} y=-\cot\psi,\qquad \psi\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,\pi{\aftergroup\egroup\originalright}]},\end{aligned}$$ in terms of which the global NHEK line element becomes $$\begin{aligned} d\tilde{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{-{\mathop{}\!\mathrm{d}}\tau^2+{\mathop{}\!\mathrm{d}}\psi^2}{\sin^2{\psi}}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\varphi-\cot{\psi}{\mathop{}\!\mathrm{d}}\tau{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]}.\end{aligned}$$ As explained in detail in App. \[app:AdS2\], the $(\tau,\psi)$ part of the metric describes the global strip of AdS$_2$ parameterized by $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}(-\infty,+\infty{\aftergroup\egroup\originalright})}$ and $\psi\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,\pi{\aftergroup\egroup\originalright}]}$. Since this two-dimensional metric is conformally flat, with lines of $\tau=\pm\psi+\tau_0$ manifestly null, it is straightforward to obtain its Carter-Penrose diagram, depicted in Fig. \[fig:PenroseDiagrams\]. Under the embedding of Poincaré NHEK into global NHEK, we see that the Poincaré coordinates $(T,R)$ cover the patch $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}(-\pi,+\pi{\aftergroup\egroup\originalright})}$ with ${{\mathopen{}\mathclose\bgroup\originalleft}|\tau{\aftergroup\egroup\originalright}|}\le\psi\le\pi$ (in particular, the future/past horizon is located at $TR=\pm1$): $$\begin{aligned} T=\frac{\sin{\tau}}{\cos{\tau}-\cos{\psi}},\qquad R=\frac{\cos{\tau}-\cos{\psi}}{\sin{\psi}}.\end{aligned}$$ Hence, the lines of constant $T$ and constant $R$ are respectively given by $$\begin{aligned} \cos{\tau}=\frac{T^2\cos{\psi}+\sqrt{1+T^2\sin^2{\psi}}}{1+T^2},\qquad \cos{\tau}=\cos{\psi}+R\sin{\psi},\end{aligned}$$ and in Fig. \[fig:GlobalStrip\], they are plotted at constant intervals of $x$, where $T=\tan{x/2}$ and $R=\tan{x/2}$. Under the embedding of near-NHEK into global NHEK obtained by composing Eqs.  and , we see that the Poincaré coordinates $(T,R)$ cover the patch $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}(0,+\pi{\aftergroup\egroup\originalright})}$ with $\pi-\psi\le\tau\le\psi$: $$\begin{aligned} T=\frac{1}{\kappa}\log\sqrt{1-\frac{2\cos{\tau}}{\cos{\tau}-\cos{\psi}}},\qquad R=\kappa\frac{\sin{\tau}}{\sin{\psi}}.\end{aligned}$$ Hence, the lines of constant $T$ and constant $R$ are respectively given by $$\begin{aligned} \cos{\tau}=\frac{e^{2\kappa T}-1}{e^{2\kappa T}+1}\cos{\psi},\qquad \sin{\tau}=\frac{R}{\kappa}\sin{\psi},\end{aligned}$$ and in Fig. \[fig:GlobalStrip\] they are plotted at constant intervals of $x$, where $T=\log{{\mathopen{}\mathclose\bgroup\originalleft}(\tan{x/2}{\aftergroup\egroup\originalright})}^{1/\kappa}$ and $R=\kappa/\sin{x}$. Note that the finite $\mathsf{SL}(2,\mathbb{R})$ transformations—Eq.  with $f(t)$ a Möbius transformation—leave the NHEK metric invariant, but not its geodesics: they are mapped into each other under the action of the global conformal group. Equivalently, we can study geodesics in global coordinates and obtain different geodesics in the Poincaré patch by using the embedding composed with the $\mathsf{SL}(2,\mathbb{R})$ transformations. This strategy can be used to map circular orbits to (slow or fast) plunges and was used to great effect in Refs. [@Hadar2014; @Hadar2015; @Hadar2017; @Compere2018]. In fact, most preexisting analyses of geodesics in NHEK have focused on equatorial circular geodesics or plunging geodesics obtainable through the above mapping from the ISCO geodesic . ![Carter-Penrose diagrams for the equatorial plane of near-extreme Kerr \[left\], extreme Kerr \[middle\] and NHEK \[right\]. (The diagrams depend on $\theta$ because these geometries are not spherically symmetric and, away from the equatorial plane, it is possible to continue past the singularity.) In the diagrams for (near-)extreme Kerr, $\mathcal{I}^\pm$ denotes future/past null infinity ($r=+\infty$ with $t=\pm\infty$), $\iota^0$ is spacelike infinity ($r=+\infty$), $r_\pm$ is the outer/inner horizon, and the singularity lies at $r=0$. The thin lines correspond to hypersurfaces of constant $r$, which are timelike outside the black hole \[red\] and spacelike inside \[gray\].[]{data-label="fig:PenroseDiagrams"}](NearExtremeKerr.pdf "fig:") ![Carter-Penrose diagrams for the equatorial plane of near-extreme Kerr \[left\], extreme Kerr \[middle\] and NHEK \[right\]. (The diagrams depend on $\theta$ because these geometries are not spherically symmetric and, away from the equatorial plane, it is possible to continue past the singularity.) In the diagrams for (near-)extreme Kerr, $\mathcal{I}^\pm$ denotes future/past null infinity ($r=+\infty$ with $t=\pm\infty$), $\iota^0$ is spacelike infinity ($r=+\infty$), $r_\pm$ is the outer/inner horizon, and the singularity lies at $r=0$. The thin lines correspond to hypersurfaces of constant $r$, which are timelike outside the black hole \[red\] and spacelike inside \[gray\].[]{data-label="fig:PenroseDiagrams"}](ExtremeKerr.pdf "fig:") ![Carter-Penrose diagrams for the equatorial plane of near-extreme Kerr \[left\], extreme Kerr \[middle\] and NHEK \[right\]. (The diagrams depend on $\theta$ because these geometries are not spherically symmetric and, away from the equatorial plane, it is possible to continue past the singularity.) In the diagrams for (near-)extreme Kerr, $\mathcal{I}^\pm$ denotes future/past null infinity ($r=+\infty$ with $t=\pm\infty$), $\iota^0$ is spacelike infinity ($r=+\infty$), $r_\pm$ is the outer/inner horizon, and the singularity lies at $r=0$. The thin lines correspond to hypersurfaces of constant $r$, which are timelike outside the black hole \[red\] and spacelike inside \[gray\].[]{data-label="fig:PenroseDiagrams"}](GlobalNHEK.pdf "fig:") ![Carter-Penrose diagrams for the equatorial plane of global NHEK \[blue\] and the Poincaré patch for NHEK \[red\] and near-NHEK \[green\]. The global NHEK strip is covered by the global coordinates $(\psi,\tau)$ with $\psi\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,\pi{\aftergroup\egroup\originalright}]}$ and $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}(-\infty,+\infty{\aftergroup\egroup\originalright})}$. The Poincaré patch for NHEK \[red\] can be embedded at any vertical position along the strip; here, it is depicted as the red patch at $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}[-\pi,+\pi{\aftergroup\egroup\originalright}]}$, corresponding to the mapping . Likewise, near-NHEK can also be placed anywhere on the strip; here, it is depicted as the green patch embedded at $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,+\pi{\aftergroup\egroup\originalright}]}$, $\psi\in{{\mathopen{}\mathclose\bgroup\originalleft}[\pi/2,\pi{\aftergroup\egroup\originalright}]}$, corresponding to the composition of the mappings and . In both the NHEK and near-NHEK patches, slices of constant Poincaré time $T$ and radius $R$ are depicted by blue and red lines, respectively. For this choice of NHEK patch, the future/past horizon is located at $\tau=\pm\psi$ (or, equivalently, $R=0$ with $TR=\pm1$).[]{data-label="fig:GlobalStrip"}](GlobalStrip.pdf){width=".8\textwidth"} Geodesics in NHEK {#sec:GeodesicsInNHEK} ================= In this section, we analyze geodesic motion in global NHEK, the Poincaré patch, and near-NHEK. For each of these spacetimes, we follow the procedure outlined for Kerr in Sec. \[sec:Kerr\] and recast the geodesic equation in first-order form. We then obtain explicit expressions for all the path integrals appearing in the equation. Finally, inverting the expression for the time-lapse allows us to solve for the radial motion as a function of coordinate time and thereby derive a complete and explicit parameterization of all NHEK geodesics. We begin by analyzing the geodesic equation in global NHEK (Sec. \[subsec:GlobalGeodesics\]), as it is a geodesically complete spacetime, unlike Poincaré NHEK (Sec. \[subsec:PoincareGeodesics\]) and near-NHEK (Sec. \[subsec:NearGeodesics\]). As a byproduct of our analysis, we elucidate the action of the NHEK isometry group $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ on the space of geodesics. We find that the group orbits are classified by the $\mathsf{SL}(2,\mathbb{R})$ Casimir $\mathcal{C}$ and angular momentum $L$, which completely determine the polar motion. Any two NHEK geodesics with the same polar motion can be mapped into each other by the action of $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ using the explicit transformations presented at the end of App. \[app:AdS2\]. Geodesics in global NHEK {#subsec:GlobalGeodesics} ------------------------ Recall from Sec. \[subsec:GlobalNHEK\] that in global coordinates, the NHEK line element is $$\begin{aligned} \label{eq:GlobalMetric} d\tilde{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}\tau^2+\frac{{\mathop{}\!\mathrm{d}}y^2}{1+y^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\varphi+y{\mathop{}\!\mathrm{d}}\tau{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ and the generators of $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ are $$\begin{aligned} L_0=i{\mathop{}\!\partial}_\tau,\qquad L_\pm=e^{\pm i\tau}{{\mathopen{}\mathclose\bgroup\originalleft}[\pm\frac{y}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\tau-i\sqrt{1+y^2}{\mathop{}\!\partial}_y\pm\frac{1}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\varphi{\aftergroup\egroup\originalright}]},\qquad W_0=\partial_\varphi.\end{aligned}$$ The Casimir of $\mathsf{SL}(2,\mathbb{R})$ is the (manifestly reducible) symmetric Killing tensor $$\begin{aligned} \mathcal{C}^{\mu\nu}=-L_0^\mu L_0^\nu+\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(L_+^\mu L_-^\nu+L_-^\mu L_+^\nu{\aftergroup\egroup\originalright})}.\end{aligned}$$ It is related to $\tilde{K}^{\mu\nu}$, the NHEK limit of the irreducible Killing tensor on Kerr , by $$\begin{aligned} \tilde{K}^{\mu\nu}=\mathcal{C}^{\mu\nu}+W_0^\mu W_0^\nu+M^2\tilde{g}^{\mu\nu}.\end{aligned}$$ The motion of a free particle of mass $\mu$ and four-momentum $P^\mu$ is described by the geodesic equation, $$\begin{aligned} P^\mu\tilde{\nabla}_\mu P^\nu=0,\qquad \tilde{g}^{\mu\nu}P_\mu P_\nu=-\mu^2.\end{aligned}$$ Geodesic motion in global NHEK is completely characterized by the three conserved quantities[^12] \[eq:GlobalConservedQuantities\] $$\begin{gathered} \triangle=iL_0^\mu P_\mu =-P_\tau,\qquad L=W_0^\mu P_\mu =P_\varphi,\\ \mathcal{C}=\mathcal{C}^{\mu\nu}P_\mu P_\nu =P_\theta^2-{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}P_\varphi^2+{{\mathopen{}\mathclose\bgroup\originalleft}(2M^2\Gamma{\aftergroup\egroup\originalright})}\mu^2 =\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(P_\tau-P_\varphi y{\aftergroup\egroup\originalright})}^2}{1+y^2}-{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})}P_y^2-P_\varphi^2,\end{gathered}$$ denoting the global energy, angular momentum parallel to the axis of symmetry, and “Casimir" constant, respectively. When connecting NHEK geodesics to the far region in Kerr, it is more convenient to work with the Carter constant $$\begin{aligned} K=\tilde{K}^{\mu\nu}P_\mu P_\nu =\mathcal{C}+L^2-\mu^2M^2,\end{aligned}$$ which is directly related to its Kerr analogue $k$. In this paper, however, we restrict our attention to motion within NHEK, which is more easily characterized by the $\mathsf{SL}(2,\mathbb{R})$ Casimir $\mathcal{C}$. By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,\triangle,L,\mathcal{C}{\aftergroup\egroup\originalright})}$, we find that a particle following a geodesic in the global NHEK geometry has an instantaneous four-momentum $P=P_\mu{\mathop{}\!\mathrm{d}}X^\mu$ of the form $$\begin{aligned} \label{eq:KerrGeodesic} P{{\mathopen{}\mathclose\bgroup\originalleft}(X^\mu,\triangle,L,\mathcal{C}{\aftergroup\egroup\originalright})}=-\triangle{\mathop{}\!\mathrm{d}}\tau\pm_y\frac{\sqrt{\mathcal{Y}(y)}}{1+y^2}{\mathop{}\!\mathrm{d}}y\pm_\theta\sqrt{\Theta_n(\theta)}{\mathop{}\!\mathrm{d}}\theta+L{\mathop{}\!\mathrm{d}}\varphi,\end{aligned}$$ where the two choices of sign $\pm_y$ and $\pm_\theta$ depend on the radial and polar directions of travel, respectively. Here, we also introduced radial and polar potentials $$\begin{aligned} \label{eq:GlobalRadialPotential} \mathcal{Y}(y)&={{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+Ly{\aftergroup\egroup\originalright})}^2-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})},\\ \label{eq:PolarPotentialNHEK} \Theta_n(\theta)&=\mathcal{C}+{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}L^2-{{\mathopen{}\mathclose\bgroup\originalleft}(2M^2\Gamma{\aftergroup\egroup\originalright})}\mu^2.\end{aligned}$$ One can then raise $P_\mu$ to obtain the equations for the geodesic trajectory, \[eq:GlobalMomentum\] $$\begin{aligned} \label{eq:GlobalRadialEquation} 2M^2\Gamma\frac{dy}{d\sigma}&=\pm_y\sqrt{\mathcal{Y}(y)},\\ 2M^2\Gamma\frac{d\theta}{d\sigma}&=\pm_\theta\sqrt{\Theta_n(\theta)},\\ 2M^2\Gamma\frac{d\varphi}{d\sigma}&=-y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}+\frac{L}{\Lambda^2},\\ \label{eq:GlobalTimeEquation} 2M^2\Gamma\frac{d\tau}{d\sigma}&=\frac{\triangle+Ly}{1+y^2}.\end{aligned}$$ The parameter $\sigma$ is the affine parameter for massless particles ($\mu=0$), and is related to the proper time $\delta$ by $\delta=\mu\sigma$ for massive particles. Following the same procedure as in Kerr, we find from Eqs.  that a geodesic labeled by $(\triangle,L,\mathcal{C})$ connects spacetime points $X_s^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(\tau_s,y_s,\theta_s,\varphi_s{\aftergroup\egroup\originalright})}$ and $X_o^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(\tau_o,y_o,\theta_o,\varphi_o{\aftergroup\egroup\originalright})}$ if $$\begin{aligned} &\fint_{y_s}^{y_o}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}}=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta_n(\theta)}},\\ \varphi_o-\varphi_s&=\fint_{y_s}^{y_o}{{\mathopen{}\mathclose\bgroup\originalleft}[-y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}}+\fint_{\theta_s}^{\theta_o}\frac{L\Lambda^{-2}(\theta)}{\pm_\theta\sqrt{\Theta_n(\theta)}}{\mathop{}\!\mathrm{d}}\theta,\\ \tau_o-\tau_s&=\fint_{y_s}^{y_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}}.\end{aligned}$$ We may rewrite these conditions as $$\begin{aligned} \label{eq:GlobalGeodesics} \tilde{I}_y=\tilde{G}_\theta,\qquad \varphi_o-\varphi_s=\tilde{G}_\varphi-\tilde{I}_\varphi,\qquad \tau_o-\tau_s=\tilde{I}_\tau,\end{aligned}$$ where we have defined the integrals $$\begin{gathered} \tilde{I}_y=\fint_{y_s}^{y_o}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}},\qquad \tilde{I}_\varphi=\fint_{y_s}^{y_o}y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}},\qquad \tilde{I}_\tau=\fint_{y_s}^{y_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}},\\ \tilde{G}_\theta=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta_n(\theta)}},\qquad \tilde{G}_\varphi=\fint_{\theta_s}^{\theta_o}\frac{L\Lambda^{-2}(\theta)}{\pm_\theta\sqrt{\Theta_n(\theta)}}{\mathop{}\!\mathrm{d}}\theta.\end{gathered}$$ ### Qualitative description of geodesic motion {#subsec:GlobalQualitative} Before solving this geodesic equation outright, it is useful to determine the qualitative behavior of the geodesics projected onto the poloidal $(y,\theta)$ plane. The analysis of the polar motion follows directly from our earlier discussion in Sec. \[sec:Kerr\] for Kerr: indeed, it suffices to note that $\Theta_n(\theta)$ takes the form $Q+P\cos^2{\theta}-\ell^2\cot^2{\theta}$ under the identification $$\begin{aligned} \label{eq:AngularIdentificationNHEK} Q=\mathcal{C}+\frac{3}{4}L^2-\mu^2M^2,\qquad P=\frac{L^2}{4}-\mu^2M^2,\qquad \ell=L.\end{aligned}$$ Therefore, the NHEK angular integral $\tilde{G}_\theta{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ takes the same form as the Kerr angular integral $G_\theta(Q,P,\ell)$ with $$\begin{aligned} \tilde{G}_\theta{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}=G_\theta{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+\frac{3}{4}L^2-\mu^2M^2,\frac{L^2}{4}-\mu^2M^2,L{\aftergroup\egroup\originalright})}.\end{aligned}$$ From Eq. , we see that NHEK geodesics (unlike their Kerr counterparts) are necessarily Type A (non-vortical) because $Q$ cannot be negative, since positivity of the angular potential requires that $$\begin{aligned} Q=\mathcal{C}+\frac{3}{4}L^2-\mu^2M^2 \ge\mathcal{C}+{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}L^2-\mu^2M^2 \ge\mathcal{C}+{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}L^2-\mu^2M^2{{\mathopen{}\mathclose\bgroup\originalleft}(1+\cos^2{\theta}{\aftergroup\egroup\originalright})} =\Theta_n \ge 0.\end{aligned}$$ Moreover, it is straightforward to check that $Q=0$ is only allowed for purely equatorial geodesics, a special case requiring a separate (and simpler) treatment. Generically, we therefore have the strict inequality $Q>0$, and the angular motion is necessarily of Type A with bounds given by Eq.  (when $P=0$) or Eq.  (when $P\neq0$). Having completed the analysis of the polar motion, we now turn to the radial motion. Its allowed range is heavily constrained by the requirement that the radial potential $\mathcal{Y}(y)$ remain positive at every point along the trajectory. As in Kerr, real zeroes of the radial potential $\mathcal{Y}(y)$ correspond to turning points $y_\pm$ in the radial motion. In global NHEK, the radial motion can be of two qualitatively different types (and is therefore simpler than in Kerr [@Bicak1989]): - Oscillatory motion between turning points $y_\pm$, corresponding to bound particles that are confined to NHEK and traverse the totality of the global strip, $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}(-\infty,+\infty{\aftergroup\egroup\originalright})}$. The global-time-lapse between successive turning points is $\Delta\tau=\pi$: each period of motion lies in a single Poincaré patch of the global strip. - Single-bounce motion (from a boundary at $y=\pm\infty$ to a turning point $y_t$ and back), corresponding to unbound particles that probe a single Poincaré patch of the global strip, $\tau\in{{\mathopen{}\mathclose\bgroup\originalleft}[\tau_0,\tau_0+2\pi{\aftergroup\egroup\originalright}]}$. The continuation of the geodesic beyond the intersection with the boundary depends on boundary conditions.[^13] The type of motion is determined by the properties of the roots of the radial potential $\mathcal{Y}(y)$. For generic values of the geodesic parameters, the (possibly complex) roots of $\mathcal{Y}(y)$ are given by $$\begin{aligned} \label{eq:GlobalTurningPoints} y_\pm=\frac{\triangle L\pm\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle^2-\mathcal{C}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}}}{\mathcal{C}}.\end{aligned}$$ As we vary the geodesic parameters, these roots move around in the complex $y$-plane. When one or more of the roots approaches or pinches the contour of integration, the radial motion of the allowed geodesics is then constrained to lie exclusively on one side of the root. Positivity of energy in the local frame of the particle, $P^\tau\ge0$, imposes the additional constraint $$\begin{aligned} \label{eq:PositiveEnergyCondition} \triangle+Ly\ge0.\end{aligned}$$ In terms of the critical radius $$\begin{aligned} y_c=-\frac{\triangle}{L},\end{aligned}$$ this constraint becomes $$\begin{aligned} y\gtrless y_c,\qquad L\gtrless 0.\end{aligned}$$ The roots $y_\pm$ can never equal $y_c$ for real values of the geodesic parameter $\triangle$, since $$\begin{aligned} \label{eq:RootSeparation} y_c=y_\pm\qquad\Longleftrightarrow\qquad \triangle^2=-L^2.\end{aligned}$$ The roots $y_\pm$ will be real when the quantity ${{\mathopen{}\mathclose\bgroup\originalleft}(\triangle^2-\mathcal{C}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}$ is positive. Positivity of $Q$ provides the bound $$\begin{aligned} \label{eq:ThetaBound} \mathcal{C}+\frac{3}{4}L^2-\mu^2M^2=Q >0\qquad\Longrightarrow\qquad \mathcal{C}>-\frac{3}{4}L^2+\mu^2M^2,\end{aligned}$$ from which it follows that $\mathcal{C}+L^2>0$. Therefore the polynomial $\mathcal{Y}(y)$ has a pair of real roots whenever $\triangle^2-\mathcal{C}\ge0$: $$\begin{aligned} y_\pm\in\mathbb{R}\qquad\Longleftrightarrow\qquad \triangle^2-\mathcal{C}\ge0.\end{aligned}$$ Generic geodesics probe both positive and negative values of $y$, so the sign of $y_\pm$ is less important than its relation to $y_c$. Since $\mathcal{Y}(y)=-\mathcal{C}y^2+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(y{\aftergroup\egroup\originalright})}}$, there are now three cases to examine separately: 1. In this case, we must demand that $y_\pm\in\mathbb{R}$, as $\mathcal{Y}(y)$ is negative outside the interval between its roots. That is, we must require that $0<\mathcal{C}<\triangle^2$ and $y\in\mathcal{I}={{\mathopen{}\mathclose\bgroup\originalleft}[y_-,y_+{\aftergroup\egroup\originalright}]}$. Moreover, one can show that $y_c<y_-<y_+$ whenever $\triangle L>0$, and $y_-<y_+<y_c$ whenever $\triangle L<0$. For all values of $L$, the condition then requires that $\triangle>0$ and the entire interval $\mathcal{I}$ is allowed. These bounded geodesics are not permitted to escape NHEK.[^14] 2. In this case, the roots $y_\pm$ are manifestly real because $\triangle^2-\mathcal{C}>0$. Moreover, $y_+<y_c<y_-$.[^15] Hence, the radial potential $\mathcal{Y}(y)$ is nonnegative if $y\le y_+$ or $y\ge y_-$. There are two further subcases to consider:[^16] 1. $L>0$, in which case the condition requires that $y>y_c$, so only the branch $y\ge y_-$ is allowed. 2. $L<0$, in which case the condition requires that $y<y_c$, so only the branch $y\le y_+$ is allowed. These unbound geodesics fall in from the boundary $(y=\pm\infty)$ of NHEK, and eventually reach a turning point before heading back towards the same boundary. 3. In this case, $\mathcal{Y}(y)$ is linear with a zero at $y_0=\frac{L^2-\triangle^2}{2\triangle L}$ and a slope given by $2\triangle L$. Therefore, for $\triangle L>0$, the allowed range of motion is $y\in[y_0,\infty)$, while for $\triangle L<0$ the allowed range is $y\in(-\infty,y_0]$. The condition then requires that $\triangle>0$ and the entire semi-infinite interval is allowed for both signs of $L$. These geodesics are a limiting example of Type II geodesics and are also unbound. To summarize, the type of radial motion is picked out by the sign of $\mathcal{C}$: ![Plot of the radial motion $(\tau,\psi(\tau))$ of geodesics in global NHEK, with $\psi=\pi/2+\arctan{y}$ where $y(\tau)$ is given in Eq. . The blue geodesic is Type I ($\mathcal{C}>0$) with ${{\mathopen{}\mathclose\bgroup\originalleft}(\triangle,L,\mathcal{C}{\aftergroup\egroup\originalright})}=(4,2,2)$. The red geodesic is Type II ($\mathcal{C}<0$) and coming in from the right boundary ($L>0$) with ${{\mathopen{}\mathclose\bgroup\originalleft}(\triangle,L,\mathcal{C}{\aftergroup\egroup\originalright})}=(4,2,-2)$. It appears twice, with a global time shift of $\Delta\tau=\pi/2$. The green geodesic is Type II ($\mathcal{C}<0$) and coming in from the left boundary ($L<0$) with ${{\mathopen{}\mathclose\bgroup\originalleft}(\triangle,L,\mathcal{C}{\aftergroup\egroup\originalright})}=(4,-2,-2)$. In the Poincaré patch, the Type I geodesics come out of the past horizon, travel to increasingly large radius until they encounter a turning point at a maximal allowed radius, and then fall back into the future horizon. The upper red geodesic is Type IIA, with $p^R\gtrless0$ in the upper/lower Poincaré patch. The lower red geodesic is Type IIB with $L>0$, while the green geodesic is Type IIB with $L<0$. Type IIA and Type IIB are related by global-time-translation symmetry.[]{data-label="fig:GeodesicTypes"}](GeodesicTypes.pdf){width=".6\textwidth"} Note that the sign of the $\mathsf{SL}(2,\mathbb{R})$ Casimir $\mathcal{C}$ determines the eventual fate of a NHEK geodesic: those with $\mathcal{C}>0$ are bound and remain in the throat forever, while those with $\mathcal{C}<0$ can reach the boundary in finite global time and escape. These different types of motion are illustrated in Fig. \[fig:GeodesicTypes\]. There is also a special class of timelike bound geodesics of constant global radius arising as a special limit of the Type I ($\mathcal{C}>0$) motion. In the limit $\mathcal{C}\to\triangle^2$, the zeroes of the radial potential coalesce, $y_-\to y_+$, and one has $\mathcal{Y}(y_\pm)=\mathcal{Y}'(y_\pm)=0$. The gravitational pull exerted by the black hole on these particles is exactly balanced by the centrifugal force they experience from its rotation. After dimensional reduction to two dimensions, NHEK geodesics coincide with Lorentz force trajectories of charged particles evolving in two-dimensional Anti-de Sitter spacetime AdS$_2$ with a background radial electric field (see App. \[app:AdS2\]). From that two-dimensional perspective, the geodesics of constant global radius arise when gravity in the AdS$_2$ box is exactly balanced out by the acceleration of the charge by the electric field. ### Geodesic integrals in global NHEK We now wish to compute the integrals $\tilde{I}_y$, $\tilde{I}_\tau$, $\tilde{I}_\varphi$, $\tilde{G}_\theta$, and $\tilde{G}_\varphi$ that appear in the NHEK geodesic equation . As previously noted, the NHEK angular integral $\tilde{G}_\theta{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ takes the same form as the Kerr angular integral $G_\theta(Q,P,\ell)$ under the indentification $$\begin{aligned} \label{eq:AngularIdentificationNHEKbis} Q=\mathcal{C}+\frac{3}{4}L^2-\mu^2M^2,\qquad P=\frac{L^2}{4}-\mu^2M^2,\qquad \ell=L.\end{aligned}$$ Similarly, given that $$\begin{aligned} \frac{L}{\Lambda^2}=L\csc^2{\theta}-\frac{L}{4}\cos^2{\theta}-\frac{3L}{4},\end{aligned}$$ the NHEK integral $\tilde{G}_\varphi{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ is a linear combination of the Kerr angular integrals $G_\theta$, $G_t$, and $G_\phi$, $$\begin{aligned} \tilde{G}_\varphi=\frac{L}{4}{{\mathopen{}\mathclose\bgroup\originalleft}(4G_\phi-G_t-3G_\theta{\aftergroup\egroup\originalright})},\end{aligned}$$ with the identification of arguments given by . Therefore, we can simply apply the results derived in Sec. \[sec:Kerr\] for the Kerr angular integrals in order to evaluate their NHEK counterparts. It remains to evaluate the radial integrals $\tilde{I}_y$, $\tilde{I}_\tau$, and $\tilde{I}_\varphi$. For this purpose, it is useful to define signs $$\begin{aligned} \nu_s=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_s^y{\aftergroup\egroup\originalright})} =(-1)^w\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_o^y{\aftergroup\egroup\originalright})},\qquad \nu_o=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_o^y{\aftergroup\egroup\originalright})} =(-1)^w\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_s^y{\aftergroup\egroup\originalright})},\end{aligned}$$ where $P_s^y$ and $P_o^y$ denote the radial momentum evaluated at the endpoints $X_s^\mu$ and $X_o^\mu$ of the geodesic, respectively, and $w\in\mathbb{N}$ denotes the number of turning points in the radial motion. Then by working through all the possible configurations, one can check that the radial path integral unpacks as follows: \[eq:GlobalType\] $$\begin{aligned} \label{eq:GlobalTypeI} \text{Type I:}\qquad& \fint_{y_s}^{y_o}={{\mathopen{}\mathclose\bgroup\originalleft}(w\pm\frac{\nu_o-\nu_s}{2}{\aftergroup\egroup\originalright})}\int_{y_-}^{y_+}-\nu_s\int_{y_\pm}^{y_s}+\nu_o\int_{y_\pm}^{y_o},\\ \label{eq:GlobalTypeII} \text{Type II, III:}\qquad& \fint_{y_s}^{y_o}=-\nu_s\int_{y_\pm}^{y_s}+\nu_o\int_{y_\pm}^{y_o},\qquad \pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}L{\aftergroup\egroup\originalright})}.\end{aligned}$$ For the Type I geodesics, Eq.  constitutes two equivalent representations that differ only in the choice of reference turning point for the integrals. For the Type II geodesics, one must select the (unique) turning point that is actually encountered along the trajectory. We can now explicitly evaluate the radial integrals. For our purposes, the relevant basis of integrals is given by $$\begin{aligned} \tilde{\mathcal{I}}_1^\pm&\equiv\int_{y_\pm}^{y_i}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(y_+-y{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(y-y_-{\aftergroup\egroup\originalright})}}},\\ \tilde{\mathcal{I}}_2^\pm&\equiv\int_{y_\pm}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(y_+-y{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(y-y_-{\aftergroup\egroup\originalright})}}},\\ \tilde{\mathcal{I}}_3^\pm&\equiv\int_{y_\pm}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{y}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(y_+-y{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(y-y_-{\aftergroup\egroup\originalright})}}},\end{aligned}$$ in terms of which $$\begin{aligned} \int_{y_\pm}^{y_i}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\tilde{\mathcal{I}}_1^\pm,\\ \int_{y_\pm}^{y_i}y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=L{{\mathopen{}\mathclose\bgroup\originalleft}(\tilde{\mathcal{I}}_1^\pm-\tilde{\mathcal{I}}_2^\pm{\aftergroup\egroup\originalright})}+\triangle\tilde{\mathcal{I}}_3^\pm,\\ \int_{y_\pm}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\triangle\tilde{\mathcal{I}}_2^\pm+L\tilde{\mathcal{I}}_3^\pm.\end{aligned}$$ In each case, for the allowed range of $y_i$, the integrals $\tilde{\mathcal{I}}_1^\pm$, $\tilde{\mathcal{I}}_2^\pm$, and $\tilde{\mathcal{I}}_3^\pm$ are all manifestly real. To evaluate them, we will use the identities with the quantities $$\begin{aligned} \label{eq:GlobalParameters} q_\pm={{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+y_\pm^2}{1+y_\mp^2}{\aftergroup\egroup\originalright})}^{1/4}>0,\qquad \cos{\alpha_\pm}=\mp\frac{1+y_-y_+}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_-^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_+^2{\aftergroup\egroup\originalright})}}}\in{{\mathopen{}\mathclose\bgroup\originalleft}[-1,1{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Observe that ${{\mathopen{}\mathclose\bgroup\originalleft}|\cos\alpha_\pm{\aftergroup\egroup\originalright}|}=1$ if and only if $y_+=y_-$, which only occurs when $\mathcal{C}=\triangle^2$. This corresponds to a degenerate limit of Type I motion in which there is no radial motion and the geodesic remains at fixed constant global radius. When $\mathcal{C}>0$, the choice of turning point (and thus the sign $\pm$ of $\tilde{\mathcal{I}}_i^\pm$) is arbitrary, so we can use either of the substitutions $$\begin{aligned} y=\frac{y_\pm+y_\mp x^2}{1+x^2}\qquad\Longrightarrow\qquad x_i^\pm={{\mathopen{}\mathclose\bgroup\originalleft}|\frac{y_i-y_\pm}{y_i-y_\mp}{\aftergroup\egroup\originalright}|}^{1/2} \in[0,+\infty).\end{aligned}$$ In terms of the quantities defined in Eq. , this results in (assuming $\alpha_-\neq0$) $$\begin{aligned} \tilde{\mathcal{I}}_1^\pm&=\mp\frac{2}{\sqrt{\mathcal{C}}}\int_0^{x_i^\pm}\frac{{\mathop{}\!\mathrm{d}}x}{1+x^2} =\mp\frac{2}{\sqrt{\mathcal{C}}}\arctan{x_i^\pm},\\ \tilde{\mathcal{I}}_2^\pm&=\mp\frac{2}{\sqrt{\mathcal{C}}}\int_0^{x_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(1+x^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}x}{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\pm^2{\aftergroup\egroup\originalright})}+2{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_-y_+{\aftergroup\egroup\originalright})}x^2+{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\mp^2{\aftergroup\egroup\originalright})}x^4} =\mp\frac{2}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_+{\aftergroup\egroup\originalright})}+\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_+{\aftergroup\egroup\originalright})}}{1+y_\mp^2}{\aftergroup\egroup\originalright}]},\\ \tilde{\mathcal{I}}_3^\pm&=\mp\frac{2}{\sqrt{\mathcal{C}}}\int_0^{x_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(y_\pm+y_\mp x^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}x}{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\pm^2{\aftergroup\egroup\originalright})}+2{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_-y_+{\aftergroup\egroup\originalright})}x^2+{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\mp^2{\aftergroup\egroup\originalright})}x^4} =\mp\frac{2}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{y_\pm\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_+{\aftergroup\egroup\originalright})}+y_\mp\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_+{\aftergroup\egroup\originalright})}}{1+y_\mp^2}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ On the other hand, when $\mathcal{C}<0$, we must choose the sign $\pm$ according to whether $L\lessgtr0$, and then use the substitution $$\begin{aligned} y=\frac{y_\pm-y_\mp x^2}{1-x^2}\qquad\Longrightarrow\qquad x_i^\pm={{\mathopen{}\mathclose\bgroup\originalleft}|\frac{y_i-y_\pm}{y_i-y_\mp}{\aftergroup\egroup\originalright}|}^{1/2} \in{{\mathopen{}\mathclose\bgroup\originalleft}[0,1{\aftergroup\egroup\originalright}]}.\end{aligned}$$ In terms of the quantities defined in Eq. , this results in $$\begin{aligned} \tilde{\mathcal{I}}_1^\pm&=\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{x_i^\pm}\frac{{\mathop{}\!\mathrm{d}}x}{1-x^2} =\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{x_i^\pm},\\ \tilde{\mathcal{I}}_2^\pm&=\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{x_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(1-x^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}x}{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\pm^2{\aftergroup\egroup\originalright})}-2{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_-y_+{\aftergroup\egroup\originalright})}x^2+{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\mp^2{\aftergroup\egroup\originalright})}x^4} =\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_-{\aftergroup\egroup\originalright})}-\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_-{\aftergroup\egroup\originalright})}}{1+y_\mp^2}{\aftergroup\egroup\originalright}]},\\ \tilde{\mathcal{I}}_3^\pm&=\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{x_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(y_\pm-y_\mp x^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}x}{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\pm^2{\aftergroup\egroup\originalright})}-2{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_-y_+{\aftergroup\egroup\originalright})}x^2+{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_\mp^2{\aftergroup\egroup\originalright})}x^4} =\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{y_\pm\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_-{\aftergroup\egroup\originalright})}-y_\mp\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,q_\pm,\alpha_-{\aftergroup\egroup\originalright})}}{1+y_\mp^2}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Finally, when $\mathcal{C}=0$, the integrals $\tilde{\mathcal{I}}_1^\pm$, $\tilde{\mathcal{I}}_2^\pm$, and $\tilde{\mathcal{I}}_3^\pm$ degenerate to $$\begin{aligned} \tilde{\mathcal{I}}_1^\pm=\int_{y_0}^{y_i}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{2\triangle L{{\mathopen{}\mathclose\bgroup\originalleft}(y-y_0{\aftergroup\egroup\originalright})}}},\qquad \tilde{\mathcal{I}}_2^\pm=\int_{y_0}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{2\triangle L{{\mathopen{}\mathclose\bgroup\originalleft}(y-y_0{\aftergroup\egroup\originalright})}}},\qquad \tilde{\mathcal{I}}_3^\pm=\int_{y_0}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{y}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{2\triangle L{{\mathopen{}\mathclose\bgroup\originalleft}(y-y_0{\aftergroup\egroup\originalright})}}}.\end{aligned}$$ To evaluate these simpler integrals, we use the substitution $$\begin{aligned} y=y_0\mp x^2\qquad\Longrightarrow\qquad x_i^0=\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|y_i-y_0{\aftergroup\egroup\originalright}|}}\in[0,+\infty),\end{aligned}$$ where we must still choose the sign $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(-L{\aftergroup\egroup\originalright})}$ since $L\lessgtr0$ corresponds to $y_i\lessgtr y_0$. This results in $$\begin{aligned} \tilde{\mathcal{I}}_1^\pm&=\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}\int_0^{x_i^0}{\mathop{}\!\mathrm{d}}x =\mp\frac{2x_i^0}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}},\\ \tilde{\mathcal{I}}_2^\pm&=\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}\int_0^{x_i^0}\frac{{\mathop{}\!\mathrm{d}}x}{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_0^2{\aftergroup\egroup\originalright})}\mp2y_0x^2+x^4} =\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^0,q_0,\alpha_0^\pm{\aftergroup\egroup\originalright})},\\ \tilde{\mathcal{I}}_3^\pm&=\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}\int_0^{x_i^0}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(y_0\mp x^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}x}{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y_0^2{\aftergroup\egroup\originalright})}\mp2y_0x^2+x^4} =\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[y_0\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^0,q_0,\alpha_0^\pm{\aftergroup\egroup\originalright})}\mp\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^0,q_0,\alpha_0^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]},\\ \label{eq:GlobalParameters0} &q_0={{\mathopen{}\mathclose\bgroup\originalleft}(1+y_0^2{\aftergroup\egroup\originalright})}^{1/4}>0,\qquad \alpha_0^\pm=\frac{\pi}{2}\mp\arctan{y_0}\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,\pi{\aftergroup\egroup\originalright}]}.\end{aligned}$$ To put everything together, it is useful to define $$\begin{aligned} \tilde{\mathcal{I}}_\varphi^\pm(x,\alpha,s)&=\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle y_\pm-L{\aftergroup\egroup\originalright})}\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_\pm,\alpha{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(s\triangle y_\mp-L{\aftergroup\egroup\originalright})}\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_\pm,\alpha{\aftergroup\egroup\originalright})}}{1+y_\mp^2},\\ \tilde{\mathcal{I}}_\varphi^0(x,\alpha,s)&={{\mathopen{}\mathclose\bgroup\originalleft}(\triangle y_0-L{\aftergroup\egroup\originalright})}\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_0,\alpha{\aftergroup\egroup\originalright})}-s\triangle\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_0,\alpha{\aftergroup\egroup\originalright})},\\ \tilde{\mathcal{I}}_\tau^\pm(x,\alpha,s)&=\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+Ly_\pm{\aftergroup\egroup\originalright})}\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_\pm,\alpha{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+sLy_\mp{\aftergroup\egroup\originalright})}\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_\pm,\alpha{\aftergroup\egroup\originalright})}}{1+y_\mp^2},\\ \tilde{\mathcal{I}}_\tau^0(x,\alpha,s)&={{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+Ly_0{\aftergroup\egroup\originalright})}\tilde{I}_-{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_0,\alpha{\aftergroup\egroup\originalright})}-sL\tilde{I}_+{{\mathopen{}\mathclose\bgroup\originalleft}(x,q_0,\alpha{\aftergroup\egroup\originalright})},\end{aligned}$$ where we remind the reader that the quantities $y_\pm$, $q_\pm$, $q_0$, and $\tilde{I}_\pm$ are defined in Eqs. , , , and , respectively. Then, we see that when $\mathcal{C}>0$, \[eq:GlobalIntegralsTypeI\] $$\begin{aligned} \int_{y_\pm}^{y_i}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{\mathcal{C}}}\arctan{x_i^\pm},\\ \int_{y_\pm}^{y_i}y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[L\arctan{x_i^\pm}+\tilde{\mathcal{I}}_\varphi^\pm{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,\alpha_+,+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]},\\ \int_{y_\pm}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{\mathcal{C}}}\tilde{\mathcal{I}}_\tau^\pm{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,\alpha_+,+{\aftergroup\egroup\originalright})},\end{aligned}$$ where the choice of sign $\pm$ is arbitrary. In particular, for Type I geodesics (with $\mathcal{C}>0$), the half-period integrals are $$\begin{aligned} \label{eq:GlobalTimeLapse} \int_{y_-}^{y_+}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}=\frac{\pi}{\sqrt{\mathcal{C}}},\qquad \int_{y_-}^{y_+}y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}=\frac{\pi L}{\sqrt{\mathcal{C}}},\qquad \int_{y_-}^{y_+}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}=\pi.\end{aligned}$$ On the other hand, when $\mathcal{C}<0$, we must choose $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(-L{\aftergroup\egroup\originalright})}$ and \[eq:GlobalIntegralsTypeII\] $$\begin{aligned} \int_{y_\pm}^{y_i}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{x_i^\pm},\\ \int_{y_\pm}^{y_i}y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[L\operatorname{arctanh}{x_i^\pm}+\tilde{\mathcal{I}}_\varphi^\pm{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,\alpha_-,-{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]},\\ \int_{y_\pm}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\tilde{\mathcal{I}}_\tau^\pm{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^\pm,\alpha_-,-{\aftergroup\egroup\originalright})}.\end{aligned}$$ Likewise, when $\mathcal{C}=0$ and $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(-L{\aftergroup\egroup\originalright})}$, \[eq:GlobalIntegralsTypeIII\] $$\begin{aligned} \int_{y_0}^{y_i}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2x_i^0}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}},\\ \int_{y_\pm}^{y_i}y{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[Lx_i^0+\tilde{\mathcal{I}}_\varphi^0{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^0,\alpha_0^\pm,\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]},\\ \int_{y_\pm}^{y_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\sqrt{\mathcal{Y}(y)}}&=\mp\frac{2}{\sqrt{2\triangle{{\mathopen{}\mathclose\bgroup\originalleft}|L{\aftergroup\egroup\originalright}|}}}\tilde{\mathcal{I}}_\tau^0{{\mathopen{}\mathclose\bgroup\originalleft}(x_i^0,\alpha_0^\pm,\pm{\aftergroup\egroup\originalright})}.\end{aligned}$$ Substituting Eqs. – into Eq.  yields the geodesic integrals for Type I motion: [align]{} \_y&=[[\[w\_s[[(-2]{})]{}\_o[[(-2]{})]{}]{}\]]{},\ \_&=L\_y[[\[\_s\_\^-\_o\_\^]{}\]]{},\ \[eq:GlobalTI\] \_&=w\_s[[\[-\_\^]{}\]]{}\_o[[\[-\_\^]{}\]]{}. The geodesic integrals for Type II and Type III motion are likewise obtained by substituting Eqs. – into Eq. , resulting in [align]{} \_y&=[[(\_s-\_o]{})]{},\ \_&=L\_y[[\[\_s\_\^-\_o\_\^]{}\]]{},\ \[eq:GlobalTII\] \_&=[[\[\_s\_\^-\_o\_\^]{}\]]{}. [align]{} \_y&=[[(\_sx\_s\^0-\_ox\_o\^0]{})]{},\ \_&=L\_y[[\[\_s\_\^0[[(x\_s\^0,\_0\^,]{})]{}-\_o\_\^0[[(x\_o\^0,\_0\^,]{})]{}]{}\]]{},\ \[eq:GlobalTIII\] \_&=[[\[\_s\_\^0[[(x\_s\^0,\_0\^,]{})]{}-\_o\_\^0[[(x\_o\^0,\_0\^,]{})]{}]{}\]]{}. For the Type II and Type III geodesics (with $\mathcal{C}<0$ and $\mathcal{C}=0$, respectively), the radial integral $\tilde{I}_y$ is in general divergent, indicating simply that the typical geodesic undergoes an infinite number of polar librations as it makes its way towards the boundary. In physical applications, this divergence is of course cut off by the finite distance to the NHEK boundary. Note however that the integral $\tilde{I}_\tau$ is well behaved as $y\to\pm\infty$, reflecting the fact that geodesics can reach the boundary of NHEK in finite global (but infinite proper) time. ### Explicit solution of the geodesic equation Up to this point, we have simply repeated the analysis performed for the Kerr integrals in Sec. \[sec:Kerr\], taking advantage of the fact that the Kerr elliptic integrals reduce to trigonometric integrals in the near-horizon scaling limit. Fortuitously, the simplicity of the global NHEK geodesic equation relative to its Kerr counterpart allows one to go further and obtain an explicit time-parameterization for the global geodesics. Comparing Eqs.  and to Eqs.  and , one sees that the disappearance of $\theta$-dependence on the right-hand-side of Eq.  allows one to reduce the $(\tau,y)$ motion in global NHEK to a single ordinary differential equation: $$\begin{aligned} \label{eq:GlobalODE} \frac{dy}{d\tau}=\pm_y\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})}}{\triangle+Ly}\sqrt{\mathcal{Y}(y)}.\end{aligned}$$ This is known as a first-order, non-linear autonomous system. Such equations are capable of displaying a number of interesting properties worth discussing. For instance, while continuity of the right-hand-side of Eq.  guarantees the existence of a solution locally, a continuous derivative is required to guarantee its uniqueness. Since this condition is violated at zeroes of the radial potential, multiple solutions may be possible for a given set of initial conditions. Indeed, these extra equilibrium solutions are the constant-radius trajectories discussed previously: a geodesic $y(\tau)=y_t$ obeys the same initial conditions as a geodesic with turning point $y_t$ whenever they meet. As we will see, although it is rarely the case for a generic non-linear autonomous system, the general solution to Eq.  includes these special equilibrium states as a subset. More interestingly, solutions to systems like Eq.  are known to be capable of exhibiting blowup in finite time. In ordinary applications, this signals an instability of the non-linear system, but in our case it represents a reasonable physical effect: geodesics in NHEK can reach the conformal boundary in finite coordinate time. It is often impossible to obtain explicit solutions to systems like Eq.  in terms of elementary functions, so typical methods focus on describing the qualitative behavior of the solutions using fixed-point theory. The equilibrium configurations (both stable and unstable), which correspond to the zeroes of the radial potential, control the late-time behavior of the geodesics. As one varies the parameters $(\triangle,L,\mathcal{C})$ of this dynamical system, the various equilibrium points move around, merge and annihilate, or emerge and separate. This qualitative analysis was performed in Sec. \[subsec:GlobalQualitative\]. When an analytic solution is possible, it is obtainable through the following procedure. Using separation of variables, one integrates Eq.  to determine $\tau(y)$, up to a constant of integration. This was done in Eqs. , , and . This procedure defines $y(\tau)$ implicitly. In global NHEK, $\tau(y)$ is in general multivalued because $y(\tau)$ is periodic. If one manages to invert $\tau(y)$ in a small neighborhood, one can then extend the solution $y(\tau)$ to the entire global strip. Indeed, because the system has time-translation symmetry (is autonomous), the general solution to the initial value problem is obtained from $y(\tau)$ by a simple shift of the argument $\tau\to\tau-\tau_\star$. Although generic autonomous systems can be difficult to integrate, Eq.  is very special. It arises from a system with extra symmetries generated by the Killing vector fields , and these symmetries can be used to reduce the differential equation to an algebraic one. While we chose to label geodesics in global NHEK by the conserved quantities associated to $iL_0^\mu$, $W_0^\mu$, and $\mathcal{C}^{\mu\nu}$, the quantities $H_+=H_+^\mu p_\mu$, $H_-=H_-^\mu p_\mu$, and $H_0=H_0^\mu p_\mu$ are all conserved along particle trajectories as well. A particularly simple combination to work with is $$\begin{aligned} \label{eq:ExtraConservation} \frac{H_+-H_-}{2H_0}=\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-\triangle y{\aftergroup\egroup\originalright})}\cos{\tau}\pm_y\sqrt{\mathcal{Y}(y)}\sin{\tau}}{{{\mathopen{}\mathclose\bgroup\originalleft}(L-\triangle y{\aftergroup\egroup\originalright})}\sin{\tau}\mp_y\sqrt{\mathcal{Y}(y)}\cos{\tau}}.\end{aligned}$$ Differentiating this relation with respect to $\tau$ and solving for $y'(\tau)$, one easily recovers . Therefore, every solution to Eq.  obeys this algebraic relation for some value of the constant ${{\mathopen{}\mathclose\bgroup\originalleft}(H_+-H_-{\aftergroup\egroup\originalright})}/2H_0$, which depends on initial conditions. In fact, one can check that different choices of this constant amount to a shift $\tau \to \tau-\tau_\star$ in , in line with our expectations for an autonomous system. For the particular choice $-L/\triangle$, further manipulation puts this equation in the form $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}[\frac{\triangle\cos{\tau}+L\sin{\tau}}{L\cos{\tau}-\triangle\sin{\tau}}{\aftergroup\egroup\originalright}]}^2=\frac{\mathcal{Y}(y)}{{{\mathopen{}\mathclose\bgroup\originalleft}(L-\triangle y{\aftergroup\egroup\originalright})}^2},\end{aligned}$$ and it is straightforward to verify that a solution to either algebraic equation satisfies Eq. . In terms of $$\begin{aligned} S(\tau)=\mathcal{C}+\frac{\triangle^2-\mathcal{C}}{\triangle^2+L^2}{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle\cos{\tau}+L\sin{\tau}{\aftergroup\egroup\originalright})}^2,\end{aligned}$$ these expressions can be inverted to give $$\begin{aligned} y(\tau)=\frac{\triangle L}{S(\tau)}{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{\triangle\sin{\tau}-L\cos{\tau}}{\triangle L}\sqrt{\frac{\triangle^2-\mathcal{C}}{\triangle^2+L^2}{{\mathopen{}\mathclose\bgroup\originalleft}[S(\tau)+L^2{\aftergroup\egroup\originalright}]}}{\aftergroup\egroup\originalright})}.\end{aligned}$$ (The quadratic equation has a second root with a $-$ sign, which is related to this root by a shift of $\tau\to\tau+\pi$.) Using global-time-translation symmetry, we can shift the argument $\tau\to\tau-\tau_\star$ to ensure that the geodesics pass through any point $(\tau_s,y_s)$ where geodesic motion is allowed: \[eq:GlobalRadialMotion\] $$\begin{aligned} y_o(\tau_o)&=\frac{\triangle L}{S(\tau_o-\tau_\star)}{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{\triangle\sin{{\mathopen{}\mathclose\bgroup\originalleft}(\tau_o-\tau_\star{\aftergroup\egroup\originalright})}-L\cos{{\mathopen{}\mathclose\bgroup\originalleft}(\tau_o-\tau_\star{\aftergroup\egroup\originalright})}}{\triangle L}\sqrt{\frac{\triangle^2-\mathcal{C}}{\triangle^2+L^2}{{\mathopen{}\mathclose\bgroup\originalleft}[S(\tau_o-\tau_\star)+L^2{\aftergroup\egroup\originalright}]}}{\aftergroup\egroup\originalright})},\\ \tau_\star&=\tau_s-\arctan{{\mathopen{}\mathclose\bgroup\originalleft}(y_s{\aftergroup\egroup\originalright})}+\arctan{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-\triangle y_s{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}}{{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle^2-\mathcal{C}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+Ly_s{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle^2+L^2{\aftergroup\egroup\originalright})}\sqrt{\mathcal{Y}(y_s)}}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ (In so doing, the values of the constants in Eq.  change, since they arise from symmetries that do not commute with global-time-translations.) This function describes the radial motion of a geodesic along the entire global strip. In the Type II case, geodesics hit the boundary at global time $$\begin{aligned} \tau_b^\pm=\tau_\star\pm\pi+\arctan{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}-\triangle^2{\aftergroup\egroup\originalright})}L\mp{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle^2+L^2{\aftergroup\egroup\originalright})}\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}\triangle}{\aftergroup\egroup\originalright}]}\mod_{2\pi}.\end{aligned}$$ Provided that one keeps track of radial turning points encountered along the way (if any), the expression allows one to obtain $\tilde{I}_y$ as a function of global time by plugging in $y_o(\tau_o)$. In turn, substitution of $\tilde{I}_y(\tau_o)$ into the inversion formulas – derived in Sec. \[sec:Kerr\] allows one to obtain the polar angle of the particle as a function of global time. For instance, in the generic case $P\neq0$ ($L\neq\pm2\mu M$), $$\begin{aligned} \cos{\theta_o(\tau_o)}=\sqrt{u_\pm}\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}(X^\pm(\tau_o){\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\qquad X^\pm(\tau_o)=F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}-\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\sqrt{-u_\mp P}\tilde{I}_y(\tau_o).\end{aligned}$$ Finally, given both $y_o(\tau_o)$ and $\theta_o(\tau_o)$, one can plug them into the expressions for $\tilde{I}_\varphi$ and $\tilde{G}_\varphi$ to obtain the azimuthal angle as a function of global time, $$\begin{aligned} \varphi_o(\tau_o)=\varphi_s+\tilde{G}_\varphi(\tau_o)-\tilde{I}_\varphi(\tau_o).\end{aligned}$$ This completes the explicit parameterization of geodesics by the global time elapsed along their trajectory. Geodesics in Poincaré NHEK {#subsec:PoincareGeodesics} -------------------------- Having solved for the most general geodesic motion in global NHEK, we now turn to the description of geodesics in the Poincaré patch. Of course, each class of geodesics in Poincaré NHEK is in principle obtainable from some geodesic in global NHEK, which is simply the geodesic completion of the Poincaré patch. However, since the Poincaré coordinates naturally arise in the near-horizon limit of the extreme Kerr black hole, the Poincaré patch description is more useful for connecting the discussion to the far region, and many expressions take a much simpler form in these coordinates. Recall from Sec. \[subsec:EmergentThroat\] that in Poincaré coordinates, the NHEK line element is $$\begin{aligned} \label{eq:PoincareMetric} d\hat{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-R^2{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\Phi+R{\mathop{}\!\mathrm{d}}T{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ and the generators of $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ are $$\begin{aligned} H_0=T{\mathop{}\!\partial}_T-R{\mathop{}\!\partial}_R,\qquad H_+={\mathop{}\!\partial}_T,\qquad H_-={{\mathopen{}\mathclose\bgroup\originalleft}(T^2+\frac{1}{R^2}{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_T-2TR{\mathop{}\!\partial}_R-\frac{2}{R}{\mathop{}\!\partial}_\Phi,\qquad W_0={\mathop{}\!\partial}_\Phi.\end{aligned}$$ The Casimir of $\mathsf{SL}(2,\mathbb{R})$ is the (manifestly reducible) symmetric Killing tensor $$\begin{aligned} \mathcal{C}^{\mu\nu}=-H_0^\mu H_0^\nu+\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(H_+^\mu H_-^\nu+H_-^\mu H_+^\nu{\aftergroup\egroup\originalright})}.\end{aligned}$$ The motion of a free particle of mass $\mu$ and four-momentum $P^\mu$ is described by the geodesic equation $$\begin{aligned} P^\mu\hat{\nabla}P^\nu=0,\qquad \hat{g}^{\mu \nu}P_\mu P_\nu=-\mu^2.\end{aligned}$$ Geodesic motion in Poincaré NHEK is completely characterized by the three conserved quantities $$\begin{gathered} E=-H_+^\mu P_\mu =-P_T,\qquad L=W_0^\mu P_\mu =P_\Phi,\\ \mathcal{C}=\mathcal{C}^{\mu\nu}P_\mu P_\nu =P_\theta^2-{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}P_\Phi^2+{{\mathopen{}\mathclose\bgroup\originalleft}(2M^2\Gamma{\aftergroup\egroup\originalright})}\mu^2 =\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(P_T-P_\Phi R{\aftergroup\egroup\originalright})}^2}{R^2}-R^2P_R^2-P_\Phi^2,\end{gathered}$$ denoting the NHEK energy, angular momentum parallel to the axis of symmetry, and $\mathsf{SL}(2,\mathbb{R})$ Casimir, respectively. By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,E,L,\mathcal{C}{\aftergroup\egroup\originalright})}$, we find that a particle following a geodesic in the Poincaré patch of the NHEK geometry has an instantaneous four-momentum $P=P_\mu{\mathop{}\!\mathrm{d}}X^\mu$ of the form $$\begin{aligned} \label{eq:GeodesicNHEK} P(X^\mu,E,L,\mathcal{C})=-E{\mathop{}\!\mathrm{d}}T\pm_R\frac{\sqrt{\mathcal{R}_n(R)}}{R^2}{\mathop{}\!\mathrm{d}}R\pm_\theta\sqrt{\Theta_n(\theta)}{\mathop{}\!\mathrm{d}}\theta+L{\mathop{}\!\mathrm{d}}\Phi,\end{aligned}$$ where the two choices of sign $\pm_R$ and $\pm_\theta$ depend on the radial and polar directions of travel, respectively. Here, we used the same polar potential $\Theta_n(\theta)$ introduced in Eq. , and additionally defined a new radial potential $$\begin{aligned} \label{eq:PoincareRadialPotential} \mathcal{R}_n(R)={{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}^2-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}R^2.\end{aligned}$$ One can then raise $P_\mu$ to obtain the equations for the geodesic trajectory, \[eq:PoincareMomentum\] $$\begin{aligned} 2M^2\Gamma\frac{dR}{d\sigma}&=\pm_R\sqrt{\mathcal{R}_n(R)},\\ 2M^2\Gamma\frac{d\theta}{d\sigma}&=\pm_\theta\sqrt{\Theta_n(\theta)},\\ 2M^2\Gamma\frac{d\Phi}{d\sigma}&=-\frac{E+LR}{R}+\frac{L}{\Lambda^2},\\ 2M^2\Gamma\frac{dT}{d\sigma}&=\frac{E+LR}{R^2}.\end{aligned}$$ The parameter $\sigma$ is the affine parameter for massless particles ($\mu=0$), and is related to the proper time $\delta$ by $\delta=\mu\sigma$ for massive particles. Following the same procedure as in Kerr and global NHEK, we find from Eqs.  that a geodesic labeled by ${{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,E,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ connects spacetime points $X_s^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(T_s,R_s,\theta_s,\Phi_s{\aftergroup\egroup\originalright})}$ and $X_o^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(T_o,R_o,\theta_o,\Phi_o{\aftergroup\egroup\originalright})}$ in Poincaré NHEK if $$\begin{aligned} &\fint_{R_s}^{R_o}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}}=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta_n(\theta)}},\\ \Phi_o-\Phi_s&=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(-\frac{E+LR}{R}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}}+\fint_{\theta_s}^{\theta_o}\frac{L\Lambda^{-2}(\theta)}{\pm_\theta\sqrt{\Theta_n(\theta)}}{\mathop{}\!\mathrm{d}}\theta,\\ T_o-T_s&=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}}.\end{aligned}$$ We may rewrite these conditions as $$\begin{aligned} \label{eq:PoincareGeodesics} \hat{I}_R=\hat{G}_\theta,\qquad \Phi_o-\Phi_s=-\hat{I}_\Phi+\hat{G}_\Phi,\qquad T_o-T_s=\hat{I}_T,\end{aligned}$$ where we have introduced new integrals $$\begin{gathered} \hat{I}_R=\fint_{R_s}^{R_o}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}},\qquad \hat{I}_\Phi=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}},\qquad \hat{I}_T=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}},\\ \hat{G}_\theta=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta_n(\theta)}},\qquad \hat{G}_\Phi=\fint_{\theta_s}^{\theta_o}\frac{L\Lambda^{-2}(\theta)}{\pm_\theta\sqrt{\Theta_n(\theta)}}{\mathop{}\!\mathrm{d}}\theta.\end{gathered}$$ ### Qualitative description of geodesic motion {#subsec:PoincareQualitative} As always, before evaluating the geodesic integrals, it is useful to determine the possible classes of geodesic motion. The angular motion is qualitatively the same as in Kerr and global NHEK, with the polar integral given by $$\begin{aligned} \label{eq:AngularIntegralsNHEK} \hat{G}_\theta=\tilde{G}_\theta =G_\theta,\end{aligned}$$ under the identifications . In particular, the angular motion is non-vortical. We now turn to the classification of the radial motion. The analysis is similar to that for global NHEK, but is slightly complicated by the presence of the Poincaré horizon. The radial potential $\mathcal{R}_n(R)=E^2+2ELR-\mathcal{C}R^2$ is a quadratic polynomial in $R$ with manifestly positive discriminant $4E^2{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}>0$. As such, it always has two real roots $$\begin{aligned} R_\pm=\frac{EL}{\mathcal{C}}{{\mathopen{}\mathclose\bgroup\originalleft}(1\pm\sqrt{1+\frac{\mathcal{C}}{L^2}}{\aftergroup\egroup\originalright})}\in\mathbb{R}.\end{aligned}$$ Positivity of energy in the local frame of the particle, $P^T\geq0$, requires that $$\begin{aligned} \label{eq:PoincarePositivity} E+LR\geq0.\end{aligned}$$ In terms of the critical radius $$\begin{aligned} R_c=-\frac{E}{L},\end{aligned}$$ this requirement becomes $$\begin{aligned} R\gtrless R_c,\qquad L\gtrless0.\end{aligned}$$ When $EL>0$, $R_c$ lies behind the Poincaré horizon and is irrelevant for motion in the Poincaré patch. For allowed values of the geodesic parameters, the roots can only coincide with $R_c$ if $E=0$, in which case $R_c=0$. Since this does not affect motion in the Poincaré patch, for all practical purposes, the roots $R_\pm$ can never equal $R_c$. Similarly, while both roots $R_\pm$ are real for all values of the geodesic parameters, it is only the positive ones that affect motion in the Poincaré patch: a Poincaré observer will never know if the geodesic reaches a turning point behind the horizon. As in Sec. \[subsec:GlobalGeodesics\], the qualitative behavior of the radial motion is determined by the sign of $\mathcal{C}$: 1. corresponds to Type I motion.[^17] In this case, one root is positive while the other is negative: if $EL>0$, then $R_-<0<R_+$, whereas if $EL<0$, then $R_+<0<R_-<R_c$. The condition requires $E>0$ in both cases. In Poincaré NHEK, particles on these geodesics come out of the past horizon at $R=0$, travel to increasingly large radius until they encounter a turning point at a maximal allowed radius ($R_\pm$ according to whether $L\gtrless0$), and then fall back into the future horizon at $R=0$.[^18] 2. corresponds to Type II motion. In this case, the roots $R_\pm$ are both simultaneously positive or negative depending on the sign of $EL$. Motion with both $E,L<0$ is not allowed, because it violates condition . This leaves us with two subtypes of geodesics: - These geodesics have both $E,L>0$. In this case, the roots are both negative, $R_+<R_-<0$, and energy positivity imposes no constraint on the radius. These geodesics all have one endpoint on the NHEK boundary and another on the future or past horizon: they explore all radii $R\in[0,\infty)$ without encountering any turning point. Those with $P^R<0$ represent particles plunging into the black hole that eventually pass through the future horizon, while those with $P^R>0$ emerge from the past horizon and appear as particles ejected by the white hole. The latter are unlikely to be of astrophysical interest since the past horizon does not exist for black holes formed by collapse. - These geodesics have $EL<0$. In this case, the roots are both positive and $0<R_-<R_c<R_+$. Hence, the radial potential $\mathcal{R}_n(R)$ is nonnegative if $R\leq R_-$ or $R\geq R_+$. There are two further subcases to consider: - in which case condition requires that $R>R_c$, so that only the branch $R_+<R$ is allowed. These geodesics fall in from the boundary of NHEK, bounce off of the radial potential, and return to the boundary. In global NHEK, these geodesics correspond to Type II motion that begins and ends on the right boundary. - in which case condition requires that $R<R_c$, so that only the branch $R<R_-$ is allowed. These geodesics enter the Poincaré patch of NHEK through the past horizon, bounce off of the radial potential and fall back into the future horizon. From the perspective of a Poincaré observer, they are qualitatively similar to the Type I Poincaré geodesics. In global NHEK, these geodesics correspond to Type II motion that begins and ends on the left boundary, passing through a single Poincaré patch. 3. corresponds to Type III motion, which is a limit of Type II. The radial potential $\mathcal{R}_n(R)=2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}$ is linear with a zero at $R_0=-E/{{\mathopen{}\mathclose\bgroup\originalleft}(2L{\aftergroup\egroup\originalright})}$. Condition then requires $E>0$. When $L>0$ (Type IIIA), the only constraint on the radial motion is that $R>0$. When $L<0$ (Type IIIB), the radial motion is constrained to the range $0\leq R\leq R_0$. To summarize, the type of radial motion is picked out by the sign of $\mathcal{C}$ and $L$: Note that since global NHEK and the Poincaré patch share a spacelike surface that connects the two boundaries of the global strip, every Type I geodesic in global NHEK also appears as a Type I geodesic in Poincaré NHEK. However, neither global NHEK nor the Poincaré patch are globally hyperbolic, and this spacelike surface is not a Cauchy surface. As a result, not all Type II geodesics in global NHEK appear as Type II geodesics in Poincaré NHEK: many simply enter and exit the strip without penetrating the Poincaré patch. Indeed, even those that do appear in the patch are not necessarily complete geodesics: Type IIA geodesics look like plunges in the Poincaré patch because they cross the horizon and only return to the boundary farther up along the global strip, while Type IIB geodesics with $L<0$ are extendible through both the past and future horizons. Only the Type IIB geodesics with $L>0$ are complete. ### Geodesic integrals in Poincaré NHEK We now wish to compute the integrals $\hat{I}_R$, $\hat{I}_T$, $\hat{I}_\Phi$, $\hat{G}_\theta$, and $\hat{G}_\Phi$ that appear in the NHEK geodesic equation . As previously noted, the Poincaré NHEK angular integral $\hat{G}_\theta{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ takes the same form as the Kerr angular integral $G_\theta(Q,P,\ell)$ under the identifications . Similarly, $$\begin{aligned} \hat{G}_\Phi{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}=\frac{L}{4}{{\mathopen{}\mathclose\bgroup\originalleft}(4G_\phi-G_t-3G_\theta{\aftergroup\egroup\originalright})},\end{aligned}$$ under the same identifications. It remains to evaluate the radial integrals $\hat{I}_R$, $\hat{I}_T$, and $\hat{I}_\Phi$. It is useful to define signs $$\begin{aligned} \nu_s=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_s^R{\aftergroup\egroup\originalright})} =(-1)^w\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_o^R{\aftergroup\egroup\originalright})},\qquad \nu_o=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_o^R{\aftergroup\egroup\originalright})} =(-1)^w\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_s^R{\aftergroup\egroup\originalright})},\end{aligned}$$ where $P_s^R$ and $P_o^R$ denote the radial momentum evaluated at the endpoints $X_s^\mu$ and $X_o^\mu$ of the geodesic, respectively, and $w\in\mathbb{N}$ denotes the number of turning points in the radial motion. As we saw in the previous subsection, radial motion in the Poincaré patch has either $w=0$ or $w=1$ turning points. For those geodesics capable of encountering a turning point (those for which at least one of $R_{\pm}>0$ or $R_0>0$), it easily follows that the radial path integral unpacks as follows:[^19] $$\begin{aligned} \label{eq:RadialIntegralsNHEK} \text{Type I, IIB, IIIB:}\qquad \fint_{R_s}^{R_o}=-\nu_s\int_{R_\pm}^{R_s}+\nu_o\int_{R_\pm}^{R_o},\qquad \pm= \begin{cases} \operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L{\aftergroup\egroup\originalright})}&\quad\text{Type I, IIB},\\ 0&\quad\text{Type IIIB}. \end{cases}\end{aligned}$$ Since only the Type I, Type IIB, and Type IIIB geodesics encounter a (positive-radius) turning point, we need to specify a different reference radius in order to compute the radial integrals for Type IIA and Type IIIA motion: integrating to $R_-$ and $R_0$ in these cases produces real, finite expressions for $\hat{I}_R$, but requires one to integrate past the non-integrable singularity at $R=0$ when computing $\hat{I}_{\Phi}$ and $\hat{I}_T$. Since $\hat{I}_{\Phi}$ and $\hat{I}_T$ are convergent at large $R$, a convenient reference radius is instead $R\to\infty$. Therefore, the radial path integral for Type IIA and Type IIIA geodesics, which always have $\nu_s=\nu_o$, unpacks as follows: $$\begin{aligned} \label{eq:RadialIntegralsNHEK2} \text{Type IIA, IIIA:}\qquad \fint_{R_s}^{R_o}=-\nu_s\int_{\infty}^{R_s}+\nu_s\int_{\infty}^{R_o}.\end{aligned}$$ To recapitulate: for Type IIA and Type IIIA motion, $\hat{I}_{\Phi}$ and $\hat{I}_T$ converge at large $R$, but are singular as the limit of integration approaches the Poincaré horizon. For these integrals, one should use the representation . Conversely, $\hat{I}_R$ is divergent for large $R$, but converges near the horizon. For this integral, one should instead resort to the representation with $R_-$ for Type IIA and $R_0$ for Type IIIA. We can now explicitly evaluate the radial integrals. For our purposes, the relevant basis of integrals is given by $$\begin{aligned} \hat{\mathcal{I}}_1^\pm&\equiv\int_{R_\pm}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}},\\ \hat{\mathcal{I}}_2^\pm&\equiv\int_{R_\pm}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E}{R}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}},\\ \hat{\mathcal{I}}_3^\pm&\equiv\int_{R_\pm}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}},\end{aligned}$$ in terms of which \[eq:PoincareIntegrals\] $$\begin{aligned} \int_{R_\pm}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{R}_n(R)}}&=\hat{\mathcal{I}}_1^\pm,\\ \int_{R_\pm}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{R}_n(R)}}&=L\hat{\mathcal{I}}_1^\pm+\hat{\mathcal{I}}_2^\pm,\\ \int_{R_\pm}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{R}_n(R)}}&=\hat{\mathcal{I}}_3^\pm.\end{aligned}$$ For Type IIA and Type IIIA geodesics, the radial turning point $R_\pm$ in the limit of integration should be replaced by $R\to\infty$ for $\hat{\mathcal{I}}_2$ and $\hat{\mathcal{I}}_3$. We define the positive quantity $$\begin{aligned} z_i^\pm={{\mathopen{}\mathclose\bgroup\originalleft}|\frac{R_i-R_\pm}{R_i-R_\mp}{\aftergroup\egroup\originalright}|}^{1/2}.\end{aligned}$$ When $\mathcal{C}>0$, motion is allowed in the range $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,R_\pm{\aftergroup\egroup\originalright}]}$, with $R_\mp<0<R_\pm$ and $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L{\aftergroup\egroup\originalright})}$. Using the substitution $$\begin{aligned} R=\frac{R_\pm+R_\mp z^2}{1+z^2},\qquad z^2=\frac{R_\pm-R}{R-R_\mp}>0,\end{aligned}$$ together with the fact that $\mathcal{C}R_+R_-=-E^2$, one finds that \[eq:PoincareTypeI\] $$\begin{aligned} \hat{\mathcal{I}}_1^\pm&=-\frac{2}{\sqrt{\mathcal{C}}}\int_0^{z_i^\pm}\frac{{\mathop{}\!\mathrm{d}}z}{1+z^2} =-\frac{2}{\sqrt{\mathcal{C}}}\arctan{z_i^\pm},\\ \hat{\mathcal{I}}_2^\pm&=-\frac{2E}{\sqrt{\mathcal{C}}}\int_0^{z_i^\pm}\frac{{\mathop{}\!\mathrm{d}}z}{R_\pm+R_\mp z^2} =-2\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{-\frac{R_\mp}{R_\pm}}z_i^\pm{\aftergroup\egroup\originalright})},\\ \hat{\mathcal{I}}_3^\pm&={\mathopen{}\mathclose\bgroup\originalleft}.-\frac{\sqrt{\mathcal{R}_n(R)}}{ER}{\aftergroup\egroup\originalright}|_{R=R_\pm}^{R=R_i} =-\frac{\sqrt{\mathcal{R}_n(R_i)}}{ER_i}.\end{aligned}$$ When $\mathcal{C}<0$, we choose $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L{\aftergroup\egroup\originalright})}$ for Type IIB geodesics. The appropriate substitution is $$\begin{aligned} \label{eq:PoincareSubstitution} R=\frac{R_\pm-R_\mp z^2}{1-z^2},\qquad z^2=\frac{R-R_\pm}{R-R_\mp}\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,1{\aftergroup\egroup\originalright}]}.\end{aligned}$$ The integrals are then given by \[eq:PoincareTypeIIB\] $$\begin{aligned} \hat{\mathcal{I}}_1^\pm&=\pm\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{z_i^\pm}\frac{{\mathop{}\!\mathrm{d}}z}{1-z^2} =\pm\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{z_i^\pm},\\ \hat{\mathcal{I}}_2^\pm&=\pm \frac{2E}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{z_i^\pm}\frac{{\mathop{}\!\mathrm{d}}z}{R_\pm-R_\mp z^2} =-2\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt\frac{R_\mp}{R_\pm}z_i^\pm{\aftergroup\egroup\originalright})},\\ \hat{\mathcal{I}}_3^\pm&={\mathopen{}\mathclose\bgroup\originalleft}.-\frac{\sqrt{\mathcal{R}_n(R)}}{ER}{\aftergroup\egroup\originalright}|_{R=R_\pm}^{R=R_i} =-\frac{\sqrt{\mathcal{R}_n(R_i)}}{ER_i}.\end{aligned}$$ On the other hand, for the Type IIA geodesics, we have $$\begin{aligned} E>0,\qquad L>0,\qquad R_+<R_-<0.\end{aligned}$$ The appropriate substitution is the same as in Eq. , with the lower sign for $\hat{\mathcal{I}}_1^-$ and the upper sign for $\hat{\mathcal{I}}_2^\infty$ and $\hat{\mathcal{I}}_3^\infty$. The integrals are given by \[eq:PoincareTypeIIA\] $$\begin{aligned} \hat{\mathcal{I}}_1^-&=\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{z_i^-}\frac{{\mathop{}\!\mathrm{d}}z}{1-z^2} =\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{z_i^-},\\ \hat{\mathcal{I}}_2^\infty&=\frac{2E}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_1^{z_i^+}\frac{{\mathop{}\!\mathrm{d}}z}{R_+-R_-z^2} =2\operatorname{arctanh}\sqrt{\frac{R_-}{R_+}}-2\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-}{R_+}}z_i^+{\aftergroup\egroup\originalright})},\\ \hat{\mathcal{I}}_3^\infty&={\mathopen{}\mathclose\bgroup\originalleft}.-\frac{\sqrt{\mathcal{R}_n(R)}}{ER}{\aftergroup\egroup\originalright}|_{R\to\infty}^{R=R_i} =\frac{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{E}-\frac{\sqrt{\mathcal{R}_n(R_i)}}{ER_i}.\end{aligned}$$ Finally, when $\mathcal{C}=0$, the quantities $E$ and $L{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}$ are separately positive, and the appropriate substitution for evaluating the integrals is $z^2=2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}$. For Type IIIB motion with $L<0$, they take the simple form \[eq:PoincareTypeIIIB\] $$\begin{aligned} \hat{\mathcal{I}}_1^0&=\int_{R_0}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =\frac{\sqrt{\mathcal{R}_n(R_i)}}{EL},\\ \hat{\mathcal{I}}_2^0&=\int_{R_0}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E}{R}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =-2\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\sqrt{\mathcal{R}_n(R_i)}}{E}{\aftergroup\egroup\originalright})},\\ \hat{\mathcal{I}}_3^0&=\int_{R_0}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =-\frac{\sqrt{\mathcal{R}_n(R_i)}}{ER_i}.\end{aligned}$$ For Type IIIA motion with $L>0$, one instead obtains \[eq:PoincareTypeIIIA\] $$\begin{aligned} \hat{\mathcal{I}}_1^0&=\int_{R_0}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =\frac{\sqrt{\mathcal{R}_n(R_i)}}{EL},\\ \hat{\mathcal{I}}_2^\infty&=\int_\infty^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E}{R}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =-2\operatorname{arcsinh}{\sqrt{-\frac{R_0}{R_i}}},\\ \hat{\mathcal{I}}_3^\infty&=\int_{\infty}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =-\frac{\sqrt{\mathcal{R}_n(R_i)}}{ER_i}.\end{aligned}$$ Substituting Eqs.  and into Eq.  yields the geodesic integrals for Type I motion: [align]{} \_R&=[[\[\_s-\_o]{}\]]{},\ \_&=L\_R+2[[\[\_s-\_o]{}\]]{},\ \_T&=\_s-\_o. The geodesic integrals for Type IIA motion are likewise obtained by substituting Eqs.  and into Eqs.  and , resulting in: [align]{} \_R&=[[\[-]{}\]]{},\ \_&=L\_R+2\_s[[\[-]{}\]]{},\ \_T&=\_s[[\[-]{}\]]{}. The integrals for Type IIB motion are obtained by substituting Eqs.  and into Eq. , resulting in: [align]{} \_R&=[[\[\_s-\_o]{}\]]{},\ \_&=L\_R+2[[\[\_s-\_o]{}\]]{},\ \_T&=\_s-\_o. Substituting Eqs.  and into Eqs.  and , we obtain the Type IIIA integrals: [align]{} \_R&=\_s[[\[-]{}\]]{},\ \_&=L\_R-2\_s[[\[-]{}\]]{},\ \_T&=\_s[[\[-]{}\]]{}. Finally, substituting Eqs.  and into Eq. , we obtain the Type IIIB integrals: [align]{} \_R&=\_o-\_s ,\ \_&=L\_R+2[[\[\_s-\_o]{}\]]{},\ \_T&=\_s-\_o. ### Explicit solution of the geodesic equation As in global NHEK, there are several complementary approaches to the solution of the geodesic equation in Poincaré NHEK. First, using Eqs. , one can recast the $(T,R)$ motion as a first-order non-linear autonomous system: $$\begin{aligned} \label{eq:PoincareODE} \frac{dR}{dT}=\pm_R\frac{R^2}{E+LR}\sqrt{\mathcal{R}_n(R)}.\end{aligned}$$ This formulation emphasizes several important properties of the problem. As in global NHEK, local existence of solutions is guaranteed, but uniqueness is not: the derivative of the right-hand side of Eq.  diverges at zeroes of the radial potential, so multiple solutions (including the physically important constant-radius trajectories) are possible for a given set of initial conditions. Similarly, the nonlinearities of the equation allow for blowup in finite coordinate time, reflecting the fact that it is possible for particles to reach the boundary of NHEK in finite time. Finally, the autonomous character of the equation allows us to obtain the general solution from a particular solution through a time translation. The qualitative fixed-point analysis of this equation was performed in Sec. \[subsec:PoincareQualitative\]. The relative simplicity of the Poincaré geodesic integrals allows for an explicit parametrization of the trajectories in terms of Poincaré time. According to the previous subsection, the time lapse along any geodesic segment is $$\begin{aligned} \label{eq:T(R)NHEK} T_o(R_o)-T_s(R_s)=\nu_s\frac{\sqrt{\mathcal{R}_n(R_s)}}{ER_s}-\nu_o\frac{\sqrt{\mathcal{R}_n(R_o)}}{ER_o}.\end{aligned}$$ However, note that it is also possible to obtain this algebraic equation for the radial motion more directly using only the symmetries of the system. Indeed, the dilation charge $$\begin{aligned} H_0\equiv H_0^\mu p_\mu =-{{\mathopen{}\mathclose\bgroup\originalleft}(ET\pm_R\frac{\sqrt{\mathcal{R}_n(R)}}{R}{\aftergroup\egroup\originalright})}\end{aligned}$$ is conserved along the trajectory of each particle. Evaluating this constant at $(T_s,R_s)$ and $(T_o,R_o)$, and then equating the results, immediately yields Eq. . In order to invert Eq. , it is useful to introduce a closely related (but not conserved) quantity $$\begin{aligned} S_n(T_o)\equiv\mathcal{C}+{{\mathopen{}\mathclose\bgroup\originalleft}(H_0+ET_o{\aftergroup\egroup\originalright})}^2 =\mathcal{C}+{{\mathopen{}\mathclose\bgroup\originalleft}[\nu_s\frac{\sqrt{\mathcal{R}_n(R_s)}}{R_s}-E{{\mathopen{}\mathclose\bgroup\originalleft}(T_o-T_s{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}^2.\end{aligned}$$ Rewriting Eq.  as $$\begin{aligned} \nu_o\frac{\sqrt{\mathcal{R}_n(R_o)}}{R_o}=\nu_s\frac{\sqrt{\mathcal{R}_n(R_s)}}{R_s}-E{{\mathopen{}\mathclose\bgroup\originalleft}(T_o-T_s{\aftergroup\egroup\originalright})} =-H_0-ET_o,\end{aligned}$$ and then squaring both sides, one finds $$\begin{aligned} \frac{E^2+2ELR_o-\mathcal{C}R_o^2}{R_o^2}={{\mathopen{}\mathclose\bgroup\originalleft}(H_0+ET_o{\aftergroup\egroup\originalright})}^2 =S_n(T_o)-\mathcal{C}.\end{aligned}$$ The solution to this quadratic equation is given by $$\begin{aligned} \label{eq:R(T)NHEK} R_o(T_o)=\frac{EL}{S_n(T_o)}{{\mathopen{}\mathclose\bgroup\originalleft}(1\pm\sqrt{1+\frac{S_n(T_o)}{L^2}}{\aftergroup\egroup\originalright})}.\end{aligned}$$ The choice of sign in Eq.  is fixed by the type of geodesic under consideration. Before we can specify it, note that whenever the time is $$\begin{aligned} T_t=T_s+\nu_s\frac{\sqrt{\mathcal{R}_n(R_s)}}{ER_s},\end{aligned}$$ we have $S_n(T_t)=\mathcal{C}$, and therefore $R_o(T_t)=R_\pm$. Hence, $T_t$ is the turning time of the radial motion (provided there is one). Also, note that $S_n(T)$ is strictly positive for Type I geodesics, while for Type II geodesics it has a pair of zeroes $$\begin{aligned} T^\pm=T_t\pm\frac{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{E}.\end{aligned}$$ We can now explicitly describe the radial motion depending on the sign of $\mathcal{C}$: 1. $0<\mathcal{C}<\infty$ corresponds to Type I motion. In this case, $R(T)$ is given for all times $T\in\mathbb{R}$ by Eq.  with the choice of sign $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L{\aftergroup\egroup\originalright})}$. The particle reaches the horizon as $T\to\pm\infty$, and encounters a turning point in between at $(T,R)=(T_t,R_\pm)$. 2. $-L^2<\mathcal{C}<0$ corresponds to Type II motion, and there are two further subcases to consider: 1. $E,L>0$ corresponds to Type IIA motion. In this case, $R(T)$ is given by Eq.  with the upper choice of sign and is defined on the domain $T\gtrless T^\pm$ according to whether $p^R\lessgtr0$. The particle reaches the boundary at $T=T^\pm$ and the horizon as $T\to\pm\infty$, without ever encountering a turning point. 2. $EL<0$ corresponds to Type IIB motion. If $L>0$, then $R(T)$ is given by Eq.  with the upper choice of sign and is defined on the domain $T\in{{\mathopen{}\mathclose\bgroup\originalleft}[T^+,T^-{\aftergroup\egroup\originalright}]}$. The particle reaches the boundary at $T=T^\pm$, and encounters a turning point in between at $(T,R)=(T_t,R_+)$. If $L<0$, then $R(T)$ is given by Eq.  with the lower choice of sign and is defined for all times $T\in\mathbb{R}$. The particle reaches the horizon as $T\to\pm\infty$, and encounters a turning point in between at $(T,R)=(T_t,R_-)$. 3. $\mathcal{C}=0$ corresponds to Type III motion, a limit of Type II motion. If $L>0$ (Type IIIA), then $R(T)$ is given by Eq.  with the upper choice of sign and is defined on the domain $T\gtrless T_t$ according to whether $p^R\lessgtr0$. The particle reaches the boundary at $T=T_t$ and the horizon as $T\to\pm\infty$, without ever encountering a turning point. If $L<0$ (Type IIIB), then $R(T)$ is given by Eq.  with the lower choice of sign and is defined for all times $T\in\mathbb{R}$. The particle reaches the horizon as $T\to\pm\infty$, and encounters a turning point in between at $(T,R)=(T_t,R_0)$. Provided that one keeps track of radial turning points encountered along the way (if any), expression allows one to obtain $\hat{I}_R$ as a function of Poincaré time by plugging in $R_o(T_o)$. In turn, substitution of $\hat{I}_R(T_o)$ into the inversion formulas – derived in Sec. \[sec:Kerr\] allows one to obtain the polar angle of the particle as a function of time. For instance, in the generic case $P\neq0$ ($L\neq\pm2\mu M$), $$\begin{aligned} \cos{\theta_o(T_o)}=\sqrt{u_\pm}\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}(X^\pm(T_o){\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\qquad X^\pm(T_o)=F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}-\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\sqrt{-u_\mp P}\hat{I}_R(T_o),\end{aligned}$$ where $P$ and $u_\pm$ are to be evaluated according to the identifications , and $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P{\aftergroup\egroup\originalright})}$. Finally, given both $R_o(T_o)$ and $\theta_o(T_o)$, one can plug them into the expressions for $\hat{I}_\Phi$ and $\hat{G}_\Phi$ to obtain the azimuthal angle, $$\begin{aligned} \Phi_o(T_o)=\Phi_s-\hat{I}_\Phi(T_o)+\hat{G}_\Phi(T_o).\end{aligned}$$ This completes the explicit parameterization of Poincaré NHEK geodesics by the time elapsed along their trajectory. ### Isometry group orbits of geodesics We can now explain how NHEK geodesics transform into one another under the action of the isometry group $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$. First, note that none of the isometries act on the polar angle $\theta$. Therefore, two NHEK geodesics can only possibly be mapped into each other if they already share the same polar motion. This requires that they share the same $\mathsf{SL}(2,\mathbb{R})$ Casimir $\mathcal{C}$ and angular momentum $L$. If that is the case, then the geodesics could potentially be related by an isometry, and indeed, this must necessarily be the case. To see why, first focus on their motion in the $(T,R)$ plane. The path $R_o(T_o)$ traced by one geodesic can always be mapped into that traced by the other using an $\mathsf{SL}(2,\mathbb{R})$ transformation, whose explicit form is given at the end of App. \[app:AdS2\]. Once their radial motion $R_o(T_o)$ and (by assumption) their polar motion $\theta_o(T_o)$ both match, it is guaranteed that their azimuthal motion $\Phi_o(T_o)$ differs by at most a constant. This last discrepancy can be eliminated by acting with the $\mathsf{U}(1)$ isometry to produce a countering constant shift. This completes the argument. Geodesics in near-NHEK {#subsec:NearGeodesics} ---------------------- Having solved for the most general geodesic motion in global and Poincaré NHEK, we now turn to the description of geodesics in the smaller patch of near-NHEK. Of course, each class of geodesics in near-NHEK is in principle obtainable from some class of geodesics in global NHEK, its geodesic completion. However, since the Poincaré coordinates naturally arise in the near-horizon limit of the near-extreme Kerr black hole, the near-NHEK description is more useful for connecting the discussion to the far region, and many expressions take a simpler form in these coordinates. Recall from Sec. \[subsec:ScalingLimits\] that the near-NHEK line element is $$\begin{aligned} \label{eq:NearMetric} d\bar{s}^2=2M^2\Gamma{{\mathopen{}\mathclose\bgroup\originalleft}[-{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2-\kappa^2}+{\mathop{}\!\mathrm{d}}\theta^2+\Lambda^2{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\Phi+R{\mathop{}\!\mathrm{d}}T{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ and the generators of $\mathsf{SL}(2,\mathbb{R})\times\mathsf{U}(1)$ are $$\begin{aligned} H_0=\frac{1}{\kappa}{\mathop{}\!\partial}_T,\qquad H_\pm=\frac{e^{\mp\kappa T}}{\sqrt{R^2-\kappa^2}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{R}{\kappa}{\mathop{}\!\partial}_T\pm{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_R-\kappa{\mathop{}\!\partial}_\Phi{\aftergroup\egroup\originalright}]},\qquad W_0={\mathop{}\!\partial}_\Phi.\end{aligned}$$ The Casimir of $\mathsf{SL}(2,\mathbb{R})$ is the symmetric Killing tensor $$\begin{aligned} \mathcal{C}^{\mu\nu}=-H_0^\mu H_0^\nu+\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(H_+^\mu H_-^\nu+H_-^\mu H_+^\nu{\aftergroup\egroup\originalright})}.\end{aligned}$$ The motion of a free particle of mass $\mu$ and four-momentum $P^\mu$ is described by the geodesic equation $$\begin{aligned} P^\mu\bar{\nabla}_\mu P^\nu=0,\qquad \bar{g}^{\mu\nu}P_\mu P_\nu=-\mu^2.\end{aligned}$$ Geodesic motion in near-NHEK is completely characterized by three conserved quantities $$\begin{gathered} E=-\kappa H_0^\mu P_\mu =-P_T,\qquad L=W_0^\mu P_\mu =P_\Phi,\\ \mathcal{C}=\mathcal{C}^{\mu\nu}P_\mu P_\nu =P_\theta^2-{{\mathopen{}\mathclose\bgroup\originalleft}(1-\frac{1}{\Lambda^2}{\aftergroup\egroup\originalright})}P_\Phi^2+{{\mathopen{}\mathclose\bgroup\originalleft}(2M^2\Gamma{\aftergroup\egroup\originalright})}\mu^2 =\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(P_T-P_\Phi R{\aftergroup\egroup\originalright})}^2}{R^2-\kappa^2}-P_R^2{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}-P_\Phi^2,\end{gathered}$$ denoting the dilation weight, angular momentum parallel to the axis of symmetry, and $\mathsf{SL}(2,\mathbb{R})$ Casimir, respectively. By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,E,L,\mathcal{C}{\aftergroup\egroup\originalright})}$, we find that a particle following a geodesic in the near-NHEK geometry has an instantaneous four-momentum $P=P_\mu{\mathop{}\!\mathrm{d}}X^\mu$ of the form $$\begin{aligned} \label{eq:GeodesicNearNHEK} P(X^\mu,E,L,\mathcal{C})=-E{\mathop{}\!\mathrm{d}}T\pm_R\frac{\sqrt{\mathcal{R}_\kappa(R)}}{R^2-\kappa^2}{\mathop{}\!\mathrm{d}}R\pm_\theta\sqrt{\Theta_n(\theta)}{\mathop{}\!\mathrm{d}}\theta+L{\mathop{}\!\mathrm{d}}\Phi,\end{aligned}$$ where the two choices of sign $\pm_R$ and $\pm_\theta$ depend on the radial and polar directions of travel, respectively. Here, we used the same polar potential $\Theta_n(\theta)$ introduced in Eq. , and additionally defined a new radial potential $$\begin{aligned} \label{eq:RadialPotentialNearNHEK} \mathcal{R}_\kappa(R)={{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}^2-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ One can then raise $P_\mu$ to obtain the equations for the geodesic trajectory, \[eq:NearMomentum\] $$\begin{aligned} 2M^2\Gamma\frac{dR}{d\sigma}&=\pm_R\sqrt{\mathcal{R}_\kappa(R)},\\ 2M^2\Gamma\frac{d\theta}{d\sigma}&=\pm_\theta\sqrt{\Theta_n(\theta)},\\ 2M^2\Gamma\frac{d\Phi}{d\sigma}&=-\frac{R{{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}}{R^2-\kappa^2}+\frac{L}{\Lambda^2},\\ 2M^2\Gamma\frac{dT}{d\sigma}&=\frac{E+LR}{R^2-\kappa^2}.\end{aligned}$$ The parameter $\sigma$ is the affine parameter for massless particles ($\mu=0$), and is related to the proper time $\delta$ by $\delta=\mu\sigma$ for massive particles. Note that $$\begin{aligned} \lim_{\kappa\to0}\mathcal{R}_\kappa(R)=\mathcal{R}_n(R).\end{aligned}$$ In fact, by comparing with Eq. , we see that geodesics in near-NHEK with temperature $\kappa$ smoothly approach geodesics in NHEK as $\kappa\to0$, even though the mapping between the geometries becomes singular. Following the same procedure as always, we find from Eqs.  that a geodesic labeled by ${{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,E,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ connects spacetime points $X_s^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(T_s,R_s,\theta_s,\Phi_s{\aftergroup\egroup\originalright})}$ and $X_o^\mu={{\mathopen{}\mathclose\bgroup\originalleft}(T_o,R_o,\theta_o,\Phi_o{\aftergroup\egroup\originalright})}$ in near-NHEK if $$\begin{aligned} &\fint_{R_s}^{R_o}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}}=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta_n(\theta)}},\\ \Phi_o-\Phi_s&=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(-\frac{R{{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}}+\fint_{\theta_s}^{\theta_o}\frac{L\Lambda^{-2}(\theta)}{\pm_\theta\sqrt{\Theta_n(\theta)}}{\mathop{}\!\mathrm{d}}\theta,\\ T_o-T_s&=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}}.\end{aligned}$$ We may rewrite these conditions as $$\begin{aligned} \label{eq:NearGeodesics} \bar{I}_R=\bar{G}_\theta,\qquad \Phi_o-\Phi_s=-\bar{I}_\Phi+\bar{G}_\Phi,\qquad T_o-T_s=\bar{I}_T,\end{aligned}$$ where we have introduced the integrals $$\begin{gathered} \bar{I}_R=\fint_{R_s}^{R_o}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}},\qquad \bar{I}_\Phi=\fint_{R_s}^{R_o}R{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}},\qquad \bar{I}_T=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}},\\ \bar{G}_\theta=\fint_{\theta_s}^{\theta_o}\frac{{\mathop{}\!\mathrm{d}}\theta}{\pm_\theta\sqrt{\Theta_n(\theta)}},\qquad \bar{G}_\Phi=\fint_{\theta_s}^{\theta_o}\frac{L\Lambda^{-2}(\theta)}{\pm_\theta\sqrt{\Theta_n(\theta)}}{\mathop{}\!\mathrm{d}}\theta.\end{gathered}$$ ### Qualitative description of geodesic motion {#subsec:NearQualitative} The qualitative behavior of the geodesics in near-NHEK closely resembles that of geodesics in the Poincaré patch, with small adjustments due to the nonzero temperature $\kappa>0$. The angular motion is the same as in global and Poincaré NHEK, with the angular integrals given by $$\begin{aligned} \bar{G}_\theta=\tilde{G}_\theta =G_\theta,\qquad \bar{G}_\Phi=\tilde{G}_\varphi =\frac{L}{4}{{\mathopen{}\mathclose\bgroup\originalleft}(4G_\phi-G_t-3G_\theta{\aftergroup\egroup\originalright})},\end{aligned}$$ under the identifications . The angular motion is non-vortical. Properties of the radial motion are determined by the properties of the zeroes of the radial potential $\mathcal{R}_\kappa(R)=E^2+{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}\kappa^2+2ELR-\mathcal{C}R^2$, which is a quadratic polynomial in $R$ with discriminant $4{{\mathopen{}\mathclose\bgroup\originalleft}(E^2+\mathcal{C}\kappa^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}$. Since $\mathcal{C}+L^2>0$, its two roots $$\begin{aligned} R_\pm=\frac{EL}{\mathcal{C}}{{\mathopen{}\mathclose\bgroup\originalleft}[1\pm\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{\mathcal{C}}{L^2}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{\mathcal{C}\kappa^2}{E^2}{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}]}\end{aligned}$$ are real whenever $\mathcal{C}>-{{\mathopen{}\mathclose\bgroup\originalleft}(E/\kappa{\aftergroup\egroup\originalright})}^2$. According to Eq. , positivity of near-NHEK energy in a local frame of the particle, $P^T\geq0$, requires that $$\begin{aligned} \label{eq:NearPositivity} E+LR>0.\end{aligned}$$ Note that although the form of the radial potential changes, the Poincaré NHEK constraint arising from positivity of energy in a local frame remains the same. In terms of the critical radius $$\begin{aligned} R_c=-\frac{E}{L},\end{aligned}$$ positivity of energy requires $$\begin{aligned} R\gtrless R_c,\qquad L\gtrless0.\end{aligned}$$ In near-NHEK, it is possible for the roots $R_\pm$ to coincide with the critical radius $R_c$ if $$\begin{aligned} \label{eq:specialcond} L^2={{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E}{\kappa}{\aftergroup\egroup\originalright})}^2.\end{aligned}$$ This set of parameters corresponds to $R_c^2=\kappa^2$. The remainder of the analysis is more intricate in the presence of a nonzero temperature $\kappa>0$ because, unlike NHEK, near-NHEK does not contain a complete spacelike surface connecting the two boundaries of the global strip. This means that, in contrast with the Poincaré patch, there exist Type I geodesics in global NHEK which never enter the near-NHEK patch. Similar issues arise in the analysis of the Type II motion, and extra care is needed in delineating all of the separate cases. Despite these complications, we still find it useful to describe the qualitative behavior of the radial motion in terms of the sign of $\mathcal{C}$, as in Sec. \[subsec:GlobalGeodesics\]: 1. still corresponds to Type I motion. In this case, one root is positive while the other is negative. The energy positivity condition rules out motion with both $E,L<0$. On the other hand: - If both $E,L>0$, then one has both $R_-<0<R_+$ and $R_c<0<\kappa<R_+$,[^20] so that geodesic motion is allowed in the range $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_+{\aftergroup\egroup\originalright}]}$. These geodesics correspond to Type I trajectories in global NHEK that enter the near-NHEK patch through the past horizon at $R=\kappa$ and continue on to a maximum radius before plunging back into the future horizon at $R=\kappa$. - If $EL<0$, then one has $R_+<0<R_-$ and motion is tentatively allowed in the range $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_-{\aftergroup\egroup\originalright}]}$, provided that $\kappa<R_-<R_c$ when $L<0$ and $R_c<\kappa<R_-$ when $L>0$. The condition for $R_-=\kappa$ is the same as the condition for $R_-=R_c$, namely $L=-E/\kappa$. If $L<0$, then one has $\kappa<R_-<R_c$ whenever $-E/\kappa<L<0$, whereas if $L>0$, one has $R_c<\kappa<R_-$ whenever $0<-E/\kappa<L$. Geodesics outside of this range of parameters do not enter the near-NHEK patch. In conclusion, Type I motion occurs in the range $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_\pm{\aftergroup\egroup\originalright}]}$ with $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(EL{\aftergroup\egroup\originalright})}$. When $EL<0$, one additionally requires $-E/\kappa<L$. Particles on these geodesics correspond to Type I geodesics in global NHEK that come out of the past horizon at $R=\kappa$, travel to increasingly large radius until they encounter a turning point at a maximal allowed radius ($R_\pm$ according to whether $EL\gtrless0$), and then fall back into the future horizon at $R=\kappa$. 2. still corresponds to Type II motion. Condition again rules out the case $E,L<0$. However, there are additional cases to consider now, since the qualitative behavior depends on whether $-(E/\kappa)^2\gtrless\mathcal{C}$. - In this case, the radial potential has two real roots, and the analysis resembles that of the radial geodesics in the Poincaré patch, with two subtypes of geodesics: - These geodesics have both $E,L>0$. In this case, both roots are negative, $R_+<R_-<0<\kappa$, and $R_c<0$. Geodesic motion is permitted in the full range $R\in[\kappa,\infty)$ and the geodesics encounter no turning points. Those with $P^R>0$ emerge from the past horizon and reach the boundary of near-NHEK, while those with $P^R<0$ plunge in from the near-NHEK boundary into the future horizon. - These geodesics have $EL<0$. In this case, both roots are positive, $0<R_-<R_+$, and $0<R_c$. When $L>0$, geodesic motion is allowed in the region $R\in[R_+,\infty)$, with $0<R_c<\kappa<R_+$ if $-E/\kappa<L$ and $0<\kappa<R_c<R_+$ if $-E/\kappa>L$. When $L<0$, geodesic motion is allowed in the region $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_-{\aftergroup\egroup\originalright}]}$, provided that $\kappa<R_-<R_c$, which requires that $-E/\kappa<L$. The geodesics with $L>0$ fall in from the boundary of near-NHEK, bounce off of the radial potential, and return to the boundary. The geodesics with $L<0$ enter near-NHEK through the past horizon, bounce off of the radial potential, and fall back into the future horizon. - The roots of the radial potential are complex and the geodesic encounters no turning points. In this case, geodesics with $E,L>0$ necessarily have $E/\kappa<L$ and are free to explore the entire near-NHEK patch $R\in[\kappa,\infty)$. These are also Type IIA geodesics. When $E<0$ but $L>0$, geodesic motion in the near-NHEK patch is only possible if $0<R_c<\kappa<R$. Therefore, geodesics with $0<-E/\kappa<L$ explore the range $R\in[\kappa,\infty)$. This is a new class of Type IIA geodesic with $EL<0$. Condition rules out the case $E>0$ with $L<0$. 3. corresponds to Type III motion, a limit of Type II motion. The radial potential $\mathcal{R}_\kappa(R)=2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}$ is linear with a zero at $R_0=-{{\mathopen{}\mathclose\bgroup\originalleft}(E^2+L^2\kappa^2{\aftergroup\egroup\originalright})}/{{\mathopen{}\mathclose\bgroup\originalleft}(2EL{\aftergroup\egroup\originalright})}$. Condition eliminates the case $E,L<0$. When $E,L>0$ (Type IIIA), the only constraint on the radial motion is that $R>\kappa$. When $EL<0$ (Type IIIB), the radial motion is allowed within the range $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_0{\aftergroup\egroup\originalright}]}$, provided that $R_c<\kappa$ when $L>0$ or that $R_c>R_0$ when $L<0$. For both signs of $L$, this happens if and only if $-E/\kappa<L$. In summary, the types of radial motion in near-NHEK are: ### Geodesic integrals in near-NHEK We now wish to compute the integrals $\bar{I}_R$, $\bar{I}_T$, $\bar{I}_\Phi$, $\bar{G}_\theta$, and $\bar{G}_\Phi$ that appear in the near-NHEK geodesic equation . The near-NHEK angular integral $\bar{G}_\theta{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}$ matches the Kerr angular integral $G_\theta(Q,P,\ell)$ under the identifications . With the same identifications, $$\begin{aligned} \bar{G}_\Phi{{\mathopen{}\mathclose\bgroup\originalleft}(\mu^2,L,\mathcal{C}{\aftergroup\egroup\originalright})}=\frac{L}{4}{{\mathopen{}\mathclose\bgroup\originalleft}(4G_\phi-G_t-3G_\theta{\aftergroup\egroup\originalright})}.\end{aligned}$$ In order to evaluate the radial integrals $\bar{I}_R$, $\bar{I}_T$, and $\bar{I}_\Phi$, we define the signs $$\begin{aligned} \nu_s=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_s^R{\aftergroup\egroup\originalright})} =(-1)^w\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_o^R{\aftergroup\egroup\originalright})},\qquad \nu_o=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_o^R{\aftergroup\egroup\originalright})} =(-1)^w\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P_s^R{\aftergroup\egroup\originalright})},\end{aligned}$$ where $P_s^R$ and $P_o^R$ denote the radial momentum evaluated at the endpoints $X_s^\mu$ and $X_o^\mu$ of the geodesic, respectively, and $w\in\mathbb{N}$ denotes the number of turning points in the radial motion. Radial motion in near-NHEK has either $w=0$ or $w=1$ turning points. Therefore, for geodesics that can encounter a turning point (those for which at least one of $R_\pm$ or $R_0$ lies outside the horizon at $R=\kappa$), the radial path integral unpacks as follows: $$\begin{aligned} \label{eq:RadialIntegralsNearNHEK} \text{Type I, IIB, IIIB:}\qquad \fint_{R_s}^{R_o}=-\nu_s\int_{R_\pm}^{R_s}+\nu_o\int_{R_\pm}^{R_o},\qquad \pm=\begin{cases} \operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(EL{\aftergroup\egroup\originalright})}&\quad\text{Type I},\\ \operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L{\aftergroup\egroup\originalright})}&\quad\text{Type IIB},\\ 0&\quad\text{Type IIIB}. \end{cases}\end{aligned}$$ Since only the Type I, Type IIB and Type IIIB geodesics encounter a turning point outside the horizon, we need to specify an alternate reference radius for the Type IIA geodesics and Type IIIA geodesics. As in Poincaré NHEK, $\bar{I}_R$ is divergent at large $R$, but converges when integrating across the horizon at $R=\kappa$: to evaluate this integral one uses the representation , with the lower endpoint of integration given by $R_-$ for Type IIA and $R_0$ for Type IIIA. The integrals $\bar{I}_{\Phi}$ and $\bar{I}_T$ are convergent at large $R$, but have a non-integrable singularity at the horizon of near-NHEK: to evaluate these integrals, one should use the alternate representation $$\begin{aligned} \label{eq:RadialIntegralsNearNHEK2} \text{Type IIA, IIIA:}\qquad \fint_{R_s}^{R_o}=-\nu_s\int_{\infty}^{R_s}+\nu_s\int_{\infty}^{R_o}.\end{aligned}$$ We can now explicitly evaluate the radial integrals. For our purposes, the relevant basis of integrals is given by $$\begin{aligned} \bar{\mathcal{I}}_1^\pm&\equiv\int_{R_\pm}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}},\\ \bar{\mathcal{I}}_2^\pm&\equiv\int_{R_\pm}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{L\kappa^2+ER}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}} =\int_{R_\pm}^{R_i}\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+L\kappa}{R-\kappa}+\frac{E-L\kappa}{R+\kappa}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}},\\ \bar{\mathcal{I}}_3^\pm&\equiv\int_{R_\pm}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}} =\int_{R_\pm}^{R_i}\frac{1}{2\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+L\kappa}{R-\kappa}-\frac{E-L\kappa}{R+\kappa}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-R{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_-{\aftergroup\egroup\originalright})}}},\end{aligned}$$ in terms of which \[eq:NearIntegrals\] $$\begin{aligned} \int^{R_i}_{R_\pm}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{R}_\kappa(R)}}&=\bar{\mathcal{I}}_1^\pm,\\ \int^{R_i}_{R_\pm}R{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{R}_\kappa(R)}}&=L\bar{\mathcal{I}}_1^\pm+\bar{\mathcal{I}}_2^\pm,\\ \int^{R_i}_{R_\pm}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{\mathcal{R}_\kappa(R)}}&=\bar{\mathcal{I}}_3^\pm.\end{aligned}$$ For Type IIA and Type IIIA geodesics, the radial turning point $R_\pm$ in the limit of integration should be replaced by $R\to\infty$ for $\bar{\mathcal{I}}_2$ and $\bar{\mathcal{I}}_3$. We define the positive quantity and signs $$\begin{aligned} w_i^\pm={{\mathopen{}\mathclose\bgroup\originalleft}|\frac{R_i-R_\pm}{R_i-R_\mp}{\aftergroup\egroup\originalright}|}^{1/2},\qquad \rho^\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L\pm\frac{E}{\kappa}{\aftergroup\egroup\originalright})}.\end{aligned}$$ When $\mathcal{C}>0$, we have the following set of inequalities for the allowed geodesic motion: $$\begin{aligned} R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_\pm{\aftergroup\egroup\originalright}]},\qquad R_\mp<0<R_\pm,\qquad R_\pm-\kappa\geq0,\qquad R_\mp+\kappa<0,\qquad L+\frac{E}{\kappa}>0,\qquad \pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(EL{\aftergroup\egroup\originalright})}.\end{aligned}$$ Using the substitution $$\begin{aligned} R=\frac{R_\pm+R_\mp w^2}{1+w^2},\qquad w^2=\frac{R_\pm-R}{R-R_\mp}>0,\end{aligned}$$ together with the fact that $$\begin{aligned} \mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(R_+\pm\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_-\pm\kappa{\aftergroup\egroup\originalright})}=-{{\mathopen{}\mathclose\bgroup\originalleft}(E\mp L\kappa{\aftergroup\egroup\originalright})}^2,\end{aligned}$$ one finds that \[eq:NearTypeI\] $$\begin{aligned} \bar{\mathcal{I}}_1^\pm&=-\frac{2}{\sqrt{\mathcal{C}}}\int_0^{w_i^\pm}\frac{{\mathop{}\!\mathrm{d}}w}{1+w^2} =-\frac{2}{\sqrt{\mathcal{C}}}\arctan{w_i^\pm},\\ \bar{\mathcal{I}}_2^\pm&=-\frac{\kappa}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}w^2}+ \int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp-\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=-\frac{\kappa}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{-{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{-\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\kappa-R_\mp{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{\kappa-R_\mp}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &=\rho^-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{-\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{\kappa-R_\mp}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})},\notag\\ \bar{\mathcal{I}}_3^\pm&=-\frac{1}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}+{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp-\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=-\frac{1}{\sqrt{\mathcal{C}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{-{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{-\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\kappa-R_\mp{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{\kappa-R_\mp}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &=-\frac{1}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\rho^-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{-\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}+\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{\kappa-R_\mp}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}.\notag\end{aligned}$$ When $\mathcal{C}<0$, we have the following set of inequalities for the allowed Type IIB motion: $$\begin{aligned} R\gtrless R_\pm,\qquad 0<\kappa<R_-<R_+,\qquad \pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(L{\aftergroup\egroup\originalright})}.\end{aligned}$$ Using the substitution $$\begin{aligned} \label{eq:SubstitutionTypeII} R=\frac{R_\pm-R_\mp w^2}{1-w^2},\qquad w^2=\frac{R_\pm-R}{R_\mp-R}\in[0,1],\end{aligned}$$ one finds that \[eq:NearTypeIIB\] $$\begin{aligned} \bar{\mathcal{I}}_1^\pm&=\pm\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{w_i^\pm}\frac{{\mathop{}\!\mathrm{d}}w}{1-w^2} =\pm\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{w_i^\pm},\\ \bar{\mathcal{I}}_2^\pm&=\pm\frac{\kappa}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp-\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=\pm\frac{\kappa}{\sqrt{\mathcal{{{\mathopen{}\mathclose\bgroup\originalleft}|C{\aftergroup\egroup\originalright}|}}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp-\kappa{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp-\kappa}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &=\mp{{\mathopen{}\mathclose\bgroup\originalleft}[\rho^-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}-\rho^+\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp-\kappa}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]},\notag\\ \bar{\mathcal{I}}_3^\pm&=\pm\frac{1}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_0^{w_i^\pm}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp-\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=\pm\frac{1}{\sqrt{\mathcal{{{\mathopen{}\mathclose\bgroup\originalleft}|C{\aftergroup\egroup\originalright}|}}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm+\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp+\kappa{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_\pm-\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_\mp-\kappa{\aftergroup\egroup\originalright})}}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp-\kappa}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &=\pm\frac{1}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\rho^-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp+\kappa}{R_\pm+\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}+\rho^+\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_\mp-\kappa}{R_\pm-\kappa}}w_i^\pm{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}.\notag\end{aligned}$$ For the Type IIA geodesics, we have two further subcases, according to whether the roots $R_\pm$ are real or complex. In the former subcase, $$\begin{aligned} -(E/\kappa)^2<\mathcal{C}<0:\qquad E>0,\qquad L>0,\qquad R_+<R_-<0,\qquad R_\pm\pm\kappa<0.\end{aligned}$$ The appropriate substitution is given in Eq. , with the lower choice of sign for $\bar{\mathcal{I}}_1$ and the upper choice of sign for $\bar{\mathcal{I}}_2$ and $\bar{\mathcal{I}}_3$: \[eq:NearTypeIIARealRoots\] $$\begin{aligned} \bar{\mathcal{I}}_1^{-}&=\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{w_i^-}\frac{{\mathop{}\!\mathrm{d}}w}{1-w^2} =\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{w_i^-},\\ \bar{\mathcal{I}}_2^\infty&=\frac{\kappa}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\int_1^{w_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_++\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_-+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_1^{w_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_--\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=\frac{\kappa}{\sqrt{\mathcal{{{\mathopen{}\mathclose\bgroup\originalleft}|C{\aftergroup\egroup\originalright}|}}}}\Bigg\{-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_++\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_-+\kappa{\aftergroup\egroup\originalright})}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}\sqrt{\frac{R_-+\kappa}{R_++\kappa}}-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}w_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &\qquad\qquad\quad+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_--\kappa{\aftergroup\egroup\originalright})}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}\sqrt{\frac{R_--\kappa}{R_+-\kappa}}-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}w_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\Bigg\}\notag\\ &=\rho^-{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}w_i^+{\aftergroup\egroup\originalright})}-\operatorname{arctanh}\sqrt{\frac{R_-+\kappa}{R_++\kappa}}{\aftergroup\egroup\originalright}]}-{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}w_i^+{\aftergroup\egroup\originalright})}-\operatorname{arctanh}\sqrt{\frac{R_--\kappa}{R_+-\kappa}}{\aftergroup\egroup\originalright}]},\notag\\ \bar{\mathcal{I}}_3^\infty&=\frac{1}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\int_1^{w_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_++\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_-+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_1^{w_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_--\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=\frac{1}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\Bigg\{\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_++\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_-+\kappa{\aftergroup\egroup\originalright})}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}\sqrt{\frac{R_-+\kappa}{R_++\kappa}}-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}w_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &\qquad\qquad\quad+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_--\kappa{\aftergroup\egroup\originalright})}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}\sqrt{\frac{R_--\kappa}{R_+-\kappa}}-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}w_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\Bigg\}\notag\\ &=-\frac{\rho^-}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}w_i^+{\aftergroup\egroup\originalright})}-\operatorname{arctanh}\sqrt{\frac{R_-+\kappa}{R_++\kappa}}{\aftergroup\egroup\originalright}]}-\frac{1}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}w_i^+{\aftergroup\egroup\originalright})}-\operatorname{arctanh}\sqrt{\frac{R_--\kappa}{R_+-\kappa}}{\aftergroup\egroup\originalright}]}.\notag\end{aligned}$$ The second subcase is $$\begin{aligned} -L^2<\mathcal{C}<-(E/\kappa)^2<0:\qquad L>0,\qquad L\pm\frac{E}{\kappa}>0,\qquad R_+=\overline{R_-}.\end{aligned}$$ Note that since $R_-$ is complex, the integral $\bar{\mathcal{I}}_1^-$ is in general also complex. However, the imaginary part cancels out in the sum , since the full contour is deformable to the real axis. Similarly, since $R_+$ is complex for this case, the integrals $\bar{\mathcal{I}}_2$ and $\bar{\mathcal{I}}_3$ can be integrated from $R_+$ to $R_i$: one avoids the non-integrable singularity at $R=\kappa$ when $R_+$ moves into the complex plane. In terms of $$\begin{aligned} W_i^\pm=\sqrt{\frac{R_i-R_\pm}{R_i-R_\mp}},\qquad \xi^\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\kappa}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L\pm E/\kappa{\aftergroup\egroup\originalright})}}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+\mp\kappa{\aftergroup\egroup\originalright})}W_{\pm\kappa}^-}{\aftergroup\egroup\originalright})} =\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+\pm\kappa{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R_-\pm\kappa{\aftergroup\egroup\originalright})}}}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+\mp\kappa{\aftergroup\egroup\originalright})}W_{\pm\kappa}^-}{\aftergroup\egroup\originalright})},\end{aligned}$$ one finds \[eq:NearTypeIIAComplexRoots\] $$\begin{aligned} \bar{\mathcal{I}}_1^{-}&=\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\int_0^{W_i^-}\frac{{\mathop{}\!\mathrm{d}}w}{1-w^2} =\frac{2}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}\operatorname{arctanh}{W_i^-},\\ \bar{\mathcal{I}}_2^+&=\frac{\kappa}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\int_0^{W_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_++\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_-+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_0^{W_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_--\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=\frac{\kappa}{\sqrt{\mathcal{{{\mathopen{}\mathclose\bgroup\originalleft}|C{\aftergroup\egroup\originalright}|}}}}{{\mathopen{}\mathclose\bgroup\originalleft}[-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{(R_++\kappa)W^-_{-\kappa}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}W_i^+{\aftergroup\egroup\originalright})}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{(R_+-\kappa)W^-_\kappa}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}W_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &=-\xi^-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}W_i^+{\aftergroup\egroup\originalright})}+\xi^+\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}W_i^+{\aftergroup\egroup\originalright})},\notag\\ \bar{\mathcal{I}}_3^+&=\frac{1}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\int_0^{W_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_++\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_-+\kappa{\aftergroup\egroup\originalright})}w^2}+\int_0^{W_i^+}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}w}{{{\mathopen{}\mathclose\bgroup\originalleft}(R_+-\kappa{\aftergroup\egroup\originalright})}-{{\mathopen{}\mathclose\bgroup\originalleft}(R_--\kappa{\aftergroup\egroup\originalright})}w^2}{\aftergroup\egroup\originalright}]}\\ &=\frac{1}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{(R_++\kappa)W^-_{-\kappa}}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}W_i^+{\aftergroup\egroup\originalright})}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{(R_+-\kappa)W^-_\kappa}\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}W_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}\notag\\ &=\frac{1}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\xi^-\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_-+\kappa}{R_++\kappa}}W_i^+{\aftergroup\egroup\originalright})}+\xi^+\operatorname{arctanh}{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{\frac{R_--\kappa}{R_+-\kappa}}W_i^+{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}.\notag\end{aligned}$$ Finally, when $\mathcal{C}=0$, $$\begin{aligned} \mathcal{R}_\kappa(R)=2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}>0,\qquad R_0=-\frac{E^2+L^2\kappa^2}{2EL},\qquad \mathcal{R}_\kappa(\pm\kappa)=\kappa^2{{\mathopen{}\mathclose\bgroup\originalleft}(L\pm E/\kappa{\aftergroup\egroup\originalright})}^2,\qquad L+\frac{E}{\kappa}>0,\end{aligned}$$ and the integrals degenerate. We use the substitution $w^2=2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}$. For Type IIIB motion with $EL<0$ and $R\in{{\mathopen{}\mathclose\bgroup\originalleft}[\kappa,R_0{\aftergroup\egroup\originalright}]}$, the integrals evaluate to \[eq:NearTypeIIIB\] $$\begin{aligned} \bar{\mathcal{I}}_1^0&=\int_{R_0}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =\frac{\sqrt{\mathcal{R}_\kappa(R_i)}}{EL},\\ \bar{\mathcal{I}}_2^0&=\int_{R_0}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{L\kappa^2+ER}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =\kappa{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(-\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0-R_i}{R_0+\kappa}}}-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0-R_i}{R_0-\kappa}}}{\aftergroup\egroup\originalright}]}\notag\\ &=\rho_-\operatorname{arctanh}\sqrt{\frac{R_0-R_i}{R_0+\kappa}}-\operatorname{arctanh}\sqrt{\frac{R_0-R_i}{R_0-\kappa}},\\ \bar{\mathcal{I}}_3^0&=\int_{R_0}^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =-{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(-\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0-R_i}{R_0+\kappa}}}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0-R_i}{R_0-\kappa}}}{\aftergroup\egroup\originalright}]}\notag\\ &=-\frac{1}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\rho_-\operatorname{arctanh}\sqrt{\frac{R_0-R_i}{R_0+\kappa}}+\operatorname{arctanh}\sqrt{\frac{R_0-R_i}{R_0-\kappa}}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ For Type IIIA motion with $EL>0$ and $R\in[\kappa,\infty)$, one has the additional relations $$\begin{aligned} R_0^2-\kappa^2={{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E^2-L^2\kappa^2}{2EL}{\aftergroup\egroup\originalright})}^2>0 \qquad\Longrightarrow\qquad R_0+\kappa<0.\end{aligned}$$ Using the substitution $w^{-2}=2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}$, the integrals take the form \[eq:NearTypeIIIA\] $$\begin{aligned} \bar{\mathcal{I}}_1^0&=\int_{R_0}^{R_i}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =\frac{\sqrt{\mathcal{R}_\kappa(R_i)}}{EL},\\ \bar{\mathcal{I}}_2^\infty&=\int_\infty^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{L\kappa^2+ER}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =\kappa{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(-\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0+\kappa}{R_0-R_i}}}-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0-\kappa}{R_0-R_i}}}{\aftergroup\egroup\originalright}]}\notag\\ &=\rho_-\operatorname{arctanh}\sqrt{\frac{R_0+\kappa}{R_0-R_i}}-\operatorname{arctanh}\sqrt{\frac{R_0-\kappa}{R_0-R_i}},\\ \bar{\mathcal{I}}_3^\infty&=\int_\infty^{R_i}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\sqrt{2EL{{\mathopen{}\mathclose\bgroup\originalleft}(R-R_0{\aftergroup\egroup\originalright})}}} =-{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L-E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(-\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0+\kappa}{R_0-R_i}}}+\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(L+E/\kappa{\aftergroup\egroup\originalright})}}{\sqrt{\mathcal{R}_\kappa(\kappa)}}\operatorname{arctanh}{\sqrt{\frac{R_0-\kappa}{R_0-R_i}}}{\aftergroup\egroup\originalright}]}\notag\\ &=-\frac{1}{\kappa}{{\mathopen{}\mathclose\bgroup\originalleft}[\rho_-\operatorname{arctanh}\sqrt{\frac{R_0+\kappa}{R_0-R_i}}+\operatorname{arctanh}\sqrt{\frac{R_0-\kappa}{R_0-R_i}}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Substituting Eqs.  and into Eq.  yields the geodesic integrals for Type I motion: [align]{} |[I]{}\_R&=[[\[\_s-\_o]{}\]]{},\ |[I]{}\_&=L|[I]{}\_R-\_s[[\[\^--]{}\]]{}\ &+\_o[[\[\^--]{}\]]{},\ |[I]{}\_T&=[[\[\^-+]{}\]]{}\ &-[[\[\^-+]{}\]]{}. Substituting Eqs.  and into Eqs.  and yields the geodesic integrals for Type IIA motion: [align]{} |[I]{}\_R&=[[\[-]{}\]]{},\ |[I]{}\_&=L|[I]{}\_R-\_s[[\[\^--]{}\]]{}\ &+\_s[[\[\^--]{}\]]{},\ |[I]{}\_T&=[[\[\^-+]{}\]]{}\ &-[[\[\^-+]{}\]]{}. [align]{} |[I]{}\_R&=[[\[-]{}\]]{},\ |[I]{}\_&=L|[I]{}\_R+\_s[[\[\^--\^+]{}\]]{}\ &-\_s[[\[\^--\^+]{}\]]{},\ |[I]{}\_T&=-[[\[\^-+\^+]{}\]]{}\ &+[[\[\^-+\^+]{}\]]{}. Substituting Eqs.  and into Eq.  yields the geodesic integrals for Type IIB motion: [align]{} |[I]{}\_R&=[[\[\_s-\_o]{}\]]{},\ |[I]{}\_&=L|[I]{}\_R\_s[[\[\^--\^+]{}\]]{}\ &\_o[[\[\^--\^+]{}\]]{},\ |[I]{}\_T&=[[\[\^-+\^+]{}\]]{}\ &[[\[\^-+\^+]{}\]]{}. Substituting Eqs.  and into Eq.  yields the geodesic integrals for Type IIIA motion: [align]{} |[I]{}\_R&=\_s[[\[-]{}\]]{},\ |[I]{}\_&=L|[I]{}\_R-\_s[[\[\_--]{}\]]{}\ &+\_s[[\[\_--]{}\]]{},\ |[I]{}\_T&=[[\[\_-+]{}\]]{}\ &-[[\[\_-+]{}\]]{}. Finally, substituting Eqs.  and into Eq.  yields the geodesic integrals for Type IIIB motion: [align]{} |[I]{}\_R&=-\_s+\_o,\ |[I]{}\_&=L|[I]{}\_R-\_s[[\[\_--]{}\]]{}\ &+\_o[[\[\_--]{}\]]{},\ |[I]{}\_T&=[[\[\_-+]{}\]]{}\ &-[[\[\_-+]{}\]]{}. ### Explicit solution of the geodesic equation We would now like to explicitly solve for the $(T,R)$ motion in near-NHEK. As always, the problem can be recast as a first-order non-linear autonomous system: $$\begin{aligned} \label{eq:NearODE} \frac{dR}{dT}=\pm_R\frac{R^2-\kappa^2}{E+LR}\sqrt{\mathcal{R}_\kappa(R)}.\end{aligned}$$ The physically relevant properties of the analogous ODEs in global and Poincaré NHEK manifest themselves in near-NHEK as well. The non-linearity of the system allows for blowup of solutions in finite coordinate time, allowing particles to reach the boundary of near-NHEK. Likewise, non-differentiability of the right-hand side of Eq.  at zeroes of the radial potential allows for multiple solutions with the same initial conditions: constant-radius trajectories also exist in near-NHEK. The qualitative fixed-point analysis of this equation was performed in Sec. \[subsec:NearQualitative\]. To proceed in solving this equation, we make use of the symmetries of near-NHEK. The quantities $H_+=H_+^\mu p_\mu$ and $H_-=H_-^\mu p_\mu$ are conserved along the particle trajectory. We can therefore evaluate the quantity $$\begin{aligned} \frac{H_+}{H_-}=e^{-2\kappa T}\frac{ER+L\kappa^2\mp_R\kappa\sqrt{\mathcal{R}_\kappa(R)}}{ER+L\kappa^2\pm_R\kappa\sqrt{\mathcal{R}_\kappa(R)}}\end{aligned}$$ at the separate points $(T_s,R_s)$ and $(T_o,R_o)$, and then set the results equal to each other to obtain $$\begin{aligned} e^{2\kappa{{\mathopen{}\mathclose\bgroup\originalleft}(T_o-T_s{\aftergroup\egroup\originalright})}}={{\mathopen{}\mathclose\bgroup\originalleft}(\frac{ER_o+L\kappa^2-\nu_o\kappa\sqrt{\mathcal{R}_\kappa(R_o)}}{ER_o+L\kappa^2+\nu_o\kappa\sqrt{\mathcal{R}_\kappa(R_o)}}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{ER_s+L\kappa^2+\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{ER_s+L\kappa^2-\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{\aftergroup\egroup\originalright})}.\end{aligned}$$ Taking the logarithm of this expression yields $$\begin{aligned} \label{eq:T(R)NearNHEK} T(R_o)-T(R_s)={\mathopen{}\mathclose\bgroup\originalleft}.\frac{1}{2\kappa}\log{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{ER+L\kappa^2\mp_R\kappa\sqrt{\mathcal{R}_\kappa(R)}}{ER+L\kappa^2\pm_R\kappa\sqrt{\mathcal{R}_\kappa(R)}}{\aftergroup\egroup\originalright}]}{\aftergroup\egroup\originalright}|_{R=R_s}^{R=R_o}.\end{aligned}$$ We would now like to invert this relation to obtain $R_o(T_o)$. In terms of the (non-conserved) quantity $$\begin{aligned} X(T_o)&\equiv e^{2\kappa T_o}{\mathopen{}\mathclose\bgroup\originalleft}.\frac{H_+}{H_-}{\aftergroup\egroup\originalright}|_{(T,R)=(T_s,R_s)} =e^{2\kappa{{\mathopen{}\mathclose\bgroup\originalleft}(T_o-T_s{\aftergroup\egroup\originalright})}}\frac{ER_s+L\kappa^2-\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{ER_s+L\kappa^2+\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}} =1+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\kappa{\aftergroup\egroup\originalright})}},\end{aligned}$$ Eq.  takes the form $$\begin{aligned} \frac{ER_o+L\kappa^2-\nu_o\kappa\sqrt{\mathcal{R}_\kappa(R_o)}}{ER_o+L\kappa^2+\nu_o\kappa\sqrt{\mathcal{R}_\kappa(R_o)}}=X(T_o).\end{aligned}$$ Isolating the terms with radicals, squaring, and solving the resulting quadratic equation, one finds $$\begin{aligned} \label{eq:R(T)NearNHEK} R_o(T_o)=\frac{EL}{S_\kappa(T_o)}{{\mathopen{}\mathclose\bgroup\originalleft}[1\pm\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{S_\kappa(T_o)}{L^2}{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1+\frac{S_\kappa(T_o)\kappa^2}{E^2}{\aftergroup\egroup\originalright})}}{\aftergroup\egroup\originalright}]},\end{aligned}$$ where $$\begin{aligned} S_\kappa(T_o)\equiv\frac{1}{X(T_o)}{{\mathopen{}\mathclose\bgroup\originalleft}[\mathcal{C}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1+X(T_o)}{2}{\aftergroup\egroup\originalright})}^2+\frac{E^2}{\kappa^2}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1-X(T_o)}{2}{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]} =S_n(T_o)+{\mathcal{O}{{\mathopen{}\mathclose\bgroup\originalleft}(\kappa^2{\aftergroup\egroup\originalright})}}.\end{aligned}$$ In the $\kappa\to0$ limit, this expression reduces to Eq. , as expected. The choice of sign in Eq.  is fixed by the type of geodesic under consideration. Before we can specify it, note that whenever the time is $$\begin{aligned} T_t=T_s-\frac{1}{2\kappa}\log{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{ER_s+L\kappa^2-\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{ER_s+L\kappa^2+\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{\aftergroup\egroup\originalright}]},\end{aligned}$$ we have $X(T_t)=1$, and therefore $S_\kappa(T_t)=\mathcal{C}$, which in turn implies $R_o(T_t)=R_\pm$. Hence, $T_t$ is the turning time of the radial motion (provided there is one). Also, note that $S_\kappa(T_t)$ is strictly positive for Type I geodesics, while for Type II geodesics, it has a pair of real zeroes $$\begin{aligned} T^\pm=T_s-\frac{1}{2\kappa}\log{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{ER_s+L\kappa^2-\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{ER_s+L\kappa^2+\nu_s\kappa\sqrt{\mathcal{R}_\kappa(R_s)}}{\aftergroup\egroup\originalright})}\frac{{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+E^2/\kappa^2{\aftergroup\egroup\originalright})}^{\pm1}}{{{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}|\mathcal{C}{\aftergroup\egroup\originalright}|}}+E/\kappa{\aftergroup\egroup\originalright})}^{\pm2}}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ We can now explicitly describe the radial motion depending on the sign of $\mathcal{C}$: 1. $0<\mathcal{C}<\infty$ corresponds to Type I motion. In this case, $R(T)$ is given for all times $T\in\mathbb{R}$ by Eq.  with the choice of sign $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(EL{\aftergroup\egroup\originalright})}$, provided that $R_\pm>\kappa$. The particle reaches the horizon at $T\to\pm\infty$, and encounters a turning point in between at $(T,R)=(T_t,R_\pm)$. 2. $-L^2<\mathcal{C}<0$ corresponds to Type II motion, and there are two further subcases to consider: 1. $E,L>0$, or $E<0<-E/\kappa<L$ with $\mathcal{C}<-{{\mathopen{}\mathclose\bgroup\originalleft}(E/\kappa{\aftergroup\egroup\originalright})}^2$, corresponds to Type IIA motion. If $E,L>0$, then $R(T)$ is given by Eq.  with the upper choice of sign. If $E<0<-E/\kappa<L$ with $\mathcal{C}<-{{\mathopen{}\mathclose\bgroup\originalleft}(E/\kappa{\aftergroup\egroup\originalright})}^2$, then $R(T)$ is given by Eq.  with the choice of sign flipping at $T={{\mathopen{}\mathclose\bgroup\originalleft}(T_++T_-{\aftergroup\egroup\originalright})}/2$. One must choose the upper/lower sign for $T<{{\mathopen{}\mathclose\bgroup\originalleft}(T_++T_-{\aftergroup\egroup\originalright})}/2$ according to whether $p^R\lessgtr0$, and the opposite sign for $T>{{\mathopen{}\mathclose\bgroup\originalleft}(T_++T_-{\aftergroup\egroup\originalright})}/2$. In both cases, $R(T)$ is defined on the domain $T\gtrless T^\pm$ according to whether $p^R\lessgtr0$. The particle reaches the boundary at $T=T^\pm$ and the horizon as $T\to\pm\infty$, without ever encountering a turning point. 2. $EL<0$ with $-{{\mathopen{}\mathclose\bgroup\originalleft}(E/\kappa{\aftergroup\egroup\originalright})}^2<\mathcal{C}$ corresponds to Type IIB motion. If $L>0$, then $R(T)$ is given by Eq.  with the upper choice of sign and is defined on the domain $T\in{{\mathopen{}\mathclose\bgroup\originalleft}[T^+,T^-{\aftergroup\egroup\originalright}]}$. The particle reaches the boundary at $T=T^\pm$, and encounters a turning point in between at $(T,R)=(T_t,R_+)$. If $L<0$, then $R(T)$ is given by Eq.  with the lower choice of sign and is defined for all times $T\in\mathbb{R}$. The particle reaches the horizon as $T\to\pm\infty$, and encounters a turning point in between at $(T,R)=(T_t,R_-)$. 3. $\mathcal{C}=0$ corresponds to Type III motion, a limit of Type II motion. If $E,L>0$ (Type IIIA), then $R(T)$ is given by Eq.  with the upper choice of sign and is defined on the domain $T\gtrless T_t$ according to whether $p^R\lessgtr0$. The particle reaches the boundary at $T=T_t$ and the horizon as $T\to\pm\infty$, without ever encountering a turning point. If $EL<0$ (Type IIIB), then $R(T)$ is given by Eq.  with the lower choice of sign and is defined for all times $T\in\mathbb{R}$. The particle reaches the horizon as $T\to\pm\infty$, and encounters a turning point in between at $(T,R)=(T_t,R_0)$. Provided that one keeps track of radial turning points encountered along the way (if any), the expression allows one to obtain $\bar{I}_R$ as a function of time by plugging in $R_o(T_o)$. In turn, substitution of $\bar{I}_R(T_o)$ into the inversion formulas – derived in Sec. \[sec:Kerr\] allows one to obtain the polar angle of the particle as a function of time. For instance, in the generic case $P\neq0$ ($L\neq\pm2\mu M$), $$\begin{aligned} \cos{\theta_o(T_o)}=\sqrt{u_\pm}\operatorname{sn}{{\mathopen{}\mathclose\bgroup\originalleft}(X^\pm(T_o){\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})},\qquad X^\pm(T_o)=F{{\mathopen{}\mathclose\bgroup\originalleft}(\arcsin{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\cos{\theta_s}}{\sqrt{u_\pm}}{\aftergroup\egroup\originalright})}{\mathopen{}\mathclose\bgroup\originalleft}|\frac{u_\pm}{u_\mp}{\aftergroup\egroup\originalright}.{\aftergroup\egroup\originalright})}-\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(p_s^\theta{\aftergroup\egroup\originalright})}\sqrt{-u_\mp P}\bar{I}_R(T_o),\end{aligned}$$ where $P$ and $u_\pm$ are to be evaluated according to the identifications , and $\pm=\operatorname{sign}{{\mathopen{}\mathclose\bgroup\originalleft}(P{\aftergroup\egroup\originalright})}$. Finally, given both $R_o(T_o)$ and $\theta_o(T_o)$, one can plug them into the expressions for $\bar{I}_\Phi$ and $\bar{G}_\Phi$ to obtain the azimuthal angle, $$\begin{aligned} \Phi_o(T_o)=\Phi_s-\bar{I}_\Phi(T_o)+\bar{G}_\Phi(T_o).\end{aligned}$$ This completes the explicit parameterization of near-NHEK geodesics by the time elapsed along their trajectory. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported in part by NSF grant 1205550 to Harvard University. DK gratefully acknowledges support from DOE grant DE-SC0009988. The authors thank Geoffrey Compère, Samuel Gralla, Shahar Hadar, Abhishek Pathak, Achilleas Porfyriadis, Andrew Strominger, and Peter Zimmerman for fruitful conversations and comments on the draft, and Yichen Shi for collaboration during early stages of the project.$\hfill{\tiny\checkmark}$ Elliptic and pseudo-elliptic integrals {#app:EllipticIntegrals} ====================================== In this appendix, we define our conventions for the elliptic integrals used throughout the text. We also present two integral identities that are needed in Sec. \[subsec:GlobalGeodesics\] to compute the radial geodesic integrals in global NHEK. ### Incomplete elliptic integrals The incomplete elliptic integral of the first kind $F$ is defined as $$\begin{aligned} F(x|k)=\int_0^x\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{1-k\sin^2{\theta}}} =\int_0^{\sin{x}}\frac{{\mathop{}\!\mathrm{d}}t}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(1-t^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1-kt^2{\aftergroup\egroup\originalright})}}}.\end{aligned}$$ The incomplete elliptic integral of the second kind $E$ is defined as $$\begin{aligned} E(x|k)=\int_0^x\sqrt{1-k\sin^2{\theta}}{\mathop{}\!\mathrm{d}}\theta =\int_0^{\sin{x}}\sqrt{\frac{1-kt^2}{1-t^2}}{\mathop{}\!\mathrm{d}}t.\end{aligned}$$ We use a $\prime$ to denote its derivative with respect to its second argument: $$\begin{aligned} E'(x|k)\equiv\frac{d}{dk}E(x|k) =-\frac{1}{2}\int_0^{\sin{x}}\frac{t^2{\mathop{}\!\mathrm{d}}t}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(1-t^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1-kt^2{\aftergroup\egroup\originalright})}}}.\end{aligned}$$ The incomplete elliptic integral of the third kind $\Pi$ is defined as $$\begin{aligned} \Pi(n;x|k)=\int_0^x\frac{1}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-n\sin^2{\theta}{\aftergroup\egroup\originalright})}}\frac{{\mathop{}\!\mathrm{d}}\theta}{\sqrt{1-k\sin^2{\theta}}} =\int_0^{\sin{x}}\frac{1}{1-nt^2}\frac{{\mathop{}\!\mathrm{d}}t}{\sqrt{{{\mathopen{}\mathclose\bgroup\originalleft}(1-t^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1-kt^2{\aftergroup\egroup\originalright})}}}.\end{aligned}$$ ### Complete elliptic integrals Elliptic integrals are said to be “complete” when the amplitude $x=\pi/2$. The complete elliptic integral of the first kind $K$ is denoted $$\begin{aligned} K(k)=F{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.\frac{\pi}{2}{\aftergroup\egroup\originalright}|k{\aftergroup\egroup\originalright})} =\frac{\pi}{2}{_2F_1}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{2},\frac{1}{2};1;k{\aftergroup\egroup\originalright})},\end{aligned}$$ where ${_2F_1}$ is Gauss’ hypergeometric function. The complete elliptic integral of the second kind $E$ is defined as $$\begin{aligned} E(k)=E{{\mathopen{}\mathclose\bgroup\originalleft}({\mathopen{}\mathclose\bgroup\originalleft}.\frac{\pi}{2}{\aftergroup\egroup\originalright}|k{\aftergroup\egroup\originalright})} =\frac{\pi}{2}{_2F_1}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{2},-\frac{1}{2};1;k{\aftergroup\egroup\originalright})}.\end{aligned}$$ The complete elliptic integral of the third kind $\Pi$ is defined as $$\begin{aligned} \Pi(n|k)=\Pi{{\mathopen{}\mathclose\bgroup\originalleft}(n;{\mathopen{}\mathclose\bgroup\originalleft}.\frac{\pi}{2}{\aftergroup\egroup\originalright}.|k{\aftergroup\egroup\originalright})} =\frac{\pi}{2}F_1{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{1}{2};\frac{1}{2},1;1;k,n{\aftergroup\egroup\originalright})},\end{aligned}$$ where $F_1$ denotes the first Appell series. ### Pseudo-elliptic integrals for global NHEK Let $x>0$, $q>0$, and $\alpha\in{{\mathopen{}\mathclose\bgroup\originalleft}(0,\pi{\aftergroup\egroup\originalright})}$. Define the manifestly real and positive quantities $$\begin{aligned} \label{eq:GlobalIntegrals} \tilde{I}_\pm(x,q,\alpha)=\frac{q^{\pm1}}{{{\mathopen{}\mathclose\bgroup\originalleft}(2q{\aftergroup\egroup\originalright})}^2\sin{\frac{\alpha}{2}}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{\pi}{2}+\arctan{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{x^2-q^2}{2qx\sin{\frac{\alpha}{2}}}{\aftergroup\egroup\originalright})}\mp\frac{1}{2}\tan{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\alpha}{2}{\aftergroup\egroup\originalright})}\log{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{x^2+2qx\cos{\frac{\alpha}{2}}+q^2}{x^2-2qx\cos{\frac{\alpha}{2}}+q^2}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ One can verify by direct calculation that[^21] $$\begin{aligned} {\mathop{}\!\partial}_x\tilde{I}_-(x,q,\alpha)=\frac{1}{q^4-2q^2x^2\cos{\alpha}+x^4},\qquad {\mathop{}\!\partial}_x\tilde{I}_+(x,q,\alpha)=\frac{x^2}{q^4-2q^2x^2\cos{\alpha}+x^4},\end{aligned}$$ and also that $$\begin{aligned} \qquad \lim_{x\to0^+}\tilde{I}_\pm(x,q,\alpha) =0.\end{aligned}$$ This implies that for all $x_i>0$, $$\begin{aligned} \label{eq:GlobalIntegralIdentities} \int_0^{x_i}\frac{{\mathop{}\!\mathrm{d}}x}{q^4-2q^2x^2\cos{\alpha}+x^4}=\tilde{I}_-(x_i,q,\alpha),\qquad \int_0^{x_i}\frac{x^2{\mathop{}\!\mathrm{d}}x}{q^4-2q^2x^2\cos{\alpha}+x^4}=\tilde{I}_+(x_i,q,\alpha).\end{aligned}$$ Charged geodesics in AdS2 {#app:AdS2} ========================= The Carter-Penrose diagram of NHEK in Fig. \[fig:PenroseDiagrams\] obviously resembles that of two-dimensional Anti de-Sitter space (AdS$_2$). This is simply explained by the observation that, for any fixed $\theta$, the NHEK geometry is a particular fibration of $\mathsf{U}(1)$ over an AdS$_2$ base. One would therefore expect the NHEK geodesics with trivial polar motion[^22] and fixed angular momentum $L$ to be related to some motion in AdS$_2$. Upon dimensional reduction, we find that this class of NHEK geodesics reduces to the Lorentz-force trajectories of an electrically charged particle moving in AdS$_2$ with a background electric field.[^23] We then solve the corresponding charged geodesic equation in global AdS$_2$, Poincaré AdS$_2$ and near-AdS$_2$, and relate the solutions to the NHEK geodesics of Sec. \[sec:GeodesicsInNHEK\]. Finally, we study the action of $\mathsf{SL}(2,\mathbb{R})$ on this space of trajectories, and show that it acts transitively, as expected. ### Dimensional reduction of NHEK to AdS2 The four-dimensional metric ansatz appropriate for the NHEK geometry is $$\begin{aligned} \label{eq:MetricAnsatz} ds_{(4)}^2=e^\varphi{{\mathopen{}\mathclose\bgroup\originalleft}[ds_{(2)}^2+{\mathop{}\!\mathrm{d}}\theta^2+e^{2\psi}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\mathrm{d}}\phi-A{\aftergroup\egroup\originalright})}^2{\aftergroup\egroup\originalright}]},\end{aligned}$$ where the metric $ds_{(2)}^2\equiv g_{ab}{\mathop{}\!\mathrm{d}}x^a{\mathop{}\!\mathrm{d}}x^b$ and one-form $A\equiv A_a{\mathop{}\!\mathrm{d}}x^a$ depend only on the coordinates $x^a$, while $\varphi(\theta)$ and $\psi(\theta)$ are scalar functions of $\theta$ only. For instance, the NHEK metric in Poincaré coordinates is of this form with $$\begin{aligned} \label{eq:MetricAnsatzNHEK} ds_{(2)}^2=-R^2{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2},\qquad A=- R{\mathop{}\!\mathrm{d}}T,\qquad e^\varphi=2M^2\Gamma(\theta),\qquad e^\psi=\Lambda(\theta),\end{aligned}$$ while for near-NHEK one has $$\begin{aligned} \label{eq:MetricAnsatznearNHEK} ds_{(2)}^2=-{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2-\kappa^2},\qquad A=- R{\mathop{}\!\mathrm{d}}T,\qquad e^\varphi=2M^2\Gamma(\theta),\qquad e^\psi=\Lambda(\theta).\end{aligned}$$ For global NHEK, the ansatz reads $$\begin{aligned} ds_{(2)}^2=-(1+y^2){\mathop{}\!\mathrm{d}}\tau^2+\frac{{\mathop{}\!\mathrm{d}}y^2}{1+y^2},\qquad A=-y{\mathop{}\!\mathrm{d}}\tau,\qquad e^\varphi=2M^2\Gamma(\theta),\qquad e^\psi=\Lambda(\theta).\end{aligned}$$ The Christoffel connection for a metric ansatz of this form is given by $$\begin{gathered} \Gamma^a_{bc}=\hat{\Gamma}^a_{bc}+\frac{1}{2}e^{2\psi}g^{ad}{{\mathopen{}\mathclose\bgroup\originalleft}[A_cF_{bd}+A_bF_{cd}{\aftergroup\egroup\originalright}]},\qquad \Gamma^a_{b\phi}=-\frac{1}{2}e^{2\psi}g^{ac}F_{bc},\qquad \Gamma^a_{b\theta}=\frac{1}{2}\dot{\phi}\delta^a_b,\\ \Gamma^\phi_{ab}=A^cg_{cd}\hat{\Gamma}^d_{ab}-\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}({\mathop{}\!\partial}_aA_b+{\mathop{}\!\partial}_bA_a{\aftergroup\egroup\originalright})}+\frac{1}{2}e^{2\psi}A^c{{\mathopen{}\mathclose\bgroup\originalleft}(A_aF_{bc}+A_bF_{ac}{\aftergroup\egroup\originalright})},\\ \Gamma^\phi_{a\theta}=-\dot{\psi}A_a,\qquad \Gamma^\phi_{a\phi}=-\frac{1}{2}e^{2\psi}A^bF_{ab},\qquad \Gamma^\phi_{\theta\phi}=\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(\dot{\varphi}+2\dot{\psi}{\aftergroup\egroup\originalright})},\\ \Gamma^\theta_{ab}=-\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}[\dot{\varphi}{{\mathopen{}\mathclose\bgroup\originalleft}(g_{ab}+e^{2\psi}A_aA_b{\aftergroup\egroup\originalright})}+2\dot{\psi}e^{2\psi}A_aA_b{\aftergroup\egroup\originalright}]},\qquad \Gamma^\theta_{a\phi}=\frac{1}{2}A_a{{\mathopen{}\mathclose\bgroup\originalleft}(\dot{\varphi}+2\dot{\psi}{\aftergroup\egroup\originalright})}e^{2\psi},\\ \Gamma^\theta_{\theta\theta}=\frac{1}{2}\dot{\varphi},\qquad \Gamma^\theta_{\phi\phi}=-\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(\dot{\varphi}+2\dot{\psi}{\aftergroup\egroup\originalright})}e^{2\psi}.\end{gathered}$$ Here and henceforth, $\hat{\Gamma}^a_{bc}$ and $D_a$ are the two-dimensional Christoffel connection and covariant derivative associated to $g_{ab}$, while $F_{ab}={\mathop{}\!\partial}_aA_b-{\mathop{}\!\partial}_bA_a$ is the curvature of $A_a$, and we use an overdot to denote a derivative with respect to $\theta$. By reducing the four-dimensional Einstein-Hilbert action $$\begin{aligned} S=\frac{1}{16\pi G_N}\int{\mathop{}\!\mathrm{d}}^4x\sqrt{-g_4}\mathcal{R}_{(4)}\end{aligned}$$ on the metric ansatz , one obtains an effective two-dimensional action for the metric $g_{ab}$ and gauge field $A_a$ with AdS$_2$ solutions supported by electric flux. In terms of these variables, the four-dimensional density is given by $$\begin{aligned} \sqrt{-g_4}\mathcal{R}_{(4)}=\sqrt{-g_2}e^{\varphi+\psi}{{\mathopen{}\mathclose\bgroup\originalleft}[\mathcal{R}_{(2)}-{{\mathopen{}\mathclose\bgroup\originalleft}(3\Ddot{\varphi}+\frac{3}{2}\dot{\varphi}^2+3\dot{\varphi}\dot{\psi}+2\dot{\psi}^2+2\Ddot{\psi}{\aftergroup\egroup\originalright})}-\frac{1}{4}e^{2\psi}F_{ab}F^{ab}{\aftergroup\egroup\originalright}]}.\end{aligned}$$ Substituting the NHEK expressions for $\varphi(\theta)$ and $\psi(\theta)$ and then performing the angular integrals yields the two-dimensional action $$\begin{aligned} S=\frac{M^2}{2G_N}\int{\mathop{}\!\mathrm{d}}^2x\sqrt{-g_2}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{R}_{(2)}+1-\frac{1}{2}F^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ This action gives rise to the Einstein-Maxwell equations of motion $$\begin{aligned} R_{ab}-\frac{1}{2}Rg_{ab}-\frac{1}{2}g_{ab}=T_{ab}^\mathrm{EM},\end{aligned}$$ where $T_{ab}^\mathrm{EM}$ is the electromagnetic stress-energy tensor $$\begin{aligned} T_{ab}^\mathrm{EM}=F_{ac}{F_b}^c-\frac{1}{4}F^2.\end{aligned}$$ In two dimensions, the Einstein tensor identically vanishes, $$\begin{aligned} G_{ab}\equiv R_{ab}-\frac{1}{2}Rg_{ab} =0,\end{aligned}$$ leaving the much simpler equation of motion $$\begin{aligned} -\frac{1}{2}g_{ab}=T_{ab}^\mathrm{EM}.\end{aligned}$$ The metric $g_{ab}$ and gauge potential $A_a$ given in Eq.  manifestly satisfy this equation. This solution corresponds to AdS$_2$ with a background electric field strength given by the $\mathsf{SL}(2,\mathbb{R})$-invariant volume form on AdS$_2$: $$\begin{aligned} \label{eq:ElectricField} F={\mathop{}\!\mathrm{d}}\tau\wedge{\mathop{}\!\mathrm{d}}y ={\mathop{}\!\mathrm{d}}T\wedge{\mathop{}\!\mathrm{d}}R.\end{aligned}$$ Note that while $F$ is $\mathsf{SL}(2,\mathbb{R})$-invariant and satisfies the two-dimensional Maxwell equations, $$\begin{aligned} {\mathop{}\!\mathrm{d}}F={\mathop{}\!\mathrm{d}}\star F =0,\end{aligned}$$ any choice of the gauge potential $A$ shifts by a gauge transformation under at least one isometry of AdS$_2$. This example exhibits the general mechanism that supports the AdS$_2$-like throats of *vacuum* near-extremal black hole solutions and reveals the obstruction to obtaining higher-dimensional analogues of AdS with such a setup. When reducing along the isometries of a spinning black hole, the frame-dragging term in the metric always appears as a gauge potential $A$ in the lower-dimensional spacetime, thereby providing a natural 2-form (namely, the electromagnetic field strength $F={\mathop{}\!\mathrm{d}}A$). The throat region outside of the black hole needs to inherit an effective cosmological constant from the four-dimensional solution, and a 2-form can only serve as a cosmological constant in two dimensions. ### Dimensional reduction of planar NHEK geodesics In this subsection, we will restrict our attention to NHEK geodesics with no polar motion. Since the reduction of NHEK along the angular directions yields AdS$_2$ with a background electric field, one expects to be able to recast the NHEK geodesic equation in terms of an effective Lorentz-force geodesic equation, with the Kaluza-Klein angular momentum $L$ in NHEK playing the role of the $\mathsf{U}(1)$ gauge charge in AdS$_2$. The NHEK momentum, denoted by $P_\mu$, is given by Eq.  in global NHEK, by Eq.  in the Poincaré patch, and by Eq.  in near-NHEK. In the two-dimensional description, the NHEK momentum plays the role of the generalized momentum $$\begin{aligned} P_a=p_a-LA_a,\qquad p^a\equiv g^{ab}{{\mathopen{}\mathclose\bgroup\originalleft}(P_a+LA_a{\aftergroup\egroup\originalright})},\end{aligned}$$ where $p_a$ is the two-dimensional momentum. The NHEK momentum satisfies the four-dimensional geodesic equation $$\begin{aligned} P^\mu\nabla_\mu P_\nu =0.\end{aligned}$$ The non-angular components of this equation take the form $$\begin{aligned} P^\mu\nabla_\mu P_a=e^{-\phi}{{\mathopen{}\mathclose\bgroup\originalleft}(p^bD_bp_a+LF_{ab}p^b{\aftergroup\egroup\originalright})} =0.\end{aligned}$$ This is the Lorentz-force geodesic equation for a charged particle propagating in AdS$_2$ with a background electric field. The four-dimensional mass, when expressed in terms of the metric ansatz , takes the form $$\begin{aligned} -\mu^2=g^{\mu\nu}P_\mu P_\mu =e^{-\phi}{{\mathopen{}\mathclose\bgroup\originalleft}(g^{ab}p_ap_b+e^{-2\psi}L^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ In terms of the effective two-dimensional mass $m^2=-g^{ab}p_ap_b$, one therefore finds $$\begin{aligned} m^2=e^\phi\mu^2+e^{-2\psi}L^2 =\mathcal{C}+L^2.\end{aligned}$$ The last equality follows from Eq.  with $P_\theta=0$, which holds for each choice of NHEK coordinate system. We have therefore reduced the problem of planar geodesic motion in NHEK to the motion of a particle with charge $q=L$ and mass $m=\sqrt{\mathcal{C}+L^2}$ in AdS$_2$ with a constant background electric field: $$\begin{aligned} \label{eq:LorentzForceLaw} p^bD_bp_a=Lp^bF_{ba},\qquad g^{ab}p_a p_b=-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ In the absence of the background field, particles are confined by the gravitational potential of AdS$_2$. The superradiant effects in the four-dimensional geometry are reflected in the electric field, which exerts a force on charged particles and can expel them from the AdS$_2$ throat. ### Charged geodesics on the global strip Global AdS$_2$ is covered by the coordinates $(\tau,y)$ with $-\infty<\tau,y<+\infty$. The line element is given by $$\begin{aligned} ds^2=-{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}\tau^2+\frac{{\mathop{}\!\mathrm{d}}y^2}{1+y^2}.\end{aligned}$$ This geometry has constant negative curvature $\mathcal{R}=-2$ and admits three Killing vector fields $$\begin{aligned} H_0=\frac{y\sin{\tau}}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\tau-\cos{\tau}\sqrt{1+y^2}{\mathop{}\!\partial}_y,\qquad H_\pm={{\mathopen{}\mathclose\bgroup\originalleft}(1\pm\frac{y\cos{\tau}}{\sqrt{1+y^2}}{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_\tau\pm\sin{\tau}\sqrt{1+y^2}{\mathop{}\!\partial}_y,\end{aligned}$$ which generate an $\mathsf{SL}(2,\mathbb{R})$ isometry group, $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}[H_0,H_\pm{\aftergroup\egroup\originalright}]}=\mp H_\pm,\qquad {{\mathopen{}\mathclose\bgroup\originalleft}[H_+,H_-{\aftergroup\egroup\originalright}]}=2H_0.\end{aligned}$$ In global coordinates, it is convenient to complexify this algebra by introducing new (complex) generators $$\begin{aligned} L_0=i{\mathop{}\!\partial}_\tau,\qquad L_\pm=e^{\pm i\tau}{{\mathopen{}\mathclose\bgroup\originalleft}[\pm\frac{y}{\sqrt{1+y^2}}{\mathop{}\!\partial}_\tau-i\sqrt{1+y^2}{\mathop{}\!\partial}_y{\aftergroup\egroup\originalright}]},\end{aligned}$$ which obey the same commutation relations: $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}[L_0,L_\pm{\aftergroup\egroup\originalright}]}=\mp L_\pm,\qquad {{\mathopen{}\mathclose\bgroup\originalleft}[L_+,L_-{\aftergroup\egroup\originalright}]}=2L_0.\end{aligned}$$ The relation between these two sets of generators is the same as in Eq. . The metric is the Casimir of $\mathsf{SL}(2,\mathbb{R})$: $$\begin{aligned} \label{eq:Casimir} g^{ab}=H_0^aH_0^b-\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(H_+^aH_-^b+H_-^aH_+^b{\aftergroup\egroup\originalright})}.\end{aligned}$$ In the global strip, the background gauge potential $$\begin{aligned} A=-y{\mathop{}\!\mathrm{d}}\tau\end{aligned}$$ gives rise to the symmetric field strength and preserves global-time-translations: $$\begin{aligned} \label{eq:GlobalInvariance} {\mathcal{L}}_{L_0}A=0.\end{aligned}$$ To solve Eq.  in this background, it is convenient to introduce the generalized momentum (NHEK momentum) $$\begin{aligned} P_a=p_a-LA_a.\end{aligned}$$ By virtue of Eq. , the global energy $$\begin{aligned} \triangle=iL_0^a P_a =-P_\tau =-p_\tau-Ly\end{aligned}$$ is conserved along the trajectory. By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(\triangle,\mathcal{C}+L^2{\aftergroup\egroup\originalright})}$, we find that a charged particle following a Lorentz-force trajectory in global AdS$_2$ with radial electric field has an instantaneous momentum $p=p_a{\mathop{}\!\mathrm{d}}x^a$ of the form $$\begin{aligned} p(x^a,\triangle,L,\mathcal{C})=-{{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+Ly{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}\tau\pm_y\frac{\sqrt{\mathcal{Y}(y)}}{1+y^2}{\mathop{}\!\mathrm{d}}y,\end{aligned}$$ where the choice of sign $\pm_y$ depends on the radial direction of travel, and we recovered the radial potential , $$\begin{aligned} \mathcal{Y}(y)={{\mathopen{}\mathclose\bgroup\originalleft}(\triangle+Ly{\aftergroup\egroup\originalright})}^2-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(1+y^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ One can then raise $p_a$ to obtain the equations for the trajectory, $$\begin{aligned} \frac{d\tau}{d\sigma}=\frac{\triangle+Ly}{1+y^2},\qquad \frac{dy}{d\sigma}=\pm_y\sqrt{\mathcal{Y}(y)}.\end{aligned}$$ Hence, a charged geodesic with global energy $\triangle$ connects spacetime points $X_s^a={{\mathopen{}\mathclose\bgroup\originalleft}(\tau_s,y_s{\aftergroup\egroup\originalright})}$ and $X_o^a={{\mathopen{}\mathclose\bgroup\originalleft}(\tau_o,y_o{\aftergroup\egroup\originalright})}$ if $$\begin{aligned} \tau_o-\tau_s&=\fint_{y_s}^{y_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{\triangle+Ly}{1+y^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}y}{\pm_y\sqrt{\mathcal{Y}(y)}}.\end{aligned}$$ This equation is solved in Sec. \[subsec:GlobalGeodesics\]. ### Charged geodesics on the Poincaré patch of AdS$_2$ The Poincaré patch of AdS$_2$ is covered by coordinates $(T,R)$ obtained from the global coordinates $(\tau,y)$ using the coordinate transformation , which results in the line element $$\begin{aligned} ds^2=-R^2{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2}.\end{aligned}$$ The generators of $\mathsf{SL}(2,\mathbb{R})$ are given by $$\begin{aligned} H_0=T{\mathop{}\!\partial}_T-R{\mathop{}\!\partial}_R,\qquad H_+={\mathop{}\!\partial}_T,\qquad H_-={{\mathopen{}\mathclose\bgroup\originalleft}(T^2+\frac{1}{R^2}{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_T-2TR{\mathop{}\!\partial}_R,\end{aligned}$$ and its Casimir reproduces the metric as in Eq. . In the Poincaré patch of AdS$_2$, the background gauge potential $$\begin{aligned} A=-R{\mathop{}\!\mathrm{d}}T\end{aligned}$$ gives rise to the symmetric field strength and preserves both time-translations and dilations (though not special conformal transformations): $$\begin{aligned} \label{eq:PoincareInvariance} {\mathcal{L}}_{H_+}A={\mathcal{L}}_{H_0}A=0.\end{aligned}$$ To solve Eq.  in this background, it is convenient to introduce the generalized momentum (NHEK momentum) $$\begin{aligned} P_a=p_a-LA_a.\end{aligned}$$ By virtue of Eq. , the Poincaré energy $$\begin{aligned} E=-H_+^a P_a =-P_T =-p_T-LR\end{aligned}$$ is conserved along the trajectory. By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(E,\mathcal{C}+L^2{\aftergroup\egroup\originalright})}$, we find that a charged particle following a Lorentz-force trajectory in a Poincaré patch of AdS$_2$ with radial electric field has an instantaneous momentum $p=p_a{\mathop{}\!\mathrm{d}}x^a$ of the form $$\begin{aligned} p(x^a,E,L,\mathcal{C})=-{{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}T\pm_R\frac{\sqrt{\mathcal{R}_n(R)}}{R^2}{\mathop{}\!\mathrm{d}}R,\end{aligned}$$ where the choice of sign $\pm_R$ depends on the radial direction of travel, and we recovered the radial potential , $$\begin{aligned} \mathcal{R}_n(R)={{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}^2-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}R^2.\end{aligned}$$ One can then raise $p_a$ to obtain the equations for the trajectory, $$\begin{aligned} \frac{dT}{d\sigma}=\frac{E+LR}{R^2},\qquad \frac{dR}{d\sigma}=\pm_R\sqrt{\mathcal{R}_n(R)}.\end{aligned}$$ Hence, a charged geodesic with Poincaré energy $E$ connects spacetime points $X_s^a={{\mathopen{}\mathclose\bgroup\originalleft}(T_s,R_s{\aftergroup\egroup\originalright})}$ and $X_o^a={{\mathopen{}\mathclose\bgroup\originalleft}(T_o,R_o{\aftergroup\egroup\originalright})}$ if $$\begin{aligned} T_o-T_s=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_n(R)}}.\end{aligned}$$ This equation is solved in Sec. \[subsec:PoincareGeodesics\]. ### Charged geodesics on near-AdS2 The coordinate transformation maps Poincaré AdS$_2$ to a smaller patch with line element $$\begin{aligned} ds^2=-{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}T^2+\frac{{\mathop{}\!\mathrm{d}}R^2}{R^2-\kappa^2}.\end{aligned}$$ As in Sec. \[subsec:ConformalSymmetryInTheSky\], this transformation implements a boundary time reparameterization $T\to e^{\kappa T}$ that introduces a small temperature $\kappa$. This smaller patch can thus be interpreted as a black hole in AdS$_2$ (see, $e.g.$, Ref. [@Maldacena2016b]). The generators of $\mathsf{SL}(2,\mathbb{R})$ are given by $$\begin{aligned} H_0=\frac{1}{\kappa}{\mathop{}\!\partial}_T,\qquad H_\pm=\frac{e^{\mp\kappa T}}{\sqrt{R^2-\kappa^2}}{{\mathopen{}\mathclose\bgroup\originalleft}[\frac{R}{\kappa}{\mathop{}\!\partial}_T\pm{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}{\mathop{}\!\partial}_R{\aftergroup\egroup\originalright}]},\end{aligned}$$ and its Casimir reproduces the metric as in Eq. . In near-AdS$_2$, the gauge potential $$\begin{aligned} A=-R{\mathop{}\!\mathrm{d}}T\end{aligned}$$ gives rise to the symmetric field strength and preserves near-AdS$_2$ time-translations:[^24] $$\begin{aligned} \label{eq:NearInvariance} {\mathcal{L}}_{H_0}A=0.\end{aligned}$$ To solve Eq.  in this background, it is convenient to introduce the generalized momentum (NHEK momentum) $$\begin{aligned} P_a=p_a-LA_a.\end{aligned}$$ By virtue of Eq. , the near-AdS$_2$ energy $$\begin{aligned} E=-\kappa H_0^a P_a =-P_T =-p_T-LR\end{aligned}$$ is conserved along the trajectory. By inverting the above relations for ${{\mathopen{}\mathclose\bgroup\originalleft}(E,\mathcal{C}+L^2{\aftergroup\egroup\originalright})}$, we find that a charged particle following a Lorentz force trajectory in near-AdS$_2$ with radial electric field has an instantaneous momentum $p=p_a{\mathop{}\!\mathrm{d}}x^a$ of the form $$\begin{aligned} p(x^a,E,L,\mathcal{C})=-{{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}{\mathop{}\!\mathrm{d}}T\pm_R\frac{\sqrt{\mathcal{R}_\kappa(R)}}{R^2-\kappa^2}{\mathop{}\!\mathrm{d}}R,\end{aligned}$$ where the choice of sign $\pm_R$ depends on the radial direction of travel, and we recovered the radial potential , $$\begin{aligned} \mathcal{R}_\kappa(R)={{\mathopen{}\mathclose\bgroup\originalleft}(E+LR{\aftergroup\egroup\originalright})}^2-{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{C}+L^2{\aftergroup\egroup\originalright})}{{\mathopen{}\mathclose\bgroup\originalleft}(R^2-\kappa^2{\aftergroup\egroup\originalright})}.\end{aligned}$$ One can then raise $p_\mu$ to obtain the equations for the trajectory, $$\begin{aligned} \frac{dT}{d\sigma}=\frac{E+LR}{R^2-\kappa^2},\qquad \frac{dR}{d\sigma}=\pm_R\sqrt{\mathcal{R}_\kappa(R)}.\end{aligned}$$ Hence, a charged geodesic with near-AdS$_2$ energy $E$ connects spacetime points $X_s^a={{\mathopen{}\mathclose\bgroup\originalleft}(T_s,R_s{\aftergroup\egroup\originalright})}$ and $X_o^a={{\mathopen{}\mathclose\bgroup\originalleft}(T_o,R_o{\aftergroup\egroup\originalright})}$ if $$\begin{aligned} T_o-T_s=\fint_{R_s}^{R_o}{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E+LR}{R^2-\kappa^2}{\aftergroup\egroup\originalright})}\frac{{\mathop{}\!\mathrm{d}}R}{\pm_R\sqrt{\mathcal{R}_\kappa(R)}}.\end{aligned}$$ This equation is solved in Sec. \[subsec:NearGeodesics\]. ### Isometry group orbits of geodesics The NHEK equatorial geodesics were classified and studied in detail in Ref. [@Compere2018]. Remarkably, it was found that all equatorial orbits can be transformed to a circular orbit by a (possibly complex) isometry. In Anti-de Sitter spacetime (which is maximally symmetric), any two geodesics can be mapped into each other by an isometry (up to parity or time-reversal transformation): all geodesics fall into a small number of orbits of the isometry group action on the manifold. In this subsection, we prove that in AdS$_2$, this statement still holds in the presence of the unique symmetric electromagnetic field . In other words, we show that any two charged geodesics with the same charge $L$ and mass $\sqrt{\mathcal{C}+L^2}$ are related by an $\mathsf{SL}(2,\mathbb{R})$ isometry. For convenience, we will only prove this explicity in the Poincaré patch of AdS$_2$, but of course it then follows that the result holds in every coordinate system. This fact is crucial for our proof in Sec. \[subsec:PoincareGeodesics\] that all NHEK geodesics with the same angular motion (and therefore the same mass $\mu$, angular momentum $L$ about the axis of symmetry, and Casimir $\mathcal{C}$) are related by an isometry of the throat. To begin with, recall from Sec. \[subsec:PoincareGeodesics\] that the general solution to the motion in the Poincaré patch is given by Eq. , which can be rearranged to obtain $$\begin{aligned} R_o(T_o)-\frac{EL}{S_n(T_o)}=\pm\frac{EL}{S_n(T_o)}\sqrt{1+\frac{S_n(T_o)}{L^2}}.\end{aligned}$$ Squaring both sides and expanding terms out yields $$\begin{aligned} R_o^2(T_o)=\frac{E^2}{S_n(T_o)}+\frac{2ELR_o(T_o)}{S_n(T_o)}.\end{aligned}$$ Multiplying by $S_n(T_o)R_o^{-2}(T_o)$ and then completing the square results in $$\begin{aligned} S_n(T_o)={{\mathopen{}\mathclose\bgroup\originalleft}[\frac{E}{R_o(T_o)}+L{\aftergroup\egroup\originalright}]}^2-L^2.\end{aligned}$$ Recalling the definition of $S_n(T)$, this is equivalent to $$\begin{aligned} \label{eq:PoincareMotion} \mathcal{C}+L^2={{\mathopen{}\mathclose\bgroup\originalleft}[\frac{E}{R_o(T_o)}+L{\aftergroup\egroup\originalright}]}^2-(H_0+ET_o)^2={{\mathopen{}\mathclose\bgroup\originalleft}[\frac{E}{R_o(T_o)}+L{\aftergroup\egroup\originalright}]}^2 -{{\mathopen{}\mathclose\bgroup\originalleft}[E{{\mathopen{}\mathclose\bgroup\originalleft}(T_o-T_s{\aftergroup\egroup\originalright})}-\nu_s\frac{\sqrt{\mathcal{R}_n(R_s)}}{R_s}{\aftergroup\egroup\originalright}]}^2,\end{aligned}$$ which is yet another form of the solution to the charged geodesic equation in Poincaré coordinates. Now consider two different geodesics: the first has energy $E_1$ and trajectory ${{\mathopen{}\mathclose\bgroup\originalleft}(T_1(\sigma),R_1(\sigma){\aftergroup\egroup\originalright})}$ with starting point ${{\mathopen{}\mathclose\bgroup\originalleft}(T_{s,1},R_{s,1}{\aftergroup\egroup\originalright})}$, while the second has energy $E_2$ and trajectory ${{\mathopen{}\mathclose\bgroup\originalleft}(T_2(\sigma),R_2(\sigma){\aftergroup\egroup\originalright})}$ with starting point ${{\mathopen{}\mathclose\bgroup\originalleft}(T_{s,2},R_{s,2}{\aftergroup\egroup\originalright})}$. However, we assume they both have the same charge $q=L$ and mass-squared $m^2=\mathcal{C}+L^2$. Then Eq.  implies that $$\begin{aligned} \label{eq:TwoTrajectories} -{{\mathopen{}\mathclose\bgroup\originalleft}(E_iT_i-c_i{\aftergroup\egroup\originalright})}^2+{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{E_i}{R_i}+L{\aftergroup\egroup\originalright})}^2=\mathcal{C}+L^2,\qquad c_i\equiv E_iT_{s,i}+\nu_{s,i}\frac{\sqrt{\mathcal{R}_n(R_{s,i})}}{R_{s,i}},\qquad i=1,2.\end{aligned}$$ Notice that the these two trajectories would match if $$\begin{aligned} T_2=\frac{E_1}{E_2}T_1+\frac{c_2-c_1}{E_2},\qquad R_2=\frac{E_2}{E_1}R_1.\end{aligned}$$ But these relations are exactly the product of two consecutive $\mathsf{SL}(2,\mathbb{R})$ transformations: first, a dilation $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}(T_1,R_1{\aftergroup\egroup\originalright})}\to{{\mathopen{}\mathclose\bgroup\originalleft}(T_1^\prime,R_1^\prime{\aftergroup\egroup\originalright})} ={{\mathopen{}\mathclose\bgroup\originalleft}(\frac{T_1}{\lambda},\lambda R_1{\aftergroup\egroup\originalright})},\qquad \lambda=\frac{E_2}{E_1},\end{aligned}$$ followed by a time-translation $$\begin{aligned} {{\mathopen{}\mathclose\bgroup\originalleft}(T_1^\prime,R_1^\prime{\aftergroup\egroup\originalright})}\to{{\mathopen{}\mathclose\bgroup\originalleft}(T_1^{\prime\prime},R_1^{\prime\prime}{\aftergroup\egroup\originalright})} ={{\mathopen{}\mathclose\bgroup\originalleft}(T_1^\prime+c,R_1^\prime{\aftergroup\egroup\originalright})},\qquad c=\frac{c_2-c_1}{E_2}.\end{aligned}$$ Therefore, the second geodesic is an $\mathsf{SL}(2,\mathbb{R})$ image of the first geodesic. To complete the demonstration, we must consider the constant radius trajectories $$\begin{aligned} R(T)=R_0,\end{aligned}$$ with $E=\mathcal{C}=0$ and $L$ arbitrary. Under the special conformal transformation generated by $H_-$, which maps $$\begin{aligned} \label{eq:SpecialConformalTransformation} T\to\frac{T-\lambda{{\mathopen{}\mathclose\bgroup\originalleft}(T^2-\frac{1}{R^2}{\aftergroup\egroup\originalright})}}{{{\mathopen{}\mathclose\bgroup\originalleft}(1-\lambda T{\aftergroup\egroup\originalright})}^2-\frac{\lambda^2}{R^2}},\qquad R\to R{{\mathopen{}\mathclose\bgroup\originalleft}[{{\mathopen{}\mathclose\bgroup\originalleft}(1-\lambda T{\aftergroup\egroup\originalright})}^2-\frac{\lambda^2}{R^2}{\aftergroup\egroup\originalright}]},\end{aligned}$$ each such curve is mapped into a new curve obeying $$\begin{aligned} -{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2\lambda^2L}{R_0}T-\frac{2\lambda L}{R_0}{\aftergroup\egroup\originalright})}^2+{{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2\lambda^2L/R_0}{R}+L{\aftergroup\egroup\originalright})}^2=L^2.\end{aligned}$$ This is of the form with arbitrary $L$ and $$\begin{aligned} E=\frac{2\lambda^2 L}{R_0},\qquad \mathcal{C}=0,\qquad ET_s+\nu_s\frac{\sqrt{\mathcal{R}_n(R_s)}}{R_s}=\frac{2\lambda L}{R_0},\end{aligned}$$ corresponding to a charged geodesic with the same mass $m^2=L^2$ and charge $q=L$, but different energy and initial position. More generally, two trajectories with the same $\mathsf{SL}(2,\mathbb{R})$ Casimir $\mathcal{C}$ and charge $L$ are related by a special conformal transformation if $$\begin{aligned} E_2=E_1-2\lambda c_1+\frac{c_1^2+\mathcal{C}}{E_1}\lambda^2,\qquad c_2=c_1-\frac{c_1^2+\mathcal{C}}{E_1}\lambda.\end{aligned}$$ In conclusion, by a combined dilation, time-translation and special conformal transformation, it is always possible to map any two charged geodesics in AdS$_2$ with the same mass and charge into each other. Note that the required transformations fully exhaust the isometries available. [^1]: A Killing tensor satisfies $\nabla_{(\lambda}K_{\mu\nu)}=0$. The antisymmetric tensor $J_{\mu\nu}=-J_{\nu\mu}$ satisfies the Killing-Yano equation $\nabla_{(\lambda}J_{\mu)\nu}=0$. [^2]: The special case $\ell=0$ needs to be treated separately, as one must account for the possibility that geodesics may climb over the black hole and pass through the rotation axis. [^3]: The Kerr principal null congruences are shear-free. By the Goldberg-Sachs theorem, this implies that the Kerr spacetime is algebraically special of Petrov Type D. This property guarantees the existence of the Killing-Yano tensor [@Stephani1978], from which many of the special properties of the Kerr geometry—including the separability of the wave equation and integrability of geodesic motion—are derived. [^4]: This is because Eq.  implies that $Q={{\mathopen{}\mathclose\bgroup\originalleft}(\Sigma p^\theta{\aftergroup\egroup\originalright})}^2-\cos^2{\theta}{{\mathopen{}\mathclose\bgroup\originalleft}(P-\ell^2\csc^2{\theta}{\aftergroup\egroup\originalright})}$. Thus $Q\le0$ is only possible if $P-\ell^2\csc^2{\theta}\ge0$, which requires that $P\ge\ell^2\csc^2{\theta}\ge\ell^2>0$. [^5]: This can be derived, for instance, from the two standard identities $\mathrm{sc}(x\pm iK(1-k)+ K(k)|k)=\pm\frac{i}{\sqrt{1-k}}\operatorname{dn}(x|k)$ and $\operatorname{sn}(ix|k)=i\,\mathrm{sc}(x|1-k)$ (Eqs. (16.8.9) and (16.20.1) of Ref. [@Abramowitz1972]) which imply that $\operatorname{dn}(x|1-k)=\pm\sqrt{k}\operatorname{sn}(ix\pm K(k)+iK(1-k)|k)$. The result is then obtained by using Eq.  together with the imaginary periodicity condition $\operatorname{sn}(x\pm2iK(1-k)|k)=\operatorname{sn}(x|k)$ whenever it is needed. [^6]: For a precisely extremal black hole, this distinction is irrelevant since all the scales we discuss lie precisely at $r=M$, so there is a single NHEK limit that resolves them. [^7]: The ISCO has finite Kerr energy given by Eq. , and retains finite energy in the NHEK limit. However, plunging trajectories that were resolved by the far region will be null in this limit, and their NHEK energy will typically diverge; see, $e.g.$, Ref. [@Gralla2016a] and Fig. 1 therein. [^8]: The precise choice of boundary conditions is delicate [@Guica2009; @Dias2009; @Amsel2009; @Compere2012]. [^9]: We thank Abhishek Pathak for help in deriving this transformation from its AdS$_3$ analogue [@Roberts2012]. [^10]: More precisely: dilations by $a^2$ are obtained by setting $b=c=0$ and $d=a^{-1}$; time-translations by $b$ are obtained by setting $a=d=1$ and $c=0$; special conformal transformations by a parameter $c$ are obtained by setting $a=d=1$ and $b=0$. [^11]: The inverse transformation on the domain $\psi\in{{\mathopen{}\mathclose\bgroup\originalleft}[0,\pi{\aftergroup\egroup\originalright}]}$ is given by $\psi=\pi/2+\arctan{y}$. [^12]: Observe that $\mathcal{C}^{\mu\nu}P_\mu P_\nu=-\mathcal{H}_0^2+\frac{1}{2}{{\mathopen{}\mathclose\bgroup\originalleft}(\mathcal{H}_+\mathcal{H}_-+\mathcal{H}_-\mathcal{H}_+{\aftergroup\egroup\originalright})}$, where $\mathcal{H}_i=H_i^\mu P_\mu$ or $\mathcal{H}_i=L_i^\mu P_\mu$ are the conserved quantities associated with the generators of $\mathsf{SL}(2,\mathbb{R})$. Since these are not independent of each other, we use the $\mathsf{SL}(2,\mathbb{R})$ Casimir, which is in involution with all the $\mathcal{H}_i$ [@AlZahrani2011]. This is exactly analogous to exploiting the conservation of $J_z$ and $J^2$, rather than $(J_x,J_y,J_z)$, in a problem with $\mathsf{SO}(3)$ symmetry. [^13]: In Anti-de Sitter space, one typically imposes reflective boundary conditions in order to describe a closed system. In the present case, such a choice of boundary conditions would lead to periodic multiple-bounce trajectories. We will not explicitly consider such solutions as they are unlikely to be relevant to astrophysical black holes, but it is trivial to construct them by gluing together a sequence of single bounces. [^14]: This bound agrees with the bound derived for equatorial geodesics in Section 2 of Ref. [@Bardeen1999], whose authors set $M=2^{-1/2}$. [^15]: By Eq. , each of the roots $y_\pm$ must lie strictly to one side of $y_c$. When $\triangle=0$, it is evident that $y_+<0=y_c<y_-$, so this ordering must hold in general. One can also prove these inequalities directly with some straightforward algebraic manipulations. [^16]: Note that $L=0$ is forbidden by Eq. , since $\mathcal{C}<0$ is incompatible with $\mathcal{C}>\mu^2M^2\ge0$. [^17]: The fact that $\mathcal{C}$ is unbounded in Poincaré NHEK does not contradict the fact that $\mathcal{C}$ is bounded above by $\triangle^2$ in global NHEK. Taking $\mathcal{C}\to\infty$ at fixed Poincaré energy $E$ sends $\triangle\to\infty$ as well, so that the global NHEK bound $\mathcal{C}<\triangle^2$ is still satisfied. [^18]: Recall that the turning points are separated by $\Delta\tau=\pi$, so there is precisely one per Poincaré patch. [^19]: This is essentially the same formula as for Type II motion in global NHEK, since there can never be more than one turning point. [^20]: To see this, note that $R_+^2-\kappa^2$ is a sum of positive terms when $\mathcal{C}>0$. [^21]: The first identity is essentially identical to Eq. (2.161.1) in Ref. [@Gradshteyn2007]. [^22]: This class of geodesics includes both the principal null congruences as well as all the equatorial geodesics, which were analyzed in Ref. [@Compere2018]. [^23]: A special subset of these geodesics was derived in Ref. [@Maldacena1999]. [^24]: Note that the gauge-equivalent connection $A=-R{\mathop{}\!\mathrm{d}}T-{\mathop{}\!\mathrm{d}}\operatorname{arctanh}(R/\kappa)$ preserves both $H_+$ and $H_0$.

--- abstract: 'In quantum magnetism, the virtual exchange of particles mediates an interaction between spins. Here we show that an inelastic Hubbard interaction fundamentally alters the magnetism of the Hubbard model due to dissipation in spin-exchange processes, leading to sign reversal of magnetic correlations in dissipative quantum dynamics. This mechanism is applicable to both fermionic and bosonic Mott insulators, and can naturally be realized with ultracold atoms undergoing two-body inelastic collisions. The dynamical reversal of magnetic correlations can be detected by using a double-well optical lattice or quantum-gas microscopy, the latter of which enhances the signal of the magnetic correlations owing to spin-charge separation in one-dimensional systems. Our results open a new avenue toward controlling quantum dissipative many-body states.' author: - Masaya Nakagawa - Naoto Tsuji - Norio Kawakami - Masahito Ueda bibliography: - 'NTMag\_ref.bib' title: Dynamical Sign Reversal of Magnetic Correlations in Dissipative Hubbard Models --- Quantum magnetism in Mott insulators is one of the central problems in strongly correlated many-body systems [@Auerbach]. A Mott insulator is described by the Hubbard model, where a strong repulsive interaction between particles precludes multiple occupation and anchors a single spin to each lattice site. While the kinetic motion of particles is frozen in Mott insulators, quantum mechanics allows particles to virtually hop between sites. A second-order process involving virtual exchange of particles leads to an effective spin-spin interaction, providing a fundamental origin of quantum magnetism [@Auerbach]. Recent developments in quantum simulations of the Hubbard model with ultracold atoms [@Esslinger10] have offered a powerful approach to unveiling low-temperature properties of quantum magnets [@Trotzky08; @Greif13; @Greif15; @Hart15; @Ozawa18]. In particular, quantum-gas microscopy has enabled site-resolved imaging of spin states [@Bakr09; @Parsons15; @Cheuk15; @Cheuk16_1], culminating in direct observation of antiferromagnetic correlations and long-range order in the Hubbard model [@Parsons16; @Cheuk16_2; @Boll16; @Mazurenko17]. The essential requirement for observing the quantum magnetism is to achieve sufficiently low temperatures comparable with the exchange coupling. In this Letter, we demonstrate that ultracold atoms undergoing inelastic collisions obey a completely different principle for realizing quantum magnetism; instead of relaxing to low-energy states, those atoms stabilize *high-energy* states due to dissipation caused by inelastic collisions. Inelastic collisions have widely been observed for atoms in excited states [@Sponselee18; @Tomita18] and molecules [@Syassen08; @Zhu14], and can be artificially induced by photoassociation [@Tomita17]. In contrast to standard equilibrium systems which favor low-energy states, the long-time behavior of dissipative systems is governed by the lifetime of each state under dissipation. We show that the spin-exchange mechanism is altered in the presence of inelastic collisions due to dissipation in an intermediate state. As a result, dissipation dramatically changes the magnetism of the Hubbard model; the magnetism is *inverted* from the conventional equilibrium one, leading to the sign reversal of spin correlations through dissipative dynamics. ![Schematic illustration of a second-order process mediating the spin-exchange interaction in the dissipative Fermi-Hubbard system. A loss in an intermediate process causes a finite lifetime of the system.[]{data-label="fig_exchange"}](Fig1.pdf){width="8.5cm"} The spin-exchange interaction in the presence of an inelastic interaction, which plays a key role in this Letter, is schematically illustrated in Fig. \[fig\_exchange\] for the Fermi-Hubbard system. Since the intermediate state in a second-order process involves a doubly occupied site, an antiferromagnetic spin configuration has a finite lifetime due to a particle loss in the intermediate state, whereas a ferromagnetic spin configuration cannot decay due to the Pauli exclusion principle. Because of this dissipative spin-exchange interaction, low-energy states gradually decay, and high-energy spin states will eventually be stabilized. Such stabilization of high-energy states cannot be achieved in conventional equilibrium systems and is reminiscent of negative-temperature states [@LandauLifshitz; @Ramsey56] realized in systems isolated from environments [@Purcell51; @Hakonen92; @Hakonen94; @Rapp10; @Rapp12; @Tsuji11; @Braun13; @Gauthier18; @Johnstone18; @Yamamoto19]. In contrast, here dissipation to an environment plays a vital role and thus offers a unique avenue towards the control of magnetism in open systems. *Model*.–  We consider a dissipative Hubbard model of two-component fermions or bosons realized with ultracold atoms in an optical lattice. The unitary part of the dynamics is governed by the Hubbard Hamiltonian which reads $$H=-t\sum_{\langle i,j\rangle,\sigma={\uparrow},{\downarrow}}(c_{i\sigma}^\dag c_{j\sigma}+\mathrm{h.c.})+U\sum_j n_{j{\uparrow}}^{(f)}n_{j{\downarrow}}^{(f)} \label{eq_FH}$$ for fermions, and $$\begin{aligned} H=&-t\sum_{\langle i,j\rangle,\sigma={\uparrow},{\downarrow}}(b_{i\sigma}^\dag b_{j\sigma}+\mathrm{h.c.})+\sum_j U_{{\uparrow}{\downarrow}}n_{j{\uparrow}}^{(b)}n_{j{\downarrow}}^{(b)}\notag\\ &+\sum_\sigma\sum_j \frac{U_{\sigma\sigma}}{2}n_{j\sigma}^{(b)}(n_{j\sigma}^{(b)}-1) \label{eq_BH}\end{aligned}$$ for bosons. Here $c_{j\sigma}$ ($b_{j\sigma}$) is the annihilation operator of a fermion (boson) with spin $\sigma$ at site $j$, and $n_{j\sigma}^{(f)}=c_{j\sigma}^\dag c_{j\sigma}$ ($n_{j\sigma}^{(b)}=b_{j\sigma}^\dag b_{j\sigma}$). We assume that hopping (with an amplitude $t$) occurs between the nearest-neighbor sites and that the on-site elastic interactions are repulsive: $U,U_{\sigma\sigma'}>0$. We also assume $t>0$ without loss of generality. Now we suppose that atoms also undergo inelastic collisions; because a large internal energy is converted to the kinetic energy, two atoms after inelastic collisions quickly escape from the trap and are lost. The dissipative dynamics of the density matrix $\rho$ of the system at time $\tau$ is described by the following quantum master equation [@BreuerPetruccione]: $$\frac{d\rho}{d\tau}=i[\rho,H]+\sum_{j,\sigma,\sigma'}\left(L_{j\sigma\sigma'}\rho L_{j\sigma\sigma'}^\dag-\frac{1}{2}\{ L_{j\sigma\sigma'}^\dag L_{j\sigma\sigma'},\rho\}\right). \label{eq_master}$$ The Lindblad operators $L_{j\sigma\sigma'}$ induce two-body losses due to the on-site inelastic collisions, and are expressed as $L_{j\sigma\sigma'}=\sqrt{2\gamma}c_{j\sigma}c_{j\sigma'}\delta_{\sigma,\uparrow}\delta_{\sigma',\downarrow}$ for fermions and $L_{j\sigma\sigma'}= \sqrt{\gamma_{\sigma\sigma'}}b_{j\sigma}b_{j\sigma'}$ for bosons. The coefficients $\gamma,\gamma_{\sigma\sigma'}>0$ are determined from the loss rates of atoms. *Spin-exchange interaction in dissipative systems*.–  We first illustrate the basic mechanism that underlies the magnetism of the dissipative Hubbard systems. We consider a strongly correlated regime ($U,U_{\sigma\sigma'}\gg t$) and assume that the initial particle density is set to unity so that a Mott insulating state is realized. For simplicity, we consider the case of the spin SU(2) invariance, i.e., $U_{{\uparrow}{\uparrow}}=U_{{\downarrow}{\downarrow}}=U_{{\uparrow}{\downarrow}}=U$. Then, if doubly occupied states and empty states are ignored, the Fermi (Bose) Hubbard model () reduces to the antiferromagnetic (ferromagnetic) Heisenberg model $H_{\mathrm{spin}}=J\sum_{\langle i,j\rangle}(\bm{S}_i\cdot\bm{S}_j-1/4)$ ($H_{\mathrm{spin}}=-J\sum_{\langle i,j\rangle}(\bm{S}_i\cdot\bm{S}_j+3/4)$) with the spin-exchange interaction $J=4t^2/U$ [@Kuklov03; @Duan03]. Here we employ a quantum-trajectory method [@Dalibard92; @Carmichael_book; @Daley14] to investigate the dynamics described by Eq.  [@supple]. The dynamics is decomposed into a non-unitary Schrödinger evolution under an effective non-Hermitian Hamiltonian $H_{\mathrm{eff}}\equiv H-\frac{i}{2}\sum_{j,\sigma,\sigma'} L_{j\sigma\sigma'}^\dag L_{j\sigma\sigma'}$ and stochastic quantum-jump processes which induce particle losses with the jump operators $L_{j\sigma\sigma}$. The non-Hermitian Hamiltonian $H_{\mathrm{eff}}$ is obtained if we replace the Hubbard interactions $U$ and $U_{\sigma\sigma'}$ with $U-i\gamma$ and $U_{\sigma\sigma'}-i\gamma_{\sigma\sigma'}$, respectively, thereby making the interaction coefficients complex-valued due to the inelastic interactions. In each quantum trajectory, the system evolves under the non-Hermitian Hubbard model during a time interval between loss events [@Ashida16; @Ashida18]. Each quantum trajectory is characterized by the number of loss events. Let us first consider a trajectory that does not involve any loss event; in this trajectory, the particle number stays constant. Since the double occupancy is still suppressed due to the large Hubbard interaction $U$, the dynamics is constrained to the Hilbert subspace of the spin Hamiltonian. The effective spin Hamiltonian, which governs the dynamics in the quantum trajectory, is derived from the non-Hermitian Hubbard model through the second-order perturbation theory, giving $$H_{\mathrm{eff}}=\eta(J_{\mathrm{eff}}+i\Gamma)\sum_{\langle i,j\rangle}\left(\bm{S}_i\cdot\bm{S}_j+\frac{1-2\eta}{4}\right), \label{eq_effHeis}$$ where $J_{\mathrm{eff}}=4Ut^2/(U^2+\gamma^2)$, $\Gamma=4\gamma t^2/(U^2+\gamma^2)$, and $\eta=+1$ ($\eta=-1$) for fermions (bosons). Here we assume spin-independent dissipation $\gamma_{\sigma\sigma'}=\gamma$ for bosonic atoms (see Supplemental Material [@supple] for a general case). Equation  shows that the spin-spin interactions are affected by dissipation even if the double occupancy is suppressed by the strong repulsion, since the virtual second-order process involves a doubly occupied site (see Fig. \[fig\_exchange\]). In fact, the energy denominators in $J_{\mathrm{eff}}=\mathrm{Re}\left[\frac{4t^2}{U-i\gamma}\right]$ and $\Gamma=\mathrm{Im}\left[\frac{4t^2}{U-i\gamma}\right]$ reflect the dissipation in the intermediate state. The eigenenergy of the Hamiltonian is given by $E_n=(J_{\mathrm{eff}}+i\Gamma)E_n^{(0)}/J$, where $E_n^{(0)}\leq 0$ is the eigenenergy of the Heisenberg model $H_{\mathrm{spin}}$. Thus, the decay rate of the $n$-th eigenstate, which is given by the imaginary part of the energy, is proportional to $E_n^{(0)}$: $-\mathrm{Im}[E_n]=-(\Gamma/J)E_n^{(0)}\geq 0$. Since $E_n^{(0)}\leq 0$, this indicates that lower-energy states have larger decay rates with shorter lifetimes. Therefore, after a sufficiently long time, only the high-energy spin states survive. ![(a) (b) Time evolution of the squared norm $\braket{\psi(\tau)|\psi(\tau)}$ \[(a)\] and the double occupancy $\bra{\psi(\tau)}\frac{1}{2}(n_{1{\uparrow}}^{(a)}n_{1{\downarrow}}^{(a)}+n_{2{\uparrow}}^{(a)}n_{2{\downarrow}}^{(a)})\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}$ ($a=f$ or $b$) \[(b)\]. Note that these quantities take the same values for the Fermi and Bose Hubbard models. (c) (d) Time evolution of the spin correlation $\bra{\psi(\tau)}\bm{S}_1\cdot\bm{S}_2\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}$ of the Fermi \[(c)\] and Bose \[(d)\] Hubbard models. The parameters are set to $U/t=10$ and $\gamma/t=3$. The unit of time is the inverse hopping rate $\tau_h=1/t$.[]{data-label="fig_2site"}](Fig2.pdf){width="8.5cm"} *Double-well systems*.–  A minimal setup to demonstrate the basic principle described above is a two-site system. It can be experimentally realized with an ensemble of double wells created by optical superlattices [@Trotzky08; @Greif13], and magnetic correlations between the left and right wells can be measured from singlet-triplet oscillations [@Greif13; @Greif15; @Ozawa18]. We consider an ensemble of double wells in which two particles with opposite spins occupy each double well. During the dissipative dynamics, a double well in which a loss event takes place becomes empty. Therefore, when a magnetic correlation is measured at time $\tau$, signals come from double wells where particles are still not lost. Such wells are faithfully described by the quantum trajectory without loss events. Figure \[fig\_2site\] shows the time evolution of the squared norm of the state $\braket{\psi(\tau)|\psi(\tau)}$, the double occupancy $\bra{\psi(\tau)}\frac{1}{2}(n_{1{\uparrow}}^{(a)}n_{1{\downarrow}}^{(a)}+n_{2{\uparrow}}^{(a)}n_{2{\downarrow}}^{(a)})\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}\ (a=f,b)$, and the spin correlation $\bra{\psi(\tau)}\bm{S}_1\cdot\bm{S}_2\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}$ obtained from a numerical solution of the Schrödinger equation $i\partial_\tau\ket{\psi(\tau)}=H_{\mathrm{eff}}\ket{\psi(\tau)}$. Here $H_{\mathrm{eff}}$ is the two-site non-Hermitian Fermi (Bose) Hubbard model and the initial state is assumed to be $c_{1{\uparrow}}^\dag c_{2{\downarrow}}^\dag\ket{0}$ ($b_{1{\uparrow}}^\dag b_{2{\downarrow}}^\dag\ket{0}$), where $\ket{0}$ is the particle vacuum. The results clearly show that the dissipative Fermi (Bose) Hubbard system develops a ferromagnetic (antiferromagnetic) correlation which is eventually saturated at $0.25$ ($-0.75$), indicating a formation of the highest-energy spin state $(\ket{{\uparrow}}_1\ket{{\downarrow}}_2+\ket{{\downarrow}}_1\ket{{\uparrow}}_2)/\sqrt{2}$ ($(\ket{{\uparrow}}_1\ket{{\downarrow}}_2-\ket{{\downarrow}}_1\ket{{\uparrow}}_2)/\sqrt{2}$) of the Heisenberg model. We note that the double occupancy in the dynamics is almost negligible and further suppressed by an increase in the dissipation $\gamma$ (see Supplemental Material for the dependence on $\gamma$ [@supple]); the latter is due to the continuous quantum Zeno effect [@Syassen08; @Zhu14; @Tomita17] by which strong dissipation inhibits hopping to an occupied site. Nevertheless, the virtual hopping process is allowed, leading to the growth in the spin correlation. Another important feature is that the squared norm stays constant after the spin correlation is saturated. Since the squared norm corresponds to the probability of the lossless quantum trajectory [@Daley14], the saturation indicates that the system enters a dark state that is immune to the dissipation. This property explains why the highest-energy spin state is realized in the long-time limit; the spin-symmetric (spin-antisymmetric) state of fermions (bosons) is actually free from dissipation and thus has the longest lifetime, since in this spin configuration both Fermi and Bose statistics dictate antisymmetry of the real-space wavefunction and hence allows no double occupancy [@FossFeig12]. *Extracting spin correlations from conditional correlators*.–  Having established the basic mechanism of the magnetism induced by dissipation, we proceed to include the effect of quantum jumps, which create holes due to particle loss. One might think that the created holes scramble the background spin configuration and disturb the development of the spin correlation. Below we show that this difficulty can be circumvented by using quantum-gas microscopy for the one-dimensional Hubbard models. We first show in Fig. \[fig\_nojump\] the time evolution of the one-dimensional dissipative Hubbard models in quantum trajectories without quantum-jump events. The system size is $N=8$ ($N=6$) for the Fermi (Bose) system. The initial states are chosen to be a Néel state $\ket{{\uparrow}{\downarrow}{\uparrow}{\downarrow}{\uparrow}{\downarrow}{\uparrow}{\downarrow}}$ for the Fermi system, and a ferromagnetic domain-wall state $\ket{{\uparrow}{\uparrow}{\uparrow}{\downarrow}{\downarrow}{\downarrow}}$ for the Bose system, in accordance with the equilibrium spin configuration of each system without dissipation. After the dissipation is switched on at $\tau=0$, the Fermi (Bose) system in Fig. \[fig\_nojump\] (a) (Fig. \[fig\_nojump\] (b)) clearly develops a ferromagnetic (antiferromagnetic) spin correlation $C(i,j;\tau)=\bra{\psi(\tau)}\bm{S}_i\cdot\bm{S}_j\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}$, whose sign is reversed from that of the initial state, and the correlation is eventually saturated at a value consistent with the highest-energy state of the antiferromagnetic (ferromagneic) Heisenberg chain. ![Dynamics of the spin correlations $C(0,j;\tau)$ for the dissipative 8-site Fermi \[(a)\] and 6-site Bose \[(b)\] Hubbard systems in the absence of quantum-jump events. The parameters are set to $U/t=10$ and $\gamma/t=10$. The unit of time is the inverse hopping rate $\tau_h=1/t$.[]{data-label="fig_nojump"}](Fig3.pdf){width="8.5cm"} Quantum-gas microscopy enables a high-precision measurement of the particle number at the single-site level [@Bakr09; @Parsons15; @Cheuk15; @Cheuk16_1]. Given a single-shot image of an atomic gas, the occupation number of each site is identified to be zero, one, or two. From this information, one can find the number of quantum jumps that have occurred by the time when the measurement is performed. Accordingly, one can take an ensemble average over quantum trajectories with a given number of quantum jumps [@Ashida18_2]. The density matrix conditioned on the number of quantum jumps from the initial time to $\tau$ is given by $\rho^{(n)}(\tau)=\mathcal{P}^{(n)}\rho(\tau)\mathcal{P}^{(n)}/\mathrm{Tr}[\mathcal{P}^{(n)}\rho(\tau)\mathcal{P}^{(n)}]$. Here $\mathcal{P}^{(n)}$ is a projection onto the sector in which $n$ quantum jumps have occurred. Then, one can calculate the correlation function $C^{(n)}(i,j;\tau)\equiv \mathrm{Tr}[\rho^{(n)}(\tau)\bm{S}_i\cdot\bm{S}_j]$ [@supple]. Figure \[fig\_withjump\](a) (\[fig\_withjump\](c)) shows the dynamics of the magnetic correlation $C^{(n)}(0,1;\tau)$ of the dissipative Fermi (Bose) Hubbard system. For comparison, we also show $C(0,1;\tau)\equiv\mathrm{Tr}[\rho(\tau)\bm{S}_0\cdot\bm{S}_1]$ where the average is taken over all quantum trajectories so as to give the solution of the master equation . The result indicates that the sign reversal of the magnetic correlations is still seen in the presence of quantum jumps, and the magnitude of the correlation increases with decreasing the number of quantum jumps. ![(a) (c) Dynamics of spin correlations $C^{(n)}(0,1;\tau)$ averaged over quantum trajectories that involve $n$ quantum jumps. The label “master” corresponds to $C(0,1;\tau)$, in which the correlation is calculated from the full density matrix of the solution to the master equation. (b) (d) Dynamics of conditional correlators $C^{(n)}_{\mathrm{proj}}(0,1;\tau)$ which eliminate the effect of holes by additional projection. (a) and (b) show the results for the dissipative Fermi-Hubbard model, and (c) and (d) show those for the dissipative Bose-Hubbard model. The parameters and the initial states are the same as in Fig. \[fig\_nojump\]. The unit of time is the inverse hopping rate $\tau_h=1/t$.[]{data-label="fig_withjump"}](Fig4.pdf){width="8.5cm"} The correlation function $C^{(n)}(i,j;\tau)$ includes the effect of holes produced by quantum jumps. However, one can remove the effect of holes and extract the contribution from spins remaining in the system by imposing a further condition with the following conditional correlator [@supple]: $$C_{\mathrm{proj}}^{(n)}(j,j+1;\tau)\equiv\frac{\mathrm{Tr}[P_jP_{j+1}\rho^{(n)}(\tau)P_jP_{j+1}\bm{S}_j\cdot\bm{S}_{j+1}]}{\mathrm{Tr}[P_jP_{j+1}\rho^{(n)}(\tau)P_jP_{j+1}]},$$ where $P_j$ is a projector onto states in which site $j$ is singly occupied. More generally, one can use a correlation function $C^{(n)}_{\mathrm{proj}}(j,j+d;d_h;\tau)\equiv\mathrm{Tr}[P_jQ_{d_h}P_{j+d}\rho^{(n)}(\tau)P_jQ_{d_h}P_{j+d}\bm{S}_j\cdot\bm{S}_{j+d}]/\mathrm{Tr}[P_jQ_{d_h}P_{j+d}\rho^{(n)}(\tau)P_jQ_{d_h}P_{j+d}]$, where $Q_{d_h}$ is another projector onto states with $d_h$ holes and $d-d_h-1$ singly occupied sites between sites $j$ and $j+d$. Such conditional correlators have been measured with quantum-gas microscopy [@Endres11; @Hilker17] by collecting images that match the conditions. Numerical results of the conditional correlators $C^{(n)}_{\mathrm{proj}}(0,1;\tau)$ for the Fermi (Bose) Hubbard system are shown in Fig. \[fig\_withjump\](b) (\[fig\_withjump\](d)). Notably, the magnetic correlations are significantly enhanced from those without projection and even saturated at the same maximum value as in the case without quantum jumps for the Fermi-Hubbard system. While saturation is not achieved in the Bose-Hubbard system since the numerical simulation is limited to $\tau/\tau_h\lesssim 10$ for sufficient statistical convergence, similar saturation behavior can be seen at a single-trajectory level [@supple]. Nevertheless, a significant increase in the antiferromagnetic correlation is clearly observed by comparing Figs. \[fig\_withjump\] (c) and \[fig\_withjump\](d). The underlying physics behind these results is spin-charge separation in one-dimensional systems [@Giamarchi_book]. In the strongly correlated Hubbard chain, the created holes move freely as if they were non-interacting, while the background spin state remains the same as that of the Heisenberg chain [@Ogata90]. In particular, given an eigenstate of the one-dimensional Hubbard chain, one can reconstruct an eigenstate of the Heisenberg model by eliminating holes involved in each particle configuration superposed in the quantum state [@Hilker17; @Ogata90; @Kruis04]. Thus, the conditional correlators $C^{(n)}_{\mathrm{proj}}(j,j+1;\tau)$ and $C^{(n)}_{\mathrm{proj}}(j,j+d;d_h;\tau)$ capture the spin correlations in the background Heisenberg model, which are equivalent to those in the case without holes at least in the highest-energy spin state achieved in the long-time limit. This explains the saturated value of the conditional spin correlation that exactly coincides with that in the trajectory without loss events shown in Fig. \[fig\_nojump\]. Although the original argument on the spin-charge separation in eigenstates of the Hubbard model was limited to the fermion case [@Hilker17; @Ogata90; @Kruis04], our numerical results indicate that this mechanism also works for the Bose-Hubbard system. *Summary and future perspectives*.–  We have shown that the inelastic Hubbard interaction alters the spin-exchange process due to a finite lifetime of the intermediate state, leading to novel quantum magnetism opposite to the conventional equilibrium magnetism. Rather than stabilizing low-energy states, high-energy spin states have longer lifetimes and are thus realized in the dissipative systems. The Hubbard models with inelastic interactions can be realized with various types of ultracold atoms in internal excited states. A possible experimental platform is a system of ytterbium atoms having long-lived excited states for which the decay to the ground state due to spontaneous emission is negligible [@Sponselee18; @Tomita18]. Furthermore, inelastic collisions can be artificially induced by using photoassociation techniques [@Tomita17], which will enable the control of quantum magnetism with dissipation. Our work raises interesting questions for future investigation. First, while we have shown that the effect of holes can be eliminated in one-dimensional systems due to spin-charge separation, it cannot in two (or higher) dimensions. Second, since the Bose-Hubbard system develops antiferromagnetic correlations due to dissipation, geometric frustration in the lattice may realize quantum spin liquids and topological order, which have not yet been realized in cold-atom experiments due to the difficulty of cooling. Third, in this Letter, we have focused on the cases with spin SU(2) symmetry. If this symmetry is relaxed, eigenstates of the non-Hermitian spin Hamiltonian with the complex-valued spin-exchange couplings are no longer the same as those of the original Hermitian spin Hamiltonian. It is therefore worthwhile to explore in these systems novel quantum magnetism with non-Hermitian spin Hamiltonians [@Lee14]. We thank Kazuya Fujimoto, Takeshi Fukuhara, and Yoshiro Takahashi for helpful discussions. This work was supported by KAKENHI (Grants No. JP16K05501, No. JP16K17729, No. JP18H01140, and No. JP18H01145) and a Grant-in-Aid for Scientific Research on Innovative Areas (KAKENHI Grant No. JP15H05855) from the Japan Society for the Promotion of Science. M.N. was supported by RIKEN Special Postdoctoral Researcher Program. N.T. acknowledges support by JST PRESTO (Grant No. JPMJPR16N7). \ **“Dynamical Sign Reversal of Magnetic Correlations in Dissipative Hubbard Models”** Non-Hermitian spin Hamiltonian ============================== We derive the non-Hermitian spin Hamiltonian that governs the time evolution in a strongly correlated regime. We start with an effective non-Hermitian Hubbard Hamiltonian $H_{\mathrm{eff}}$ and decompose it into the kinetic part $H'$ and the interaction part $H_0$, where $$\begin{aligned} H'&=-t\sum_{\langle i,j\rangle}\sum_{\sigma=\uparrow,\downarrow}(c_{i\sigma}^\dag c_{j\sigma}+\mathrm{h.c.}),\notag\\ H_0&=(U-i\gamma)\sum_j n_{j\uparrow}^{(f)}n_{j\downarrow}^{(f)},\notag\end{aligned}$$ for fermions, and $$\begin{aligned} H'&=-t\sum_{\langle i,j\rangle}\sum_{\sigma=\uparrow,\downarrow}(b_{i\sigma}^\dag b_{j\sigma}+\mathrm{h.c.}),\notag\\ H_0&=(U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow})\sum_j n_{j\uparrow}^{(b)}n_{j\downarrow}^{(b)}+\sum_j\sum_{\sigma=\uparrow,\downarrow}\frac{U_{\sigma\sigma}-i\gamma_{\sigma\sigma}}{2}n_{j\sigma}^{(b)}(n_{j\sigma}^{(b)}-1),\notag\end{aligned}$$ for bosons. In the strongly correlated regime $U,U_{\sigma\sigma'}\gg t$, the kinetic term $H'$ can be treated as a perturbation. For simplicity, we consider a Mott insulating state and ignore holes. According to the second-order perturbation theory, an effective Hamiltonian is given by $$H_{\mathrm{spin}}=E_0+\mathcal{P}H'\frac{1}{E_0-H_0}H'\mathcal{P},$$ where $\mathcal{P}$ is a projector onto the Hilbert subspace in which each lattice site is occupied by one atom. Here the energy $E_0$ of the unperturbed state is set to $E_0=0$. In the simplest two-site case, the Hilbert subspace is spanned by four spin configurations $\{\ket{\uparrow\uparrow},\ket{\uparrow\downarrow},\ket{\downarrow\uparrow},\ket{\downarrow\downarrow}\}$. In this case, the spin Hamiltonian reads $$\begin{aligned} H_{\mathrm{spin}}=-\frac{2t^2}{U-i\gamma}(\ket{\uparrow\downarrow}\bra{\uparrow\downarrow}+\ket{\downarrow\uparrow}\bra{\downarrow\uparrow}-\ket{\downarrow\uparrow}\bra{\uparrow\downarrow}-\ket{\uparrow\downarrow}\bra{\downarrow\uparrow}),\end{aligned}$$ for fermions, and $$\begin{aligned} H_{\mathrm{spin}}=-2t^2\Bigl(&\frac{2}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}\ket{\uparrow\uparrow}\bra{\uparrow\uparrow}+\frac{2}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}\ket{\downarrow\downarrow}\bra{\downarrow\downarrow}+\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\ket{\uparrow\downarrow}\bra{\uparrow\downarrow}+\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\ket{\downarrow\uparrow}\bra{\downarrow\uparrow}\notag\\ &+\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\ket{\downarrow\uparrow}\bra{\uparrow\downarrow}+\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\ket{\uparrow\downarrow}\bra{\downarrow\uparrow}\Bigr),\end{aligned}$$ for bosons. Hence, for fermions, the spin Hamiltonian is given by the non-Hermitian Heisenberg model $$H_{\mathrm{spin}}=\frac{4t^2}{U-i\gamma}\sum_{\langle i,j\rangle}\left(\bm{S}_i\cdot\bm{S}_j-\frac{1}{4}\right), \label{eq_Hspin_fermion}$$ and for bosons it is given by $$\begin{aligned} H_{\mathrm{spin}}=&\sum_{\langle i,j\rangle}\Bigl[-\frac{4t^2}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}(S_i^xS_j^x+S_i^yS_j^y)-4t^2\Bigl(\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}+\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}-\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\Bigr)S_i^zS_j^z\notag\\ &-t^2\Bigl(\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}+\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}+\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\Bigr)-2t^2\Bigl(\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}-\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}\Bigr)(S_i^z+S_j^z)\Bigr]\notag\\ =&\sum_{\langle i,j\rangle}\Bigl[(J_{\mathrm{eff}}^\perp+i\Gamma^\perp)(S_i^xS_j^x+S_i^yS_j^y)+(J_{\mathrm{eff}}^z+i\Gamma^z)S_i^zS_j^z+C\Bigr]+(h_r+ih_i)\sum_j S_j^z, \label{eq_Hspin_boson}\end{aligned}$$ where $$\begin{aligned} J_{\mathrm{eff}}^\perp&=-\mathrm{Re}\left[\frac{4t^2}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\right],\\ \Gamma^\perp&=-\mathrm{Im}\left[\frac{4t^2}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\right],\\ J_{\mathrm{eff}}^z&=-4t^2\mathrm{Re}\left[\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}+\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}-\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\right],\\ \Gamma^z&=-4t^2\mathrm{Im}\left[\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}+\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}-\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\right],\\ C&=-t^2\Bigl(\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}+\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}+\frac{1}{U_{\uparrow\downarrow}-i\gamma_{\uparrow\downarrow}}\Bigr),\\ h_r&=-2zt^2\mathrm{Re}\left[\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}-\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}\right],\\ h_i&=-2zt^2\mathrm{Im}\left[\frac{1}{U_{\uparrow\uparrow}-i\gamma_{\uparrow\uparrow}}-\frac{1}{U_{\downarrow\downarrow}-i\gamma_{\downarrow\downarrow}}\right].\end{aligned}$$ Here, $z$ denotes the coordination number of the lattice. For $U_{\uparrow\uparrow}=U_{\downarrow\downarrow}=U_{\uparrow\downarrow}=U$ and $\gamma_{\uparrow\uparrow}=\gamma_{\downarrow\downarrow}=\gamma_{\uparrow\downarrow}=\gamma$, the model for bosons reduces to the non-Hermitian Heisenberg model considered in the main text. If a bosonic system does not respect the spin SU($2$) symmetry, the non-Hermitian spin model is an XXZ model with complex-valued spin-spin interactions and a magnetic field. We note that the effective magnetic field has an imaginary part $h_i$ in general. In the Hermitian case, the real magnetic field $h_r$ can be compensated by an additional external magnetic field [@Duan03]. However, the imaginary magnetic field cannot be compensated by any real external field and thus inevitably affects the behavior of dissipative spin systems. Dependence of the dynamics on dissipation ========================================= In Fig. \[fig\_Jeff\], we show how the real and imaginary parts of the effective spin-exchange interactions, which are respectively given by $J_{\mathrm{eff}}=4Ut^2/(U^2+\gamma^2)$ and $\Gamma=4\gamma t^2/(U^2+\gamma^2)$, depend on the inelastic collision rate $\gamma$. The imaginary part reaches the maximum $\Gamma=0.5J$ at $\gamma/U=1$ and then decreases with increasing $\gamma$. The suppression of the effective dissipation rate $\Gamma$ at large $\gamma$ is attributed to the continuous quantum Zeno effect [@Syassen08; @Zhu14; @Tomita17], which freezes the hopping of atoms due to a large dissipation. On the other hand, the real part $J_{\mathrm{eff}}$ of the spin-exchange interaction monotonically decreases as a function of $\gamma$. The dependence of the dynamics of the Hubbard model on dissipation is shown in Fig. \[fig\_2site\_supple\]. Here we calculate the dynamics of the two-site non-Hermitian Fermi and Bose Hubbard models which can be realized with a double-well optical lattice as mentioned in the main text. When small dissipation is introduced to the system \[Figs. \[fig\_2site\_supple\](a)-(d)\], fast oscillations of the double occupancy and the spin correlation due to a large on-site repulsion $U$ are damped by dissipation. As the strength of dissipation is increased \[Figs. \[fig\_2site\_supple\](e)-(h)\], the development of the ferromagnetic (antiferromagnetic) spin correlation in the Fermi (Bose) system is accelerated by an increase in the imaginary part of the spin-exchange interaction $\Gamma$, which governs the time scale of the dissipative spin dynamics. At the optimal value $\gamma/U=1$ \[Figs. \[fig\_2site\_supple\](i)-(l)\], the fastest formation of the spin correlation is observed. We note that the double occupancy is gradually suppressed with increasing the dissipation \[see Figs. \[fig\_2site\_supple\](b), (f), and (j)\]. This behavior is a consequence of the continuous quantum Zeno effect, as mentioned in the main text. ![(a) $\gamma$-dependence of the imaginary part of the effective spin-exchange interaction. (b) $\gamma$-dependence of the real part of the effective spin-exchange interaction. Here $J$ is given by $J=4t^2/U$.[]{data-label="fig_Jeff"}](FigS1.pdf){width="10cm"} ![Dependence of the dynamics of the two-site dissipative Hubbard model on dissipation. (a), (e), (i) Time evolution of the squared norm $\braket{\psi(\tau)|\psi(\tau)}$. (b), (f), (j) Time evolution of the double occupancy $\bra{\psi(\tau)}\frac{1}{2}(n_{1{\uparrow}}^{(a)}n_{1{\downarrow}}^{(a)}+n_{2{\uparrow}}^{(a)}n_{2{\downarrow}}^{(a)})\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}$ ($a=f$ or $b$). The squared norm and the double occupancy take the same values for the Fermi and Bose Hubbard systems. (c), (g), (k) Time evolution of the spin correlation $\bra{\psi(\tau)}\bm{S}_1\cdot\bm{S}_2\ket{\psi(\tau)}/\braket{\psi(\tau)|\psi(\tau)}$ of the non-Hermitian Fermi-Hubbard model. (d), (h), (l) Time evolution of the spin correlation of the non-Hermitian Bose-Hubbard model. The strength of the interaction is set to $U/t=10$ in all figures, and the strength of dissipation is set to $\gamma/t=0.1$ in (a)-(d), $\gamma/t=1$ in (e)-(h), and $\gamma/t=10$ in (i)-(l). The unit of time is the inverse hopping rate $\tau_h=1/t$.[]{data-label="fig_2site_supple"}](FigS2.pdf){width="17.5cm"} Details of the quantum-trajectory method ======================================== The dynamics of a dissipative Hubbard model is simulated by using a quantum trajectory method [@Dalibard92; @Carmichael_book; @Daley14]. According to a random number $R_1$ taken from an interval $0\leq R_1\leq 1$, the system evolves under the nonunitary Schrödinger equation $i\partial_\tau\ket{\tilde{\psi}(\tau)}=H_{\mathrm{eff}}\ket{\tilde{\psi}(\tau)}$ up to a time $\tau_1$ when the squared norm $\braket{\tilde{\psi}(\tau_1)|\tilde{\psi}(\tau_1)}$ is equal to $R_1$. Here $H_{\mathrm{eff}}$ is the $N$-site non-Hermitian Fermi or Bose Hubbard Hamiltonian and we assume the periodic boundary condition. The Schrödinger dynamics is numerically calculated by exact diagonalization of $H_{\mathrm{eff}}$. At time $\tau_1$, a loss event takes place and the state is acted on by the quantum-jump operator $L_{j\sigma\sigma'}$ and then normalized: $$\ket{\tilde{\psi}(\tau_1+0)}=\frac{L_{j\sigma\sigma'}\ket{\tilde{\psi}(\tau_1-0)}}{\sqrt{\bra{\tilde{\psi}(\tau_1-0)}L_{j\sigma\sigma'}^\dag L_{j\sigma\sigma'}\ket{\tilde{\psi}(\tau_1-0)}}}.$$ The quantum-jump operator $L_{j\sigma\sigma'}$ for the loss event at $\tau_1$ is chosen according to the probability distribution $$\frac{\bra{\tilde{\psi}(\tau_1-0)}L_{j\sigma\sigma'}^\dag L_{j\sigma\sigma'}\ket{\tilde{\psi}(\tau_1-0)}}{\sum_{j,\sigma,\sigma'}\bra{\tilde{\psi}(\tau_1-0)}L_{j\sigma\sigma'}^\dag L_{j\sigma\sigma'}\ket{\tilde{\psi}(\tau_1-0)}}.$$ After $\tau_1$, we take another random number $R_2$ and repeat the above procedure. When a sufficiently large number $\mathcal{N}$ of quantum trajectories are sampled, the density matrix of the solution of the master equation is given by $$\rho(\tau)\simeq \frac{1}{\mathcal{N}}\sum_{a=1}^{\mathcal{N}}\ket{\psi_a(\tau)}\bra{\psi_a(\tau)}, \label{eq_trajrho}$$ where $\ket{\psi_a(\tau)}=\ket{\tilde{\psi}_a(\tau)}/\sqrt{\braket{\tilde{\psi}_a(\tau)|\tilde{\psi}_a(\tau)}}$ is a normalized state of the $a$-th quantum trajectory $(a=1,\cdots,\mathcal{N})$. The approximate equality becomes the exact one in the $\mathcal{N}\to\infty$ limit. Each quantum trajectory can be characterized by the number of quantum jumps. Let $N^{(n)}(\tau)$ be the number of quantum trajectories that involve $n$ quantum jumps between the initial time and $\tau$. Then, from Eq. , the density matrix conditioned on the number of quantum jumps is given by $$\begin{aligned} \rho^{(n)}(\tau)=&\frac{\mathcal{P}^{(n)}\rho(\tau)\mathcal{P}^{(n)}}{\mathrm{Tr}[\mathcal{P}^{(n)}\rho(\tau)\mathcal{P}^{(n)}]}\notag\\ \simeq&\frac{1}{N^{(n)}(\tau)}\sum_{a=1}^{N^{(n)}(\tau)}\ket{\psi^{(n)}_a(\tau)}\bra{\psi^{(n)}_a(\tau)},\end{aligned}$$ where $\ket{\psi^{(n)}_a(\tau)}\ (a=1,\cdots,N^{(n)}(\tau))$ denotes the normalized state of the $a$-th quantum trajectory that includes $n$ quantum jumps. The correlation function is thus calculated as $$\begin{aligned} C^{(n)}(i,j;\tau)=&\mathrm{Tr}[\rho^{(n)}(\tau)\bm{S}_i\cdot\bm{S}_j]\notag\\ \simeq&\frac{1}{N^{(n)}(\tau)}\sum_{a=1}^{N^{(n)}(\tau)}\bra{\psi^{(n)}_a(\tau)}\bm{S}_i\cdot\bm{S}_j\ket{\psi^{(n)}_a(\tau)}.\end{aligned}$$ Similarly, the conditional correlator $C^{(n)}_{\mathrm{proj}}(j,j+1;\tau)$ can also be calculated as $$\begin{aligned} C_{\mathrm{proj}}^{(n)}(j,j+1;\tau)=&\frac{\mathrm{Tr}[P_jP_{j+1}\rho^{(n)}(\tau)P_jP_{j+1}\bm{S}_j\cdot\bm{S}_{j+1}]}{\mathrm{Tr}[P_jP_{j+1}\rho^{(n)}(\tau)P_jP_{j+1}]}\notag\\ \simeq&\frac{\sum_{a=1}^{N^{(n)}(\tau)}\bra{\psi^{(n)}_a(\tau)}P_jP_{j+1}\bm{S}_j\cdot\bm{S}_{j+1}P_jP_{j+1}\ket{\psi^{(n)}_a(\tau)}}{\sum_{a=1}^{N^{(n)}(\tau)}\bra{\psi^{(n)}_a(\tau)}P_jP_{j+1}\ket{\psi^{(n)}_a(\tau)}}.\end{aligned}$$ In the numerical simulation, we use $\mathcal{N}=10000$ trajectories for the $8$-site dissipative Fermi-Hubbard model and $\mathcal{N}=40000$ trajectories for the $6$-site dissipative Bose-Hubbard model. In Fig. \[fig\_Ntraj\], we show the time evolution of the number of quantum trajectories $\mathcal{N}^{(n)}(\tau)$. For the case of the Fermi-Hubbard system, $N^{(n)}(\tau)$ for each $n$ remains a finite value even after the long time since the Fermi-Hubbard system has dark states, which are spin-symmetric Dicke states [@FossFeig12], in each particle-number sector. In particular, we have $N^{(n=0)}(\tau)\simeq 100$ trajectories with no quantum jump at $\tau/\tau_h=40$. On the other hand, the Bose-Hubbard system does not have a dark state except for the two-particle sector which corresponds to $n=2$ case in Fig. \[fig\_Ntraj\](b), since $N$ spins cannot form a perfect antisymmetric state except for $N=2$. As a result, $N^{(n=0)}(\tau)$ and $N^{(n=1)}(\tau)$ in Fig. \[fig\_Ntraj\](b) decay and vanish in the long-time limit. To achieve sufficient statistical convergence, we restrict the time to $\tau/\tau_h\lesssim 10$, for which we have $N^{(n=1)}(10\tau_h)\simeq 400$ trajectories. ![Time evolution of the number of quantum trajectories $N^{(n)}(\tau)$ that include $n$ quantum jumps between the initial time and $\tau$ for (a) the $8$-site dissipative Fermi-Hubbard model and (b) the $6$-site dissipative Bose-Hubbard model. The parameters and the initial states are the same as in Fig. \[fig\_withjump\] in the main text.[]{data-label="fig_Ntraj"}](FigS3.pdf){width="12cm"} Dynamics in single quantum trajectories ======================================= Figure \[fig\_traj\](a) (\[fig\_traj\](c)) shows the dynamics of the spin correlation $C(j,j+1;\tau)=\bra{\tilde{\psi}(\tau)}\bm{S}_j\cdot\bm{S}_{j+1}\ket{\tilde{\psi}(\tau)}/\braket{\tilde{\psi}(\tau)|\tilde{\psi}(\tau)}$ of the dissipative Fermi (Bose) Hubbard model calculated from a single quantum trajectory which involves a loss event. The parameters and the initial states are the same as in Fig. \[fig\_nojump\]. In Fig. \[fig\_traj\](a), a quantum-jump event takes place at $\tau/\tau_h\simeq 3$ and creates a hole at site $j=0$. In Fig. \[fig\_traj\](c), a quantum-jump event at $\tau/\tau_h\simeq 1.7$ annihilates one spin-up boson and one spin-down boson at site $j=0$. In both cases, the spin correlations after the quantum jump oscillate since the created holes move among the lattice sites and disturb the background spin configuration. After the ensemble average is taken, the oscillation disappears, and the spin correlation of the Fermi (Bose) system shows the formation of ferromagnetic (anfiferromagnetic) correlations, while the magnitude is reduced due to the effect of holes \[see Figs. \[fig\_withjump\](a) and (c) in the main text\]. In contrast, Fig. \[fig\_traj\](b) (\[fig\_traj\](d)) shows the conditional correlator $$C_{\mathrm{proj}}(j,j+1;\tau)=\frac{\bra{\psi(\tau)}P_jP_{j+1}\bm{S}_j\cdot\bm{S}_{j+1}P_jP_{j+1}\ket{\psi(\tau)}}{\bra{\psi(\tau)}P_jP_{j+1}\ket{\psi(\tau)}},$$ which is calculated from the same trajectories as those in Figs. \[fig\_traj\](a) (\[fig\_traj\](c)). Remarkably, although the ferromagnetic (antiferromagnetic) correlation, which develops through the dissipative spin-exchange mechanism, is disturbed once by a quantum jump, it starts to grow again and is finally saturated at the same value as in the case of no quantum jump. This indicates that the spin configuration after removing holes in the long-time limit is equivalent to that of the highest-energy state of the Heisenberg models as a consequence of spin-charge separation. ![(a) (c) Dynamics of spin correlations $C(j,j+1;\tau)$ in a single quantum trajectory which involves a quantum-jump event. (b) (d) Dynamics of conditional spin correlations $C_{\mathrm{proj}}(j,j+1;\tau)$ in the same trajectories. (a) and (b) show the results for the dissipative Fermi-Hubbard model, and (c) and (d) show the results for the dissipative Bose-Hubbard model. The parameters are the same as in Fig. \[fig\_nojump\]. The arrows indicate the time at which the quantum-jump event takes place. The unit of time is the inverse hopping rate $\tau_h=1/t$.[]{data-label="fig_traj"}](FigS4.pdf){width="12cm"} Figure \[fig\_jump2\] shows the dynamics of the spin correlation functions along a quantum trajectory with two jump events. Here the dissipative Fermi-Hubbard model with 8 sites is studied. The initial state is chosen to be the Néel state as in the main text. The first quantum-jump event at $\tau/\tau_h\simeq 3$ occurs at site $j=0$ and decreases the particle number from eight to six. Subsequently, the second two-body loss event takes place at site $j=4$ at time $\tau/\tau_h\simeq 4.7$, leaving four atoms in the system. As shown in Fig. \[fig\_jump2\](b), the conditional correlators involving sites at which the loss events take place are considerably affected by the quantum jumps (see $j=0$ and $j=3$ lines). Remarkably, the conditional correlators at the other sites are not quite disturbed (see $j=1$ and $j=2$ lines) and eventually saturated at the completely ferromagnetic value $C_{\mathrm{proj}}(j,j+1;\tau)=0.25$ in a time scale comparable with that along the quantum trajectory without loss events shown in Fig. \[fig\_nojump\](a) in the main text. Such a feature is not clearly observed in the standard correlators \[Fig. \[fig\_jump2\](a)\] and can be probed by the conditional correlators through quantum-gas microscopy. ![(a) Dynamics of spin correlations $C(j,j+1;\tau)$ of the dissipative Fermi-Hubbard model in a quantum trajectory involving two quantum jumps. (b) Dynamics of conditional spin correlations $C_{\mathrm{proj}}(j,j+1;\tau)$ along the same trajectory as that in (a). The parameters are the same as in Fig. \[fig\_nojump\]. The arrows indicate the times at which the quantum-jump events occur. The unit of time is the inverse hopping rate $\tau_h=1/t$.[]{data-label="fig_jump2"}](FigS5.pdf){width="12cm"}

--- abstract: | Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are subject to a non-unitary evolution $X$. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more non-unitary evolutions. We propose an alternative method by which unitarity of the evolution may be regained, by extending $X$ to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localised in regions spacelike separated from the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal. address: - | Department of Applied Mathematics and Theoretical Physics, University of Cambridge,\ Silver Street, Cambridge CB3 9EW, U.K.\ and Institut für theoretische Physik, Universität Bern,\ Sidlerstrasse 5, CH-3012 BERN, Switzerland.[^1] - | Department of Applied Mathematics and Theoretical Physics, University of Cambridge,\ Silver Street, Cambridge CB3 9EW, U.K. author: - 'C.J. Fewster[^2]' - 'C.G. Wells[^3]' date: '15 November 1994, Revised ' title: | Unitarity of Quantum Theory and\ Closed Time-like Curves --- Introduction ============ Various recent studies [@Boul; @FPS1; @Pol] of perturbative interacting quantum field theory in the presence of a compact region of closed timelike curves (CTCs) have concluded that the evolution from initial states in the far past of the CTCs to final states in their far future fails to be unitary, in contrast with the situation for free fields [@Boul; @FPS2; @GPPT]. The same conclusion has also been reached non-perturbatively for a model quantum field theory [@Pol2]. This presents many problems for the usual Hilbert space framework of quantum theory: as we describe in Section \[sect:nuqm\], the Schrödinger and Heisenberg pictures are inequivalent and ambiguities arise in assigning probabilities to events occurring before [@FPS1], or spacelike separated from [@Jacobson], the region of non-unitary evolution. The main reaction to these difficulties has been to abandon the Hilbert space formulation in favour of a sum over histories approach such as the generalised quantum mechanics of Gell-Mann and Hartle (see, e.g., [@GMH]). In particular, Hartle [@Hartle] has addressed the issue of non-unitary evolutions in generalised quantum mechanics. Nonetheless, it is of interest to see if the Hilbert space approach can be ‘repaired’ by restoring unitarity. Recently, Anderson [@Arley] has proposed that this be done as follows. Suppose a non-unitary evolution operator $X$ is defined on Hilbert space ${\cal H}$ with inner product $\langle\cdot\mid\cdot\rangle$. We assume that $X$ is bounded with bounded inverse. Anderson defines a new inner product $\langle\cdot\mid\cdot\rangle^\prime$ on ${\cal H}$ by $\langle\psi\mid\varphi\rangle^\prime = \langle X^{-1}\psi \mid X^{-1}\varphi \rangle$, and denotes ${\cal H}$ equipped with the new inner product as ${\cal H}^\prime$. Regarded as a map from ${\cal H}$ to ${\cal H}^\prime$, $X$ is clearly unitary.[^4] The essence of Anderson’s proposal is to restore unitarity by regarding $X$ in this way. Of course, one also needs to be able to represent observables as self-adjoint operators on both Hilbert spaces; Anderson has shown how this may be done by establishing a correspondence (depending on the evolution) between self-adjoint operators on ${\cal H}$ and those on ${\cal H}^\prime$. When only one non-unitary evolution is considered, this proposal is equivalent to remaining in the Hilbert space ${\cal H}$ and replacing $X$ by $U_X = (XX^*)^{-1/2}X$, i.e., the unitary part of $X$ in the sense of the polar decomposition [@RSi]. A curious feature of Anderson’s proposal emerges when one considers the composition of two or more consecutive periods of non-unitary evolution [@Anote]. If an evolution $Y$ is followed by $X$, one might expect that the combined evolution would be represented by the composition of the unitary parts, i.e., $U_X U_Y$. However, this does not generally agree with the unitary part of the composition, $U_{XY}$, and so there would be an ambiguity depending on whether one thought of the full evolution as a one-stage or two-stage journey. Anderson’s response to this is to argue that the second evolution should be treated in a different way, essentially (as we show in Section \[sect:And\]) by replacing $X$ by the unitary part of $X(YY^*)^{1/2}$. This removes the ambiguity mentioned above, but has the undesirable feature that the treatment of the second evolution depends on the first. In Section \[sect:And\], we will show that this leads to an operational problem for physicists living in a universe containing CTC regions. It is therefore prudent to seek other means by which unitarity can be restored. In this paper, we propose a method of unitarity restoration using the mathematical technique of [*unitary dilations*]{}. This is motivated by the simple geometric observation that any linear transformation of the real line is the projection of an orthogonal transformation (called an [*orthogonal dilation*]{} of the original mapping) in a larger (possibly indefinite) inner product space. To see this, note that any linear contraction on the line may be regarded as the projection of a rotation in the plane: the contraction in length along the $x$-axis, say, being balanced by a growth in the $y$-component. Similarly, a linear dilation on the line may be regarded as the projection of a Lorentz boost in two dimensional Minkowski space. This observation may be extended to operators on Hilbert spaces: it was shown by Sz.-Nagy [@Nagy] that any contraction (i.e., an operator $X$ such that $\|X\psi\|\le\|\psi\|$ for all $\psi$) has a unitary dilation acting on a larger Hilbert space. The theory was subsequently extended to non-contractive operators by Davis [@Davis] at the cost of introducing indefinite inner product spaces. Unitary dilations have previously found physical applications in the quantum theory of open systems [@Davies], and have also been employed by one of us in an inverse scattering construction of point-like interactions in quantum mechanics [@F1; @F2]. Put concisely, starting with a non-unitary evolution $X$, we pass to a unitary dilation of $X$, mapping between enlarged inner product spaces whose inner product may (possibly generically) be indefinite. The signature of the inner product is determined by the operator norm $\|X\|$ of $X$: if $\|X\|\le 1$, the enlarged inner product spaces are Hilbert spaces, whilst for $\|X\|>1$, they are indefinite inner product spaces (Krein spaces). Within the context of our proposal, it is therefore important to determine $\|X\|$ for any given CTC evolution operator. Essentially, the unitary dilation proposal performs the minimal book-keeping required to restore unitarity by asserting the presence of a hidden component of the wavefunction, which is naturally associated with the CTC region. These ‘extra dimensions’ are not accessible to experiments conducted outside the CTC region, but provide somewhere for particles to hide from view, whilst maintaining global unitarity. We will see that our proposal thereby circumvents the problems associated with non-unitary evolutions mentioned above. Of course, it is a moot point whether or not one should require a unitary evolution of quantum fields in the presence of CTCs; one might prefer a more radical approach such as that advocated by Hartle [@Hartle]. Our philosophy here is to determine the extent to which the conventional formalism of quantum theory can be repaired. The plan of the paper is as follows. We begin in Section \[sect:nuqm\] by describing the implications of non-unitarity for the Hilbert space formulation of quantum mechanics and then give a rigorous description of Anderson’s proposal in Section \[sect:And\], where we also note the operational problem mentioned above. In Section \[sect:udp\], we introduce our proposal for unitarity restoration, and show how composition may be treated within this context in Section \[sect:comp\]. In Section \[sect:conc\], we conclude by discussing the physical significance of our proposal. There are two appendices: Appendix A contains the proof of two results required in the text, whilst Appendix B describes yet another proposal for unitarity restoration based on tensor products. However, this proposal (in contrast to that advocated by Anderson, and our own) fails to remove the ambiguity noted by Jacobson [@Jacobson]. Non-Unitary Quantum Mechanics {#sect:nuqm} ============================= As we mentioned above, a non-unitary evolution raises many problems for the standard formalism and interpretation of quantum theory, some of which we now discuss. Firstly, the usual equivalence of the Schrödinger and Heisenberg pictures is lost. Given an evolution $X$ of states and an observable $A$, we would naturally define the evolved observable $A^\prime$ so that for all initial states $\psi$, the expectation value of $A^\prime$ in state $\psi$ equals the expectation of $A$ in the evolved state $X\psi$. Explicitly, we require $$\frac{\langle{\psi}\mid{A^\prime\psi}\rangle}{\langle\psi\mid\psi \rangle}= \frac{\langle X\psi\mid AX\psi\rangle}{\langle X\psi\mid X\psi \rangle} \label{eq:exp}$$ for all $\psi$ in the Hilbert space ${\cal H}$. If $X$ is unitary up to a scale (i.e., $X^*X=XX^*=\lambda\openone$, $\lambda\in{\Bbb R}^+$), then equation (\[eq:exp\]) is uniquely solved by the Heisenberg evolution $A^\prime=X^{-1}AX$. On the other hand, if $X$ is not unitary up to scale, then there is no operator $A^\prime$ satisfying (\[eq:exp\]) unless $A$ is a scalar multiple of the identity. For completeness, we give a proof of this fact. Defining $f(\psi)$ to equal the RHS of (\[eq:exp\]), and taking $\psi$ and $\varphi$ to be any orthonormal vectors, we note that linearity of $A^\prime$ entails $$f(\psi)+f(\varphi)=f(\psi+\varphi)+f(\psi-\varphi),$$ whilst linearity of $A$ implies $$\begin{aligned} f(\psi)\|X\psi\|^2 + f(\varphi)\|X\varphi\|^2 &=&\frac{1}{2}\left\{f(\psi+\varphi)\|X(\psi+\varphi)\|^2 \right. \nonumber \\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\left. f(\psi-\varphi)\|X(\psi-\varphi)\|^2\right\}.\end{aligned}$$ Multiplying $\varphi$ by a phase to ensure that $\langle X\psi\mid X\varphi\rangle$ is imaginary (and hence that $\|X(\psi\pm\varphi)\|^2=\|X\psi\|^2+\|X\varphi\|^2$), we combine these relations to obtain $$(f(\psi)-f(\varphi))(\|X\psi\|^2-\|X\varphi\|^2)=0 ,$$ which is clearly insensitive to the phase of $\varphi$ and therefore holds for all orthonormal vectors $\psi$ and $\varphi$. If $X$ is not unitary up to scale, we choose $\varphi$ and $\psi$ so that $\|X\psi\|\not=\|X\varphi\|$. Thus $f(\psi)=f(\varphi)=F$ for some $F$. It follows that $f(\chi)=F$ for all $\chi\perp{\rm span}\,\{\psi,\varphi\}$ (because $\|X\chi\|$ cannot equal both $\|X\psi\|$ and $\|X\varphi\|$) and hence for all $\chi\in{\cal H}$. Thus $A$ is a scalar multiple of the identity. Thus, the conventional equivalence of the Schrödinger and Heisenberg pictures is radically broken. If there are evolved states, there are no evolved operators, and [*vice versa*]{}. In addition, the Heisenberg picture places restrictions on the class of allowed observables. In order to preserve the canonical commutation relations, we take the evolution to be $A\rightarrow X^{-1}AX$; however, we also want to preserve self-adjointness of observables under evolution. Combining these two requirements, we conclude that $A$ must commute with $XX^*$ and therefore with $(XX^*)^{1/2}$ – the non-unitary part of the evolution in the sense of the polar decomposition. Thus, the claim attributed to Dirac [@Rovelli] that ‘Heisenberg mechanics is the good mechanics’ carries the price of a restricted class of observables when the evolution is non-unitary. A second problem with non-unitary evolutions, noted by Jacobson [@Jacobson] (see also Hartle’s elaboration [@Hartle]) is that one cannot assign unambiguous values to expectation values of operators localised in regions spacelike separated from the CTC region. Let $\cal R$ be a compact region spacelike separated from the CTCs, and which is contained in two spacelike slices $\sigma_+$ and $\sigma_-$, such that $\sigma_-$ passes to the past of the CTCs and $\sigma_+$ to their future. If $A$ is an observable which is localised within $\cal R$, one can measure its expectation value with respect to the wavefunction on either spacelike surface. In order for these values to agree, equation (\[eq:exp\]) must hold with $A^\prime=A$. If $X$ is unitary up to scale, this is satisfied by any observable which commutes with $X$ – in particular by all observables localised in $\cal R$. However, if $X$ is not unitary up to scale, our arguments above show that there is no observable (other than multiples of the identity) for which unambiguous expectation values may be calculated. Jacobson concludes that a breakdown of unitarity implies a breakdown of causality. Thirdly, Friedman, Papastamatiou and Simon [@FPS1] have pointed out related problems with the assignment of probabilities for events occurring before the region of CTCs. They consider a microscopic system which interacts momentarily with a measuring device before the CTC region and which is decoupled from it thereafter. The microscopic system passes through the CTC region, whilst the measuring device does not. However, the probability that a certain outcome is observed on the measuring device depends on whether it is observed before or after the microscopic system passes through the CTCs. This is at variance with the Copenhagen interpretation of quantum theory. The Anderson Proposal {#sect:And} ===================== We begin by giving a rigorous description of Anderson’s proposal [@Arley]. Let ${\cal H}$ be a Hilbert space with inner product $\langle\cdot \mid\cdot \rangle$ and suppose that the non-unitary evolution operator $X:{\cal H} \rightarrow {\cal H}$ is bounded with bounded inverse. We now define a quadratic form on ${\cal H}$ by $$q(\psi,\varphi) = \langle X^{-1}\psi \mid X^{-1}\varphi \rangle ,$$ which (because $(X^{-1})^*X^{-1}$ is positive and $X$ and $X^{-1}$ are bounded) defines a positive definite inner product on ${\cal H}$ whose associated norm is complete. Replacing $\langle\cdot\mid\cdot\rangle$ by this inner product, we obtain a new Hilbert space which we denote by ${\cal H}^\prime$. Because ${\cal H}^\prime$ coincides with ${\cal H}$ as a vector space, there is an identification mapping $\i:{\cal H} \rightarrow{\cal H}^\prime$ which maps $\psi\in{\cal H}$ to $\psi\in{\cal H}^\prime$. The inner product of ${\cal H}^\prime$ is $$\langle\psi\mid\varphi\rangle^\prime = \langle X^{-1}\i^{-1}\psi \mid X^{-1}\i^{-1}\varphi \rangle , \label{eq:IPprime}$$ for $\psi,\varphi\in{\cal H}^\prime$. The identification mapping is present because $X^{-1}$ is not, strictly speaking, defined on ${\cal H}^\prime$. As a minor abuse of notation, one can omit these mappings provided that one takes care of which inner product and adjoint are used in any manipulations. This is the approach adopted by Anderson. The advantage of writing in the identifications is that one cannot lose track of the domain or range of any operator, and adjoints automatically take care of themselves. From equation (\[eq:IPprime\]), it is clear that $\i X:{\cal H}\rightarrow {\cal H}^\prime$ (i.e., “$X$ regarded as a map from ${\cal H}$ to ${\cal H}^\prime$”) is unitary – the non-unitarity of $X$ is cancelled by that of $\i$. Anderson therefore adopts $\i X$ as the correct unitary evolution: in the Schrödinger picture, an initial state $\psi\in{\cal H}$ is evolved unitarily to $\i X\psi\in {\cal H}^\prime$. The next component in Anderson’s proposal concerns observables. Given an observable (e.g., momentum or position) represented as a self-adjoint operator $A$ on ${\cal H}$, one needs to know how this observable is represented on ${\cal H}^\prime$ in order to evolve expectation values in the Schrödinger picture. At first, one might imagine that $A$ should be carried over directly using the identification mapping to form $A^\prime = \i A\i^{-1}$. However, this idea fails because $\i A\i^{-1}$ is not self-adjoint in ${\cal H}^\prime$ unless $A$ commutes with $XX^*$: an unacceptable restriction on the class of observables. Instead, Anderson proposes that $A^\prime$ should be defined by $$A^\prime = \i R_X A R_X^{-1} \i^{-1} \label{eq:otra}$$ where $R_X = (XX^*)^{1/2}$ is self-adjoint and positive on ${\cal H}$. The operator $\i R_X$ is easily seen to be unitary, and it follows that $A^\prime$ is self-adjoint on ${\cal H}^\prime$. With this definition, the expectation value of $A$ in (normalised) state $\psi$ evolves as $$\langle\psi\mid A\psi\rangle \longrightarrow \langle \i X\psi \mid A^\prime \i X\psi \rangle^\prime = \langle U_X\psi\mid AU_X\psi\rangle ,$$ where $U_X=R_X^{-1}X$ is the unitary part of $X$ in the sense of the polar decomposition [@RSi]. So far, it appears that Anderson’s proposal is equivalent to Schrödinger picture evolution using $U_X$ in the original Hilbert space, or Heisenberg evolution $A\rightarrow U_X^{-1}AU_X$. However, one must be careful with this statement when one considers the composition of two consecutive periods of evolution, say $Y$ followed by $X$. We take both operators to be maps of ${\cal H}$ to itself, as required by Anderson [@Anote; @Aques]. Proceeding naïvely, we encounter the following problem: taking the unitary parts and composing, we obtain $U_XU_Y$, whilst composing and taking the unitary part (i.e., considering the evolution as a whole, rather than as a two stage process) we find $U_{XY}$. For consistency, we would require that these evolutions should be equal up to a complex phase $\lambda$. As we show in Appendix A, this is possible if and only if $X^*X$ commutes with $YY^*$ and $\lambda = 1$. Composition would therefore fail in general. In response to this, Anderson has proposed that composition be treated as follows [@Anote]. Suppose $Y:{\cal H}\rightarrow {\cal H}$ is the first non-unitary evolution, and apply Anderson’s proposal to form a Hilbert space ${\cal H}^\prime$ and an identification map $\j:{\cal H}\rightarrow {\cal H}^\prime$ so that $\j Y$ is unitary. The next step is to form the ‘push-forward’ $X^\prime$ of the operator $XR_Y$ to ${\cal H}^\prime$, which is defined by $$X^\prime = \j R_Y (XR_Y) R_Y^{-1}\j^{-1} = \j R_Y X \j^{-1}.$$ $X^\prime$ is decomposed as $R_{X^\prime} U_{X^\prime}$ in ${\cal H}^\prime$, and $U_{X^\prime}$ is ‘pulled back’ to ${\cal H}$ as $\widetilde{U_{X^\prime}} = R_Y^{-1}\j^{-1}U_{X^\prime}\j R_Y$. Anderson states that the correct composition law is to form the product $\widetilde{U_{X^\prime}}U_Y$. In fact, we can simplify this slightly, because $$\widetilde{U_{X^\prime}} = R_Y^{-1} \j^{-1} U_{X^\prime}\j R_Y = U_{R_Y^{-1}\j^{-1}X^\prime\j R_Y} = U_{X R_Y}$$ where we have used the fact that $U_{VXW} = V U_X W$ if $V$ and $W$ are unitary. Thus we can eliminate ${\cal H}^\prime$ from the discussion, and the composition rule is essentially to replace the second evolution by $U_{X R_Y}$ rather than $U_X$. This is certainly consistent: for $U_{X R_Y} = U_{XY U_Y^{-1}}= U_{XY} U_Y^{-1}$, and so $U_{X R_Y} U_Y = U_{XY}$. However, although this prescription is consistent, it has the drawback that one must know about the first non-unitary evolution in order to treat the second correctly (i.e., one must use $U_{X R_Y}$ rather than $U_X$). More generally, it is easy to see that, given $n$ consecutive evolutions $X_1,\ldots,X_n$, one should replace each $X_r$ by $U_{X_r R_{X_{r-1}\ldots X_2 X_1}}$ for $r\ge 1$, so one needs to know about all previous evolutions at each step. This gives rise to the following operational problem: suppose two physicists, $A$ and $B$ live in a universe with two isolated compact CTC regions corresponding to evolutions $Y$ and $X$ respectively. Suppose that $A$ knows about both evolutions, but $B$ only knows about $X$. Thus, if $A$ follows Anderson’s proposal, she replaces these evolutions by $U_Y$ and $U_{XR_Y}$ respectively. But $B$ would surely replace $X$ by $U_X$, which differs from $U_{XR_Y}$ unless $X^*X$ commutes with $YY^*$ (as a corollary of the Theorem in Appendix A). [*The two physicists treat the second evolution in different ways and will therefore compute different values for expectation values of physical observables in the final state.*]{}[^5] This shows that, in Anderson’s proposal, it is necessary to know about all non-unitary evolutions in one’s past in order to treat non-unitary evolutions in one’s future correctly. For completeness, let us see how this composition law appears in the formulation of Anderson’s proposal in which one modifies the Hilbert space inner product. Again we start with the evolution $Y$, and form the identification map $\j:{\cal H}\rightarrow {\cal H}^\prime$. In addition, we can treat the combined evolution $Z =XY$ using Anderson’s proposal to form a Hilbert space ${\cal H}^{\prime\prime}$ and identification map ${\rm k}:{\cal H}\rightarrow {\cal H}^{\prime\prime}$, such that ${\rm k} Z$ is unitary. The wavefunction is evolved from ${\cal H}$ to ${\cal H}^\prime$ using $\j Y$, and from ${\cal H}$ to ${\cal H}^{\prime\prime}$ using ${\rm k} Z$. Thus it evolves from ${\cal H}^\prime$ to ${\cal H}^{\prime\prime}$ under ${\rm k}Z(\j Y)^{-1} = \i \j X\j^{-1}$, where $\i = {\rm k}\j^{-1}$ is clearly the identification mapping between ${\cal H}^{\prime}$ and ${\cal H}^{\prime\prime}$. This evolution, which is forced upon us by the requirement that the wavefunction be evolved consistently, is exactly what arises from Anderson’s proposal applied to the operator $\j X\j^{-1}$ in ${\cal H}^\prime$. One might expect that observables would be transformed from ${\cal H}^\prime$ to ${\cal H}^{\prime\prime}$ using the rule (\[eq:otra\]) applied to this evolution. However, we will now show that this is not the case. An observable $A$ on ${\cal H}$ is represented as $A^\prime = \j R_Y A R_Y^{-1}\j^{-1}$ on ${\cal H}^\prime$, and by $A^{\prime\prime}= {\rm k} R_Z A R_Z^{-1}{\rm k}^{-1}$ on ${\cal H}^{\prime\prime}$. Thus, the transformation between $A^\prime$ and $A^{\prime\prime}$ is $$A^{\prime\prime} = {\rm k} R_Z R_Y^{-1}\j^{-1} A^\prime \j R_Y R_Z^{-1}{\rm k}^{-1}. \label{eq:A12}$$ Let us note that this is [*not*]{} the transformation law which follows from a naïve application of Anderson’s proposal to $\j X \j^{-1}$ in ${\cal H}^{\prime}$, which would be of form $$A^{\prime\prime} = \i R_W A^\prime R_W^{-1}\i^{-1} \label{eq:A12fm}$$ with $W = \j X \j^{-1}$. Indeed, the expression (\[eq:A12\]) cannot generally be put into this form for any $W$. For suppose that there exists some $W$ such that (\[eq:A12\]) and (\[eq:A12fm\]) are equivalent for all self-adjoint $A^\prime$. Then $R_W = \lambda\j R_Z R_Y^{-1} \j^{-1}$ for some $\lambda \in {\Bbb C}$ which may be re-written as $\j^{-1}R_W(\j^{-1})^* = \lambda R_Z R_Y$ using the unitarity of $\j R_Y$. The LHS is self-adjoint, so the lemma in Appendix A entails that $ZZ^*$ and $YY^*$ must commute, which is a non-trivial condition on $X$ and $Y$ when both are non-unitary. Hence in general, the transformation (\[eq:A12\]) is not of the form (\[eq:A12fm\]). Thus, for consistency to be maintained, the transformation rule for observables between ${\cal H}^{\prime}$ and ${\cal H}^{\prime\prime}$ takes a different form from that which holds between ${\cal H}$ and ${\cal H}^\prime$ or ${\cal H}^{\prime\prime}$. We regard this as an undesirable feature of Anderson’s proposal. The Unitary Dilation Proposal {#sect:udp} ============================= We begin by describing the theory of unitary dilations [@Nagy; @Davis]. Let ${\cal H}_1,\ldots,{\cal H}_4$ be Hilbert spaces and let $X$ be a bounded operator from ${\cal H}_1$ to ${\cal H}_2$. Then an operator $\hat{X}$ from ${\cal H}_1\oplus{\cal H}_3$ to ${\cal H}_2\oplus{\cal H}_4$ is called a [*dilation*]{} of $X$ if $X=P_{{\cal H}_2}\hat{X}|_{{\cal H}_1}$ where $P_{{\cal H}_2}$ is the orthogonal projector onto ${\cal H}_2$. In block matrix form, $\hat{X}$ takes form $$\hat{X} = \left(\begin{array}{cc} X & P \\ Q & R \end{array}\right). \label{eq:dilfm}$$ Our nomenclature follows that of Halmos [@Halmos]. Given $X:{\cal H}_1\rightarrow{\cal H}_2$, one may ask whether $X$ possesses a [*unitary*]{} dilation. It turns out that such a dilation always exists, although one must pass to indefinite inner product spaces if the operator norm $\|X\|$ of $X$ exceeds unity. One may construct a unitary dilation of $X$ as follows. Firstly, its departure from unitarity may be quantified with the operators $M_1=\openone-XX^*$ and $M_2=\openone-X^*X$. As a consequence of the spectral theorem, we have the intertwining relations $$X^*f(M_1)=f(M_2)X^*;\qquad X f(M_2)=f(M_1)X \label{eq:inter}$$ for any continuous Borel function $f$. The closures of the ranges of $M_1$ and $M_2$ are denoted ${\cal M}_1$ and ${\cal M}_2$ respectively. For $i=1,2$, we now define ${\cal K}_i={\cal H}_i\oplus{\cal M}_i$, equipped with the (possibly indefinite) inner product $[\cdot,\cdot]_{{\cal K}_i}$ given by $$\left[\left(\begin{array}{c} \varphi \\ \Phi\end{array}\right), \left(\begin{array}{c} \psi \\ \Psi\end{array}\right)\right]_{{\cal K}_i}= \langle\varphi\mid\psi\rangle + \langle\Phi\mid {\rm sgn}\, M_i\Psi\rangle,$$ where the inner products on the right are taken in ${\cal H}$ and ${\rm sgn}\, M_i=|M_i|^{-1}M_i$ where $|M_i|=(M_i^*M_i)^{1/2}$. It is easy to show that ${\rm sgn}\, M_i$ is positive if $\|X\|\le 1$, in which case $[\cdot,\cdot]_{{\cal K}_i}$ is positive definite; however, for $\|X\|>1$, the inner products above are indefinite, and ${\cal K}_1$ and ${\cal K}_2$ are [*Krein spaces*]{} (for details on the theory of operators in indefinite inner product spaces, see the monographs [@Bognar; @Azizov]). It is important to remember that the ${\cal K}_i$ also have a positive definite inner product from their original definition as a direct sum of Hilbert spaces.[^6] Thus a bounded linear operator $A$ from ${\cal K}_1$ to ${\cal K}_2$ has two adjoints: the Hilbert space adjoint $A^*$, and the Krein space adjoint, which we denote $A^\dagger$. It is a simple exercise to show that $A^\dagger$ is given by $$A^\dagger = J_1 A^* J_2 , \label{eq:adj}$$ where the operators $J_i$ defined on ${\cal K}_i$ are unitary involutions given by $J_i=\openone_{{\cal H}_i}\oplus {\rm sgn}\, (M_i)$. Next, we define a dilation $\hat{X}:{\cal K}_1\rightarrow{\cal K}_2$ of $X$ by $$\hat{X} = \left(\begin{array}{cc} X & -{\rm sgn}\,(M_1)|M_1|^{1/2} \\ |M_2|^{1/2} & X^*|_{{\cal M}_1} \end{array} \right), \label{eq:dil}$$ which has adjoint $\hat{X}^\dagger$ given by (\[eq:adj\]) as $$\hat{X}^\dagger = \left(\begin{array}{cc} X^* & {\rm sgn}\,(M_2)|M_2|^{1/2} \\ -|M_1|^{1/2} & {\rm sgn}\,(M_1)X|_{{\cal M}_2}{\rm sgn}\,(M_2) \end{array}\right).$$ It is then a matter of computation using the intertwining relations to show that $\hat{X}^\dagger\hat{X}=\openone_{{\cal K}_1}$ and $\hat{X}\hat{X}^\dagger=\openone_{{\cal K}_2}$. $\hat{X}$ is therefore a unitary dilation of $X$. The construction we have given is not unique. For suppose that ${\cal N}_1$ and ${\cal N}_2$ are Krein spaces, and that $U_i:{\cal M}_i\rightarrow{\cal N}_i$ are unitary (with respect to the indefinite inner products). Then $$\widetilde{X}= \left(\begin{array}{cc} \openone & 0 \\ 0 & U_2 \end{array}\right) \hat{X} \left(\begin{array}{cc} \openone & 0 \\ 0 & U_1^\dagger \end{array}\right) \label{eq:free}$$ is also a unitary dilation of $X$, mapping between ${\cal H}\oplus{\cal N}_1$ and ${\cal H}\oplus{\cal N}_2$. Because this just amounts to a redefinition of the auxiliary spaces, it carries no additional physical significance. One may show that all other unitary dilations of $X$ require the addition of larger auxiliary spaces than the ${\cal M}_i$ (for example, one could dilate $\hat{X}$ further). Thus $\hat{X}$ is the minimal unitary dilation of $X$ up to unitary equivalence of the above form. Having described the general theory, let us now apply it to the case of interest. For simplicity, we assume that the Hilbert spaces of initial and final states are identical, so ${\cal H}_1={\cal H}_2={\cal H}$. We also assume that the evolution operator $X$ is bounded with bounded inverse. If the initial hypersurface contains regions which are causally separate from the CTC region, we assume that $X$ has been normalised to be unitary on states localised in such regions. We point out that such exterior regions may not exist – even if the CTC region is itself compact. Consider, for example, a spacetime that is asymptotically (the universal cover of) anti-de Sitter space. In such a spacetime, hypersurfaces sufficiently far to the future and far to the past of the CTC region will be entirely contained within the CTC region’s light cone and there will be no exterior region on which to set up our normalisation. We may normalise the evolution operator on hypersurfaces for which an exterior region may be identified and extend arbitrarily to those surfaces where no such region exists. Indeed, it is entirely possible that every point in spacetime is contained in the light cone of the CTC region; in this case we give up any attempt to find a ‘physical’ normalisation for the evolution operator. The spaces ${\cal M}_1$ and ${\cal M}_2$ are defined as above. Note that we have the polar decomposition $X=(XX^*)^{1/2}U$, where $U$ is a [*unitary*]{} operator because $X$ is invertible. As a consequence of the intertwining relations, we have $$U M_2 = M_1 U$$ and hence that ${\cal M}_1=U{\cal M}_2$. Thus the ${\cal M}_i$ are isomorphic as Hilbert spaces. Moreover, $U$ is also unitary with respect to the indefinite inner products on the auxiliary spaces ${\cal M}_1$ and ${\cal M}_2$, which follows from the identity $U{\rm sgn}\, (M_2)={\rm sgn}\, (M_1)U$. We can therefore use the freedom provided by equation (\[eq:free\]) to arrange that the same auxiliary space is used both before and after the evolution. Our proposal is the following. Given a non-unitary evolution $X$, there exists an (indefinite) auxiliary space ${\cal M}$ (isomorphic to the ${\cal M}_i$) and a unitary dilation $\widetilde{X}:{\cal K}\rightarrow{\cal K}$ of $X$, where ${\cal K}={\cal H}\oplus{\cal M}$. We regard this as describing the full physics of the situation: on ${\cal K}$, the evolution is unitary, whilst its restriction to the original Hilbert space ${\cal H}$ yields the non-unitary operator $X$. The auxiliary space ${\cal M}$ represents degrees of freedom localised within the CTC region, not directly accessible to experiments outside.[^7] Observables are defined as follows. Given any self-adjoint operator $A$ on ${\cal H}$, we define the corresponding observable on ${\cal K}$: $$\widetilde{A} = \left(\begin{array}{cc} A & 0 \\ 0 & 0 \end{array}\right).$$ The form of $\widetilde{A}$ is chosen to prevent the internal degrees of freedom being probed from outside. Let us point out that many features of this proposal can only be determined in the context of a particular evolution $X$ and therefore a particular CTC spacetime. There are, however, various model independent features of our proposal, which we discuss below. [**Predictability**]{} Because the initial state involves degrees of freedom not present on the initial hypersurface (i.e., the component of the wavefunction in ${\cal M}$), it is clear that – as far as physical measurements are concerned – there is some loss of predictability in the final state. This problem can be circumvented by the requirement that the initial state should have no component in ${\cal M}$. [**Expectation Values**]{} Let us examine the evolution of the expectation value of $\widetilde{A}$. On the premise that the initial state has no component in ${\cal M}$ and takes the vector form $(\psi,0)^T$, the initial expectation value of $\widetilde{A}$ is $$\frac{\left[\left(\begin{array}{c} \psi \\ 0\end{array}\right), \widetilde{A}\left(\begin{array}{c} \psi \\ 0\end{array}\right) \right]_{{\cal K}_2}}{\left[ \left(\begin{array}{c} \psi \\ 0\end{array}\right), \left(\begin{array}{c} \psi \\ 0\end{array}\right) \right]_{{\cal K}_2}}= \frac{\langle \psi\mid A\psi\rangle}{\langle\psi\mid\psi\rangle},$$ i.e., the expectation value of $A$ in state $\psi$. After evolution, the expectation value is $$\frac{\left[\widetilde{X}\left(\begin{array}{c} \psi \\ 0\end{array}\right), \widetilde{A}\widetilde{X}\left(\begin{array}{c} \psi \\ 0\end{array}\right) \right]_{{\cal K}_2}}{\left[\widetilde{X} \left(\begin{array}{c} \psi \\ 0\end{array}\right), \widetilde{X}\left(\begin{array}{c} \psi \\ 0\end{array}\right) \right]_{{\cal K}_2}}= \frac{\langle X\psi\mid AX\psi\rangle}{\langle\psi\mid\psi\rangle}.$$ It is important to note that both denominators are equal to $\|\psi\|^2$ (because the full evolution is unitary) – this removes many of the problems encountered in Section \[sect:nuqm\]. In particular, let us return to the problem noted by Jacobson [@Jacobson], writing ${\cal R}$ for the region spacelike separated from the CTC region, and taking $X$ to be the evolution from states on $\sigma_-$ to states on $\sigma_+$. We assume (as in [@Jacobson]) that $X$ acts as the identity on ${\cal H}_{\cal R}$, the subspace of states supported in ${\cal R}$. Any local observable associated with ${\cal R}$ should vanish on the orthogonal complement of ${\cal H}_{\cal R}$ in ${\cal H}$: accordingly, it follows that $X^*AX=A$, and hence that the expectation value is independent of the choice of hypersurface ($\sigma_+$ or $\sigma_-$) on which it is computed. Thus Jacobson’s ambiguity is avoided for all local observables associated with regions spacelike separated from the causality-violating region. More generally, it is avoided for all observables $A$ such that $A=X^*AX$. This is satisfied if the range of $A$ is contained in ${\cal U}=\ker M_1\cap\ker M_2\subset{\cal H}$ and $A$ commutes with the restriction $X|_{\cal U}$ of $X$ to ${\cal U}$. In addition, the breakdown of the Copenhagen interpretation noted in [@FPS1] is avoided as a direct consequence of the unitarity of $\widetilde{X}$. [**Time Reversal**]{} Let us suppose the existence of an anti-unitary involution $T$ on ${\cal H}$ implementing time reversal. The [*time reverse*]{} $X_{\rm rev}$ of $X$ is given by $X_{\rm rev}=TXT^{-1}$; $X$ is said to be [*time reversible*]{} if $X_{\rm rev}=X^{-1}$. We would like to understand how the time reversal properties of $\hat{X}$ are related to those of $X$. For convenience we will work in terms of $\hat{X}$; the discussion may be rephrased in terms of $\widetilde{X}$ by inserting suitable unitary operators between the ${\cal M}_i$ and ${\cal M}$. First, we must define the time reversal of $\hat{X}$. The natural definition is $$(\hat{X})_{\rm rev} = \left(\begin{array}{cc} T & 0 \\ 0 & T|_{{\cal M}_2} \end{array}\right) \hat{X} \left(\begin{array}{cc} T^{-1} & 0 \\ 0 & (T|_{{\cal M}_1})^{-1} \end{array}\right),$$ which entails that time reversal and dilation commute in the sense that $(\hat{X})_{\rm rev} = \widehat{X_{\rm rev}}$. However, because dilation and inversion do not commute (i.e., $(\hat{X})^{-1} \not = \widehat{X^{-1}}$) unless $X$ is unitary, we find that a time reversible evolution $X$ does not generally yield a time reversible dilation: $$(\hat{X})_{\rm rev} = \widehat{X_{\rm rev}} = \widehat{X^{-1}} \not = (\hat{X})^{-1}.$$ Thus if $X$ is non-unitary and time reversible, then $\hat{X}$ is not time reversible. On the other hand, suppose that $\hat{X}$ is time reversible. Then $\widehat{X_{\rm rev}}=\widehat{X^*}$ from which it follows that $X$ would obey the modified reversal property $X_{\rm rev}=X^*$. It would be interesting to determine, for concrete CTC models, whether $X$ obeys the usual time reversal property $X_{\rm rev}=X^{-1}$ or the modified property $X_{\rm rev}=X^*$ (of course it might not obey either property). To summarise this section, we have seen how unitarity can be restored using the method of unitary dilations, thereby removing the problems associated with non-unitary evolutions. Any observable on ${\cal H}$ defines an observable in our proposal. Composition {#sect:comp} =========== We have described how a single non-unitary evolution may be dilated to a unitary evolution between enlarged inner product spaces. In what sense does our proposal respect the composition of two (or more) non-unitary evolutions? Let us consider two evolutions $X$ and $Y$ on ${\cal H}$ and their composition $XY$. We define the $M_i$ and ${\cal M}_i$ as before and introduce $N_1=\openone-YY^*$, $N_2=\openone-Y^*Y$ and ${\cal N}_i=\overline{{\rm Ran}\, N_i}$ to be the closure of the range of $N_i$ for $i=1,2$. As before, we can construct dilations $\hat{X}$ and $\hat{Y}$. However, because $\hat{X}: {\cal H}\oplus{\cal M}_1\rightarrow {\cal H}\oplus{\cal M}_2$ and $\hat{Y}: {\cal H}\oplus{\cal N}_1\rightarrow {\cal H}\oplus{\cal N}_2$, it is not immediately apparent how the dilations may be composed. The solution is to dilate both $\hat{X}$ and $\hat{Y}$ further, as follows: $\check{Y}:{\cal H}\oplus{\cal M}_1\oplus{\cal N}_1\rightarrow {\cal H}\oplus{\cal M}_1\oplus{\cal N}_2$ is given by $$\check{Y}=\left(\begin{array}{ccc} Y & 0 & -{\rm sgn}\, N_1 |N_1|^{1/2} \\ 0 & \openone_{{\cal M}_1} & 0 \\ |N_2|^{1/2} & 0 & Y^*|_{{\cal N}_1} \end{array}\right),$$ and $\check{X}:{\cal H}\oplus{\cal M}_1\oplus{\cal N}_2\rightarrow {\cal H}\oplus{\cal M}_2\oplus{\cal N}_2$ is given by $$\check{X}=\left(\begin{array}{ccc} X & -{\rm sgn}\, M_1 |M_1|^{1/2} & 0 \\ |M_2|^{1/2} & X^*|_{{\cal M}_1} & 0\\ 0 & 0 & \openone_{{\cal N}_2} \end{array}\right) .$$ The product $\check{X}\check{Y}$ is given by $$\check{X}\check{Y}=\left(\begin{array}{ccc} XY & -{\rm sgn}\, M_1 |M_1|^{1/2} & -X{\rm sgn}\, N_1 |N_1|^{1/2}\\ |M_2|^{1/2}Y & X^*|_{{\cal M}_1} & -|M_2|^{1/2}{\rm sgn}\, N_1 |N_1|^{1/2} \\ |N_2|^{1/2} & 0 & Y^*|_{{\cal N}_1} \end{array}\right),$$ and is a unitary dilation of $XY$, mapping from ${\cal H}\oplus{\cal M}_1\oplus{\cal N}_1$ to ${\cal H}\oplus{\cal M}_2\oplus{\cal N}_2$. This state of affairs is quite natural: we have argued that each CTC region carries with it its own auxiliary space (isomorphic to the ${\cal M}_i$ and the ${\cal N}_i$); one would therefore expect that the combined evolution should be associated with the direct sum of these auxiliary spaces. However, in order to show how our proposal respects composition, we need to show how the product $\check{X}\check{Y}$ is related to the dilation $\widehat{XY}$ arising from the prescription (\[eq:dil\]). To this end, we introduce $P_1=\openone-XYY^*X^*$, $P_2=\openone-Y^*X^*XY$ and ${\cal P}_i=\overline{{\rm Ran}\, P_i}$. Note that $$P_1=M_1+XN_1X^* \quad {\rm and}\quad P_2=N_2+Y^*M_2Y. \label{eq:iden}$$ Now let $$Q_1=\left(\begin{array}{c} |M_1|^{1/2} \\ |N_1|^{1/2}X \end{array} \right) \quad {\rm and}\quad Q_2=\left(\begin{array}{c} |M_2|^{1/2}Y \\ |N_2|^{1/2} \end{array} \right),$$ and define $U_i$ ($i=1,2$) on ${\rm Ran}\, |P_i|^{1/2}\subset{\cal P}_i$ by $U_i=Q_i|P_i|^{-1/2}$. The $U_i$ are easily seen to be isometries (with respect to the appropriate inner products) from their domains into ${\cal M}_i\oplus{\cal N}_i$ such that $Q_i|_{\overline{{\rm Ran}\, P_i}}=U_i|P_i|^{1/2}$. Provided that ${\cal Q}_i=\overline{Q_i\overline{{\rm Ran}\, P_i}}$ is orthocomplemented in ${\cal M}_i\oplus{\cal N}_i$ (in the indefinite inner product), one may then show that $${\rm P}_{{\cal H}\oplus {\cal Q}_2}\check{X}\check{Y}|_{{\cal H}\oplus{\cal Q}_1}= \left(\begin{array}{cc} \openone & 0 \\ 0 & U_2 \end{array}\right) \left(\begin{array}{cc} XY & -{\rm sgn}\, P_1 |P_1|^{1/2} \\ |P_2|^{1/2} & (XY)^*|_{{\cal P}_1} \end{array}\right) \left(\begin{array}{cc} \openone & 0 \\ 0 & U_1^\dagger \end{array}\right),$$ where ${\rm P}_{{\cal H}\oplus{\cal Q}_2}$ is the orthoprojector onto ${\cal H}\oplus{\cal Q}_2$. Thus $\check{X}\check{Y}$ is a dilation of an operator isometrically equivalent to $\widehat{XY}$. The isometries act non-trivially only on the auxiliary spaces and have no physical significance. The extra dimensions introduced by the dilation are also to be expected because the combined evolution $Z=XY$ may be factorised in many different ways; hence the two individual evolutions carry more information than their combination. The assumption that the ${\cal Q}_i$ are orthocomplemented is easily verified if the operators $U_i$ are bounded, for in this case, they may be extended to unitary operators on the whole of ${\cal P}_i$. Then ${\cal Q}_i$ is the unitary image of a Krein space and is orthocomplemented by Theorem VI.3.8 in [@Bognar]. $U_1$ is bounded if there exists $K$ such that $\|P_1\psi\|<\epsilon$ only if $\|M_1\psi\|+\|N_1X\psi\|<K\epsilon$ for all sufficiently small $\epsilon>0$. Similarly, $U_2$ is bounded if $\|P_1\psi\|<\epsilon$ only if $\|M_2Y\psi\|+\|N_2\psi\|<K\epsilon$ for all sufficiently small $\epsilon>0$. Physically, this equates to the reasonable condition that the combined evolution can be ‘almost unitary’ on a given state only if the individual evolutions are also ‘almost unitary’. As a particular instance of the above, we consider the case where $Y$ is unitary. The $N_i$ therefore vanish and the ${\cal N}_i$ are trivial; in addition, $P_1=M_1$ and $P_2=Y^*M_2Y$. The operator $\check{Y}$ is $$\check{Y}=\left(\begin{array}{cc} Y & 0 \\ 0 & \openone_{{\cal M}_1} \end{array}\right)$$ and $\check{X}=\hat{X}$. The combined evolution is thus $$\check{X}\check{Y} = \hat{X}\left(\begin{array}{cc} Y & 0 \\ 0 & \openone_{{\cal M}_1} \end{array}\right)$$ which is unitarily equivalent to $\widehat{XY}$ in the sense that $$\hat{X}\left(\begin{array}{cc} Y & 0 \\ 0 & \openone_{{\cal M}_1} \end{array}\right)= \left(\begin{array}{cc} \openone & 0 \\ 0 & Y \end{array}\right)\widehat{XY}.$$ We emphasise that the first factor on the RHS has no physical significance and is merely concerned with mapping the auxiliary spaces ${\cal P}_2$ to ${\cal M}_2$ in a natural way. To conclude this section, we make three comments. Firstly, note that if $A$ belongs to the class of observables which avoid the Jacobson ambiguity for each CTC region individually, then it also avoids this ambiguity for the combined evolution; for if $A=X^*AX=Y^*AY$, then certainly $A=Y^*X^*AXY$. Thus there is no ‘multiple Jacobson ambiguity’. Secondly, in our proposal one does not need to know the past history of the universe in order to evolve forward from any given time, because the auxiliary degrees of freedom associated with one CTC region are essentially passive ‘spectators’ during the evolution associated with any other such region. This is in contrast with the composition rule proposed by Anderson [@Anote]. Thirdly, one might ask [@Aques] what would happen if the non-unitary evolution was continuous rather than occurring in discrete steps. This question could be tackled using a suitable generalisation of the theory of unitary dilations of semi-groups discussed by Davies [@Davies]. Conclusion {#sect:conc} ========== We have examined Anderson’s proposal [@Arley] for restoring unitarity to quantum evolution in CTC spacetimes, and noted an operational problem arising when one considers the composition of two or more non-unitary evolutions. Instead, we have advocated a new method for the restoration of unitarity, based on the mathematical theory of unitary dilations, which does respect composition under certain reasonable conditions. Because unitarity is restored on the full inner product space, problems associated with non-unitary evolutions such as Jacobson’s ambiguity are avoided. Our philosophy here has been to regard the non-unitarity of $X$ as a signal that the full physics (and a unitary evolution) is being played out on a larger state space than is observed. This bears some resemblance to the situation in special relativity, where time dilation signals that one must pass to spacetime (and an indefinite metric) in order to restore an orthogonal transformation between reference frames. (Indeed, the Lorentz boost in two dimensional Minkowski space is precisely an orthogonal dilation of the time dilation effect). For our case of interest, the physical picture is that the auxiliary space ${\cal M}$ corresponds to degrees of freedom within the CTC region. Non-unitarity of the evolution signals that a particle cannot pass through the CTC region unscathed: part of the initial state becomes trapped in the auxiliary space corresponding to the CTCs. A similar conclusion is espoused by three of the authors of [@GPPTa]. In the case in which $X$ has norm less than or equal to unity (so that the full space ${\cal K}$ has a positive definite inner product), this effect has a relatively simple interpretation. Namely, there is a non-zero probability that an incident particle will never emerge from the CTC region. To see how this can occur, we note that computations of the propagator (see particularly [@Pol2]) proceed by requiring consistency of the evolution round the CTCs. We suggest that part of the incident state becomes trapped in order to achieve this consistency. On the other hand, perturbative calculations in $\lambda\phi^4$ theory by Boulware [@Boul] suggest that $\|X\|$ could well exceed unity. In this case, ${\cal K}$ is an indefinite Krein space, and it would apparently be possible that the ‘probability’ of the particle escaping from the CTC region could be greater than one. In principle, one might try to avoid this by seeking natural positive definite subspaces of the initial and final Krein spaces. The obvious choice would be to take the initial Hilbert space to be ${\cal H}$ and the final Hilbert space to be the image of ${\cal H}$ under $\widetilde{X}$. However, this may lead to some problems in defining observables on the final Hilbert space. If one decides to face the problem directly (which seems preferable), one would be forced to conclude that CTCs are incompatible with the twin requirements of unitarity [*and*]{} a Hilbert space structure. The initial and final state spaces would naturally be Krein spaces. This would not be entirely unexpected: studies of quantum mechanics on the ‘spinning cone’ spacetime [@JdSG] have concluded that the inner product becomes indefinite precisely inside the region of CTCs. ‘Probabilities’ greater than unity would denote the breakdown of the theory in a manner analogous to the Klein paradox (see the extensive discussion in the monograph of Fulling [@Full]) in which strong electrostatic fields force the Klein-Gordon inner product to be indefinite. In our case, it is the geometry of spacetime which leads us to an indefinite inner product. We expect that particle creation would occur in this case, as it does in the usual Klein paradox. The Klein paradox can be resolved by treating the electromagnetic field as a dynamic field, rather than as a fixed external field. Particles are created in a burst as the field collapses (unless it is maintained by some external agency). In our case it seems reasonable that, in the context of a full quantum theory of gravity, a burst of particle creation occurs and the CTC region collapses. This is essentially the content of Hawking’s Chronology Protection Conjecture [@Hawking]. Thus the emergence of Krein spaces in our proposal may be interpreted as a signal for the instability of the CTC spacetime. Finally, our treatment has been entirely in terms of states and operators; it would be interesting to see how it translates into density matrices and the language of generalised quantum mechanics [@GMH]. We thank Lloyd Alty, Arlen Anderson, Mike Cassidy, Andrew Chamblin, Atsushi Higuchi, Seth Rosenberg and John Whelan for valuable discussions. We also thank Malcolm Perry for useful discussions concerning ref. [@GPPTa]. In addition, CJF thanks Churchill College, Cambridge, the Royal Society and the Schweizerischer Nationalfonds for financial support. In this appendix, we prove the following [**Theorem**]{} [*Suppose $X$ and $Y$ are bounded with bounded inverses. Then $U_{XY}=\lambda U_X U_Y$ if and only if $X^*X$ commutes with $YY^*$ and $\lambda =1$.*]{} [*Proof:*]{} Starting with the sufficiency, we note that $Z=(X^*)^{-1} (X^*X)^{1/2}(YY^*)^{-1/2}X^{-1}$ is positive and squares to give $(XYY^*X^*)^{-1}$ (using the commutation property). It follows that $Z$ is equal to the unique positive square root of $(XYY^*X^*)^{-1}$ and hence that $$U_{XY} = (XYY^*X^*)^{-1/2}XY = (X^*)^{-1} (X^*X)^{1/2} (YY^*)^{-1/2}Y.$$ Using the fact that $(X^*)^{-1} (X^*X)^{1/2}=U_X$, we have proved sufficiency. To demonstrate necessity, we note that $U_{XY}=\lambda U_X U_Y$ only if $$X^*(XYY^*X^*)^{-1/2}X = \lambda (X^*X)^{1/2}(YY^*)^{-1/2}. \label{eq:only}$$ It follows that the RHS must be self-adjoint and positive. We now apply the following Lemma: [**Lemma**]{} [*Suppose that $A$ and $B$ are bounded with bounded inverses and self-adjoint, and suppose that $\alpha AB$ is self-adjoint and positive for some $\alpha\in{\Bbb C}$, $\alpha\not=0$. Then $\alpha=\pm 1$ and $A$ and $B$ commute.*]{} [*Proof:*]{} Because $\alpha AB$ is self-adjoint, we have $$\alpha AB = \alpha^* BA. \label{eq:comm}$$ Now note that $$\begin{aligned} \alpha^*(\alpha AB- z)^{-1} &=& \alpha^*(\alpha^* BA-z)^{-1} \nonumber \\ &=& \alpha B (\alpha AB -z\alpha^*/\alpha)^{-1}B^{-1}.\end{aligned}$$ Because $AB$ has non-empty spectrum on the positive real axis and because the resolvent $(\alpha AB- z)^{-1}$ is an analytic operator valued function of $z$ in ${\Bbb C}\backslash{\Bbb R}^+$, we conclude that $\alpha^*/\alpha$ must be real and positive. Accordingly, $\alpha=\pm 1$ and equation (\[eq:comm\]) implies that $A$ and $B$ commute. $\square$ In our case, this implies that $\lambda = \pm 1$ and that $X^*X$ commutes with $YY^*$. Moreover, because the two square roots on the RHS of equation (\[eq:only\]) are positive and commute, we conclude that $\lambda = 1$ in order that the RHS be positive. $\square$ Here, we consider another possible method for the restoration of unitarity which, however, suffers from problems related to Jacobson’s ambiguity. Instead of focussing on direct sums of Hilbert spaces, this proposal uses tensor products and always maintains a positive definite inner product. We start with $X:{\cal H}\rightarrow {\cal H}$, bounded with bounded inverse and non-unitary as before, and define a new Hilbert space ${\cal H}_X=(\openone\otimes X)\Sigma$, where $\Sigma\subset{\cal H}\otimes{\cal H}$ is the closure of the space of finite linear combinations of terms of form $\psi\otimes\psi$ for $\psi\in{\cal H}$. Similarly, we define ${\cal H}_{X^{-1}}=(\openone\otimes X^{-1})\Sigma$. Now define the operator $\tilde{X}=X\otimes X^{-1}$ restricted to ${\cal H}_X$. Clearly, $\tilde{X}(\psi\otimes X\psi)= \varphi\otimes X^{-1}\varphi$ where $\varphi=X\psi$, and so $\tilde{X}:{\cal H}_X\rightarrow{\cal H}_{X^{-1}}$. Moreover, $$\begin{aligned} \langle\tilde{X}(\psi\otimes X\psi)\mid\tilde{X}(\varphi\otimes X\varphi)\rangle &=& \langle X\psi\otimes\psi\mid X\varphi\otimes\varphi\rangle\nonumber\\ &=& \langle X\psi\mid X\varphi\rangle \langle\psi\mid\varphi\rangle \nonumber \\ &=& \langle\psi\otimes X\psi\mid \varphi\otimes X\varphi\rangle\end{aligned}$$ and therefore $\tilde{X}$ is a unitary operator from ${\cal H}_X$ to ${\cal H}_{X^{-1}}$. Let us examine the structure of this proposal in more detail. First, there is a natural transposition operation ${\cal T}$ on ${\cal H}\otimes{\cal H}$: ${\cal T} (\varphi\otimes\psi) =\psi\otimes\varphi$. It is easy to see that $\tilde{X}$ is the restriction of ${\cal T}$ to ${\cal H}_X$: hence all the information about $X$ is encoded into the definition of ${\cal H}_X$. Have we lost any information in this process? Suppose ${\cal H}_X={\cal H}_Y$ for two distinct operators $X$ and $Y$. Then $\openone\otimes Z$ is a bounded invertible linear map (though not necessarily unitary) of $\Sigma$ onto itself, where $Z=X^{-1}Y$. Because ${\cal T}$ restricts to the identity on $\Sigma$, we require $\psi\otimes Z\psi = Z\psi\otimes\psi$ for each $\psi\in{\cal H}$. Taking an inner product with $\phi\otimes\psi$ for some $\phi$, we obtain $$\langle\phi\mid\psi\rangle\langle \psi\mid Z\psi\rangle= \langle\phi\mid Z\psi\rangle \langle\psi\mid\psi\rangle.$$ Because $\phi$ is arbitrary, $\psi$ is therefore an eigenvector of $Z$. But $\psi$ was also arbitrary and therefore $Z=\lambda\openone$ for some constant $\lambda\in{\Bbb C}\backslash\{0\}$. Thus $Y=\lambda X$, so this construction loses exactly one scalar degree of freedom. Effectively, we have lost the (scalar) operator norm $\|X\|$ of $X$, but no other information. We have therefore restored unitarity at the price of introducing a second Hilbert space and correlations between the two. The evolution on the large space is unitary. This fits well with the picture of acausal interaction between the initial space and the CTC region in its future. The physical interpretation is as follows: the ‘time machine’ contains a copy of the external universe, which evolves backwards in time, starting with the final state of the quantum fields and ending with their initial state. It is impossible to prepare the initial state of the CTC region independently from the initial state of the exterior quantum fields. However, problems arise when observables are defined. Here, observables on the initial space are naturally defined to be self-adjoint operators on ${\cal H}\otimes{\cal H}$ with ${\cal H}_X$ as an invariant subspace (observables on the final space would have ${\cal H}_{X^{-1}}$ invariant). An operator of form $A\otimes B$ maps ${\cal H}_X$ to itself only if $B=XAX^{-1}$; combining this with the requirement of self-adjointness, one finds that $A$ must commute with $X^*X$ and its powers. Thus this proposal places restrictions on the class of allowed observables. The requirement that ${\cal H}_X$ be an invariant subspace for all observables was adopted so that our space of initial states is invariant under the unitary groups generated by observables (e.g. translations). If we relax this, and define observables to be self-adjoint operators on ${\cal H}\otimes{\cal H}$, it appears that $A\otimes\openone$ corresponds naturally to the operator $A$ on ${\cal H}$. However, this suffers from the ambiguity pointed out by Jacobson [@Jacobson]. D.G. Boulware, Phys. Rev. D[**46**]{} (1992) 4421 J.L. Friedman, N.J. Papastamatiou and J.Z. Simon, Phys. Rev. D[**46**]{} (1992) 4456 H.D. Politzer, Phys. Rev. D[**46**]{} (1992) 4470 J.L. Friedman, N.J. Papastamatiou and J.Z. Simon, Phys. Rev. D[**46**]{} (1992) 4442 D.S. Goldwirth, M.J. Perry, T. Piran and K.S. Thorne, Phys. Rev. D[**49**]{} (1994) 3951 H.D. Politzer, Phys. Rev. D[**49**]{} (1994) 3981 T. Jacobson, in [*Conceptual Problems of Quantum Gravity*]{}, ed. A. Ashtekar and J. Stachel, (Birkhäuser, Boston (1991)) J.B. Hartle, in [*Quantum Cosmology and Baby Universes: Proceedings of the 1989 Jerusalem Winter School for Theoretical Physics*]{}, ed. S. Coleman, J.B. Hartle, T. Piran and S. Weinberg, (World Scientific, Singapore, 1991) J.B. Hartle, Phys. Rev. D[**49**]{} (1994) 6543 A. Anderson, Phys. Rev. D[**51**]{} (1995) 5707 M. Reed and B. Simon, [*Methods of Modern Mathematical Physics Vol I: Functional Analysis*]{} (Academic Press, New York, 1972) Note added in proof to Ref. [@Arley] B. Sz.-Nagy, Acta Sci. Math. (Szeged) [**15**]{} (1953) 87 C. Davis, Acta Sci. Math. (Szeged) [**31**]{} (1970) 75 E.B. Davies, [*Quantum Theory of Open Systems*]{}, (Academic Press, London, 1976) C.J. Fewster, J. Phys. A [**28**]{} (1995) 1107 C.J. Fewster, [*Thin Wire Idealisations of the Dirac Equation in the Presence of a Flux Tube*]{} in preparation C. Rovelli, in [*Conceptual Problems of Quantum Gravity*]{}, ed. A. Ashtekar and J. Stachel, (Birkhäuser, Boston (1991)) P.R. Halmos, [*A Hilbert Space Problem Book*]{} (Van Nostrand, Princeton, 1967) J. Bognár, [*Indefinite Inner Product Spaces*]{} (Springer Verlag, Berlin, 1974) T.Ya. Azizov and I.S. Iokhvidov, [*Linear Operators in Spaces with an Indefinite Metric*]{} (Wiley, Chichester, 1989) A. Anderson, Private communication Preprint version of Ref. [@GPPT] P. de Sousa Gerbert and R. Jackiw, Commun. Math. Phys. [**124**]{} (1989) 229 S.A. Fulling, [*Aspects of Quantum Field Theory in Curved Space-Time*]{} LMS Student Texts 17 (Cambridge University Press, Cambridge, 1989) S.W. Hawking, Phys. Rev. D[**46**]{} (1992) 603 [^1]: Current Address [^2]: E-mail address: fewster@butp.unibe.ch [^3]: E-mail address: C.G.Wells@damtp.cam.ac.uk [^4]: We will give a more rigorous formulation of this statement in Section \[sect:And\]. [^5]: One [*can*]{} arrange that $A$ and $B$ agree if $A$ replaces $Y$ and $X$ by $U_{R_X Y}$ and $U_X$ respectively, because $U_{R_X Y} = U_X^{-1}U_{XY}$. However, this would require $A$ to know about $B$’s existence and ignorance of the first evolution. [^6]: In fact, this inner product determines the topology of ${\cal K}_i$. [^7]: [*Indirectly*]{}, we can infer their presence by analysing $X$.

--- author: - Nagisa Oi - Hideo Matsuhara - Kazumi Murata - Tomotsugu Goto - Takehiko Wada - Toshinobu Takagi - Youichi Ohyama - Matthew Malkan - Myungshin Im - Hyunjin Shim - Stephen Serjeant - Chris Pearson date: 'Received ; accepted xxx xx, xxxx' title: 'Optical – Near-Infrared catalogue for the $AKARI$ North Ecliptic Pole Deep Field.' --- INTRODUCTION ============ An understanding of the cosmic history of star formation is one of the most important aspects in the study of galaxies. It has been shown that the star formation rate density derived from UV and optical wavelengths at $z\sim1$ is one order of magnitude higher than in the local Universe [@1995ApJ...441...18M; @2001AJ....122..288H; @1997ApJ...486L..11C; @1999ApJ...519....1S; @2007ApJ...657..738L]. These results highlight the importance of studying the cosmic star formation activity at high redshift ($z\sim1$). Studies of the extragalactic background have suggested at least one third (or half) of the luminous energy generated by stars has been reprocessed into the infrared by dust , and that the contribution of infrared luminous sources to the cosmic infrared luminosity density increases with redshift, especially at $z>1$ . For luminous infrared galaxies ($L_{\rm IR}\hspace{0.3em}\raisebox{0.4ex}{$>$}\hspace{-0.65em}\raisebox{-.7ex}{$\sim$}\hspace{0.3em} 10^{11}L_{\odot}$), the SFR derived from UV continuum or optical diagnostics (H$\alpha$ emission line) is smaller than that inferred from the infrared or radio luminosity, which are free from dust extinction. The tendency becomes more conspicuous with increasing infrared luminosity. This suggests that more luminous galaxies, i.e., galaxies with larger SFR, are more heavily obscured by dust [@2001AJ....122..288H; @2001ApJ...558...72S; @2002ApJ...581..205H; @2006ApJ...637..227C; @2007ApJS..173..404B]. Therefore, to understand the star-formation history, it is essential to investigate star-formation in highly dust-obscured galaxies at high redshift ($z$1) in the mid-infrared region where dust extinction is less severe. The infrared space telescope $AKARI$ [@2007PASJ...59S.369M] was launched in February 2006 and the $AKARI$/$IRC$ [InfraRed Camera: @2007PASJ...59S.401O] obtained higher quality near – mid infrared data than the previous infrared space missions such as Infrared Astronomical Satellite [@1984ApJ...278L...1N] and Infrared Space Observatory . The $IRC$ has three channels (NIR, MIR-S, and MIR-L) with nine filters ($N2, N3, N4, S7, S9W, S11, L15, L18W$, and $L24$), and $AKARI$ carried out a deep survey program in the direction of North Ecliptic Pole (NEP), so-called $NEP$-$Deep$ $survey$ [hereafter, NEP-Deep: @2006PASJ...58..673M] with all the nine $IRC$ bands. Each single band observed $\sim$0.6 square degrees, and $\sim$ 0.5 square degrees circular area centered on $\alpha$ = 17$^h$55$^m$24$^s$, $\delta$ = $+$66$^{\circ}$37$^{'}$32$^{''}$ are covered by all the nine bands. The full width at half maximum (FWHM) of each IRC channel is $\sim$ 4 arcsec in NIR, $\sim$5 arcsec in MIR-S, and $\sim$6 arcsec in MIR-L. constructed a revised near to mid infrared catalogue of the $AKARI$ NEP-Deep. The 5$\sigma$ detection limits in the $AKARI$ NEP-Deep catalogue are 13, 10, 12, 34, 38, 64, 98, 105 and 266 $\mu$Jy at the $N2, N3, N4, S7, S9W, S11, L15, L18W$, and $L24$ bands, respectively. The real strength of the NEP-Deep survey lies in the unprecedented photometric coverage from 2 $\mu$m to 24 $\mu$m, critically including the wavelength domain between Spitzer$^{'}$s $IRAC$ [@2004ApJS..154...10F] and MIPS [@2004ApJS..154...25R] instruments from 8 $\mu$m to 24 $\mu$m, where only limited coverage is available from the peak-up camera on the IRS [@2004ApJS..154...18H]. Therefore, NEP-Deep survey has a great advantage to be able to study infrared galaxies without uncertainties of interpolation over the mid-infrared wavelength range. Furthermore, the NEP-Deep field is a unique region because many astronomical satellites have accumulated many deep exposures covering this location due to the nature of its position on the sky. X-ray observations by $Chandra$, UV observations by $GALEX$, and far-infrared observations by $Herschel$ were made toward the NEP-Deep field. In addition, sub-mm data from $SCUBA$-2 on the James Clerk Maxwell Telescope ($JCMT$) and radio data from Westerbork Synthesis Radio Telescope ($WSRT$) were also obtained in the NEP-Deep field . Thus this multi-band $AKARI$ NEP-Deep field data set provides a unique opportunity to study star formation of infrared luminous galaxies. Deep ancillary optical and near-infrared data covering the entire NEP-Deep region are essential for finding counterparts of sources detected with $AKARI$. In addition, multi-wavelength data are needed for accurate photometric redshift estimation, especially in covering the rest-frame wavelength range around the 4000$\AA$ break. Many studies have been made of the optical counterparts of the $AKARI$ sources. used the Subaru/Suprime cam (S-cam) to obtain deep optical images (28.4 mag for $B$-band, 28.0 mag for $V$, 27.4 mag for $R$, 27.0 mag for $i^{'}$, and 26.2 mag for $z^{'}$ in the AB magnitude system), but these data only cover one half of the NEP-Deep area. [@2007ApJS..172..583H] observed the entire NEP-Deep field with Canada France Hawaii Telescope (CFHT) in the $g^{'}$, $r^{'}$, $i^{'}$, and $z^{'}$-bands with limiting magnitudes of 26.1, 25.6, 24.7, and 24.0 mag (4$\sigma$) with 1 arcsec diameter apertures, however these images have insensitive areas resulting from gaps between CCD chips. [@2007AJ....133.2418I] observed the area with KPNO-2.1 m/FLAMINGOS in the $J$- and $K_{\rm s}$-bands, but the 3$\sigma$ detection limits were shallow (21.85 mag and 20.15 mag in the $J$ and $K_{\rm s}$-bands, respectively). Moreover, these data cover only about one third of the NEP-Deep field ($25^{'}\times30^{'}$). Figure \[fig:SEDexampleOptAKARI\] is an example of SEDs for star-forming galaxies, Mrk231, M82, and Arp220 if their redshifts are at $z=0.5$ with infrared luminosity of 10$^{11}L_{\odot}$ (left) or $z=1.0$ with infrared luminosity of 2$\times$10$^{12}L_{\odot}$. In order to optically identify Arp220 at $z=1$ (blue SED) which can be detected by $AKARI$, 26 – 23 mag \[AB\] for $g^{'}$–$z^{'}$ bands and 22 – 23 mag \[AB\] for near-infrared bands are required. In this study, we present deep optical $g^{'}$, $r^{'}$, $i^{'}$, and $z^{'}$ (orange square in Figure \[fig:NEPD\]) and near infrared $Y$, $J$, and $K_{\rm s}$ imaging data (red square in Figure \[fig:NEPD\]) in the NEP-Deep field using MegaCam [@2003SPIE.4841...72B] and the Wide-field InfraRed Camera [WIRCam; @2004SPIE.5492..978P] on the CFHT. We also combine existing $u^{*}$-band data to provide a band-merged catalogue based on the $z^{'}$-band. The catalogue will be used to analyze the $AKARI$ NEP sources. This paper is organized as follows; we describe our observations and data reduction procedures in $\S$2, and source extraction in each band in $\S$3. We present our band-merged catalogue in $\S$4. In $\S$5, we describe our method of star-galaxy separation based on colour-colour-criteria and the SExtractor stellarity index. We calculate photometric redshifts for the sources in our catalogue by Spectral Energy Distribution (SED) fitting which is presented in $\S$6. $\S$7 provides properties of $AKARI$ sources identified their optical counterparts in our catalogue. We summarize our results in the final section. ![Example of galaxies SEDs. Left panel shows the SEDs with redshift of 0.5 with infrared luminosity of 10$^{11}L_{\odot}$, while right panel is for the SEDs with redshift of 1.0 with infrared luminosity of 2$\times$10$^{12}L_{\odot}$. Red, Green and Blue solid lines represent SEDs of Mrk231, M82, and Arp220. Thick horizontal lines in pink and sky blue show the 5$\sigma$ detection limits of $AKARI$ $N2$ – $L24$ filters and ground based $g^{'}$ – $K_{\rm s}$ band 4 $\sigma$ limiting magnitudes which we newly observed (see Table \[tb:photometry\]). Filter response curves of CFHT/MegaCam, WIRCam, and $AKARI/IRC$ are shown at the bottom of the panels. Red, green, blue, pink, cyan, orange, and black dotted-lines represent $g^{'}$, $r^{'}$, $i^{'}$, $z^{'}$, $Y$, $J$, and $K_{\rm s}$, while red, green, blue, pink, cyan, yellow, black, orange, and gray solid lines are $AKARI/IRC$ $N2$ – $L24$ bands, respectively.[]{data-label="fig:SEDexampleOptAKARI"}](SEDexampleOptAKARI_z0.5_1_v2.ps){width="90mm"} ![Coverages of various surveys in the NEP-Deep field. The colour image in the middle is the three-colour near-infrared image made with the $IRC$ data (blue–2 $\mu$m, green–3 $\mu$m, red–4 $\mu$m). The orange box is the survey coverage of MegaCam $g^{'}$, $r^{'}$, $i^{'}$, and $z^{'}$, and the red box is that of WIRCam $Y$, $J$, and $K_{\rm s}$. The purple box represents the MegaCam $u^{*}$-band observation field .[]{data-label="fig:NEPD"}](NEPD.ps){width="90mm"} OBSERVATION and DATA REDUCTION {#sec:OBSERVATION and DATA REDUCTION} ============================== CFHT/MegaCam (optical: $g^{'}$, $r^{'}$, $i^{'}$, and $z^{'}$-bands) -------------------------------------------------------------------- We obtained optical ($g^{'}$, $r^{'}$, $i^{'}$, and $z^{'}$) imaging of nearly the entire NEP-Deep field with CFHT/MegaCam at the telescope prime focus. The observations were carried out in queue mode, spread over 22 photometric nights from 2011 April 27 to 2011 September 4, resulting in a total of 166 frames (PI: T. Goto). MegaCam is a wide-field imager composed of 36 2048$\times$4612 pixel CCDs, covering a full 1$\times$1 deg$^2$ field of view with a pixel scale of 0.187 arcsec. The MegaCam $g^{'}$, $r^{'}$, $i^{'}$, and $z^{'}$-band filters follow the original Sloan Digital Sky Survey [SDSS; @2000AJ....120.1579Y] filters[^1]. The pre-processing, including bias subtraction and flat-fielding of raw images, was carried out using the Elixir pipeline [@2004PASP..116..449M]. Elixir is a collection of programs and databases, specializing for MegaCam data. By stacking the Elixir-processed images, we produced a final mosaiced image using the AstrOmatic softwares including WeightWatcher[^2] [@2008ASPC..394..619M] for creating a weight map, SExtractor[^3] for extraction and photometry of sources, SCAMP[^4][@2006ASPC..351..112B] for astronomical calibration with 2MASS as an astrometric reference catalogue, and modified SWarp[^5] [@2002ASPC..281..228B] with a 3-sigma clipping function[^6] for stacking images. The images were scaled to have a photometric zero-point of 30.000 in AB magnitudes. Before the stacking, images were re-sampled using a Lanczos-3 6-tap filter with a 128 pixels mesh by Swarp. A left panel of Figure \[fig:FWHMDistribution\] shows the FWHM distribution of the individual images used for the stacking measured by the Elixir. We can see that the FWHM ranges for $r^{'}$ – $z^{'}$-bands are 0.60 – 1.20, and that for $g^{'}$-band is slightly worse (0.75 – 1.25) compared with the others. The FWHMs of final stacked images for $r^{'}$ – $z^{'}$ bands are $\sim$ 0.85 – 0.9 arcsec, and 0.96 for $g^{'}$-band. ![FWHM distribution of each flame we used in optical (left) and infrared (right). The solid, dashed, dotted, and dashed-dotted lines in the left panel show $g^{'}r^{'}i^{'}z^{'}$-bands, respectively, while the solid, dashed, and dotted lines in the right panel represent $YJK_{\rm s}$-bands. []{data-label="fig:FWHMDistribution"}](FWHM_megacam.ps "fig:"){width="50mm"} ![FWHM distribution of each flame we used in optical (left) and infrared (right). The solid, dashed, dotted, and dashed-dotted lines in the left panel show $g^{'}r^{'}i^{'}z^{'}$-bands, respectively, while the solid, dashed, and dotted lines in the right panel represent $YJK_{\rm s}$-bands. []{data-label="fig:FWHMDistribution"}](FWHM_wircam.ps "fig:"){width="50mm"} In this work we carefully exclude a false source contamination of our final catalogue. During this data reduction, we removed peaked isolated noise using the normalized Median Absolute Deviation [MAD; @1990AJ....100...32B]. We considered 15 pixels surrounding each pixel in each image, and calculated a median and MAD values. If the original pixel value deviated by more than 3 MAD from the median value, it was regarded as bad, and was replaced with the median value. Since fake sources lying astride over a few pixels or along artificial structures close to bright sources cannot be removed by the MAD method, we use those weight maps during source extraction by SExtractor (see $\S$\[sec:SExtractor\]) and carry out band-merging procedure (see $\S$\[sec:Band Merging and Quality Frag\]) to reduce the false source contaminations. CFHT/WIRCam (Near-Infrared: $Y$, $J$, and $K_{\rm s}$-bands) {#sec:wircam} ------------------------------------------------------------ We used the CFHT/WIRCam to obtain deep near-infrared ($Y$, $J$, and $K_{\rm s}$) images at the NEP-Deep field. WIRCam uses four $2048 \times 2048$ HAWAII2RG CCD arrays. The camera provides 0.3 arcsec per pixel sampling, and the close packing enables coverage of an almost contiguous field of view of $20.5^{'} \times 20.5^{'}$. Observations in $Y$, $J$, and $K_{\rm s}$-bands were conducted in queue mode spread over 19 photometric nights from 2010 March 31 to 2010 June 29, resulting in a total of 2009 frames (PI: T. Goto). We observed four positions with dithering, covering completely a 47$^{'}\times44^{'}$ field of view in each band. The pre-processing, removal of the instrumental signature and astrometry for raw WIRCam images, was carried out by the $^{'}$I$^{'}$iwi[^7] preprocessing pipeline. In general, photometric calibration by SCAMP is based on both internal pairing among frames of overlapping regions with SExtracted source catalogues of WIRCam itself and external pairing with reference catalogues with SExtracted source catalogues of WIRCam. However, we found the internal pairing method did not work well when cross checking the uniformity of the photometric zero point across the SWarp stacked images by using the SCAMP photometric solution. Therefore, in this paper, we only used the external pairing method. Photometric calibration of the $J$ and $K_{\rm s}$-band observations were performed by matching to the corresponding 2MASS catalogue, the weighted images were then stacked using the modified SWarp. We compared 2MASS sources with photometric errors of less than 0.05 mag, $J<16.48$ mag and $K_{\rm s}<15.90$ mag, respectively, with the extracted sources of WIRCam. We found the zero point and its uniformity within the 47$\times$44 arcmin$^2$ area for the $J$ and $K_{\rm s}$-bands to be 25.964 mag (0.072 mag) and 26.927 mag (0.073 mag), respectively. Since the 2MASS catalogue does not have $Y$-band magnitudes, we did the photometric calibration of the $Y$-band by comparing with the predicted $Y$-band magnitudes estimated by SED fitting of stars. For the purpose of creating the predicted $Y$-band magnitude catalogue as the external reference, first we created a band-merged catalogue (see §\[sec: z base catalogue\]) and selected stars with a stellarity index given by SExtractor, CLASS\_STAR parameter in the $z^{'}$-band image, and appropriate $u^{*}-J$ and $g^{'}-K_{\rm s}$ colours (see §\[S-G separation\]). We calculated the predicted $Y$-band magnitude based on the $\chi^2$ fitting analysis of the SED using a publicly available $LePhare$ code, with 154 star templates, which include main sequence stellar SEDs from [@1998PASP..110..863P] and white dwarf SEDs from [@1995AJ....110.1316B]. For the SED fitting, we input the photometric data sets from $u^{*}$ to $K_{\rm s}$-bands omitting the $Y$-band. Then we calibrated the photometry with the predicted $Y$-band magnitude as the external reference catalogue of SCAMP, and made the final images with the modified SWarp. In Figure \[fig:Yzeromag\], we compared the observed $Y$-band magnitudes from the final image with the predicted magnitudes based on the SED fitting for the selected stars. The figure shows that both values are agreed with each other. We obtained the photometric zero point and the root mean square (RMS) in the $Y$-band as 25.715 mag and 0.095 ($Y<23$mag). Before the final stack, we re-samplied WIRCam images using the Lanczos-3 6-tap filter with a 128 pixels mesh by Swarp as well as MegaCam imagers. And then we adopted the threshold of FWHM $<$ 1.2 arcsec estimated by SExtractor to removed the bad quality images. The FWHM distributions we used for the final stacked images are shown in the right panel of Figure \[fig:FWHMDistribution\], and the seeing of the final stacked images of $Y$, $J$, and $K_{\rm s}$-bands are all $\sim$ 0.7 – 0.8 arcsec. PROCEDURE for SOURCE EXTRACTION {#sec:PROCEDURE for SOURCE EXTRACTION} =============================== Source Extraction {#sec:SExtractor} ----------------- Name Value ------------------ ---------------------- DETECT\_THRESH 1.5 DETECT\_MINAREA 5 DEBLEND\_MINCONT 0.001 FILTER\_NAME gauss\_3.0\_7x7.conv WEIGHT\_TYPE MAP\_WEIGHT : Parameters of SExtractor. \[tb:para-sex\] ![Examples of weight maps for MegaCam $z^{'}$-band (left) and $K_{\rm s}$-band (right), respectively. The pixel values are inversely proportional to the local variance of the image, and the areas of the pixel value of 0 are the regions without data. Crosses of clear less-weighted areas in MegaCam data are present due to the gaps between detectors, while WIRCam data are relatively homogeneous. []{data-label="fig:WeightMaps"}](untitled.eps){width="90mm"} ------------------------------- --------- --------- --------- --------- -- -------- -------- ------------- $g^{'}$ $r^{'}$ $i^{'}$ $z^{'}$ $Y$ $J$ $K_{\rm s}$ FWHM (arcsec) 0.96 0.88 0.84 0.86 0.80 0.81 0.66 Total integration time \[ks\] 23.2 17.6 27.6 23.2 21.2 19.9 18.3 Detection number 224,449 178,191 143,318 88,666 39.063 34,138 38,053 Limiting mag \[AB\] 26.7 25.9 25.1 24.1 23.4 23.0 22.7 50% Completeness \[AB\] 26.1 25.4 24.9 23.7 22.8 22.4 22.3 MAG\_APER – MAG\_AUTO \[AB\] 0.55 0.56 0.49 0.53 0.48 0.47 0.38 ------------------------------- --------- --------- --------- --------- -- -------- -------- ------------- \[tb:photometry\] \ For the detection of objects in MegaCam and WIRCam images, we used SExtractor. Seeing is not homogeneous across bands (see Figure \[fig:FWHMDistribution\] and in Table \[tb:photometry\]). In order to carry out photometry for each band with optimal aperture, we ran SExtractor with each band image individually with same parameter set. The SExtractor parameter set for all bands is summarized in Table \[tb:para-sex\]. A 7pix$\times$7pix Gaussian Point Spread Function (PSF) convolution filter with FWHM = 3.0 pixels was used for helping to detect faint sources. Fluxes of extracted sources were measured using SExtractor with elliptical apertures of 2.5 times of the Kron radius [@1980ApJS...43..305K], i.e., SExtractor’s MAG\_AUTO. We used weight maps created during SWarp process to avoid false detections. Figure \[fig:WeightMaps\] is an example of weight maps (left: MegaCam $z^{'}$, right: WIRCam $K_{\rm s}$). The MegaCam $z^{'}$-band weight map shows four less weighted regions in a cross direction and eight minor less weighted regions in longitudinal direction due to the detector gaps. This trend for all our MegaCam data are very similar to that of $z^{'}$-band. While the WIRCam $K_{\rm s}$-band weight map looks nicely uniform and no distinct less weighted gaps comparing with the MegaCam data. The trend for $Y$ and $J$-ban data are almost the same as that of $K_{\rm s}$-band. Data at the less weighted area are relatively noisier than the other area, and noise causes false detections. Weight maps affect error estimates and reduce the number of false detections at the noisy region. From extracted source catalogue for each band, we removed sources with a photometric flag of 4 or larger from the detected source list because these sources can be unreliable indicating that at least one pixel is saturated or close to the image boundary or both. The number of sources in the extracted catalogue for each band is given in Table \[tb:photometry\]. Figure \[fig:NumbercountCompleteness\] (except bottom right panel) shows number counts for all our seven band data. Number counts from [@2007ApJS..172..583H] at NEP-Wide field and those from 2MASS data around NEP region are over plotted in the figure as references of total (star + galaxy) number counts (blue open squares). We found that our total number counts are consistent with those surveys’ results. We also calculated the number counts of stars at around NEP-region using a TRILEGAL galaxy model [@2012rgps.book..165G] in each band, and then we subtracted it from the total number counts to estimate galaxy number counts (cyan filled circles). The number counts are compared with CFHTLS T0007[^8] for MegaCam data and with WIRCam Deep survey for WIRCam $J$ and $K_{\rm s}$-bands. At the bright magnitude range, our galaxy number counts in $g^{'}$-band is slightly larger compared with those from CFHTLS, while in $z^{'}$, and $J$-bands our results are slightly smaller than that of the comparison fields. These differences are about 15% and it can be explained an uncertainly of the TRILEGAL models. At the fainter magnitude range, where the effect from star number counts is weak, our results and their comparisons are excellently agreed with each other. Detection Limits and Completeness --------------------------------- We determined the 4$\sigma$ detection limits over a circular aperture of 1 arcsec radius. The detection limit changes depending on the source position on the mosaiced images since the exposure map is not uniform. We present the median values over the entire field for all MegaCam and WIRCam bands in Table \[tb:photometry\]. The limiting magnitudes are 26.7, 25.9, 25.1 and 24.1 mag for MegaCam $g^{'}$, $r^{'}$, $i^{'}$ and $z^{'}$-bands, and 23.4, 23.0, 22.7 mag for WIRCam $Y$, $J$, $K_{\rm s}$-bands, respectively. The completeness of each band was estimated via simulations with artificial sources. We produced 2,000 artificial sources per 0.2 magnitude with the averaged radial profile of 20 point sources in each band. We separated the artificial sources by at least 60 pixels from each other. We ran SExtractor on each image with the artificial sources by utilizing exactly the same extraction parameters as for the original images. A success of the detection of those artificial sources is judged based on source location (within 1 arcsec from the true location) and magnitude (within 0.5 mag from the true magnitude). The 50% completeness limits are listed in Table \[tb:photometry\] and the completeness curves are shown in right panel of Figure \[fig:NumbercountCompleteness\] as a function of magnitude. The completeness curves show that the detection probability begins to drop rapidly at around 90% value, and the magnitude difference between 90% and 10% completeness is about 2 mag. The limiting magnitudes listed in Table \[tb:photometry\] are actually for point sources because the values represent the 4$\sigma$ detection within an 1 arcsec aperture radius. Over 99% of the flux of a point source is included within the aperture (FWHM = 0.8–0.9 arcsec), while an extended galaxy at low-redshift with magnitude equal to the limiting magnitude cannot be detected with 4 $\sigma$ since such galaxies are spatially resolved in our images. We measured the difference between the 1 arcsec radius aperture magnitude (MAG\_APER) and the total magnitude (MAG\_AUTO). In $\S$\[sec:PHOTOMETRIC REDSHIFT\], we calculate the photometric redshift via SED fitting and find the peak redshift distribution of our data to be $z\sim0.5$. We then compared the MAG\_APER with MAG\_AUTO for galaxies at $z\sim0.5$ ($0.45<z<0.55$) as a typical example in our data. The differences in the all images is around 0.38–0.56 mag. We, therefore, note a limiting magnitude for extended sources is typically brighter by $\sim$0.5 mag. BAND-MERGED catalogue based on $z^{'}$-BAND {#sec: z base catalogue} =========================================== In this section, we describe the processes of making the band-merged catalogue. One of our aims of this work is to construct a $z^{'}$-band based band-merged catalogue and find counterparts of $AKARI$ infrared sources . We use optical data as a base-band for the band-merged catalogue since the optical data cover the $AKARI$ NEP-Deep field wider than the near-infrared ones. Since a redder band is much more useful to identify $AKARI$ sources because bluer bands can be affected by heavy dust extinction, the $z^{'}$-band is adopted as the base-band, although it is shallower than the bluer bands. Since $u^{*}$-band photometric data are important for the photometric redshift estimation in §\[sec:PHOTOMETRIC REDSHIFT\], in this work we merged not only the newly obtained MegaCam 4 bands and WIRCam 3 bands, but also the $u^{*}$-band catalogue constructed by . This $u^{*}$-band image is centered at R.A. = 17$^h$55$^m$24$^s$, Dec. = $+$66$^{\circ}$37$^{'}$32$^{"}$ \[$J$2000\] and has a limiting magnitude of 24.6 mag \[5$\sigma$ ; AB\]. Details of the source extraction and the photometry for $u^{*}$-band are described in §2.2.2 of . Astrometric Accuracy {#sec:Astrometry} -------------------- We cross-matched sources in all the bands to create the band-merged catalogue based on the $z^{'}$-band and checked the astrometric accuracy by examining the coordinate offset with respect to 2MASS and our other band catalogues. Counterparts in each catalogue for $z^{'}$-band sources were searched within an 1 arcsec radius. We found 2334 counterpart of 2MASS sources. The spatial distribution of the offset, mean offsets and RMS offset, are 0.013 arcsec and 0.32 arcsec, respectively. For the other bands, the RMS for $u^{*}$-band is 0.18 arcsec, and 0.08–0.11 arcsec in the other bands. The large RMS in $u^{*}$-band is probably due to a slightly degraded PSF. All the offsets and the RMS offsets are summarized in Table \[tb:3sigma-RADECoffset\], and Figure \[fig:RADEC-Z-GRIu\] shows examples of the relative positions between the $z^{'}$-band and 2MASS, $u^{*}$, and $K_{\rm s}$-bands. ------------------------- ------------- -------- -- ----------- -------- Band $z^{'}$ vs. 2MASS 0.0018 0.2067 0.0129 0.2420 $u^{*}$ vs. $z^{'}$ $-$0.0043 0.1114 0.0098 0.1438 $g^{'}$ vs. $z^{'}$ $-$0.0041 0.0733 0.0107 0.0821 $r^{'}$ vs. $z^{'}$ $-$0.0024 0.0645 0.0032 0.0641 $i^{'}$ vs. $z^{'}$ $-$0.0019 0.0591 0.0016 0.0583 $Y$ vs. $z^{'}$ $-$0.000003 0.0613 0.0002 0.0630 $J$ vs. $z^{'}$ 0.0002 0.0620 0.0003 0.0626 $K_{\rm s}$ vs. $z^{'}$ 0.0003 0.0623 $-$0.0001 0.0628 ------------------------- ------------- -------- -- ----------- -------- : Accuracy of astrometry in arcsec. Column(1): Pair of bands. Column(2) and (3): R.A. offset and RMS, respectively. Column(4) and (5): DEC. offset and RMS, respectively. \[tb:3sigma-RADECoffset\] \ Band Merging {#sec:Band Merging and Quality Frag} ------------ To identify counterparts with cross-matching between $z^{'}$-band and the other bands, we adopted a 0.5 arcsec search radius that corresponds to 3 times the RMS of astrometric accuracy between $z^{'}$-band and $u^{*}$-band, i.e., the pair that gave the worst coordinate matching accuracy. This small search area size helped us to minimize the chance coincidence rate (see $\S$\[sec: false detection\]). After the source matching, we compiled all the catalogues into a single band-merged catalogue based on the $z^{'}$-band sources. To ensure a robust catalogue against false sources and transients (e.g., time variable objects and asteroids), we excluded sources with only $z^{'}$-band detection. As a result, the catalogue includes 85$\hspace{0.1em}$797 out of 88$\hspace{0.1em}$666 $z^{'}$-band sources. The number of matched sources between $z^{'}$-band and the other bands are summarized in Table \[tb:matching\]. In Table \[tb:final-catalogue\], we summarize the statistics of sources with multiple band detections for all 8 bands in the optical and near-infrared together. ----------------- --------- --------- --------- --------- -- -------- -------- ------------- $u^{*}$ $g^{'}$ $r^{'}$ $i^{'}$ $Y$ $J$ $K_{\rm s}$ Total 55,145 75,961 79,454 83,295 34,162 31,002 31,634 Case A (flag=0) 55,063 75,959 79,452 83,286 34,160 31,000 31,634 Case B (flag=1) 82 0 2 5 0 0 0 Case C (flag=2) 0 2 0 4 2 2 0 ----------------- --------- --------- --------- --------- -- -------- -------- ------------- \[tb:matching\] \# of detected bands 2 3 4 5 6 7 8 total ---------------------- ------- ------- -------- -------- ------- ------- -------- -------- \# of sources 2,583 3,725 17,214 28,727 6,108 8,550 18,890 85,797 fraction 0.03 0.04 0.20 0.34 0.07 0.10 0.22 1 \[tb:final-catalogue\] During the matching of the sources, we identified three types of counterpart identifications: (case A) a single $z^{'}$-band source has a unique counterpart, (case B) a single $z^{'}$-band source has multiple counterpart candidates, and (case C) multiple $z^{'}$-band sources have one common counterpart candidate. Due to the small searching area (0.5 arcsec radius), a significant fraction of the counterparts of $z^{'}$-band sources are categorized into the case A. We assigned these unique pairs the $^{"}$0$^{"}$ flag to indicate that the pair is reliable in our band-merged catalogue. Both of the cases B and C are very rare for all bands ($<$ 0.01%) except the case B of $u^{*}$-band. We visually checked the $u^{*}$-band image and suspect that distorted PSF in $u^{*}$-band causes such multiple detections of a single source. We treated the case B by summing fluxes of the multiple counterpart candidates and give them a flag $^{"}$1$^{"}$, while for the case C we listed the common source in other band as a counterpart for both of the $z^{'}$-band sources with flag $^{"}$2$^{"}$ in the catalogue. The number of pairs found in those case are summarized in Table \[tb:matching\]. False Matching Rate {#sec: false detection} ------------------- Reducing the risk of contamination by false sources is one of important points for catalogue construction. For this purpose, we cleaned up bad pixels ($\S$\[sec:OBSERVATION and DATA REDUCTION\]), used weight maps during extracting sources in each band image individually ($\S$\[sec:SExtractor\]), and cross-matched sources among bands and excluded sources detected only in $z^{'}$-band ($\S$\[sec:Band Merging and Quality Frag\]). However, there is a small, but non-zero possibility of false matching in the merged catalogue. In this section we estimate the false matching rate in the catalogue. First, we estimated the false detection rate in each image. We multiplied each image by –1 to make a positive-negative reversed image, and then ran SExtractor on a negative image with the same parameters as for a positive image to detect false sources. The main locations of the false sources in the negative image are along the radial structures of bright stars and at edges of the image. The numbers of detections in the negative image for each band are summarized in Table \[tb:negative-matching\]. In the $z^{'}$-band case, we detected 88$\hspace{0.1em}$666 and 943 sources in the positive and negative images, respectively. Thus the false detection rate for $z^{'}$-band is $\sim$1.06%. In our worst case, $r^{'}$-band, the false detection rate is $\sim$1.65%. Second, we estimated a false–false matching rate between $z^{'}$ and other band spurious sources in the negative images with 0.5 arcsec search radius. Even in the worst case, between $z^{'}$ and $g^{'}$-bands, the number of spuriously matched sources is only 15 (15/943$\sim$1.60%). This small percentage is probably due to the fairly tight matching radius criterion. Next, we measured a real–false matching rate between the sources in the positive $z{'}$-band image and false sources in other band negative images. Even in the worst case, $z^{'}$ and $u^{*}$-bands, the rate is very low ($\sim$0.002%). The results of false matching are also shown in Table \[tb:negative-matching\]. Surprisingly, the real–false matching rate is smaller than the false–false matching rate in all bands except the $u^{*}$-band. This is because most false sources detected in the negative images locate around strong artifacts around bright stars and the sky noise itself in each image randomly created less spurious sources. These results suggest that the real–false matching rate is extremely small in the final catalogue when we use the 0.5 arcsec radius search area, nevertheless many of the faint sources appear around the artifacts and the edges of images. Finally, we calculated the probability of chance coincidences between the real sources. If their source positions are completely uncorrelated and randomly distributed, the chance coincidence rate can be calculated with $n\times S$, where $n$ is number density of images and $S$ is search area. In the case of the $g^{'}$-band, $n$ is 224$\hspace{0.1em}$450 per square degree and $S$ is $\pi\times(0.5$ arcsec)$^{2}$, giving a predicted chance coincidence of 1.36%. However these positions are actually correlated when the sources are detected in the same field of view, because a source at a given band should be present in the other bands. Therefore the real chance coincidence rate in the final catalogue is expected to be much smaller than the predictions. We thus conclude that the false matching probability is negligibly small. ------------- --------- --------------- ------------ ------------ ------- $u^{*}$ 149,168 1,475 (0.99%) 0 (0.00%) 2 (0.002%) 0.90% $g^{'}$ 224,449 2,361 (1.05%) 15 (1.60%) 1 (0.001%) 1.36% $r^{'}$ 178,191 2,944 (1.65%) 13 (1.38%) 1 (0.001%) 1.08% $i^{'}$ 143,318 911 (0.64%) 9 (0.95%) 0 (0.00%) 0.87% $z^{'}$ 88,666 943 (1.06%) N/A N/A N/A $Y$ 39,063 456 (1.17%) 0 (0.00%) 0 (0.00%) 0.41% $J$ 34,138 183 (0.54%) 0 (0.00%) 0 (0.00%) 0.36% $K_{\rm s}$ 38,053 98 (0.26%) 0 (0.00%) 0 (0.00%) 0.40% ------------- --------- --------------- ------------ ------------ ------- \[tb:negative-matching\] STAR – GALAXY SEPARATION {#S-G separation} ======================== In this section, we attempt to separate stars securely from galaxies using stellarity index of SExtractor (CLASS\_STAR) and colour. CLASS\_STAR results from a supervised neural network that is trained to perform a star/galaxy classification with values between 0 and 1. Sources with CLASS\_STAR $>$ 0.95 (as stars) and those with the value $<$ 0.2 (as galaxies) in the catalogue are plotted separately in the $u^{*}-r^{'}$ versus $g^{'}-z^{'}$ colour plane (left panel of Figure \[fig:urgz-uJgKs-stellar-09502\]). The figure shows that they are clearly separated with some small fraction of exceptions by a three-straight-line boundary connecting ($u^{*}-r^{'}$, $g^{'}-z^{'}$) = ($-$1.0, 0.5), (0.5, 0.5), (3.0, 1.7), and (3.0, 5.0) [@2007ApJS..172..583H]. However, contamination by extended sources becomes larger at $u^{*}-r^{'}>3$ which is most likely due to red galaxies at $z<0.4$ [Figure 1 of @2001AJ....122.1861S]. In contrast, we found that the $u^{*}-J$ versus $g^{'}-K_{\rm s}$ colour diagram (right panel of Figure \[fig:urgz-uJgKs-stellar-09502\]) enables us to separate stars from galaxies with less contamination. The boundary of these regions can be drawn by the combination of straight lines connecting at ($u^{*}-J$, $g^{'}-K_{\rm s}$)=($-$1.0,$-$1.4), (4.3, 2.1), (5.4, 3.5), and (9.0, 6.0). \ We also investigated colour dependence on the stellarity index ranges. We plotted sources with 0.9 $<$ CLASS\_STAR $<$ 1.0, 0.8 $<$ CLASS\_STAR $<$ 0.9, 0.7 $<$ CLASS\_STAR $<$ 0.8, and 0.6 $<$ CLASS\_STAR $<$ 0.7 in the (a)–(d) panels of Figure \[fig:uJgKs-stellar-1.0-0.9-0.8-0.7-0.6\], respectively. The lines in all panels are the boundary shown in Figure \[fig:urgz-uJgKs-stellar-09502\]. The figure shows that most of the sources with CLASS\_STAR $>$ 0.8 in panels (a) and (b) fall under the boundary suggesting stars, while sources with CLASS\_STAR $<$ 0.7 in panel (d) are mainly located above the boundary indicating extended sources. The sources at 0.7 $<$ CLASS\_STAR $<$ 0.8 seem to be a mixture of the two classes. If we use only this colour criterion and stellarity index to separate stars from galaxies, quasars are likely to be misclassified as stars. In order to avoid this, we excluded sources bluer than $u^{*}-g^{'}$ = 0.4 from the star candidates [@2000AJ....120.2615F]. In summary, the following criteria are for secure identification of stars; (1) $g^{'}-K_{\rm s}$ is bluer than the boundary in the $u^{*}-J$ versus $g^{'}-K_{\rm s}$ colour diagram, (2) CLASS\_STAR $>$ 0.8, and (3) $u^{*}-g^{'}>0.4$. 5441 star candidates satisfy those criteria in our catalogue with a flag of “STAR”. We note that the flagging of sources with at least one band non-detection in the $u^{*}$, $g^{'}$, $J$, and $K_{\rm s}$-bands is represented as “\*\*\*\*” in the catalogue because the colour criteria cannot be used for them. PHOTOMETRIC REDSHIFT DERIVATION {#sec:PHOTOMETRIC REDSHIFT} =============================== Photometric Redshift Computing with $Le Phare$ ---------------------------------------------- We computed photometric redshifts ($z_{\rm p}$) using $Le Phare$ code . We used 62 galaxy templates and the 154 star templates that we used for predicting $Y$-band magnitude as the photometric reference during the earlier data reduction (§\[sec:wircam\]). The galaxy templates are empirical ones from [@1980ApJS...43..393C] with two starburst templates from [@1996ApJ...467...38K]. These templates are interpolated and adjusted to match the VIMOS VLT Deep Survey (VVDS) spectra better . For galaxy templates of Scd and later spectral types, we considered the dust extinction reddening. Since considerable changes in the extinction curve are expected from galaxy to galaxy, we used two extinction laws. We adopted an interstellar extinction law determined from starburst galaxies [@2000ApJ...533..682C] and the Small Magellanic Cloud extinction law with varying $E(B-V)$ of 0.0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5. A broad absorption excess at 2175 $\AA$ (UV bump) sometimes necessary to explain rest-frame UV flux in some starburst galaxies . Therefore we allowed an additional UV bump at 2175 $\AA$ for Calzetti extinction law if it produces a smaller $\chi^2$. Systematic offset are often found between the best fit SED templates and the observed apparent magnitudes in a given filter . Uncertainties in the zero point calibration of the photometric data as well as imperfect knowledge of galaxy SEDs are responsible these offsets. They may produce biases in the photometric redshift measurements. In order to correct it, for each filter, $f$, we computed the systematic offset ($s^f$) between the photometry in a given filter and the photometry for the best SED fitting at the fixed spectroscopic redshift ($z_{\rm s}$) of the $AKARI$ detected galaxies with the option AUTO\_ADAPT in the $LePhare$ algorithm. 483 sources excluding Type 1 AGN taken with Keck/DEIMOS (Takagi et al., in prep.), MMT/Hectospec and WIYN/HYDRA [@2013ApJS..207...37S] are used as an input catalogue for the training and estimation of the $z_{\rm p}$ accuracy. The all $z_{\rm s}$ of 483 galaxies were measured by using two or more emission lines or absorption lines. Many objects with $z_{\rm s}$ distribute in the local universe at $z<1$ (453 sources $\sim$ 94%) and only $\sim$ 6% of the galaxy are at $z>1$. Iteratively, we then searched for the best set of corrections that minimized the offsets. Once found, the offset was applied to the SEDs when computing the photometric redshift for the entire catalogue. The $z_{\rm p}$ is the redshift value which minimizes the merit function $\chi^2(z,T,A)$: $$\begin{aligned} \chi2=\Sigma^{N_f}_{f=1}\left(F^f_{obs}-A\times F^f_{pred}(z,T)10^{-0.4s^f} / \sigma^f_{obs}\right),\end{aligned}$$ where $F^f_{pred}(z,T)$ is the predicted flux for a template $T$ at redshift $z$. $F^f_{obs}$ the observed flux, $\sigma^f_{obs}$ is the associated error, and $A$ is the normalization factor. The grid spacing in redshift is $\delta z = 0.02$. A redshift probability distribution function (PDFz) is computed for each object using the $\chi^2$ merit function, PDFz $\propto$ exp(–$\chi^2$($z$)/2) every grid space. The best redshift is derived by parabolic interpolation of the PDF with galaxy library. If a second peak is found in the PDFz with a height larger than 5% of the main peak, the corresponding redshift is given as a second solution. In addition to the best $\chi^2$ derived from the galaxy library (hearafter $\chi^2_{gal}$), a best $\chi^2$ computed using the stellar liberally is derived for each object (hearafter $\chi^2_{star}$). Photometric Redshift Accuracy ----------------------------- We assess the photometric redshift accuracy by comparing photometric redshifts against spectroscopic redshifts. Figure \[fig:zszp\] shows the comparison between $z_{\rm p}$ with $z_{\rm s}$ from the 483 sources. Objects, whose second solution in the PDFz are found, are shown with pink triangles. Since $z_{\rm p}$ of those sources are not reliable, we excluded for the measurement of $z_{\rm p}$ accuracy. We obtained an accuracy of $\sigma_{\Delta z/(1+z)}\sim$ 0.032 and a catastrophic error rate of $\eta=$ 5.8% ($\frac{\Delta z}{1+z}>0.15$) at $z<1$, while $\sigma_{\Delta z/(1+z)}\sim$ of 0.117 and $\eta=$ 16.6% at $z>1$. The median offset between the photometric redshift and spectroscopic redshift is $\Delta z/(1+z)=$ 0.008 and 0.018 for $z<1$ and $z>1$, respectively. used the $LePhare$ with the CFHTLS Deep and Wide T0004 catalogues ($u^{*}, g^{'}, r^{'}, i^{'}, z^{'}$) to calculate $z_{\rm p}$ and measured the accuracy comparing with VVDS-deep and -wide, and DEEP2 (Davis et al 2003 2007) $z_{\rm s}$ samples. They found a dispersion of 0.028-0.030 and an outlier rate of about 3-4% in the range $17.5<i^{'}_{AB}<22.5$ at $0<z<2$. In our catalogue, there are 382 sources with $z_{\rm s}$ in the range of $17.5<i^{'}_{AB}<22.5$, and  97% of them are $z_{\rm s}<1$. We compared the $z_{\rm p}$ with $z_{\rm s}$ of them and found an accuracy of $\sigma_{\Delta z/(1+z)}\sim$ 0.031 with a catastrophic error rate of $\eta=$ 3.5%, which are comparable to the CFHTLS results. Since evaluation of the $z_{\rm p}$ accuracy from the comparison with $z_{\rm s}$ is limited to specific range of magnitude and redshift, we use the 1$\sigma$ uncertainty in the derived $z_{\rm p}$ probability distribution to extend the uncertainty estimates over the full magnitude/redshift space. For the estimate of the accuracy, we exclude stars. In $\S$\[S-G separation\], we securely flagged star using ($u^{*}-J$) – ($g^{'}-K_{\rm s}$) colour in addition to the stellarity index. During SED fitting using $Le Phare$, we also calculated $\chi^2_{gal}$ and $\chi^2_{star}$ for each object. If $\chi^2_{star}<\chi^2_{gal}$, it suggests that the object is more likely to be a star than a galaxy. We compared the SED $\chi^2$ classification and colour with the stellarity index, and found that both classifications are excellently agreed with each other : 5381 out of 5441 star-flagged sources with the color-morphology classification method are classified as stars with the SED $\chi^{2}$ criterion. It suggests that both criteria work well, and then we excluded stars flagged by either SED $\chi^2$ classification or the color-morphology classification for measure of the $z_{\rm p}$ accuracy below. Following , we plotted the 1$\sigma$ negative and positive uncertainties derived from the PDFz as a function of redshift and $z^{'}$-band magnitude (Figure \[fig:1sigma\]). The uncertainties are derived from at 68% of the probability distribution. This figure shows that the accuracy is inevitably degraded for fainter galaxies at all redshift and the $z_{\rm p}$ have significantly higher uncertainty at $z_{\rm p}$ 1. From $z_{\rm p}>0$ up to $z_{\rm p}$ = 1, the 1$\sigma$ errors do not dependent significantly on the redshift at $z^{'}<$ 23. With $z^{'}$ = 23 – 24, the accuracy is $\sim$ 0.7, which is consistent with the accuracy estimated from the $z_{\rm p}$ – $z_{\rm s}$ comparison (see Figure \[fig:zszp\]). At $z_{\rm p}>1$, the accuracy with $z^{'}<$ 22 is continuously small, while the accuracy for fainter sources goes high. Therefore, our photometric redshifts are highly accurate up to at least $z_{\rm p}<1$ with $z^{'}<24$ or $z_{\rm p}<2.5$ with $z^{'}<$ 22. Figure \[fig:hist-zp\] shows the photometric redshift distribution for all objects classified as galaxy with at least five band detections. The peak of the distribution is located at redshifts between at $z\sim$ 0.3 – 0.4, and the number of sources monotonically decreases with increasing redshift. $AKARI$ SOURCES =============== constructed a revised near to mid infrared catalogue for the $AKARI$ NEP-Deep survey. by combining revised $AKARI$ near-infrared (NIR) a sub-catalogue ($N2$–$N4$) and mid-infrared (MIR) sub-catalogue ($S7$–$L24$).They compared the $AKARI$ NEP-Deep catalogue with our final catalogue to identify optical counterparts. In total, 23$\hspace{0.1em}$345 $AKARI$ sources have their optical counterparts, and 8916 of them have mid-infrared (any of $S7$ – $L24$) fluxes. In this section, we will focus on the sources detected by $AKARI$ with optical counterparts and demonstrate the properties of the $AKARI$ sources. $g^{'}z^{'}K_{\rm s}$ Diagram {#sec:gzk} ----------------------------- [@2004ApJ...617..746D] developed a highly popular technique that allows both the selection and classification of $1.4<z<2.5$ star-forming galaxies, passively evolved galaxies and stars using a simple $BzK_{\rm s}$ colour-colour diagram. In the $BzK_{\rm s}$ technique, high redshift objects are uniquely located in regions of the $BzK_{\rm s}$ diagram. The validity of this technique has been supported by many authors. Original paper of [@2004ApJ...617..746D] used the VLT Bessel $B$ band filter, the VLT Gunn $z$ filter, and the ISAAC $K_{\rm s}$ band filter. Although the $z^{'}$ and $K_{\rm s}$-band filters of CFHT are similar to those Daddi et al. used, the CFHT MegaCam $g^{'}$-band filter has offset from the $B$ band filter. [@2013arXiv1307.6094L] considered the positions of the [@2004ApJ...617..746D] models in the $g^{'}z^{'}K_{\rm s}$ colour-colour diagram to modify the $BzK_{\rm s}$ technique for use with MegaCam and WIRCam $g^{'}z^{'}K_{\rm s}$ filters. They found that even the $g^{'}z^{'}K_{\rm s}$ and $BzK_{\rm s}$ filter systems are different, the main qualitative features of the colour-colour diagram of models are similar enough. Hence, $g^{'}z^{'}K_{\rm s}$ diagram can be used for classification of $1.4<z<2.5$ star-forming galaxies, passively evolved galaxies and stars. [@2013arXiv1307.6094L] defined new colour-cats to distinguish those properties as $$\begin{aligned} (z^{'}-K_{\rm s}) - 1.27(g^{'}-z^{'})&=& -0.022\\ (z^{'}-K_{\rm s}) - 1.27(g^{'}-z^{'})&=& -0.022\cap(z^{'}-K_{\rm s}) =2.55\\ (z^{'}-K_{\rm s})-0.45(g^{'}-z^{'})&=& -0.57.\end{aligned}$$ As the same as $BzK_{\rm s}$ technique, high redshift star-forming galaxies are located in the left of the Eq. (2) and stars are separated by Eq. (4) in the $g^{'}z^{'}K_{\rm s}$ diagram. We first plotted the $z_{\rm p} <1.4$ $AKARI$ sources (left panel of Figure \[fig:gzkslgt4\]). Almost all stars ($\sim$ 98.9%) classified in $\S$\[S-G separation\] are lying under the Eq. (4), while extended galaxies are distributed enclosed area by the three equations. It suggests that the criteria of star–galaxy classification discussed in $\S$\[S-G separation\] work well. On the other hand, we plotted galaxies at $1.4<z_{\rm p}<2.5$ in the $g^{'}z^{'}K_{\rm s}$ diagram in right panel of Figure \[fig:gzkslgt4\]. All points in the left panel are plotted in gray. We found that the $1.4<z_{\rm p}<2.5$ galaxies are located at upper-left of galaxies of $z_{\rm p} <1.4$ systematically although more than 50% of $1.4<z_{\rm p}<2.5$ galaxies are scattered into $z_{\rm p} <1.4$ region. This is probably a consequence of the increased scatter of the photometric redshift higher than $z_{\rm p}>1$. When we use the $g^{'}z^{'}K_{\rm s}$ diagram, we can distinguish 759 $AKARI$ sources ($\sim$ 3.2%) as high-$z$ galaxies independent of $z_{\rm p}$ with large uncertainty at $z_{\rm p}>1$. The average $z^{'}$ magnitude of the high-$z$ galaxies of $1.4<z_{\rm p}$ is 23.2 ($\pm$ 0.4) mag, which is consistent with the $z^{'}$-band 50% completeness, suggesting that it is possible there are $AKARI$ sources of high-$z$ star-forming galaxies without optical counterparts. We also found that only less than 0.1% of $AKARI$ sources come into the passive galaxies’ area where is above the Eq. (3). Seen from the $g^{'}z^{'}K_{\rm s}$ diagram, the $AKARI$ sample is dominated by sources at $z<1.4$, and $<$ 5% of the $AKARI$ sources with optical counterparts are located at the high-$z$ universe and most of them seem to be star-forming active galaxies. Mid Infrared Sources {#sec:AKARI} -------------------- We compared $z_{\rm p}$ between sources with detections and non-detections in $AKARI$ MIR bands. We divided $AKARI$ sources into two groups, one is detected in any of MIR bands ($S7$–$L24$) and the other is not. After excluding star-flagged sources, MIR detected and non-detected groups include 8085 sources and 11$\hspace{0.1em}$616 sources, respectively. Figure \[fig:histwwoAKARI\] shows $z_{\rm p}$ histograms for each group. This figure shows that MIR non-detected group is distributed at higher redshift systematically compared with MIR detected group. The distribution of MIR detected group has sharp peak around $z\sim 0.5$ and suddenly decreases at higher redshift, while MIR non-detected sources have gentle peak around $z_{\rm p} \sim 0.5 - 1$. This tendencies are explained by the difference of sensitivities of $AKARI$ near and mid infrared detectors. Since mid-infrared band sensitivities are relatively poorer than those of near-infrared bands, there is a bias that MIR bands detect lower-redshift bright sources. Finally we plotted the $AKARI$ sources in the $g^{'}z^{'}K_{\rm s}$ plane in Figure \[fig:reddening\]. This figure shows that some sources with non-detections in $AKARI$ MIR bands ($\sim$ 22%) are plotted in the stellar region under the Eq. (4), while stellar contamination to the sources with detections in the MIR bands are negligible small ($\sim$ 4.0%). Very similar fraction of sources with and without MIR detections ($\sim$ 5.3% and $\sim$ 6.4%, respectively) are distributed in the $1.4<z<2.5$ star-forming galaxy region. For these sources located in the $1.4<z<2.5$ star-forming galaxy region, MIR detected sources seem to be systematically redder along the line of Eq. (2) compared with MIR non-detected group. Since galaxies become redder along the line of Eq. (2) depending on the $E(B-V)$ values in [@2003MNRAS.344.1000B] model as shown in Figure \[fig:gzkslgt4\], here we defined an axis along the line of Eq. (2) as “Reddening axis”. The difference between MIR detected and non-detected groups can be more clearly seen along the “Reddening axis” as shown in right panel of Figure \[fig:reddening\]. Although the colour can be redder both from the old age and the star formation along the “Reddening axis”, the effect from the age is significantly small. Therefore, the redder colour of MIR detected sources probably indicates that they are dust attenuated star-forming galaxies. In conclusion, when we select sources detected in $AKARI$ MIR bands, then dusty, star-forming galaxies at around $z$0.5 are preferentially-chosen with less stellar contamination. SUMMARY and CONCLUSION ====================== We obtained $g^{'}$, $r^{'}$, $i^{'}$, $z^{'}$, $Y$, $J$, and $K_{\rm s}$-band images with MegaCam and WIRCam at CFHT, and cross-matched each of all the band catalogues with existing $u^{*}$-band catalogue to construct an optical – near infrared 8-band merged photometric catalogue covering the $AKARI$ NEP-Deep field. The 4$\sigma$ detection limits over an 1 arcsec radius aperture are 26.7, 25.9, 25.1, 24.1, 23.4, 23.0, and 22.7 AB mag for $g^{'}$, $r^{'}$, $i^{'}$, $z^{'}$, $Y$, $J$, and $K_{\rm s}$-bands, respectively. The astrometry of the band-merged catalogue is offset by 0.013 arcsec to the 2MASS with an RMS offset of 0.32 arcsec, while the RMS offsets between $z^{'}$-band and the other bands are 0.08–0.11 arcsec, but that between $z^{'}$ and $u^{*}$-bands is worse (0.18 arcsec) because of a slightly degraded PSF in $u^{*}$-band. We securely distinguished stars from galaxies with ($u^{*}-J$) versus ($g^{'}-K_{\rm s}$) colour-colour diagram with CLASS\_STAR of $<$ 0.8. We derived photometric redshifts based of a SED template fitting procedure using $LePhare$. The errors comparison with $z_{\rm s}$ for our spectroscopic sub-sample reveals the dispersion of $\sigma_{\Delta z/(1+z)}\sim0.032$ with $\eta=5.8$% catastrophic failures at $z_{\rm p}<1$. We extrapolate this result to fainter magnitudes using the 1$\sigma$ uncertainties in the $z_{\rm p}$ probability distribution functions. At $z_{\rm p}<1$, we estimate a $z_{\rm p}$ accuracy of $\sigma_{\Delta z} = 0.7$ with $z^{'}<24$, while the accuracy is strongly degraded at $z_{\rm p}>1$ or $z^{'}>24$. We investigated properties of $AKARI$ sources with optical counterparts from our catalogue. We plotted the sources with $g^{'}z^{'}K_{\rm s}$ diagram and found that most of the $AKARI$ sources are located in the low-$z$ ($z<1.4$) and $<$ 5% of them are classified as high-$z$ ($1.4<z<2.5$) star-forming galaxies. Among the high-$z$ star-forming galaxies, MIR detected sources seems to be affected stronger dust extinction compared with sources with non-detections in $AKARI$ MIR bands. Our final catalogue contains 85$\hspace{0.1em}$797 sources with $z_{\rm p}$, which are detected in $z^{'}$-band and at least one of the other bands. This optical to near-infrared photometric data with photometric redshift in the $AKARI$ NEP-Deep field derived here are crucial to study star-formation history at the era of active universe ($z\sim1$) using multi-wavelength data sets ($Chandra$, $GALEX$, $AKARI$, $Herschel$, $SCUBA$-2, and $WSRT$). This work is supported by the Japan Society for the Promotion of Science (JSPS; grant number 23244040) and grant NSC 100-2112-M-001-001-MY3 (Y.O.). MI acknowledges the support from a grant No. 2008-0060644 of CRI/NRFK/MSIP of Korea. This work is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, and WIRCam, a joint project of CFHT, Taiwan, Korea, Canada, France, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. S. Foucaud (National Taiwan Normal University) is acknowledged for this modified SWarp in Section \[sec:OBSERVATION and DATA REDUCTION\] [57]{} natexlab\#1[\#1]{} , S., [Walcher]{}, C. J., [Le F[è]{}vre]{}, O., [et al.]{} 2007, , 476, 137 , T. C., [Flynn]{}, K., & [Gebhardt]{}, K. 1990, , 100, 32 , E. 2006, in Astronomical Society of the Pacific Conference Series, Vol. 351, Astronomical Data Analysis Software and Systems XV, ed. C. [Gabriel]{}, C. [Arviset]{}, D. [Ponz]{}, & S. [Enrique]{}, 112 , E. & [Arnouts]{}, S. 1996, , 117, 393 , E., [Mellier]{}, Y., [Radovich]{}, M., [et al.]{} 2002, in Astronomical Society of the Pacific Conference Series, Vol. 281, Astronomical Data Analysis Software and Systems XI, ed. D. A. [Bohlender]{}, D. [Durand]{}, & T. H. [Handley]{}, 228 , R., [Hudelot]{}, P., [McCracken]{}, H. J., [et al.]{} 2012, , 545, A23 , R. C., [Colina]{}, L., & [Finley]{}, D. S. 1995, , 110, 1316 , O., [Charlot]{}, X., [Abbon]{}, P., [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]{}, 72–81 , M., [Brown]{}, M. J. I., [Ashby]{}, M. L. N., [et al.]{} 2006, , 651, 791 , G. & [Charlot]{}, S. 2003, , 344, 1000 , V., [Takeuchi]{}, T. T., [Iglesias-P[á]{}ramo]{}, J., [et al.]{} 2007, , 173, 404 , D., [Armus]{}, L., [Bohlin]{}, R. C., [et al.]{} 2000, , 533, 682 , P. I., [Yan]{}, L., [Im]{}, M., [et al.]{} 2006, , 637, 227 , G. D., [Wu]{}, C.-C., & [Weedman]{}, D. W. 1980, , 43, 393 , A. J., [Szalay]{}, A. S., [Dickinson]{}, M., [Subbarao]{}, M. U., & [Brunner]{}, R. J. 1997, , 486, L11 , J., [Ilbert]{}, O., [Kilbinger]{}, M., [et al.]{} 2009, , 500, 981 , E., [Cimatti]{}, A., [Renzini]{}, A., [et al.]{} 2004, , 617, 746 , G. G., [Hora]{}, J. L., [Allen]{}, L. E., [et al.]{} 2004, , 154, 10 , K., [Ivezi[ć]{}]{}, [Ž]{}., [Fan]{}, X., [et al.]{} 2000, , 120, 2615 , A., [Rodighiero]{}, G., & [Vaccari]{}, M. 2008, , 487, 837 , L., [Barbieri]{}, M., [Groenewegen]{}, M. A. T., [et al.]{} 2012, [TRILEGAL, a TRIdimensional modeL of thE GALaxy: Status and Future]{}, ed. A. [Miglio]{}, J. [Montalb[á]{}n]{}, & A. [Noels]{}, 165 , T., [Takagi]{}, T., [Matsuhara]{}, H., [et al.]{} 2010, , 514, A6 , E. K. S., [Malkan]{}, M. A., [Teplitz]{}, H. I., [McCarthy]{}, P. J., & [Yan]{}, L. 2002, , 581, 205 , A. M., [Connolly]{}, A. J., [Haarsma]{}, D. B., & [Cram]{}, L. E. 2001, , 122, 288 , J. R., [Roellig]{}, T. L., [van Cleve]{}, J., [et al.]{} 2004, , 154, 18 , N., [Lee]{}, M. G., [Lee]{}, H. M., [et al.]{} 2007, , 172, 583 , O., [Arnouts]{}, S., [McCracken]{}, H. J., [et al.]{} 2006, , 457, 841 , O., [Capak]{}, P., [Salvato]{}, M., [et al.]{} 2009, , 690, 1236 , K., [Matsuhara]{}, H., [Oyabu]{}, S., [et al.]{} 2007, , 133, 2418 , M. F., [Steinz]{}, J. A., [Anderegg]{}, M. E., [et al.]{} 1996, , 315, L27 , A. L., [Calzetti]{}, D., [Bohlin]{}, R. C., [et al.]{} 1996, , 467, 38 , R. G. 1980, , 43, 305 , G., [Abergel]{}, A., [Boulanger]{}, F., [D[é]{}sert]{}, F. X., & [Puget]{}, J.-L. 1999, , 344, 322 , A.-O. & [Marcin]{}, S. 2013, ArXiv e-prints , C., [Malkan]{}, M. A., [Kashikawa]{}, N., [et al.]{} 2007, , 657, 738 , P. 1995, , 441, 18 , E. A. & [Cuillandre]{}, J.-C. 2004, , 116, 449 , C. & [Bertin]{}, E. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 394, Astronomical Data Analysis Software and Systems XVII, ed. R. W. [Argyle]{}, P. S. [Bunclark]{}, & J. R. [Lewis]{}, 619 , M., [Iovino]{}, A., [Buzzoni]{}, A., & [Valls-Gabaud]{}, D. 2001, , 380, 425 , H., [Wada]{}, T., [Matsuura]{}, S., [et al.]{} 2006, , 58, 673 , H., [Baba]{}, H., [Barthel]{}, P., [et al.]{} 2007, , 59, 369 , K., [Matsuhara]{}, H., [Wada]{}, T., [et al.]{} 2013, , 559, A132 , G., [Habing]{}, H. J., [van Duinen]{}, R., [et al.]{} 1984, , 278, L1 , T., [Matsuhara]{}, H., [Wada]{}, T., [et al.]{} 2007, , 59, 401 , A. J. 1998, , 110, 863 , M. L., [Lequeux]{}, J., [Prevot]{}, L., [Maurice]{}, E., & [Rocca-Volmerange]{}, B. 1984, , 132, 389 , J.-L., [Abergel]{}, A., [Bernard]{}, J.-P., [et al.]{} 1996, , 308, L5 , P., [Stadler]{}, E., [Doyon]{}, R., [et al.]{} 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5492, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. A. F. M. [Moorwood]{} & M. [Iye]{}, 978–987 , G. H., [Young]{}, E. T., [Engelbracht]{}, C. W., [et al.]{} 2004, , 154, 25 , H., [Im]{}, M., [Ko]{}, J., [et al.]{} 2013, , 207, 37 , F. W., [Malkan]{}, M. A., & [Scully]{}, S. T. 2006, , 648, 774 , C. C., [Adelberger]{}, K. L., [Giavalisco]{}, M., [Dickinson]{}, M., & [Pettini]{}, M. 1999, , 519, 1 , I., [Ivezi[ć]{}]{}, [Ž]{}., [Knapp]{}, G. R., [et al.]{} 2001, , 122, 1861 , M., [Mobasher]{}, B., [Chan]{}, B., [et al.]{} 2001, , 558, 72 , T., [Matsuhara]{}, H., [Goto]{}, T., [et al.]{} 2012, , 537, A24 , T., [Matsuhara]{}, H., [Oyabu]{}, S., [et al.]{} 2008, , 60, 517 , D. G., [Adelman]{}, J., [Anderson]{}, Jr., J. E., [et al.]{} 2000, , 120, 1579 \[tb:example\] [^1]: http://www3.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/megapipe/docs/filters.html [^2]: http://www.astromatic.net/software/weightwatcher [^3]: http://www.astromatic.net/software/sextractor [^4]: http://www.astromatic.net/software/scamp [^5]: http://www.astromatic.net/software/swarp [^6]: This version of SWarp is modified to have 3-sigma clipping function by S. Foucaud (private communication). [^7]: [http://cfht.hawaii.edu/Instruments/Imaging/WIRCam/IiwiVersion1 Doc.html]{} [^8]: http://www.cfht.hawaii.edu/Science/CFHTLS/T0007/ CFHTLS\_T0007-TechnicalDocumentation.pdf

--- abstract: 'Dynamic properties of a one-dimensional probabilistic cellular automaton are studied by monte-carlo simulation near a critical point which marks a second-order phase transition from a active state to a effectively unique absorbing state. Values obtained for the dynamic critical exponents indicate that the transition belongs to the universality class of directed percolation. Finally the model is compared with a previously studied one to show that a difference in the nature of the absorbing states places them in different universality classes.' author: - |  \  \ [Pratip Bhattacharyya]{} [^1]\  \ Low Temperature Physics Section\ Saha Institute of Nuclear Physics\ Sector - 1, Block - AF, Bidhannagar\ Calcutta 700 064, India. title: '**Dynamic critical properties of a one-dimensional probabilistic cellular automaton**' --- = 23.0cm = 16.5cm = -1.0cm = -0.1cm = 20 true pt Introduction ============= Discrete models of nonequilibrium stochastic processes form a class of interacting particle systems [@Ligget1985]. Of the models studied with short range and translationally invariant interactions in space and time, the ones exhibiting a continuous phase transition from an active steady state to an absorbing state fall into two universality classes [@Grassberger1995] : \(1) the class of directed percolation (DP),\ (2) the class of parity conservation (PC). Models with a unique absorbing state have been conjectured to belong to the DP class [@Janssen1981; @Grassberger1982]. This is yet the larger of the two classes and includes, for example, lattice models of directed percolation in $d + 1$ dimensions [@Kinzel1983], the contact process for an epidemic [@Harris1974], Schlögl’s first and second models of autocatalytic reactions [@Schlogl1972], the Domany-Kinzel automaton [@Domany1984; @Kinzel1985], a lattice version of reggeon field theory [@Grassberger1979] and branching annihilating random walks with an odd number of offsprings [@Jensen1993-1]. The order parameter is usually the density of particles (occupied sites, kinks). The models mentioned above are one-component systems and therefore the order parameter is scalar, a requirement of the DP-conjecture [@Janssen1981; @Grassberger1982]. The DP-conjecture was generalised to include multicomponent systems such as the ZGB model of heterogeneous catalysis [@Ziff1986; @Grinstein1989; @Jensen1990] and interacting dimer-trimer models [@Albano1995]. The class of DP was shown to further include systems with an infinite number of absorbing states [@Jensen1993-2]; these absorbing states are frozen configurations characterised by unique statistical properties, a lack of long range correlations and a general lack of symmetry among them. On the other hand, models with degenerate, mutually symmetric absorbing states are believed to belong to the parity-conserving class where the number of particles are conserved modulo 2. Prominent examples are the probabilistic cellular automaton models $A$ and $B$ of [@Grassberger1984; @Grassberger1989-1], an interacting monomer- dimer model [@Kim1994] and branching annihilating random walks (BAW) with an even number of offsprings [@Jensen1994]. According to one point of view [@Grassberger1995; @Grassberger1989-1] the mechanism that puts these models in a class different from that of DP is the conservation of particle number modulo 2. This point was proved for BAW with an even number of offsprings: introduction of spontaneous annihilation of particles in the model destroyed the conservation of their number modulo 2 and the critical behaviour of this modified model was observed to be in the class of DP [@Jensen1993-3]. According to a second point of view the critical behaviour of the models in the PC class is due to the symmetry among its absorbing states  [@Park1995; @Hinrichsen1997]. To prove the point the interacting monomer-dimer model of [@Kim1994], in the presence of a weak parity-conserving field that destroyed the symmetry among the absorbing states, was shown to exhibit critical behaviour in the DP class  [@Park1995]. This view was further emphasized in the generalised versions of the Domany-Kinzel automaton and the contact process  [@Hinrichsen1997]. In these generalised models there was no explicit parity-conservation law and with two symmetric absorbing states the critical behaviour was in the PC class. In the presence of a symmetry breaking field the critical behaviour of these models changed to the DP class. However in the case of the probabilistic cellular automata of [@Grassberger1984; @Grassberger1989-1] both the conservation of particle number modulo 2 and mutual symmetry among the absorbing states are present and therefore it it is not apparent which of these two features is responsible for the models to be in the PC class. In this paper I shall study, using time-dependent monte-carlo simulations, the dynamic critical properties of a one-dimensional probabilistic cellular automaton [@Bhattacharyya1996] which has three absorbing states and exhibits a phase transition from a active state to one of them only. The simulations provide values for the critical point (more accurate than previous estimates [@Bhattacharyya1996; @Bhattacharyya1997]) and the dynamic critical exponents that decide the universality class to which the phase transition belongs. The model ========== The probabilistic cellular automaton studied here is elementary (in the sense of Wolfram [@Wolfram1983]) with two states per site and translationally invariant nearest neighbour interactions. The probabilistic behaviour enters the model through two mutually symmetric components of the evolution rule (like specific noise added to a otherwise deterministic system), the rest of the rule components being deterministic in nature. Formally the model [@Bhattacharyya1996] is defined as a line of sites with a binary variable $x_i \in \{0, 1\}$ assigned to each site $i$. A site is said to be occupied if $x_i = 1$ and unoccupied otherwise. Starting from a given configuration $\{x^{(0)}_i\}$ the system evolves by parallel update of the variable $x_i$ at all lattice sites following a local rule of evolution. With nearest neighbour interactions the evolution rule is defined by a set of eight components $[x^{(t)}_{i-1},~x^{(t)}_i, ~x^{(t)}_{i+1}] \mapsto x^{(t+1)}_i$ corresponding to $2^3$ distinct three-site neighbourhoods : $${t : \over {t+1 :}}~~~~~~{111 \over 0}~~~~{110 \over 0}~~~~{011 \over 0}~~~~{101 \over 0}~~~~{010 \over 1}~~\underbrace{{100 \over }~~~~{001 \over }}_{\begin{array}{lll} 1 & \mbox{with probability} & p\\ 0 & \mbox{with probability} & 1 - p \end{array}}~~{000 \over 0} \label{eq:model}$$ The evolution rule (\[eq:model\]) thus follows, according to Wolfram’s nomenclature scheme [@Wolfram1983], elementary rule $4$ with probability $1 - p$ and elementary rule $22$ with probability $p$. Clearly, the components of the evolution rule are of two kinds: $(1)$ active components, where the central site changes its value, and $(2)$ passive components, where the value of the central site remains unchanged. The dynamic evolution of the system is due to the active components. This involves two opposing processes : $(a)$ Annihilation of adjacent occupied sites (multi-particle annihilation) due to the rule components $111 \mapsto 0,~110 \mapsto 0,~011 \mapsto 0$ prevents the survival of occupied pairs of nearest neighbours; this is a deterministic process. $(b)$ Creation of an occupied site $(100 \mapsto 1,~001 \mapsto 1)$ requires an unoccupied site to have exactly one occupied neighbour; this process of creation occurs only with a probability $p$. The multi-particle annihilation and creation of particles  can, in effect, give rise to a diffusion process $-$ if a particle at site $i$ gets annihilated after creating another particle at a neighbouring site $i + r$, it has effectively taken a step from $i$ to $i + r$. The passive components determine the absorbing states of the model. Since there is no spontaneous creation of occupied sites $(000 \mapsto 0)$ the vacuum (all sites unoccupied) is always an absorbing state : $$\makebox[5cm][l]{Absorbing state I :} x_i = \left . \begin{array}{ll} 0 & \mbox{for all $i$.} \end{array} \right . \label{eq:absorb1}$$ Again, it is evident from the rule components $010 \mapsto 1$ and $101 \mapsto 0$ that an occupied site with unoccupied neighbours remains occupied and vice versa. These features of the evolution rule lead to two mutually symmetric absorbing states : $$\makebox[5cm][l]{Absorbing State II :} x_i = \left\{ \begin{array}{ll} 0 & \mbox{for $i$ = even,} \\ 1 & \mbox{for $i$ = odd.} \end{array} \right . \label{eq:absorb2}$$ $$\makebox[5cm][l]{Absorbing State III :} x_i = \left\{ \begin{array}{ll} 1 & \mbox{for $i$ = even,} \\ 0 & \mbox{for $i$ = odd.} \end{array} \right . \label{eq:absorb3}$$ The main point of interest in this paper is a phase transition exhibited by the model [@Bhattacharyya1996]. For $p$ less than a critical value $p_c$ there exists three distinct steady states given by the three absorbing states of the model. For all initial states but two, it has been observed in computer simulations [@Bhattacharyya1996] that the only steady state is the vacuum (absorbing state I). The two cases of exception occur when the initial state is either absorbing state II or absorbing state III, which must also be respectively the steady states of the system. Evolving from any other initial state the mutually symmetric absorbing states II and III are never reached. This can be understood by the fact that these states are fixed points that repel $-$ a damage introduced in these two states by flipping only a single bit spreads through the entire lattice and eventually the vacuum is reached. Above $p_c$ there is another steady state called the active state with a constant non-zero density of occupied sites. The active state is stable only on an infinite lattice; for finite lattices it occurs as a metastable state that will decay to the vacuum if allowed to evole for sufficiently long time. On an infinite lattice, in the supercritical region $(p > p_c)$, all possible initial states other than the three absorbing states evolve into the active steady state. The density of occupied sites $\rho_\infty$ in the steady state acts as the order parameter for this phase transition $-$ in the supercritical region $\rho_\infty$ goes to zero continuously as $p$ approaches $p_c$ [@Bhattacharyya1996] : $$\rho_\infty \sim (p - p_c)^{\beta},~~~~~~~p \rightarrow p_{c \, +} , \label{eq:order}$$ where $\beta$ is the critical exponent for the order parameter. For random initial states the model therefore makes a continuous (second order) phase transition, at $p = p_c$, from a active state with $\rho_\infty > 0$ to a effectively unique absorbing state, the vacuum. Because of the conjecture of [@Janssen1981; @Grassberger1982] this phase transition is expected to belong to the universality class of DP. Dynamic properties at the critical point ======================================== The phase transition is characterised here by critical exponents describing the dynamic properties of the model at the point of transition (the critical point $p_c$). To do so dynamic properties of the model are studied by monte-carlo simulations on a computer, only close to $p_c$. While the study of steady state properties require simulations starting from disordered states, the dynamic properties are studied using initial states with a single occupied site. Following the evolution rule (\[eq:model\]) an initial occupied site grows into a cluster; the position of this initial occupied site is called the origin of the cluster. For each value of $p$, $10^4$ clusters were simulated. Each cluster was allowed to evolve for $5000$ time steps, unless it had died out earlier. Typical examples of evolution near the critical point are shown in figure 1. The quantities measured are : $(1)$ the survival probability $P(t)$, which gives the chance that there is at least one occupied site after $t$ time steps, $(2)$ the average number of occupied sites $N(t)$ after $t$ time steps, and $(3)$ the mean square radius $R^2 (t)$ of the cluster (or, the mean square displacement from the origin  of the cluster) after $t$ time steps. At the critical point $p = p_c$, these quantities are expected to follow power-type scaling laws in the long-time limit $(t \rightarrow \infty)$ : $$\begin{aligned} P(t) &\sim& t^{- \delta}, \nonumber \\ N(t) &\sim& t^{\eta}, \label{eq:dynamic} \\ R^2 (t) &\sim& t^z, \nonumber\end{aligned}$$ where $\delta$, $\eta$ and $z$ are dynamic critical exponents. In the case of $N(t)$ the average is taken over all clusters including those which have died out, while $R^2 (t)$ is averaged over the occupied sites in the surviving clusters only. Results for the three quantities $P(t)$, $N(t)$ and $R^2 (t)$, obtained from computer simulations of the model (\[eq:model\]), are shown in figure 2. On log-log plot curves in the subcritical region bend downward while those in the supercritical region bend upward; at the critical point the curves are expected, if the scaling laws (\[eq:dynamic\]) are true, to be straight lines as $t \rightarrow \infty$. It is obvious from figure 2 that $0.75 < p_c < 0.753$. More precise estimates for $p_c$ and the critical exponents are obtained by the method of effective exponents due to Grassberger  [@Grassberger1989-2]. Effective exponents $\delta_t$, $\eta_t$ and $z_t$ are defined as the local slopes of the curves shown in figure 2 : $$- \, \delta_t = {\Delta [\log P(t)] \over \Delta [\log t]}~,~~~~~~ \eta_t = {\Delta [\log N(t)] \over \Delta [\log t]}~,~~~~~~ z_t = {\Delta [\log R^2(t)] \over \Delta [\log t]}~, \label{eq:effectdef}$$ which are measured by using the formulae [@Grassberger1989-2] : $$- \, \delta_t = {\log[P(t) / P(t/m)] \over \log m}~, \label{eq:effective1}$$ and similar expressions for $\eta_t$ and $z_t$. Results for $m = 10$ are shown in figure 3. Like the curves in figure 2, curves for $p < p_c$ bend downward while those for $p > p_c$ bend upward. For finite times the dynamic quantities do not have a pure power-law scaling form like (\[eq:dynamic\]); in general there exists correction-to-scaling of the type : $$P(t) \sim t^{- \delta} \left(1 + {a \over t} + {b \over t^{\delta '}} + \cdots \right), \label{eq:correction}$$ and similar expressions for $N(t)$ and $R^2 (t)$ with correction-to-scaling exponents $\delta '$, $\eta '$ and $z '$ respectively. Consequently the behaviour of the effective exponents defined in (\[eq:effectdef\]) are given by [@Grassberger1989-2] : $$\delta_t = \delta + {a \over t} + {\delta ' b \over t^{\delta '}} + \cdots , \label{eq:effective2}$$ and similar expressions for $\eta_t$ and $z_t$. The critical exponents of the model appear as the asymptotic values of the corresponding effective exponents : $$\begin{array}{lll} \makebox[3.2cm][l]{$\delta = \lim_{t \rightarrow \infty} \delta_t$~,} & \makebox[3.2cm][l]{$\eta = \lim_{t \rightarrow \infty} \eta_t$~,} & \makebox[3.2cm][l]{$z = \lim_{t \rightarrow \infty} z_t$~.} \end{array} \label{eq:asymptote}$$ Therefore in a plot of an effective exponent (as the ordinate) versus $1/t$ (as the abscissa) the corresponding critical exponent is obtained as the intercept of the curve for $p = p_c$ on the ordinate axis. Using this method the following estimates for the critical characteristics of the model were obtained from computer simulation data : $$p_c = 0.7515 \pm 0.0005 \label{eq:criticalpoint}$$ and $$\begin{aligned} \delta &=& 0.16 \pm 0.01, \nonumber \\ \eta &=& 0.32 \pm 0.02, \label{eq:criticalvalue} \\ z &=& 1.27 \pm 0.01. \nonumber\end{aligned}$$ The value of the critical point agrees closely with previous estimates [@Bhattacharyya1996; @Bhattacharyya1997] and improves upon them. The values of the critical exponents are also found to satisfy the scaling relation [@Grassberger1979] : $$d \: z = 2 \eta + 4 \delta, \label{eq:scale1}$$ where $d$ is the number of spatial dimensions of the system (here $d = 1$). Evolving from disordered initial states, the density of occupied sites $\rho_t$ was observed to follow the same dynamic scaling as the survival probability. In computer simulations of [@Bhattacharyya1996] the exponent characterising the critically slow relaxation $\rho_t \sim t^{- \alpha}$ at $p = p_c$ was found to be $\alpha \approx 0.16 \approx \delta$. The values of all the three dynamic critical exponents agree, within the limits of error, with the corresponding values for DP in $1 + 1$ dimensions [@Grassberger1979; @Essam1988; @Dickman1991]. In a previous work [@Bhattacharyya1997] the critical exponents $\nu_\perp$ and $\nu_\parallel$ for the correlation length and correlation time respectively at this particular phase transition, obtained by finite-size scaling methods, were also found to belong to the DP class. However, the order parameter exponent $\beta$ was observed to disagree with the DP value [@Bhattacharyya1996]; this was an error arising out of finite-size effects and fluctuations due to the small sample size used for averaging. Since $\delta$ and $\nu_\parallel$ are already in the class of DP, the exponent $\beta$, by virtue of the relation $\beta = \delta \, \nu_\parallel$ [@Grassberger1979], must also agree with the directed percolation value for $1 + 1$ dimensions. The last result is concerned with the fractal dimension of the clusters in the single space dimension at $p = p_c$. The average number of occupied sites per surviving cluster is given by $N_s(t) = N(t) \, / \, P(t)$. The fractal dimension $d_F$ of the clusters at fixed time is defined by : $$N_s \sim R \, ^{d_F}. \label{eq:fractal}$$ Following the definition of the dynamic exponents (\[eq:dynamic\]) the fractal dimension is expected to satisfy the relation : $$d_F \: z = 2(\eta + \delta). \label{eq:scale2}$$ Figure 4 shows a log-log plot of $N_s$ versus $R$ at $p = p_c$. The slope of the curve is given by : $$d_F = 0.74 \pm 0.02,$$ which satisfies relation (\[eq:scale2\]) within the limits of error. Discussion ========== In this concluding section I shall compare the model defined by (\[eq:model\]) with another that has the same set of absorbing states and that belongs to a different university class. The probabilistic cellular automaton (\[eq:model\]) studied here was found to have three absorbing states given by (\[eq:absorb1\]), (\[eq:absorb2\]) and (\[eq:absorb3\]). For $p < p_c$ the vacuum (absorbing state I) is the only attractor of the model while the other two absorbing states (II and III) are never reached from disordered initial states. Consequently the vacuum appears, in effect, to be the unique absorbing state in the subcritical region. The dynamic critical exponents characterising the phase transition in this model indicate that the transition belongs to the class of DP, in agreement with the conjecture of [@Janssen1981; @Grassberger1982]. On the other hand, the phase transition occuring in model $A$ of [@Grassberger1984] belongs to the PC class. This model is yet another one-dimensional elementary probabilistic cellular automaton defined by the evolution rule : $${t : \over {t+1 :}}~~~~~~{111 \over 0}~~\underbrace{{110 \over }~~~~{011 \over }}_{\begin{array}{lll} 0 & \mbox{with probability} & p\\ 1 & \mbox{with probability} & 1 - p \end{array}}~~{101 \over 0}~~~~{010 \over 1}~~~~{100 \over 1}~~~~{001 \over 1}~~~~{000 \over 0} \label{eq:modelA}$$ It is remarkable that the three absorbing states of this model are exactly the same as those of model (\[eq:model\]). However, contrary to their nature in model (\[eq:model\]), the mutually symmetric absorbing states II and III occur as attractors of this model for $p < p_c$ while aborbing state I (the vacuum) is never reached from disordered initial states. It appears that the contrast in the nature of the absorbing states between the two models places them in different universality classes. In that case the non-DP behaviour of model (\[eq:modelA\]) must be due to the degeneracy in the absorbing state in the subcritical region, thus supporting the view of [@Park1995; @Hinrichsen1997]. Acknowledgment {#acknowledgment .unnumbered} ============== I am grateful to Professor Bikas K. Chakrabarti for discussions and for critically reading the manuscript. The work was supported by CSIR, Government of India. [99]{} Ligget T M 1985 [*Interacting Particle Systems*]{} (New York: Springer-Verlag). Grassberger P 1995 [*J. Stat. Phys.*]{} [**79**]{} 13. Janssen H K 1981 [*Z. Phys.*]{} B [**42**]{} 151. Grassberger P 1982 [*Z. Phys.*]{} B [**47**]{} 365. Kinzel W 1983 Directed Percolation [*Percolation Structures and Processes, Annals of Israel Physical Society*]{} Vol. 5, ed G Deutscher, R Zallen and J Adler (Bristol: Adam Hilger). Harris T E 1974 [*Ann. Prob.*]{} [**2**]{} 969. Schlögl F 1972 [*Z. Phys.*]{} B [**253**]{} 147. Domany E and Kinzel W 1984 [*Phys. Rev. Lett.*]{} [**53**]{} 311. Kinzel W 1985 [*Z. Phys.*]{} B [**58**]{} 229. Grassberger P and de la Torre A 1979 [*Ann. Phys.*]{} [**122**]{} 373. Jensen I 1993 [*Phys. Rev.*]{} E [**47**]{} R1. Ziff R M, Gulari E and Barshad Y 1986 [*Phys. Rev. Lett.*]{} [**56**]{} 2553. Grinstein G, Lai Z W and Browne D A 1989 [*Phys. Rev.*]{} A [**40**]{} 4820. Jensen I, Folgedby H C and Dickman R 1990 [*Phys. Rev.*]{} A [**41**]{} 3411. Albano E V 1995 [*Physica*]{} A [**214**]{} 426. Jensen I 1993 [*Phys. Rev. Lett.*]{} [**70**]{} 1465. Grassberger P, Krause F and von der Twer T 1984 [*J. Phys. A: Math. Gen.*]{} [**17**]{} L105. Grassberger P 1989 [*J. Phys. A: Math. Gen.*]{} [**22**]{} L1103. Kim H H and Park H 1994 [*Phys. Rev. Lett.*]{} [**73**]{} 2579. Jensen I 1994 [*Phys. Rev.*]{} E [**50**]{} 3623. Jensen I 1993 [*J. Phys. A: Math. Gen.*]{} [**26**]{} 3921. Park H and Park H 1995 [*Physica*]{} A [**221**]{} 97. Hinrichsen H 1997 [*Phys. Rev.*]{} E [**55**]{} 219. Wolfram S 1983 [*Rev. Mod. Phys.*]{} [**55**]{} 601. Bhattacharyya P 1996 [*Physica*]{} A [**234**]{} 427. Bhattacharyya P [*Physica*]{} A (in press). Grassberger P 1989 [*J. Phys. A: Math. Gen.*]{} [**22**]{} 3673. Essam J W, Guttmann A J and de Bell K 1988 [*J. Phys. A: Math. Gen.*]{} [**21**]{} 3815. Dickman R and Jensen I 1991 [*Phys. Rev. Lett.*]{} [**67**]{} 2391. Figure Captions {#figure-captions .unnumbered} =============== [**Figure 1.**]{}  Typical examples of evolution from a single occupied site following rule (\[eq:model\]) : (a) a case in the subcritical region ($p = 0.74$) and (b) a case in the supercritical region ($p = 0.76$). The first $200$ time-steps if the evolution process are shown. [**Figure 2.**]{}  Results for three dynamic properties from monte-carlo simulations of the model : (a) survival probability, (b) average number of occupied sites and (c) mean square radius of the evolving cluster. Each of the three panels contains five curves that correspond to $p$ = $0.753$ (top), $0.752$, $0.7515$, $0.751$, and $0.75$ (bottom) respectively. [**Figure 3.**]{}  The effective exponents measured as the local slopes of the curves shown in figure 2. For large $t$, data have been averaged over many time-steps in order to suppress fluctuations. [**Figure 4.**]{}  The log-log plot of the average number of occupied sites as a function of the root-mean-square radius of the clusters at $p = p_c$. The slope of the curve gives the fractal dimension of the clusters at fixed time. The straight line drawn below the curve is the graph of $N_s = {\rm const.}~R^{0.74}$. [^1]: E-mail : pratip@hp1.saha.ernet.in

--- abstract: 'We model a tiny heat engine as a Brownian particle that moves in a viscous medium in a sawtooth potential (with or without load) assisted by $\it {alternately}$ placed hot and cold heat baths along its path. We find closed form expression for the steady state current as a function of the model parameters. This enables us to deal with the energetics of the model and evaluate either its efficiency or its coefficient of performance depending upon whether the model functions either as a heat engine or as a refrigerator, respectively. We also study the way current changes with changes in parameters of interest. When we plot the phase diagrams showing the way the model operates, we not only find regions where it functions as a heat engine and as a refrigerator but we also identify a region where the model functions as neither of the two.' author: - Mesfin Asfaw - Mulugeta Bekele title: Exploring the operation of a tiny heat engine --- **Introduction** ================= Understanding the physical properties of devices of micron- and nano-meters sizes is of interest these days since the trend is in miniaturizing them. Such devices may need to be kept under certain environmental condition such as a particular temperature in order for them to optimally operate. We envision heat pumps (or refrigerators) of similar sizes that provide such favorable condition. On the other hand, heat engines of similar sizes may provide energy to these devices in order for them to perform different tasks. This paper deals with a model of such tiny heat engine or, conversely, heat pump and explores the details. The idea of microscopic (or Brownian) heat engine working due to non-uniform temperature first came up with the works of B[ü]{}ttiker [@but], van kampen [@van], and Landuaer [@lan1] while they were involved in exposing the significance of the now influential papers of Landauer on blowtorch effect [@lan2; @lan3]. Later works took models of Brownian heat engine and dealt with the energetics only at the quasi-static limit [@miki3; @Aus1]. Energetics considerations for such engine operating in a [finite time]{} were first addressed by Derènyi et al. [@Ast2] and recently by us [@mesfin1; @mesfin2]. These two works of us also revealed that the quasi-static limits of either the heat engine or the heat pump behave exactly like that of a Carnot’s heat engine or heat pump. In our first paper [@mesfin1], we considered a Brownian particle moving through a highly viscous medium in a periodic sawtooth potential (with or without load) assisted by the thermal kick it gets from [*alternately*]{} placed hot and cold heat reservoirs along its path. The heat reservoirs were placed in such a way that the whole left side of each sawtooth from its barrier top is coupled to the hot reservoir while the whole right side is coupled to the cold reservoir. In the present work, instead of coupling the whole left side of each sawtooth to the hot reservoir we let it be hot within a narrower but varying width located at a position that can also vary as a parameter. This will introduce two additional parameters to the model and, thereby, address a more general problem. By evaluating the way energy exchange takes place between the particle and the medium, we explore the conditions under which the model works either as a heat engine or as a heat pump. One new result is the existence of a third region in the parameter space where the model works neither as a heat engine nor as a heat pump with rich phase diagrams. The paper is organized as follows: In section II we will consider our model in the absence of external load. In this case we will show that the model works only as a heat engine and, at quasistatic limit, the efficiency goes to that of Carnot efficiency. In section III we will consider our model in the presence of external load and explore how current, efficiency and coefficient of performance of the model vary with change in model parameters of interest. We will identify the different operating regions (as a heat engine, as a refrigerator or neither of the two) of the model in phase diagrams. Section IV deals with summary and conclusion. The model with no external load =============================== The model consists of a Brownian particle which moves in a viscous medium in a periodic sawtooth potential where the background temperature is also periodic but piecewise constant as shown in Fig. 1. Here U(x) denotes the potential of the sawtooth with a barrier height of $U_{0}$. One of the maxima of the potential is located at x=0 and the two minima on the left and right sides of this maximum potential are located at $x=-L_{1}$ and $x=L_{2}$, respectively. Two dimensionless parameters, $\alpha $ and $\delta $, specify the position and width of each hot locality such that $\alpha $ describes the position of the left side of the hot locality from the nearest barrier top while $\delta $ describes the width of the hot locality, both in units of $L_{1}$. Here one should note that $\alpha$ and $\delta$ are limited to take values between zero and one: $0<\alpha<1$ and $0<\delta <1$. On the other hand, in our previous work [@mesfin1] both $\alpha$ and $\delta$ are taken to have a fixed value of one. The quantities $T_{h}$ and $T_{c}$ in the figure denote the temperature values of the hot and cold localities, respectively. A third parameter, $\tau$, relates the temperature of the hot locality with the cold locality such that $T_{h}=T_{c}(1+\tau ) $. The potential, $U(x)$, for our model is a series of identical sawtooth potentials. One sawtooth potential around x=0 is described by $$U(x)=\cases{ U_{0}[{x\over L_{1}}+1],&if $-L_{1}< x \le 0$;\cr U_{0}[{-x\over L_{2}}+1],&if $0 < x \le L_{2}$;\cr }$$ and this potential profile repeats itself such that $U(x+L)=U(x)$ where $L=L_{1}+L_{2}$. Note that $U(x)$ is determined by three parameters: $U_{0}$, $L_{1}$ and $L_{2}$. On the other hand, the temperature profile is given by $$T(x)=T_{c}+\tau T_{c} (\Theta (x-\alpha L_{1})- \Theta (x-(\alpha+\delta) L_{1}),$$ for $-L_{1}<x \le L_{2}$ and $\Theta$ is the Heavyside function. This temperature profile repeats itself with the same period as that of the potential: $T(x+L)=T(x)$. The presence of hot and cold regions coupled to the different parts of the sawtooth potential determines the Brownian particle to be driven unidirectionally and attain a steady state current, $J$. A general expression for the steady state current of the Brownian particle in any periodic potential (with or without load) is derived in Appendix A. Using the particular potential and temperature profiles of Eqs. (1) and (2), we have evaluated the expressions for $F$, $G_1$, $G_2$ and $H$ in Appendix A, Eqs. (A10, A11,A12, and A13), and found an exact closed form expression for the steady state current $J$ $$J={-F\over G_{1}G_{2} +H F}.$$ The expressions for $F$, $G_{1}$, $G_{2},$ and $H$ are given in Appendix B. The drift velocity, $v$, is then given by $$v=J(L_{1}+L_{2}).$$ As an approximation, we neglect the energy transfer via kinetic energy due to the particle recrossing of the boundary between the regions [@Aus1; @Ast2] and find the energetics of the particle as it exchanges energy with the heat reservoirs. In one cycle the Brownian particle is in contact with the hot region of width $\delta L_{1}$ while the particle is in contact with the cold region of width $L_{2}+L_{1}(1-\delta)$. When the particle moves through the hot region, it takes an energy $Q_{h}$ which enables it not only to climb up the potential of magnitude $\delta U_{0}$ but also acquire energy $\delta v\gamma L_{1}$ to overcome the viscous drag force within the hot interval. Hence during one cycle, the amount of heat flow out of the hot reservoir, $Q_{h}$, is given by $$Q_{h}=\delta U_{0}+\gamma v L_{1}\delta.$$ The net heat flow per cycle from the Brownian particle to the cold reservoir, $Q_{c}$, is by a similar argument given by $$Q_{c}=\delta U_{0}-\gamma v (L_{2}+L_{1}(1-\delta)).$$ The difference between $Q_{h}$ and $Q_{c}$ is the useful work: $$W=\gamma v(L_{1}+L_{2}).$$ This is exactly the amount of work required to transport the Brownian particle moving with a drift velocity $v$ through one cycle in the viscous medium. It is important to emphasize at this point that in the absence of external load the model works only as heat engine. Following the definition of generalized efficiency introduced in [@Ast2], the efficiency of the engine, $\eta$, is given by $$\eta={Q_{h}-Q_{c}\over Q_{h}}={\gamma v (L_{1}+L_{2})\over \delta (U_{0}+\gamma v L_{1})}.$$ One can notice that the magnitude of the current approaches to zero when $U_{0}$ goes to zero. This corresponds to the quasistatic limit of the engine; i.e. $J \to 0$. Evaluating the efficiency at quasistatic limit, we found that $$\displaystyle \lim _{{U_{0}\to 0}}{\eta }={(T_{h}-T_{c})\over T_{h}},$$ which is exactly equal to the efficiency of a Carnot engine. In order for us to explore how current and efficiency vary with change in one of the newly introduced parameters $\alpha$ and $\delta$, we first introduce three scaled parameters: scaled length $\ell=L_{2}/ L_{1} $, scaled barrier height $ u=U_{0}/ k_BT_{c}$ and scaled current $j=J/J_{0}$ where $J_{0}=k_BT_{c}/\gamma L_{1}^2$; $k_B$ being Boltzmann’s constant. For fixed values of $\alpha=1$, $u=4$ and $\tau=1$, the current $j$ is plotted as a function of $\delta$ as shown in Fig. 2. When $\delta=0$, the whole sawtooth potential is coupled with cold reservoir. Hence there is no net current as can be seen in the figure. Increasing the width of the hot locality leads to an increase in the current. We also plot the engine efficiency, $\eta$, as a function of $\delta$ as shown in Fig. 3. The figure shows that $\eta$ increases with increase in $\delta$. One can fix the width of the hot locality, $\delta$, and investigate the influence of shifting the position of the hot locality along one side of the sawtooth potential. Figure 4 is a plot of $j$ versus $\alpha$ for a given values of $\delta=0.2$, $u=4$ and $\tau=1$. The figure shows that the current $j$ increases as $\alpha$ increases which implies that positioning the hot locality near the potential minimum produces the highest possible value of current than putting it anywhere else. Likewise, plot of $\eta$ versus $\alpha$ in Fig. 5 shows similar feature as in Fig. 4 in the sense that one gets highest possible efficiency if the hot locality is put near the potential minimum than anywhere else. In the next section we will consider our model to also have external load. The model with constant external load ===================================== Consider the model in the presence of a constant external load $f$ as shown in Fig. 6. The potential profile will then change from $U(x)$ described by Eq. (1) to $U(x)+fx$. Similar to the previous section, the closed form expression for steady state current $J^{L}$ in the presence of external load is derived and given by $$J^{L}={-F^{L}\over G_{1}^{L}G_{2}^{L} +H^{L}F^{L}}.$$ The expressions for $F^{L}$, $G_{1}^{L}$, $G_{2}^{L},$ and $H^{L}$ are given in Appendix C. The drift velocity, $v^{L}$, is then given by $$v^{L}=J^{L}(L_{1}+L_{2}).$$ Before we begin exploring various properties of the model, we introduce a scaled quantity associated with the load: $\lambda=fL_{1}/T_{c}$. We will then have a total of six parameters that can be controlled independently: $u$, $l$, $\tau$, $\delta$, $\alpha$ and $\lambda$. Fig. 7 is a plot of (scaled) current $j$ versus $\delta$ for fixed values of all the other parameters. The figure shows that the current monotonously increases with $\delta$ being negative for small values of $\delta$ (hot locality width) and then becoming positive beyond a certain value of $\delta$. This implies that in the interval where the current is negative external work is being done [*on*]{} the engine while in the interval where the current is positive the engine [*does*]{} work not only against the viscous medium but also against the load. In order to find whether the engine works either as a heat engine, or as a refrigerator or other wise, we need to find energy exchange between the particle and the heat reservoirs. When the model behaves as a heat engine, the current takes postive value (see Fig. 7) and in one cycle the particle takes heat energy $Q_{h}^L$ from the hot reservoir of amount $$Q_{h}^{L}=\delta(U_{0}+fL_{1}+\gamma v^L L_{1}),$$ and gives off an amount of heat $$Q_{c}^{L}=\delta U_{0}-[((L_{1}(1-\delta)+L_{2}))(f+\gamma v^L)]$$ to the cold reservoir. The net work, $W$, done by the heat engine in one cycle is given by the difference between $Q_{h}^L$ and $Q_{c}^L$: $$W=(\gamma v^L+f)(L_{1}+L_{2}).$$ The generalized efficiency of the heat engine will then be given by $$\eta={W\over Q_h^L}={(\gamma v^L+f)(L_{1}+L_{2})\over \delta(U_{0}+fL_{1}+\gamma v^LL_{1})}.$$ On the other hand, when the model works as a refrigerator heat flows out of the cold reservoir and enters the hot reservoir driven by the external work of amount $W^L = f(L_1 + L_2)$ done on the refrigerator per cycle. In the region where the model works as a refrigrator, the coefficient of performance (COP) of the refrigerator, $P_{ref}$ , is given by $$P_{ref} = { Q_c^L\over W^L}={\delta U_{0}-[((L_{1}(1-\delta)+L_{2}))(f+\gamma v^L)]\over f(L_{1}+L_{2})}.$$ The quasi-static limit of the engine corresponds to the case where the current approaches to zero either from the heat engine side or from the refrigerator side. We have found that this limit is satisfied when $$f={\delta\tau U_{0}\over ((1+l)(1+\tau)-\delta\tau)L_{1}}.$$ This is the boundary demarcating the domain of the operation of the engine as a refrigerator from that as a heat engine. Evaluating the expression for both $\eta$ and $P_{ref}$ as we approach this boundary, we analytically found that they are exactly equal to that of Carnot efficiency and Carnot COP, respectively: $$\displaystyle \lim _{J^+\to 0}{\eta }={(T_{h}-T_{c})\over T_{h}},$$ and $$\displaystyle \lim _{J^-\to 0}{P_{ref} }={T_{c}\over (T_{h}-T_{c})}.$$ We now explore how the $\eta$ and $P_{ref}$ behave as a function of $\delta$ fixing all other parameters. Within the region where the model works as a heat engine, we find the efficiency to monotonously decrease from its maximum value (Carnot efficiency) as the width, $\delta$, of the hot locality increases ( see Fig. 8). On the other hand, within the region where the model works as a refrigerator the COP linearly increases with increase in $\delta$ until it attains its maximum possible value (Carnot refrigerator) as shown in Fig. 9. Before we plot the phase diagrams showing regions where the model works as a heat engine, as a refrigerator and as neither of the two we think that it is illustrative to know how $Q_h$ and $Q_c$ behave with scaled load, $\lambda$. We introduce two scaled quantities $q_h=Q_h/k_BT_c$ and $q_c=Q_c/k_BT_c$ corresponding to $Q_h$ and $Q_c$, respectively. Figs. 10a and 10b show plots of $q_h$ versus $\lambda$ and $q_c$ versus $\lambda$, respectively, after fixing the other parameters. Note that Fig. 10a shows that $q_{h}$ is always positive while Figure 10b shows that $q_{c}$ takes a negative value within some interval values of $\lambda$. The quantity $q_{h}-q_{c}$ is always positive. In the region that $q_{c}$ is negative, the model works neither as a heat engine nor as a refrigerator since under this situation external work is supplying energy to [*both*]{} the hot and cold reservoirs. We have found that when $$0<\lambda <{\delta\tau u\over ((1+l)(1+\tau)-\delta\tau)}$$ the model works as a heat engine while the model works as a refrigerator when $${\delta\tau u\over ((1+l)(1+\tau)-\delta\tau)}<\lambda<\left({\delta u \over (1-\delta)+l}-j(1+l)\right).$$ The model works neither as a heat engine nor as a refrigerator if $$\lambda>\left({\delta u \over (1-\delta)+l}-j(1+l)\right).$$ We plot the three operation regions of the model in Fig. 11. The figure demonstrates that in the parameter space of $\lambda - \tau$, the model works as a heat engine, as a refrigerator or as neither of the two for fixed $\delta=.6$, $\ell=3$, $u=4$ and $\alpha=1$. One can see in the phase diagram that for $\tau=4$, the model works neither as a heat engine nor as a refrigerator when $1.21875<\lambda< 3.156$. In fact in this region, $q_{c}$ is negative as can be seen in Fig.10b. Figure 12 shows the three regions in $\lambda - \delta$ parameters space in which the model operates as a heat engine, as a refrigerator and as neither of the two for fixed $l=3$, $\alpha=1$, $\tau=1$and $u=4$. summary and conclusion ====================== We considered a simple model of a Brownian heat engine and explored how current, efficiency and performance of a refrigerator behave as model parameters of interest vary. In the absence of external load, the model works as a heat engine while in the presence of external load the model works as a heat engine, as a refrigerator or neither of the two depending on the values in the parameter space. At a quasistatic limit the efficiency as well as the COP of the engine goes to that of Carnot efficiency and Carnot COP. In this paper we reported an extended version of the work [@mesfin1]. The result we obtained shows that unlike the previous work, there are regions that the model may work neither as a heat engine nor as a refrigerator determined by the width of hot locality. In this work, we found a closed expression for the steady state current. This enabled us to explore the energetics of the Brownian heat engine not only at the quasistatic limits but also while operating at any finite time. This is a clear exposition of the power of having analytic expression for the concerned physical quantities. Derivation of the steady state current ====================================== Consider motion of a Brownian particle in any periodic potential, $U(x)$, and temperature, T(x), profiles where both have the same periodic length of $L_{1}+L_{2}=L$ as shown in Fig. 13. In the high friction limit, the dynamics of the Brownian particle is governed by the Smoluchowski equation [@san]: $${\partial P(x,t)\over \partial t}={\partial\over \partial x} ({1\over \gamma }[U'(x)P(x,t)+{\partial \over \partial x} (T(x)P(x,t)) ])$$ where P(x,t) is the probability density of finding the particle at position x at time t, $U'(x)={d\over dx}U$ and $\gamma $ is the coefficient of friction of the particle. Boltzman’s constant, $k_{B}$, and the mass of the particle are taken to be unity. The steady state solution, $P^{s}(x)$, of the Smoluchowski equation implies a constant current, J, given by $$-{1\over \gamma}[U'(x)P^{s}(x)+{d \over dx}(T(x) P^{s}(x))]=J.$$ Periodic boundary condition implies $$P^{s}(x+L)=P^{s}(x).$$ Equation A3 is valid for any periodic potential $even$ $ in$ $the$ $presence$ $of$ $additional$ $constant$ $external$ $load$. The stationary probability distribution can be then given by $$P^{s}(x)= {B\over T(x)}exp(-\phi (x))$$ where B is the normalization constant and $ \phi (x)=\int _{-L_{1} }^{x} {U'(x)dx'\over T(x')}$ with $-L_{1}<x\le L_{2}$. After multiplying both sides of Eq. (A4) by $ exp(\phi (x))$, we can write it as $${d\over dx}[e^{\phi (x)}T(x)P^{s}(x)]={-\gamma J}e^{\phi (x)}.$$ Integrating Eq. (A5) from $ -L_{1}$ to $ x $ and, after some algebra, one gets $$P^{s}(x)={e^{-\phi (x)}\over T(x)}{\brack T(-L_{1})P^{s}(-L_{1}) - \gamma J \int _{-L_{1}}^{x} e^{\phi (x)}dx} .$$ Taking x=$L_{2}$ in Eq. (A6), we get $$(e^{\phi (L_{2})}-1)P^{s}(L_{2})T(L_{2})=-\gamma J \int _{-L_{1}}^{L_{2}} {e^{\phi (x)}}dx.$$ Applying normalizing condition, $ \int _{-L_{1}}^{L_{2}}P^{s}(x)dx=1$, on Eq. (A6) gives us the relation $$\begin{aligned} 1& = &P^{s}(L_{2})T(L_{2})\int _{-L_{1}}^{L_{2}}{e^{-\phi (x)}\over T(x)}dx- \nonumber \\ & & \gamma J \int _{-L_{1}}^{L_{2}} {e^{-\phi (x)}\over T(x)} \int _{-L_{1}}^{x} {e^{\phi (x')}}dx'.\end{aligned}$$ Using Eqs. (A7) and (A8) it is simple to get the steady state current $$J={-F\over G_{1}G_{2} + HF},$$ where$$\begin{aligned} F&=&e^{\phi (L_{2})}-1,\\ G_{1}&=&\int _{-L_{1}}^{L_{2}}{e^{-\phi (x) }\over T(x)}dx,\\ G_{2}&=& \int _{-L_{1}}^{L_{2}}{e^{\phi (x') }\gamma (x')}d'x,\\ H&=&\int _{-L_{1}}^{L_{2}}{e^{-\phi (x) }\over T(x)}dx \int _{-L_{1}}^{x}{e^{\phi (x') }\gamma (x')}d'x.\end{aligned}$$ In this Appendix, we will give the expressions for $F$, $G_{1}$, $G_{2}$, and $H$ which define the value of the steady state current, $J$, for load free case $$\begin{aligned} F& = &e^{\delta U_{0}[{1\over T_{h}}-{1\over T_{c}}]}-1 \\ G_{1}& = &{L_{1}\over U_{0}}[1-e^{{(\alpha -1)U_{0}\over T_{c}}}]+{L_{1}\over U_{0}} e^{{(\alpha -1)U_{0}\over T_{c}}}[1-e^{{-\delta U_{0}\over T_{h}}}]+\nonumber \\ & & {L_{1}\over U_{0}}e^{{(\alpha -1)U_{0}\over T_{c}}-{U_{0}\delta \over T_{h}}} [1-e^{{(\delta-\alpha) U_{0}\over T_{c}}}] +{L_{2}\over U_{0}}e^{{(\alpha -1)U_{0}\over T_{c}}-{U_{0}\delta \over T_{h}}- {(\alpha -\delta) U_{0}\over T_{c}}}[e^{{U_{0}\over T_{c}}}-1] \\ G_{2}& = & {\gamma T_{c}L_{1}\over U_{0}}[e^{{(1-\alpha )U_{0}\over T_{c}}}-1]+ {\gamma T_{c}L_{1}\over U_{0}}e^{{(1-\alpha )U_{0}\over T_{c}}+{U_{0}\delta \over T_{h}}} [e^{{(\alpha-\delta) U_{0}\over T_{c}}}-1] +{\gamma T_{h}L_{1}\over U_{0}}e^{{(1-\alpha )U_{0}\over T_{c}}}[e^{{\delta U_{0}\over T_{h}}}-1]+{\gamma T_{c}L_{2}\over U_{0}}\nonumber \\ & & e^{{(1-\alpha )U_{0}\over T_{c}}+{U_{0}\delta \over T_{h}}+ {(\alpha-\delta) U_{0}\over T_{c}}} [1-e^{{-U_{0}\over T_{c}}}]\\ H& = &t_{1}+t_{2}+t_{3}+t_{4}+t_{5}+t_{6}+t_{7}+t_{8}+t_{9}+t_{10}\\ t_{1}& = &{L_{1}\gamma \over U_{0}}[(1-\alpha )L_{1}+{T_{c}L_{1}\over U_{0}}[ e^{{U_{0}(\alpha -1)\over T_{c}}}-1]],\\ t_{2}& = &{L_{1}^{2}T_{c}\gamma \over U_{0}^{2}}e^{{U_{0}(\alpha -1)\over T_{c}}} [e^{{(1-\alpha )U_{0}\over T_{c}}}-1][1-e^{{-U_{0}\delta \over T_{h}}}],\\ t_{3}& = &{L_{1}\gamma \over U_{0}}[\delta L_{1}+{T_{h}L_{1}\over U_{0}}[e^{{-U_{0}\delta \over T_{h}}} -1]],\\ t_{4}& = &{L_{1}^{2}T_{c}\gamma \over U_{0}^{2}}[1-e^{{U_{0}(1-\alpha )\over T_{c}}}] e^{{U_{0}(\alpha -1)\over T_{c}}-{\delta U_{0}\over T_{h}}} [e^{{-U_{0}(\alpha -\delta )\over T_{c}}}-1],\\ t_{5}& = &{L_{1}^{2}T_{h}\gamma \over U_{0}^{2}}[e^{{-U_{0}(\alpha -\delta )\over T_{c}}}-1] [1-e^{{U_{0}\delta \over T_{h}}}]e^{{-U_{0}\delta \over T_{h}}},\\ t_{6}& = &{L_{1}\gamma \over U_{0}}[(\alpha -\delta )L_{1}+{T_{c}L_{1}\over U_{0}} [e^{{-U_{0}(\alpha -\delta )\over T_{c}}}-1]],\\ t_{7}& = &{L_{1}L_{2}T_{c}\gamma \over U_{0}^{2}}[e^{{U_{0}(1-\alpha )\over T_{c}}}-1] e^{{-U_{0}\over T_{c}}+{U_{0}\delta \over T_{c}}-{U_{0}\delta \over T_{h}}} [e^{{U_{0}\over T_{c}}}-1],\\ t_{8}& = &{L_{1}L_{2}T_{h}\gamma \over U_{0}^{2}}[e^{{U_{0}\delta \over T_{h}}}-1] [e^{{U_{0}\over T_{c}}}-1]e^{{-U_{0}(\alpha -\delta )\over T_{c}}-{U_{0}\delta \over T_{h}} },\\ t_{9}& = &{L_{1}L_{2}T_{c}\gamma \over U_{0}^{2}}e^{{-U_{0}(\alpha -\delta )\over T_{c}}} [e^{{U_{0}(\alpha -\delta )\over T_{c}}}-1][e^{{U_{0}\over T_{c}}}-1],\\ t_{10} & = &{-L_{2}\gamma \over U_{0}}[L_{2}-{L_{2}T_{c}\over U_{0}}[ e^{{U_{0}\over T_{c}}}-1]].\end{aligned}$$ In this Appendix we will give the expressions for $F^{L}$, $G_{1}^{L}$, $G_{2}^{L}$ and $H^{L}$ which define the value of the steady state current, $J^{L}$, for nonzero external load case. $$\begin{aligned} F^{L}& = &e^{{U_{0}+fL_{1}\over T_{c}}+{-U_{0}+fL_{2}\over T_{c}}+ {\delta (U_{0}+fL_{1})\over T_{h}}-{\delta (U_{0}+fL_{1})\over T_{c}}}-1,\\ G_{1}^{L}& = &{L_{1}\over (U_{0}+fL_{1})}[1-e^{{(\alpha -1)(U_{0}+fL_{1})\over T_{c}}}], +{L_{1}\over (U_{0}+fL_{1})} e^{{(\alpha -1)(U_{0}+fL_{1})\over T_{c}}}[1-e^{{-\delta (U_{0}+fL_{1})\over T_{h}}}] +{L_{1}\over(U_{0}+fL_{1})}e^{{(\alpha -1)(U_{0}+fL_{1})\over T_{c}}-{(U_{0}+fL_{1})\delta \over T_{h}}}\nonumber \\ & & [1-e^{{(\delta-\alpha) (U_{0}+fL_{1})\over T_{c}}}] +{L_{2}\over (U_{0}-fL_{2})}e^{{(\alpha -1)(U_{0}+fL_{1})\over T_{c}}-{(U_{0}+fL_{1})\delta \over T_{h}}- {(\alpha -\delta) (U_{0}+fL_{1})\over T_{c}}}[e^{{(U_{0}-fL_{2})\over T_{c}}}-1]\\ G_{2}^{L}& = &{\gamma T_{c}L_{1}\over (U_{0}+fL_{1})}[e^{{(1-\alpha ) (U_{0}+fL_{1})\over T_{c}}}-1] +{\gamma T_{c}L_{1}\over (U_{0}+fL_{1})}e^{{(1-\alpha ) (U_{0}+fL_{1})\over T_{c}} +{ (U_{0}+fL_{1})\delta \over T_{h}}} [e^{{(\alpha-\delta) (U_{0}+fL_{1})\over T_{c}}}-1] +{\gamma T_{h}L_{1}\over (U_{0}+fL_{1})}e^{{(1-\alpha ) (U_{0}+fL_{1})\over T_{c}}}\nonumber \\ & &[e^{{\delta (uo+fL_{1})\over T_{h}}}-1]+{\gamma T_{c}L_{2}\over (U_{0}-fL_{2})} e^{{(1-\alpha ) (U_{0}+fL_{1})\over T_{c}}+{ (U_{0}+fL_{1})\delta \over T_{h}}+ {(\alpha-\delta) (uo+fL_{1})\over T_{c}}} [1-e^{{-U_{0}+fL_{2}\over T_{c}}}].\\ H^{L}& = &T_{1}+T_{2}+T_{3}+T_{4}+T_{5}+T_{6}+T_{7}+T_{8}+T_{9}+T_{10} \\ T_{1}& = &{L_{1}\gamma \over (U_{0}+fL_{1}) }[(1-\alpha )L_{1}+{T_{c}L_{1}\over (U_{0}+fL_{1})}[ e^{{ (U_{0}+fL_{1})(\alpha -1)\over T_{c}}}-1]],\\ T_{2}& = &{L_{1}^{2}T_{c}\gamma \over (U_{0}+fL_{1}) ^{2}}e^{{ (U_{0}+fL_{1})(\alpha -1)\over T_{c}}} [e^{{(1-\alpha ) (U_{0}+fL_{1})\over T_{c}}}-1][1-e^{{- (U_{0}+fL_{1})\delta \over T_{h}}}],\\ T_{3}& = &{L_{1}\gamma \over (U_{0}+fL_{1})}[\delta L_{1}+{T_{h}L_{1}\over (U_{0}+fL_{1})}[e^{{-(U_{0}+fL_{1})\delta \over T_{h}}} -1]],\\ T_{4}& = &{L_{1}^{2}T_{c}\gamma \over (U_{0}+fL_{1})^{2}}[1-e^{{(U_{0}+fL_{1})(1-\alpha )\over T_{c}}}] e^{{(U_{0}+fL_{1})(\alpha -1)\over T_{c}}-{\delta (U_{0}+fL_{1})\over T_{h}}} [e^{{-(U_{0}+fL_{1})(\alpha -\delta )\over T_{c}}}-1],\\ T_{5}& = &{L_{1}^{2}T_{h}\gamma \over (U_{0}+fL_{1})^{2}}[e^{{-(U_{0}+fL_{1})(\alpha -\delta )\over T_{c}}}-1] [1-e^{{(U_{0}+fL_{1})\delta \over T_{h}}}]e^{{-(U_{0}+fL_{1})\delta \over T_{h}}},\\ T_{6}& = &{L_{1}\gamma \over (U_{0}+fL_{1})}[(\alpha -\delta )L_{1}+{T_{c}L_{1}\over (U_{0}+fL_{1})} [e^{{-(U_{0}+fL_{1})(\alpha -\delta )\over T_{c}}}-1]],\\ T_{7}& = &{L_{1}L_{2}T_{c}\gamma \over (U_{0}+fL_{1})(U_{0}-fL_{2})}[e^{{(U_{0}+fL_{1})(1-\alpha )\over T_{c}}}-1] e^{{-(U_{0}+fL_{1})\over T_{c}}+{(U_{0}+fL_{1})\delta \over T_{c}}-{(U_{0}+fL_{1})\delta \over T_{h}}} [e^{{(U_{0}-fL_{2})\over T_{c}}}-1],\\ T_{8}& = &{L_{1}L_{2}T_{h}\gamma \over (U_{0}-fL_{2}) (U_{0}+fL_{1})}[e^{{ (U_{0}+fL_{1})\delta \over T_{h}}}-1] [e^{{ (U_{0}-fL_{2})\over T_{c}}}-1]e^{{- (U_{0}+fL_{1})(\alpha -\delta )\over T_{c}}-{ (U_{0}+fL_{1})\delta \over T_{h}} },\\ T_{9}& = &{L_{1}L_{2}T_{c}\gamma \over (U_{0}+fL_{1})(U_{0}-fL_{2})}e^{{-(U_{0}+fL_{1})(\alpha -\delta )\over T_{c}}} [e^{{(U_{0}+fL_{1})(\alpha -\delta )\over T_{c}}}-1][e^{{(U_{0}-fL_{2})\over T_{c}}}-1],\\ T_{10}& = &{-L_{2}\gamma \over(U_{0}-fL_{2})}[L_{2}-{L_{2}T_{c}\over (U_{0}-fL_{2})}[ e^{{(U_{0}-fL_{2})\over T_{c}}}-1]].\end{aligned}$$ We would like to thank The Intentional Program in Physical Sciences, Uppsala University, Uppsala, Sweden for the support they have been providing for our research group. MB would also like to thank the Kavli Institute for Theoretical Physics, University of california at Santa Barbara, for providing a conducive environment to write the major part of the paper during his visit there. [12]{} Büttiker M., *Z. Phys. B*, [**68**]{} (1987) 161. Van Kampen N. G., *IBM J.Res. Dev.*, [**32**]{} (1988) 107. Landauer R., *J. Stat. Phys.*, [**53**]{} (1988) 233. Landauer R., *Phys. Rev. A*, [**12**]{} (1975) 636. Landauer R., *Helv. Phys. Acta*, [**56**]{} (1983) 847. Miki Matsuo and Shin-ichi Sasa, *Physica A*, [**276**]{} (1999) 188. Derènyi I. and Astumian R. D., *Phys. Rev. E*, [**59**]{} (1999) R6219. Derènyi I., Bier M. and Astumian R. D., *Phys. Rev. Lett*, [**83**]{} (1999) 903. J.M. Sancho, M. San Miguel, D. Dürr, J. Stat. Phys. [**28**]{}, (1982) 291 .

--- abstract: 'We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic $n$–space for $n$ greater than $3$. For instance, every nonuniform arithmetic lattice in $SO(n, 1)$ is noncoherent, provided that $n$ is at least $6$.' address: - | Department of Mathematics\ University of California, Davis\ 1 Shields Ave\ CA 95616\ USA - | UFR de Mathématiques\ Université de Lille 1\ 59655 Villeneuve d’Ascq cedex\ France\ Department of Mechanics and Mathematics\ Lomonosov Moscow State University\ Vorob’evy Gory\ Moscow 119992, GSP-2\ Russia author: - Michael Kapovich - Leonid Potyagailo - Ernest Vinberg bibliography: - 'link.bib' title: 'Noncoherence of some lattices in ${\rm Isom}(\mathbb{H}^n)$' --- We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6. We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n&ndash;space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6. Introduction {#intro} ============ The aim of this paper is to prove noncoherence of certain families of lattices in the isometry group $\isom$ of the hyperbolic $n$–space $\H^n$ ($n>3$). We recall that a group $G$ is called *coherent* if every finitely generated subgroup of $G$ is finitely presented. It is well known that all lattices in ${\operatorname{Isom}}(\H^2)$ and ${\operatorname{Isom}}(\H^3)$ are coherent. Indeed, it is easy to prove that every finitely generated Fuchsian group is finitely presented. The coherence of $3$–manifold groups was proved by PScott [@Sc1]. First examples of geometrically finite noncoherent discrete subgroups of $\iso4$ were constructed by the first and second author [@KaP] and the second author [@P1; @P2]. An example of noncoherent uniform lattice in $\iso4$ was given by Bowditch and Mess [@BM]. In what follows we will identify $\H^n$ with a connected component of the hyperboloid $$\{ x: f(x)=-1\}\subset \R^{n+1},$$ where $f$ is a real quadratic form of signature $(n,1)$ in $n+1$ variables. Then the group $\isom$ is identified with the index $2$ subgroup $O'(f, \R)\subset O(f, \R)$ preserving $\H^n$. Let $f$ and $g$ be quadratic forms on finite-dimensional vector spaces $V$ and $W$ over $\Q$. It is said that $f$ *represents* $g$ if the vector space $V$ admits an orthogonal decomposition (with respect to $f$) $$V= V'\oplus V''$$ so that $f|V'$ is isometric to $g$. In other words, after a change of coordinates, the form $f$ can be written as $$f(x_1,...,x_n)=g(x_1,...,x_k)+ h(x_{k+1},..., x_{n})$$ where $n=\dim(V)$ and $k=\dim(W)$. Whenever $f$ represents $g$, a finite index subgroup of $O(g, \Z)$ is naturally embedded into $O(f, \Z)$. The main result of this paper is: \[Theorem A\] For every $n\ge 4$ and every rational quadratic form $f$ of signature $(n,1)$ which represents the form $$q_3=-x_0^2+x_1^2+x_2^2+x_3^2,$$ the lattice $O(f, \Z)$ is noncoherent. \[C1\] For every $n\ge 4$ there are infinitely many commensurability classes of nonuniform noncoherent lattices in $\isom$. We refer the reader to for the discussion of uniform lattices. By combining with some standard facts on rational quadratic forms, we prove: \[Theorem B\] For $n\ge 6$ every nonuniform arithmetic lattice in $\isom$ is noncoherent. As a by-product of the proof, in , we obtain a simple proof of the following result of independent interest (which was proven by Agol, Long and Reid [@ALR] in the case $n=3$). Recall that a subgroup of a group $\Gamma$ is called *separable* if it can be represented as the intersection of a family of finite index subgroups of $\Gamma$. For instance, separability of the trivial subgroup is nothing else than residual finiteness of $\Gamma$. \[Theorem C\] In every nonuniform arithmetic lattice in $\isom$ ($n\le 5$), every geometrically finite subgroup is separable. We refer the reader to Bowditch [@Bowditch] for the definition of geometrically finite discrete subgroups of $\isom$. Recall only that every discrete group which admits a finitely-sided convex fundamental polyhedron is geometrically finite. In we adopt the method of Gromov and Piatetski-Shapiro [@GP] to obtain examples of nonarithmetic noncoherent lattices: \[Theorem D\] For each $n\ge 4$ there exist both uniform and nonuniform noncoherent nonarithmetic lattices in $\isom$. The above results provide a strong evidence for the negative answer to the following question in the case of nonuniform lattices: \[question\] \[DWise\] Does there exist a coherent lattice in $\isom$ for any $n>3$? In we provide some tentative evidence for the negative answer to this question in the uniform case as well. Our proof of the noncoherence in the nonuniform case is different from the one by Bowditch and Mess [@BM]: The finitely generated infinitely presented subgroup that we construct is generated by four subgroups stabilizing 4 distinct hyperplanes in $\H^n$, while in the construction used in [@BM] two hyperplanes were enough. Direct repetition of the arguments used in [@BM] does not seem to work in the nonuniform case. During this work the first author was partially supported by the NSF grants DMS-04-05180 and DMS-05-54349. A part of this work was done when the first and the second authors were visiting the Max Planck Institute for Mathematics in Bonn. The work of the third author was partially supported by the SFB 701 at Bielefeld University. The second author is deeply grateful to Heiner Zieschang who was his host during his Humboldt Fellowship at the University of Bochum in 1991–1992. The third author is thankful to Heiner for many years of his generous friendship. Preliminaries {#prelim} ============= We refer the reader to Kapovich [@Kapovich] and Maskit [@Maskit] for the basics of discrete groups of isometries of the hyperbolic spaces $\H^n$. Given a convex polyhedron $Q\subset \H^n$ let $G(Q)$ denote the subgroup of $\isom$ generated by the reflections in the walls of $Q$. We will frequently use the quadratic forms $$q_n=-x_0^2+x_1^2+...+x_n^2.$$ Let $f$ be a quadratic form $$f=\sum_{i, j} a_{ij} x_i x_j$$ defined over a number field $K\subset \R$, and $\si$ be an embedding $K\to \R$. Then $f^\si$ will denote the form $$\sum_{i, j} \si(a_{ij}) x_i x_j.$$ Arithmetic groups {#ar} ----------------- Let $f$ be a quadratic form of signature $(n,1)$ in $n+1$ variables with coefficients in a totally real algebraic number field $K\subset \R$ satisfying the following condition: $$\label{*}\begin{array}{l} \mbox{For every nontrivial (ie, different from the identity) embedding $\si \co K\to \R$}\\ \mbox{the quadratic form $f^\si$ is positive definite.}\end{array}\tag{$*$}$$ Below we discuss discrete subgroups of $\isom$ defined using the form $f$. Let $A$ denote the ring of integers of $K$. We define the group $\Gamma:=O(f, A)$ consisting of matrices with entries in $A$ preserving the form $f$. Then $\Gamma$ is a discrete subgroup of $O(f, \R)$. Moreover, it is a *lattice*, ie, its index 2 subgroup $$\Gamma'=O'(f, A):=O(f, A)\cap O'(f, \R)$$ acts on $\H^n$ so that $\H^n/\Gamma'$ has finite volume. Such groups $\Gamma$ (and subgroups of $\isom$ commensurable to them) are called *arithmetic subgroups of the simplest type* in $O(n,1)$; see Vinberg and Shvartsman [@VS88]. If $\Gamma\subset \isom$ is an arithmetic lattice so that either $\Gamma$ is nonuniform or $n$ is even, then it follows from the classification of rational structures on $\isom$ that $\Gamma$ is commensurable to an arithmetic lattice of the simplest type. For odd $n$ there is another family of arithmetic lattices given as the groups of units of appropriate skew-Hermitian forms over quaternionic algebras. Yet other families of arithmetic lattices exist for $n=3$ and $n=7$. See, for example, Vinberg and Shvartsman [@VS88] or Millson and Li [@MillsonLi]. A lattice $\Gamma\subset \isom$ is called *uniform* if $\H^n/\Gamma$ is compact and *nonuniform* otherwise. An arithmetic lattice $O(f, A)$ of the simplest type is nonuniform if and only if $K=\Q$ and $f$ is *isotropic*, ie, there exists a nonzero vector $v\in \Q^{n+1}$ such that $f(v)=0$. Meyer’s theorem (which follows from the Hasse–Minkowski principle; see [@BoSh pp61–62] or [@Ca Corollary 1, p75]) states that every indefinite rational quadratic form of rank $\ge 5$ is isotropic. Thus, for each rational quadratic form $f$ of signature $(n,1)$, $n\ge 4$, the lattice $O(f, \Z)$ is nonuniform. Conversely, every nonuniform arithmetic lattice in $\isom$ is commensurable to $O(f, \Z)$, where $f$ is a rational quadratic form. In particular, the groups $O'(q_n, \Z)\subset \isom$ are nonuniform arithmetic lattices. The group $O'(q_3, \Z)$ coincides with the group $G(\Delta)$, where $\Delta\subset \H^3$ is the simplex with the Coxeter diagram $1$ \[b\] at 3 6 $2$ \[b\] at 74 6 $3$ \[b\] at 146 6 $4$ \[b\] at 219 6 {width="2in"} \[f1.fig\] (see [@VS88 Chapter 6, 2.1] and references therein). \[fiber\] The group $G(\Del)$ contains a finite index subgroup $\Gamma$ such that $\H^3/\Ga$ fibers over the circle. Let $v_{4}\in \De$ denote the (finite) vertex of $\Del$ disjoint from the $4$–th face. Consider the union $O$ of the images of $\Del$ under the stabilizer of $v_{4}$ in $G(\Del)$. Then $O$ is a regular right-angled ideal hyperbolic octahedron in $\H^3$ [@Po-Vi; @VS88]. The group $G(\Delta)$ contains $G(O)$ as a finite index subgroup. It is well known that $G(O)$ is commensurable with the fundamental group of the Borromean rings complement which fibers over the circle [@Th]. The property of being the fundamental group of a surface bundle over the circle is hereditary with respect to subgroups of finite index. Thus $G(\Del)$ contains a subgroup $\Gamma$ of finite index so that $\H^3/\Ga$ fibers over the circle. Rational quadratic forms {#rational} ------------------------ The following proposition is well-known in the theory of rational quadratic forms; see Cassels [@Ca Exercise 8, Page 101]. We present a proof for the sake of completeness. \[forms\] Let $f$ and $g$ be nonsingular rational quadratic forms having respectively the signatures $(r,s)$ and $(p,q)$ such that $r\geq p$ and $s\geq q.$ If ${\rm rank}(f)-{\rm rank} (g) \geq 3$ then $f$ represents $g$. Recall that a rational quadratic form $f$ on a rational vector space $V$ *represents* $b\in \Q$ if there exists a vector $v\in V\setminus \{0\}$ such that $f(v)=b$. We use the following lemma. \[number\] Suppose that $f$ is a nonsingular rational quadratic form in $n\ge 4$ variables and $b$ is an arbitrary nonzero rational number. 1. If $f$ is positive definite and $b>0$ then $f$ represents $b$. 2. If $f$ is indefinite then $f$ represents $b$. The form $$F(y_1,...,y_n, y_{n+1}):=f(y_1,...,y_n)-b y_{n+1}^2$$ is an indefinite nonsingular form of rank $\ge 5$. By Meyer’s theorem the form $F$ represents $0$. Hence by [@BoSh Theorem 6, p393], the form $f$ represents $b$. Let $n:=r+s, k:=p+q$ be the ranks of $f$ and $g$ respectively. After changing coordinates in $\Q^k$ we may assume that $g$ has the diagonal form $$g=b_1x_1^2+...+b_kx_k^2,$$ where $b_i\in \Q_+$, if $i\le p$ and $b_i\in \Q_-$, if $i>p$. The form $f$ is isomorphic to $b_1y_1^2+f_1(y_2,..., y_n)$ since $f$ represents $b_1$ by . By applying the same procedure to $f_1$ and arguing inductively we obtain, after $k$ steps, $$f=b_1y_1^2+...+b_ky_k^2+f_k$$ where $f_k$ is a form in $n-k$ variables. Note that the argument works as long as $n-k\geq 3$. Indeed, if $n=k+3$ we will have $$f=b_1y_1^2+...+b_{k-1}y_{k-1}^2+f_{k-1}(y_k, y_{k+1}, y_{k+2}, y_{k+3})$$ and therefore we can apply the above argument the last time to $f_{k-1}$. We now use the above proposition to prove stated in . We will use the following result proven by PScott in [@Sc] for the convex–cocompact subgroups and by Agol, Long and Reid [@ALR] for the geometrically finite subgroups: Let $\Ga$ be a nonuniform arithmetic lattice in ${\operatorname{Isom}}(\H^k)$, $k\le 5$. Then $\Ga$ is commensurable to $O(g, \Z)$ where $g$ is a nonsingular rational quadratic form of signature $(k, 1)$. According to [@Po-Vi] there exists a right-angled noncompact convex polyhedron of finite volume $P^8\subset \H^8$. Moreover, the group $G(P^8)$ is a finite index subgroup in $O'(q_8, \Z)$, see [@VS88]. Since ${\rm rank}(q_8)-{\rm rank}(g)\ge 3$, it follows that $q_8$ represents $g$, see . Hence we have a natural embedding of a finite index subgroup of $\Gamma$ into $G(P^8)$. As $P^8$ is right-angled, every geometrically finite subgroup of $G(P^8)$ is separable. Since subgroup separability is hereditary with respect to passing to a subgroup, we conclude that every geometrically finite subgroup of $\Ga$ is separable. Hyperplane separability {#separability} ----------------------- In we will need the following variation on subgroup separability. Suppose that $\Gamma=O'(f, \Z)$ is an arithmetic subgroup of $\isom$, where $f$ is a rational quadratic form of signature $(n,1)$. Let $V_i\subset \R^{n+1}$, $i=0, 1,...,k$ be rational vector subspaces of codimension 1, so that $V_i\otimes \R$ intersects $\H^n$ along the hyperplane $H_i$, $i=0, 1,...,k$. We assume that $$\label{eq1} H_0\cap H_i=\emptyset, \quad i=1,...,k.$$ The following proposition is a generalization of Long [@Long]; its proof follows the lines of the proof of Margulis and Vinberg [@MV Lemma 10]. \[separ\] There exists a finite index subgroup $\Gamma'\subset \Gamma$ so that for every $\ga\in \Ga'$ either $\ga(H_0)=H_0$ or $$\ga(H_0)\cap (H_0\cup H_1\cup ...\cup H_k)=\emptyset .$$ Let $(\cdot , \cdot)$ denote the symmetric bilinear form on $\R^{n+1}$ corresponding to $f$. Suppose that $V, V'\subset \R^{n+1}$ are codimension 1 vector subspaces which intersect $\H^n$ along hyperplanes $H, H'$. Let $e, e'\in \R^{n+1}$ be nonzero vectors orthogonal to $V, V'$ respectively. Then $H$ intersects $H'$ transversally iff $$|(e, e')|< \sqrt{( e, e) (e', e')}.$$ For each $V_i$ ($i=0, 1, ...,k$) choose an orthogonal primitive integer vector $e_i$. Then implies that $$|(e_i, e_0)|\ge \sqrt{( e_i, e_i) (e_0, e_0)}, \quad i=1,...,k.$$ Choose a natural number $N$ which is greater than $$2\mskip-3mu\max_{i=0, 1, ...,k} |( e_0, e_i)|.$$ Let $\Gamma'=\Gamma(N)$ denote the level $N$ congruence subgroup in $\Gamma$, ie, the kernel of the natural homomorphism $$\Gamma \to GL(n+1, \Z/N\Z).$$ Then for every $\gamma\in \Gamma'$, $i=0, 1, ...,k$, $$(\gamma(e_i), e_0)\equiv (e_i, e_0) \hbox{~~(mod}~~ N)$$ and therefore either $$\begin{aligned} |(\ga(e_i), e_0)|&=|(e_i, e_0)| \\ |(\gamma(e_i), e_0)| &> |(e_i, e_0)| \ge \sqrt{( e_i, e_i) (e_0, e_0)}= \sqrt{( \gamma(e_i), \gamma(e_i)) (e_0, e_0)},\tag*{\hbox{or}}\end{aligned}$$ hence either $\gamma(H_0)=H_i$ or $\gamma(H_0)\cap H_i=\emptyset$. Lastly, we have to ensure that $\ga(e_0)\ne \pm e_i$ for $i=1,...,k$ and all $\ga\in \Ga'$. This is achieved by taking $N$ which does not divide some nonzero entries of $e_0+e_i$ and of $e_0- e_i$ for all $i=1,...,k$. A construction of noncoherent groups {#nonco} ------------------------------------ Let $L\subset {\operatorname{Isom}}(\H^3)$ be a subgroup commensurable to the reflection group $G(\Delta)$ defined in . We embed $\H^3$ in $\H^4$ as a hyperplane $H$ and naturally extend the action of $L$ from $H$ to $\H^4$. Let $p_1, p_2\in \partial H$ be distinct parabolic points of $L$. Let $\Pi_1, \Pi_2$ be perpendicular hyperplanes in $\H^4$ which are parallel to $H$ and asymptotic to $p_1, p_2$, respectively. Let $\tau_i$ denote the (commuting) reflections in $\Pi_i, i=1, 2$. Set $\tau_3:=\tau_1\tau_2$. Let $G$ denote the subgroup of $\iso4$ generated by $L, \tau_1, \tau_2$. [[@KaP]]{}\[kptheorem\] For every choice of the group $L$, hyperplane $H$, points $p_1, p_2$ and hyperplanes $\Pi_1, \Pi_2$ as above, the group $G$ is noncoherent. We will need the following: \[kpcor\] Suppose that $L_0, L_1, L_2, L_3$ are arbitrary finite index subgroups in $$L,~~ \tau_1 L\tau_1, ~~\tau_2 L \tau_2, ~~\tau_3 L \tau_3, \quad \hbox{respectively}.$$ Then the subgroup $S$ of $G$ generated by $L_0, L_1, L_2, L_3$ is noncoherent. The intersection $$L':=L_0\cap \tau_1 L_1\tau_1\cap \tau_2 L_2 \tau_2\cap \tau_3 L_3 \tau_3$$ is a finite index subgroup in $L$. Let $S'$ denote the subgroup of $S$ generated by $$\label{eq2} L', ~~\tau_1 L'\tau_1, ~~\tau_2 L' \tau_2, ~~\tau_3 L' \tau_3.$$ It is clear that $S'$ has index $4$ in the group generated by $L', \tau_1, \tau_2$. Since the latter is noncoherent by , it follows that $S'$, and thus $S$, is noncoherent as well. Note that the groups in have the invariant hyperplanes $H$, $\tau_1(H)$, $\tau_2(H)$, $\tau_3(H)$, respectively. See , where we use the projective model of $\H^4$. (0,0)![[]{data-label="f2.fig"}](\figdir/f6 "fig:") \#1\#2\#3\#4\#5[ @font ]{} (5147,5162)(3566,-5486) (3566,-2894)[(0,0)\[lb\]]{} (6445,-3696)[(0,0)\[lb\]]{} (5091,-3085)[(0,0)\[lb\]]{} (6274,-5440)[(0,0)\[lb\]]{} (5151,-3956)[(0,0)\[lb\]]{} (6835,-2723)[(0,0)\[lb\]]{} (6425,-4698)[(0,0)\[lb\]]{} (5431,-1520)[(0,0)\[lb\]]{} (6856,-1540)[(0,0)\[lb\]]{} (5502,-2291)[(0,0)\[lb\]]{} (7021,-3976)[(0,0)\[lb\]]{} Construction of noncoherent arithmetic lattices {#proofA} =============================================== Our strategy is to embed a noncoherent group $G$ (of the type described in ) into the lattice $O(f, \Z)$. Then it would follow that $O(f, \Z)$ is noncoherent. Let $q_3$ be the quadratic form of rank 4 on the rational vector space $U$ as in . Then $O'(q_3, \Z)=G(\Delta)$, see . We can change the coordinates in $U$ to $y_i$ ($i=1,2,3,4$) so that $q_3$ takes the form: $$g=2y_1y_2+ y^2_3+y_4^2.$$ Let $\{e_1, e_2, e_3, e_4\}$ be the corresponding basis of $U$. Note that the group $O(g, \Z)$ is commensurable to $O(q_3, \Z)$. Let $(U, g)\to (V,f)$ be a rational embedding. Pick a nonzero vector $e_5\in V$ orthogonal to $U$. Then $$a:=f(e_5)>0.$$ Define a $5$–dimensional vector space $W$ spanned by the vector $e_5$ and $U$. Let $h$ be the restriction of the form $f$ to $W$; hence we have $(U,g)\subset (W,h)\subset (V,f)$. It therefore suffices to embed some noncoherent group $G$ (as in ) into the group $O'(h, \Z)$. We let $(\cdot , \cdot)$ denote the bilinear form on $W$ corresponding to $h$. The space $W$ splits as the orthogonal direct sum $U\oplus \Q e_5$. We will consider $\H^4$ canonically embedded in $W\otimes \R$ and identify $\H^3$ with the hyperplane $H:=U\otimes \R\cap \H^4\subset \H^4$. After replacing $e_2$ with $ae_2$ we obtain $(e_1, e_2)=a$. Set $$u_1:= e_1+e_5, u_2:= -e_2+ e_5.$$ Thus $$(u_1, u_1)= (u_2, u_2)=a,\ \ (u_1, u_2)=(u_1, e_1)=(u_2, e_2)=0,\ \ (u_1, e_2)=-(u_2,e_1)=a.$$ Let $U_i\subset W$ ($i=1, 2$) be the $4$–dimensional vector subspace orthogonal to $u_i$. Since $a>0$, it follows that each $U_i\otimes \R$ ($i=1, 2$) intersects $\H^4$ along a hyperplane $\Pi_i$. The reflection $$\tau_i\co w\mapsto w- 2\frac{(w,u_i)}{(u_i, u_i)} u_i$$ in the subspace $U_i$ is represented by a matrix with integer coefficients in the basis $\{e_1,...,e_5\}$. Thus $\tau_i\in O'(h, \Z)$, $i=1, 2$. Because $g(e_i)=0$, the vector $e_i$ corresponds to a parabolic point $p_i\in \partial \H^4$ of the group $O(g, \Z)$, $i=1, 2$. Since $(u_1, u_2)=0$, it follows that $\Pi_1$ is perpendicular to $\Pi_2$. Moreover, since $e_i\in U_i$, we conclude that $\partial \Pi_i$ contains $p_i$, $i=1, 2$. Since $$(u_i, e_5)=\sqrt{(u_i, u_i) (e_5, e_5)},$$ the hyperplane $\Pi_i$ is parallel to $H$; see the proof of . Let $L$ be a finite index subgroup of $O'(g, \Z)$ contained in $O'(h, \Z)$. The group $G$, generated by $L, \tau_1, \tau_2$, is contained in $O'(h, \Z)$. then implies that the lattice $O'(h, \Z)$ is noncoherent. Hence $O(f, \Z)$ is noncoherent as well. follows. For any number $a\in \N$ consider the quadratic form $$f_a(x_0, x_1,...,x_n)=q_3(x_0, x_1, x_2, x_3) + ax_4^2 + x_5^2+...+x_n^2.$$ Each $f_a$ defines a nonuniform arithmetic lattice $O'(f_a, \Z)\subset \isom$. Moreover, for infinitely many appropriately chosen primes $a$ these lattices are not commensurable. Since each form $f_a$ represents $q_3$, follows from . \[bm\] For each $n\ge 4$ there exist uniform noncoherent arithmetic lattices in $\isom$. Moreover, for each $n\ge 5$ there are infinitely many commensurability classes of such lattices. The assertion is a rather direct corollary of the result of Bowditch and Mess [@BM], but we present a proof for the sake of completeness. We start with a review of the example of Bowditch and Mess [@BM] which is a noncoherent uniform arithmetic lattice in $\iso4$. Consider the right-angled regular $120$–cell $D\subset \H^4$. It is a compact regular polyhedron; see for instance Davis [@Davis] or Vinberg and Shvartsman [@VS88]. It appears that it was first discovered by Schlegel in 1883 [@Schlegel], who was interested in classifying *honeycombs* in the spaces of constant curvature; see Coxeter [@Coxeter]. Each facet of $D$ is a right-angled regular dodecahedron. Let $\Gamma=G(D)\subset \iso4$ be the reflection group determined by $D$. The group $\Gamma$ is commensurable to $O(q, A)$, where $q(x_0, x_1, x_2, x_3, x_4)$ is the quadratic form given by the matrix $$\label{eq3} \left[ \begin{array}{ccccc} 1& -\cos(\pi/5) & 0 & 0 & 0 \\ -\cos(\pi/5) & 1& -1/2 & 0 & 0 \\ 0& -1/2 & 1 & -1/2 & 0 \\ 0& 0 & -1/2 & 1 & -\cos(\pi/5)\\ 0 & 0 & 0& -\cos(\pi/5)&1 \end{array} \right]$$ and $A$ is the ring of integers of the field $K=\Q(\sqrt{5})$. Thus $\Gamma$ is a (uniform) arithmetic lattice. Consider the facets $F_1, F_2$ of $D$ which share a common $2$–dimensional face $F$. There is a canonical isomorphism $\varphi\co G(F_1)\to G(F_2)$ fixing $G(F)$ elementwise. The reflection group $G(F_1)$ contains a finite index subgroup isomorphic to the fundamental group of a hyperbolic $3$–manifold $M^3$ which fibers over $S^1$; see Thurston [@Th]. Let $N_1\subset \pi_1(M^3)$ be a normal surface subgroup and set $N_2:=\varphi(N_1)\subset G(F_2)$. In particular, both $N_1, N_2$ are finitely generated. On the other hand, $N_i\cap G(F)$ is a free group $E$ of infinite rank, $i=1, 2$. One then verifies that the subgroup of $\Gamma$ generated by $N_1$ and $N_2$ is isomorphic to $ N_1*_E N_2$ and therefore is not finitely presentable [@Ne]. Hence $\Gamma$ is a noncoherent uniform arithmetic lattice in $\iso4$. In order to construct lattices in $\isom$ consider the quadratic forms $$f_a(x_0, x_1,..., x_n)=q(x_0, x_1, x_2, x_3, x_4)+ ax_5^2+ x_6^2+...+x_n^2,$$ where $a\in \N$ are primes. Since $q^\si$ is positive definite for the (unique) nontrivial embedding $\sigma \co K\to \R$, it follows that each $O(f_a, A)$ is a uniform arithmetic lattice in $O(f_a, \R)$. As in the noncompact case, the groups $O(f_a, \R)$ are not commensurable for infinitely many primes $a$. As $O(q, A)\subset O(f_a, A)$, the assertion follows. \[R\] Clearly, the subgroup generated by any finite index subgroups of $G(F_1)$ and $G(F_2)$ is noncoherent as well. The above construction produces only one commensurability class of noncoherent lattices in $\iso4$. Using noncommensurable arithmetic lattices in $\iso4$ containing $G(F_1)$, one can construct infinitely many commensurability classes of uniform noncoherent arithmetic lattices in $\iso4$. Let $\Gamma$ be a nonuniform arithmetic lattice in $\isom$ where $n\geq 6$. Then $\Gamma$ is commensurable to $O(f, \Z)$ for some rational form $f$ of signature $(n,1)$. Since $n+1\ge 7$ and $q_3$ has rank $4$, it follows from that $f$ represents $q_3$. Therefore, by , the group $O(f,\Z)$ is noncoherent. Thus $\Gamma$ is noncoherent as well. Nonarithmetic noncoherent lattices {#non-arithmetic} ================================== We produce these noncoherent examples by using the construction of nonarithmetic lattices in ${\operatorname{Isom}}(\H^n)$ due to Gromov and Piatetski-Shapiro [@GP]. We begin with a review of their construction. Let $f$ be a quadratic form of signature $(n-1,1)$ in $n$ variables with coefficients in a totally real algebraic number field $K\subset \R$. Let $A$ denote the ring of integers of $K$. We assume that $f$ satisfies Condition from . We let $K_+$ denote the set of $a\in K$ such that for each embedding $\si\co K\to \R$ we have $\si(a)>0$. For $a\in K_+$ we consider the quadratic form $$h_a(x_0, x_1, ..., x_n)=f(x_0, x_1,..., x_{n-1})+ ax_n^2.$$ It has signature $(n,1)$ and satisfies Condition (\*). Then $\Gamma_a:=O'(h_a, A)$ is a lattice in $\isom$. Similarly, $\Gamma_0:=O'(f,A)$ is a lattice in ${\operatorname{Isom}}(\H^{n-1})$. In what follows we will consider pairs of groups $\Gamma_a, \Gamma_1$, where $a\in \N$. Observe that both groups contain the subgroup $\Gamma_0$. Let $\Gamma_a'\subset \Gamma_a, \Gamma_1'\subset \Gamma_1$ be torsion-free finite index subgroups such that $$\Gamma_1'\cap \Gamma_0= \Gamma_a'\cap \Gamma_0.$$ We let $\Gamma_0'$ denote this intersection and set $M_1:=\H^n/\Gamma_1', M_a:=\H^n/\Gamma_a'$. Without loss of generality (after passing to deeper finite index subgroups), we may assume that $\H^{n-1}/\Gamma_0'$ isometrically embeds into $M_1$ and $M_a$ as a nonseparating totally geodesic hypersurface; see Millson [@Millson]. Cut $M_1$ and $M_a$ open along these hypersurfaces. The resulting manifolds $M_1^+, M_a^+$ both have totally geodesic boundaries isometric to the disjoint union of two copies of $M_0=\H^{n-1}/\Gamma_0'$. Let $M$ be the connected hyperbolic manifold obtained by gluing $M_1^+, M_a^+$ via the isometry of their boundaries. It is easy to see that $M$ is complete. Then there exists a lattice $\Gamma\subset {\operatorname{Isom}}(\H^n)$ such that $M=\H^n/\Gamma$. Note that $M$ is compact iff both $M_1, M_a$ are. It is proven in [@GP] that Note that there exist infinitely numbers $a$ which are not squares in $K$. Indeed, it is well known that square roots of prime numbers are linearly independent over $\Q$. Therefore only finitely many of them belong to $K$. We now prove by working with specific examples. For $n\ge 5$ take $K=\Q(\sqrt{5})$ and consider the quadratic form $$f=q(x_0, x_1, x_2, x_3, x_4)+ x_5^2+...+x_{n-1}^2,$$ where the form $q$ is given by the matrix . The quadratic form $q$ yields a uniform arithmetic lattice $O'(q, A)$ in ${\operatorname{Isom}}(\H^4)$ which is commensurable to the reflection group $G(D)$ defined in the proof of . The group $O'(q, A)$ is noncoherent according to . On the other hand, by applying the Gromov–Piatetski-Shapiro construction to $f$ and taking any prime number $a\ne 5$, we obtain a uniform nonarithmetic lattice in $\isom$ which contains $O'(q, A)$ and, hence, is noncoherent. It remains to analyze the case $n=4$. Take facets $F_1, F_2, F_3$ of $D$ so that $F_1$ and $F_2$ intersect along a $2$–dimensional face and $$F_3\cap F_1=F_3\cap F_2=\emptyset.$$ Then the group generated by the reflections in the facets of $F_1$ and $F_2$ is noncoherent; see the proof of . By taking an appropriate finite index subgroup $\Gamma_1\subset G(D)$, we obtain a hyperbolic $4$–manifold $M_1=\H^4/\Gamma_1$ which contains embedded totally geodesic hypersurfaces $S_i$ corresponding to the facets $F_i$, $i=1, 2,3$, so that $$S_3\cap S_1=S_3\cap S_2=\emptyset, \quad S_1\cap S_2\ne \emptyset .$$ Now cut $M_1$ open along $S_3$ and apply the gluing construction of Gromov and Piatetski–Shapiro. In this way one can obtain a nonarithmetic compact hyperbolic manifold $M$ whose fundamental group contains the subgroup of $\Gamma_1$ generated by some finite index subgroups of $G(F_1)$ and $G(F_2)$ and, hence, is noncoherent (see ). For $n\ge 5$ take $K=\Q$ and consider the quadratic form $f=q_{n-1}$. Taking any prime number for $a$, apply the same argument as in the compact case. Consider $n=4$. We will imitate the proof in the compact case. However we will appeal to the results of instead of using a particular fundamental domain. Let $\Gamma:=O'(q_4, \Z)$. Clearly, $q_4$ represents the form $q_3$. Set $L:=O'(q_3,\Z)\subset \Gamma$, and let $\tau_1, \tau_2 \in O'(q_4,\Z)$ be the commuting reflections constructed in the proof of . Set $\tau_3:=\tau_1 \tau_2$ and $L_0:=L, L_i:=\tau_i L \tau_i$, $i=1, 2, 3$. By passing to any finite index subgroups $L_i'\subset L_i$, we obtain a noncoherent subgroup $G'$ in $\Gamma$ generated by $\smash{L_i'}, i=0, 1, 2, 3$; see . Since $\Gamma$ is a linear group, we can assume without loss of generality that $\Gamma$ is torsion-free. Let $H_0=H\subset \H^4$ be the $L$–invariant hyperplane. Then $H_i:=\tau_i (H)$ ($i=1,2,3$) is the $L_i$–invariant hyperplane. There exists a finite index subgroup $\Gamma'\subset \Gamma$ so that for the groups $L_i':=L_i\cap \Gamma'$ we have: 1. $H_0/L_0'$ embeds as a hypersurface $S_0$ into $\H^4/\Gamma'$. 2. Let $M^+$ denote the manifold obtained by cutting $\H^4/\Gamma'$ along $S_0$. Then $G'$ embeds into $\pi_1(M^+)$. We have to find a subgroup $\Ga'$ so that: (a)For all $\ga\in \Ga'$ either $\ga(H_0)=H_0$ or $$\ga(H_0) \cap( H_0\cup H_1 \cup H_2 \cup H_3)=\emptyset.$$ (b)For all $\ga\in \Ga'$, the hyperplane $\ga(H_0)$ does not separate the above hyperplanes from each other. This is achieved by applying to the hyperplanes $H_0, H_1, H_2, H_3$ and $H_4:=\Pi_1$, $H_5:=\Pi_2$. We now glue an appropriately chosen manifold $M_a^+$ along the boundary of $M^+$. Let $M$ be the resulting complete hyperbolic manifold. Then, as in the case $n\ge 5$, the fundamental group of $M$ is nonarithmetic. On the other hand, $$G'\subset \pi_1(M^+)\subset \pi_1(M).$$ Therefore $\pi_1(M)$ is noncoherent. Noncoherence and Thurston’s conjecture {#spec} ====================================== We recall the following conjecture: \[Thurston’s virtual fibration conjecture\] Suppose that $M$ is a hyperbolic $3$–manifold of finite volume. Then there exists a finite cover over $M$ which fibers over the circle. We expect that all lattices in $\isom$ are noncoherent for $n\ge 4$. Proving this for nonarithmetic lattices is clearly beyond our reach. Therefore we restrict to the arithmetic case. Even in this case our discussion will be rather speculative. We restrict to the arithmetic groups of the simplest type $\Gamma=O(f, A)$, where $f$ is a quadratic form on $V=K^{n+1}$ and $K\subset \R$ is a totally real algebraic number field (see ). Choose a basis $\{e_0, e_1,...,e_n\}$ in which the form $f$ is diagonal: $$f=a_0 x_0^2+ a_1 x_1^2+... + a_n x_n^2.$$ Here $a_0<0$, $a_1,...,a_n>0$ and for all nontrivial embeddings $\si\co K\to \R$ we have $\si(a_i)>0$, for all $i=0, 1, ...,n$. To simplify the discussion, we will assume that $\Gamma$ is uniform (the nonuniform lattices were discussed in Theorems \[Theorem A\] and \[Theorem B\]). For a $4$–element subset $I=\{0, i, j, k\}\subset \{0, 1,...,n\}$ let $V_I\subset V$ denote the linear span of the basis vectors $e_l, l\in I$. Set $H_I:=V_I\otimes \R \cap \H^n$. Then $f|V_I$ determines a lattice $\Gamma_{I}$ in ${\operatorname{Isom}}(H_I)$, which is naturally embedded into $\Ga$. Assuming Thurston’s conjecture, up to taking finite index subgroups, each $\Gamma_{I}$ contains an (infinite) normal finitely generated surface subgroup $N_{I}$. Moreover, by taking $I$ and $J$ such that $I\cap J$ consists of 3 elements, we obtain subgroups $\Gamma_{I}, \Gamma_{J}$ whose intersection is a Fuchsian group $F$. It now follows from the separability of $F$ in $\Gamma$ (see Bowditch and Mess [@BM], Long [@Long] or of this paper) that, after passing to certain finite index subgroups $\Ga_I'\subset \Gamma_{I}, \Ga_J'\subset \Gamma_{J}$, we get the inclusion $$\label{eq4} \Ga_I' *_F \Ga_J'\subset \Gamma.$$ Set $N_{I}':= \Ga_{I}'\cap N_{I}, N'_{J}:= \Ga_J'\cap N_{J}$. Then $E:=N_I'\cap N_J'$ is a free group of infinite rank. Now implies that $\Gamma$ is noncoherent since the subgroup $$N_I' *_E N_J'\subset \Gamma$$ is finitely generated but not finitely presented [@Ne]. Therefore we obtain: Thus we expect the negative answer to asked by Dani Wise.

--- abstract: 'Despite the success of modern nuclear parton distribution functions (nPDFs) in describing nuclear hard-process data, they still suffer from large uncertainties. One of the poorly constrained features is the possible asymmetry in nuclear modifications of valence $u$ and $d$ quarks. We study the possibility of using pion–nucleus Drell–Yan dilepton data as a new constraint in the global analysis of nPDFs. We find that the nuclear cross-section ratios from the NA3, NA10 and E615 experiments can be used without imposing significant new theoretical uncertainties and, in particular, that these datasets may have some constraining power on the $u$/$d$ -asymmetry in nuclei.' address: - 'University of Jyvaskyla, Department of Physics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland' - 'Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland' - 'Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, E-15782 Galicia, Spain' author: - Petja Paakkinen - 'Kari J. Eskola' - Hannu Paukkunen bibliography: - 'piA-DY-article.bib' title: 'Applicability of pion–nucleus Drell–Yan data in global analysis of nuclear parton distribution functions' --- Drell-Yan process ,Pion-nucleus scattering ,Nuclear parton distribution functions Introduction ============ Since the discovery of the EMC effect in 1983 [@Aubert:1983xm] the nuclear effects in bound-hadron partonic structure have been under active study [@Arneodo:1992wf; @Malace:2014uea]. For collinearly factorizable hard processes this phenomenon can be described by nuclear modifications of parton distribution functions (PDFs), the latest global extractions being EPS09 [@Eskola:2009uj], DSSZ [@deFlorian:2011fp] and nCTEQ15 [@Kovarik:2015cma], see Refs. [@Eskola:2012rg; @Paukkunen:2014nqa] for reviews. Despite the success of nPDFs in describing also nuclear hard-process data from the LHC [@Armesto:2015lrg], they still suffer from large uncertainties. One of the shortcomings is the lack of data which would constrain the nuclear effects of all parton flavours simultaneously without any a priori assumptions. For example, it has been customary to assume that nuclear modifications for both valence quarks $u$ and $d$ are the same. While this assumption has been consistent e.g. with the available LHC data [@Armesto:2015lrg] and neutrino-nucleus deep inelastic scattering [@Paukkunen:2013grz], the two are not expected to be exactly the same [@Brodsky:2004qa]. It is only recently that an attempt to fit these separately has been carried out [@Kovarik:2015cma] but due to the lack of constraining data inconclusive results are obtained. Among other possibilities [@Chang:2011ra; @Cloet:2012td] it has been also suggested [@Dutta:2010pg] that Drell–Yan dilepton data from pion–nucleus collision experiments could be used in nPDF global analyses to constrain the $u$/$d$ -asymmetry. In this Letter, we provide a detailed study of this possibility in terms of the available data and next-to-leading order (NLO) cross-section computations with the EPS09 and nCTEQ15 nPDFs. Dependence on pion PDFs ======================= The NA3 [@Badier:1981ci], NA10 [@Bordalo:1987cs] and E615 [@Heinrich:1989cp] experiments provide pion–nucleus ($\pi^\pm + A$) Drell–Yan dilepton $(l^- l^+)$ production data in the following per-nucleon cross-section ratios: $$\begin{aligned} R^{+/-}_A(x_2) &\equiv \frac{\mathrm{d}\sigma(\pi^+ + A \rightarrow l^- l^+ + X) / \mathrm{d}x_2}{\mathrm{d}\sigma(\pi^- + A \rightarrow l^- l^+ + X) / \mathrm{d}x_2}, \\ R^{-}_{A_1/A_2}(x_2) &\equiv \frac{\frac{1}{A_1} \mathrm{d}\sigma(\pi^- + A_1 \rightarrow l^- l^+ + X) / \mathrm{d}x_2}{\frac{1}{A_2} \mathrm{d}\sigma(\pi^- + A_2 \rightarrow l^- l^+ + X) / \mathrm{d}x_2}.\end{aligned}$$ Here, $x_2 \equiv \frac{M}{\sqrt{s}} \mathrm{e}^{-y}$, where $M$ and $y$ are the invariant mass and rapidity of the lepton pair. The pion–nucleon center-of-mass energy is denoted by $\sqrt{s}$. At leading order (LO), the Drell–Yan cross section reads $$\begin{aligned} &\frac{\mathrm{d}\sigma(\pi^\pm + A \rightarrow l^- l^+ + X)}{\mathrm{d}x_2} \\&\overset{\text{LO}}{=} \!\int_{\Delta M}\!\mathrm{d}M \frac{8\pi\alpha^2}{9s x_2 M} \sum_q e_q^2 [q_{\pi^\pm}(x_1) \bar{q}_A(x_2) + \bar{q}_{\pi^\pm}(x_1) q_A(x_2)] \notag ,\end{aligned}$$ where $\alpha$ is the fine-structure constant, $x_1 \equiv \frac{M}{\sqrt{s}} \mathrm{e}^{y} = \frac{M^2}{s x_2}$, and the sum goes over the quark flavors $q$ with $e_q$ being the quark charge.The quark/antiquark distributions in a pion (nucleus) at factorization scale $Q \sim M$ are denoted by $q_{\pi^\pm(A)}$/$\bar{q}_{\pi^\pm(A)}$. The range of the mass integral $(\Delta M)$ as well as $\sqrt{s}$ depend on the experiment and are $4.1~\mathrm{GeV} < M < 8.5~\mathrm{GeV}$ and $\sqrt{s} = 16.8~\mathrm{GeV}$ for NA3. The NA10 experiment provides data at two different beam energies, 286 GeV ($\sqrt{s} = 23.2~\mathrm{GeV}$) and 140 GeV ($\sqrt{s} = 16.2~\mathrm{GeV}$), with a mass range $4.2~\mathrm{GeV} < M < 15~\mathrm{GeV}$ for the higher and $4.35~\mathrm{GeV} < M < 15~\mathrm{GeV}$ for the lower energy, but in both cases excluding the $\Upsilon$ peak region $8.5~\mathrm{GeV} < M < 11~\mathrm{GeV}$.[^1] In the E615 data the mass range is $4.05~\mathrm{GeV} < M < 8.55~\mathrm{GeV}$ at $\sqrt{s} = 21.7~\mathrm{GeV}$, but with an additional kinematical cut $x_1 > 0.36$, which was imposed by the experiment to reduce contributions from the pion sea quarks. Assuming the isospin and charge conjugation symmetry we have $u_{\pi^{+}} = d_{\pi^{-}} = \bar{d}_{\pi^{+}} = \bar{u}_{\pi^{-}}$ and $d_{\pi^{+}} = u_{\pi^{-}} = \bar{u}_{\pi^{+}} = \bar{d}_{\pi^{-}}$. Hence, in the limit where the pion sea quarks can be neglected and assuming that the mass integration range is narrow enough so that the scale evolution of the PDFs does not play a role, the LO approximation gives $$\begin{aligned} R^{+/-}_A(x_2) &\approx \frac{4\bar{u}_A(x_2) + d_A(x_2)}{4u_A(x_2) + \bar{d}_A(x_2)}, \label{eq:rpmappr} \\ R^{-}_{A_1/A_2}(x_2) &\approx \frac{4u_{A_1}(x_2) + \bar{d}_{A_1}(x_2)}{4u_{A_2}(x_2) + \bar{d}_{A_2}(x_2)}, \label{eq:rnmappr}\end{aligned}$$ where $u_A$ and $d_A$ are the per-nucleon distributions of $u$ and $d$ quarks in a nucleus $A$ with $Z$ protons, $$\begin{aligned} u_A &\equiv \frac{Z}{A} u_{p/A} + \frac{A-Z}{A} d_{p/A},\\ d_A &\equiv \frac{Z}{A} d_{p/A} + \frac{A-Z}{A} u_{p/A}.\end{aligned}$$ Here, $u_{p/A}$, $d_{p/A}$ are the parton distribution functions of a bound proton and we have again used the isospin symmetry to write $u_{n/A} = d_{p/A}$, $d_{n/A} = u_{p/A}$. As the dependence on the pion PDFs essentially cancels in $R^{-}_{A_1/A_2}$ and $R^{+/-}_A$, these quantities promise to be good candidates for global nPDF analyses, where the objective is to probe the nuclear modifications without being significantly sensitive to (possibly poorly known) pion structure. By comparing Equations  and we see that while $R^{-}_{A_1/A_2}$ probes dominantly the valence quarks, $R^{+/-}_A$ carries more sensitivity to sea quarks as well. ![Comparison of NLO predictions with the E615, NA10 and NA3 data. In all panels, we use the GRV (blue) and SMRS (red) PDFs for the pion, and the EPS09 nuclear modifications with the CT14 proton PDFs for the nuclei. In the upper-left panel we have taken into account the kinematical cut $x_1 > 0.36$ and in the right-hand-side panels an isospin correction as described in Section \[sec:isospin\] has been applied to the theory predictions.[]{data-label="fig:pion_pdfs"}](figures/piA-DY-article-figure0.pdf){width="\columnwidth"} The above approximative cancellation of the pion PDFs in cross-section ratios has to be tested explicitly in a NLO calculation to avoid including any biased constraints to nPDF analysis. In Figure \[fig:pion\_pdfs\], we plot the NA3, NA10 and E615 data along with our NLO results using the GRV [@Gluck:1991ey] and SMRS [@Sutton:1991ay] pion PDFs together with EPS09 nuclear modifications and CT14 [@Dulat:2015mca] free-proton PDFs.[^2] For hydrogen and deuterium we use the unmodified CT14 PDFs. In the upper-left panel we have taken into account the kinematical cut $x_1 > 0.36$ and in the right-hand-side panels an isospin correction as described in the next section has been applied. The NLO calculations were done using MCFM 7.0.1 [@Campbell:2015qma]. For the data points only statistical errors are available, but these are in any case expected to be dominant in comparison to the systematical errors (except the normalization error of the NA10 data discussed in the next section). The SMRS pion PDFs provide three different sets to account for the uncertainty in the fraction of pion momentum carried by the sea quarks.We find that the NLO predictions are largely insensitive to the choice of pion PDFs. Especially the SMRS $15\%$ sea set which is to be considered as their central prediction is almost indistinguishable from the GRV results. A slight separation between the different SMRS sets is observed towards large $x_2$ in $R^{+/-}_\text{W}$, but in comparison to the data uncertainties this is insignificant. Isospin correction and normalization of NA10 datasets {#sec:isospin} ===================================================== ![As Figure \[fig:pion\_pdfs\], but showing the error estimates from the CT14 PDFs as shaded blue bands for the results obtained with EPS09 and GRV pion PDFs. In the right-hand-side panels we show both the isospin corrected (solid) and uncorrected (dashed) NLO results.[]{data-label="fig:proton_pdf"}](figures/piA-DY-article-figure1.pdf){width="\columnwidth"} The NA10 collaboration has corrected their data for the isospin effects. The exact form of correction was obtained from a LO Monte Carlo simulation but is not quoted point by point along with the data [@Bordalo:1987cs].[^3] To mimic these corrections and compare with the data the best we can, we apply an isospin correction by computing the theory predictions as $$\begin{split} &(R^{-}_\text{W/D})^\text{NLO}_\text{isospin corrected} \\&\qquad\qquad= (R^{-}_\text{isocalar-W/W})^\text{LO}_\text{no nPDFs} \times (R^{-}_\text{W/D})^\text{NLO} , \end{split}$$ where “isoscalar-W" is the isospin-symmetrized W nucleus ($Z=A/2$) and where the LO correction factor $(R^{-}_\text{W/isocalar-W})^\text{LO}_\text{no nPDFs}$ is evaluated with the central set of CT14 without nuclear modifications in PDFs. This correction has been applied on the right-hand-side panels of Figure \[fig:pion\_pdfs\] and the effect can be seen in Figure \[fig:proton\_pdf\], where we plot both the corrected and uncorrected predictions using GRV pion PDFs. In Figure \[fig:proton\_pdf\], we also show the error bands from the CT14 proton PDFs (using the asymmetric prescription [@Nadolsky:2001yg] to combine the uncertainties from the error sets) which are typically rather small in comparison to the data uncertainties except, perhaps, the E615 data at smallest values of $x_2$. To some extent, the isospin corrected NA10 data also contain input from the proton PDFs used by the experiment in their Monte Carlo code, but we do not study such a source of uncertainty here further. We observe that our isospin corrected theory prediction overshoots especially the low-energy NA10 data. This can be accounted for by the systematic overall normalization uncertainty of the data, quoted in [@Bordalo:1987cs] to be $\sigma_{\mathcal{N}^\text{data}} = 6\%$. To compare the predictions from different nPDFs with the NA10 data in shape and not in overall normalization, we normalize the results as follows: We fix the optimal normalization factor $\mathcal{N}^\text{data}$ for each data set and theory prediction separately by minimizing $$\chi^2(\mathcal{N}^\text{data}) = \sum_i \frac{(\mathcal{N}^\text{data} R_i^\text{data} - R_i^\text{theory})^2}{(\sigma_i^\text{data})^2} + \frac{(\mathcal{N}^\text{data} - 1)^2}{(\sigma_{\mathcal{N}^\text{data}})^2}$$ with respect to data normalization $\mathcal{N}^\text{data}$ [@Stump:2001gu]. In the above equation $R_i^{\rm data}$ and $R_i^{\rm theory}$ are the experimental and theoretical values for $i$th bin in a data set, and $\sigma_i^{\rm data}$ is the data uncertainty (here statistical). We then obtain the theory predictions normalized to data as $$(R_i^\text{theory})_\text{normalized} = \frac{R_i^\text{theory}}{\mathcal{N}^\text{data}}.$$ ----------------- -------------- -------------- (r)[2-3]{} nPDF 286 GeV data 140 GeV data EPS09 1.044 1.125 nCTEQ15 1.058 1.141 ----------------- -------------- -------------- : Normalization factors for the NA10 data sets.[]{data-label="tbl:norm"} ![A comparison of the uncertainty bands obtained using the EPS09 (blue lines and bands) and nCTEQ15 (green lines and bands) nuclear PDFs. In the right-hand-side panels we show both the unnormalized (dashed) and results normalized to the data (solid).[]{data-label="fig:nuclear_pdfs"}](figures/piA-DY-article-figure2.pdf){width="\columnwidth"} The values for $\mathcal{N}^\text{data}$ are given in Table \[tbl:norm\] and the normalized results as well as the unnormalized ones are presented in Figure \[fig:nuclear\_pdfs\] for the EPS09 and nCTEQ15 nuclear PDFs.[^4] For predictions with nCTEQ15 PDFs we use their own free proton set for hydrogen and deuterium (and CT14 for EPS09). When calculating the nPDF errors, we have also normalized each error set separately. We observe that the optimal normalization for the NA10 286 GeV dataset is within the given $6\%$ overall normalization uncertainty, but for the 140 GeV dataset it is more than twice the suggested uncertainty limit. Such a large normalization issue is not unheard of: For example, while the carbon-to-deuteron and lead-to-deuteron nuclear ratios in deep inelastic scattering measured by the E665 collaboration [@Adams:1995is] are individually largely apart from other measurements, the lead-to-carbon ratio formed from these two agrees well with other experiments [@Arneodo:1996rv]. A similar normalization issue may be in question here as well. Compatibility with nuclear PDFs =============================== Comparing the results obtained with the EPS09 and nCTEQ15 nuclear PDFs in Figure \[fig:nuclear\_pdfs\] we find that both these sets are in a fairly good agreement with the data, but display a large difference in their uncertainty estimates. To understand this, let us study the $R^{-}_\text{W/D}$ ratio measured by NA10. For large $x_2$, only the valence quarks in nuclei contribute and in the LO approximation we have $$R^{-}_\text{W/D} \overset{x_2 \rightarrow 1}{\approx} R^\text{W}_\text{V-isoscalar} + R^\text{W}_\text{V-nonisoscalar}, \label{eq:totalR}$$ where $$R^A_\text{V-isoscalar} \equiv \frac{u^\text{V}_{p/A} + d^\text{V}_{p/A}}{u^\text{V}_{p} + d^\text{V}_{p}}$$ is the nuclear modification factor for an average valence quark in an isoscalar nucleus and $$R^A_\text{V-nonisoscalar} \equiv \left(\frac{2Z}{A} - 1\right)\frac{u^\text{V}_{p/A} - d^\text{V}_{p/A}}{u^\text{V}_{p} + d^\text{V}_{p}}$$ the corresponding non-isoscalarity correction. For neutron-rich nuclei this correction is negative and typically small in comparison to the isoscalar contribution. In Figure \[fig:nuclear\_valence\_mods\], we plot these two components for tungsten along with the nuclear modification factors $$R^{W}_{u_\text{V}} \equiv \frac{u^\text{V}_{p/A}}{u^\text{V}_{p}}, \qquad R^{W}_{d_\text{V}} \equiv \frac{d^\text{V}_{p/A}}{d^\text{V}_{p}}$$ at factorization scale $Q = 5~\mathrm{GeV}$. We find that EPS09 and nCTEQ15 agree on $R^\text{W}_\text{V-isoscalar}$, which is well constrained in both analyses, but there is a slight disagreement on $R^\text{W}_\text{V-nonisoscalar}$. In addition, we see that nCTEQ15 has significantly larger error bands in both of these components. To study this difference in more detail, we plot in Figure \[fig:nuclear\_valence\_mods\] also the nCTEQ15 error sets 25 and 26, which give the largest deviations from the central-set predictions. We can make two observations: First, from the lower panels in Figure \[fig:nuclear\_valence\_mods\], we see that these two error sets are related to the nuclear modifications of $u$ and $d$ valence quarks with set 25 giving the most extreme difference, and set 26 being closer to uniform modifications. Second, from the upper panels in Figure \[fig:nuclear\_valence\_mods\], we find that the deviations from the central prediction are in the same direction for both $R^\text{W}_\text{V-isoscalar}$ and $R^\text{W}_\text{V-nonisoscalar}$ (upwards for set 25, downwards for set 26), and combine additively in Equation (\[eq:totalR\]) thereby explaining the larger error bands seen in Figure \[fig:nuclear\_pdfs\]. ![The different LO valence-quark contributions to $R^{-}_\text{W/D}$ (upper panels) and the valence quark nuclear modification factors (lower panels) at factorization scale $Q = 5~\mathrm{GeV}$. Solid lines correspond to the EPS09 (blue) and nCTEQ15 (green) central sets and dotted lines indicate the error sets 25 and 26 of the nCTEQ15. The uncertainty bands are shown as light green (nCTEQ15) and light blue (EPS09) bands.[]{data-label="fig:nuclear_valence_mods"}](figures/piA-DY-article-figure3.pdf){width="\columnwidth"} ![As Figure \[fig:nuclear\_pdfs\], but with only normalized results shown and the nCTEQ15 error sets 25 and 26 (dotted lines) plotted.[]{data-label="fig:nuclear_pdfs_final"}](figures/piA-DY-article-figure4.pdf){width="\columnwidth"} It is now evident that the studied observables are sensitive to the mutual differences between $u$ and $d$ valence quark nuclear modifications. On one hand, the EPS09 error sets underestimate the true uncertainty because flavor dependence of valence quark nuclear modifications was not allowed in that particular analysis. On the other hand, the nCTEQ15 error bands are large since the flavor dependence was allowed, but not well constrained in their analysis. The size of nCTEQ15 error bands suggest that the pion–nucleus Drell–Yan data can have some constraining power on the difference of valence modifications. Indeed, in Figure \[fig:nuclear\_pdfs\_final\] we plot the predictions using the nCTEQ15 error sets 25 and 26, and observe that the most extreme deviation from identical nuclear modifications of $u$ and $d$ quarks given by set 25 is disfavored by NA3 and NA10 data. ![Comparison of the Omega data with predictions using the GRV (blue) and SMRS (red) pion parton distributions together with the EPS09 nuclear modifications combined to the CT14 proton PDFs and also from using the nCTEQ15 (green) nuclear PDFs with the GRV pion PDFs.[]{data-label="fig:omega"}](figures/piA-DY-article-figure5.pdf) In addition to the NA3, NA10 and E615 data we have studied also the results from the Omega experiment [@Corden:1980xf]. The data at $\sqrt{s} = 8.7~\mathrm{GeV}$ as a function of the lepton pair invariant mass are shown in Figure \[fig:omega\] for $x_\mathrm{F} \equiv \frac{2p_\mathrm{L}^*}{\sqrt{s}} > 0$, where $p_\mathrm{L}^*$ is the longitudinal momentum of the lepton pair along the beam line in the center-of-mass frame. We find that the data disagree with theory predictions in bins around the J/$\psi$ peak. Furthermore, at low invariant masses the choice of pion PDFs becomes significant and that especially towards larger invariant masses the data are not precise enough to discriminate between the nuclear PDFs. Hence it is not reasonable to include this dataset into a global nPDF analysis. Conclusions =========== We have studied the prospects of including NA3, NA10, E615 and Omega pion–nucleus Drell–Yan data to global analyses of nuclear parton distribution functions. The NA3, NA10 and E615 data are compatible (modulo NA10 normalization at lower beam energies) with modern nPDFs and can thus be used in a global analysis without causing significant tension. The Omega data is not compatible with the NLO theory predictions and not precise enough to be useful in the nPDF analysis. The cross-section ratios used in the experiments are largely independent of pion parton distributions and hence the inclusion of these data will not impose significant new theoretical uncertainties to the analysis. Some sensitivity to baseline proton PDFs however still persists. When implementing these data to a global analysis, one needs to take into account the isospin correction and normalization uncertainty in the NA10 datasets. This can be done as described above. These pion–nucleus Drell–Yan data will be included in the successor of the EPS09 analysis [@EPPS16]. The considered nuclear ratios are sensitive to the possible $u$/$d$ -asymmetry of nuclear modification factors but the data are not precise enough to pin down this difference completely. Regarding this matter we seem to reach a somewhat different conclusion than Dutta *et al.* [@Dutta:2010pg] who claimed that NA3 data would favor flavor-dependent nuclear PDFs. We, in our analysis, find a very good agreement between the data and $u$/$d$ -symmetric (EPS09) nuclear modifications. Moreover, our analysis suggests that the most extreme differences in $u$ and $d$ quark nuclear modifications as given by particular nCTEQ15 error sets are disfavored by the NA3 and NA10 datasets. Acknowledgements {#acknowledgements .unnumbered} ================ This research was supported by the Academy of Finland, Project 297058 of K.J.E., and by the European Research Council grant HotLHC ERC-2011-StG-279579 and by Xunta de Galicia (Conselleria de Educacion) – H.P. is part of the Strategic Unit AGRUP2015/11. P.P. gratefully acknowledges the financial support from the Magnus Ehrnrooth Foundation. [^1]: Dutta *et al.* [@Dutta:2010pg] used the NA10 data combined from the two different beam energies. We take these as separate datasets. [^2]: The NA3 data is originally given as $R^-_{\rm H/Pt}$ which we have inverted as it is customary to take the ratio with respect to the lighter nucleus. [^3]: We thank P. Bordalo for discussion on this matter. [^4]: Since nCTEQ15 grids for platinum have not been available for us, we have used their grids for gold instead in $R^{-}_\text{Pt/H}$. Since the mass numbers are very close, $A_\text{Pt} = 195$ and $A_\text{Au} = 197$, this should be an excellent approximation.

--- author: - 'Shayan Modiri Assari,  Haroon Idrees,  and Mubarak Shah, ' bibliography: - 'PSECrowds.bib' title: | Re-identification of Humans in Crowds using\ Personal, Social and Environmental Constraints --- [Shell : Bare Demo of IEEEtran.cls for Computer Society Journals]{} Introduction ============ {width="100.00000%"} Human re-identification is a fundamental and crucial problem for multi-camera surveillance systems [@wang2013intelligent; @gong2014person]. It involves re-identifying individuals after they leave field-of-view (FOV) of one camera and appear in FOV of another camera (see Fig \[fig:teaser\](a)). The investigation process of the Boston Marathon bombing serves to highlight the importance of re-identification in crowded scenes. Authorities had to sift through a mountain of footage from government surveillance cameras, private security cameras and imagery shot by bystanders on smart phones [@cnn_boston_2013]. Therefore, automatic re-identification in dense crowds will allow successful monitoring and analysis of crowded events. Since re-identification involves associating object hypotheses, it is possible to draw some parallels to tracking as well. For non-overlapping cameras, it can be viewed as long-term occlusion handling, albeit relatively more difficult because, a) objects disappear for much longer durations of time and the observed appearances change significantly. The appearance changes due to different scene illuminations and cameras properties, and because of the objects undergoing transformations in apparent sizes and poses as well. b) Simple motion models, such as the constant velocity, also break down for longer durations of time. For instance, a person walking in a particular direction may appear as walking in a very different direction in another camera with non-overlapping FOV. Also, c) the number of possible hypotheses increases many folds since the local neighborhood priors typically employed for tracking are not applicable for re-identification. Dense crowds are the most challenging scenario for human re-identification. For large number of people, appearance alone provides a weak cue. Often, people in crowds wear similar clothes that makes re-identification even harder. This is particularly true for the Grand Central dataset which is characterized by severe visual and appearance ambiguity among targets (Fig. \[fig:teaser\]c). Unlike standard surveillance scenarios previously tackled in literature, we address this problem for thousands of people where at any $30$ second interval, hundreds of people concurrently enter a single camera’s FOV. To handle the broad field-of-view, we divide the scene into multiple gates and learn transitions between them. Traditionally, re-identification has been primarily concerned with matching static snapshots of people from multiple cameras. Although there have been a few works that modeled social effects for re-identification such as grouping behavior [@zheng2009associating; @cai2010matching; @bialkowski2013person], they mostly deal with static images. In this paper, we study the use of time and video information for this task, and propose to consider the dynamic spatio-temporal context of individuals and the environment to improve the performance of human re-identification. The primary contribution of our work is to explicitly address the influence of personal goals, neighboring people and environment on human re-identification through high-order relationships. We complement appearance, typically employed for re-identification, with multiple personal, social and environmental (PSE) constraints, many of which are applicable without knowledge of camera topology. The PSE constraints include *preferred speed and destination*, as well as *social grouping* and *collision avoidance*. The environmental constraints are modeled by learning the repetitive patterns that occur in surveillance networks, as individuals exiting camera from a particular location (gate) are likely to enter another camera from another specific location. The travel times between the gates are estimated as well. These are employed both as soft (*spatial grouping*) and hard constraints (*travel times and destination*). The PSE constraints that are linear in nature, i.e. occur between objects, are shown with black lines in Fig. \[fig:teaser\](b), while quadratic ones occur between matching hypotheses, i.e., pairs of objects, are shown with red lines in Fig. \[fig:teaser\](b). Thus, if there are $N_a$ and $N_b$ number of people in two cameras, then the total number of possible matching hypotheses is $N_aN_b$, and there are $(N_aN_b)^2$ possible quadratic hypotheses. The time limits naturally reduce some of the hypotheses, nonetheless for large number of people these constraints and costs can be overwhelming. Therefore, we propose to iteratively prune possible re-identification hypotheses in an EM-like approach, where travel times and destination probability distributions are refined using high scoring hypotheses, and hypotheses improved using updated transition information between the gates in different cameras. We evaluate the re-identification performance using the Cumulative Matching Curve (CMC) [@loy2013person; @zhao2013unsupervised] which quantify the rankings for each query person and average them over all queries. In addition to producing rankings for different queries, we also generate the more useful $1-1$ correspondences for individuals across cameras through joint optimization over all individuals using the proposed linear and quadratic PSE constraints. In [@modiri2016human], we employed a stochastic local search algorithm to optimize the objective function simultaneously for all people. In this extension to [@modiri2016human], we optimize the objective function using the efficient Frank-Wolfe algorithm [@frank1956algorithm; @lacoste2015global] on a convex approximation of the original function with linear relaxation on the binary variables during the computation of conditional gradient, and show that the algorithm can solve the re-identification problem for hundreds of people and give an order of magnitude speed up with minor loss in performance over the stochastic local search method. To the best of our knowledge, this is the first paper to address human re-identification using personal, social and environmental constraints in crowded scenes, and perform joint optimization to re-identify all the subjects in non-overlapping cameras. The evaluation is performed on three datasets, PRID [@hirzer2011person], DukeMTMC [@ristani2016performance] and the challenging Grand Central dataset [@yi2015understanding] which depicts dense crowds[^1]. Compared to our ECCV paper [@modiri2016human], we make several improvements and extensions: 1) We automatically learn multi-modal distributions on transition times between gates in non-overlapping cameras, and use them to improve re-identification. 2) We train discriminative appearance model per-person using positive and negative samples mined from both cameras, and show that this boosts performance. Furthermore, 3) we tailor Frank-Wolfe algorithm to solve the re-identification problem giving an order of magnitude speed up with minor loss in performance over stochastic search algorithm. Finally, 4) we add results on DukeMTMC [@ristani2016performance] dataset as well, and perform a detailed analysis of contribution of different PSE constraints to overall performance of our approach. The rest of the paper is organized as follows. We discuss related work in Sec. \[secRelatedWork\], and present the proposed personal, social and environmental (PSE) constraints in Sec. \[secPSEConstraints\]. The Stochastic Local Search and the computationally efficient Frank-Wolfe approaches for joint optimization over all subjects in the cameras are presented in Sec. \[sec:optimization\]. The results of our experiments are reported in Sec. \[secExperiments\], and we conclude with some directions for future research in Sec. \[secConclusion\]. Related Work {#secRelatedWork} ============ Our approach is at the crossroads of human re-identification in videos, dense crowd analysis and social force models. Next, we provide a brief literature review of each of these areas. **Person Re-identification** is an active area of research in computer vision, with some of the recent works including [@li2014deepreid; @liao2015person; @paisitkriangkrai2015learning; @chen2015similarity; @zheng2015scalable; @zhang2015beyond; @zhang2015group; @ahmed2015improved; @lisanti2015person] applicable to static images. In videos, several methods have been developed for object handover across cameras [@wang2013intelligent; @javed2008modeling; @song2008robust; @chakraborty2015network; @das2014consistent]. Most of them focus on non-crowd surveillance scenarios with emphasis on modeling color distortion and learning brightness transfer functions that relate different cameras [@porikli2003inter; @prosser2008multi; @javed2005appearance; @gilbert2006tracking], whereas others relate objects by developing illumination-tolerant representations [@madden2007tracking] or comparing possible matches to a reference set [@chen2015multitarget]. Similarly, Kuo [@kuo2010inter] used Multiple Instance Learning to combine complementary appearance descriptors. The spatio-temporal relationships across cameras [@makris2004bridging; @stauffer2005learning; @tieu2005inference; @ardeshir2016ego2top; @ardeshir2016egocentric] or prior knowledge about topology has also been used for human re-identification. Chen [@chen2008adaptive] make use of prior knowledge about camera topology to adaptively learn appearance and spatio-temporal relationships between cameras, while Mazzon [@mazzon2012person] use prior knowledge about relative locations of cameras to limit potential paths people can follow. Javed [@javed2008modeling] presented a two-phase approach where transition times and exit/entrance relationships are learned first, which are later used to improve object correspondences. Fleuret [@fleuret2008multicamera] predicted occlusions with a generative model and a probabilistic occupancy map. Dick and Brooks [@dick2005stochastic] used a stochastic transition matrix to model patterns of motion within and across cameras. These methods have been evaluated on non-crowded scenarios, where observations are sparse and appearance is distinctive. In crowded scenes, hundreds of people enter a camera simultaneously within a small window of few seconds, which makes learning transition times during an unsupervised training period virtually impossible. Furthermore, since it is not always possible to obtain camera topology information, our approach is applicable whether or not the camera topology is available. **Dense Crowds** studies [@ali2008floor; @zhou2015learning] have shown that walking behavior of individuals in crowds is influenced by several constraints such as entrances, exits, boundaries, obstacles; as well as preferred speed and destination, along with interactions with other pedestrians whether moving [@mehran2009abnormal; @ge2012vision] or stationary [@yi2015understanding]. Wu [@wu2011efficient] proposed a two-stage network-flow framework for linking tracks interrupted by occlusions. Alahi  [@alahi2014socially] identify origin-destination (OD) pairs using trajectory data of commuters which is similar to grouping. In contrast, we employ several PSE constraints besides social grouping. **Social Force Models** have been used for improving tracking performance [@leal2011everybody; @pellegrini2009you; @yamaguchi2011you]. Pellegrini [@pellegrini2009you] were the first to use social force models for tracking. They modeled collision avoidance, desired speed and destination and showed its application for tracking. Yamaguchi [@yamaguchi2011you] proposed a similar approach using a more sophisticated model that tries to predict destinations and groups based on features and classifiers trained on annotated sequences. Both methods use agent-based models and predict future locations using techniques similar to crowd simulations. They are not applicable to re-identification, as our goal is not to predict but to associate hypotheses. Therefore, we use social and contextual constraints for re-identification in an offline manner. Furthermore, both these methods require observations to be in metric coordinates, which for many real scenarios might be impractical. For re-identification in static images, group context was used by Zheng [@zheng2009associating; @gong2014person], who proposed ratio-occurrence descriptors to capture groups. Cai [@cai2010matching] use covariance descriptor to match groups of people, as it is invariant to illumination changes and rotations to a certain degree. For re-identifying players in group sports, Bialkowski [@bialkowski2013person] aid appearance with group context where each person is assigned a role or position within the group structure of a team. In videos, Qin [@qin2013social] use grouping in non-crowded scenes to perform hand over of objects across cameras. They optimize track assignment and group detection in an alternative fashion. On the other hand, we refrain from optimizing over group detection, and use multiple PSE constraints (speed, destination, social grouping etc.) for hand over. We additionally use group context in space, i.e., objects that take the same amount of time between two gates are assigned a cost similar to grouping, when in reality they may not be traveling together in time. Mazzon and Cavallaro [@mazzon2013multi] presented a modified social force multi-camera tracker where individuals are attracted towards their goals, and repulsed by walls and barriers. They require a surveillance site model beforehand and do not use appearance. In contrast, our formulation avoids such assumptions and restrictions. In summary, our approach does not require any prior knowledge about the scene nor any training phase to learn patterns of motion. Ours is the first work to incorporate multiple personal, social and environmental constraints simultaneously for the task of human re-identification in crowd videos. The Personal, Social and Environmental (PSE) Constraints {#secPSEConstraints} ======================================================== {width="100.00000%"} In this section, we present our approach to re-identify people across non-overlapping cameras. We employ personal (*appearance*, *preferred speed*), social (*spatial* and *social grouping*), as well as environmental (*destination*, *travel time*) constraints, that are designed to be applicable even when the knowledge about camera topology is unavailable. However, when topology information is provided, additional PSE constraints (*collision avoidance*, or *speed in the invisible region*) become computable and are then used in our formulation (Sec. \[subsec:topology\]). Since environmental constraints which capture transition probability distributions between different camera regions are not known a priori, we therefore solve re-identification and estimation of transition probability distributions in an alternative fashion. This is explained in Fig. \[fig:figPipeline\] which describes the overall pipeline of our approach. Let $O_{i_a}$ represent an observation of an object $i$ in camera $a$. Its trajectory (track) is given by a set of points $[{\ensuremath{\mathbf{p}}}_{i_a}(t^{\eta}_{i_a}), \dots, {\ensuremath{\mathbf{p}}}_{i_a}(t^{\chi}_{i_a})]$, where $t^{\eta}_{i_a}$ and $t^{\chi}_{i_a}$ represent the time it entered and exited the camera $a$, respectively. Given another observation of an object $j$ in camera $b$, $O_{j_b}$, a possible match between the two is denoted by ${M_{i_a}^{j_b}} = {\langle O_{i_a}, O_{j_b} \rangle}$. To simplify notation, we drop the symbol for time $t$ and use it only when necessary, thus, ${\ensuremath{\mathbf{p}}}^{\chi}_{i_a} \equiv {\ensuremath{\mathbf{p}}}_{i_a}(t^{\chi}_{i_a})$ and ${\ensuremath{\mathbf{p}}}^{\eta}_{j_b} \equiv {\ensuremath{\mathbf{p}}}_{j_b}(t^{\eta}_{j_b})$. The entrances and exits in each camera are divided into multiple gates. These gates are virtual in nature, and correspond to different regions in the fields-of-view of cameras. For the case of two cameras $a$ and $b$, the gates are given by ${\ensuremath{\mathbf{G}}}_{1_a},\ldots,{\ensuremath{\mathbf{G}}}_{U_a}$ and ${\ensuremath{\mathbf{G}}}_{1_b},\ldots,{\ensuremath{\mathbf{G}}}_{U_b}$, where $U_a$ and $U_b$ are the total number of gates in both cameras, respectively. Furthermore, we define a function $g({\ensuremath{\mathbf{p}}}(t))$, which returns the nearest gate when given a point in the camera. For instance, for a person $i_a$, $g({\ensuremath{\mathbf{p}}}^{\chi}_{i_a})$ returns the gate from which the person $i$ exited camera $a$, by computing the distance of ${\ensuremath{\mathbf{p}}}^{\chi}_{i_a}$ to each gate. Mathematically, this is given by: $$g({\ensuremath{\mathbf{p}}}^{\chi}_{i_a}) = {\operatornamewithlimits{\arg\,\min}}_{{\ensuremath{\mathbf{G}}}_{u_a}}\|{\ensuremath{\mathbf{G}}}_{u_a} - {\ensuremath{\mathbf{p}}}^{\chi}_{i_a}\|^2, \;\; \forall u_a=1,\ldots,U_a.$$ Next, we describe the costs for different PSE linear, $\phi(.)$, and quadratic, $\varphi(.)$, constraints employed in our framework for re-identification. Since all costs have their respective ranges, we use a logistic function, $\hat{\phi}(.) = \alpha(1 + \exp(-\beta \phi(.))^{-1}$, to balance them. Most of the constraints do not require knowledge about camera topology, and are described next. Personal Constraints {#subsec:personal} -------------------- The personal constraints, which are linear in nature, capture the individual characteristics in the form of appearance and motion of each person in the different cameras. **Appearance:** To compute appearance similarity between observations $O_{i_a}$ and $O_{j_b}$, we use features from Convolutional Neural Networks [@simonyan2014very]. In particular, we extract features from Relu6 and Fc7 layers, followed by homogenous kernel mapping [@vedaldi2012efficient] and linear kernel as the the similarity metric. However, computing appearance similarity using single snapshot (bounding box) is suboptimal, as overtly visible background in the detection bounding box, or occlusions and noise can cause a drop in performance. To handle this, we sample five snapshots per track, i.e., multiple detections along the track, and then take the median of the appearance similarity between all possible 5x5 detection pairs as overall similarity between two observations $O_{i_a}$ and $O_{j_b}$. Since median is less sensitive to outliers, we found it to outperform minimum and maximum functions. This yields appearance similarity, $\phi_{\textrm{app}}(O_{i_a}, O_{j_b})$ between objects $O_{i_a}$ and $O_{j_b}$. However, an important observation regarding videos is that they provide spatio-temporal information about the observed individuals which can be used to learn strong discriminative appearance models. This stems from the fact that it is always possible to find a set of people which the individual under consideration can *never* match within the same camera, and more importantly, in the other cameras. For instance, consider $O_{i_a}$ in camera $a$ for which we intend to learn a discriminative model. Let its appearance descriptors be given by $[{\ensuremath{\mathbf{x}}}_{i_a}(t^{\eta}_{i_a}), \dots, {\ensuremath{\mathbf{x}}}_{i_a}(t^{\chi}_{i_a})]$ corresponding to the points $[{\ensuremath{\mathbf{p}}}_{i_a}(t^{\eta}_{i_a}), \dots, {\ensuremath{\mathbf{p}}}_{i_a}(t^{\chi}_{i_a})]$ from its track. To simplify notation, we drop time and represent multiple snapshots of an individual available in its track with ${\ensuremath{\mathbf{x}}}_{i_a}$. The discriminative model requires both positive and instances samples, and with multiple cameras, we get four cases: - [$S^{+}_a$: This includes all the samples from the track of individual $O_{i_a}$ in camera $a$, i.e., ${\ensuremath{\mathbf{x}}}_{i_a}$ with label $y_{i_a}=1$.]{} - [$S^{-}_a$: These include all the samples from tracks of other individuals in camera $a$, i.e., $\{{\ensuremath{\mathbf{x}}}_{i'_a}, \forall i'_a | i'_a \neq i_a$} with corresponding labels $y_{i'_a}=-1$.]{} - [$S^{-}_b$: These include individuals in camera $b$, $\{O_{j_b}\}$ that can never match to $O_{i_a}$. For instance, individuals from the past, or way into the future, or individuals that are extremely poor matches using other easy-to-compute costs. When the number of people is large, only hard negatives are used. Each instance in this set has label $y_{j_b}=-1$.]{} - [$S^{+}_b$: Since we do not know the true positive or re-identification match in camera $b$, we relax this constraint by ensuring a single positive label over multiple possible matches, i.e., individuals $O_{j_b}$ which have low appearance and speed costs and lie within the expected time frame. Only one of the instances in this set can have a positive label, therefore $\sum_{j_b \in S^{+}_b} (y_{j_b}+1)/2 = 1$.]{} Thus, the objective function for the soft-margin classifier becomes: $$\begin{aligned} & \min_{{\ensuremath{\mathbf{w}}}, v, \xi} \frac{1}{2}\|{\ensuremath{\mathbf{w}}}\|^2 + C\sum_{i}\xi_{i} & \label{eqMISVMObj} \\ \textmd{s.t.}\;\;\; & y_{i}(\langle {\ensuremath{\mathbf{w}}}, {\ensuremath{\mathbf{x}}}_{i} \rangle + v) \geq 1 - \xi_{i}, & \forall i \in S^{+}_a, S^{-}_a, S^{-}_b, \label{eqMISVMConst1}\\ & \max(\langle {\ensuremath{\mathbf{w}}}, {\ensuremath{\mathbf{x}}}_{i} \rangle + v) \geq 1 - \xi_{i}, & \forall i \in S^{+}_b, \label{eqMISVMConst2}\\ & \xi_{i} \geq 0,\;\; y_i \in \{-1,1\}, \label{eqMISVMConst3}\end{aligned}$$ where ${\ensuremath{\mathbf{w}}}, v$ are SVM weight vector and bias respectively, and $\xi$ represents the slack variables. Note that in Eq. \[eqMISVMConst2\], $y_i = +1$ and is therefore omitted. This forms a special case of Multiple Instance Learning [@andrews2002support], solved by imputing the labels of $S^{+}_b$ and solving the SVM objective, alternatively. In essence, the above formulation takes advantage of video information using semi-labeled data, and also handles the issue of domain adaptation by forcing the classifier to perform well on hard negatives from the other camera. The cost between two objects using discriminative appearance model is given by: $$\phi_{\textrm{disc}}(O_{i_a}, O_{j_b}) = \langle {\ensuremath{\mathbf{w}}}_{i_a}, {\ensuremath{\mathbf{x}}}_{j_b} \rangle + v_{i_a}.$$ **Preferred Speed:** The walking speed of humans has been estimated to be around $1.3$ m/s [@robin2009specification]. Since, we do not assume the availability of metric rectification information, we cannot use this fact directly in our formulation. However, a consequence of this observation is that we can assume the walking speed of individuals, *on average*, in different cameras is constant. We assume a Normal distribution, $\mathcal N(.)$, on observed speeds in each camera. The variation in walking speeds of different individuals is captured by the variance of the Normal distribution. Let $\mathcal N(\mu_a, \sigma_a)$ and $\mathcal N(\mu_b, \sigma_b)$ denote the distribution modeled in the two cameras. Since a particular person is being assumed to walk with the same speed in different cameras, the cost for preferred speed using the exit speed of person $i_a$, $\dot{{\ensuremath{\mathbf{p}}}}^{\chi}_{i_a}$, and the entrance speed of person $j_b$, $\dot{{\ensuremath{\mathbf{p}}}}^{\eta}_{j_b}$ is given by: $$\begin{gathered} \label{eqSpeedCost} \dot{{\ensuremath{\mathbf{p}}}}^{\chi}_{i_a} = \sigma_a^{-1} ( \|{\ensuremath{\mathbf{p}}}^{\chi}_{i_a} - {\ensuremath{\mathbf{p}}}^{\chi - 1}_{i_a} \| - \mu_a),\;\;\\ \dot{{\ensuremath{\mathbf{p}}}}^{\eta}_{j_b} = \sigma_b^{-1} ( \|{\ensuremath{\mathbf{p}}}^{\eta+1}_{j_b} - {\ensuremath{\mathbf{p}}}^{\eta}_{j_b} \| - \mu_b),\\ \phi_{\textrm{spd}}(O_{i_a}, O_{j_b}) = |\dot{{\ensuremath{\mathbf{p}}}}^{\chi}_{i_a} - \dot{{\ensuremath{\mathbf{p}}}}^{\eta}_{j_b}|.\end{gathered}$$ Environmental Constraints {#subsec:environmental} ------------------------- Next, we describe the environmental constraints, which are linear in nature and predict the most probable paths and travel times between gates across cameras. **Destination and Travel Time:** For re-identification in multiple cameras, the knowledge about probable destination gives a prior for an individual’s location in another camera. Furthermore, since people disappear between cameras, the consistency in time required to travel between a particular set of gates in two different cameras for different individuals serves as an implication to their correctness. We capture these observations by modeling the transition probability distributions between gates in different cameras, as well the time required to travel between them. {width="100.00000%"} Assuming we have a set of putative matches $\{M_{i_a}^{j_b}\}$ (Fig. \[fig:figPipeline\] (d)), we estimate the probability distribution of transition between exit gate $G_{u_a}$ and entrance gate $G_{u_b}$ as: $$\label{eqDestinationCost} p(G_{u_a},G_{u_b}) = \frac{| g({\ensuremath{\mathbf{p}}}^{\chi}_{i_a})=G_{u_a} \wedge g({\ensuremath{\mathbf{p}}}^{\eta}_{j_b})=G_{u_b} |} {\sum_{u'_b} | g({\ensuremath{\mathbf{p}}}_{i_a}^{\chi})=G_{u_a} \wedge g({\ensuremath{\mathbf{p}}}_{j_b}^{\eta})=G_{u'_b}| },$$ while the travel times are modeled using Mixture-of-Gaussians [@figueiredo2002unsupervised] for each pair of gates. We use up to $K=5$ components, where exact number is automatically determined using data by [@figueiredo2002unsupervised]. Thus, the probability of travel time is given by, $$\label{eqTravelTimesCost} q(\Delta t | G_{u_a},G_{u_b}) = \sum_{k=1}^{K} w_k\mathcal{N}(\mu_k, \Sigma_k).$$ Thus, the cost for destination and travel times between gates for the match $\langle O_{i_a}, O_{j_b} \rangle$ is given by: $$\begin{aligned} \label{eqDestTimeCost} \hspace{3em}&\hspace{-3em}\phi_{\textrm{tr}}(O_{i_a}, O_{j_b}) = \nonumber\\ &- p\big(g({\ensuremath{\mathbf{p}}}_{i_a}^{\chi}), g({\ensuremath{\mathbf{p}}}_{j_b}^{\eta}) \big) \cdot q(t^{\eta}_{j_b} - t^{\chi}_{i_a} | g({\ensuremath{\mathbf{p}}}_{i_a}^{\chi}), g({\ensuremath{\mathbf{p}}}_{j_b}^{\eta}).\end{aligned}$$ Since the transition probability distributions in Eq. \[eqDestinationCost\] and travel times in Eq. \[eqTravelTimesCost\] are not known in advance, we use an EM-like approach that iterates between solving $1-1$ correspondences using the linear and quadratic constraints, and estimating transition information using those correspondences (Fig. \[fig:figPipeline\] (d-f)). Fig. \[fig:figTransition\] shows some intermediate results when computing transition time and destination probability distributions. For travel times, we initialize with the uniform distribution, and update the travel time distribution with a momentum of 0.15 at each iteration. As can be seen with blue curves in Fig. \[fig:figTransition\] (a,b), the estimation of travel times improves across iterations. Social Constraints {#subsec:social} ------------------ The quadratic social constraints are computed between pairs of re-identification hypotheses, i.e., between possible matches that have the same destinations and travel times. **Spatial Grouping:** The distance traveled by different individuals between two points (or gates) across cameras should be similar. Since the camera topology is not available in this case, the distance can be implicitly computed as a product of velocity and time. This is a quadratic cost computed between every two possible matches, $M_{i_a}^{j_b}$ and $M_{i'_a}^{j'_b}$, given by: $$\begin{aligned} \label{eqSpatial} \hspace{3em}&\hspace{-3em}\varphi_{\textrm{spt}}(M_{i_a}^{j_b}, M_{i'_a}^{j'_b}) = \nonumber\\ &\exp(-|{\ensuremath{\mathbf{p}}}^{\chi}_{i_a} - {\ensuremath{\mathbf{p}}}^{\chi}_{i'_a}|) \cdot \exp(-|{\ensuremath{\mathbf{p}}}^{\eta}_{j_b} - {\ensuremath{\mathbf{p}}}^{\eta}_{j'_b}|)\nonumber\\ &\cdot | (\dot{{\ensuremath{\mathbf{p}}}}^{\chi}_{i_a}+\dot{{\ensuremath{\mathbf{p}}}}^{\eta}_{j_b})(t^{\eta}_{j_b} - t^{\chi}_{i_a}) - (\dot{{\ensuremath{\mathbf{p}}}}^{\chi}_{i'_a}+\dot{{\ensuremath{\mathbf{p}}}}^{\eta}_{j'_b})(t^{\eta}_{j'_b} - t^{\chi}_{i'_a})|.\end{aligned}$$ Effectively, if the exit and entrance locations are nearby (the first two terms in Eq. \[eqSpatial\]), then we compute the distance traveled by each match in the pair using the product of mean velocity and time required to travel between those locations (the third term). It is evident from Eq. \[eqSpatial\] that the exponentiation in first two terms will allow this cost to take effect only when the entrance and exit locations are both proximal. If so, the third term will then measure the difference in distance traveled by the two possible matches (tracks), and penalize using that difference. If the distance is similar, the cost will be low suggesting both matches (tracks) should be included in the final solution. If the difference is distance is high, then at least one or both of the matches are incorrect. **Social Grouping:** People tend to walk in groups. In our formulation, we reward individuals in a *social group* that exit and enter together from the same locations at the same times, $$\begin{aligned} \label{eqGroup} \hspace{1em}&\hspace{-1em}\varphi_{\textrm{grp}}(M_{i_a}^{j_b}, M_{i'_a}^{j'_b}) =\nonumber\\ &\exp(-|{\ensuremath{\mathbf{p}}}^{\chi}_{i_a} - {\ensuremath{\mathbf{p}}}^{\chi}_{i'_a}|-|{\ensuremath{\mathbf{p}}}^{\eta}_{j_b} - {\ensuremath{\mathbf{p}}}^{\eta}_{j'_b}|-|t^{\eta}_{j_b} - t^{\eta}_{j'_b}|- |t^{\chi}_{i_a} - t^{\chi}_{i'_a}|).\end{aligned}$$ Here, the first two terms capture the difference in exit and entrance locations, respectively, and the third and fourth terms capture the difference in exit and entrance times, respectively. PSE Constraints with Camera Topology {#subsec:topology} ------------------------------------ The PSE constraints presented in the previous subsections are applicable when the spatial relations between the cameras are not known. However, if the inter-camera topology is available, then it can be used to infer the motion of people as they travel in the invisible or unobserved regions between the cameras. The quality of paths in the invisible region can be subject to constraints such as *preferred speed* or *direction of movement*, which can be quantified and introduced into the framework. Furthermore, collision avoidance is another social constraint that can only be applied when inter-camera topology is known. Given two objects in cameras $a$ and $b$, $O_{i_a}$ and $O_{i_b}$, in the same reference of time, we predict the possible trajectory between the object hypotheses. This is obtained by fitting a spline, given by $\bm \gamma_{i_a}^{j_b}$, in both $x$ and $y$ directions using cubic interpolation between the points ${\ensuremath{\mathbf{p}}}_{i_a}$ and ${\ensuremath{\mathbf{p}}}_{j_b}$ parameterized with their respective time stamps. **Collision Avoidance:** Let the point of closest approach between two interpolated trajectories be given by: $$d(\bm \gamma_{i_a}^{j_b}, \bm \gamma_{i'_a}^{j'_b}) = \min_{\max(t_{i_a}^{\chi}, t_{i'_a}^{\chi} ),\ldots,\min(t_{j_b}^{\eta}, t_{j'_b}^{\eta})} \|\bm \gamma_{i_a}^{j_b}(t) - \bm \gamma_{i'_a}^{j'_b}(t)\|,$$ we quantify the collision avoidance as a quadratic cost between pairs of possible matches: $$\begin{gathered} \label{eqAccCostTopology} \phi_{\textrm{invColl}}(M_{i_a}^{j_b}, M_{i'_a}^{j'_b}) = \\\big( 1 - \varphi_{\textrm{grp}}(M_{i_a}^{j_b}, M_{i'_a}^{j'_b}) \big) .\exp \big(- d(\bm \gamma_{i_a}^{j_b}, \bm \gamma_{i'_a}^{j'_b}) \big).\end{gathered}$$ Since people avoid collisions with others and change their paths, this is only applicable to trajectories of people who are not traveling in a group (first term in Eq. \[eqAccCostTopology\]), i.e., the cost will be high if two people not walking in a group come very close to each other when traveling through the invisible region between the cameras. **Speed in Invisible Region:** The second constraint we compute is an improved version of the *preferred speed* - a linear constraint which now also takes into account the direction is addition to speed of the person in the invisible region. If the velocity of a person within visible region in cameras and while traveling through the invisible region is similar, this cost would be low. However, for an incorrect match, the difference between speed in visible and invisible regions will be high. Let $\dot{\bm \gamma}$ denote the velocity at respective points in the path, both in the visible and invisible regions. Then, the difference of maximum and minimum speeds in the entire trajectory quantifies the quality of a match, given by, $$\label{eqSpeedCostTopology} \phi_{\textrm{invSpd}}(O_{i_a}, O_{j_b}) = | \max_{t_{i_a}^{\eta} \ldots t_{j_b}^{\chi}}{\dot{\bm \gamma}_{i_a}^{j_b}(t)} - \min_{t_{i_a}^{\eta} \ldots t_{j_b}^{\chi}}{\dot{\bm \gamma}_{i_a}^{j_b}(t)}|.$$ When the inter-camera topology is available, these constraints are added to the Eq. \[eqL\] and the method described in the Sec. \[sec:optimization\] is used to re-identify people across cameras. Optimization with PSE Constraints {#sec:optimization} ================================= In this section, we present the optimization techniques which use the aforementioned constraints. Let $z_{i_a}^{j_b}$ be the variable corresponding to a possible match $M_{i_a}^{j_b}$. Our goal is to optimize the loss function over all possible matches and pairs of matches, which is the weighted sum of linear and quadratic terms. When knowledge about topology is not available, the loss function is given by: $$\begin{aligned} \label{eqL} & \nonumber \mathcal{L}({\ensuremath{\mathbf{z}}}) = \sum_{\substack{{i_a, j_b}\\ {i'_a, j'_b}}} z_{i_a}^{j_b} z_{i'_a}^{j'_b} \underbrace{ \Big( \hat{\varphi}_{\textrm{spt}}(M_{i_a}^{j_b}, M_{i'_a}^{j'_b}) + \hat{\varphi}_{\textrm{grp}}(M_{i_a}^{j_b}, M_{i'_a}^{j'_b}) \Big)}_{{\ensuremath{\mathbf{Q}}} (\text{Quadratic Terms})} + \\ & \sum_{i_a,j_b} z_{i_a}^{j_b} \underbrace{\big( \hat{\phi}_{\textrm{app}}(M_{i_a}^{j_b}) + \hat{\phi}_{\textrm{disc}}(M_{i_a}^{j_b}) + \hat{\phi}_{\textrm{spd}}(M_{i_a}^{j_b}) + \hat{\phi}_{\textrm{tr}}(M_{i_a}^{j_b})\big)}_{{\ensuremath{\mathbf{L}}} (\text{Linear Terms})},\end{aligned}$$ subject to the following conditions: $$\label{eqConstraints} \sum_{i_a}z_{i_a}^{j_b} \leq 1, \forall j_b, \sum_{j_b}z_{i_a}^{j_b} \leq 1, \forall i_a, z_{i_a}^{j_b} \in \{0,1\}.$$ The problem in Eq. \[eqL\] is non-convex due to the nature of PSE constraints, and due to binary and combinatorial nature of variables in Eq. \[eqConstraints\], it is NP-hard. The first solution we present solves the non-convex optimization in its original form (presented in our ECCV paper [@modiri2016human]) through Stochastic Local Search, and a new approach which uses convex approximation on the loss function and linear relaxation on the binary constraints when computing conditional gradient, and is solved using the Frank-Wolfe algorithm [@frank1956algorithm; @lacoste2015global]. Stochastic Local Search Optimization {#subsec:StochasticSearch} ------------------------------------ **Input**: $O_{i_a}, O_{j_b} \;\;\; \forall i_a, j_b$, $R$ (\# steps)\ **Output**: $\mathcal{L}^*, {\ensuremath{\mathbf{z}}}^*; \;\; \;\; 0 \leq |t^{\eta}_{j_b} - t^{\chi}_{i_a}| \leq \tau, \forall z_{i_a}^{j_b}$ ------------------------------------------------------------------------ Initialize $[\mathcal{L}^*, {\ensuremath{\mathbf{z}}}^*]$ for Linear Constraints with $\textproc{Munkres}$ [@munkres1957algorithms] $[\mathcal{L}^-, {\ensuremath{\mathbf{z}}}^-] = \textproc{RemoveMat}(\mathcal{L}^*, {\ensuremath{\mathbf{z}}}^*, r)$ $\mathcal{L}' = \mathcal{L}^-, {\ensuremath{\mathbf{z}}}' = {\ensuremath{\mathbf{z}}}^-$ $[\mathcal{L}^+, {\ensuremath{\mathbf{z}}}^+] = \textproc{AddMat}(\mathcal{L}', {\ensuremath{\mathbf{z}}}', s)$ $\mathcal{L}' = \mathcal{L}^+, {\ensuremath{\mathbf{z}}}' = {\ensuremath{\mathbf{z}}}^+$ $\mathcal{L}^* = \mathcal{L}', {\ensuremath{\mathbf{z}}}^* = {\ensuremath{\mathbf{z}}}'$ Algorithm \[alg:optimization\] presents an approach which optimizes Eq. \[eqL\] subject to the conditions in Eq. \[eqConstraints\] through Stochastic Local Search [@spall2005introduction]. The solution is initialized using linear terms with Munkres [@munkres1957algorithms]. This is followed by stochastic removal and addition of matches into the current solution. However, the solution is updated whenever there is a decrease in the loss function in Eq. \[eqL\], as can be seen from Line $13$. Once the change in loss is negligible or a maximum pre-defined number of iterations is reached, the algorithm stops and returns the best solution obtained. In Alg. \[alg:optimization\], the sub-procedure $\textproc{RemoveMat}(\mathcal{L}, {\ensuremath{\mathbf{z}}}, r)$ removes $r$ hypotheses from the solution as well as their respective linear and quadratic costs by assigning probabilities (using respective costs) for each node in the current solution ${\ensuremath{\mathbf{z}}}$. In contrast, the sub-procedure $\textproc{AddMat}(\mathcal{L}, {\ensuremath{\mathbf{z}}},s)$ adds new hypotheses to the solution using the following steps: - Populate a list of matches for which $z_{i_a}^{j_b}$ can be $1$ such that Eq. \[eqConstraints\] is satisfied. - Generate combinations of cardinality $s$ using the list. - Remove combinations which dissatisfy Eq. \[eqConstraints\]. - Compute new $\mathcal{L}$ in Eq. \[eqL\] for each combination. This is efficiently done by adding $|{\ensuremath{\mathbf{z}}}|*s$ quadratic values and $s$ linear values. - Pick the combination with lowest loss $\mathcal{L}$, add it to ${\ensuremath{\mathbf{z}}}$ and return. Fig. \[fig:intermediateOptimization\] shows the intermediate results for this optimization approach using Alg. \[alg:optimization\]. The $x$-axis is the step number, whereas the left $y$-axis shows the value of loss function in Eq. \[eqL\] (blue curve), and the right $y$-axis shows the F-Score in terms of correct matches (orange curve). We also show results of Hungarian Algorithm (Munkres) [@munkres1957algorithms] in dotted orange line using linear constraints, which include appearance and speed similarity. These curves show that Alg. \[alg:optimization\] simultaneously improves the loss function in Eq. \[eqL\] and the accuracy of the matches as the number of steps increases. ![The graph shows the performance of Algorithm \[alg:optimization\] using both linear and quadratic constraints, compared against Hungarian Algorithm [@munkres1957algorithms] using only the linear costs shown with orange dotted line. The loss function in Eq. \[eqL\] is shown in blue, whereas the accuracy is shown in red. Quadratic PSE constraints in conjunction with Alg. \[alg:optimization\] yield an improvement of **$\sim8\%$** over linear constraints.[]{data-label="fig:intermediateOptimization"}](./IterationCostsAccuracy_v3.pdf){width="50.00000%"} Frank-Wolfe Optimization {#subsec:FW} ------------------------ The Stochastic Local Search algorithm presented in the previous subsection optimizes over a non-convex quadratic function with binary variables. Rewriting the loss function in matrix form in Eq. \[eqL\], we get $$\label{eqL2} \mathcal{L}({\ensuremath{\mathbf{z}}}) = {\ensuremath{\mathbf{z}}}^T {\ensuremath{\mathbf{Q}}} {\ensuremath{\mathbf{z}}} + {\ensuremath{\mathbf{L}}} {\ensuremath{\mathbf{z}}},$$ subject to the linear and binary constraints in Eq. \[eqConstraints\]. Due to the linear nature of the constraints, the convex hull or polytope $\mathcal{D}$ from Eq. \[eqConstraints\] is convex. Furthermore, the quadratic function in Eq. \[eqL2\] can be made convex by taking the normalized Laplacian of ${\ensuremath{\mathbf{Q}}}$, $\hat{{\ensuremath{\mathbf{Q}}}} = {\ensuremath{\mathbf{I}}} - {\ensuremath{\mathbf{D}}}^{\frac{1}{2}} {\ensuremath{\mathbf{Q}}} {\ensuremath{\mathbf{D}}}^{\frac{1}{2}}$, where the diagonal matrix ${\ensuremath{\mathbf{D}}}$ contains the row sums of ${\ensuremath{\mathbf{Q}}}$ and ${\ensuremath{\mathbf{I}}}$ is the identity matrix. This allows the use of Frank-Wolfe algorithm (also known as Conditional Gradient Method) which approximates Eq. \[eqL2\] with linear subproblems and iteratively minimizes the following objective: $$\label{eqFWhComplete} {\ensuremath{\mathbf{h}}}_{k+1} = \underset{{\ensuremath{\mathbf{h}}} \in \mathcal{D}}{\textrm{arg\;min}}\;\; \mathcal{L}({\ensuremath{\mathbf{z}}}_{k}) + \nabla\mathcal{L}({\ensuremath{\mathbf{z}}}_{k})^{T}({\ensuremath{\mathbf{h}}} - {\ensuremath{\mathbf{z}}}_{k}).$$ Since the minimization in Eq. \[eqFWhComplete\] does not depend on ${\ensuremath{\mathbf{z}}}_k$, we get the following equivalent optimization problem: $$\label{eqFWh} {\ensuremath{\mathbf{h}}}_{k+1} = \underset{{\ensuremath{\mathbf{h}}} \in \mathcal{D}}{\textrm{arg\;min}}\;\; \langle {\ensuremath{\mathbf{h}}}, \nabla\mathcal{L}({\ensuremath{\mathbf{z}}}_{k}) \rangle,$$ and the solution is updated as a weighted average of previous and new solution: $$\label{eqFWUpdate} {\ensuremath{\mathbf{z}}}_{k+1} = (1 - \lambda_{k+1}) {\ensuremath{\mathbf{z}}}_k + \lambda_{k+1} {\ensuremath{\mathbf{h}}}_{k+1},$$ where $$\label{eqFWLambda} \lambda_{k+1} = \underset{\lambda \in [0,1]}{\textrm{arg\;min}} \;\; \mathcal{L}({\ensuremath{\mathbf{z}}}_k + \lambda({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k)).$$ Typically, $\lambda_{k+1} = 2/(2+k)$ such that the weight of new solution reduces as number of iterations increase, or it can be computed through line search. Since Eq. \[eqL2\] is quadratic, a closed-form solution to $\lambda_{k+1}$ exists. Using the fact that $\nabla \mathcal{L}({\ensuremath{\mathbf{z}}}) = -{\ensuremath{\mathbf{Q}}} {\ensuremath{\mathbf{z}}} - {\ensuremath{\mathbf{L}}}$, the minimum values of Eq. \[eqFWLambda\] occurs when, $$\begin{gathered} \frac{\partial}{\partial \lambda} \mathcal{L}\big({\ensuremath{\mathbf{z}}}_k + \lambda_{k+1}({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k)\big) = 0,\\ \nabla \mathcal{L}\big({\ensuremath{\mathbf{z}}}_k + \lambda_{k+1}({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k)\big)^T({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k) = 0,\\ \lambda_{k+1} = \frac{\nabla \mathcal{L}({\ensuremath{\mathbf{z}}}_k)^T ({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k)}{({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k)^T {\ensuremath{\mathbf{Q}}} ({\ensuremath{\mathbf{h}}}_{k+1} - {\ensuremath{\mathbf{z}}}_k)}.\end{gathered}$$ The algorithm begins by randomly initializing a point ${\ensuremath{\mathbf{h}}}_0$ from the solution space $\mathcal{D}$. Given the current solution ${\ensuremath{\mathbf{z}}}_{k}$, Eq. \[eqFWh\] finds the point where the gradient of the loss function is minimum, and the solution is updated through Eq. \[eqFWUpdate\]. For the case of linear program, the solution lies on the boundary of the polytope. To avoid the occasional zig-zag behavior of optimizer for solutions near boundary of the polytope, an *away* step is computed within the convex hull of points in $\mathcal{D}$ visited till iteration $k$, i.e. $\mathcal{S}_{k}$: $$\begin{gathered} {\ensuremath{\mathbf{f}}}_{k} \leftarrow \underset{{\ensuremath{\mathbf{h}}} \in \mathcal{S}_{k}}{\textrm{argmax}} \langle {\ensuremath{\mathbf{h}}}, \nabla\mathcal{L}({\ensuremath{\mathbf{z}}}) \rangle\label{eqFWf},\end{gathered}$$ where $\mathcal{S}_{k}$ contains the *active corners* or previously seen integer solutions. Then, at each iteration, the step that gives the steepest descent between Eq. \[eqFWh\] and Eq. \[eqFWf\] is selected. Since the solution ${\ensuremath{\mathbf{z}}}_{k+1}$ of Frank-Wolfe at each iteration does not satisfy the binary constraints in Eq. \[eqConstraints\], we round the solution in ${\ensuremath{\mathbf{z}}}_{k+1}$ by transforming the vector ${\ensuremath{\mathbf{z}}}_{k+1}$ into a matrix, where rows correspond to people in the first camera, and columns to people in second camera. The new cost matrix is solved through <span style="font-variant:small-caps;">Munkres</span> for $1-1$ matching and the solution satisfies Eq. \[eqConstraints\]. For the next iteration of the Frank-Wolfe, we transform the binary matrix again into vector form where each element represents a re-identification hypothesis. This process of transforming the variables to binary is termed as *rounding*. The algorithm is run till the duality gap is larger than a fixed threshold or a pre-defined number of iterations is reached. Compared to Stochastic Local Search (Alg. \[alg:optimization\]), Frank-Wolfe solves a convex approximation of the original objective function in Eq. \[eqL\], and relaxes the binary constraints in Eq. \[eqConstraints\] when computing the conditional gradient. As we will see in Sec. \[secExperiments\], this results in a slight drop in performance with an order of magnitude gain in computation speed. Experiments {#secExperiments} =========== Since PSE constraints depend on time and motion information in the videos, many commonly evaluated datasets such as VIPeR [@gray2008viewpoint] and ETHZ [@ess2008mobile] cannot be used for computing PSE constraints. We evaluate the proposed approach on the PRID [@hirzer2011person], DukeMTMC [@ristani2016performance] and the challenging Grand Central Dataset [@yi2015understanding]. First, we introduce the datasets and the ground truth that was generated for evaluation, followed by detailed analysis of our approach as well as contribution of different personal, social and environmental (PSE) constraints to the overall performance. Datasets and Experimental Setup {#subsection:Dataset} ------------------------------- ### Grand Central Grand Central is a dense crowd dataset that is particularly challenging for the task of human re-identification. The dataset contains $120,000$ frames, with a resolution of $1920 \times 1080$ pixels. Recently, Yi [@yi2015understanding] used a portion of the dataset for detecting stationary crowd groups. They released annotations for trajectories of $12,684$ individuals for $6,000$ frames at $1.5$ fps. We rectified the perspective distortion from the camera and put bounding boxes at correct scales using the trajectories provided by [@yi2015understanding]. However, location of annotated points were not consistent for any single person, or across different people. Consequently, we manually adjusted the bounding boxes for $1,500$ frames at $1.5$ fps, resulting in ground truth for $17$ minutes of video data. We divide the scene into three horizontal sections, where two of them become separate cameras and the middle section is treated as invisible or unobserved region. The locations of people in each camera are in independent coordinate systems. The choice of dividing the scene in this way is meaningful, as both cameras have different illuminations due to external lighting effects, and the size of individuals is different due to perspective effects. Furthermore, due to the wide field of view in the scene, there are multiple entrances and exits in each camera, so that a person exiting the first camera at a particular location has the choice of entering from multiple different locations. Figure \[fig:teaser\](c) shows real examples of individuals from the two cameras and elucidates the fact that due to the low resolution, change in brightness and scale, the incorrect nearest neighbors matches using the appearance features often rank higher than the correct ones for this dataset. ### DukeMTMC Recently, the DukeMTMC dataset was released to quantify and evaluate the performance of multi-target, multi-camera tracking systems. It is high resolution 1080p, 60fps dataset and includes surveillance footage from 8 cameras with approximately 85 minutes of videos for each camera. There are cameras with both overlapping and non-overlapping fields-of-view. The dataset is of low density with 0 to 54 people per frame. Since only the ground truth for training set has been released so far, which constitutes first 50 minutes of video for each camera, we report performance on the training set only. Cameras 2 and 5 which are disjoint, and have the most number of people (934 in total, with 311 individuals appearing in both cameras), were selected for experiments. To remain consistent with the other datasets, we perform evaluation in terms of Cumulative Matching Curves (CMC) and F-Score on 1-1 assignment. ### PRID PRID 2011 is a camera network re-identification dataset containing 385 pedestrians in camera ‘*a*’ and $749$ pedestrians in camera ‘*b*’. The first $200$ pedestrians from each camera form the ground truth pairs while the rest appear in one camera only. The most common evaluation method on this dataset is to match people from cam ‘*a*’ to the ones in cam ‘*b*’. We used the video sequences and the bounding boxes provided by the authors of [@hirzer2011person] so we can use the PSE constraints in our evaluation. Since the topology of the scene is unknown, we have used the constraints which do not need any prior knowledge about the camera locations. We evaluated on the entire one hour sequences and extract visual features in addition to various PSE constraints. In accordance with previous methods, we evaluate our approach by matching the $200$ people in cam ‘*a*’ to $749$ people in cam ‘*b*’ and quantify the ranking quality of matchings. ![This figure illustrates the CMC evaluation procedure with quadratic constraints. Given object tracks in the two cameras $O_{1_a}, O_{2_a}, O_{3_a}$ and $O_{1_b}, O_{2_b}, O_{3_b}$, (a) the linear constraints are computed between objects, and (b) quadratic constraints between each possible pair of matches. Adding a new match (shown with amber) requires adding one linear value and number of quadratic values equal to the size of current solution.[]{data-label="fig:figQuadraticExplanation"}](./figQuadraticExplanation.pdf){width="50.00000%"} **Parameters:** Since there are multiple points / zones of entrances and exits, we divide the boundaries in each camera into $U_a=U_b=11$ gates. The weights used in Eq. \[eqL\] are approximated using grid search on a separate set and then used for all the datasets. They are $\alpha_{\textrm{spt}}=\alpha_{\textrm{invColl}}=.2$, $\alpha_{\textrm{tr}}=1$, and $\alpha_{\textrm{spd}}=\alpha_{\textrm{invSpd}}=-\alpha_{\textrm{grp}}=5$. Note that, social grouping is rewarded in our formulation, i.e. people who enter and exit together in space and time are more likely to be correct matches when re-identifying people across cameras. Evaluation Measures ------------------- Cumulative Matching Characteristic (CMC) curves are typically used evaluating performance of re-identification methods. For each person, all the putative matches are ranked according to similarity scores, i.e. for each person $O_{i_a}$, the cost of assignment ${M_{i_a}^{j_b}} = {\langle O_{i_a}, O_{j_b} \rangle}$ is calculated for every possible match to $O_{j_b}$. Then, the accuracy over all the queries is computed for each rank. Area Under the Curve (AUC) for CMC gives a single quantified value over different ranks and an evaluation for overall performance. The advantage of CMC is that it does not require $1-1$ correspondence between matches, and is the optimal choice for evaluating different cost functions or similarity measures. The CMC curves are meaningful only for linear constraints. Unlike linear constraints which penalize or reward matches (pair of objects), quadratic constraints penalize or reward pairs of matches. Figure \[fig:figQuadraticExplanation\] illustrates the idea of quantifying both linear and quadratic costs through CMC, since this measure quantifies quality of costs independent of optimization. Given three objects $O_{1_a}, O_{2_a}, O_{3_a}$ and $O_{1_b}, O_{2_b}, O_{3_b}$ in cameras $a$ and $b$, respectively, the black lines in Fig. \[fig:figQuadraticExplanation\] (a) show linear constraints / matchings. Let us assume we intend to evaluate quadratic constraints for the match between $O_{1_a}$ and $O_{2_b}$. For this, we assume that all other matches are correct (red lines), and proceed with adding relevant quadratic (Fig. \[fig:figQuadraticExplanation\]) and linear costs. For evaluating match between $O_{1_a}$ and $O_{2_b}$, we add linear costs between them, as well as quadratic costs between other matches (shown with red circles in Fig. \[fig:figQuadraticExplanation\](b)), and pair-wise costs of the match under consideration with all other matches (shown with orange circles). This is repeated for all possible matches. Later, the matches are sorted and evaluated similar to standard CMC. Note that, this approach gives an optimization-independent method of evaluating quadratic constraints. Nonetheless, the explicit use of ground truth during evaluation of quadratic constraints makes them only comparable to other quadratic constraints. To evaluate $1-1$ correspondence between matches, we use F-score which is defined as $2 \times (\textsf{precision} \times \textsf{recall}) / (\textsf{precision} + \textsf{recall})$ on the output of optimization. We used Hungarian Algorithm (Munkres) [@munkres1957algorithms] for comparison as it provides a globally optimal solution for linear costs. For the proposed PSE constraints, we use Stochastic Local Search (Sec. \[subsec:StochasticSearch\]) and Frank-Wolfe algorithm (Sec. \[subsec:FW\]) since we use both linear and quadratic costs. ![This graph shows the CMC for different PSE constraints proposed in this paper on Grand Central Dataset. The results of random assignment are shown with blue curve, while appearance features yield the orange curve. Incorporating PSE constraints such as preferred speed (amber), transition probabilities (purple), discriminative appearance (green), social and spatial grouping (cerulean and maroon, respectively) further improve the performance. Given the topology, we can additionally incorporate collision avoidance (blue) and preferred speed in the invisible region (orange), which give the best performance. The numbers in parentheses are the AUCs between ranks 1:50 for each CMC.[]{data-label="fig:ranking"}](./CMC_GC_with_topology_FINAL.pdf){width=".5\textwidth"} Results and Comparison ---------------------- In Table \[tbl:resultsGC\], we present the results on Grand Central dataset of our approach with several baselines and existing methods. The first five columns show values of Cumulative Matching Characteristic curves at ranks $1, 5, 10$, $20$ and $50$. We also report Area Under the Curve (AUC) for CMC between ranks $1$ and $100$. The values of CMC are computed before any optimization. The last column shows the F-Score of $1-1$ assignments after optimization. In Table \[tbl:resultsGC\], the first row shows the results of random assignment, whereas next seven rows show results using existing re-identification methods. These include LOMO-XQDA [@liao2015person], SDALF [@farenzena2010person], SAM [@alahi2014socially], eSDC-knn [@zhao2013unsupervised], Manifold Learning [@loy2013person] - normalized (Ln) and unnormalized (Lu), as well as CNN features [@simonyan2014very] which use VGG-19 deep network. Finally, the last two rows show the results of our approach both for the case when camera topology is not known and when it is known. For 1-1 assignment, we present results for both the Stochastic Local Search as well as Frank-Wolfe algorithm in the last column. Frank-Wolfe algorithm drops the F-Score slightly compared to Stochastic Local Search. Overall, these results show that PSE constraints - both linear and quadratic - significantly improve the performance of human re-identification especially in challenging scenarios such as dense crowds. The results on Cameras 2 and 5 of DukeMTMC dataset are shown in Table \[tbl:resultsDUKE\]. The first row shows the results of random assignment, while results of Manifold Learning [@loy2013person] - normalized (Ln) and unnormalized (Lu), as well as CNN features [@simonyan2014very] are presented in the next three rows. The results from our approach are shown in the final row. Despite being an easier dataset compared to Grand Central, the PSE constraints with the proposed optimizations outperform the appearance features by a large margin. Next, we present results on PRID dataset in Table \[tbl:resultsPRID\]. The first three rows show Reranking [@leng2015person] on KissME [@koestinger2012large], LMNN [@weinberger2008fast], and Mahalanobis distance learning [@roth2014mahalanobis] for re-identification. Next two rows show the performance of non-linear Metric Learning [@paisitkriangkrai2015learning] and Descriptive & Discriminative features [@hirzer2011person]. The last row shows the performance of our method which is better than existing unsupervised approaches for human re-identification. For this dataset, the spatial grouping did not improve the results since the dataset captures a straight sidewalk and does not involve decision makings and different travel times between different gates. Analysis -------- ![This figure shows the improvement over iterations while learning the transition probability distributions of *destination* and *travel time* between gates in different cameras.[]{data-label="fig:iterationsPerformance"}](./Iterations_GCDirect.pdf){width=".5\textwidth"} We performed several experiments to gauge the performance of different PSE constraints and components of the proposed approach on Grand Central dataset. The comparison of different constraints using Cumulative Matching Characteristics (CMC) is shown in Figure \[fig:ranking\]. In this figure, the $x$-axis is the rank, while $y$-axis is accuracy with corresponding rank on $x$-axis. First, we show the results of randomly assigning objects between cameras (blue curve). Then, we use appearance features (Convolutional Neural Network) for re-identification and do not use any personal, social or environmental constraints, which we also use to compute the appearance similarity for our method (shown with orange curve). The low performance highlights the difficult nature of this problem in crowded scenes. Next, we introduce linear constraints of preferred speed shown with amber curve which gives an improvement of $\sim26\%$ in terms of area under CMC between ranks $1$ and $50$. Similarly, transition data learned between the two cameras adds another $\sim26\%$ in terms of AUC. Figure \[fig:iterationsPerformance\] shows the improvement of performance through estimation of more accurate transition distributions over different iterations. Similarly, Figure \[fig:evidenceSpeedTransition\] shows real qualitative results of preferred speed, destination and travel time, and elucidates the function of each constraint for the problem of re-identification. {width=".95\textwidth"} {width=".95\textwidth"} Next, we study the impact of quadratic constraints of social (cerulean curve) and spatial (maroon curve) grouping, both of which make slight improvement to the matching performance, with combined effect of $\sim3\%$ improvement in Rank-1 and $\sim8\%$ in Rank 5 performance. Note that both these quadratic constraints are antipodal in the sense that former rewards while latter penalizes the loss function. The last two curves in Figure \[fig:ranking\] show the performance using constraints computable if camera topology is known. Given topology, we employ collision avoidance shown in blue (diamond markers), whereas the constraint capturing the desire of people to walk with preferred speed between cameras is shown in orange (diamond markers), which gives the maximum AUC of $96.27\%$ in conjunction with other PSE constraints. Beyond $90\%$, the increments in AUC appears small, however a noticeable improvement in performance is visible between Ranks 1:10 which is crucial for 1-1 assignment. This study shows that except for collision avoidance, all PSE constraints contribute significantly to the performance of human re-identification. We provide real examples of collision avoidance and social grouping in Fig. \[fig:Grouping\_Collision\](a) and (b), respectively. In Fig. \[fig:Grouping\_Collision\], the bounding boxes are color-coded with time using colormap shown on left. White-to-Yellow indicate earlier time stamps while Red-to-Black indicate later ones. The person under consideration is shown with dashed white line, while the track of two other people in each image are color-coded with costs using colormap on the right. Here, blue indicates low cost whereas red means high cost. Collision avoidance which has been shown to work for tracking in non-crowded scenes [@pellegrini2009you] deteriorates the results slightly in crowded scenes. Fig. \[fig:Grouping\_Collision\](a) shows a case where collision avoidance constraint assigns a high cost to a pair of correct matches. Due to limitation in space in dense crowds, people do not change their path significantly. Furthermore, any slight change in path between cameras is unlikely to have any effect on matching for re-identification. On the other hand, the grouping constraint yields a noticeable increase in performance as evident in Fig. \[fig:Grouping\_Collision\](b) This is despite the fact that the Grand Central dataset depicts dense crowd of commuters in a busy subway station, many of whom walk alone. Conclusion {#secConclusion} ========== This paper addresses the problem of re-identifying people across non-overlapping cameras in crowded scenes. Due to the difficult nature of the problem, the appearance similarity alone gives poor performance. We employ several personal, social and environmental constraints in the form of *appearance, preferred speed, destination, travel time*, and *spatial and social grouping*. These constraints do not require knowledge about camera topology, however if available, it can be incorporated into our formulation. Since the problem with PSE constraints is NP-hard, we proposed stochastic local search, and the computationally efficient Frank-Wolfe algorithm that can handle both quadratic and linear constraints. The crowd dataset used in the paper brings to light the difficulty and challenges of re-identifying and associating people across cameras in crowds, while the personal, social and environmental constraints highlight the importance and utility of extra-appearance information available in videos for the task of re-identification. **Acknowledgment:** This material is based upon work supported in part by, the U.S. Army Research Laboratory, the U.S. Army Research Office under contract/grant number W911NF-14-1-0294. [Shayan Modiri Assari]{} received his B.S. degree in Electrical Engineering from Sharif University of Technology in 2011 and the M.S. degree in Computer Engineering from University of Central Florida in 2015. He is currently a PhD candidate in University of Central Florida’s Center for Research in Computer Vision (CRCV). He has published papers in European Conference on Computer Vision, Computer Vision and Pattern Recognition, and Journal of Machine Vision and Applications. His research interests include event detection, action recognition, object tracking, person re-identification, video surveillance and graph theory. [Haroon Idrees]{} is a Postdoctoral Associate at the Center for Research in Computer Vision at University of Central Florida. He has published several papers in conferences and journals such as CVPR, ICCV, ECCV, Journal of Image and Vision Computing, Computer Vision and Image Understanding, and IEEE Transactions on Pattern Analysis and Machine Intelligence. His research interests include crowd analysis, person detection and re-identification, visual tracking, multi-camera and airborne surveillance, action recognition and localization, and multimedia content analysis. He received the BSc (Hons) degree in Computer Engineering from the Lahore University of Management Sciences, Lahore, Pakistan in 2007, and the PhD degree in Computer Science from the University of Central Florida in 2014. [Mubarak Shah,]{} the Trustee chair professor of computer science, is the founding director of the Center for Research in Computer Vision at the University of Central Florida (UCF). He is an editor of an international book series on video computing, editor-in-chief of Machine Vision and Applications journal, and an associate editor of ACM Computing Surveys journal. He was the program cochair of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) in 2008, an associate editor of the IEEE Transactions on Pattern Analysis and Machine Intelligence, and a guest editor of the special issue of the International Journal of Computer Vision on Video Computing. His research interests include video surveillance, visual tracking, human activity recognition, visual analysis of crowded scenes, video registration, UAV video analysis, and so on. He is an ACM distinguished speaker. He was an IEEE distinguished visitor speaker for 1997-2000 and received the IEEE Outstanding Engineering Educator Award in 1997. In 2006, he was awarded a Pegasus Professor Award, the highest award at UCF. He received the Harris Corporation�s Engineering Achievement Award in 1999, TOKTEN awards from UNDP in 1995, 1997, and 2000, Teaching Incentive Program Award in 1995 and 2003, Research Incentive Award in 2003 and 2009, Millionaire�s Club Awards in 2005 and 2006, University Distinguished Researcher Award in 2007, Honorable mention for the ICCV 2005 Where Am I? Challenge Problem, and was nominated for the Best Paper Award at the ACM Multimedia Conference in 2005. He is a fellow of the IEEE, AAAS, IAPR, and SPIE. [^1]: Data and ground truth available at: <http://crcv.ucf.edu/projects/Crowd-Reidentification>

--- abstract: 'Starting from the Argonne V18 nucleon-nucleon interaction and using the Unitary Correlation Operator Method, a correlated interaction $\mathrm{v}_{\mathrm {UCOM}}$ has been constructed, which is suitable for calculations within restricted Hilbert spaces. In this work we employ the $\mathrm{v}_{\mathrm {UCOM}}$ in Hartree-Fock, perturbation-theory and RPA calculations and we study the ground-state properties of various closed-shell nuclei, as well as some excited states. The present calculations provide also important feedback for the optimization of the $\mathrm{v}_{\mathrm {UCOM}}$ and valuable information on its properties. The above scheme offers the prospect of ab initio calculations in nuclei, regardless of their mass number. It can be used in conjunction with other realistic NN interactions as well, and with various many-body methods (Second RPA, QRPA, Shell Model, etc.).' author: - 'P. Papakonstantinou' - 'R. Roth' - 'H. Hergert' - 'N. Paar' title: | Nuclear Structure and Response based on\ Correlated Realistic NN Interactions --- [ address=[Institut für Kernphysik, Technische Universität Darmstadt,\ Schlossgartenstr. 9, D-64289 Darmstadt, Germany]{} ]{} [ address=[Institut für Kernphysik, Technische Universität Darmstadt,\ Schlossgartenstr. 9, D-64289 Darmstadt, Germany]{} ]{} [ address=[Institut für Kernphysik, Technische Universität Darmstadt,\ Schlossgartenstr. 9, D-64289 Darmstadt, Germany]{} ]{} [ address=[Institut für Kernphysik, Technische Universität Darmstadt,\ Schlossgartenstr. 9, D-64289 Darmstadt, Germany]{} ]{} The Unitary Correlation Operator Method (UCOM) provides a novel scheme for carrying out nuclear structure calculations starting from realistic nucleon-nucleon (NN) interactions [@FNR98; @NeF03; @RNH04; @RHP05]. The major short-range correlations, induced by the strong repulsive core and the tensor part of the NN potential, are described by a state-independent unitary correlation operator $C$. This can be used to introduce correlations into an uncorrelated $A-$body state or, alternatively, to perform a similarity tranformation of an operator of interest. Applied to a realistic NN interaction, in particular, the method produces a “correlated” interaction, $\mathrm{v}_{\mathrm {UCOM}}$, which can be used as a universal effective interaction, for calculations within simple Hilbert spaces. The same transformation can then be applied to any other operator under study, as is needed for a consistent UCOM treatment. The utilization of the UCOM involves a cluster expansion of the correlated operators and, currently, a truncation at the two-body (2B) level. The latter is justified by the short range of the correlations treated by the method. The aim is to treat explicitly only the state-independent short-range correlations (SRC); long-range correlations (LRC) should be described by the model space. The correlation functions are determined for each $(S,T)$ channel by minimizing the energy of the two-nucleon system. Three-body interactions are currently not included in the formulation; one way to account for these is by adding a simple phenomenological non-local 2B correction to the correlated Hamiltonian. The introduction of such a correction (the same for all nuclei) has allowed the UCOM to successfully describe properties of nuclei up to mass numbers $A\approx 60$, in the framework of variational calculations within Fermionic Molecular Dynamics [@RNH04]. In this work we study nuclear structure and response, based on realistic interactions, without restricting ourselves to light-to-medium systems. This is made possible by employing the $\mathrm{v}_{\mathrm {UCOM}}$ in Hartree-Fock (HF)- and RPA-based models. The 2B matrix elements of the ${\mathrm v}_{\rm UCOM}$ (without correction terms), Coulomb interaction and (intrinsic) kinetic energy are calculated in a harmonic-oscillator (HO) basis. These are used as input to subsequent HF and RPA calculations, in configuration space. Only spherical, closed-shell nuclei have been considered so far. The following methods have been used: [HF]{} : - a spherical-HF method, to estimate at “zeroth order” the nuclear ground state (gs) properties. The HF single-particle states are expanded in the HO basis. [HF+PT]{} : - second order perturbation theory (PT) is performed on the HF basis to obtain a correction to the gs energy. [HF+RPA]{} : - a self-consistent model, which allows us to estimate a correction to the gs energy due to LRC, and in addition to study collective excitations. [ERPA]{} : - an extended RPA [@CGP98], which is built on top of the true RPA gs and involves an iterative solution of the RPA equations. Corrected single-particle energies and occupation numbers can be obtained, as well as excitation properties. The PT and (E)RPA allow us to account for LRC, in addition to the SRC introduced via the UCOM. The optimal separation of the two types of correlations remains an important task. It is expressed primarily by a constraint on the range of the tensor correlation functions, imposed during their parameterization. By varying this range (more precisely, the “correlation volume” $I_{\vartheta}$ [@RHP05]), a family of correlators and respective correlated interactions are obtained. Our present results provide important feedback for the optimization of the ${\mathrm v}_{\mathrm{UCOM}}$ and valuable information on its properties. The maximum HO-energy and angular-momentum quantum numbers used here are $N_{\max}=2n+\ell=12$ and $\ell_{\max}=10$, respectively, providing a satisfactory degree of convergence. We use the $\mathrm{v}_{\mathrm{UCOM}}$ parameterizations discussed in Ref. [@RHP05], based on the Argonne V18 interaction. The results presented in Figs. 1 and 2 were obtained with $I_{\vartheta}^{(S=1,T=0)}=0.09$ fm$^3$. This value best reproduces, in exact no-core shell model calculations, the binding energy of the light systems $^{4}$He, $^3$H [@RHP05]. {height=".13\textheight" width="70.00000%"} ![Isoscalar monopole resonance of $^{40}$[Ca]{}, in HF+RPA (blue lines) and ERPA (red lines). An arrow indicates the experimental centroid.](PapakonstantinouP_fig2.eps){width="75.00000%"} #### Ground state properties Binding is achieved already at the HF level. The inclusion of LRC corrections to the gs energy via PT brings the gs energy very close to the experimental data, as shown in Fig. 1. The RPA correlations result in overbinding [@PPH05], mainly due to the double-counting of second-order corrections inherent in the model. In general, larger-$I_{\vartheta}$ tensor correlators provide stronger binding both at the HF and the RPA level. The HF single-particle levels (not shown) are too sparse [@PPH05]. The omission of a three-body interaction and of LRC are responsible for this effect. Occupation numbers $n_i=\langle a^{\dagger}_i a_i\rangle$ (in standard notation) have been calculated within ERPA for $^{16}$O and $^{40}$Ca. In principle, correlated operators $C^{\dagger} a^{\dagger}_i a_i C$ should be used. The small depletion of the Fermi sea that we obtain using uncorrelated operators reflects the effect of the gs LRC [@AHP93]. #### Collective excitations Our RPA results on the isoscalar (IS) giant monopole resonance (ISGMR), for various medium and heavy nuclei, are in good agreement with the experimental data. An example is shown in Fig. 2, where we see also that the gs correlations taken into account in ERPA have a relatively small effect. In general, lower $I_{\vartheta}$ values result in lower ISGMR energies [@PPH05]. The isovector (IV) dipole strength (not shown) is distributed at energies which are too high compared with experiment; ERPA is not able to correct for this result. [Inclusion of $2p2h$ configurations within Second RPA is expected to bring the IVD strength to lower energies.]{} In conclusion, the performance of the $\mathrm{v}_{\mathrm{UCOM}}$ is very encouraging. Certain aspects of the model (e.g., optimal tensor correlators, three-body effects) are still open to improvement. The above scheme can be used in conjunction with other realistic (local or non-local) NN interactions, as well as with various many-body methods (Second RPA, QRPA, Shell Model, etc), and offers the prospect of ab initio calculations across the nuclear chart. [ Work supported by the Deutsche Forschungsgemeinschaft through contract SFB 634. ]{} [9]{} H. Feldmeier, T. Neff, R. Roth, J. Schnack, *Nucl. Phys.* **A 632**, 61 (1998). T. Neff, H. Feldmeier, *Nucl. Phys.* **A 713**, 311 (2003). R. Roth, T. Neff, H. Hergert, H. Feldmeier, *Nucl. Phys.* **A 745**, 3 (2004). R. Roth, H. Hergert, P. Papakonstantinou, T. Neff, H. Feldmeier, arXiv:nucl-th/0505080. F. Catara, M. Grasso, G. Piccitto, M. Sambataro, *Phys. Rev.* **B 58**, 16070 (1998). N. Paar, P. Papakonstantinou, H. Hergert, R. Roth, Proc. LV National Conference on Nuclear Physics, “Frontiers in the Physics of Nucleus”, St. Petersburg, Russia, 2005; arXiv:nucl-th/0506076. A.N. Antonov, P.E. Hodgson and I.Zh. Petkov, [*Nucleon Correlations in Nuclei*]{}, Springer-Verlag, 1993.

--- abstract: 'The non-resonant two-photon one-electron ionization of neutral atoms is studied theoretically in the framework of relativistic second-order perturbation theory and independent particle approximation. In particular, the importance of relativistic and screening effects in the total two-photon ionization cross section is investigated. Detailed computations have been carried out for the $K$-shell ionization of neutral Ne, Ge, Xe, and U atoms. The relativistic effects significantly decrease the total cross section, for the case of U, for example, they reduce the total cross section by a factor of two. Moreover, we have found that the account for the screening effects of the remaining electrons leads to occurrence of an unexpected minimum in the total cross section at the total photon energies equal to the ionization threshold, for the case of Ne, for example, the cross section drops there by a factor of three.' author: - 'J. Hofbrucker' - 'A. V. Volotka' - 'S. Fritzsche' title: ' Relativistic calculations of the non-resonant two-photon ionization of neutral atoms ' --- Introduction {#Sec.Introduction} ============ Two-photon one-electron ionization is one of the fundamental non-linear processes in the light-matter interaction with various spectroscopic applications. In the past, however, most experiments were focused on the two-photon ionization of atomic outer shell electrons [@Duncanson/PRL:1976; @Siegel/JPB:1983; @Dodhy/PRL:1985; @Wang/PRA:2000]. Only with the recent advancements in free-electron lasers (FEL), the study of non-linear processes in atoms and molecules at extreme ultraviolet and x-ray energies became feasible [@Pellegrini/RMP:2016] and renewed an interest in inner-shell excitation and ionization processes. One of the first experiments utilising FEL facilities used electron or ion spectrometers to study the two-photon ionization of 4$d$ electron of neutral Xe atom [@Richardson/PRL:2010] and 1$s$ electron of heliumlike Ne$^{8+}$ ion [@Doumy/PRL:2011]. Meanwhile, modern FEL facilities reach beam intensities of about $I \approx 10^{20}$ W/cm$^2$, although the non-resonant two-photon ionization of inner-shell electrons still remains a challenge due to small cross sections. Experimentally, the two-photon ionization can be measured by collecting the $K$-fluorescence. This fluoresce radiation emitted by bound electrons decaying into the $K$-shell vacancy is a direct signature of the two-photon ionization process. This experimental approach has been utilized in the case of $K$-shell ionization of neutral Ge [@Tamasaku/NP:2014] and Zr [@Ghimire] atoms. First detailed calculations of the two-photon ionization of atomic hydrogen were performed by Zernik within non-relativistic dipole approximation [@Zernik/PR:1964] more than 50 years ago. In this work, he also introduced the well-known $Z^{-6}$ scaling of the total cross section with the nuclear charge $Z$ for non-resonant two-photon ionization and, hence, provided an estimate of a total cross section for all hydrogenlike ions. However, later in Refs. [@Koval/JPB:2003; @Koval/JPB:2004; @Koval/Dissertation], complete relativistic calculations were carried out which demonstrated that quite strong deviation from this scaling occurs due to relativistic effects. Recently, in Refs. [@Florescu/PRA:2011; @Florescu/PRA:2012], the relativistic effects have also been investigated in the two-photon above-threshold ionization of low-$Z$ hydrogenlike ions. Although the relativistic effects have been found rather important, no systematic relativistic calculations have been carried out until now for the two-photon ionization of neutral atoms. In this paper, we investigate the relativistic and screening effect contributions to the total two-photon $K$-shell ionization cross section of neutral atoms. In particular, we consider photon energies below the ionization threshold so that no single photon $K$-shell ionization is possible. In Sec. \[Sec.Theory\], we first formulate the relativistic second-order perturbation theory, based on the Dirac equation for describing the non-resonant two-photon ionization. By using, in addition, the independent particle approximation and particle-hole formalism, we are able to reduce the many-electron transition amplitude to an effective *single-electron* amplitude, from which an expression for the total two-photon ionization cross section is obtained. In section \[Sec.Computation\], we then outline the numerical procedure that is employed in this work. Detailed calculations are carried out for the non-resonant two-photon $K$-shell ionization of neutral Ne, Ge, Xe, and U atoms. The total cross section as a function of energy is compared for hydrogenlike and neutral systems in Sec. \[Sec.ResultsAndDiscussion\]. In this section, we also demonstrate that the relativistic effects need to be taken into account in the two-photon ionization cross section calculations of neutral atoms, especially for high-$Z$ atoms. Moreover, we show that although screening effects tend to increase the cross section, they result in an unexpected minimum of the total cross section in the near-threshold ionization energy region. Finally, a summary is given in Sec. \[Sec.SummaryAndOutlook\]. Relativistic units ($\hbar=c=m=1$) are used throughout the paper, unless stated otherwise. Theoretical background {#Sec.Theory} ====================== We here consider the two-photon one-electron ionization of neutral atoms. This process can be expressed as follows $$\begin{aligned} \ketm{\alpha_i J_i M_i}+\gamma_1+\gamma_2 \rightarrow \ketm{\alpha_f J_f M_f}+\ketm{\bm{p}_e m_e},\end{aligned}$$ where the atom is initially in the many-electron state $\ketm{\alpha_i J_i M_i}$, with total angular momentum $J_i$, its projection $M_i$, and where $\alpha_i$ denotes all further quantum numbers that are needed for unique characterization of the state. After the simultaneous interaction of the atom with two photons $\gamma_1$ and $\gamma_2$ with energies $\omega_1$ and $\omega_2$, respectively, the system is in a final state $\ketm{\alpha_f J_f M_f, \bm{p}_e m_e}$. The system now consists of a singly charged ion $\ketm{\alpha_f J_f M_f}$ with a hole in the substate $\ketm{a}$, as well as a continuum electron $\ketm{\bm{p}_e m_e}$, with well-defined asymptotic momentum $\bm{p}_e$ and spin projection $m_e$. In the following subsection, we will use the particle-hole formalism and the independent particle approximation in order to reduce the many-electron transition amplitude to a one-electron transition amplitude. Then, employing the density matrix formalism, we derive an expression for the total non-resonant two-photon ionization cross section. Evaluation of transition amplitude {#Subsec.TransitionAmplitude} ---------------------------------- In second-order perturbation theory, the transition amplitude for the two-photon single-electron photoionization of an atom in the initial state $\ketm{\alpha_iJ_iM_i}$ into a final state $\ketm{\alpha_f J_f M_f, \bm{p}_e m_e}$ under the simultaneous absorption of two photons with wave vectors $\bm{k}_1,\bm{k}_2$ and polarization vectors $\bm{\hat{\varepsilon}}_{\lambda_1}, \bm{\hat{\varepsilon}}_{\lambda_2}$ can be written as $$\begin{aligned} \label{GeneralTransitionAmplitude} M_{J_i M_i J_f M_f m_e}^{\lambda_1 \lambda_2}&=&~\int\kern-1.5em\sum_{\nu}\frac{ \mem{\alpha_fJ_fM_f,\bm{p}_e m_e} {\hat{\mathcal{R}}(\bm{k}_2,\bm{\hat{\varepsilon}}_{\lambda_2})} {\alpha_{\nu}J_{\nu}M_{\nu}} \mem{\alpha_{\nu}J_{\nu}M_{\nu}} {\hat{\mathcal{R}}(\bm{k}_1,\bm{\hat{\varepsilon}}_{\lambda_1})} {\alpha_iJ_iM_i} } {E_{i}+\omega_1-E_{\nu}}\\ \nonumber &&\hspace{4cm}+(\bm{k}_1\leftrightarrow \bm{k}_2,\bm{\hat{\varepsilon}}_{\lambda_1}\leftrightarrow \bm{\hat{\varepsilon}}_{\lambda_2},\omega_1 \leftrightarrow \omega_2 ).\end{aligned}$$ For the general case of two inequivalent photons, the additional term $(\bm{k}_1\leftrightarrow \bm{k}_2,\bm{\hat{\varepsilon}}_{\lambda_1}\leftrightarrow \bm{\hat{\varepsilon}}_{\lambda_2},\omega_1 \leftrightarrow \omega_2 )$ arises from the interchange of the interaction sequence of the two photons with the atom. The evaluation of expression (\[GeneralTransitionAmplitude\]) requires a summation to be carried out over the complete spectrum of intermediate states $\ketm{\alpha_{\nu}J_{\nu}M_{\nu}}$. The operator $\hat{\mathcal{R}}$ denotes the one-particle transition operator describing the electron-photon interaction. This operator can be represented in the second quantization formalism (see, e.g., [@Johnson/Book]) as $$\begin{aligned} \label{OperatorExpansion} \hat{\mathcal{R}}(\bm{k},\bm{\hat{\varepsilon}}_{\lambda})= \sum_{lm} \mem{l}{\alpha_{\mu} A^{\mu}_{\lambda}(\omega)}{m} a^{\dag}_{l} a_{m},\end{aligned}$$ where $\ketm{m}, \ketm{l}$ are the single-electron initial and final states, $a^{\dag}_l$ and $a_m$ are the corresponding electron creation and annihilation operators, $\alpha_{\mu}$ denotes the four-vector of the Dirac matrices and $A^{\mu}_{\lambda}=({\phi_{\lambda},\bm{A}_\lambda})$ is the photon wavefunction. Due to the interaction of the atom with the two photons, an electron from a substate $\ketm{a}\equiv\ketm{n_a j_a l_a m_a}$ of the atom is promoted into a continuum state, leaving a hole (or vacancy) in the atomic subshell. Here, $n_a$ is the principal quantum number, $l_a$ is the orbital angular momentum, $j_a$ and $m_a$ are the total angular momentum and its projection, respectively. According to the particle-hole formalism, a state with a hole in a substate $\ketm{n_a j_a l_a m_a}$ has angular momentum properties of a particle with angular momentum $j_a$ and its projection $-m_a$. Then, within the independent particle approximation, the final state after an ionization process is obtained by applying the electron creation ($a^{\dag}_{p_e m_e}$) and annihilation ($a_{n_a j_a l_a m_a}$) operators to the initial state and coupling the initial atom and hole angular momenta using a Clebsh-Gordan coefficient $\sprm{.. ..}{..}$. Hence, the final state of the system can be expressed as $$\begin{aligned} \label{FinalMultiElectronWaveFunction} \ketm{\alpha_f J_f M_f, \bm{p}_e m_e}&=& \sum_{m_a M} \sprm{j_a,-m_a,J_i, M}{J_f M_f} \\ \nonumber &\times & (-1)^{j_a-m_a}a^{\dag}_{p_e m_e}a_{n_a j_a l_a m_a}\ketm{\alpha_i J_i M}.\end{aligned}$$ If we therefore insert expressions (\[OperatorExpansion\]) and (\[FinalMultiElectronWaveFunction\]) into Eq. (\[GeneralTransitionAmplitude\]), apply the electron creation and annihilation operators and carry out the summation over the magnetic quantum number $M$, the many-electron transition amplitude reduces to an amplitude which only depends on one-electron wavefunctions of the active electron $$\begin{aligned} \label{SimplifiedTransitionAmplitude} &M&^{\lambda_1 \lambda_2}_{J_i M_i J_f M_f m_e}\\\nonumber &=&\sum_{m_a}(-1)^{j_a-m_a} \nonumber \sprm{j_a, -m_a, J_i, M_i}{J_f, M_f} \\\nonumber &\times& \int\kern-1.5em\sum_{~n} \frac{ \mem{\mathbf{p}_e m_e} {\alpha_{\mu} A^{\mu}_{\lambda_2}(\omega_2)} {n} \mem{n} {\alpha_{\mu} A^{\mu}_{\lambda_1}(\omega_1)} {a} } {E_{n_a j_a}+\omega_1-E_{n_n j_n}}\\ &+&(\bm{k}_1\leftrightarrow \bm{k}_2,\bm{\hat{\varepsilon}}_{\lambda_1}\leftrightarrow \bm{\hat{\varepsilon}}_{\lambda_2},\omega_1 \leftrightarrow \omega_2 ) \nonumber ,\end{aligned}$$ where a summation is carried out over the complete energy spectrum of the single-electron intermediate states $\ketm{n}$. Employment of the independent particle approximation allows us to turn from many-electron wavefunctions to one-electron ones. Further simplification of the one-electron transition amplitude can be achieved using the multipole decomposition of the photon field $\bm{A}_{\lambda}(\omega)$ into spherical tensors [@Varshalovich/Book:1988] $$\begin{aligned} \label{MultipoleExpansion} \bm{A}_{\lambda}(\omega)= 4\pi \sum_{J M p} i^{J-p} [\bm{\hat{\varepsilon}}_{\lambda}\cdot\bm{Y}_{JM}^{(p)*}(\bm{\hat{k}})] \bm{a}^{(p)}_{JM}(\bm{r}),\end{aligned}$$ where $\bm{Y}_{JM}^{(p)}$ is a vector spherical harmonics and the index $p$ describes the electric ($p=1$) and magnetic ($p=0$) components of the electromagnetic field. In addition, we also perform an expansion of the continuum electron wavefunction into its partial waves [@Eichler/PR:2007] $$\begin{aligned} \label{PartialWaveExpansion} \ketm{\mathbf{p}_e m_e}&=& \displaystyle{\frac{1}{\sqrt{\varepsilon_e |\bm{p}_e|}}} \sum_{jm_j}\sum_{lm_l} i^{l} e^{-i \Delta_{jl}} \hspace{3cm}\\ \nonumber &\times& \sprm{l,m_l,1/2, m_e}{j,m_j} \ketm{\varepsilon_e j l m_j} Y^*_{l m_l}(\hat{p}_e),\end{aligned}$$ with $\varepsilon_e=\sqrt{\bm{p}_e^2+m^2}$ being the electron energy, $\Delta_{jl}$ the phase factor [@Eichler/PR:2007] and $Y^*_{l m_l}(\hat{p}_e)$ the spherical harmonics that depends specifically on the direction of the emitted electron. In the expansion, the summation runs over all total and orbital angular momentum quantum numbers $j$ and $l$, and $\ketm{\varepsilon_e j l m_j}$ are partial waves of the free electron with well-defined electron energy $\varepsilon_e$ and quantum numbers $j,l$, and $m_j$. The transition amplitude $M^{\lambda_1 \lambda_2}_{J_i M_i J_f M_f m_e}$ from equation (\[SimplifiedTransitionAmplitude\]) can be further simplified using the expansions (\[MultipoleExpansion\]) and (\[PartialWaveExpansion\]). Moreover, by assuming two identical photons, we can write their momenta as $\bm{k}_1=\bm{k}_2=\bm{k}$ and polarization vectors as $\bm{\hat{\varepsilon}}_{\lambda_1}=\bm{\hat{\varepsilon}}_{\lambda_2}= \bm{\hat{\varepsilon}}_{\lambda}$. Then by choosing $\bm{\hat{k}}$ as the quantization axis, the dot product of the polarization vector and the spherical harmonics in the multipole expansion (\[MultipoleExpansion\]) can be written as $\bm{\hat{\varepsilon}}_{\lambda}\cdot\bm{Y}_{JM}^{(p)}(\bm{\hat{k}})=\sqrt{[J]/8\pi}(-\lambda)^{p}\delta_{\lambda M}$, where $[J]=2J+1$. Furthermore, by employing the Wigner-Eckart theorem, the amplitude (\[SimplifiedTransitionAmplitude\]) can be expressed in terms of the reduced transition amplitude, which describes the two-photon interaction with the electron independently of the magnetic quantum numbers $m_a, m_n$, and $m_j$. By carrying out all the above simplifications, we can express the many-electron two-photon amplitude (\[GeneralTransitionAmplitude\]) within the independent particle approximation by $$\begin{aligned} \label{FinalTransitionAmplitude} M^{\lambda_1 \lambda_2}_{J_i M_i J_f M_f m_e}&=& \nonumber \sum_{p_1 J_1} \sum_{p_2 J_2} \sum_{n_n j_n l_n m_n} i^{J_1-p_1+J_2-p_2} \sqrt{\frac{[J_1, J_2]}{[j_n, j_a]}} (-\lambda_1)^{p_1} (-\lambda_2)^{p_2} \sum_{jm_j} \sum_{lm_l}(-i)^l e^{i\Delta_{jl}} \sprm{j,m_j, l, m_l}{1/2 , m_e}\\ \nonumber &\times& Y_{l,m_l}(\hat{p}_e) (-1)^{j-m_j} \sprm{j,m_j,J_1,-\lambda_1}{j_n,m_n} \sum_{m_a} \sprm{j_a, -m_a, J_i, M_i}{J_f, M_f} \\ &\times& \sprm{j_n, m_n, J_2,-\lambda_2}{j_a, m_a} \frac{ \redmem{\varepsilon_e j l} {\bm{\alpha} \cdot \bm{ a}^{(p_2)}_{J_2M_2}} {n_n j_n l_n} \redmem{n_n j_n l_n} {\bm{\alpha} \cdot \bm{ a}^{(p_1)}_{J_1M_1}} {n_a j_a l_a} }{E_{n_a j_a}+\omega_1-E_{n_n j_n}}.\end{aligned}$$ Having the two-photon transition amplitude for the interaction of the atom with the radiation field, we can employ the density matrix theory to obtain the corresponding two-photon ionization cross section. Here, the density matrix of the overall system (ion + outgoing electron) is applied to deal efficiently with the degrees of freedom of the two subsystems and to easily trace-out all those degrees, which are not observed experimentally. Total cross section {#Subsec.TotalCrossSection} ------------------- The density matrix of the final system state contains complete information about both the singly ionized atom and the free electron, and can be expressed in terms of the transition amplitude (\[FinalTransitionAmplitude\]) is given by $$\begin{aligned} \label{FinalDensityMatrix} &&\mem{\alpha_f J_f M_f, \bm{p}_e m_e}{\hat{\rho_f}}{\alpha_f J_f M_f', \bm{p}_e m_e'} \nonumber \\ \nonumber &&=\sum_{M_i \lambda_1 \lambda_2} \sum_{M_i' \lambda_1' \lambda_2'} \mem{\alpha_i J_i M_i, \bm{k} \lambda_1 \bm{k} \lambda_2}{\hat{\rho}}{\alpha_i J_i M_i', \bm{k} \lambda_1' \bm{k} \lambda_2'} \\ &&\times M^{\lambda_1 \lambda_2}_{J_i M_i J_f M_f m_e} M^{\lambda_1' \lambda_2'*}_{J_i M_i' J_f M_f' m_e'},\end{aligned}$$ where $\mem{\alpha_i J_i M_i, \bm{k} \lambda_1 \bm{k} \lambda_2}{\hat{\rho}}{\alpha_i J_i M_i', \bm{k} \lambda_1' \bm{k} \lambda_2'}$ refers to the density matrix of the initial state of the system. As the atom and the incident radiation are initially independent, the initial-state density matrix can be written as a direct product of the neutral atom and the two photon density matrices as follows [@Blum/Book:1981] $$\begin{aligned} \label{InitialDensityMatrix} &&\mem{\alpha_i J_i M_i, \bm{k} \lambda_1 \bm{k} \lambda_2}{\hat{\rho}}{\alpha_i J_i M_i', \bm{k} \lambda_1' \bm{k} \lambda_2'}\hspace{1.8cm}\\\nonumber &&= \mem{\alpha_i J_i M_i}{\hat{\rho_i}}{\alpha_i J_i M_i'} \mem{\bm{k}\lambda_1}{\hat{\rho}_{\gamma}}{\bm{k}\lambda_1'} \mem{\bm{k}\lambda_2}{\hat{\rho}_{\gamma}}{\bm{k}\lambda_2'}.\end{aligned}$$ Here, the $\mem{\bm{k}\lambda}{\hat{\rho}_{\gamma}}{\bm{k}\lambda'}$ are the photon helicity density matrices which allow us to conveniently parametrize the polarization of the photons by means of Stokes parameter $$\begin{aligned} \mem{\bm{k}\lambda}{\hat{\rho}_{\gamma}}{\bm{k}\lambda'}=\frac{1}{2}\twobytwo{1+P_3}{P_1-iP_2}{P_1+iP_2}{1-P_3}.\end{aligned}$$ In this formalism, it is indeed easy to express any degree of polarization with the linear ($P_1, P_2$) and circular ($P_3$) Stokes parameters and to calculate the corresponding total cross section. As mentioned before, we assume equal momenta of the two photons, however, the photon helicities $\lambda$ (spin projections onto the $\bm{\hat{k}}$ direction) may still differ. Below, we shall assume that the atom is initially unpolarized and that the density matrix of the neutral atom is simply given by $$\begin{aligned} \label{InitialDensityMatrix} \mem{\alpha_i J_i M_i}{\hat{\rho}_i}{\alpha_i J_i M_i'} = \frac{1}{[J_i]}\delta_{M_i M_i'}.\end{aligned}$$ To extract the observable quantity from the density matrix (\[FinalDensityMatrix\]), we can define a (so called) “detector operator” $\hat{P}$ which characterizes the experimental detector system as a whole. This operator determines the probability for an “event” to be recorded at the detector. Then, the probability is simply given by the trace of the product of the detector operator and the density matrix. Here, we consider an electron detector insensitive to the electron polarization detecting electrons in $4\pi$ solid angle. The detector can be thus described by the operator $\hat{P}=\int d\hat{p}_e \sum_{m_e} \ketm{\bm{p}_e m_e}\bram{\bm{p}_e m_e}$. Moreover, as we do not observe the final ionic state, we have to sum over the corresponding quantum numbers $J_f$ and $M_f$. Then, the total cross section for non-resonant ionization of an atom by two photons with $\bm{k}_1=\bm{k}_2=\bm{k}$ and $\bm{\hat{k}}||\bm{\hat{z}}$ is given by $$\begin{aligned} \label{TotalCrossSection} \sigma(\omega)&=&\frac{32 \pi^5 \alpha^2}{\omega^2} \sum_{J_f M_f}\mathrm{Tr}(\hat{P} \hat{\rho}_f)\\\nonumber &=&\frac{32 \pi^5 \alpha^2}{\omega^2} \frac{1}{[J_i]} \sum_{ \lambda_1 \lambda_2 \lambda_1' \lambda_2'} \mem{\bm{k}\lambda_1} {\hat{\rho}_{\gamma}} {\bm{k}\lambda_1'} \mem{\bm{k}\lambda_2} {\hat{\rho}_{\gamma}} {\bm{k}\lambda_2'}\\\nonumber &\times& \int d\hat{p}_e \sum_{J_f M_i M_f m_e} M^{\lambda_1 \lambda_2}_{J_i M_i J_f M_f m_e} M^{\lambda_1' \lambda_2'*}_{J_i M_i J_f M_f m_e}.\end{aligned}$$ As this expression represents second-order cross section, it has the units of $[L^4T]$. Computations {#Sec.Computation} ============ From the theoretical description above, it can be seen that the main computation challenge lies in the infinite summations of the reduced matrix elements (\[FinalTransitionAmplitude\]) over all multipole orders and infinite number of intermediate states. To deal with this numerically, the infinite summations over the multipoles of each of the two photons were restricted to a maximum value of $J_{\mathrm{max}}=5$. This limit is sufficient to obtain convergence of the corresponding total cross section at less than 0.001% level. To sum over the infinite number of intermediate states, finite basis-set [@Sapirstein/JPB:1996] constructed from $B$-splines by applying the dual-kinetic-balance approach [@Shabaev/PRL:2004] was employed. This technique allows us to reduce infinite sum over the intermediate states in (\[FinalTransitionAmplitude\]) to finite sum over pseudospectrum. This approach has been previously successfully applied, for example, in the calculations of two-photon decay rates of heliumlike ions [@Volotka/PRA:2011; @Surzhykov/PRA:2010] or cross sections of x-ray Rayleigh scattering [@Volotka/PRA:2016]. The continuum-state wavefunctions were obtained by numerical solutions of the Dirac equation with help of the RADIAL package [@Salvat/CPC:1995]. In order to account for the screening effects, we solve the Dirac equation with a screening potential, which partially accounts for the interelectronic interaction. We use the core-Hartree potential, which corresponds to a potential created by all bound electrons except of the active electron. The core-Hartree potential reproduces the electron binding energies in excellent agreement to the experimental values within $\pm 0.2\%$ error for all atoms under consideration. To analyse the sensitivity to the choice of potential, in addition to the core-Hartre potential, two different screening potentials were also used. The potential taken from Ref. [@Salvat/PRA:1987], to which we refer as the “Salvat” potential and Salvat potential modified in a way to reproduce experimental binding energies $E_{\mathrm{bind}}^{\mathrm{exp}}$, referred to as “Salvat $E_{\mathrm{bind}}^{\mathrm{exp}}$”. All results presented were calculated using the core-Hartree potential, except for Fig. \[Fig.PotentialComparison\] where results from the different potentials are compared. In addition, in order to check the consistency of our results, we carried out the calculations in length and velocity gauges. The results for both gauges were in a perfect agreement as expected for any local potential. Even though the agreement of the two gauges does not prove validity of the results, it shows that the effective single-electron amplitudes (\[FinalTransitionAmplitude\]) are properly implemented in our codes. Results and discussion {#Sec.ResultsAndDiscussion} ====================== {width="0.97\linewidth"} {width="0.97\linewidth"} {width="0.97\linewidth"} {width="0.97\linewidth"} Even though the formalism derived in Sec. \[Sec.Theory\] applies generally for neutral atoms as well as ions, detailed calculations have been carried out for $K$-shell two-photon ionization of neutral neon, germanium, xenon, and uranium atoms. Specifically, the contributions of relativistic and screening effects to the total cross section have been investigated. In this section, we will compare cross sections for $K$-shell ionization of hydrogenlike and neutral atoms. Since there are two electrons in the $K$-shell of neutral atoms but only one electron in the $K$-shell hydrogenlike ions, we introduce an additional factor of two in the hydrogenlike calculation for the sake of comparison. We begin by comparing the ionization of hydrogenlike and neutral atoms in terms of the total cross section as a function of so called excess energy. Excess energy represents the factor by which the combined photon energy exceeds the ionization threshold, i.e., $\varepsilon=2\omega/E_{\mathrm{bind}}$. Figure \[Fig.PotentialComparison\] presents the total cross sections for ionization of neutral (solid black) as well as for H-like (dashed-dotted red) neon, germanium, xenon, and uranium by linearly polarized photons. We can notice that the first resonant behaviour in the total cross section occurs in lower energy for H-like ions than for neutral atoms. This resonant behaviour occurs when the single photon energy reaches the $1s \rightarrow 2p$ transition energy. Although the $2p$ state is generally occupied for neutral atoms, the resonant two-photon ionization can be understood as follows. The $2p$ electron is ionized by the first photon and the corresponding vacancy is then filled by excitation of the $1s$ electron by the second photon. Since the present work is devoted to the non-resonant ionization, the $1s \rightarrow 2p$ resonant energy and the ionization threshold define the energy range of current interest. The more significant difference between neutral and H-like systems lies in the decrease of the total cross section near the ionization threshold. This cross section reduction is strongest for elements with nuclear charge $Z=7-12$ and becomes much less significant for heavy atoms. In the case of H-like ions, no such behaviour has been predicted and the total two-photon ionization cross section is slowly decreasing in non-resonant energy regions [@Koval/Dissertation; @Koval/JPB:2003], which we also confirm by the present calculations. This means that the change of the total cross section for light neutral elements close to the ionization threshold can be directly linked to the deviation of the binding potential from the Coulomb potential created by the nucleus. In the next subsection, we will investigate these effects (which we refer to as “screening effects”) further by looking at the $s-$ and $d-$ partial waves of the free electron. These partial waves strongly dominate others as they are the only allowed by two electric dipole ($E1E1$) transitions. {width="0.97\linewidth"} {width="0.97\linewidth"} {width="0.97\linewidth"} {width="0.97\linewidth"} Figure \[Fig.PotentialComparison\] shows also the comparison between the three screening potentials (solid black, dashed green, and blue curves) introduced in Sec. \[Sec.Computation\]. We see that for low-$Z$ and medium-$Z$ atoms, the core-Hartree and Salvat potentials differ in the magnitudes of the total cross section by less than $25\%$. This is partially caused by the calculated value of the binding energy. When the Salvat potential was modified to reach perfect agreement with the experimental binding energies, the cross section difference from the core-Hartree calculation was reduced to about $10\%$. Therefore, even though part of the difference between the cross sections as predicted by each potential arises from the difference of binding energies, the distinct potential formulations also result in a deviation. Despite the small magnitude differences, all screening potentials predict similar energy dependence of the total cross section. The agreement of these potentials justifies that the obtained behaviour and magnitude is not very sensitive to the choice of potential. We ascribe the difference between the core-Hartree and Salvat calculations as an uncertainty of presented results. The uncertainty decreases from 25$\%$ for Ne to 10$\%$ for U. We restrict all further discussion to the use of core-Hartree potential. Partial wave analysis {#Subsec.PartialWave} --------------------- To gain deeper understanding of the total cross section results, we now wish to look at the dominant $E1E1$ ionization channels. In this approximation, only the $J=1$ multipole of each of the two photons is considered. Since we are interested in ionization of a $1s$ electron with zero orbital angular momentum $l=0$, $E1E1$ transition allows only two possible ionization channels; $s\rightarrow p \rightarrow s$ and $s \rightarrow p \rightarrow d$. Therefore, only $s-$ and $d-$ partial waves of the free electron are allowed in dipole approximation. While both of these channels are open for linear and unpolarized light, only the $s\rightarrow p \rightarrow d$ channel is open for circularly polarized light. This restriction comes from the conservation of the angular momentum projection. Since we are considering two equally circularly polarized photons, the angular momentum projection must change by $\pm 2$, then $|m_j|>1/2$ is always the case, making the final $s-$state forbidden. This is a point worth remembering. As we will soon see, the absence of the $s\rightarrow p\rightarrow s$ channel leads to a magnification of the screening effects, which increases the probability of experimental detection of these effects. Figure \[Fig.PartialWave\] shows the plots of the partial-wave cross sections considering only the $s-$ or $d-$ partial-waves of the continuum electron as a function of excess energy. Results are presented for ionization of H-like (dash-dotted red and short-dashed orange) and neutral (solid black and long-dashed green) neon, germanium, xenon, and uranium atoms by linearly polarized light. We can see that the energy dependence of the partial-wave cross section of H-like ions fulfils our expectation we gained from Fig. \[Fig.PotentialComparison\]. The cross sections of $s\rightarrow p\rightarrow d$ channel always dominates the $s\rightarrow p\rightarrow s$ channel and the two curves remain approximately parallel. Analogously to the total cross section, both channels can be considered constant in non-resonant energy region up to the proximity of the $1s\rightarrow 2p$ resonant energy. Similar behaviour can be seen in the case of neutral uranium. However, for neutral atoms with lower nuclear charge, we observe a competition of the two partial waves in near-threshold energy region. A drop of the dominant channel occurs and creates a minimum of the cross section, analogous to the Cooper minimum in single photon ionization process. The minimum is most pronounced for neon, for which the cross section of the $s\rightarrow p\rightarrow s$ channel is greater than the dominant $s\rightarrow p\rightarrow d$ channel in an energy region from the ionization threshold up to a crossing point of the channels at $\varepsilon=1.12$. This crossing of the ionization channels is present for atoms with nuclear charges in the range $Z=5-13$. Although for elements in this range other than neon, the crossing point lies in lower energies and the effects are thus weaker. In the top left part of each of the figures \[Fig.PartialWave\], the ratio of total cross section for ionization by circularly $\sigma^{\mathrm{circ}}$ and linearly $\sigma^{\mathrm{lin}}$ light are also presented. According to the known estimate $\sigma^{\mathrm{circ}} / \sigma^{\mathrm{lin}} \approx 3/2$ [@Lambropoulos/PRL:1972], the ratio should be always approximately equal to $3/2$ in non-resonant energy region. While this holds true for the H-like ions (dash-dotted red curve), in the case of neutral atoms (solid black curve), the screening effects result in a strong deviation from the estimated value. This follows directly from the discussion of partial waves above. Screening and relativistic effects {#Subsec.Effects} ---------------------------------- ![The scaling factor $\zeta$ as a function of nuclear charge for ionization of a $1s$ electron of neutral atoms ($Z=4-92$) by two linearly (top), circularly (middle), and unpolarized (bottom) photons at the excess energies $\varepsilon=1.05$ and $\varepsilon=1.40$. According to the non-relativistic scaling of H-like ions (dashed green), the cross section scales with $Z^{-6}$. The deviation from this scaling due to relativistic effects is clearly visible for H-like (dash-dotted red) as well as neutral (solid black and short-dashed blue) atoms. Moreover, further deviation of the scaling factor in low-$Z$ region is present for neutral atoms due to screening effects.[]{data-label="Fig.Effects"}](ScaleFactor_linear){width="\linewidth"} ![The scaling factor $\zeta$ as a function of nuclear charge for ionization of a $1s$ electron of neutral atoms ($Z=4-92$) by two linearly (top), circularly (middle), and unpolarized (bottom) photons at the excess energies $\varepsilon=1.05$ and $\varepsilon=1.40$. According to the non-relativistic scaling of H-like ions (dashed green), the cross section scales with $Z^{-6}$. The deviation from this scaling due to relativistic effects is clearly visible for H-like (dash-dotted red) as well as neutral (solid black and short-dashed blue) atoms. Moreover, further deviation of the scaling factor in low-$Z$ region is present for neutral atoms due to screening effects.[]{data-label="Fig.Effects"}](ScaleFactor_circular){width="\linewidth"} ![The scaling factor $\zeta$ as a function of nuclear charge for ionization of a $1s$ electron of neutral atoms ($Z=4-92$) by two linearly (top), circularly (middle), and unpolarized (bottom) photons at the excess energies $\varepsilon=1.05$ and $\varepsilon=1.40$. According to the non-relativistic scaling of H-like ions (dashed green), the cross section scales with $Z^{-6}$. The deviation from this scaling due to relativistic effects is clearly visible for H-like (dash-dotted red) as well as neutral (solid black and short-dashed blue) atoms. Moreover, further deviation of the scaling factor in low-$Z$ region is present for neutral atoms due to screening effects.[]{data-label="Fig.Effects"}](ScaleFactor_unpolarised){width="\linewidth"} In previous subsections, we already saw that one needs to take screening effects into account for low- and medium-$Z$ elements. Moreover, in Ref. [@Koval/JPB:2003] it is shown that in two-photon ionization of H-like ions, the relativistic effects cannot be neglected for heavy atoms. It is, therefore, reasonable to expect similar behaviour for ionization of neutral atoms. It is the purpose of this subsection to show the relative strengths and nuclear charge dependences of both these effects as well as their contributions to the total cross section. In non-relativistic theory, the non-resonant cross section for the two-photon ionization of H-like ions in dipole approximation scales with the nuclear charge as $\sigma(Z,\omega Z^2)=\sigma(Z=1,\omega)Z^{-6}$ [@Zernik/PR:1964]. We will use the same way as in Ref. [@Koval/JPB:2003] and introduce so called scaling factor $\zeta$ to the above expression, i.e., $\sigma(Z,\omega Z^2)=\zeta(Z)\sigma(Z=1,\omega)Z^{-6}$. The deviation of the scaling factor from the value $1$ then represents various effects arising from the full relativistic description and/or the interelectronic interaction. For non-relativistic $E1E1$ calculation in Coulomb potential, the scaling factor is $\zeta(Z)=1$ for all $Z$ values and is almost independent of the excess energy in the non-resonant region. Figure \[Fig.Effects\] shows the plot of the scaling factor $\zeta (Z)$ as a function of nuclear charge for two-photon ionization by linearly, circularly, and unpolarized light. The results are shown for non-relativistic (dashed green) and relativistic (dash-dotted red) calculations for ionization of H-like ions as well as relativistic calculation for ionization of neutral atoms at $\varepsilon=1.05$ (solid black) and $\varepsilon=1.40$ (long-dashed blue) excess energies. We can see that for neutral atoms, there are two distinct deviations of the scaling factor from the constant non-relativistic value. One of the deviations stretches between the medium- and high-$Z$ region and is also present for the case of hydrogenlike atoms. The second deviation lies in the low-$Z$ region and is present only for the ionization of neutral atoms. Let us start with the deviation in the low-$Z$ region. This deviation results from the interelectronic interaction, which decrease the electron binding energies and as a result, increase the total cross section. We can see, that this is indeed the case for the $\varepsilon=1.40$ excess energy, where the screening effects increase the total cross section in the low-$Z$ region. This increase rapidly weakens with increasing nuclear charge as we would expect. However, for $\varepsilon=1.05$, the screening effects result in decrease of the cross section, with a maximum at $Z=10$. This trough in the scaling factor directly reflects the decrease of cross section we have seen in Figs. \[Fig.PotentialComparison\] and \[Fig.PartialWave\]. The sharpness of the trough is a result of the discrete values of the nuclear charge $Z$ values. For photon energies exceeding the ionization threshold by more than $15\%$, i.e. $\varepsilon > 1.15$, the trough disappears. From figure \[Fig.Effects\], we can see that the screening effects are strongest for the case of ionization by circularly polarized light. We can understand this from the partial-wave analysis in Sec. \[Subsec.PartialWave\]. If we look at the partial wave cross sections for neon in Fig. \[Fig.PartialWave\], we can see that the dominant $s\rightarrow p \rightarrow d$ channel drops strongly near the ionization threshold. For ionization by linearly and unpolarized light, it is partially balanced by the increase of the $s\rightarrow p\rightarrow s$ channel. However, as explained before, for ionization by circularly polarized light the $E1E1$ transition allows only the dominant $s\rightarrow p \rightarrow d$ channel to be open. Therefore, due to the lack of the final $s-$ partial wave, the drop of the cross section does not get balanced out and the screening effects become stronger. ![The total cross section for two-photon ionization for linearly (top) and circularly (bottom) polarized light as a function of nuclear charge. The cross section is plotted for two excess energy values, $\varepsilon=1.05$ (solid black) and $\varepsilon=1.4$ (long-dashed green). The $Z^{-6}$ scaling law (short-dashed red) and experimental values for germanium [@Tamasaku/NP:2014] and zirconium [@Ghimire] atoms are also shown.[]{data-label="Fig.ZScaling"}](Scaling_linear){width="\linewidth"} ![The total cross section for two-photon ionization for linearly (top) and circularly (bottom) polarized light as a function of nuclear charge. The cross section is plotted for two excess energy values, $\varepsilon=1.05$ (solid black) and $\varepsilon=1.4$ (long-dashed green). The $Z^{-6}$ scaling law (short-dashed red) and experimental values for germanium [@Tamasaku/NP:2014] and zirconium [@Ghimire] atoms are also shown.[]{data-label="Fig.ZScaling"}](Scaling_circular){width="\linewidth"} The second deviation of the scaling factor in medium- and high-$Z$ region in the Fig. \[Fig.Effects\] comes from the relativistic effects. The importance of these effects continuously grows with increasing nuclear charge $Z$. We can also see that unlike screening effects, relativistic effects are independent of polarization. This means that relativistic effects influence all partial waves in a same way. For ionization of uranium by light of any polarization, the relativistic effects decrease the total cross section by about a factor of two. We would expect, that the relativistic effects would be stronger for the ionization of hydrogenlike ions than for ionization of neutral atoms, since the electron binding energies of hydrogenlike ions are higher. However, from Fig. \[Fig.Effects\], we can see, that the deviation of the scaling factor (and therefore the cross section itself) for neutral atoms due to relativistic effects is similar as for H-like ions. Comparison with experiment {#Subsec.Zdependence} -------------------------- Due to the relativistic and screening effects, corrections to the non-resonant two-photon ionization scaling law $\sigma(Z)=\sigma(Z=1)Z^{-6}$ increase in complexity. The magnitude of these effects depends mainly on the nuclear charge but screening effects also depend on the incident photon energies and polarizations. That is why we present the $Z$-dependence of the total cross section in addition to the scaling factor given in previous subchapter. Figure \[Fig.ZScaling\] shows calculated cross sections for elements in the range $Z=4-92$ for two energies; $\varepsilon=1.05$ (solid black) and $\varepsilon=1.40$ (long-dashed green) as well as the scaling law (short-dashed red). Total cross sections for other photon energies $1.05 < \varepsilon < 1.40$ lay in between the two corresponding lines in Fig. \[Fig.ZScaling\]. The cross section difference between the two energies arises due to the screening effects as explained before. Figure \[Fig.ZScaling\] also shows experimental values for the $K$-shell ionization of neutral Ge and Zr atoms. We can see that our result for Ge is close to the experimental value as well as for Zr, which lies within the experimental uncertainty. However, in another experiment, Doumy, *et al.* [@Doumy/PRL:2011] measured the two-photon ionization of heliumlike Ne to be $7\times 10^{-54}$ cm$^4$ s. Theoretical calculations [@Sytcheva/PRA:2012; @Novikov/JPB:2001; @Koval/Dissertation] of this cross section resulted in a discrepant value, lower by about three orders of magnitude. We applied our formalism for the case of Ne$^{8+}$ as well, and obtained a cross section of 3.1$\times 10^{-57}$ cm$^4$ s which is in an agreement with previous calculations [@Sytcheva/PRA:2012; @Koval/Dissertation; @Novikov/JPB:2001]. Thus, the three orders of magnitude deviation obtained suggests a resonant enhancement of the cross section and can be explained by broader spectral bandwidth of the FEL employed. Summary and outlook {#Sec.SummaryAndOutlook} =================== The non-resonant two-photon ionization of neutral atoms has been described in fully relativistic theory based on second-order perturbation theory and Dirac equation. Using the independent particle approximation and particle-hole formalism, the many-electron transition amplitude describing the electron-photon interaction has been simplified to one-electron amplitude. An expression of the total two-photon ionization cross section has been obtained using the framework of density matrix theory and the transition amplitude. Detailed calculations of the total cross section have been carried out for ionization of neon, germanium, xenon, and uranium atoms using three screening potentials. Our results show that both relativistic as well as screening effects need to be considered in the calculation of two-photon ionization cross section. Relativistic effects significantly decrease the total cross section for heavy atoms, for the case of uranium, they decrease the cross section by a factor of two. Screening effects are highly sensitive to the photon energy and polarization as well as to the nuclear charge of the atom. In general, screening effects increase the cross section for low-$Z$ atoms by a factor of up to 1.5. However, for near-threshold photon energies, we observe a minimum in the total cross section which has pure screening origin. Due to a single allowed ionization channel, screening effects are most pronounced for ionization by circularly polarized light. For ionization of Ne, the cross section drops by a factor of three in the near-threshold energy region. Both, the relativistic as well as the screening effect will likely affect the photoelectron angular distribution of the two-photon ionization of neutral atoms. Therefore, it would be of great interest to use the theoretical formalism described above to also investigate the angular distribution, especially in the case, where the minimum of the cross section occurs. This will be the concern of our further study.\ J.H. acknowledges the support from the Helmholtz Institute Jena. This work has been supported by the BMBF (Grant No. 05K13VHA). [28]{} J. A. Duncanson, Jr., M. P. Strand, A. Lindgard, and R. S. Berry, Phys. Rev. Lett. **37**, 987 (1976). A. Siegel, J. Ganz, W. Bußert, and H. Hotop, J. Phys. B **16**, 2945 (1983). A. Dodhy, R. N. Compton, and J. A. D. Stockdale, Phys. Rev. Lett. **54**, 422 (1985). Z.-M. Wang and D. S. Elliot, Phys. Rev. A **62**, 053404 (2000). C. Pellegrini, A. Marinelli, and S. Reiche, Rev. Mod. Phys. **88**, 015006 (2016). V. Richardson, *et al.*, Phys. Rev. Lett. **105**, 013001 (2010). G. Doumy, *et al.*, Phys. Rev. Lett. **106**, 083002 (2011). K. Tamasaku, *et al.*, Nat. Photon. **8**, 313 (2014). S. Ghimire, *et al.*, accepted in Phys. Rev. A. W. Zernik, Phys. Rev. **135**, A51 (1964). P. Koval, S. Fritzsche, and A. Surzhykov, J. Phys. B **36**, 873 (2003). P. Koval, S. Fritzsche, and A. Surzhykov, J. Phys. B **37**, 375–388 (2004). P. Koval, Two-photon ionization of atomic inner-shells, University of Kassel, (2004). V. Florescu, O. Budriga, and H. Bachau, Phys. Rev. A **84**, 033425 (2011). V. Florescu, O. Budriga, and H. Bachau, Phys. Rev. A **86**, 033413 (2012). W. R. Johnson, Atomic Structure Theory Lectures on Atomic Physics (Springer, Berlin Heidelberg, 2007). D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988). J. Eichler and T. Stöhlker, Phys. Rep. **439** 1 (2007). K. Blum, Physics of Atoms and Molecules (Plenum Press, New York, 1981). J. Sapirstein and W. R. Johnson, J. Phys. B **29**, 5213 (1996). V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. Lett. **93**, 130405 (2004). A. Surzhykov, *et al.*, Phys. Rev. A **81**, 042510 (2010). A. V. Volotka, A. Surzhykov, V. M. Shabaev, and G. Plunien, Phys. Rev. A **83**, 062508 (2011). A. V. Volotka, V. A. Yerokhin, A. Surzhykov, Th. Stöhlker, and S. Fritzsche, Phys. Rev. A **93**, 023418 (2016). F. Salvat, J. M. Fernandez-Varea, and W. Williamson Jr., Comput. Phys. Commun. **90**, 151 (1995). F. Salvat, J. D. Martinez, R. Mayol, and J. Parellada, Phys. Rev. A **36**, 467, (1987). P. Lambropoulos, Phys. Rev. Lett. **28**, 585 (1972). S. Novikov and A. Hopersky, J. Phys. B **34**, 4857 (2001). A. Sytcheva, S. Pabst, S.-K. Son, and R. Santra, Phys. Rev. A **85**, 023414 (2012).

--- address: '$^{1}$University of Sheffield, $^{2}$Imperial College London, $^{3}$University of Bern' author: - 'P. Stowell$^{1}$, S. Cartwright$^{1}$, L. Pickering$^{2}$, C. Wret$^{2}$, C. Wilkinson$^{3}$\' title: 'Neutrino Monte-Carlo Event Generators and Cross-section Data' --- Introduction ============ There are currently multiple neutrino event generators available, providing a large range of different interaction models to choose from when trying to construct a complete nuclear scattering model. The NUISANCE framework [@nuisance] has recently been released to try and provide the neutrino community with the necessary tools to help select and tune these different generators by making comparisons to existing cross-section data. The structure of NUISANCE allows multiple generators to be read into the same analysis routines, enforcing consistency of the signal selections applied and ensuring any differences observed are due to the underlying physics assumptions in each model being compared. Providing an interface between generator reweighting engines and ROOT’s minimiser libraries [@ref:ROOT] has created a powerful tool that can be used to automatically tune generator models to multiple datasets by scanning the parameter space and minimising a likelihood test statistic. In these proceedings preliminary tuning results of several different components of the NEUT [@neutgen] and NuWro [@nuwrogen] generator models are compared. Bubble Chamber Tunings ====================== Fermi motion and binding energy effects are small for deuterium targets. This allows good constraints to be placed on neutrino-nucleon interaction models through fits to deuterium-filled bubble chamber data. Cross-section and event rate distributions from charged-current (CC) neutrino quasi-elastic (QE) and charged pion production ($1\pi^{+}$) measurements from the ANL, BNL, BEBC, and FNAL experiments, were chosen for these studies [@ref:anlccqe; @ref:anlccres1; @ref:anlccres2; @ref:bnlccqe; @ref:bnlccres1; @ref:bnlccres2; @ref:fnalccqe; @ref:bebcccqe]. The measured cross-section and event rate distributions were digitised and added as dataset comparison routines into the NUISANCE framework so the data could be included in joint likelihood fits of the generator models. The nominal NEUT and NuWro free nucleon models for QE and $1\pi^{+}$ scattering of free nucleons were chosen as candidate models to be tuned with the NUISANCE framework. These generators both use the Llewellyn-Smith[@ref:llewellynsmith] (LS) model to describe quasi-elastic scattering, and the Rein-Sehgal[@reinseghal] (RS) model to describe pion production, their main difference being that NEUT simulates multiple nuclear resonances using the RS model, whereas NuWro simulates only the $\Delta(1232)$ component, relying on a $\Delta$/DIS extrapolation to populate the higher order resonances. Both generators have “reweight engines” allowing the user to make model predictions over a range of model parameters after event generation [@nubros]. These reweight engines were used to study variations in a set of free model parameters. For the QE model only the quasi-elastic axial mass parameter ($M_{A}^{QE}$) was treated as free. In the pion production model both the resonant axial mass ($M_{A}^{1\pi}$), and the axial coupling constant ($C_A^5$) were treated as free. The overall similarity between the two generators provides an additional validation test of the tuning results. Since both generators use the LS and RS models it is expected they should obtain similar best fit results when tuning to distributions where the majority of events originate from low hadronic mass events ($W < 1.4$ GeV). The published flux distributions were used to generate charged-current events in each generator. The target in each case was considered to be a free proton and neutron to give a combined deuteron cross-section without binding energy or Fermi motion effects. From these Monte-Carlo (MC) samples, events were selected that matched the published signal selections and normalised to give matching cross-section predictions. In the cases where only event rate information was given the predictions were normalised to match the integrated event rate in the data. An additional correction was applied to the QE model predictions to convert them from free nucleon predictions to that for a bound deuteron [@ref:singh]. This correction, applied as a function of true $Q^{2}$, was found to have a negligible effect on the fits but was left in to maintain consistency with tuning studies shown in the original publications [@ref:anlccqe]. ------------ -------------------- ---------------- ---------------------- ----------------- ---------------- Model $M_{A}^{QE}$ (GeV) $\chi^{2}$/DOF $M_{A}^{1\pi}$ (GeV) $C_{A}^{5}$ $\chi^{2}$/DOF NEUT 5.3.6 $1.04 \pm 0.03$ 159.8 / 146 $0.89 \pm 0.04$ $1.02 \pm 0.05$ 102.8 / 102 NuWro v12 $1.03 \pm 0.03$ 154.4 / 146 $0.92 \pm 0.04$ $1.04 \pm 0.05$ 111.2 / 102 ------------ -------------------- ---------------- ---------------------- ----------------- ---------------- : \[tab:tuningbc\] Tuning results for the free nucleon interaction models in the NEUT and NuWro Monte-Carlo generators. ![\[fig:ccqebc\]Comparisons of the best fit predictions in the NEUT and NuWro generators to CCQE data. (left) ANL $E_{\nu}$ cross-section data used to constrain the ANL CCQE normalisation. (right) BNL $Q^{2}$ event rate data used to place an additional shape constraint on the BNL cross-section predictions. ](plot_BCCCQE_ANL_CCQE_XSec_1DEnu_nu_comp_stats.pdf "fig:"){width="40.00000%"} ![\[fig:ccqebc\]Comparisons of the best fit predictions in the NEUT and NuWro generators to CCQE data. (left) ANL $E_{\nu}$ cross-section data used to constrain the ANL CCQE normalisation. (right) BNL $Q^{2}$ event rate data used to place an additional shape constraint on the BNL cross-section predictions. ](plot_BCCCQE_BNL_CCQE_Evt_1DQ2_nu_comp_stats.pdf "fig:"){width="40.00000%"} A joint likelihood was formed within the NUISANCE framework by first selecting a single distribution from each publication to place a constraint on the normalisation for that measurement (e.g. CCQE $\sigma(E_{\nu}))$. An additional shape-only likelihood was then added for each remaining distribution in the measurement itself (e.g. CCQE $Q^{2}$ Event Rates) to form a total likelihood for that measurement. The purpose of adding these shape-only terms is to minimise any bias that may be introduced when a model is tuned to only a single distribution, whilst trying to avoid issues with over-counting placing too strong a constraint on the total cross-section. These likelihoods for each dataset were added uncorrelated to form a total likelihood for the chosen models in the study. The NUISANCE tuning framework was set up to automatically scan the parameter space until a best fit parameter set was found. The results can be seen in Table \[tab:tuningbc\], with examples of the best fit predictions for both generators in Figs. \[fig:ccqebc\] and \[fig:ccresbc\]. Both generators were found to be capable of describing the data with an acceptable goodness-of-fit. The disagreement seen between the generators in the quasi-elastic $\chi^2$ results are due to slight differences in the generated MC statistics, whereas the differences in the $1\pi^{+}$ fits arise from fundamental differences in the generator models themselves. This can be seen in Fig. \[fig:ccresbc\] where higher order resonances can be seen contributing to the NEUT prediction at high angles introducing a difference between the two generator predictions. In both cases the differences are not large enough to significantly shift the tuning results, with both generators finding best fit results in agreement with one another, providing a set of free nucleon parameters suitable for propagation to future nuclear tuning studies. ![\[fig:ccresbc\] Comparisons of the best fit predictions in the NEUT and NuWro generators to CC1$\pi$ data. (left) BNL $E_{\nu}$ cross-section data used to constrain the BNL CC1$\pi$ normalisation (right) ANL Adler Angle event rate data used to place an additional shape constraint on the BNL cross-section predictions. Effects of higher order resonances can be seen in the difference between NEUT and NuWro at low angles. ](plot_BCCCQE_BNL_CC1ppip_XSec_1DEnu_nu_comp.pdf "fig:"){width="40.00000%"} ![\[fig:ccresbc\] Comparisons of the best fit predictions in the NEUT and NuWro generators to CC1$\pi$ data. (left) BNL $E_{\nu}$ cross-section data used to constrain the BNL CC1$\pi$ normalisation (right) ANL Adler Angle event rate data used to place an additional shape constraint on the BNL cross-section predictions. Effects of higher order resonances can be seen in the difference between NEUT and NuWro at low angles. ](plot_BCCCQE_ANL_CC1ppip_Evt_1DcosmuStar_nu_comp.pdf "fig:"){width="40.00000%"} NuWro/NEUT LFG Tunings ====================== When extending neutrino interaction models to nuclear targets, an inclusive generator model must also consider how the presence of the nuclear medium can modify the interaction. We consider the latest model available in the NEUT 5.3.6 generator, consisting of a Relativistic Fermi Gas[@smithmoniz] (RFG) with relativistic RPA correction and a Nieves multi-nucleon model [@nieves] (NEUT RFG+Nieves). The choice of nuclear spectral function to model nucleon binding energy and Fermi motion introduces a problem in generator model tuning, since multiple discrete models are available. It is believed that any direct measurements of quasi-elastic scattering are also likely to be sensitive to additional multi-nucleon interaction channels (2p2h) that can produce final states of similar topologies to true quasi-elastic scattering inside the nucleus [@nieves]. For comparison we also consider two alternative models in the NuWro generator, a model with a local Fermi gas RPA correction and a Nieves 2p2h model (NuWro LFG+Nieves), and a RFG with a transverse enhancement model[@temmodel] (NuWro RFG+TEM). Each model was tested against MiniBooNE and MINERvA CCQE data in both neutrino and anti-neutrino runs [@ref:mbnu; @ref:mbnub; @ref:minnu; @ref:minnub]. Although the collaborations define their signal as “true CCQE interactions”, experience suggests that all four measurements are in fact sensitive to both CCQE and 2p2h interaction channels. Model predictions corresponding to each dataset were therefore produced by generating events with the published flux distribution and selecting only those events which originated from one of these two interaction channels. A joint sample likelihood for the study was defined using MiniBooNE $T_{\mu}-\cos\theta_{\mu}$ data with shape-only uncorrelated errors and a floating normalisation, and MINERvA $Q^{2}_{QE}$ data with full covariance between neutrino and antineutrino distributions, matching the method used in Ref. 2. To look at variations in both of the interaction channels, the quasi-elastic axial mass (alters only CCQE interactions) and 2p2h normalisation (alters only 2p2h interactions) were treated as free parameters that could could be changed to improve agreement between the data and MC. The $\chi^{2}$ values shown in Table \[tab:ccqeresults\] are unrealistically small, because the MiniBooNE 2D distribution public data release does not provide bin-to-bin correlations. When varying both parameters freely similar results were found for all three models, an inflation of the axial mass away from the bubble chamber tuning result, and a large suppression of the 2p2h cross-section normalisation compared to the nominal prediction. Both parameters were estimated to be highly correlated when using MINUIT’s HESSE[@ref:hesse] routine to estimate parameter errors, a feature of the strong shape-constraint that the MINERvA dataset places on the fit. The use of a local Fermi gas model was insufficient to relieve the tensions seen in previous joint fit studies to this data and a significant model variation is likely needed to relieve the tensions whilst still maintaining consistency with other theoretical and experimental constraints. One significant problem with this method of tuning individual interaction channels to this data is that an unknown fraction of pion-less delta decay events was subtracted from the each distribution, directly by the MiniBooNE collaboration in their background subtraction procedure, and indirectly by MINERvA in their cut on their recoil energy deposited inside the detector outside the interaction vertex. If reliable constraints on free cross-section parameters for nuclear targets are to be extracted, a series of dedicated tuning studies using more inclusive signal definitions with minimal model-dependent background corrections is required. Model $M_A$ (GeV) 2p2h Norm (%) $\chi^{2}$/DOF ------------------ ----------------- ----------------- ---------------- -- NuWro LFG+Nieves $1.16 \pm 0.03$ $8.3 \pm 11.9$ 100.74 / 229 NuWro RFG+TEM $1.15 \pm 0.03$ $21.3 \pm 12.5$ 93.62 / 229 NEUT RFG+Nieves $1.14 \pm 0.03$ $25.5 \pm 12.4$ 106.25 / 229 : \[tab:ccqeresults\] Tuning results for the NEUT and NuWro CCQE+2p2h models when compared in joint fits to MiniBooNE and MINERvA quasi-elastic cross-section data. {width="35.00000%"} {width="35.00000%"} MINERvA CC-inclusive comparisons ================================ The MINERvA collaboration has attempted to study the presence of nuclear effects in neutrino carbon interactions directly through the extraction of both the 3-momentum transfer ($q_3$) and hadronic recoil energy for a given event [@minervadata]. The variable “energy available” ($E_{av}$) is defined as the sum of kinetic energy of protons and charged pions, and the total energy of neutral pions, electrons, and photons, leaving the nucleus. Subtracting the muon energy from the observed energy deposited around the vertex allows a CC-inclusive event selection to be unfolded into a differential cross-section measurement in terms of $E_{av}$ and $q_{3}$. Comparisons between this data and GENIE have shown disagreement in the “dip” region at high $q_{3}$ between the quasi-elastic and resonance peaks ($0.4 < q_{3}/\mbox{GeV} < 0.6$) . Similar differences between model predictions and data have been observed by the NOvA collaboration when studying hadronic recoil energy, and it has been suggested that changes to how we model 2p2h interactions could relieve this tension. For comparison the best fit results from the NEUT and NuWro tunings to both bubble chamber and carbon measurements are compared to this “recoil energy data” in Fig. \[fig:minervaneutnuwro\]. Simple variations in the axial mass and 2p2h normalisation are found to be incapable of filling in the disagreement between the data and MC, but the large shape disagreement in the “dip” region is significantly smaller for NuWro as a result of using a local Fermi gas model. Since the signal definition is CC-inclusive and extremely sensitive to final state particle multiplicities, it is also possible to create similar predictions through multiple smaller variations of different features of the inclusive generator model. For example, Fig. \[fig:neutpnnn\] shows the different contributions to the NEUT and 2p2h cross-section from pn and nn pairs. This fraction of these pairs currently has a reasonably large uncertainty assigned, and is just one of many examples of free parameters that could be used to sculpt the total CC-inclusive prediction to better match the data. The major complication of trying to use such a measurement on its own to understand where models may be deficient is that final state “recoil energy” variables will be extremely sensitive to final state interaction models. Changes in these models can cause events to migrate in “recoil energy” space making it difficult to disentangle which exact features of the model may be problematic. Since no measurements have been made in these kinematic variables in the past, it is difficult to tell in which of the many interaction channels or FSI model the tensions may really lie, and a full CC-inclusive model tuning with additional constraints from CC$0\pi$/CC$1\pi$ data may be required to extract reliable results from these recoil energy measurements. ![\[fig:minervaneutnuwro\] Comparison of the previous NEUT and NuWro tuning results to MINERvA low recoil scattering data. Shown are the NEUT predictions using the bubble chamber tuning (blue), the NEUT predictions using the CCQE tuning (red), the NuWro LFG+Nieves predictions with bubble chamber tuning (purple), and CCQE tuning (green). The dashed lines of matching colour show the predicted 2p2h contribution to the cross-section in each bin.](nufact_proc1_neutnuwro.pdf){width="68.00000%"} ![\[fig:neutpnnn\] Comparison of NEUT 2p2h pp/nn pair contributions in the $E_{av}$ variable. Each prediction has been scaled up by a factor of 10 so that its shape is clear. Shown in red and blue are the nn and pn contributions respectively. ](nufact_proc2_neutmec.pdf){width="68.00000%"} Conclusion ========== The NUISANCE tuning group has defined a publicly accessible framework that supports neutrino event generator tuning. Early studies of the NEUT and NuWro generator free nucleon models have found that both generators obtain consistent results when tuning to deuterium bubble chamber data. One weakness of these tunings is the lack of correlations across bubble chamber experiments, and future studies will look at correlating flux uncertainties in the studies to obtain more reliable best fit parameters. When compared to nuclear CCQE data from MINERvA and MiniBooNE, both generators were also found to produce very similar tuning results despite much clearer differences between the models investigated. The use of alternative spectral function definitions was found to make little difference in the observed suppression of the 2p2h normalisation, hinting that the difficulties in achieving good agreement between these experiments may lie with the cross-section extraction methods used to obtain the data itself. Finally, comparisons of the fit results to MINERvA CC-inclusive showed that more significant modifications to the full cross-section model are required to obtain a reasonable agreement with newer CC-inclusive cross-section data. Acknowledgments {#acknowledgments .unnumbered} =============== The author wishes to thank the UK STFC for supporting this work. References {#references .unnumbered} ========== [unsrt]{} P. Stowell, L. Pickering, C. Wilkinson, C. Wret, “NUISANCE Framework”, http://nuisance.hepforge.org/ Brun, Rene, and Fons Rademakers. Nucl. Instrum. Meth. A [**389.1**]{} (1997) Y. Hayato, Acta Phys. Polon. B [**40**]{}, 2477 (2009). C. Juszczak, Acta Phys. Polon. B [**40**]{}, 2507 (2009) S. J. Barish [*et al.*]{}, Phys. Rev. D [**16**]{}, 3103 (1977). Radecky et al. Phys Rev D, 3rd series, volume 25, number 5, 1 March 1982, p 1161-1173 Thpr: Derrick et al. Phys Rev D, Vol 23, Number 3, 1 Feb 1981, p 569-575 N. J. Baker [*et al.*]{}, Phys. Rev. D [**23**]{}, 2499 (1981). Kitagaki et al. Phys Rev D, Vol 34, Number 9, 1 November 1986, p 2554-2565 K. Furuno et al., (NuInt02), UC Irvine, U.S.A., KEK Preprint 2003-48 T. Kitagaki [*et al.*]{}, Phys. Rev. D [**28**]{}, 436 (1983). D. Allasia [*et al.*]{}, Nucl. Phys. B [**343**]{}, 285 (1990). Smith, CH Llewellyn Physics Reports 3 5 1972 D. Rein and L. M. Sehgal, Annals Phys.  [**133**]{}, 79 (1981). L. Pickering, P. Stowell and J. Sobczyk, arXiv:1610.07053 \[hep-ex\]. S. K. Singh, Nucl. Phys. B [**36**]{}, 419 (1972). R. A. Smith and E. J. Moniz, Nucl. Phys. B [**43**]{}, 605 (1972) J. Nieves, I. Ruiz Simo and M. J. Vicente Vacas, Phys. Rev. C [**83**]{}, 045501 (2011) A. Bodek, H. S. Budd and M. E. Christy, Eur. Phys. J. C [**71**]{}, 1726 (2011) A. A. Aguilar-Arevalo [*et al.*]{} \[MiniBooNE\], Phys. Rev. D [**81**]{}, 092005 (2010) G. A. Fiorentini [*et al.*]{} \[MINERvA\], Phys. Rev. Lett.  [**111**]{}, 022502 (2013) A. A. Aguilar-Arevalo [*et al.*]{} \[MiniBooNE\], Phys. Rev. D [**88**]{}, no. 3, 032001 (2013) L. Fields [*et al.*]{} \[MINERvA\], Phys. Rev. Lett.  [**111**]{}, no. 2, 022501 (2013) J., F., and M. I. N. U. I. T. Roos. Comp. Physics Comm. 10.6 (1975): 343-367. P. A. Rodrigues [*et al.*]{} \[MINERvA Collaboration\], Phys. Rev. Lett.  [**116**]{}, 071802 (2016)

--- abstract: 'We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex/holomorphic structures on them introduced by A. Schwarz. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommutative tori. Also, we present the explicit composition formula of morphisms, which in fact depends on the noncommutativity.' --- YITP-05-63\ October, 2005 2.5 cm [**Categories of holomorphic line bundles on higher dimensional noncommutative complex tori**]{}\ [ ]{}\ [ Hiroshige Kajiura [^1]\ [ ]{}\ *Yukawa Institute for Theoretical Physics, Kyoto University\ Kyoto 606-8502, Japan\ *]{} Introduction ============ In this paper, we propose a way to construct differential graded (DG) categories of finitely generated projective modules over higher dimensional noncommutative complex tori. Also, we give explicit examples of this construction as noncommutative deformations of the DG categories of holomorphic line bundles over higher dimensional complex tori. The motivation of the present paper is to construct an explicit example of the noncommutative deformations of a complex manifold which may be thought of as one of the extended deformations of a complex manifold proposed by Barannikov-Kontsevich [@BK]. For a $n$-dimensional complex or a Calabi-Yau manifold $M$, the extended deformation is defined by a deformation $\epsilon\in\g^1$ of the Dolbeault operator $\bpart:\g^k\to\g^{k+1}$ such that $(\bpart+\epsilon)^2=0$, where $\g:=\oplus_{k=0}^n\g^k$ is the graded vector space given by $\g^k:=\oplus_{k=p+q} \Gamma(M,\wedge^p TM\otimes \wedge^q {\bar TM}^*)$. The degree one graded piece consists of $\g^1=\Gamma(M,\wedge^2 TM)\oplus \Gamma(M,TM\otimes {\bar TM}^*)\oplus \Gamma(M,\wedge^2 {\bar TM}^*)$. Namely, it defines an extended deformation of the usual complex structure deformation $\epsilon\in\Gamma(M,TM\otimes {\bar TM}^*)$. In particular, the deformation corresponding to $\epsilon\in\Gamma(M,\wedge^2 TM)$ is called the [*noncommutative deformation*]{} of the complex manifold $M$. On the other hand, examples of the models of noncommutative deformation should be constructed so that we can see how it is noncommutative, as in the case of the deformation quantization [@Ko]. A candidate of it may be to consider an algebra deformation of $M$. Let $V:=\oplus_{k=0}^n V^k$ be a graded vector space given by $V^k:=\Gamma(M,\wedge^k {\bar TM})$. This $V$ has a natural graded commutative product $\cdot: V^k\otimes V^l\to V^{k+l}$ together with a differential given by the Dolbeault operator $\bpart:V^k\to V^{k+1}$. Then, $(V,\bpart,\cdot)$ forms a DG algebra. A deformation of this DG algebraic structure may describe the noncommutative deformation corresponding to $\epsilon\in\Gamma(M,\wedge^2 TM)$. We can also replace this DG algebra with the DG category of holomorphic vector bundles or coherent sheaves on $M$, where the DG algebra $V$ should be included in the DG category as the endomorphism algebra of the structure sheaf on $M$. This algebraic or categorical approach can be thought of as a part of the spirits of [*homological mirror symmetry*]{} by Kontsevich [@mirror]. Actually, (topological) string theory suggests considering the graded vector spaces $\g$ and $V$; the algebraic structures on their cohomologies are defined as the [*closed [*[@W1]*]{} and open [*[@W2]*]{} string amplitudes*]{}, respectively, in the B-twisted topological string theory (B-model). The DG category above corresponds to physically what is called a [*D-brane category*]{} (see [@Dbcat]); the objects are the D-branes, the morphisms are the open strings between the D-branes in the B-model. Thus, the approach above is physically to construct an open string model instead of the closed string model. It is natural from viewpoints of both string theory (see [@infty; @thesis]) and deformation theory (see [@SS; @FOOO] and references therein) that DG categories constructed as above should be treated in the category of $A_\infty$-categories, where equivalence between $A_\infty$-categories should be defined by the homotopy equivalence. Now, let us concentrate on a $n$-dimensional complex tori $(M:=T^{2n},\bpart)$. It would be easy if its noncommutative deformation corresponding to $\epsilon\in\Gamma(T^{2n},\wedge^2 TM)$ can be described by the DG algebra $(V,\bpart,*)$, where $ * :V^k\otimes V^l\to V^{k+l}$ is the natural extension of the Moyal product on $V^0$, the space of functions on $T^{2n}$, defined by the Poisson bivector $\epsilon\in\Gamma(T^{2n},\wedge^2 TM)$. However, as far as one identifies homotopy equivalent DG algebras with each other, all these DG algebras turn out to be equivalent, being independent of the noncommutative parameter $\epsilon$. In fact, one can easily show that the DG algebra $(V,\bpart,*)$ is formal, , homotopy equivalent to a graded algebra on the cohomology $H(V,\bpart)$, and in particular the product on the cohomology $H(V,\bpart)$ is independent of $\epsilon$. These results follow from the fact that one can take a Hodge-Kodaira decomposition of the complex $(V,\bpart)$ so that the harmonic form is closed with respect to the product $*$. Therefore, in the same way as in the complex one-tori (= real two-tori) case [@foliation; @PoSc; @KimKim; @nchms], we should include nontrivial vector bundles which are compatible with the complex structure in some sense. In the real two-tori case, one can construct a DG category of holomorphic vector bundles, in the sense of [@Stheta], over a noncommutative two-torus [@PoSc; @nchms], where holomorphic vector bundles are described by DG-modules. In particular, the derived category of the DG-category is independent of the noncommutativity parameter $\theta\in{{\mathbb{R}}}$ [@PoSc]. Though noncommutative deformation of complex tori in this approach is relatively well understood for complex one-tori case, its higher dimensional extension is quite nontrivial and interesting especially from the viewpoint of the extended deformation [@BK; @Kap; @gual; @oren]. However, in this higher dimensional case, a different problem will arise. Even though we start with a DG-module of a holomorphic vector bundle, its noncommutative deformation might not be described by a DG module. There may be several ways to resolve this problem. Our idea in this paper is that we treat the deformed holomorphic vector bundles as [*curved differential graded (CDG)*]{}-modules over $V$. The important point is that even though the deformed ones are not DG modules, the space of morphisms may be equipped with a differential. Namely, in the context of DG categories, the cohomology should be defined not on the objects but on the [*morphisms*]{} between the objects. Thus, one may be able to extract finite dimensional graded vector spaces as the cohomologies of the morphisms. According to such a spirit, we construct DG categories consisting of some of these CDG modules of deformed holomorphic vector bundles on higher dimensional noncommutative tori. We remark that this procedure is just the same as the DG categories of B-twisted topological Landau-Ginzburg model by [@kl:1; @kl:2; @at] and also similar to the procedure by FOOO [@FOOO] in the mirror dual A-model side. It would also be interesting to construct a triangulated category via the twisted complexes as is done in [@at; @at-wdg; @Block]. Our starting point is based on A. Schwarz’s framework of noncommutative supergeometry [@S-Qalg] and noncommutative complex tori [@Stheta; @Stensor]. In [@S-Qalg], a CDG-algebra [@Pos] is re-studied and applied to noncommutative geometry under the name a [*Q-algebra*]{}, where modules over a Q-algebra is discussed. On the other hand, in [@Stheta], a complex structure is introduced on a real $2n$-dimensional noncommutative torus $T^{2n}_\theta$, and a holomorphic structure on the [*Heisenberg modules*]{}, noncommutative analogs of vector bundles, over $T^{2n}_\theta$ is defined. Then, our set-up can be thought of as an application of the noncommutative supergeometry [@S-Qalg] to the theory of holomorphic Heisenberg modules [@Stheta]. This set-up provides us with explicit descriptions of noncommutative models. Though one of our motivation comes from Fukaya’s noncommutative model of Lagrangian foliations on symplectic tori [@f-nc] and their mirror dual, our approach in this paper is different from the one since we deal with the Heisenberg modules which are [*finitely generated*]{} projective modules over noncommutative tori. For recent papers, see [@BBK] for another approach to noncommutative complex tori and the set-up in [@Block] which should be closer to ours. The construction of this paper is as follows. In section \[sec:CDG\], we recall the definitions of CDG algebras [@Pos], CDG modules and CDG categories. The notion of modules over a Q-algebra is more general than that of the CDG-modules over a CDG algebra. However, for our purpose, it is enough to consider CDG modules since we begin with Heisenberg modules with a [*constant*]{} curvature connections. In section \[sec:CDGtori\], we construct CDG categories of Heisenberg modules over noncommutative tori with complex structures. In particular, we propose a way to obtain DG categories as full subcategories of the CDG categories. Along the general strategy in section \[sec:CDGtori\], we construct CDG categories of holomorphic line bundles over noncommutative complex tori and the DG categories as their full subcategories in section \[sec:three\]. In subsection \[ssec:comm\], we construct the CDG category on a commutative complex tori. In this case, the CDG category is exactly a DG category. In subsection \[ssec:nc\], we consider three types of noncommutative deformations of the DG category as CDG categories. Then, we obtain DG categories as the full subcategories of the CDG categories. Furthermore, we present the composition formula of the zero-th cohomologies of the DG categories explicitly. The structure constants of the compositions in fact depend on the noncommutative parameters, which implies that the DG categories or the triangulated/derived categories of them depend on the noncommutative parameters. These results can be thought of as generalizations of complex one-dimensional case [@foliation; @PoSc; @KimKim; @nchms] and also a complex two-tori case [@KimLee; @KimKim2] (in the case that the structure constant of the composition is not deformed by the noncommutative parameter). Also, from a string theory or homotopy algebraic point of view, these deformations should corresponds to deformations of an $A_\infty$-structure as weak $A_\infty$-algebras discussed in the context of open-closed homotopy algebras (OCHAs) [@ocha] (see also [@HLL]). [**Notations**]{}: In this paper, any (graded) vector space stands for the one over the field $k={{\mathbb{C}}}$. We use indices $i,j,\cdots$ for both the ones which run over $1,\cdots,d=2n$ and the ones which run over $1,\cdots,n$, where $n$ and $d=2n$ are the complex and the real dimension of a noncommutative torus. [**Acknowledgments**]{}:  I would like to thank O. Ben-Bassat, A. Kato, A. Takahashi and Y. Terashima for valuable discussions and useful comments. The author is supported by JSPS Research Fellowships for Young Scientists. CDG algebras, CDG modules and CDG categories {#sec:CDG} ============================================ A [*curved differential graded (CDG) algebra*]{} $(V, f, d, m)$ consists of a ${{\mathbb{Z}}}$ (or ${{\mathbb{Z}}}_2$) graded vector space $V=\oplus_{k\in{{\mathbb{Z}}}} V^k$, where $V^k$ is the degree $k$ graded piece, equipped with a degree two element $f\in V^2$, a degree one differential $d :V^{k}\raw V^{k+1}$ and a degree preserving bilinear map $m : V^k\otimes V^{l}\raw V^{k+l}$ satisfying the following relations: $$\begin{aligned} & d(f)=0\ ,\label{df} \\ & (d)^2(v)= m(f,v) - m(v,f)\ ,\label{d2=f}\\ & d m(v,v')=m(d(v),v') +(-1)^{|v|} m(v,d(v'))\ , \label{leib}\\ & m(m(v,v'),v'') = m(v,m(v',v''))\ , \label{asso}\end{aligned}$$ where $|v|$ is the degree of $v$, that is, $v\in V^{|v|}$. Suppose that we have in addition a nondegenerate symmetric inner product $$\eta:V^k\otimes V^l\raw {{\mathbb{C}}}$$ of fixed degree $|\eta|\in{{\mathbb{Z}}}$ on $V$. Namely, the $\eta$ is nondegenerate, nonzero only if $k+l+|\eta|=0$, and satisfies $\eta(v,v')=(-1)^{kl}\eta(v',v)$ for $v\in V^k$ and $v'\in V^l$. Then, we call $(V,f,\eta,d,m)$ a [*cyclic CDG algebra*]{} if the following conditions hold: $$\begin{aligned} &\eta(d(v),v')+(-1)^{|v|}\eta(v,d(v'))=0\ , &\eta(m(v,v'),v'') =(-1)^{|v|(|v'|+|v''|)} \eta(m(v',v''),v)\ .\end{aligned}$$ \[defn:CDG\] A CDG algebra is identified with a weak $A_\infty$-algebra $(V,\{m_k:V^{\otimes k}\to V\}_{k\ge 0})$ with $m_0=f$, $m_1=d$, $m_2=\cdot$ and $m_3=m_4=\cdots=0$. This algebraic structure is what is called a Q-algebra introduced in the framework of noncommutative supergeometry in [@S-Qalg]. Also, a CDG algebra $(V,f,d,\cdot)$ with $f=0$ is a DG algebra, which is a (strict) $A_\infty$-algebra $(V,\{m_k\}_{k\ge 1})$ with $m_3=m_4=\cdots =0$. \[rem:CDG\] A [*right CDG module*]{} $(\cE,d^\cE,m^\cE)$ over a CDG algebra $(V,-f,d,m)$ is a ${{\mathbb{Z}}}$-graded vector space $\cE$ equipped with a degree one linear map $d^\cE:\cE\to \cE$ and a right action $m^\cE:\cE\otimes V\to \cE$ satisfying the following condition: for any $v,v'\in V$ and $v^\cE\in\cE$, $$\begin{split} & (d^\cE)^2(v^\cE)=m^\cE(v^\cE, f)\ ,\\ & d^\cE m^\cE(v^\cE,v)=m^\cE(d^\cE(v^\cE),v) +(-1)^{|\xi|}m^\cE(v^\cE,d(v))\ ,\\ & m^\cE(v^\cE,m(v,v'))=m^\cE(m^\cE(v^\cE,v),v')\ . \end{split}$$ In particular, if $f=0$, then $(\cE,d^\cE,m^\cE)$ is called a [*DG-module*]{} over a DG algebra $(V,d,m)$. The third condition is nothing but the condition that $\cE$ is a (graded) right module over $V$. In the category extension of CDG algebras, elements of a CDG algebra turns out to be morphisms in the CDG-category. A [*CDG category*]{} $\cC$ consists of a set of objects $\Ob(\cC)=\{a, b, \cdots\}$, a ${{\mathbb{Z}}}$-graded vector space $V_{ab}=\oplus_{k\in{{\mathbb{Z}}}} V^k_{ab}$ for each two objects $a$, $b$ and the grading $k\in{{\mathbb{Z}}}$, $f_a: {{\mathbb{C}}}\raw V^2_{aa}$ for each $a$, a differential $d :V^k_{ab}\raw V^{k+1}_{ab}$ and a composition of morphisms $m: V^k_{bc}\otimes V^l_{ab}\to V^{k+l}_{ac}$ satisfying the following relations: $$\begin{aligned} & d(f_a)=0\ ,\label{df-cat} \\ & (d)^2(v_{ab})=m(f_b,v_{ab}) - m(v_{ab},f_a)\ ,\label{d2=f-cat}\\ & d m(v_{bc},v_{ab})=m(d(v_{bc}),v_{ab}) +(-1)^{|v_{bc}|} m(v_{bc},d(v_{ab}))\ , \label{leib-cat}\\ & m(m(v_{cd},v_{bc}),v_{ab}) = m(v_{cd},m(v_{bc},v_{ab}))\ , \label{asso-cat}\end{aligned}$$ where $|v_{ab}|$ is the ${{\mathbb{Z}}}$-grading of $v_{ab}$, that is, $v_{ab}\in V_{ab}^{|v_{ab}|}$. Let $\eta$ be a nondegenerate symmetric inner product of fixed degree $|\eta|\in{{\mathbb{Z}}}$ on $V:=\oplus_{a,b}V_{ab}$. Namely, for each $a$ and $b$, $$\eta: V^k_{ba}\otimes V^l_{ab}\raw {{\mathbb{C}}}$$ is nondegenerate, nonzero only if $k+l+|\eta|=0$, and satisfies $\eta(V^k_{ba},V^l_{ab})=(-1)^{kl}\eta(V^l_{ab},V^k_{ba})$. In this situation, we call a CDG category with inner product $\eta$ a [*cyclic CDG category*]{} $\cC$ if the following conditions hold: $$\begin{aligned} &\eta(dv_{ab},v_{ab})+(-1)^{|v_{ab}|}\eta(v_{ab},dv_{ab})=0\ , \label{d-int}\\ &\eta(m(v_{bc}\otimes v_{ab}),v_{ca}) =(-1)^{{|v_{bc}|}(|v_{ab}|+|v_{ca}|)} \eta(m(v_{ab}\otimes v_{ca}),v_{bc})\ . \label{cyclic}\end{aligned}$$ Also, we call a cyclic CDG category $\cC$ a [*cyclic DG category*]{} if $f_a=0$ for any $a\in\Ob(\cC)$. \[defn:cDGcat\] A CDG category $\cC$ consisting of one object only is a CDG algebra. Similarly, for a fixed object $a\in\Ob(\cC)$, the CDG category structure of $\cC$ reduces to a CDG algebra $(V_{aa},f_a,d,m)$. On the other hand, if the space of morphisms $\cV=\oplus_{a,b}V_{ab}$ is thought of as a ${{\mathbb{Z}}}$-graded vector space, $(\cV,\eta,\oplus_{a\in\Ob(\cC)}f_a,d,m)$ can be regarded as a cyclic CDG algebra (see also [@S-Qalg]). Suppose that a CDG category $\cC$ has an object $o\in\Ob(\cC)$ such that $(V_{oo},\eta,d,m)$ forms a DG algebra and for any object $a\in\Ob(\cC)$ there exists a center ${\hat f_a}$ in $V_{oo}$ such that $$m(f_a,v_{oa})=m(v_{oa},{\hat f_a})\ .$$ Then, $(V_{oa}=:\cE_a,d,m)$ can be regarded as a CDG module over the cyclic CDG algebra $(V_{oo},\eta,-{\hat f_a},d,m)$. CDG modules and CDG categories on noncommutative tori {#sec:CDGtori} ===================================================== Higher dimensional noncommutative tori {#ssec:nctori} -------------------------------------- Let us consider an algebra generated by $U_i$, $i=1,\cdots, d$, with relations $$\label{ui} U_jU_k = e^{-2\pi\i\theta^{jk}}U_{k}U_{j} \ ,\qquad j,k=1,\cdots, d$$ for an antisymmetric $d\times d$ matrix $\theta:=\{\theta^{jk}\}$. Any element is spanned over ${{\mathbb{C}}}$ by elements $U_{\vec{m}}$, ${\vec{m}}= (m_1,\dots, m_d)\in{{\mathbb{Z}}}^d$, which are defined by $$U_{\vec{m}}:= U_{1}^{m_{1}}U_{2}^{m_{2}} \dots U_{d}^{m_{d}}e^{\pi\i\sum_{j<k} m_{j}m_{k}\theta^{jk}}\ .$$ The relation between $U_{\vec{m}}$ and $U_{\vec{m}'}$ becomes $$\label{un} U_{\vec{m}}U_{\vec{m}'} = e^{\pi\i\sum_{j,k}m_{j}\theta^{jk}m'_{k}} U_{\vec{m}+\vec{m}'}\ .$$ One can represent any element of this algebra as $$u=\sum_{\vec{m}\in{{\mathbb{Z}}}^d} u_{\vec{m}}U_{\vec{m}}\ ,\qquad u_{\vec{m}}\in{{\mathbb{C}}}\ .$$ For any element $u$ represented as above, an involution $*$ is defined by $$u^*:=\sum_{\vec{m}\in{{\mathbb{Z}}}^d} u_{\vec{m}}^* U_{\vec{m}}^*\ ,$$ where $u_{\vec{m}}^*$ is the complex conjugate of $u_{\vec{m}}$ and $U_{\vec{m}}^{*}:= U_{-\vec{m}}$. One can consider a subalgebra $T_\theta^d$ such that any element, again represented as $u=\sum_{{\vec{m}}} u_{\vec{m}}U_{\vec{m}}$, belongs to the Schwartz space $\cS({{\mathbb{Z}}}^d)$, that is, the coefficients $\{u_{\vec{m}}\}$ as a function on ${{\mathbb{Z}}}^d$ tend to zero faster than any power of $||{\vec{m}}||$. This algebra $T_\theta^d$ is in fact a $C^*$-algebra and called (the smooth version of) a [*noncommutative torus*]{} [@Rhigh; @KS]. There is a canonical normalized [*trace*]{} on $T_{\theta}^{d}$ specified by the rule $$\label{Tr} \operatorname{Tr}(u)=u_{\vec{m}=0}\ ,\qquad u=\sum_{{\vec{m}}} u_{\vec{m}}U_{\vec{m}}\ .$$ For $\theta = 0$ we can realize the algebra $T_{\theta}^{d}$ as an algebra of functions on a $d$-dimensional torus $T^{d}$. Then the trace (\[Tr\]) corresponds to an integral over $T^{d}$ provided the volume of $T^{d}$ is one. In order to define a connection on a module $E$ over noncommutative torus $T_{\theta}^{d}$ we shall first define a natural Lie algebra of shifts $\cL_\theta$ acting on $T_{\theta}^{d}$. The shortest way to define this Lie algebra is by specifying a basis consisting of derivations $\delta_j:T^d_\theta\to T^d_\theta$, $j=1,\dots, d$, satisfying $$\label{delta} \delta_{j} (U_{\vec{m}}) = 2\pi\i m_{j}U_{\vec{m}} \, .$$ For the multiplicative generators $U_{j}$ the above relation reads as $$\label{deltaij} \delta_{j}U_{k} = 2\pi\i \delta_{jk}U_{k} \, .$$ This derivations then span a $d$-dimensional abelian Lie algebra (over ${{\mathbb{C}}}$) that we denote $\cL_\theta$. A [*connection*]{} on a (right) module $E$ over $T_{\theta}^d$ is a set of operators $\nabla_{X}:E\to E$, $X\in\cL_\theta$ depending linearly on $X$ and satisfying $$\nabla_X(\xi\cdot u)=\nabla_X(\xi)\cdot u+\xi\cdot X(u)$$ for any $\xi\in E$ and $u\in T_{\theta}^{d}$. In general, connections whose curvature equals the identity endomorphism times a numerical tensor are called constant curvature connections. We express the [*constant curvature*]{} of a constant curvature connection $\nabla_i:=\nabla_{\delta_i}$, $i=1,\cdots, d$, as $$\label{F} [\nabla_i,\nabla_j] =-2\pi\sqrt{-1}F_{ij}\cdot\1_{\End_{T_{\theta}^{d}}(E)}\ , \qquad F_{ij}=-F_{ji}\in{{\mathbb{R}}}\ .$$ On a noncommutative torus, one can construct a class of finitely generated projective modules called [*Heisenberg modules*]{} (see [@Rhigh; @KS]). In fact, on any Heisenberg module there exists a constant curvature connection. They play a special role. It was shown by Rieffel ([@Rhigh]) that if the matrix $\theta^{ij}$ is irrational in the sense that at least one of its entries is irrational then any projective module over $T_{\theta}^{d}$ is isomorphic to a direct sum of Heisenberg modules. Heisenberg modules are applied to discuss the Morita equivalence of noncommutative tori. A noncommutative tori $T^d_\theta$ is Morita equivalent to $T^d_{\theta'}$ if [@RS] and only if [@S] $\theta'=g(\theta)$, $g\in SO(d,d,{{\mathbb{Z}}})$ (for more recent papers, see [@TW; @hLi; @Ell-Li]). Here, the $SO(d,d,{{\mathbb{Z}}})$ action on the space of skew symmetric matrices in $\Mat_d({{\mathbb{R}}})$ is defined by $$g(\theta):= (\cA\theta+\cB)(\cC\theta+\cD)^{-1}\ , \quad g:={\begin{pmatrix}}\cA & \cB \\ \cC & \cD {\end{pmatrix}}\in SO(d,d,{{\mathbb{Z}}})\ ,$$ where $SO(d,d,{{\mathbb{Z}}}):=\{g\in\Mat_{2n}({{\mathbb{Z}}})\ |\ g^tJg=J \}$ for $J:=\left({\begin{smallmatrix}}\0_n & \1_n \\ \1_n & \0_n{\end{smallmatrix}}\right)$. To establish this Morita equivalence, one may construct a $T^d_{\theta}$-$T^d_{g(\theta)}$ [*Morita equivalence bimodule*]{}, denoted by $P_{\theta\text{-}g(\theta)}$ (see [@Rhigh; @RS]). One can in fact construct the Morita equivalence bimodule $P_{\theta\text{-}g(\theta)}$ for any $g\in SO(d,d,{{\mathbb{Z}}})$ as a left Heisenberg module $E$ over $T^d_\theta$. In this case, the algebra $\End_{T^d_\theta}(E)$, the algebra of endomorphisms of $E$ which commute with the left action of $T_\theta^d$, coincides with the noncommutative torus $T^d_{g(\theta)}$. This implies that one can construct a $T^d_{g(\theta)}$-$T^d_{\theta}$ Morita equivalence bimodule $P_{g(\theta)\text{-}\theta}$ as a right Heisenberg module over $T^d_\theta$. We denote it by $E_{g,\theta}$; we have $\End_{T^d_\theta}(E_{g,\theta})\simeq T_{g(\theta)}^d$. Also, the $T^d_\theta$-$T^d_{g(\theta)}$ Morita equivalence bimodule is given by the right Heisenberg module $E_{g^{-1},g(\theta)}$. On $T_{g(\theta)}^d$, a [*trace*]{} $\operatorname{Tr}_{T_{g(\theta)}^d}:T_{g(\theta)}^d\to{{\mathbb{C}}}$ and [*derivations*]{} $\delta_i:T_{g(\theta)}^d\to T_{g(\theta)}^d$, $i=1,\cdots, d$, are defined by appropriate rescaling of those for $T^d_\theta$ as $$\label{Tr-End} \operatorname{Tr}_{T_{g(\theta)}^d}(u) =\sqrt{|\det(\cC\theta+\cD)|}\, u_{{\vec{m}}=0} \,\qquad u:=\sum_{{\vec{m}}\in{{\mathbb{Z}}}^n} u_{\vec{m}} Z_{{\vec{m}}}\in T_{g(\theta)}^d$$ and $$\delta_j(Z_{{\vec{m}}}) =\frac{2\pi\i m_j}{\sqrt{|\det(\cC\theta+\cD)|}}\, Z_{{\vec{m}}}\ ,$$ where $g=\left({\begin{smallmatrix}}\cA & \cB \\ \cC & \cD{\end{smallmatrix}}\right)$ and $Z_1,\cdots,Z_d$ are the generators of $T^d_{g(\theta)}$ with relations $Z_j Z_k = e^{-2\pi\i(g\theta)^{jk}}Z_k Z_j$, $j,k=1,\cdots, d$. For $X\in\cL_\theta$, a linear map $\nabla:T_{g(\theta)}^d\otimes\cL_\theta\to T_{g(\theta)}^d$ is defined by extending linearly $$\label{delta-End} \nabla_{\delta_i}(u):=\delta_i(u)\ ,\quad i=1,\cdots,d\ ,\qquad u\in T_{g(\theta)}^d\ .$$ When we have a $T^d_\theta$-$T^d_{\theta'}$ Morita equivalence bimodule, one can consider the following [*tensor product*]{} (see [@S; @S-Qalg]): $$E_{g_a,\theta}\otimes_{T^d_\theta}P_{\theta\text{-}g(\theta)} \simeq E_{g_a g^{-1},g(\theta)}\ \label{MEtensor}$$ for a Heisenberg module $E_{g_a,\theta}$ with any $g_a\in SO(d,d,{{\mathbb{Z}}})$, where the tensor product $\otimes_{T^d_\theta}$ is defined by the standard tensor product $\otimes$ over ${{\mathbb{C}}}$ with the identification $(\xi_a\cdot u)\otimes p\sim \xi_a\otimes (u\cdot p)$ for any $\xi_a\in E_{g_a,\theta}$, $u\in T_\theta^d$ and $p\in P_{\theta\text{-}g(\theta)}$. Let us denote $\theta_a:=g_a(\theta)$. For a right Heisenberg module $E_{g_a,\theta}$ and a Morita equivalence bimodule $P_{\theta_b\text{-}\theta_a}$ with a $SO(d,d,{{\mathbb{Z}}})$ element $g_b$, the existence of the tensor product (\[MEtensor\]) implies that we have the following tensor product: $$P_{\theta_b\text{-}\theta_a}\otimes_{T_{\theta_a}^d}E_{g_a,\theta} \simeq E_{g_b,\theta}\ .$$ Thus, we may identify $P_{\theta_b\text{-}\theta_a}$ with the space of homomorphisms from $E_{g_a,\theta}$ to $E_{g_b,\theta}$. Hereafter we write $$\Hom(E_{g_b,\theta},E_{g_a,\theta}) :=P_{g_b(\theta)\text{-}g_a(\theta)}\ .$$ On a Morita equivalence $T^d_{\theta_b}$-$T^d_{\theta_a}$ bimodule $\Hom(E_{g_a,\theta},E_{g_b,\theta})$, we define a connection $\nabla:\Hom(E_{g_a,\theta},E_{g_b,\theta})\otimes\cL_\theta \to \Hom(E_{g_a,\theta},E_{g_b,\theta})$ by a linear map satisfying the following relation: $$\nabla_X(u_b\cdot \xi)=\nabla_X(u_b)\cdot\xi +u_b\cdot\nabla_X(\xi)\ ,\quad \nabla_X(\xi\cdot u_a)= \nabla_X(\xi)\cdot u_a+\xi\cdot\nabla_X(u_a)\ , \quad u_a\in T_{\theta_a}^d\ ,\quad u_b\in T_{\theta_b}^d\ ,$$ where $\nabla_X(u_b)$ and $\nabla_X(u_a)$ are defined by eq.(\[delta-End\]). Since these Morita equivalence bimodules are Heisenberg modules, they can be equipped with constant curvature connections. A Heisenberg module $E_{g,\theta}$ attached to an element $g\in SO(d,d,{{\mathbb{Z}}})$ as above is called a [*basic module*]{}. Since any Heisenberg module is constructed by a direct sum of basic modules, in this paper we concentrate on categories of basic modules. Let $\Ob:=\{a,b,\cdots\}$ be a collection of labels and consider a map $g:\Ob\to SO(d,d,{{\mathbb{Z}}})$, $g(a):=g_a\in SO(d,d,{{\mathbb{Z}}})$ for $a\in\Ob$. For the collection of Heisenberg modules $\{E_{g_a,\theta}\ |\ a\in\Ob\}$, assume we have an associative [product]{} $$m: \Hom(E_{g_b,\theta},E_{g_c,\theta})\otimes \Hom(E_{g_a,\theta},E_{g_b,\theta}) \to\Hom(E_{g_a,\theta},E_{g_c,\theta})$$ for any $a,b,c\in\Ob$. Namely, for any $a,b,c,d\in\Ob$ and $\xi_{ab}\in\Hom(E_{g_a,\theta},E_{g_b,\theta})$, $\xi_{bc}\in\Hom(E_{g_b,\theta},E_{g_c,\theta})$, $\xi_{cd}\in\Hom(E_{g_c,\theta},E_{g_d,\theta})$, we assume that the product $m$ satisfies $$\label{nc-associativity} m(m(\xi_{cd},\xi_{bc}),\xi_{ab}) =m(\xi_{cd},m(\xi_{bc},\xi_{ab}))\ .$$ Such a product $m: \Hom(E_{g_b,\theta},E_{g_c,\theta})\otimes \Hom(E_{g_a,\theta},E_{g_b,\theta}) \to\Hom(E_{g_a,\theta},E_{g_c,\theta})$ is essentially the tensor product; $m$ is constructed by fixing a map inducing the isomorphism $$\Hom(E_{g_b,\theta},E_{g_c,\theta})\otimes_{T^d_{\theta_b}} \Hom(E_{g_a,\theta},E_{g_b,\theta}) \simeq\Hom(E_{g_a,\theta},E_{g_c,\theta})\ .$$ There exists a choice of the map so that the associativity holds (see [@S-Qalg]). For $a\in\Ob$, suppose that a constant curvature connection $\nabla_a$ is defined on $E_{g_a,\theta}$. Also, for $b\in\Ob$, $a\ne b$, define a constant curvature connection $\nabla:\Hom(E_{g_a,\theta},E_{g_b,\theta})\otimes\cL_\theta\to \Hom(E_{g_a,\theta},E_{g_b,\theta})$ whose constant curvature $F_{ab}:=\{F_{ab,ij}\}_{i,j=1,\cdots,d}$ is defined by $$F_{ab,ij}\cdot\xi_{ab}:=\frac{\i}{2\pi}[\nabla_i,\nabla_j](\xi_{ab}) \ ,\qquad F_{ab,ij}=-F_{ab,ji}\in{{\mathbb{R}}}$$ for any $\xi\in\Hom(E_{g_a,\theta},E_{g_b,\theta})$. Then, a constant curvature connection $\nabla_b:E_{g_b,\theta}\otimes\cL_\theta\to E_{g_b,\theta}$ can be induced as follows [@S; @S-Qalg]: $$\nabla_b(m(\xi_{ab},\xi_a)):= m(\nabla(\xi_{ab}),\xi_a)+m(\xi_{ab},\nabla_a(\xi_a))$$ for any $\xi_a\in E_{g_a,\theta}$ and $\xi_{ab}\in\Hom(E_{g_a,\theta},E_{g_b,\theta})$, where the relation between the curvatures of $E_{g_a,\theta}$, $E_{g_b,\theta}$ and that of $\Hom(E_{g_a,\theta},E_{g_b,\theta})$ is given by $$F_b-F_a=F_{ab}\ .$$ Thus, repeating this procedure leads to the following category $\cC_{\theta,E}^{pre}$: \[defn:pre-C\] For a noncommutative torus $T^d_\theta$, let $\Ob:=\{a,b,\cdots\}$ be a collection of labels and $E$ a map from $\Ob$ to the space of basic modules with constant curvature connections; for $a\in\Ob$, we denote $E(a)=(E_{g_a,\theta},\nabla_a)=:E_a$. A [*category*]{} $\cC_{\theta,E}^{pre}$ is defined by the following data. $\bullet$ The collection of objects is $$\Ob(\cC_{\theta,E}^{pre}):=\Ob\ .$$ Each object $a\in\Ob$ is associated with a basic module with a constant curvature connection $E_a$ whose constant curvature is denoted by a skewsymmetric matrix $F_a\in\Mat_{2n}({{\mathbb{R}}})$. $\bullet$  For any $a,b\in\Ob(\cC_{\theta,E}^{pre})$, the space of morphisms is $$\Hom_{\cC_{\theta,E}^{pre}}(a,b):=\Hom(E_{g_a,\theta},E_{g_b,\theta})\ ,$$ which is equipped with a constant curvature connection $\nabla:\Hompre(a,b)\otimes\cL_\theta\to \Hompre(a,b)$ with its constant curvature $F_{ab}=F_b-F_a$. $\bullet$  For any $a,b,c\in\Ob$, there exists an associative product (eq.(\[nc-associativity\])) $$m:\Hom_{\cC_{\theta,E}^{pre}}(b,c)\otimes \Hom_{\cC_{\theta,E}^{pre}}(a,b)\to\Hom_{\cC_{\theta,E}^{pre}}(a,c)\ .$$ $\bullet$  For any $a,b,c\in\Ob$ and $\xi_{ab}\in\Hom_{\cC_{\theta,E}^{pre}}(a,b)$, $\xi_{bc}\in\Hom_{\cC_{\theta,E}^{pre}}(b,c)$, the Leibniz rule holds: $$\label{nc-leibniz} \nabla m(\xi_{bc},\xi_{ab}) = m(\nabla(\xi_{bc}),\xi_{ab}) +m(\xi_{bc},\nabla(\xi_{ab}))\ .$$ $\bullet$  For any $a\in\Ob$, a trace $\operatorname{Tr}_a:\Hompre(a,a)\to{{\mathbb{C}}}$ is given by eq.(\[Tr-End\]): $$\operatorname{Tr}_a (u) =\sqrt{|\det(\cC_a\theta+\cD_a)|}\, u_{{\vec{m}}=0} \,\qquad u:=\sum_{{\vec{m}}\in{{\mathbb{Z}}}^n} u_{\vec{m}} Z_{{\vec{m}}}\in\Hompre(a,a)$$ for $g_a=\left({\begin{smallmatrix}}\cA_a & \cB_a \\ \cC_a & \cD_a{\end{smallmatrix}}\right)$. In particular, for any $a,b\in\Ob$, $\operatorname{Tr}_a m: \Hom_{\cC_{\theta,E}^{pre}}(b,a)\otimes \Hom_{\cC_{\theta,E}^{pre}}(a,b)\to{{\mathbb{C}}}$ gives a nondegenerate bilinear map such that $$\label{pre-cyclic} \operatorname{Tr}_a m(\xi_{ba},\xi_{ab})=\operatorname{Tr}_b m(\xi_{ab},\xi_{ba})\ ,\qquad \xi_{ab}\in\Hom_{\cC_{\theta,E}^{pre}}(a,b)\ ,\quad \xi_{ba}\in\Hom_{\cC_{\theta,E}^{pre}}(b,a)\ .$$ The last identity (\[pre-cyclic\]) together with the nondegeneracy of $\operatorname{Tr}_a m$ is a typical property of Morita equivalence bimodules (see [@Rhigh; @KS], and for noncommutative two-tori case [@nchms]). Noncommutative complex tori and CDG structures on them {#ssec:nctoriCDG} ------------------------------------------------------ Let us consider a complex structure on the noncommutative torus $T_\theta^{2n}$ as introduced by A. Schwarz [@Stheta]. We take a different notation which fits our arguments, though it is equivalent to the one in [@Stheta]. When we define a complex structure on a commutative torus $T^{2n}$, we may take a ${{\mathbb{C}}}$-valued $n$ by $n$ matrix $\tau=\{\tau_j^{\, i}\}$, $i,j=1,\cdots,n$, whose imaginary part $\tau_I:=\Im(\tau)$ is positive definite. A commutative complex torus is then described by ${{\mathbb{C}}}^n/({{\mathbb{Z}}}^n+\tau^t{{\mathbb{Z}}}^n)$, where $\tau^t$ is the transpose of $\tau$. The complex coordinates of ${{\mathbb{C}}}^n$ are given by $(z_1,\cdots,z_n)$, $z^i=x^i+\sum_j y^j\tau_j^{\, i}$, $i=1,\cdots, n$. The corresponding Dolbeault operator $\bpart$ is given by $$\bpart=\sum_{i=1}^nd\zb^i\fpartial{\zb^i}\ ,\quad \fpartial{\zb^i} :=\ov{2\sqrt{-1}}\sum_{j=1}^n \(((\tau_I)^{-1}\tau)_i^j\fpartial{x^j}-((\tau_I)^{-1})_i^j\fpartial{y^j}\)\ ,$$ where we denote $\Im(\tau)=:\tau_I$ which is by definition positive definite. Based on these formula, for a noncommutative torus $T^{2n}_\theta$ and a fixed complex structure $\tau$, let us define $\bpart_i\in\cL_\theta$, $i=1,\cdots, n$, by $$\bpart_i:=\ov{2\sqrt{-1}}\sum_{j=1}^n \(((\tau_I)^{-1}\tau)_i^j\delta_j-((\tau_I)^{-1})_i^j\delta_{n+j}\)\ .$$ Also, for $E_a:=(E_{g_a,\theta},\nabla_a)$, a Heisenberg module $E_{g_a,\theta}$ over $T^{2n}_\theta$ with a constant curvature connection $\nabla_{a,i}$, $i=1,\cdots, 2n$, define a [*holomorphic structure*]{} $\nabb_{a,i}:E_{g_a,\theta}\to E_{g_a,\theta}$, $i=1,\cdots,n$, by $$\label{hol} \nabb_{a,i}:=\ov{2\sqrt{-1}}\sum_{j=1}^n \(((\tau_I)^{-1}\tau)_i^j\nabla_{a,j}-((\tau_I)^{-1})_i^j\nabla_{a,n+j}\)\ .$$ For each pair $(E_a,E_b)$, we define a [*holomorphic structure*]{} $\nabb_i:\Hom(E_{g_a,\theta},E_{g_b,\theta}) \to\Hom(E_{g_a,\theta},E_{g_b,\theta})$, $i=1,\cdots,n$, by the same formula: $$\label{hol-hom} \nabb_{i}:=\ov{2\sqrt{-1}}\sum_{j=1}^n \(((\tau_I)^{-1}\tau)_i^j\nabla_{j}-((\tau_I)^{-1})_i^j\nabla_{n+j}\)\ .$$ Let $\Lambda$ be the Grassmann algebra generated by $d\zb^1,\cdots,d\zb^n$ of degree one. Namely, they satisfy $d\zb^id\zb^j=-d\zb^jd\zb^i$ for any $i,j=1,\cdots,n$, so in particular $(d\zb^i)^2=0$. These generators are thought of as the formal basis of the antiholomorphic one forms on the complex noncommutative torus $T_\theta^{2n}$. By $\Lambda^k$ we denote the degree $k$ graded piece of $\Lambda$. The graded vector space $V:=T^{2n}_\theta\otimes\Lambda$ is then thought of as the space of the smooth antiholomorphic forms on the complex noncommutative torus $T_\theta^{2n}$, which also has the the graded piece decomposition: $$V=\oplus_{k=0}^n V^k\ .$$ Any element in $V^k$ can be written as $$v=\sum_{{\vec{m}}\in{{\mathbb{Z}}}^n}\sum_{i_1,\cdots,i_k} v_{{\vec{m}};i_1\cdots i_k}U_{{\vec{m}}}\cdot (d\zb^{i_1}\cdots d\zb^{i_k})\ ,$$ where $v_{{\vec{m}};i_1\cdots i_k}\in{{\mathbb{C}}}$ is skewsymmetric with respect to the indices $i_1\cdots i_k$. A product $m:V^k\otimes V^l\to V^{k+l}$ is defined naturally by combining the product on $T_{\theta}^{2n}$ with the one on the Grassmann algebra $\Lambda$, and then $(V,m)$ forms a graded algebra. One can define a differential $d:V^k\to V^{k+1}$, $$d:=\sum_{i=1}^nd\zb^i\cdot\bpart_i\ ,$$ which satisfies the Leibniz rule with respect to the product $m$. An inner product $\eta: V^k\otimes V^l\raw{{\mathbb{C}}}$ of degree $-n$ is defined by the composition of the product $m$ with a trace $\int_{T_\theta^{2n}} :V\raw{{\mathbb{C}}}$: $$\eta=\int_{T_\theta^{2n}} m\ ,\qquad \int_{T_\theta^{2n}} v = v_{{\vec{m}}=0;i_1\cdots i_k} \epsilon^{i_1\cdots i_k}_{1\cdots n}\ .$$ Here $\epsilon$ is defined by $$\epsilon^{i_1\cdots i_k}_{1\cdots n}= \begin{cases} 0 & (k\ne n) \\ \sum_{\sigma\in\S_n}\epsilon(\sigma)\delta^{i_1}_{\sigma(1)}\cdots \delta^{i_n}_{\sigma(n)}& k=n \ , \end{cases}$$ where $\epsilon(\sigma)$ is the signature of the permutation of $n$ elements $\sigma\in\S_n$. Namely, $\int_{T_\theta^d} : V^k\raw{{\mathbb{C}}}$ is thought of as the integration of differential forms over $T^{2n}_\theta$, as an extension of the trace map $\operatorname{Tr}:T_\theta^{2n}\to{{\mathbb{C}}}$ in eq.(\[Tr\]), and hence gives a nonzero map only if $k=n$. $(V,\eta,d,m)$ forms a cyclic DG algebra. For $E_a:=(E_{g_a,\theta},\nabla_a)$ a Heisenberg module over $T_{\theta}^{2n}$ with a constant curvature connection, we lift $E_a$ to a ${{\mathbb{Z}}}$-graded right module $\cE_a:=E_{g_a,\theta}\otimes\Lambda$ over $V$. We denote by $m_a:\cE_a\otimes V\to\cE_a$ the right action of $V$ on $\cE_a$. The connection $\nabla_a:E_{g_a,\theta}\otimes\cL_\theta\to E_{g_a,\theta}$ is then lifted to a degree one linear map $d_a:\cE_a\to\cE_a$ defined by $$d_a:=\sum_{i=1}^nd\zb^i\cdot\nabb_{a,i} \ ,$$ where $\nabb_{a,i}$ is the holomorphic structure (\[hol\]). This $d_a$ is not a differential in general. Namely, the graded module has its curvature: $$(d_a)^2 v^{\cE_a}= \fh_a\cdot v^{\cE_a}\ ,\qquad \fh_a:=-\pi\i \left( d\zb^t\tau_I^{-1}\right) {\begin{pmatrix}}\tau & -\1_n{\end{pmatrix}}F_a {\begin{pmatrix}}\tau^t \\ -\1_n {\end{pmatrix}}\left(\tau_I^{t,-1}d\zb \right)\in\Lambda^2$$ for any $v^{\cE_a}\in\cE_a$, where $d\zb^t:=(\zb^1,\cdots,\zb^n)$. This $d_a$ defines a differential on $\cE_a$, that is, $\fh_a=0$ if and only if $${\begin{pmatrix}}\tau & -\1_n {\end{pmatrix}}F_a{\begin{pmatrix}}\tau^t \\ -\1_n {\end{pmatrix}}=0\ .$$ In this case, $(\cE_a,d_a,m_a)$ forms a DG module over $V$. In the commutative case ($\theta=0$), this condition on $F_a$ is nothing but the condition that the corresponding (line) bundle is holomorphic, , the curvature is of $(1,1)$-form with respect to the complex structure defined by $\tau$. However, for general $\theta$, $\fh_a$ can not be zero even if it is zero when $\theta$ is set to be zero. On the other hand, since $\fh_a\in\Lambda^2\subset V^2$ is a center in $V$ with respect to the product $m$, $(d)^2(v)=m(\fh_a,v)-m(v,\fh_a)\ (=0)$ holds and then $(V,\eta,-\fh_a,d,m)$ forms a cyclic CDG algebra. Thus: \[lem:H-CDG\] $(\cE_a, \fh_a, d_a,m_a)$ forms a CDG module over the cyclic CDG algebra $(V,\eta,-\fh_a,d,m)$. We call this $\fh_a\in V^2$ the [*potential two-form*]{} of $\cE_a$. Now, for a category $\cC^{pre}_{\theta,E}$ given in Definition \[defn:pre-C\], we construct a CDG category $\cC_{\theta,\tau,E}=:\cC$. \[defn:C\] For a fixed category $\cC^{pre}_{\theta,E}$, a [*category*]{} $\cC$ is defined as follows. $\bullet$  The collection of objects is $$\Ob(\cC):=\Ob\ ,$$ where any object $a\in\Ob$ is associated with a CDG module $(\cE_a,\fh_a,d_a,m_a)$ over the CDG algebra $(V,\eta,-\fh_a,d,m)$ corresponding to $E_a$ as in Lemma \[lem:H-CDG\]. $\bullet$  For any $a,b\in\Ob(\cC)$, the space of morphisms is the graded vector space $$\HomC(a,b):=\Hompre(a,b)\otimes\Lambda=:V_{ab}=\oplus_{k=1}^n V_{ab}^k\ ,$$ which is equipped with a degree one linear map $d:V_{ab}^k\to V_{ab}^{k+1}$, $$d:=\sum_{i=1}^nd\zb^i\nabb_i\ ,$$ where $\nabb_i$ is the holomorphic structure (\[hol-hom\]) corresponding to the constant curvature connection $\nabla:\Hompre(a,b)\otimes\cL_\theta\to\Hompre(a,b)$. $\bullet$ For any $a,b,c\in\Ob(\cC)$, an associative product $m:V_{bc}^l\otimes V_{ab}^k\to V_{ac}^{k+l}$ is given by the lift of the product $m$ on $\Hompre(*,*)$. $\bullet$  For any two objects $a,b\in\Ob(\cC)$, a nondegenerate graded symmetric inner product $\eta: V_{ba}\otimes V_{ab}\to{{\mathbb{C}}}$ of degree $-n$ is defined by $$\eta=\int_{T_{\theta_a}^{2n}} m\ ,\qquad m:V_{ba}\otimes V_{ab}\to V_{aa}\ .$$ Here $\int_{T_{\theta_a}^{2n}}:V_{aa}\to{{\mathbb{C}}}$ is defined by $$\int_{T_{\theta_a}^{2n}} v =\sqrt{|\det(\cC_a\theta+\cD_a)|}\, v_{{\vec{m}}=0;i_1\cdots i_k} \epsilon^{i_1\cdots i_k}_{1\cdots n}\ \,\quad v:=\sum_{{\vec{m}}\in{{\mathbb{Z}}}^n}\sum_{i_1,\cdots,i_k} v_{{\vec{m}};i_1\cdots i_k}U_{{\vec{m}}}\cdot (d\zb^{i_1}\cdots d\zb^{i_k})\in V_{aa}$$ for $g_a=\left({\begin{smallmatrix}}\cA_a & \cB_a \\ \cC_a & \cD_a{\end{smallmatrix}}\right)$ as an extension of the trace map $\operatorname{Tr}_a\to{{\mathbb{C}}}$. Due to the Leibniz rule (\[nc-leibniz\]), it is clear that $d:V_{ab}^k\to V_{ab}^{k+1}$ is a derivation: $$d m(v_{bc}\otimes v_{ab}) = m(d(v_{bc})\otimes v_{ab}) +(-1)^{|v_{bc}|}m(v_{bc}\otimes d(v_{ab}))\ . \label{v-Leibniz}$$ Let us define $\fh_{ab}\in\Lambda^2$ by $$\fh_{ab}:=d^2\ ,\qquad d:V_{ab}^k\to V_{ab}^{k+1}\ .$$ Then, the Leibniz rule (\[v-Leibniz\]) and $F_{ab}=F_b-F_a$ imply $\fh_{ab}=\fh_b-\fh_a$. For each $a\in\Ob(\cC)$, let $f_a:=\fh_a\cdot\1_a\in V_{aa}$, where $\1_a$ is the identity in $T^{2n}_{\theta_a}$. The following is the main claim of this paper. -   For a given category $\cC^{pre}_{\theta,E}$ in Definition \[defn:pre-C\], $\cC:=\cC_{\theta,\tau,E}$ forms a cyclic CDG category. -   Let $\cC^\fh$ be the full subcategory of $\cC$ such that any $a\in\Ob(\cC^\fh)\subset\Ob(\cC)$ is a set of CDG modules over a cyclic CDG algebra $(V,\eta,-\fh,d,m)$ for a fixed $\fh\in\Lambda^2\subset V^2$. Then $\cC^\fh$ forms a cyclic DG category. \[prop:CDGcat\] This implies that one can construct a kind of DG categories of holomorphic vector bundles over a noncommutative tori. Three examples of noncommutative deformations {#sec:three} ============================================= Now, we construct examples of various noncommutative deformations of the DG categories of Heisenberg modules described by CDG modules over a cyclic CDG algebra of a noncommutative torus. The set-up given in the previous subsection allows us to deform both the complex structure $\tau$ and the noncommutativity $\theta$ or either of them. In this paper, starting from a commutative ($\theta=0$) $n$ dimensional complex torus with the standard complex structure $\tau=\i\1_n$ in subsection \[ssec:comm\], we deform the noncommutative parameter $\theta$ with preserving the standard complex structure in subsection \[ssec:nc\]. Also, we give the composition formula on the zero-th cohomologies of the DG categories explicitly. We show that the structure constants of the compositions in fact depends on $\theta$. Commutative case {#ssec:comm} ---------------- Let us begin with the commutative torus $T^{2n}:=T^{2n}_{\theta=0}$. The generators $U_1,\cdots,U_n,U_{{\bar 1}}:=U_{n+1},\cdots, U_{{\bar n}}:=U_{2n}$ then commute with each other. The arguments in subsection \[ssec:nctoriCDG\] show that it is enough to construct a category $\cC^{pre}_{\theta=0,E}$ in order to construct a cyclic CDG category $\cC$. A category $\cC^{pre}_{\theta=0,E}$ is constructed as follows. Any object $a\in\Ob(\cC^{pre}_{\theta=0,E})$ is associated with a pair $E_a:=(E_{g_a,\theta=0},\nabla_a)$ of basic module $E_{g_a,\theta=0}$ with a constant curvature connection $\nabla_a$. The basic module is defined by $$E_{g_a,\theta=0}:=\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_{a}{{\mathbb{Z}}}^n))$$ for a fixed nondegenerate symmetric matrix $A_a\in\Mat_n({{\mathbb{Z}}})$, where $g_a\in SO(d,d,{{\mathbb{Z}}})$ is given by $$g_a= {\begin{pmatrix}}\1_{2n} & \0_{2n} \\ F_a & \1_{2n}{\end{pmatrix}}\ ,\qquad F_a :={\begin{pmatrix}}\0_n & A_{a}\\ -A_{a} & \0_n{\end{pmatrix}}\ .$$ The right action of $T^{2n}$ on $E_{g_a,\theta=0}$ is defined by specifying the right action of each generator; for $\xi_a\in E_{g_a,\theta=0}$, it is given by $$\begin{split} (\xi_aU_i)(x;\mu)& =\xi_a(x;\mu) e^{2\pi\sqrt{-1}(x_i+(A_{a}^{-1}\mu)_i))} \ ,\\ (\xi_aU_\ib)(x;\mu)& =\xi_a(x+A_{a}^{-1}t_i;\mu-t_i) \ ,\qquad i=1,\cdots, n \ , \end{split}$$ where $x:=(x_1\cdots x_n)^t\in{{\mathbb{R}}}^n$, $\mu\in{{\mathbb{Z}}}^n/A_{a}{{\mathbb{Z}}}^n$ and $t_i\in{{\mathbb{R}}}^n$ is defined by $(t_1 \cdots t_n)=\1_n$. A constant curvature connection $\nabla_a:E_{g_a,\theta=0}\otimes\cL_\theta\to E_{g_a,\theta=0}$ is given by $$(\nabla_{a,1}\cdots \nabla_{a,2n})^t= {\begin{pmatrix}}\1_n & \\ & -A_{a} {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}\ , \label{4dim-conn}$$ where $\partial_x:=\left({\begin{smallmatrix}}\fpartial{x_1}& \cdots & \fpartial{x_n}{\end{smallmatrix}}\right)^t$, and the curvature (defined by eq.(\[F\])) is $F_a$ above. The generators of the endomorphism algebra is the same as $U_i, U_\ib$: $$\begin{split} (Z_i\xi_a)(x;\mu)& =e^{2\pi\sqrt{-1}(x_i+(A_{a}^{-1}\mu)_i))}\xi_a(x;\mu)\ ,\\ (Z_\ib\xi_a)(x;\mu)& =\xi_a(x+A_{a}^{-1}t_i;\mu-t_i) \ ,\qquad i=1,\cdots, n \ . \end{split}$$ Namely, the endomorphism algebra also forms a commutative torus $T^{2n}$. This $E_a:=(E_{g_a},\nabla_a)$ is lifted to a CDG module $(\cE_a,\fh_a,d_a,m_a)$ over the cyclic CDG algebra $(V=T^{2n}\otimes\Lambda,\eta,-\fh_a,d,m)$ by the procedure in the previous subsection, where the complex structure is taken to be the standard one: $\tau=\i\1_n$. Then, one obtains $(d_a)^2=\fh_a=0$, that is, $\cE_a$ in fact forms a DG-module corresponding to a [*holomorphic line*]{} bundle. For any $a,b\in\Ob(\cC^{pre}_{\theta=0,E})$, the space $\Homp0(a,b)$ is defined as follows. If $A_{ab}:=A_b-A_a$ is nondegenerate, then it is again the Schwartz space $\Homp0(a,b):=\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n))$. For $\xi_{ab}\in\Homp0(a,b)$, the right action of $T^{2n}$, generated by $U_i$ and $U_\ib$, and the left action of $T^{2n}$, generated by $Z_i$ and $Z_\ib$, are defined by $$\begin{split} (\xi_{ab} U_i )(x;\mu)& = \xi_{ab}(x;\mu) e^{2\pi\sqrt{-1}(x_i+(A_{ab}^{-1}\mu)_i))} \ ,\\ (\xi_{ab} U_\ib)(x;\mu)& =\xi_{ab}(x+A_{ab}^{-1}t_i;\mu-t_i)\ ,\\ (Z_i \xi_{ab})(x;\mu)& = e^{2\pi\sqrt{-1}(x_i+(A_{ab}^{-1}\mu)_i))} \xi_{ab}(x;\mu)\ ,\\ (Z_\ib \xi_{ab})(x;\mu)& =\xi_{ab}(x+A_{ab}^{-1}t_i;\mu-t_i)\ \end{split}$$ for $i=1,\cdots,n$, where $\mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n$. In fact, all these generators $U_i$, $U_\ib$, $Z_i$ and $Z_\ib$ commute with each other. On the other hand, if $A_a=A_b$, we define $\Homp0(a,b):=T^{2n}$, on which the left and right actions of $T^{2n}$ are defined just by the commutative product on $T^{2n}$. In general, the way of constructing the space $\Hompre(a,b)=\Hom(E_{g_a,\theta=0},E_{g_b,\theta=0})$ depends on the rank of $A_{ab}:=A_b-A_a$. In rank $n$ case (nondegenerate case), one has $\Homp0(a,b):=\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n))$ and in rank $0$ case, one has $\Homp0(a,b):=T^{2n}$ as above. In rank $1<r<n$ case, we should combine these two constructions with each other appropriately. In order to avoid such case-by-case arguments, in this paper we assume that $A_{ab}$ is nondegenerate for any $a,b\in\Ob(\cC^{pre}_{\theta=0,E})$ such that $a\ne b$. The constant curvature connection $\nabla_i:\Homp0(a,b)\to\Homp0(a,b)$, $i=1,\cdots, 2n$, is given by $$(\nabla_1 \cdots \nabla_{2n})^t := {\begin{pmatrix}}\1_n & \\ & -A_{ab} {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}$$ if $a\ne b$, and if $a=b$, it is defined by the derivation $\nabla$ on the noncommutative torus $T^{2n}_{\theta_a}=T^{2n}_{\theta_b}$ in eq.(\[delta-End\]) with $\theta_a=\theta_b=0$. For $a,b,c\in\Ob(\cCp0)$ and $\xi_{ab}\in\Homp0(a,b)$, $\xi_{bc}\in\Homp0(b,c)$, the product $m:\Homp0(b,c)\otimes\Homp0(a,b)\to\Homp0(a,c)$ is given as follows: For $a=b$, it is the right action of $T^{2n}$ on $\Homp0(b,c)$. For $b=c$, it is the left action of $T^{2n}$ on $\Homp0(a,b)$. For $a=c$, the product $m:\Homp0(b,a)\otimes\Homp0(a,b)\to T^{2n}$ is given by $$m(\xi_{ba},\xi_{ab})(x,\rho) =\sum_{\vec{m}\in{{\mathbb{Z}}}^n}\sum_{\mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n} U_{\vec{m}}\int_{{{\mathbb{R}}}^n} dx^n \xi_{ba}(x,\mu)\cdot(\xi_{ab}(x,-\mu)U_{-\vec{m}})$$ for $\xi_{ab}\in\Homp0(a,b)$ and $\xi_{ba}\in\Homp0(b,a)$, where $U_{\vec{m}}\in\Homp0(a,a)=T^{2n}$. For the remaining general case, it is given by $$m(\xi_{bc},\xi_{ab})(x,\rho) =\sum_{u\in{{\mathbb{Z}}}^n} \xi_{bc}(x+A_{bc}^{-1}(u-A_{ab}A_{ac}^{-1}\rho),-u+\rho)\cdot \xi_{ab}(x-A_{ab}^{-1}(u-A_{ab}A_{ac}^{-1}\rho),u)\ . \label{tensor-comm}$$ These structures together with the trace map defined by eq.(\[Tr\]) forms a category $\cCp0$ and the corresponding cyclic CDG category $\cC_{\theta=0}$. In particular, we have $d^2=0$ for $d:V_{ab}\to V_{ab}$ with any pair $a,b\in\Ob(\cC_{\theta=0})$. Thus, $\cC_{\theta=0}$ is a cyclic DG category. For any $a,b\in\Ob(\cC_{\theta=0})$, $a\ne b$, the bases of the zero-th cohomology of $V_{ab}$ are given by gaussians [@Stheta] (see also [@DKL]) and called the [*theta vector*]{}, though here we are discussing the $\theta=0$ case. We shall give examples of these theta vectors in noncommutative case $\theta\ne 0$ in the next subsection. The mirror dual $\Th^{2n}$ of this complex torus $T^{2n}:={{\mathbb{C}}}^n/({{\mathbb{Z}}}^n\oplus \i{{\mathbb{Z}}}^n)$ is the real $2n$-dimensional torus with a symplectic structure $\omega:=\left({\begin{smallmatrix}}\0_n & -\1_n \\ \1_n & \0_n {\end{smallmatrix}}\right)$. In this mirror dual torus $\Th^{2n}$, a line bundle specified by $A_a$ corresponds to an affine lagrangian submanifold $L_a$. Then, the intersection of $L_a$ and $L_b$ is a point ${\hat v}_{ab}$ on $\Th^{2n}$, which defines the set ${\tilde}{V}_{ab}$ of the infinite copies of the points on the covering space ${{\mathbb{C}}}^n$. The structure constant $C_{abc,\rho}^{\mu\nu}\in{{\mathbb{C}}}$ can be identified with the sum of the exponentials of the symplectic areas of the triangles ${\tilde}{v}_{ab}{\tilde}{v}_{bc}{\tilde}{v}_{ac}$ for any ${\tilde}{v}_{ab}\in{\tilde}{V}_{ab}$, ${\tilde}{v}_{bc}\in{\tilde}{V}_{bc}$ and ${\tilde}{v}_{ac}\in{\tilde}{V}_{ac}$ with respect to $\omega$, where the triangles related by a parallel translations on the covering space are identified with each other and not overcounted (see [@nctheta]). Noncommutative deformations of holomorphic line bundles {#ssec:nc} ------------------------------------------------------- Let us consider a real $2n$-dimensional noncommutative torus $T^{2n}_\theta$ with its generators $U_1,\cdots, U_{2n}$ with the following relation: $$U_iU_j=e^{-2\pi\sqrt{-1}\theta^{ij}}U_jU_i\ ,\qquad \theta:={\begin{pmatrix}}\theta_1 & -\theta_2 \\ \theta_2^t & \theta_3 {\end{pmatrix}}\ .$$ Since $\theta\in\Mat_{2n}({{\mathbb{R}}})$ is antisymmetric, $\theta_1,\theta_3\in\Mat_n({{\mathbb{R}}})$ are antisymmetric and $\theta_2\in\Mat_n({{\mathbb{R}}})$ can be an arbitrary $n$ by $n$ matrix. A Heisenberg module $E_{g,\theta}$, $g\in SO(2n,2n,{{\mathbb{Z}}})$, on this noncommutative torus $T^{2n}_\theta$ is associated with two notions, the $K_0$ group element and the Chern character (see [@KS]). The Chern character of $E_{g,\theta}$ is defined by its constant curvature $F$, a skewsymmetric $2n$ by $2n$ matrix with entries in ${{\mathbb{R}}}$. On the other hand, the $K_0$ group element of $E_{g,\theta}$ is defined by $F_0$, the constant curvature of $E_{g,\theta}$ when we set $\theta=0$. Thus, the $K_0$ group element is independent of the noncommutativity $\theta$. A Heisenberg module $E_{g,\theta}$ is thought of as a noncommutative analog of a line bundle if $g\in SO(2n,2n,{{\mathbb{Z}}})$ is of the form: $$g={\begin{pmatrix}}\1_{2n} & \0_{2n} \\ F_0 & \1_{2n}{\end{pmatrix}}$$ for a skew symmetric matrix $F_0\in\Mat_{2n}({{\mathbb{Z}}})$. In fact, $F_0$ corresponds to the first Chern character of a line bundle if $\theta=0$. Since we shall discuss noncommutative deformations of the line bundles in the previous subsection, let us consider in particular the case that $F_0$ is of the following form $$F_0={\begin{pmatrix}}\0_n & A \\ -A & \0_n {\end{pmatrix}}\ ,$$ where $A\in\Mat_n({{\mathbb{Z}}})$ is a nondegenerate symmetric matrix. For the Heisenberg modules $E_{g,\theta}$ with $g$ given as above, we shall consider noncommutative tori of the following three cases: [**Type**]{} $\theta_1$: $\theta_2=\theta_3=0$, $\theta_2$: $\theta_1=\theta_3=0$, $\theta_3$: $\theta_1=\theta_2=0$ . In each case, the endomorphism algebra, $T_{\theta'}$, $\theta':=(\1_n\theta + \0_n)(F_0\theta + \1_n)^{-1}$, turns out to be as follows. In Type $\theta_1$ case and $\theta_3$ case: $$\theta={\begin{pmatrix}}\theta_1 & \0_n \\ \0_n & \0_n{\end{pmatrix}}\ ,\qquad \theta={\begin{pmatrix}}\0_n & \0_n \\ \0_n & \theta_3{\end{pmatrix}}\ ,$$ we have $\theta'=\theta$. However, in Type $\theta_2$ case, one obtains $$\theta'={\begin{pmatrix}}\0_n & -\theta_2(\1_n+A\theta_2)^{-1} \\ \theta_2^t(\1_n+A\theta_2^t)^{-1} & \0_n {\end{pmatrix}}\ , \qquad \theta:={\begin{pmatrix}}\0_n & -\theta_2 \\ \theta_2^t & \0_n{\end{pmatrix}}\ . \label{theta_2-transf}$$ Now, for the cyclic DG category $\cC_{\theta=0}$ in the previous subsection, we construct its noncommutative deformations of three types above explicitly as CDG categories. Namely, we construct noncommutative deformations of $\cCp0$ in the previous subsection. [**Type $\theta_1$**]{} A category $\cCpre$ is constructed as follows. Any object $a\in\Ob(\cCpre)$ is associated with a pair $E_a:=(E_{g_a,\theta},\nabla_a)$. The basic module $E_{g_a,\theta}$ is defined by $$E_{g_a,\theta}=\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_a{{\mathbb{Z}}}^n))$$ for a nondegenerate symmetric matrix $A_a\in\Mat_n({{\mathbb{Z}}})$, where $$g_a= {\begin{pmatrix}}\1_{2n} & \0_{2n} \\ F_{a,0} & \1_{2n}{\end{pmatrix}}\ ,\qquad F_{a,0}:={\begin{pmatrix}}\0_n & A_{a}\\ -A_{a} & \0_n{\end{pmatrix}}\ .$$ For $\xi_a\in E_{g_a,\theta}$, the action of each generator is defined by $$\begin{split} (\xi_aU_i )(x;\mu)& = \xi_a(x;\mu)* e^{2\pi\sqrt{-1}(x_i+(A_{a}^{-1}\mu)_i))} \ ,\\ (\xi_aU_\ib)(x;\mu)& =\xi_a(x+A_{a}^{-1}t_i;\mu-t_i) \ ,\qquad i=1,\cdots, n \ . \end{split}$$ Here $*: C^\infty({{\mathbb{R}}}^n)\otimes C^\infty({{\mathbb{R}}}^n)\to C^\infty({{\mathbb{R}}}^n)$ is the Moyal star product ([@Moyal]) defined by $$(f * g)(x):=f(x) e^{\frac{\sqrt{-1}}{4\pi}\lpartial{x}\theta_1\rpartial{x}} g(x)\ ,$$ where $\lpartial{x}\theta_1\rpartial{x}:=\sum_{p,q=1}^n \lpartial{x_p}\theta^{pq}_1\rpartial{x_q}$. The generator of the endomorphism is then given by $$\begin{split} (Z_i\xi_a)(x;\mu)& = e^{2\pi\sqrt{-1}(x_i+(A_{a}^{-1}\mu)_i))} * \xi_a(x;\mu)\ ,\\ (Z_\ib\xi_a)(x;\mu)& =\xi_a(x+A_{a}^{-1}t_i;\mu-t_i)\ . \end{split}$$ A constant curvature connection $\nabla_a: E_{g_a,\theta}\otimes\cL_\theta\to E_{g_a,\theta}$ is given as $$(\nabla_{a,1},\cdots,\nabla_{a,n})^t = {\begin{pmatrix}}\1_n & \\ \ov{2}A_a\theta_1 & -A_a {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}\ ,$$ whose the constant curvature is $$F_a = {\begin{pmatrix}}\0_n & A_a\\ -A_a & -A_a\theta_1 A_a{\end{pmatrix}}\ .$$ We assume that $A_{ab}$ is nondegenerate for any $a,b\in\Ob(\cCpre)$, $a\ne b$. For any $a,b\in\Ob(\cCpre)$, the space $\Hompre(a,b)$ is defined as follows. If $a\ne b$, $\Hompre(a,b):=\Hom(E_{g_a,\theta},E_{g_b,\theta}) =\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n))$; the right action of $T_{\theta_a}$, generated by $U_i$ and $U_\ib$, and the left action of $T_{\theta_b}$, generated by $Z_i$ and $Z_\ib$, are defined by $$\begin{split} (\xi_{ab} U_i )(x;\mu)& = \xi_{ab}(x;\mu)* e^{2\pi\sqrt{-1}(x_i+(A_{ab}^{-1}\mu)_i))} \ ,\\ (\xi_{ab} U_\ib)(x;\mu)& =\xi_{ab}(x+A_{ab}^{-1}t_i;\mu-t_i)\ ,\\ (Z_i \xi_{ab})(x;\mu)& = e^{2\pi\sqrt{-1}(x_i+(A_{ab}^{-1}\mu)_i))} * \xi_{ab}(x;\mu)\ ,\\ (Z_\ib \xi_{ab})(x;\mu)& =\xi_{ab}(x+A_{ab}^{-1}t_i;\mu-t_i)\ . \end{split}$$ If $a=b$, then $\Hompre(a,b)=T_{\theta_a}=T_{\theta_b}$ and these actions are defined by the usual product of noncommutative torus $T_{\theta_a}=T_{\theta_b}$. The constant curvature connection $\nabla:\Hompre(a,b)\otimes\cL_\theta\to\Hompre(a,b)$ is given by $$(\nabla_{1}\cdots \nabla_{2n})^t:={\begin{pmatrix}}\1_n & \\ \ov{2}A_{ab}^+\theta_1 & -A_{ab} {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}\ , \qquad A_{ab}^+:=A_a+A_b$$ if $a\ne b$, and if $a=b$, it is defined by the derivation $\nabla$ of the noncommutative torus $T_{\theta_a}=T_{\theta_b}$ in eq.(\[delta-End\]). For any $a,b,c\in\Ob(\cCpre)$ and $\xi_{ab}\in\Hompre(a,b)$, $\xi_{bc}\in\Hompre(b,c)$, the product $m:\Hompre(b,c)\otimes\Hompre(a,b)\to\Hompre(a,c)$ is given as follows: For $a=b$, it is the right action of $T_{\theta_a}=T_{\theta_b}$ on $\Hompre(b,c)$. For $b=c$, it is the left action of $T_{\theta_b}=T_{\theta_c}$ on $\Hompre(a,b)$. For $a=c$, the product $m:\Hompre(b,a)\otimes\Hompre(a,b)\to T_{\theta_a}$ is given by $$m(\xi_{ba},\xi_{ab})(x,\rho) =\sum_{\vec{m}\in{{\mathbb{Z}}}^n}\sum_{\mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n} U_{\vec{m}}\int_{{{\mathbb{R}}}^n} dx^n \xi_{ba}(x,\mu) * (\xi_{ab}(x,-\mu)U_{-\vec{m}})\ .$$ For the remaining general case, it is given by $$m(\xi_{bc},\xi_{ab})(x,\rho) =\sum_{u\in{{\mathbb{Z}}}^n} \xi_{bc}(x+A_{bc}^{-1}(u-A_{ab}A_{ac}^{-1}\rho),-u+\rho) * \xi_{ab}(x-A_{ab}^{-1}(u-A_{ab}A_{ac}^{-1}\rho),u)\ .$$ The trace map is then given by eq.(\[Tr-End\]). [**Type $\theta_2$**]{} A category $\cCpre$ is constructed as follows. Any object $a\in\Ob(\cCpre)$ is associated with a pair $E_a:=(E_{g_a,\theta},\nabla_a)$, where we assume $\det(\1_n+\theta_2 A_a)\ne 0$, which is always satisfied if one of the entries of $\theta_2$ is irrational. The basic module $E_{g_a,\theta}$ is defined by $$E_{g_a,\theta}=\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_a{{\mathbb{Z}}}^n))$$ for a nondegenerate symmetric matrix $A_a\in\Mat_n({{\mathbb{Z}}})$, where $$g_a= {\begin{pmatrix}}\1_{2n} & \0_{2n} \\ F_{a,0} & \1_{2n}{\end{pmatrix}}\ ,\qquad F_{a,0} ={\begin{pmatrix}}\0_n & A_{a}\\ -A_{a} & \0_n{\end{pmatrix}}\ .$$ For $\xi_a\in E_{g_a,\theta}$, the action of each generator is defined by $$\begin{split} (\xi_a U_i )(x;\mu)& = \xi_a(x;\mu) e^{2\pi\sqrt{-1}x_i+(A_a^{-1}\mu)_i))} \ ,\\ (\xi_aU_\ib)(x;\mu)& = \xi_a(x+(\1_n+\theta_2 A_a)A_a^{-1}t_i;\mu-t_i)\ . \end{split}$$ The action of the generators of the endomorphism is then given by $$\begin{split} (Z_i\xi_a)(x;\mu)& = e^{2\pi\sqrt{-1}(((\1_n+\theta_2 A_a)^{-1}x)_i+(A_a^{-1}\mu)_i))} \xi_a(x;\mu)\ ,\\ (Z_\ib\xi_a)(x;\mu)& = \xi_a(x+A_a^{-1}t_i;\mu-t_i)\ , \end{split}$$ where the relation is $$Z_iZ_j=e^{-2\pi\sqrt{-1}\theta_a}Z_jZ_i\ ,\qquad \theta_a ={\begin{pmatrix}}\0_n & -(\1_n+\theta_2 A_a)^{-1}\theta_2 \\ \theta^t_2(\1_n+A_a\theta_2^t)^{-1} & \0_n {\end{pmatrix}}\ .$$ A constant curvature connection $\nabla_a: E_{g_a,\theta}\otimes\cL_\theta\to E_{g_a,\theta}$ is given as $$(\nabla_{a,1}\cdots \nabla_{a,2n})^t = {\begin{pmatrix}}\1_n & \0_n \\ \0_n & -(A_a^{-1}+\theta_2)^{-1} {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}\ ,$$ with its curvature $$F_a = {\begin{pmatrix}}\0_n & (A_a^{-1}+\theta_2^t)^{-1} \\ -(A_a^{-1}+\theta_2)^{-1} & \0_n {\end{pmatrix}}\ .$$ We assume that $A_{ab}$ is nondegenerate for any $a,b\in\Ob(\cCpre)$, $a\ne b$. For any $a,b\in\Ob(\cCpre)$, the space $\Hompre(a,b)$ is defined as follows. If $a\ne b$, $\Hompre(a,b):=\Hom(E_{g_a,\theta},E_{g_b,\theta}) =\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n))$; the right action of $T_{\theta_a}$, generated by $U_i$ and $U_\ib$, and the left action of $T_{\theta_b}$, generated by $Z_i$ and $Z_\ib$, are defined by $$\begin{split} (\xi_{ab} U_i )(x;\mu)& = \xi_{ab}(x;\mu) e^{2\pi\sqrt{-1}((\1_n+\theta_2 A_a)^{-1}x)_i+(A_{ab}^{-1}\mu)_i))} \ ,\\ (\xi_{ab} U_\ib)(x;\mu)& = \xi_{ab}(x+(\1_n+\theta_2 A_b)A_{ab}^{-1}t_i;\mu-t_i)\ ,\\ (Z_i \xi_{ab})(x;\mu)& = e^{2\pi\sqrt{-1}(((\1_n+\theta_2 A_b)^{-1}x)_i+(A_{ab}^{-1}\mu)_i))} \xi_{ab}(x;\mu)\ ,\\ (Z_\ib \xi_{ab})(x;\mu)& = \xi_{ab}(x+(\1_n+\theta_2 A_a)A_{ab}^{-1}t_i;\mu-t_i)\ . \end{split}$$ If $a=b$, then $\Hompre(a,b):=T_{\theta_a}=T_{\theta_b}$ and these actions are defined by the usual product of noncommutative torus $T_{\theta_a}=T_{\theta_b}$. Note that $\Hompre(a,b)=\Hom(E_{g_a,\theta},E_{g_b,\theta})$ is isomorphic to $E_{g_bg_a^{-1},\theta}$; for $a\ne b$, an element $\xi_{ab}\in\Hom(E_{g_a,\theta},E_{g_b,\theta})$ is identified with $\xi'_{ab}\in E_{g_bg_a^{-1},\theta}$ by the following relation: $$\xi'_{ab}(x',\mu) =\xi'_{ab}( (\1_n+\theta_2 A_a)^{-1}x,\mu) =\xi_{ab}(x,\mu)\ .$$ A constant curvature connection $\nabla: \Hompre(a,b)\otimes\cL_\theta\to\Hompre(a,b)$ is given by $$(\nabla_{1}\cdots \nabla_{2n})^t= {\begin{pmatrix}}\1_n & \0_n \\ \0_n & -(A_b^{-1}+\theta_2)^{-1}+(A_a^{-1}+\theta_2) {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}$$ if $a\ne b$, and if $a=b$, it is defined by the derivation $\nabla$ of noncommutative torus $T_{\theta_a}=T_{\theta_b}$ in eq.(\[delta-End\]). For any $a,b,c\in\Ob(\cCpre)$ and $\xi_{ab}\in\Hompre(a,b)$, $\xi_{bc}\in\Hompre(b,c)$, the product $m:\Hompre(b,c)\otimes\Hompre(a,b)\to\Hompre(a,c)$ is given as follows: For $a=b$, it is the right action of $T_{\theta_a}=T_{\theta_b}$ on $\Hompre(b,c)$. For $b=c$, it is the left action of $T_{\theta_b}=T_{\theta_c}$ on $\Hompre(a,b)$. For $a=c$, the product $m:\Hompre(b,a)\otimes\Hompre(a,b)\to T_{\theta_a}$ is given by $$m(\xi_{ba},\xi_{ab})(x,\rho) =\sum_{\vec{m}\in{{\mathbb{Z}}}^n}\sum_{\mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n} U_{\vec{m}}\int_{{{\mathbb{R}}}^n} dx^n \xi_{ba}(x,\mu)\cdot (\xi_{ab}(x,-\mu)U_{-\vec{m}})\ .$$ For the remaining general case, it is given by $$\begin{split} & m(\xi_{bc},\xi_{ab})(x,\rho) = \\ &\quad \sum_{u\in{{\mathbb{Z}}}^n} \xi_{bc}(x+(\1_n+\theta_2A_c)A_{bc}^{-1}(u-A_{ab}A_{ac}^{-1}\rho), -u+\rho) \cdot \xi_{ab}(x-(\1_n+\theta_2A_a)A_{ab}^{-1}(u-A_{ab}A_{ac}^{-1}\rho),u)\ . \end{split}$$ The trace map is then given by eq.(\[Tr-End\]). [**Type $\theta_3$**]{} A category $\cCpre$ is constructed as follows. Any object $a\in\Ob(\cCpre)$ is associated with a pair $E_a:=(E_{g_a,\theta},\nabla_a)$. The basic module $E_{g_a,\theta}$ is defined by $$E_{g_a,\theta}=\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_a{{\mathbb{Z}}}^n))$$ for a nondegenerate symmetric matrix $A_a\in\Mat_n({{\mathbb{Z}}})$, where $$g_a= {\begin{pmatrix}}\1_{2n} & \0_{2n} \\ F_{a,0} & \1_{2n}{\end{pmatrix}}\ ,\qquad F_{a,0} ={\begin{pmatrix}}\0_n & A_{a}\\ -A_{a} & \0_n{\end{pmatrix}}\ .$$ For $\xi_a\in E_{g_a,\theta}$, the action of each generator is defined by $$\begin{split} (\xi_aU_i)(x;\mu)& =\xi_a(x;\mu) e^{2\pi\sqrt{-1}(x_i+(A_a^{-1}\mu)_i))} \ ,\\ (\xi_aU_\ib)(x;\mu)& =\xi_a(x+A_a^{-1}t_i;\mu-t_i) e^{-\pi\sqrt{-1}x^tA_a\theta_3 t_i} \ ,\qquad i=1,\cdots, n \ , \end{split}$$ and the endomorphisms are generated by $$\begin{split} (Z_i\xi_a)(x;\mu)& = e^{2\pi\sqrt{-1}(x_i+(A_a^{-1}\mu)_i))} \xi_a(x;\mu) \ ,\\ (Z_\ib \xi_a)(x;\mu)& = e^{\pi\sqrt{-1}x^tA_a\theta_3 t_i}\xi_a(x+A_a^{-1}t_i;\mu-t_i) \ . \end{split}$$ A constant curvature connection $\nabla_a: E_{g_a,\theta}\otimes\cL_\theta\to E_{g_a,\theta}$ is given as $$(\nabla_{a,1}\cdots \nabla_{a,2n})^t = {\begin{pmatrix}}\1_n & -\ov{2}A_a\theta_3 A_a \\ \0_n & -A_a {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}\ ,$$ with its curvature $$F_a = {\begin{pmatrix}}-A_a\theta_3 A_a & A_a \\ -A_a & \0_n {\end{pmatrix}}\ .$$ We assume that $A_{ab}$ is nondegenerate for any $a,b\in\Ob(\cCpre)$, $a\ne b$. For any $a,b\in\Ob(\cCpre)$, the space $\Hompre(a,b)$ is defined as follows. If $a\ne b$, $\Hompre(a,b):=\Hom(E_{g_a,\theta},E_{g_b,\theta}) =\cS({{\mathbb{R}}}^n\times({{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n))$; the right action of $T_{\theta_a}$, generated by $U_i$ and $U_\ib$, and the left action of $T_{\theta_b}$, generated by $Z_i$ and $Z_\ib$, are defined by $$\begin{split} (\xi_{ab} U_i)(x;\mu)& =\xi_{ab}(x;\mu) e^{2\pi\sqrt{-1}(x_i+(A_{ab}^{-1}\mu)_i))} \ ,\\ (\xi_{ab} U_\ib)(x;\mu)& =\xi_{ab}(x+A_{ab}^{-1}t_i;\mu-t_i) e^{-\pi\sqrt{-1}x^tA_{ab}\theta_3 t_i}\ ,\\ (Z_i \xi_{ab})(x;\mu)& = e^{2\pi\sqrt{-1}(x_i+(A_{ab}^{-1}\mu)_i))} \xi_{ab}(x;\mu) \ ,\\ (Z_\ib \xi_{ab})(x;\mu)& = e^{\pi\sqrt{-1}x^tA_{ab}\theta_3 t_i} \xi_{ab}(x+A_{ab}^{-1}t_i;\mu-t_i) \ . \end{split}$$ If $a=b$, then $\Hompre(a,b):=T_{\theta_a}=T_{\theta_b}$ and the left and right actions on it are defined by the usual product of noncommutative torus $T_{\theta_a}=T_{\theta_b}$. A constant curvature connection $\nabla_a: \Hompre(a,b)\otimes\cL_\theta\to\Hompre(a,b)$ is given by $$(\nabla_{1}\cdots \nabla_{2n})^t = {\begin{pmatrix}}\1_n & -\ov{2}A_{ab}^+\theta_3 A_{ab} \\ \0_n & -A_{ab} {\end{pmatrix}}{\begin{pmatrix}}\partial_x \\ 2\pi\sqrt{-1}x {\end{pmatrix}}$$ if $a\ne b$, and if $a=b$, it is defined by the usual derivation $\nabla$ of noncommutative torus $T_{\theta_a}=T_{\theta_b}$ (\[delta-End\]). For any $a,b,c\in\Ob(\cCpre)$ and $\xi_{ab}\in\Hompre(a,b)$, $\xi_{bc}\in\Hompre(b,c)$, the product $m:\Hompre(b,c)\otimes\Hompre(a,b)\to\Hompre(a,c)$ is given as follows: For $a=b$, it is the right action of $T_{\theta_a}=T_{\theta_b}$ on $\Hompre(b,c)$. For $b=c$, it is the left action of $T_{\theta_b}=T_{\theta_c}$ on $\Hompre(a,b)$. For $a=c$, the product $m:\Hompre(b,a)\otimes\Hompre(a,b)\to T_{\theta_a}$ is given by $$m(\xi_{ba},\xi_{ab})(x,\rho) =\sum_{\vec{m}\in{{\mathbb{Z}}}^n}\sum_{\mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n} U_{\vec{m}}\int_{{{\mathbb{R}}}^n} dx^n \xi_{ba}(x,\mu)\cdot (\xi_{ab}(x,-\mu)U_{-\vec{m}})\ .$$ For the remaining general case, it is given by $$m(\xi_{bc},\xi_{ab})(x,\rho) =\sum_{u\in{{\mathbb{Z}}}^n} \xi_{bc}(x',-u+\rho)\cdot \exp\(-\pi\sqrt{-1}{x'}^tA_{bc}\theta_3 A_{ab}x''\)\cdot \xi_{ab}(x'',u)\ ,$$ where $x':=x+A_{bc}^{-1}(u-A_{ab}A_{ac}^{-1}\rho)$ and $x'':=x-A_{ab}^{-1}(u-A_{ab}A_{ac}^{-1}\rho)$. The trace map is then given by eq.(\[Tr-End\]). By direct calculations, one obtains the followings. For a fixed noncommutative parameter of type $\theta_1$, $\theta_2$ or $\theta_3$, (Associativity): the composition $m$ of morphisms is associative. (Leibniz rule): the constant curvature connections on morphisms satisfy the Leibniz rule. Then, in any case of the Type $\theta_s$, $s=1,2,3$, $\cCpre$ forms a category in Definition \[defn:pre-C\] and then the corresponding category $\cC$ forms a cyclic CDG category. In particular, by looking at the condition that $d:V_{ab}\to V_{ab}$ satisfies $d^2=0$ explicitly, one can see the followings: For Type $\theta_1$, Type $\theta_2$ such that $\theta_2^t=-\theta_2$ and Type $\theta_3$, two objects $a,b\in\Ob(\cC)$ in the CDG category $\cC$ forms a full sub-DG category $\cC^\fh$ of $\cC$ for some $\fh\in\Lambda^2\subset V^2$ if and only if $$A_a\theta_s A_a=A_b\theta_s A_b\ ,\qquad s=1,2,3\ \label{cd}$$ holds. \[prop:cd\] Thus, we have obtained the cyclic DG categories $\cC^\fh$ on noncommutative tori with three types of noncommutativities $\theta_1$, $\theta_2$ and $\theta_3$. Let us calculate the compositions of the morphisms of the zero-th cohomologies $H^0(\cV):=\oplus_{a,b\in\Ob(\cC^\fh)} H^0(V_{ab})$ of these DG categories $\cC^\fh$. They define ring structures on $H^0(\cV)$, which are sub-rings of the full cohomologies $H^*(\cV)$. We shall observe that the ring $H^0(\cV)$ actually depends on the noncommutative parameter $\theta_s$, as opposed to the complex one-tori case [@foliation; @KimKim; @PoSc; @nchms]. We remark that, from a homotopy algebraic viewpoint, a DG category is homotopy equivalent to a [*minimal*]{} $A_\infty$ category. Then, by forgetting the higher compositions of the minimal $A_\infty$ category, one obtains the ring $H^*(\cV)$. Thus, if at least the sub-ring $H^0(\cV)$ is deformed, the minimal $A_\infty$-structure is also deformed. Recall that $H^0(V_{ab})$ is given by $\Ker(d:V_{ab}^0\to V_{ab}^1)= \cap_{i=1}^n\Ker(\nabb_i:V_{ab}^0\to V_{ab})$. [**Type $\theta_1$**]{}For $a,b\in\Ob(\cCpre)$, $a\ne b$, the holomorphic structure $\nabb_i:\Hompre(a,b)\to\Hompre(a,b)$ is given by $$(\nabb_1\cdots\nabb_n)^t = \left(\1_n + \frac{\sqrt{-1}}{2}A_{ab}^+\theta\right)\partial_x +2\pi A_{ab}x\ .$$ The cohomology $H^0(V_{ab})$ is spanned by the bases of the form $$\label{theta-vector_1} e_{ab}^\mu(x;\rho) :=C_{ab}\delm{A_{ab}}^\mu_\rho\exp\( -\pi\( x^t M_{ab} x \) \)\ , \qquad \mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n\ ,$$ where $C_{ab}\in{{\mathbb{C}}}$ is an appropriate rescaling and $M_{ab}\in\Mat_n({{\mathbb{C}}})$ should be a symmetric matrix satisfying $$(\nabb_1\cdots\nabb_n)\, \left(\exp\( -\pi\( x^t M_{ab} x \) \)\right)=0\ . \label{theta_1-cd}$$ The condition (\[theta\_1-cd\]) turns out to be $$-\left(\1_n+\frac{\sqrt{-1}}{2}A_{ab}^+\theta_1\right)M_{ab} +A_{ab}=0\ .$$ This $M_{ab}$ is symmetric if and only if the condition (\[cd\]) holds: $$A_a\theta_1 A_a=A_b\theta_1 A_b\ ,$$ and then the explicit form of $M_{ab}$ is given by $$M_{ab}=A_{ab} \(A_{ab}+\frac{\sqrt{-1}}{2}(A_a\theta_1 A_b-A_b\theta_1 A_a)\)^{-1}A_{ab}\ .$$ Here the real part of $M_{ab}\in\Mat_n({{\mathbb{C}}})$ should be positive definite in order for $e_{ab}^\mu$ to exist in $H^0(V_{ab})$. Note that the part of $M_{ab}$ is positive definite if and only if $A_{ab}$ is positive definite. This $e_{ab}^\mu\in H^0(V_{ab})$ in eq.(\[theta-vector\_1\]) is a [*theta vector*]{}. Thus, for $a,b\in\Ob(\cC)$ such that $A_a\theta_1 A_a=A_b\theta_1 A_b$ and $A_{ab}$ is positive definite, $\dim(H^0(V_{ab}))=\sharp({{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n)=\det(A_{ab})$. For the rescaling $C_{ab}$ in eq.(\[theta-vector\_1\]), we set $$C_{ab}:= \frac{\det(\1_n+\i A_a\theta)^{\ov{4}} \det(\1_n+\i A_b\theta)^{\ov{4}}} {\det(\1_n+\2i A_{ab}^+\theta)^{\ov{2}}}\ .$$ Assume now that $A_{ab}$ and $A_{bc}$ are positive definite. Then we get the product formula: $$\begin{split} & m(e^\mu_{ab},e^\nu_{bc}) =\frac{}{} \sum_{\rho\in{{\mathbb{Z}}}^n/A_{ac}{{\mathbb{Z}}}^n} \sum_{u\in{{\mathbb{Z}}}^n}\delm{A_{ab}}^\mu_{-u+\rho}\delm{A_{bc}}^\nu_u \\ &\quad \exp\(-\pi(u-A_{bc}A_{ac}^{-1}\rho)^t \((A_{ab}^{-1}+A_{bc}^{-1})(\1_n+\sqrt{-1}A_b\theta)^{-1}\) (u-A_{bc}A_{ac}^{-1}\rho)\) \cdot e_{ac}^\rho\ . \end{split}$$ Note that the $n$ by $n$ matrix $(A_{ab}^{-1}+A_{bc}^{-1})(\1_n+\sqrt{-1}A_b\theta)^{-1}$ is automatically symmetric due to the condition $A_a\theta_1 A_a=A_b\theta_1 A_b=A_c\theta_1 A_c$. In this Type $\theta_1$ case, these theta vectors $\{e_{ab}^\mu\}$ can be described by theta functions, where the product of two theta vectors just corresponds to the Moyal product of two theta functions [@nctheta]. [**Type $\theta_2$**]{}For $a,b\in\Ob(\cCpre)$, $a\ne b$, the holomorphic structure $\nabb_i:\Hompre(a,b)\to\Hompre(a,b)$ is given by $$(\nabb_1\cdots\nabb_n)^t = \partial_x +2\pi \((A_b^{-1}+\theta_2)^{-1}-(A_a^{-1}+\theta_2)^{-1}\)x\ .$$ The theta vectors are of the form $$e_{ab}^\mu(x,\rho)=\delm{A_{ab}}^\mu_\rho\, \exp\(-\pi x^t M_{ab} x\)\ ,\qquad \mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n\ ,$$ where $M_{ab}\in\Mat_n({{\mathbb{R}}})\subset\Mat_n({{\mathbb{C}}})$ is given by $$\label{Mab-theta_2} M_{ab}:=(A_b^{-1}+\theta_2)^{-1}-(A_a^{-1}+\theta_2)^{-1} =(\1_n+A_b\theta_2)^{-1}A_{ab}(\1_n+\theta_2A_a)^{-1} \ ,$$ which should be a symmetric positive definite matrix. Then, one has $\dim(H^0(V_{ab}))=\det(A_{ab})$. Assume that $M_{ab}$ and $M_{bc}$ are positive definite. Then, $M_{ac}$ is also positive definite. The product of $e_{ab}^\mu$ with $e_{bc}^\nu$ is then $$\begin{split} & m(e_{ab}^\mu,e_{bc}^\nu) = \sum_{\rho\in{{\mathbb{Z}}}^n/A_{ac}{{\mathbb{Z}}}^n} \sum_{u\in{{\mathbb{Z}}}^n}\delm{A_{ab}}^\mu_{-u+\rho}\delm{A_{bc}}^\nu_u \\ & \exp\(-\pi(u-A_{bc}A_{ac}\rho)^t ((A_{ab}^{-1}+A_{bc}^{-1})(\1_n+A_b\theta_2^t)(\1_n+A_b\theta_2)^{-1}) (u-A_{bc}A_{ac}\rho)\)\cdot e_{ac}^\rho\ . \end{split}$$ In particular, one can see that the structure constant does not depend on $\theta_2$ if and only if $theta_2$ is symmetric: $\theta_2^t=\theta_2$. This gives the reason that the structure constant of the product does not depend on the noncommutative parameter in the case of noncommutative real two-tori [@foliation; @PoSc; @nchms]. See also [@KimLee; @KimKim2], where for a complex two-tori with noncommutativity of Type $\theta_2$ with symmetric $\theta_2$, such structure constants are computed and checked to be independent of the noncommutativity $\theta_2$. When $\theta_2$ is antisymmetric, $M_{ab}$ in eq.(\[Mab-theta\_2\]) is symmetric if and only if $A_a\theta_2 A_a= A_b\theta_2 A_b$, where the $n$ by $n$ matrix $(A_{ab}^{-1}+A_{bc}^{-1})(\1_n+A_b\theta_2^t)(\1_n+A_b\theta_2)^{-1} \in\Mat_n({{\mathbb{C}}})$ is symmetric: $$(A_{ab}^{-1}+A_{bc}^{-1})(\1_n+A_b\theta_2^t)(\1_n+A_b\theta_2)^{-1} =(\1_n-\theta_2 A_b)^{-1}(\1_n-\theta_2^tA_b)(A_{ab}^{-1}+A_{bc}^{-1})\ .$$ [**Type $\theta_3$**]{}For $a,b\in\Ob(\cCpre)$, $a\ne b$, the holomorphic structure $\nabb_i:\Hompre(a,b)\to\Hompre(a,b)$ is given by $$(\nabb_1\cdots\nabb_n)^t = \partial_x + 2\pi\(\1_n-\frac{\sqrt{-1}}{2}A_{ab}^+\theta_3\)A_{ab} x \ .$$ The theta vectors are of the form $$e_{ab}^\mu(x,\rho)=\delm{A_{ab}}^\mu_\rho\, \exp\(-\pi x^t M_{ab} x\)\ ,\qquad \mu\in{{\mathbb{Z}}}^n/A_{ab}{{\mathbb{Z}}}^n\ ,$$ where $M_{ab}\in\Mat_n({{\mathbb{C}}})$ is given by $$M_{ab}:=\(\1_n-\frac{\sqrt{-1}}{2}A_{ab}^+\theta_3\)A_{ab}\ ,$$ which should be a symmetric matrix whose real part is positive definite. Here, again, the real part of $M_{ab}$ is positive definite if and only if $A_{ab}$ is positive definite. Then, one has $\dim(H^0(V_{ab}))=\det(A_{ab})$. The condition that $M_{ab}$ above is symmetric is equal to $$A_{ab}^+\theta_3A_{ab}=(A_{ab}^+\theta_3A_{ab})^t\ ,$$ which is in fact equivalent to $A_a\theta_3 A_a=A_b\theta_3 A_b$. Assume now that $A_{ab}$ and $A_{bc}$ are positive definite. The product of two theta vectors is given by $$\begin{split} & m(e_{ab}^\mu, e_{bc}^\nu) =\sum_{\rho\in{{\mathbb{Z}}}^n/A_{ac}{{\mathbb{Z}}}^n} \sum_{u\in{{\mathbb{Z}}}^n}\delm{A_{ab}}^\mu_{-u+\rho}\delm{A_{bc}}^\nu_{u} \\ &\quad \exp\(-\pi(u-A_{bc}A_{ac}^{-1}\rho)^t \((A_{ab}^{-1}+A_{bc}^{-1})(\1_n-\sqrt{-1}A_b\theta_3)\) (u-A_{bc}A_{ac}^{-1}\rho)\) \cdot e_{ac}^\rho\ . \end{split}$$ One can show that the matrix $(A_{ab}^{-1}+A_{bc}^{-1})(\1_n-\sqrt{-1}A_b\theta_3)$ defining a quadratic form in the expression above is symmetric: $$(A_{ab}^{-1}+A_{bc}^{-1})(\1_n-\sqrt{-1}A_b\theta_3) =(\1_n+\sqrt{-1}\theta_3 A_b)(A_{ab}^{-1}+A_{bc}^{-1})\ .$$ Thus, we have seen that the structure constants $C_{abc,\rho}^{\mu\nu}$ depend on the noncommutative parameter $\theta$ in all these three cases. Though in this paper we have fixed a constant curvature connection for a Heisenberg module, we can also take all the constant curvature connections on a Heisenberg module into account in a similar way as in the noncommutative complex one-tori case [@foliation; @nchms]. It might also be interesting to investigate the details of the moduli space of the constant curvature connections on a Heisenberg module on $T^{2n}_\theta$ which is known to form a commutative torus $T^{2n}$. We end with showing an example for the case of noncommutative complex two-torus ($n=2$). In this case, for any fixed $\theta_s$, $s=1,2,3$, the condition eq.(\[cd\]) reduces to $$\det(A_a)=\det(A_b)\ .$$ Thus, for the objects of $\cC^f$, one can in general have infinite number of objects. For instance, diagonal matrices $A\in\Mat_n({{\mathbb{Z}}})$ with $\det(A)=-4$ are $$A_a={\begin{pmatrix}}1 & 0 \\ 0 & -4{\end{pmatrix}}\ ,\quad A_b={\begin{pmatrix}}2 & 0 \\ 0 & -2{\end{pmatrix}}\ ,\quad A_c={\begin{pmatrix}}4 & 0 \\ 0 & -1{\end{pmatrix}}\ ,$$ and $A_{a'}:=-A_a$, $A_{b'}:=-A_b$, $A_{c'}:=-A_c$. Since the zero-th cohomologies of morphisms between $\{a,b,c\}$ and $\{a',b',c'\}$ are absent, let us concentrate on the one side $\{a,b,c\}$. Then, all $A_{ab}, A_{bc}$ and $A_{ac}$ are positive definite, and the dimensions of the zero-th cohomologies are $$\dim(H^0(V_{ab}))=2\ ,\qquad \dim(H^0(V_{bc}))=2\ ,\qquad \dim(H^0(V_{ac}))=9\ .$$ However, there exist infinite symmetric matrices $A\in\Mat_n({{\mathbb{Z}}})$ with $\det(A)=-4$, since the matrix $g^tA g$ has $\det(A)=-4$ for any $SL(2,{{\mathbb{Z}}})$ element $g$. [99]{} S. Barannikov, M. Kontsevich, “Frobenius manifolds and formality of Lie algebras of polyvector fields,” Internat. Math. Res. Notices 1998, 201–215. O. Ben-Bassat, “Mirror symmetry and generalized complex manifolds,” math.AG/0405303. O. Ben-Bassat, J. Block and T. Pantev, “Non-commutative tori and Fourier-Mukai duality,” math.AG/0509161. J. Block, “Duality and equivalence of module categories in noncommutative geometry I,” math.QA/0509284. M. Dieng and A. Schwarz, “Differential and complex geometry of two-dimensional noncommutative tori,” math.QA/0203160. L. Dabrowski, T. Krajewski and G. Landi, “Some properties of non-linear sigma models in noncommutative geometry,” Int. J. Mod. Phys. B [**14**]{} (2000) 2367, hep-th/0003099. “Non-linear sigma-models in noncommutative geometry: Fields with values in finite spaces,” Mod. Phys. Lett. A [**18**]{} (2003) 2371, math.QA/0309143. G. A. Elliott, H. Li, “Morita equivalence of smooth noncommutative tori,” math.OA/0311502; “Strong Morita equivalence of higher-dimensional noncommutative tori. II,” math.OA/0501030. K. Fukaya, “Mirror symmetry of abelian varieties and multi theta functions,” J. Algebraic Geom. [**11**]{} (2002), 393–512, preprint, Kyoto University, 1998, available at http://www.kusm.kyoto-u.ac.jp/ $\tilde{}$ fukaya/fukaya.html. K. Fukaya, “Floer homology of Lagrangian foliation and non-commutative mirror symmetry,” Preprint 98-08, Kyoto University, 1998, available at http://www.kusm.kyoto-u.ac.jp/ $\tilde{}$ fukaya/fukaya.html. K. Fukaya, Y. G. Oh, H. Ohta and K. Ono, “Lagrangin intersection Floer theory - anomaly and obstructon -,” preprint 2000, available at http://www.kusm.kyoto-u.ac.jp/ $\tilde{}$ fukaya/fukaya.html. M. Gualtieri, “Generalized complex geometry,” Oxford University DPhil thesis, math.DG/0401221. H. Kajiura, “Homotopy algebra morphism and geometry of classical string field theories,” Nucl. Phys. B [**630**]{} (2002) 361, hep-th/0112228. H. Kajiura, “Kronecker foliation, D1-branes and Morita equivalence of noncommutative two-tori,” JHEP [**0208**]{} (2002) 050, hep-th/0207097. H. Kajiura, “Noncommutative homotopy algebras associated with open strings,” math.QA/0306332. H. Kajiura, “Homological mirror symmetry on noncommutative two-tori,” hep-th/0406233. H. Kajiura, “Star product formula of theta functions,” math.QA/0510307. H. Kajiura and J. Stasheff, “Homotopy algebras inspired by classical open-closed string field theory,” arXiv:math.QA/0410291; “Open-closed homotopy algebras in mathematical physics,” hep-th/0510118. A. Kapustin, “Topological strings on noncommutative manifolds,” Int. J. Geom. Meth. Mod. Phys.  [**1**]{} (2004) 49, hep-th/0310057. A. Kapustin and Y. Li, “D-branes in Landau-Ginzburg models and algebraic geometry,” JHEP [**0312**]{}, 005 (2003), hep-th/0210296. A. Kapustin and Y. Li, “Topological Correlators in Landau-Ginzburg Models with Boundaries,” Adv. Theoret. Math. Phys. [**7**]{} (2003) 727-749, hep-th/0305136. E. Kim and H. Kim, “Moduli Spaces of Standard Holomorphic Bundles on a Noncommutative Complex Torus,” math.QA/0312228. E. Kim and H. Kim, “Noncommutative Riemann conditions,” hep-th/0410108. H. Kim and C. Y. Lee, “Theta functions on noncommutative T(4),” J. Math. Phys.  [**45**]{} (2004) 461, hep-th/0303091. A. Konechny and A. Schwarz, “Introduction to M(atrix) theory and noncommutative geometry,” hep-th/0012145, and references therein. M. Kontsevich, “Homological algebra of mirror symmetry,” Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995. M. Kontsevich “Deformation Quantization Of Poisson Manifolds,” Lett. Math. Phys. [**66**]{} (2003), 157–216, math.QA/9709040. C. I. Lazaroiu, “String field theory and brane superpotentials,” JHEP [**0110**]{} (2001) 018, hep-th/0107162. C. I. Lazaroiu, “D-brane categories,” Int. J. Mod. Phys. A [**18**]{} (2003) 5299, hep-th/0305095. M. Herbst, C. I. Lazaroiu and W. Lerche, “Superpotentials, A(infinity) relations and WDVV equations for open topological strings,” JHEP [**0502**]{} (2005) 071, hep-th/0402110. C. I. Lazaroiu and R. Roiban, “Holomorphic potentials for graded D-branes,” JHEP [**0202**]{} (2002) 038, arXiv:hep-th/0110288; “Gauge-fixing, semiclassical approximation and potentials for graded Chern-Simons theories,” JHEP [**0203**]{} (2002) 022, hep-th/0112029. H. Li, “Strong Morita equivalence of higher-dimensional noncommutative tori,” J. Reine Angew. Math. [**576**]{} (2004), 167–180, math.OA/0303123. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Phil. Soc. [**45**]{} (1949), 99–124. A. Polishchuk and A. Schwarz, “Categories of holomorphic vector bundles on noncommutative two-tori,” Commun. Math. Phys.  [**236**]{} (2003) 135, math.QA/0211262. A. Polishchuk and E. Zaslow, “Categorical mirror symmetry: the elliptic curve,” Adv. Theor. Math. Phys. [**2**]{} (1998), 443–470, math.AG/9801119. L. E. Positsel’skii, “Nonhomogeneous quadratic duality and curvature,” (Russian) Funktsional. Anal. i Prilozhen. [**27**]{} (1993), 57–66, 96; translation in Funct. Anal. Appl. [**27**]{} (1993), 197–204. M. Rieffel, “Projective modules over higher-dimensional non-commutative tori,” Canadian J. Math. [**40**]{}(1988), 257–338. M. Rieffel and A. Schwarz, “Morita equivalence of multidimensional noncommutative tori,” Internat. J. Math.[**10**]{} (2) (1999) 289-299, math.QA/9803057. J. D. Stasheff, “On the homotopy associativity of $H$-spaces, I, II,” Trans. Amer. Math. Soc. [**108**]{} (1963) 275, 293. M. Schlessinger and J. Stasheff, “Deformaion theory and rational homotopy type,” U. of North Carolina preprint, 1979; short version: “The Lie algebra structure of tangent cohomology and deformation theory,” J. Pure Appl. Alg. [**38**]{} (1985), 313–322. A. Schwarz, “Morita equivalence and duality,” Nucl. Phys. B [**534**]{} (1998) 720, hep-th/9805034. A. Schwarz, “Theta functions on noncommutative tori,” Lett. Math. Phys. [**58**]{} (2001), 81–90. math.QA/0107186. A. Schwarz, “Noncommutative supergeometry, duality and deformations,” Nucl.Phys. B [**650**]{} (2003) 475-496, hep-th/0210271, Extended version of hep-th/9912212. A. Takahashi, “Matrix Factorizations and Representations of Quivers I,” math.AG/0506347. A. Takahashi, “Homological algebra of weak differential graded algebras,” in preparation. X. Tang, A. Weinstein, “Quantization and Morita equivalence for constant Dirac structures on tori,” math.QA/0305413. E. Witten, “Mirror manifolds and topological field theory,” Essays on mirror manifolds, 120–158, Internat. Press, Hong Kong, 1992, hep-th/9112056. E. Witten, “Chern-Simons gauge theory as a string theory,” Prog. Math.  [**133**]{} (1995) 637, hep-th/9207094. [^1]: e-mail address: kajiura@yukawa.kyoto-u.ac.jp