Content-Type: multipart/mixed; boundary="-------------0007101547622" This is a multi-part message in MIME format. ---------------0007101547622 Content-Type: text/plain; name="00-288.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-288.keywords" Parabolic Anderson model, almost-sure asymptotics, large deviations, Dirichlet eigenvalues, screening effect. ---------------0007101547622 Content-Type: application/x-tex; name="1DpAm-final.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="1DpAm-final.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % On the screening effect of heavy lower tails % % in one-dimensional parabolic Anderson model % % %%%%%%%%%% % % Marek Biskup and Wolfgang Koenig % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \def\version{June 15, 2000} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\documentclass[11pt]{article} \documentclass[reqno,12pt]{amsart} %\usepackage[notref,notcite]{showkeys} %\pagestyle{plain} %\usepackage{babel} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{srcltx} \usepackage{euscript} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% DO YOU PREFER THE C.M. 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\vspace{0.2cm} {\flushleft\bf\arabic{section}.\arabic{subsection}~\bf #1 } \vspace{3mm}\nopagebreak} \newcommand{\sbsect}[1]{\vspace{0.1cm}\noindent {\bf #1}\vspace{0.1cm}} \renewcommand{\subsubsection}{% \secdef \subsubsect\sbsbsect} \newcommand{\subsubsect}[2][default]{% \refstepcounter{subsubsection} \nopagebreak \vspace{0.1\baselineskip} \nopagebreak {\flushleft \sffamily\slshape \arabic{section}.\arabic{subsection}.\arabic{subsubsection} \ % \sffamily #1\/.}\ } \newcommand{\sbsbsect}[1]{\vspace{0.1cm}\noindent {\bf #1}\ } %\newenvironment{proof}{{\bf Proof:\ }\nopagebreak}% %{\nopagebreak {\hspace*{\fill}\rule{2mm}{2mm}}\\ } %\newenvironment{example}{\refstepcounter{theorem} %{\bf Example \thetheorem\ }\nopagebreak }% %{\nopagebreak {\hfill\rule{2mm}{2mm}}\\ } %\newenvironment{remark}{\refstepcounter{theorem} %{\bf Remarks \thetheorem\ }\nopagebreak }% %{\nopagebreak {\hfill\rule{2mm}{2mm}}\\} %\newtheorem{theorem}{Theorem}[section] %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{proposition}[theorem]{Proposition} %\newtheorem{remark}[theorem]{Remarks} %\newtheorem{cor}[theorem]{Corollary} %\newtheorem{prop}[theorem]{Proposition} %\newtheorem{step}[theorem]{Step} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{formula}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %\begin{quote} %{\small %\rightline{Technical Report} %\rightline{MSR-TR-2000-36}} %\end{quote} %\vspace{10mm} \title[Screening in 1D parabolic Anderson model] {\large Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model} \author[Marek Biskup and Wolfgang K\"onig]{} \maketitle \thispagestyle{empty} \vspace{0.2cm} \centerline{\sc Marek Biskup$^1$\/ and Wolfgang K\"onig\small$^2$} \vspace{0.8cm} \centerline {\em $^1$Microsoft Research, One Microsoft Way, Redmond WA 98052, U.S.A.} \centerline{{\em email: }{\sf biskup@microsoft.com}} \centerline{\small and} \centerline{\em $^2$Fachbereich Mathematik~MA7-5, Technische Universit\"at Berlin,} \centerline{\em Stra\ss e des 17.~Juni~136, 10623 Berlin, Germany} \centerline{{\em email: }{\sf koenig@math.tu-berlin.de}} \vspace{0.5cm} \centerline{\small(\version)} \vspace{0.5cm} \begin{quote} \begin{quote} {\footnotesize {\bf Abstract:} } \footnotesize We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u\/\/$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d.\ potentials $(\xi(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $\xi(0)$; too heavy lower tails of $\xi(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $\xi(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study \cite{BK00} of this model in general dimension. \end{quote} \end{quote} \vfill \begin{tabular}{lp{13cm}} \multicolumn{2}{l} {{\footnotesize\it AMS Subject Classification: }\footnotesize Primary---60F10, %Large deviations 82B44; %Disordered systems (random Ising models, random Schrödinger % operators, etc.) Secondary---35B40, %Asymptotic behavior of solutions (of PDE....added M.B.) %82C44, %Dynamics of disordered systems (random Ising systems, etc.) 35K15. %Initial value problems for second-order, parabolic equations }\\ {\footnotesize\it Key words and phrases: } \footnotesize Parabolic Anderson model, almost-sure asymptotics, large deviations, Dirichlet \\ \footnotesize eigenvalues, screening effect. \end{tabular} \eject \section{Introduction} \subsection{Model and main aim} \label{Mass} \noindent In a recent paper \cite{BK00}, we have studied the asymptotic behavior of the solution $u(t,z)$ to the so-called parabolic Anderson model for non-positive i.i.d.\ potentials. Here we answer an open question concerning the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in dimension one for potentials lacking the first logarithmic moment. Interestingly, a new phenomenon arises: too heavy tails of the potential at $-\infty$ hamper the mass flow to remote areas, thus rendering the more favorable regions inaccessible. This effect is unique to $d=1$, since only in one-dimensional topology particles are not able to bypass deep broad valleys in the potential landscape. The general model is defined as follows. Let $u\colon[0,\infty) \times\Z^d\to[0,\infty)$ be the solution to the parabolic problem \begin{equation} \label{Anderson} \begin{array}{rcll} \displaystyle \partial_t \,u(t,z)\!\!\! &=&\!\!\!\kappa \DeltaD u(t,z)+\xi(z) u(t,z),\qquad &(t,z)\in(0, \infty)\times \Z^d,\\ u(0,z)\!\!\!&=&\!\!\!1,&z\in\Z^d, \end{array} \end{equation} where $\partial_t$ is the time derivative, $\kappa>0$ is the diffusion constant, $\DeltaD$ is the discrete Laplacian on~$\Z^d$, $[\DeltaD f](z)=\sum_{x\sim z}[f(x)-f(z)]$, and $\xi=(\xi(z))_{z\in\Z^d}$ is an i.i.d.\ field. We use $\langle\,\cdot\,\rangle$ to denote the expectation with respect to $\xi$ and $\prob(\cdot)$ to denote the underlying probability measure. One interpretation of the quantity $u(t,z)$ is the total expected mass accumulated at time $t$ by a particle starting at $z$ at time $0$ and diffusing through a random field of sources (sites $x$ with $\xi(x)>0$) and sinks (sites $x$ with $\xi(x)<0$). The references \cite{GM90}, \cite{CM94} and \cite{K00} provide more explanation and other interpretations. Besides \cite{BK00}, the large-$t$ behavior of the solution to \eqref{Anderson} has extensively been studied (in general dimension) for various other classes of distributions: see \cite{GM90,GM98,GH99} for $\xi$ having the so-called double-exponential upper tail, and \cite{GK98,GKM99} for a continuous variant of \eqref{Anderson} with $\xi$ either Gaussian or (smeared) Poissonian field. %Closely related is also the recent %work on the ground state energy of a random Hamiltonian \cite{MW00}. The techniques used in these studies go back to the pioneering work of Donsker and Varadhan \cite{DV75,DV79}; however, there is also an intimate relation to Sznitman's method of enlargement of obstacles \cite{S98}. We refer to \cite{K00} for a comprehensive discussion of these relations and a unified presentation of the above results. Henceforth, we shall focus on the almost-sure behavior of $u(t,0)$ in the non-positive case, i.e., $\xi\in[-\infty,0]^{\Z^d}$. In dimensions $d\ge2$, the analysis in \cite{BK00} produced a fairly complete picture. Indeed, interesting behavior occurs only when $p=\prob(\xi(0)>-\infty)>p_{\text{c}}(d)$, the threshold for site percolation on $\Z^d$, and when conditioned on the event that the origin lies in the infinite cluster of sites $x$ with $\xi(x)>-\infty$. Below and, provided there is no critical percolation (which is rigorously known for $d=2$ \cite{R78} and $d\ge19$ \cite{HS90}), also at $p_{\text{c}}(d)$, and also when the origin lies in a finite cluster for $p>p_{\text{c}}(d)$, the quantity $u(t,0)$ decays exponentially in $t$ with a $\xi$-dependent rate. In dimension $d=1$, we have $p_{\text{c}}(d)=1$, which necessitated setting $\prob(\xi(0)=-\infty)=0$ in \cite{BK00}. However, the latter condition was not sufficient because the existence of the first logarithmic moment, i.e., $\langle \log(-\xi(0)\vee1)\rangle<\infty$, also had to be assumed in order to establish an asymptotics analogous to the supercritical case in $d\ge2$. In particular, two intriguing questions remained unanswered: \begin{enumerate} \item[$\bullet$] Is $\langle \log(-\xi(0)\vee1)\rangle<\infty$ optimal in the sense that $\langle \log(-\xi(0)\vee1)\rangle=\infty$ implies a strictly different asymptotic behavior of $u(t,0)$? \item[$\bullet$] What is the precise decay rate when the finiteness of $\langle \log(-\xi(0)\vee1)\rangle$ is robustly violated (keeping however the restriction to ``no atom at $-\infty$'')? \end{enumerate} In this paper we give answers to these questions under mild regularity conditions on the lower tail of the distribution of $\xi$. In particular, we show that $\langle \log(-\xi(0)\vee1)\rangle<\infty$ is only marginally non-optimal for the behavior described in \cite{BK00}, see Remark~\ref{rem2} after Theorem~\ref{mainres}. As it turns out, the decay of $u(t,0)$ is determined solely by upper {\it and\/} lower tails of $\prob(\xi(0)\in\cdot)$. The reason why the intermediate part of the distribution does not play any role is that these tails give rise to two dominant and mutually competing mechanisms (field-shape optimization in the upper tail {\sl versus} screening effect in the lower tail) whose balancing determines the decay rate. See Subsection~\ref{heuristics} for more precise heuristic explanation. \subsection{Our assumptions} \label{ass} \noindent We proceed by stating precisely the needed assumptions, both on upper and lower tails of $\xi(0)$. First we restrict ourselves to dimension $d=1$ for the sequel of this paper. In accord with \cite{BK00}, we consider the distributions with the upper tail of the form \begin{equation} \label{perfect} \prob\bigl(\xi(0)\ge-x\bigr)= \exp\bigl\{- x^{-\frac\gamma{1-\gamma}+o(1)}\bigr\},\qquad x\downarrow0, \end{equation} for some $\gamma\in[0,1)$. However, instead of the distribution function, it is more convenient to work with the cumulant generating function \begin{equation} \label{Hdef} H(\ell)=\log\langle e^{\ell\xi(0)}\rangle,\qquad \ell\ge0. \end{equation} The regime in \eqref{perfect} corresponds to the behavior $H(\ell)=-\ell^{\gamma+o(1)}$ as $\ell\to\infty$. \vspace{0.1cm} \noindent{\bf Assumption~(H).\/} {\it Let\/ $\esssup \xi(0)=0$ and suppose there are constants\/ $A>0$ and\/ $\gamma\in[0,1)$, and a positive increasing function\/ $t\mapsto\alpha_t$ such that \begin{equation} \label{mainass} \lim_{t\to\infty}\frac{\alpha_t^{3}}{t}H\Bigl(\frac t{\alpha_t} y\Bigr)=-A y^\gamma,\qquad y>0. \end{equation} } \vspace{0.1cm} The limit in \eqref{mainass} is necessarily uniform on compact sets in $(0,\infty)$, the pair $(A,\alpha_t)$ is unique up to a scaling transformation. Moreover, $t\mapsto\alpha_t$ is regularly varying and $\alpha_t=t^{\nu+o(1)}$ as $t\to\infty$ where $\nu=(1-\gamma)/(3-\gamma)\in(0,1/3]$. In particular, $t/\alpha_t\to\infty$, i.e., Assumption~(H) indeed controls the upper tails of $\xi(0)$. We say that $H$ {\it is in the $\gamma$-class\/} if \eqref{mainass} holds. Next we formulate our assumption on the lower tails of $\xi(0)$ at $\essinf\xi(0)=-\infty$. As the opposite case has already been handled in \cite{BK00}, we shall focus on the case where $\log(-\xi(0)\vee1)$ is not integrable. Central to our attention are lower tails of the form \begin{equation} \label{lowertails} \prob\bigl(\log(-\xi(0)\vee1)>x\bigr)=x^{-\zeta+o(1)},\qquad x\to\infty, \end{equation} with some $\zeta\in[0,1]$. In terms of the modified cumulant generating function \begin{equation} \label{G} G(\ell)=-\log\bigl\langle (-\xi(0)\vee1)^{-1/\ell}\bigr\rangle, \qquad \ell> 0, \end{equation} the behavior \eqref{lowertails} roughly corresponds to $G(\ell)=\ell^{-\zeta+o(1)}$ as $\ell\to \infty$. Note that $G$ is positive and decreasing since $\essinf\xi(0)<-1$. The following is a weak regularity condition for $G$ at infinity. \vspace{0.1cm} \noindent{\bf Assumption~(G).\/} {\it Let\/ $\langle\log(-\xi(0)\vee1)\rangle=\infty$ but\/ $\prob(\xi(0)=-\infty)=0$. Suppose that for each\/ $\eta\in(0,1)$\/ there is a function\/ $\widetilde G_\eta \colon(0,\infty)\to(0,\infty)$ with the following properties: \begin{enumerate} \item[(i)] $\widetilde G_\eta(\ell)\le G(\ell)^{\eta+o(1)}$ as $\ell\to\infty$. \item[(ii)] $\ell\mapsto1/\widetilde G_\eta(\ell)$ is increasing and concave for $\ell$ large enough. \item[(iii)] The random variable $1/\widetilde G_\eta(\log(-\xi(0)\vee1))$ has the first moment. \end{enumerate} } \vspace{0.1cm} \begin{remark} \label{rem1} As it turns out, Assumption~(G) is needed only for the proof of the lower bound in our main result (see Theorem~\ref{mainres} below); the upper bound requires no assumptions at all. The role of Assumption~(G) and particularly of its part (i) is the following: Abbreviate $Y=\log(-\xi(0)\vee1)$ and note that, for any $\delta\in(0,1]$, $G(\ell)\le\langle Y^\delta\rangle \ell^{-\delta}$. Therefore, $G(\ell)\le\ell^{-\zeta_*+o(1)}$ where $\zeta_*=\sup\{\delta\ge0\colon\langle Y^\delta\rangle<\infty\}$. However, a lower bound of the same (even asymptotic) form requires some regularity of $\ell\mapsto G(\ell)$ as $\ell\to\infty$, which is the essence of (i--iii). \end{remark} \begin{remark} \label{rem1'} In the view of Remark~\ref{rem1}, it is immediate that Assumption~(G) holds for $G(\ell)= \ell^{-\zeta+o(1)}$ with some $\zeta\in(0,1]$. The reason why we prefer the above (little cumbersome) setting as opposed to simple regularity of $G$ is that many cases with $G(\ell)=\ell^{o(1)}$ are automatically included. Indeed, consider the following example: Let $\theta>0$ and $\prob(\log(-\xi(0)\vee1)\in dx)\sim C/[x\log^{1+\theta}(x)]dx$ as $x\to\infty$, where $C$ is the normalizing constant. Then $G(\ell)\sim C'(\log\ell)^{-\theta}$ and Assumption~(G) holds with $\widetilde G_\eta(\ell)=G(\ell)[\log\log(\ell\vee e)]^{1+\theta'}$ for any $\eta<1$ and any $\theta'>0$. \end{remark} \subsection{Main result} \label{result} \noindent We begin by defining the scale function of the almost-sure asymptotics: \begin{equation} \label{b_t} \frac{b_t}{\alpha_{b_t}^2}=-\log G(t),\qquad t>0. \end{equation} In other words, $t\mapsto b_t$ is the inverse of the function $t\mapsto t\alpha_t^{-2}$ (which we may and shall assume to be strictly increasing), evaluated at $-\log G(t)$. Note that, since $\lim_{\ell\to\infty}G(\ell)=0$, we have $b_t\to\infty$ as $t\to\infty$. If $G(\ell)=\ell^{-\zeta+o(1)}$ as $\ell\to\infty$ for some $\zeta\in(0,1]$, then $\alpha_{b_t}^2= \zeta^{\beta}(\log t)^{\beta+o(1)}$, where $\beta=2\nu/(1-2\nu)=2(1-\gamma)/(1-3\gamma)\in(0,2]$. In the case $\zeta=0$, $\alpha_{b_t}=o(\log^\beta t)$ as $t\to\infty$. Here is our main result. The constant $\widetilde\chi$ appearing in \eqref{mainreseq} depends only on $A$, $\gamma$ and $\kappa$ and will be defined in Subsection~\ref{defs}. \begin{theorem} \label{mainres} Let $d=1$ and suppose that Assumption~(H) and Assumption~(G) hold. Define $t\mapsto b_t$ as in \eqref{b_t}, and let $\widetilde\chi$ be the constant in Theorem~1.5 of\/\/ \cite{BK00}. Then \begin{equation} \label{mainreseq} \lim_{t\to\infty}\frac{\alpha_{b_t}^2}t\log u(t,0)=-\widetilde\chi, \qquad \prob\text{\rm-almost surely.} \end{equation} \end{theorem} \vspace{2mm} Interestingly, if $\ell\mapsto G(\ell)$ has a power-law decay as $\ell\to\infty$, the lower-tail dependence of the rate can explicitly be computed. This allows for an easy comparison with the assertion in Theorem~1.5 of \cite{BK00}. Let $t\mapsto b_t^*$ be the scale function introduced in \cite{BK00}: \begin{equation} \label{b_t*} \frac{b_t^*}{\alpha_{b_t^*}^2}=\log t. \end{equation} Recall $\zeta_*=\sup\{\delta\ge0\colon\langle[\log(-\xi(0)\vee1)] ^\delta\rangle<\infty\}$. For $G$ decaying with a power law, necessarily, $G(\ell)=\ell^{-\zeta_*+o(1)}$. The following is an immediate consequence of Theorem~\ref{mainres} and the regularity of $t\mapsto\alpha_t$: \begin{cor}\label{special} Let $d=1$, suppose $\prob(\xi(0)=-\infty)=0$ and suppose that Assumption~(H) holds. Assume that either $\zeta_*=1$ or $\zeta_*\in(0,1)$ and $G(\ell)=\ell^{-\zeta_*+o(1)}$ as $\ell\to\infty$. Then \begin{equation} \label{correseq} \lim_{t\to\infty}\frac{\alpha_{b_t^*}^2}t\log u(t,0)= -\zeta_*^{-\beta}\widetilde\chi, \qquad \prob \text{\rm-almost surely.} \end{equation} where $\beta=2(1-\gamma)/(1-3\gamma)$. \end{cor} \begin{remark} \label{rem2} By comparison of Corollary~\ref{special} and Theorem~1.5 of \cite{BK00}, $\zeta_*=1$ is necessary and sufficient for the assertion of the latter to hold, at least in the class of distribution with $G$ decaying as a positive power. In particular, the condition that $\langle\log(-\xi(0)\vee 1)\rangle<\infty$ in \cite{BK00} is only marginally non-optimal because Theorem~1.5 also literally holds if we just assume that $[\log(-\xi(0)\vee 1)]^\delta$ be integrable for any $\delta<1$. This answers the first of the questions above. \end{remark} \begin{remark} \label{rem2'} The cases with $\zeta_*>0$ have a different absolute size of the rate while the time dependence remains as for $\zeta_*=1$. However, when $\zeta_*=0$, also the time dependence changes. For instance, in the aforementioned example $\prob(\log(-\xi(0)\vee 1)\in dx)\sim C/[x\log^{1+\theta}(x)]\,dx$ as $x\to\infty$ (see Remark~\ref{rem1'}), $\alpha_{b_t}^2=[\log\log t]^{\beta+o(1)}$, which grows much slower than in the case $\zeta_*>0$. For yet thicker lower tails, even slower growths are possible. We conclude that the result of Theorem~1.5 of \cite{BK00} qualitatively changes only when $-\xi(0)$ lacks all positive logarithmic moments. \end{remark} The remainder of this paper is organized as follows: In the next section we define some important objects and use them to give a heuristic outline of the proof. The actual proof comes in Section~\ref{proof}. Since many steps can almost literally be taken over from \cite{BK00}, we stay as terse as possible. The essentially novel part are Lemmas~\ref{trivbound}, \ref{LLN}, and~\ref{last}. \section{Definitions and heuristics} \subsection{Auxiliary objects} \label{defs} \noindent For the sake of both completeness and later reference, we will now introduce the objects needed to define the quantity $\widetilde\chi$ in Theorem~\ref{mainres}. Then we proceed by recalling the Feynman-Kac representation and some formulas for Dirichlet eigenvalues. \subsubsection{Definition of $\widetilde\chi$} Let $\F_R$ be the set of continuous functions $f\colon\R\to[0,\infty)$ satisfying $\supp f\subset [-R,R]$ and having total integral equal to one. Let $C^+(R)$ (resp., $C^-(R)$) be the set of continuous functions $[-R,R]\to [0,\infty)$ (resp.\ $[-R,R]\to (-\infty,0]$). For $H$ in the $\gamma$-class, let $\skrih_R\colon C^+(R)\to(-\infty,0]$ be the functional defined by \begin{equation} \skrih_R(f)=-A\int_{[-R,R]} f^\gamma\, \1_{\{f>0\}}\, dx, \end{equation} where $A$ is as in \eqref{mainass}. Let $\L_R\colon C^-(R)\to[0,\infty]$ be the Legendre transform of $\skrih_R$: \begin{equation}\label{calLdef} \L_R(\psi)=\sup\bigl\{(f,\psi)-{\mathcal H}_R(f)\colon f\in C^+(R),\,\supp f\subset\supp\psi\bigr\}, \end{equation} where $(f,\psi)=\int f(x)\psi(x)dx$. Conventionally, $\L_R(0)=\infty$. If $H$ is in the $\gamma$-class with a $\gamma\in[0,1)$, $\L_R(\psi)$ can explicitly be computed: for any $\psi\in C^-(R)$, $\psi\not\equiv0$, \begin{equation} \label{L_R} \L_R(\psi)=\begin{cases}(1-\gamma^{-1})(A\gamma)^{\frac{1}{1-\gamma}} \int |\psi(x)|^{-\frac\gamma{1-\gamma}}\,dx, \quad&\text{if } \gamma\in(0,1),\\-A|\supp \psi|,&\text{if }\gamma=0, \end{cases} \end{equation} where $|\supp \psi|$ is the Lebesgue measure of $\supp \psi$. (Here $\L_R(\psi)=\infty$ whenever $\gamma\in(0,1)$ and the integral diverges.) The last object we need is the principal eigenvalue of the operator $\kappa\Delta+\psi$ on $L^2([-R,R])$ with Dirichlet boundary conditions: \begin{equation} \label{conteigen} \lambda_R(\psi)=\sup\bigl\{(\psi, g^2)-\kappa\|\nabla g\|_2^2\colon g\in C_{\rm c}^\infty(\supp\psi,\R),\|g\|_2 =1\bigr\}, \end{equation} with the interpretation $\lambda_R(0)=-\infty$. Then \begin{equation} \label{chitildedef} \widetilde\chi=-\sup_{R>0}\,\sup\left\{\lambda_R(\psi)\colon\psi\in C^-(R),\, {\mathcal L}_R(\psi)\leq 1\right\}. \end{equation} As was proved in \cite{BK00}, $\widetilde\chi\in(0,\infty)$. \begin{remark} In $d=1$, the minimizer of an associated variational problem (namely, that for the annealed or moment asymptotics) can explicitly be computed, see \cite{BK98}. Proposition~1.4 of \cite{BK00} then allows $\widetilde \chi$ to be evaluated in a closed form. Except for $\gamma=0$, no such expression is known in higher dimensions. \end{remark} \subsubsection{Feynman-Kac formula, Dirichlet eigenvalues} Let $(X(s))_{s\in[0,\infty)}$ be the continuous-time simple random walk on $\Z$ with generator $\kappa\Delta^{\rm d}$. We use $\E_x$ to denote the expectation with respect to the walk starting at $x$. The Feynman-Kac representation for $u(t,\cdot)$ then reads \begin{equation} \label{FKF} u(t,x)=\E_x\Bigl[\exp\Bigl\{\int_0^t\xi\bigl(X(s)\bigr)ds\Bigr\}\Bigr]. \end{equation} Given $R>0$, let $Q_R=[-R,R]\cap\Z$ and let $u_R(t,x)$ be the solution to the system \eqref{Anderson} in $Q_R$ and Dirichlet boundary condition $u_R(\cdot,x)=0$ for $x\not\in Q_R$. Let $\tau_R$ be the first exit time from $Q_R$, i.e., $\tau_R=\inf\{s>0\colon X(s)\not\in Q_R\}$. Then \begin{equation} \label{FKbox} u_R(t,x)=\E_x\Bigl[\exp\Bigl\{\int_0^t\xi\bigl(X(s)\bigr)ds\Bigr\} \1\{\tau_R>t\}\Bigr] \end{equation} Note that $R\mapsto u_R(t,x)$ is increasing. In the forthcoming developments we will also need the principal Dirichlet eigenvalue of the operator $\kappa\DeltaD+\xi$ in the box $z+Q_R$ centered at $z$: \begin{equation} \label{eigenval} \lambdaD_{z;R}(\xi)=\sup\biggl\{\,\sum_{x\in Q_R}\xi(x+z)g(x)^2+\kappa\sum_{x\in Q_R} g(x)[\DeltaD g](x)\colon g\in\ell^2(Q_R),\,\Vert g\Vert_2=1\biggr\}. \end{equation} Note that, by the standard eigenvalue expansion (see \cite{BK00}), \begin{equation} \label{ulambda} {\rm e}_R(z)^2 e^{t\lambdaD_{z;R}(\xi)}\le u_R(t,z)\le \#Q_R \,e^{t\lambdaD_{z;R}(\xi)}, \end{equation} where ${\rm e}_R(\cdot)$ is the $\ell^2$-normalized principal eigenvector in $Q_R$. In particular, the logarithmic asymptotics of $u_R(t,z)$ and the asymptotics of $t\lambdaD_{z;R}(\xi)$ coincide provided $R=R(t)$ does not grow too fast with $t$ (which ensures that $t\mapsto{\rm e}_{R(t)}(z)^2$ does not decay too fast). \subsection{Heuristic explanation} \label{heuristics} \noindent As alluded to in the introduction, \eqref{mainreseq} results from the competition of two mechanisms: (1)~searching for optimal shapes of the potential by the walk in \eqref{FKF} and (2)~screening off far away sites by regions of strongly negative potential. Let us describe this interplay in detail. To avoid cluttering of indices we often use $\alpha(b_t)$ in the place of $\alpha_{b_t}$. Consider a ``macrobox'' $Q_{r(t)}=[-r(t),r(t)]\cap\Z$ with $r(t)\approx\exp[b_t\alpha(b_t)^{-2}]$, where we think of $b_t$ as of a yet undetermined scale function. Fix $R>0$ and a shape function $\psi\in C^-(R)$ satisfying ${\mathcal L}_R(\psi)< 1$. A Borel-Cantelli argument shows that there exists a randomly located microbox in $Q_{r(t)}$, with diameter $2R\alpha(b_t)$, where $\xi$ is shaped like $\psi_t(\cdot)\approx\psi(\cdot/\alpha(b_t))/ \alpha(b_t)^2$. Let us assume that $R$ and $\psi$ approximately maximize \eqref{chitildedef}, i.e., $\lambda_R(\psi)\approx -\widetilde \chi$. Then the dominating strategy for the walk is to move in a short time to that favorable microbox and spend the rest of the time until $t$ in it. The contribution coming from the long stay in the microbox is roughly $\exp[t\lambda_{R\alpha(b_t)}(\psi_t)]$, which can be approximated by $\exp[t\alpha(b_t)^{-2}\lambda_R(\psi)]\approx \exp[-t\alpha(b_t)^{-2}\widetilde\chi]$, using the scaling properties of the Laplace operator. The size of the macrobox is determined by the amount of mass the walk loses on the way from the origin to the favorable microbox, while traveling through long stretches of large negative potential. A calculation shows that the penalty it pays is roughly of order $\exp[-\sum_{x=1}^{r(t)} \log(-\xi(x)\vee1)]$. (An optimal strategy is not to spend more than $(-\xi(x)\vee1)^{-1}$ time units at each site $x$ on the way.) Under our assumptions on the lower tails of $\xi(0)$, a Borel-Cantelli argument shows that this penalty is roughly $\exp[-G^{-1}(1/r(t))]$ where $G^{-1}$ denotes the inverse function of $G$. As it turns out, the two mechanisms run at optimal ``speed'' when the two exponents are roughly of the same order, i.e., $G^{-1}(1/r(t))\approx t\alpha(b_t)^{-2}\approx t$, because $\alpha_{b_t}\ll t$. Recalling that $r(t)\approx\exp[b_t\alpha(b_t)^{-2}]$, this reasoning leads to \eqref{b_t}. A fine tuning of $r(t)$ makes the contribution from the travel to the microbox negligible compared to the contribution from the stay in it, i.e., we shall in fact have $G^{-1}(1/r(t))=o(t\alpha(b_t)^{-2})$. Hence, we obtain \eqref{mainreseq} with $\widetilde\chi$ as in \eqref{chitildedef}. \section{Proof of Theorem~\ref{mainres}} \label{proof} \noindent As in \cite{BK00}, the main result will be proved by separately proving upper and lower bounds in \eqref{mainreseq}. The proof of Corollary~\ref{special} comes at the very end of this section. \subsection{The upper bound} \label{upperbd} \noindent Recall the notation of Subsection~\ref{defs}, in particular that $Q_R=[-R,R]\cap\Z$. Let \begin{equation} \label{r(t)} r(t)=-\frac3{G(t)}\log G(t). \end{equation} Note that $r(t)= t^{\zeta+o(1)}$ as $t\to\infty$ if $\widetilde G(\ell)=\ell^{-\zeta+o(1)}$ as $\ell\to\infty$. Abbreviate $B_R(t)=Q_{r(t)+2\lfloor R\rfloor}$. The crux of the proof of the upper bound in Theorem~\ref{mainres} is the following generalization of Proposition~4.4 of \cite{BK00} adapted to the new definition of $r(t)$. \vbox{ \begin{proposition} \label{propcompact} There exists a constant $C=C(\kappa)>0$ and a random variable $C_\xi\in(0,\infty)$ such that, $\prob$-almost surely, for all $R,t>C$, \begin{equation} \label{compact} u(t,0)\leq C_\xi e^{-t}+e^{Ct/R^2}3r(t) \exp\Bigl\{t\max_{z\in B_R(t)}\lambdaD_{z;2R}(\xi)\Bigr\}. \end{equation} \end{proposition} } \begin{proofsect}{Proof of Theorem~\ref{mainres}, upper bound} With Proposition~\ref{propcompact} in the hand, the proof goes along very much the same lines as in \cite{BK00}. Indeed, let $r(t)$ be as in \eqref{r(t)} and set $R$ in \eqref{compact} to be $R\alpha(Kb_t)$, where $K>0$ will be chosen later and $R$ will tend to $\infty$. Let $H$ be in the $\gamma$-class and recall that $\alpha_t=t^{\nu+o(1)}$ with $\nu=(1-\gamma)/(3-\gamma)$. Abbreviate $B(t)=B_{R\alpha(Kb_t)}(t)$ and $\lambda(z)=\lambdaD_{z;{2R\alpha(Kb_t)}}(\xi)$, and note that $r(t)\le e^{o(t\alpha(b_t)^{-2})}$. Then, using also that $\lim_{t\to\infty}\alpha(Kb_t)/\alpha(b_t)=K^\nu$, we have from \eqref{compact} that \begin{equation} \label{upperas} \limsup_{t\to\infty}\frac {\alpha_{b_t}^2}t\log u(t,0)\leq \frac C{K^{2\nu} R^2}+\limsup_{t\to\infty}\Bigl[\alpha_{b_t}^2 \max_{z\in B(t)}\lambda(z)\Bigr], \end{equation} $\prob$-almost surely. Abbreviating $M(t)=\max_{z\in B(t)}\lambda(z)$, we have to prove that, for any $\eps>0$, \begin{equation}\label{aim1} \limsup_{t\to\infty}\alpha_{b_t}^2 M(t)\leq -\widetilde \chi+\eps,\qquad \prob\text{-almost surely}, \end{equation} for some appropriate $K\in(0,\infty)$ and sufficiently large $R$. Note that the eigenvalues $\lambda(z)$ have identical distribution. Furthermore, their exponential moments can be estimated by \begin{equation}\label{expmom} \limsup_{R\to\infty}\,\limsup_{t\to\infty}\frac{\alpha_{b_t}^2}{b_t} \log\bigl\langle e^{Kb_t\lambda(z)}\bigr\rangle\le -K^{1-2\nu}\chi, \end{equation} where $\chi\in(0,\infty)$ is a constant related to $\widetilde\chi$, see \cite{BK00}. Since $t\mapsto M(t)$ is increasing and $t\mapsto\alpha_{b_t}$ slowly varying, it suffices to prove \eqref{aim1} for $t$ taking only a discrete set of values; the main difference compared to \cite{BK00} is that now we take \begin{equation} \frac1{G(t)}\in\{e^n\colon n\in\N\}. \end{equation} Let $G(t)=e^{-n}$ and note that \eqref{b_t} implies that $b_t\alpha_{b_t}^{-2}=n$. The proof now proceeds exactly as in \cite{BK00}: We let $p_n(\eps)=\prob(M(t)\alpha_{b_t}^2\ge-\widetilde\chi+\eps)$ and use the Chebyshev inequality and \eqref{expmom} to derive that $p_n(\eps)$ is summable on $n$ for all $\eps>0$, provided $K$ is chosen appropriately and $R$ is sufficiently large. The claim is finished using the Borel-Cantelli lemma. \qed \end{proofsect} It remains to prove Proposition~\ref{propcompact}. In \cite{BK00}, the choice $t\log t$ for $r(t)$ allowed us to use a simple probability estimate for the simple random walk; in particular, the corresponding bound \eqref{compact} held true uniformly in all non-positive potentials. In our present cases, $r(t)$ is typically much smaller than $t\log t$ and the potential has to cooperate to get the bound \eqref{compact}. Unlike in \cite{BK00}, the role of the potential is actually dominant in the cases of our present interest. \begin{lemma} \label{trivbound} For any $b\in(2\kappa,\infty)$ there is a random variable $C(\xi)\in(0,\infty)$ such that, $\prob$-almost surely, \begin{equation} \label{aprioribd} u(t,0)-u_R(t,0)\le C(\xi)\biggl(\,\prod_{x=0}^R\frac{b}{-\xi(x)\vee b} +\prod_{x=-R}^{0}\frac{b}{-\xi(x)\vee b}\biggr),\qquad R\in\N, t\geq 0. \end{equation} \end{lemma} \begin{proofsect}{Proof} Let $(X_k)_{k\in\N_0}$ be the embedded discrete-time simple random walk on $\Z$ and let $\ell_n(x)$ be its local times defined by $\ell_n(x)=\sum_{k=1}^n \1\{X_k=x\}$. Let $\ED_y$ denote the expectation with respect to the discrete-time walk, starting at $y\in\Z$. Abbreviate $\xi_k=\xi(X_k)$ and $\widehat u_R(t,0)=u(t,0)-u_R(t,0)$. Then, by \eqref{FKF} and \eqref{FKbox}, \begin{equation} \label{discreterep} \widehat u_R(t,0)=e^{-2\kappa t}\sum_{n\ge R} (2\kappa)^{n} \,\ED_0\biggl[\int_{\vartriangle_n\!(t)} dt_1\dots dt_{n}\, \exp\Bigl\{\sum_{k=0}^n\xi_kt_k\Bigr\} \1\{\supp\ell_n\not\subset Q_R\}\biggr], \end{equation} where $\vartriangle_n\!\!(t)=\{(t_1,\dots,t_{n})\in(0,\infty)^n\colon t_1+\dots+t_{n}\le t\}$, and $t_0$ is a shorthand for $t-(t_1+\dots+t_{n})$. Fix $b>2\kappa$ and define \begin{equation} \mathcal A_n=\bigl\{x\in\supp\ell_n\colon \xi(x)\le -b\bigr\}. \end{equation} Let \begin{equation} {\mathcal I}_n=\bigl\{k\in\{1,\dots, n\}\colon X_k\notin\mathcal A_n\bigr\} \end{equation} be the set of all the times at which the walk visits a point $x$ with $\xi(x)> -b$. By relaxing the constraint $t_1+\dots+t_n\le t$ in $\vartriangle_n\!\!(t)$ to $t_k\le t$ for every $k\in {\mathcal I}_n$, neglecting the terms with $k\in{\mathcal I}_n\cup\{0\}$ in the exponential, and integrating out $t_1,\dots,t_{n}$, we get \begin{equation} \label{bd} \widehat u_R(t,0)\le e^{-2\kappa t}\sum_{n\ge R}\sum_{m=0}^{n} \frac{(2\kappa t)^m}{m!}\ED_0\biggl[ \1\bigl\{\# {\mathcal I}_n=m\bigr\}\, \1\{\supp\ell_n\not\subset Q_R\}\prod_{0-b)\in(0,1]$ and note that by Cram\'er's theorem we have $\prob(\#(\mathcal A_n\cap I)\le \frac {p}2\#I)\le e^{-c\# I}$ for every bounded interval $I\subset \Z$ and some $c>0$ independent of $I$. A routine application of the Borel-Cantelli lemma implies that \begin{equation}\label{Abig} \forall\,\text{interval } I\subset[-n,n]\cap\Z\colon\qquad \# I\geq n^{1/4}\,\Longrightarrow\, \# (\mathcal A_n\cap I)> \frac {p}2\#I, \end{equation} for $n$ large enough, $\prob$-almost surely. Now we prove that with high probability, there are sufficiently large intervals which are traversed from one end to the other at least twice by the random walk $(X_k)_{k=0,\dots,n}$. Fix $K_n=\lfloor 3\log n\rfloor$ and abbreviate $k_n=\lfloor n/K_n\rfloor$. We divide the walk into $K_n$ pieces $(X^{(i)}_k)_{k=0,\dots,k_n}$ (neglecting a small overshoot) with $X^{(i)}_k=X_{(i-1)k_n+k}-X_{(i-1)k_n}$ for $i=1,\dots,K_n$. Note that these $K_n$ walks are independent copies of each other. Let us introduce the events \begin{equation} B_n=\bigcap_{i=1}^{K_n-1}\bigl\{\sgn X_{k_n}^{(i)}=\sgn X_{k_n}^{(i+1)}\bigr\} \qquad\mbox{and}\qquad C_n=\bigcup_{i=1}^{K_n}\bigl\{\max_{1\le k\le k_n} \bigl|X_k^{(i)}\bigr|\le L_n\bigr\}, \end{equation} where $L_n=\sqrt{k_n/(\eta\log n)}$. It is elementary that $\P_0^{\rm d}(B_n)\leq 2^{-K_n+1}\leq n^{-2+o(1)}$ as $n\to\infty$. Furthermore, with the help of a concatenation argument and convergence of simple random walk to Brownian motion we derive that $\P_0^{\rm d}(C_n)\leq n^{-2+o(1)}$, whenever $\eta>0$ is large enough. Now we estimate \begin{equation} \label{3.17} \ED_0\biggl[\,\prod_{x\in{\mathcal A}_n}\Bigl(\frac{2\kappa}{b}\Bigr)^{\ell_n(x)-1}\biggr]\leq \P_0^{\rm d}(B_n)+ \P_0^{\rm d}(C_n)+\ED_0\biggl[\,\1_{B_n^{\rm c}\cap C_n^{\rm c}}\prod_{x\in{\mathcal A}_n}\Bigl(\frac{2\kappa}{b}\Bigr)^{\ell_n(x)-1}\biggr]. \end{equation} Note that, on $B_n^{\rm c}\cap C_n^{\rm c}$, there is an interval $I\subset[-n,n]\cap\Z$ with $\# I\geq L_n$ such that every point of $I$ is visited by at least two of the subwalks, i.e., we have $\ell_n(x)\geq 2$ for any $x\in I$. If $n$ is sufficiently large, we deduce from \eqref{Abig} that there are at least $pL_n/2$ points $x$ with $\ell_n(x)\geq2$. By using this in \eqref{3.17}, we have \begin{equation} \ED_0\biggl[\,\prod_{x\in{\mathcal A}_n}\Bigl(\frac{2\kappa}{b}\Bigr)^{\ell_n(x)-1}\biggr]\le n^{-2+o(1)}+\Bigl(\frac{2\kappa}{b}\Bigr)^{L_n p/2},\qquad n\to\infty. \end{equation} The right hand side is clearly summable on $n\in\N$ since $2\kappa/b<1$. This finishes the proof. \qed \end{proofsect} Our next task is to get a good estimate on the size of the products in \eqref{aprioribd}. \begin{lemma} \label{LLN} Suppose that $\langle \log(-\xi(0)\vee1)\rangle=\infty$. Then, for all $b\ge1$, \begin{equation} \label{LLNbd} \lim_{n\to\infty} \frac 1{G^{-1}(1/n)}\sum_{x=1}^{\lfloor 2n\log n\rfloor} \log\Bigl(\frac{-\xi(x)\vee b}{b}\Bigr)=\infty\qquad \prob\text{\rm -almost surely.} \end{equation} \end{lemma} \begin{proofsect}{Proof} Abbreviate $N_n=\lfloor 2n\log n\rfloor$ and let $b\ge1$. Then \begin{equation} \log\Bigl(\frac{-\xi(x)\vee b}{b}\Bigr)\ge\log(-\xi(x)\vee1) -\log b. \end{equation} Using this estimate and the Chebyshev inequality, we have for any $\theta>0$ that \begin{equation}\label{esti} \!\!\!\prob\biggl(\sum_{x=1}^{N_n} \textstyle{\log\bigl({\frac{-\xi(x)\vee b}b}\bigr)\le \theta G^{-1}(1/n)\biggr) \le \exp\bigl\{-N_n G(1/\lambda)+N_n\lambda\log b +\lambda\theta G^{-1}(1/n)\bigr\}}, \end{equation} for any $\lambda>0$. Set $\lambda=1/G^{-1}(1/n)$ and note that we have $G(1/\lambda)/\lambda\to\infty$ as $\lambda\downarrow 0$, due to $\langle\log(-\xi(0)\vee1)\rangle=\infty$. Consequently, the term with $\log b$ is negligible and the right-hand side of \eqref{esti} is bounded by $n^{-2+o(1)}$. The claim is finished by the Borel-Cantelli lemma. \qed \end{proofsect} \begin{proofsect}{Proof of Proposition~\ref{propcompact}} Pick any $b\in(2\kappa,\infty)$. Let $t_0$ be so large such that the sum in \eqref{LLNbd} with $n=\lceil 1/G(t)\rceil$ for all $t\geq t_0$ exceeds $G^{-1}(1/n)$. Note that $r(t)\ge\lfloor 2n\log n\rfloor$. Combining the results of Lemma~\ref{trivbound} for $R=r(t)$ and Lemma~\ref{LLN}, we derive, for sufficiently large $n$ resp.\ $t$, the bound \begin{equation} u(t,0)-u_{r(t)}(t,0)\le 2C(\xi) \exp\bigl(-G^{-1}(1/n)\bigr), \end{equation} where $C(\xi)$ is the constant from \eqref{aprioribd}. But $G^{-1}(1/n)\ge t$ by our choice of $n$, which means that $u(t,0)-u_{r(t)}(t,0)\le C_\xi e^{-t}$, where $C_\xi=2C(\xi)\vee e^{t_0}$. The rest of the argument does not involve the particular form of $r(t)$ and can directly be taken over from \cite{BK00}. \qed \end{proofsect} \subsection{The lower bound} \label{lowerbd} \noindent Unlike the upper bound, the lower bound was basically proved already in \cite{BK00}, up to a change of the spatial scale and Lemma~\ref{last} below. For this reason, we shall only indicate the necessary changes. First we prove the following converse of Lemma~\ref{LLN}: \begin{lemma} \label{last} Fix $\eta\in(0,1)$ and let $\widetilde G_\eta$ satisfy (ii) and (iii) in Assumption~(G). Then there exists a $\varrho\in(0,\infty)$ such that \begin{equation} \limsup_{n\to\infty}\frac 1{\widetilde G_\eta^{-1}(\varrho/n)} \sum_{x=1}^n\log\bigl(-\xi(x)\vee1\bigr)\leq 1 \qquad\prob\text{\rm-almost surely.} \end{equation} \end{lemma} \begin{proofsect}{Proof} The argument is based on the asymptotic sublinearity of $1/\widetilde G_\eta$ at infinity. However, in order to have sublinearity on the whole interval $(0,\infty)$, we first construct an auxiliary modification of $\widetilde G_\eta$. Let $x_0>0$ be such that $1/\widetilde G_\eta$ is positive, increasing, and concave on $[x_0,\infty)$. Let $D_0$ to be the right derivative of $1/\widetilde G_\eta$ at $x_0$. Define $\widehat G_\eta\colon(0,\infty)\to(0,\infty)$ by the formula \begin{equation} 1/\widehat G_\eta(x)=\begin{cases} D_0 x\quad&\text{if }x\le x_0,\\ 1/\widetilde G_\eta(x)+D_0 x_0-1/\widetilde G_\eta(x_0)\quad&\text{if }x>x_0. \end{cases} \end{equation} Note that $1/\widehat G_\eta$ is positive, increasing, concave and hence sublinear on $(0,\infty)$. Moreover, Assumption~(G)(iii) holds true for $\widetilde G_\eta$ replaced by $\widehat G_\eta$. For $a\ge 1$, abbreviate $Y_a(x)=\log(-\xi(x)\vee a)$. Choose $a=e^{x_0}$ and estimate, for $n\to\infty$, \begin{equation} \label{esticonv} \frac 1{\widetilde G_\eta\Bigl(\sum_{x=1}^n Y_a(x)\Bigr)}\leq\frac{1+o(1)}{ \widehat G_\eta\Bigl(\sum_{x=1}^n Y_a(x)\Bigr)}\le (1+o(1)) \sum_{x=1}^n \frac 1{\widehat G_\eta(Y_a(x))}, \end{equation} where we used the fact that $\sum_{x=1}^n Y_a(x)\to\infty$ almost surely, and sublinearity of $1/\widehat G_\eta$. Since we have that $\langle 1/\widehat G_\eta(Y_a(x))\rangle<\infty$, the Strong Law of Large Numbers tells us that the right-hand side of \eqref{esticonv} is almost surely no more than $\varrho n$, where for $\varrho$ we can take, for instance, \begin{equation} \varrho = 2\bigl\langle 1/\widehat G_\eta(Y_a(0))\bigr\rangle. \end{equation} Hence, we derive \begin{equation} \sum_{x=1}^n Y_1(x)\le \sum_{x=1}^n Y_a(x) \le \widetilde G_\eta^{-1}(\varrho/n), \end{equation} which directly yields the desired claim. \qed \end{proofsect} Another important ingredient is the following adaptation of the crucial Proposition~5.1 of \cite{BK00} to the present situation. For $\eta\in(0,1)$, choose $\varrho$ as in Lemma~\ref{last} and let this time \begin{equation}\label{gammatlower} \gamma_t=\frac{\varrho}{\widetilde G_\eta(t\alpha_{b_t}^{-3})} \end{equation} be the size of the macrobox $Q_{\gamma_t}$ (see Subsection~\ref{heuristics}). Note that $t^{\eta\zeta+o(1)}\leq\gamma_t\leq t^{\zeta+o(1)}$ as $t\to\infty$ if $G(\ell)=\ell^{-\zeta+o(1)}$ as $\ell\to\infty$. Suppose without loss of generality that $t\mapsto\gamma_t$ is increasing. Define for each $\psi\in C^-(R)$ a ``microbox'' \begin{equation} \label{Qdef} Q^{(t)}=\begin{cases}Q_{R\alpha(b_t)}\quad&\text{if }\gamma\not=0,\\ Q_{R\alpha(b_t)}\cap\supp\psi_t\quad&\text{if }\gamma=0,\\ \end{cases} \end{equation} where $\psi_t\colon \Z\to (-\infty,0]$ is the function $\psi_t(\cdot)=\psi(\cdot/\alpha_{b_t})/\alpha_{b_t}^2$. The crucial input for the lower bound is the following claim, which says that, with probability one provided $\L_R(\psi)<1$ and $t$ is large, there is at least one microbox $Q^{(t)}$ in $Q_{\gamma_t}$, where $\xi$ is no less than (the accordingly shifted) $\psi_t$. \begin{proposition} \label{ximicrobox} Let $R>0$ and fix $\psi\in C^-(R)$ satisfying $\L_R(\psi)< 1$. Let $\eps>0$ and suppose Assumptions~(G) and (H) hold. Then the following holds almost surely: For each $\eta\in(\L_R(\psi),1)$, there is a\/ $t_0=t_0(\xi,\psi,\eps,R,\eta)<\infty$ such that for each $t\ge t_0$, there is a\/ $y_t\in Q_{\gamma_t}$ with \begin{equation}\label{xilowerbound} \xi(z+y_t)\geq\psi_t(z)-\eps\alpha_{b_t}^{-2} \qquad \forall z\in Q^{(t)}. \end{equation} \end{proposition} \begin{proofsect}{Proof} % of Proposition~\ref{ximicrobox}} We begin by formalizing the event in \eqref{xilowerbound}; in order to later approximate continuous $t$ by a discrete variable, we write $\eps/2$ instead of $\eps$: \begin{equation} \label{Adef} A_y^{(t)}=\bigcap_{z\in Q^{(t)}} \bigl\{\xi(y+z)\geq \psi_t(z)-\textstyle{\frac {\eps}{2\alpha(b_t)^2}}\bigr\}. \end{equation} Note that the probability of $A_y^{(t)}$ does not depend on $y$ and note that different $A_y^{(t)}$'s are independent if the $y$'s have distance larger than $3R\alpha(b_t)$ from each other. The proof of Lemma~5.5 in \cite{BK00} shows that $\prob(A_0^{(t)})\geq G(t)^{\L_R(\psi)+o(1)}$ as $t\to\infty$ (the only modification required is to replace every occurrence of $t$ in the meaning $\exp\{b_t\alpha(b_t)^{-2}\}$ by $1/ G(t)$). In order to prove our claim, it is sufficient to show the summability of \begin{equation} \label{ptdef} p_t=\prob\Bigl(\bigcap_{y\in Q_{\gamma_t}}\bigl(A_y^{(t)}\bigr)^{\rm c}\Bigr) \end{equation} over all $t>0$ such that $1/G(t)\in \{e^n\colon n\in\N\}$. (The sufficiency follows from the facts that $\alpha(b_t)/\alpha(b_{et})\to1$ as $t\to\infty$ and that $t\mapsto b_t$ and $t\mapsto\gamma_t$ are increasing. The error terms are absorbed into an extra $\eps/2$ in \eqref{xilowerbound} compared to \eqref{Adef}, see \cite{BK00}.) Using the independence of $A_y^{(t)}$ for $y\in B(t)=Q_{\gamma_t}\cap \lfloor 3R\alpha(b_t) \rfloor\Z$ and the bound $\prob(A_0^{(t)})\geq G(t)^{\L_R(\psi)+o(1)}$, we easily derive \begin{equation} \label{ptesti} p_t\leq \bigl(1-G(t)^{\L_R(\psi)+o(1)}\bigr)^{\#B(t)} \leq\exp\biggl\{-\frac{ G(t)^{\L_R(\psi)+o(1)}}{\alpha(b_t) \widetilde G_\eta(t\alpha(b_t)^{-3})}\biggr\}, \end{equation} where we used that $\#B(t)\ge 2\gamma_t/(3R\alpha(b_t))$ and then applied the definition of $\gamma_t$. Use concavity of $1/\widetilde G_\eta$ to estimate $1/\widetilde G_\eta(t\alpha(b_t)^{-3})\geq \alpha(b_t)^{-3}/\widetilde G_\eta(t)$ and use Assumption~(G)(i) to bound $\widetilde G_\eta(t)$ by $G(t)^{\eta+o(1)}$. Furthermore, since $\alpha(b_t)$ is bounded from above by a positive power of $b_t\alpha(b_t)^{-2}$, we see from \eqref{b_t} that $\alpha(b_t)=G(t)^{o(1)}$. Applying all this reasoning on the right-hand side of \eqref{ptesti}, we see that $p_t\leq\exp(-G(t)^{\L_R(\psi)-\eta+o(1)})$ as $t\to\infty$, which is summable on the sequence of $t$ such that $1/G(t)\in\{e^n\colon n\in \N\}$. This finishes the proof. \qed \end{proofsect} Now we finish the proof of our main result. \begin{proofsect}{Proof of Theorem~\ref{mainres}, lower bound} Let $\eps>0$ and fix $R>0$ and $\psi\in C^-(R)$ such that $\L_R(\psi)<1$. Let $\eta\in(\L_R(\psi),1)$, define $\gamma_t$ as in \eqref{gammatlower} and let $y_t$ be as in Proposition~\ref{ximicrobox}; suppose $y_t\ge0$ without loss of generality. Let $r_x=[-\xi(x)\vee1]^{-1}$. As in \cite{BK00}, the lower bound will be obtained by restricting the walk in \eqref{FKF} to perform the following: The walk keeps jumping toward $y_t$, spending at most time $r_x$ at each site $x$ such that it reaches $y_t$ before time $\gamma_t$. Then it stays at $y_t$ until time $\gamma_t$ and then within $y_t+Q^{(t)}$ for the remaining time $t-\gamma_t$. Inserting this event into \eqref{FKF} and invoking Markov property at time $\gamma_t$ we get \begin{equation} u(t,0)\geq \text{II}\times\text{III}, \end{equation} where the same argument as in \cite{BK00} shows that $\text{III}\ge e^{t\alpha(b_t)^{-2}[\lambda_R(\psi)-\eps]}$ for large $t$, while for $\text{II}$ we have \begin{equation} \begin{aligned} \text{II}=\int_{\vartriangle_{y_t}\!(\gamma_t)} &dt_0\dots dt_{y_t-1}\, e^{-2\kappa\gamma_t}\exp\Bigl\{\sum_{k=0}^n \xi_kt_k\Bigr\}\prod_{x=0}^{y_t-1}\1\{t_x\le r_{x-1}\} \\ &\ge e^{-2\kappa\gamma_t}\prod_{x=0}^{y_t-1}\Bigl[r_xe^{r_x\xi(x)}\Bigr]\ge e^{-(2\kappa+1)\gamma_t}\exp\Bigl\{-\sum_{x=0}^{y_t-1} \log\bigl(-\xi(x)\vee1\bigr)\Bigr\}, \end{aligned} \end{equation} where we recalled the notation of \eqref{discreterep}. Now $y_t\le\gamma_t$, so using Lemma~\ref{last} we have that \begin{equation} \text{II}\ge e^{-(2\kappa+1)\gamma_t}\exp\bigl\{- \widetilde G_\eta^{-1}(\varrho/\gamma_t)(1+o(1))\bigr\}=e^{-(2\kappa+1)\gamma_t- t\alpha(b_t)^{-3}(1+o(1))}, \end{equation} where we used the definition of $\gamma_t$. Since $1/\widetilde G_\eta$ is asymptotically concave, $\gamma_t=\varrho/\widetilde G_\eta(t\alpha_{b_t}^{-3})\le O(t\alpha_{b_t}^{-3})$ and the exponent is $o(t\alpha_{b_t}^{-2})$. Consequently, \begin{equation} u(t,0)\ge e^{t\alpha(b_t)^{-2}[\lambda_R(\psi)-\eps+o(1)]}, \end{equation} where $o(1)$ still depends on $\eta$. The proof is finished by letting $t\to\infty$ (which eliminates the dependence on $\eta$), optimizing over $\psi$ and $R$ with $\L_R(\psi)<1$ and letting $\eps\downarrow0$. \qed \end{proofsect} \section*{Acknowledgments} \nopagebreak M.B. would like to thank Yimin Xiao and Oded Schramm for discussions about the behavior of sums of i.i.d.\ random variables with infinite mean. \vspace{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{GKM98} \bibitem[BGT87]{BGT87} %{\sc Bingham, N.H., Goldie, C.M.} and {\sc Teugels, J.L.} (1987). 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource dvidict begin % textext encoding /textext[/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon /Phi/Psi/Omega/ff/fi/fl/ffi/ffl /dotlessi/dotlessj/grave/acute/caron/breve/macron/ring /cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash /suppress/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright /parenleft/parenright/asterisk/plus/comma/hyphen/period/slash /zero/one/two/three/four/five/six/seven /eight/nine/colon/semicolon/exclamdown/equal/questiondown/question /at/A/B/C/D/E/F/G /H/I/J/K/L/M/N/O /P/Q/R/S/T/U/V/W /X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent /quoteleft/a/b/c/d/e/f/g /h/i/j/k/l/m/n/o /p/q/r/s/t/u/v/w /x/y/z/endash/emdash/hungarumlaut/tilde/dieresis 128 notdef]def end % dvidict dvidict begin % TEXNANSI encoding /TEXNANSI[ n/Euro n n/fraction/dotaccent/hungarumlaut/ogonek /fl/fraction/cwm/ff/fi/fl/ffi/ffl /dotlessi/dotlessj/grave/acute/caron/breve/macron/ring /cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash /space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright /parenleft/parenright/asterisk/plus/comma/hyphen/period/slash /zero/one/two/three/four/five/six/seven /eight/nine/colon/semicolon/less/equal/greater/question /at/A/B/C/D/E/F/G /H/I/J/K/L/M/N/O /P/Q/R/S/T/U/V/W /X/Y/Z/bracketleft/backslash/bracketright/circumflex/underscore /quoteleft/a/b/c/d/e/f/g /h/i/j/k/l/m/n/o /p/q/r/s/t/u/v/w /x/y/z/braceleft/bar/braceright/tilde/dieresis /Lslash/quotesingle/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl /circumflex/perthousand/Scaron/guilsinglleft/OE/Zcaron/asciicircum/minus /lslash/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash /tilde/trademark/scaron/guilsinglright/oe/zcaron/asciitilde/Ydieresis /nbspace/exclamdown/cent/sterling/currency/yen/brokenbar/section /dieresis/copyright/ordfeminine/guillemotleft/logicalnot/sfthyphen/registered/macron /degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered /cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown /Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis /Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply /Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls /agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla /egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis /eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide /oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def end % dvidict %%EndProlog %%BeginSetup dvidict begin %%BeginResource: font cmr8 %!PS-AdobeFont-1.1: CMR8 001.003 %%CreationDate: 1992 Feb 19 19:55:11 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.003) readonly def /Notice (Copyright (c) 1988 - 1992 Blue Sky Research and Y&Y, Inc. http://www.YandY.com) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMR8) def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 171 /sfthyphen put dup 172 /nbspace put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put readonly def /FontBBox{-267 -250 1070 968}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FCA78881823C5537FE6C3DDA8EE5B897E651A592E4F5 56BEF9359FC62408332EBCEA130F7B8EC0669941AAFAE641FAD7D88DD4ECF7A242DCFA717D34D4 C60059FBA9A1623373EE8C93009FD5DA54B37B1B48962BA909E289A3264FA7BB8348760BB3C5AA B5C7AC22980AC562528D2206EBF93AF39013E278B7FDFEA34364982F0B2F75172FD5B35A7FA624 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font cmmi10 %!PS-AdobeFont-1.1: CMMI10 001.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1988 - 1996 Blue Sky Research and Y&Y, Inc. (978) 371-3286) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /BaseFontName (CMMI10) def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta 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2698701896B39D26FE9D2FF2BA0BAACA77A75FF905B3CAF7C7C8A5243C54469EBA42D0ECB15AFF 53770C93A32B18205D2FD601F88A36F53B93CDB0A247C4E575 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font cmr10 %!PS-AdobeFont-1.1: CMR10 001.004 %%CreationDate: 1994 Jun 08 05:28:48 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.004) readonly def /Notice (Copyright (c) 1988 - 1994 Blue Sky Research and Y&Y, Inc. http://www.YandY.com) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMR10) def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 171 /sfthyphen put dup 172 /nbspace put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put readonly def /FontBBox{-40 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FCA78881823C5537FE6C3DDA8EE5B897E651A592E4F5 56BEF9359FC62408332EBCEA130F7B8EC0669941AAFAE641FAD7D88DD4ECF7A242DCFA717D34D4 C60059FBA9A1623373EE8C93009FD5DA54B37B1B48962BA909E289A3264FA7BB8348760BB3C5AA B5C7AC22980AC562528D2206EBF93AF39013E278B7FDFEA34364982F0B2F75172FD5B35A7FA624 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font cmsy10 %!PS-AdobeFont-1.1: CMSY10 001.100 %%CreationDate: 1997 Aug 23 16:21:30 % Copyright (C) 1997 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. http://www.YandY.com) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMSY10) def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5096651 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font msbm10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Copyright (C) 1991, 1992 Y&Y, Inc. All rights reserved. % Copyright (C) 1991, 1992 Blue Sky Research, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (2.1) readonly def /Notice (Copyright (c) 1991, 1992 Y&Y, Inc. All rights reserved. (978) 371-3286) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /BaseFontName (MSBM10) def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 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cleartomark %%EndResource %%BeginResource: font cmmi7 %!PS-AdobeFont-1.1: CMMI7 001.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1988 - 1996 Blue Sky Research and Y&Y, Inc. (978) 371-3286) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /BaseFontName (CMMI7) def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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cleartomark %%EndResource %%BeginResource: font cmsy7 %!PS-AdobeFont-1.1: CMSY7 001.100 %%CreationDate: 1997 Aug 23 16:21:31 % Copyright (C) 1997 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. http://www.YandY.com) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMSY7) def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5096648 def currentdict end currentfile eexec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cleartomark %%EndResource %%BeginResource: font msbm7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Copyright (C) 1991, 1992 Blue Sky Research, Inc. All rights reserved. % Copyright (C) 1991, 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (2.1) readonly def /Notice (Copyright (c) 1991, 1992 Y&Y, Inc. All rights reserved. (978) 371-3286) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /BaseFontName (MSBM7) def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 188 /follownotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 191 /notsimilar put dup 192 /notapproxequal put dup 193 /upslope put dup 194 /downslope put dup 195 /notsubsetoreql put dup 196 /epsiloninv put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndResource %%BeginResource: font cmcsc10 %!PS-AdobeFont-1.1: CMCSC10 001.100 %%CreationDate: 1996 Aug 06 10:25:56 % Copyright (C) 1996 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. http://www.YandY.com) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMCSC10) def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /arrowup put dup 12 /arrowdown put dup 13 /quotesingle put dup 14 /exclamdown put dup 15 /questiondown put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /less put dup 61 /equal put dup 62 /greater put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 171 /sfthyphen put dup 172 /nbspace put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put readonly def /FontBBox{0 -250 1077 969}readonly def /UniqueID 5087402 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FFD20326947B1065649B533EB5E9E3A88DD77004CD4F ABF149D6C8D56292D3B45879C69986C03ED4B6B9E5994D6A23138375D9A8118817361F21201BE4 98FC13671B01C995B90C8C26E88A39F062C9EEB1EFFD60138701300DD9F1A6070DF9A8F3F1D1EF 12004D9E25F4776A471728A4B0F77FF3EAADF16830490108D0BD04DC0DF4E81FC4EAC68C2887D3 1B383587E8F5CB726231787CFFFFBFCC4553D0D3BC2E86F70DB73DC2DBAB69068BD569668EE828 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%%EndResource %%BeginResource: font cmmi12 %!PS-AdobeFont-1.1: CMMI12 001.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1988 - 1996 Blue Sky Research and Y&Y, Inc. (978) 371-3286) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /BaseFontName (CMMI12) def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font cmr12 %!PS-AdobeFont-1.1: CMR12 001.003 %%CreationDate: 1992 Jun 07 22:11:29 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.003) readonly def /Notice (Copyright (c) 1988 - 1992 Blue Sky Research and Y&Y, Inc. http://www.YandY.com) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMR12) def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 171 /sfthyphen put dup 172 /nbspace put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put readonly def /FontBBox{-246 -251 988 968}readonly def /UniqueID 5000794 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FCA78881823C5537FE6C3DDA8EE5B897E651A592E4F5 56BEF9359FC62408332EBCEA130F7B8EC0669941AAFAE641FAD7D88DD4ECF7A242DCFA717D34D4 C60059FBA9A1623373EE8C93009FD5DA54B37B1B48962BA909E289A3264FA7BB8348760BB3C5AA B5C7AC22980AC562528D2206EBF93AF39013E278B7FDFEA34364982F0B2F75172FD5B35A7FA624 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font cmmi8 %!PS-AdobeFont-1.1: CMMI8 1.10 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1988 - 1992 Blue Sky Research and Y&Y, Inc. (978) 371-3286) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /BaseFontName (CMMI8) def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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%%BeginResource: font cmex10 %!PS-AdobeFont-1.1: CMEX10 001.101 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1992, 1997 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.101) readonly def /Notice (Copyright (c) 1992, 1997 Y&Y, Inc. All Right Reserved. (978) 371-3286) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMEX10) def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 4 /floorleftbig put dup 5 /floorrightbig put dup 6 /ceilingleftbig put dup 7 /ceilingrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 14 /slashbig put dup 15 /backslashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 24 /ceilingleftbigg put dup 25 /ceilingrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 30 /slashbigg put dup 31 /backslashbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 36 /floorleftBigg put dup 37 /floorrightBigg put dup 38 /ceilingleftBigg put dup 39 /ceilingrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 44 /slashBigg put dup 45 /backslashBigg put dup 46 /slashBig put dup 47 /backslashBig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 63 /arrowvertex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 70 /unionsqtext put dup 71 /unionsqdisplay put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 74 /circledottext put dup 75 /circledotdisplay put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 85 /unionmultitext put dup 86 /logicalandtext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 93 /unionmultidisplay put dup 94 /logicalanddisplay put dup 95 /logicalordisplay put dup 96 /coproducttext put dup 97 /coproductdisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put dup 108 /ceilingleftBig put dup 109 /ceilingrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 119 /arrowvertexdbl put dup 120 /arrowtp put dup 121 /arrowbt put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put dup 126 /arrowdbltp put dup 127 /arrowdblbt put dup 128 /parenleftBigg put dup 159 /radical put dup 160 /space put dup 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 165 /floorleftbig put dup 166 /floorrightbig put dup 167 /ceilingleftbig put dup 168 /ceilingrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 173 /angbracketleftbig put dup 174 /angbracketrightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 177 /slashbig put dup 178 /backslashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 185 /floorleftbigg put dup 186 /floorrightbigg put dup 187 /ceilingleftbigg put dup 188 /ceilingrightbigg put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 191 /angbracketleftbigg put dup 192 /angbracketrightbigg put dup 193 /slashbigg put dup 194 /backslashbigg put dup 195 /parenleftBigg put dup 196 /arrowdblbt put dup 208 /Gamma put dup 209 /Delta put dup 210 /Theta put dup 211 /Lambda put dup 212 /Xi put dup 213 /Pi put dup 214 /Sigma put dup 215 /Upsilon put dup 216 /Phi put dup 217 /Psi put dup 218 /Omega put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5092766 def currentdict end currentfile eexec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cleartomark %%EndResource %%BeginResource: font cmsy8 %!PS-AdobeFont-1.1: CMSY8 001.100 %%CreationDate: 1997 Aug 23 16:21:32 % Copyright (C) 1997 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. http://www.YandY.com) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMSY8) def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 /follows put dup 195 /arrowleft put dup 196 /spade put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5096649 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font msbm8 %!PS-AdobeFont-1.1: MSBM8 001.100 %%CreationDate: 1996 Aug 12 11:18:20 % Copyright (C) 1996 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. (978) 371-3286) readonly def /FullName (MSBM8) readonly def /FamilyName (AMS Mathematical Symbol) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (MSBM8) def end readonly def /FontName /MSBM8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /lessornotequal put dup 1 /greaterornotequal put dup 2 /notlessequal put dup 3 /notgreaterequal put dup 4 /notless put dup 5 /notgreater put dup 6 /notprecedes put dup 7 /notfollows put dup 8 /lessornotdbleql put dup 9 /greaterornotdbleql put dup 10 /notlessorslnteql put dup 11 /notgreaterorslnteql put dup 12 /lessnotequal put dup 13 /greaternotequal put dup 14 /notprecedesoreql put dup 15 /notfollowsoreql put dup 16 /precedeornoteqvlnt put dup 17 /followornoteqvlnt put dup 18 /lessornotsimilar put dup 19 /greaterornotsimilar put dup 20 /notlessdblequal put dup 21 /notgreaterdblequal put dup 22 /precedenotslnteql put dup 23 /follownotslnteql put dup 24 /precedenotdbleqv put dup 25 /follownotdbleqv put dup 26 /lessnotdblequal put dup 27 /greaternotdblequal put dup 28 /notsimilar put dup 29 /notapproxequal put dup 30 /upslope put dup 31 /downslope put dup 32 /notsubsetoreql put dup 33 /notsupersetoreql put dup 34 /notsubsetordbleql put dup 35 /notsupersetordbleql put dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 38 /subsetornoteql put dup 39 /supersetornoteql put dup 40 /subsetnoteql put dup 41 /supersetnoteql put dup 42 /notsubseteql put dup 43 /notsuperseteql put dup 44 /notparallel put dup 45 /notbar put dup 46 /notshortbar put dup 47 /notshortparallel put dup 48 /notturnstile put dup 49 /notforces put dup 50 /notsatisfies put dup 51 /notforcesextra put dup 52 /nottriangeqlright put dup 53 /nottriangeqlleft put dup 54 /nottriangleleft put dup 55 /nottriangleright put dup 56 /notarrowleft put dup 57 /notarrowright put dup 58 /notdblarrowleft put dup 59 /notdblarrowright put dup 60 /notdblarrowboth put dup 61 /notarrowboth put dup 62 /dividemultiply put dup 63 /emptyset put dup 64 /notexistential put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put dup 93 /tildewide put dup 94 /tildewider put dup 96 /Finv put dup 97 /Gmir put dup 102 /Omegainv put dup 103 /eth put dup 104 /equalorsimilar put dup 105 /beth put dup 106 /gimel put dup 107 /daleth put dup 108 /lessdot put dup 109 /greaterdot put dup 110 /multicloseleft put dup 111 /multicloseright put dup 112 /barshort put dup 113 /parallelshort put dup 114 /integerdivide put dup 115 /similar put dup 116 /approxequal put dup 117 /approxorequal put dup 118 /followsorequal put dup 119 /precedesorequal put dup 120 /archleftdown put dup 121 /archrightdown put dup 122 /Digamma put dup 123 /kappa put dup 124 /k put dup 125 /planckover2pi put dup 126 /planckover2pi1 put dup 127 /epsiloninv put dup 128 /notsubsetoreql put dup 160 /space put dup 161 /lessornotequal put dup 162 /greaterornotequal put dup 163 /notlessequal put dup 164 /notgreaterequal put dup 165 /notless put dup 166 /notgreater put dup 167 /notprecedes put dup 168 /notfollows put dup 169 /lessornotdbleql put dup 170 /greaterornotdbleql put dup 173 /notlessorslnteql put dup 174 /notgreaterorslnteql put dup 175 /lessnotequal put dup 176 /greaternotequal put dup 177 /notprecedesoreql put dup 178 /notfollowsoreql put dup 179 /precedeornoteqvlnt put dup 180 /followornoteqvlnt put dup 181 /lessornotsimilar put dup 182 /greaterornotsimilar put dup 183 /notlessdblequal put dup 184 /notgreaterdblequal put dup 185 /precedenotslnteql put dup 186 /follownotslnteql put dup 187 /precedenotdbleqv put dup 188 /follownotdbleqv put dup 189 /lessnotdblequal put dup 190 /greaternotdblequal put dup 191 /notsimilar put dup 192 /notapproxequal put dup 193 /upslope put dup 194 /downslope put dup 195 /notsubsetoreql put dup 196 /epsiloninv put readonly def /FontBBox{-59 -474 2492 974}readonly def /UniqueID 5092737 def currentdict end currentfile eexec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cleartomark %%EndResource %%BeginResource: font cmmi6 %!PS-AdobeFont-1.1: CMMI6 1.10 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1988 - 1992 Blue Sky Research and Y&Y, Inc. (978) 371-3286) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /BaseFontName (CMMI6) def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /dotlessi put dup 124 /dotlessj put dup 125 /weierstrass put dup 126 /vector put dup 127 /tie put dup 128 /psi put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndResource %%BeginResource: font cmr6 %!PS-AdobeFont-1.1: CMR6 001.003 %%CreationDate: 1992 Feb 19 19:55:05 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.003) readonly def /Notice (Copyright (c) 1988 - 1992 Blue Sky Research and Y&Y, Inc. http://www.YandY.com) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMR6) def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 171 /sfthyphen put dup 172 /nbspace put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put readonly def /FontBBox{-307 -250 1193 968}readonly def /UniqueID 5000789 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FCA78881823C5537FE6C3DDA8EE5B897E651A592E4F5 56BEF9359FC62408332EBCEA130F7B8EC0669941AAFAE641FAD7D88DD4ECF7A242DCFA717D34D4 C60059FBA9A1623373EE8C93009FD5DA54B37B1B48962BA909E289A3264FA7BB8348760BB3C5AA B5C7AC22980AC562528D2206EBF93AF39013E278B7FDFEA34364982F0B2F75172FD5B35A7FA624 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndResource %%BeginResource: font cmsy6 %!PS-AdobeFont-1.1: CMSY6 001.100 %%CreationDate: 1997 Aug 23 16:21:31 % Copyright (C) 1997 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. http://www.YandY.com) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMSY6) def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 29 /greatermuch put dup 30 /precedes put dup 31 /follows put dup 32 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 36 /arrowboth put dup 37 /arrownortheast put dup 38 /arrowsoutheast put dup 39 /similarequal put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 44 /arrowdblboth put dup 45 /arrownorthwest put dup 46 /arrowsouthwest put dup 47 /proportional put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 53 /triangleinv put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 62 /latticetop put dup 63 /perpendicular put dup 64 /aleph put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 93 /unionmulti put dup 94 /logicaland put dup 95 /logicalor put dup 96 /turnstileleft put dup 97 /turnstileright put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 108 /arrowbothv put dup 109 /arrowdblbothv put dup 110 /backslash put dup 111 /wreathproduct put dup 112 /radical put dup 113 /coproduct put dup 114 /nabla put dup 115 /integral put dup 116 /unionsq put dup 117 /intersectionsq put dup 118 /subsetsqequal put dup 119 /supersetsqequal put dup 120 /section put dup 121 /dagger put dup 122 /daggerdbl put dup 123 /paragraph put dup 124 /club put dup 125 /diamond put dup 126 /heart put dup 127 /spade put dup 128 /arrowleft put dup 160 /space put dup 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 165 /divide put dup 166 /diamondmath put dup 167 /plusminus put dup 168 /minusplus put dup 169 /circleplus put dup 170 /circleminus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 192 /greatermuch put dup 193 /precedes put dup 194 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cleartomark %%EndResource %%BeginResource: font cmss12 %!PS-AdobeFont-1.1: CMSS12 001.003 %%CreationDate: 1992 Feb 19 19:55:34 % Copyright (C) 1988 Blue Sky Research. All rights reserved. % Copyright (C) 1992 Y&Y, Inc. All rights reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.003) readonly def /Notice (Copyright (c) 1988 - 1992 Blue Sky Research and Y&Y, Inc. http://www.YandY.com) readonly def /FullName (CMSS12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (CMSS12) def end readonly def /FontName /CMSS12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put dup 128 /suppress put dup 160 /space put dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 171 /sfthyphen put dup 172 /nbspace put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put readonly def /FontBBox{-238 -251 978 944}readonly def /UniqueID 5000804 def currentdict end currentfile eexec 8053514D28EC28DA1630165FAB262882D3FCA78881823C5537FE6C3DDA8EE5B897E651A592E4F5 56BEF9359FC62408332EBCEA130F7B8EC0669941AAFAE641FAD7D88DD4ECF7A242DCFA717D34D4 C60059FBA9A1623373EE8C93009FD5DA54B37B1B48962BA909E289A3264FA7BB8348760BB3C5AA B5C7AC22980AC562528D2206EBF93AF39013E278B7FDFEA34364982F0B2F75172FD5B35A7FA624 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cleartomark %%EndResource %%BeginResource: font msam10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Copyright (C) 1991, 1992 Y&Y, Inc. All rights reserved. % Copyright (C) 1991, 1992 Blue Sky Research, Inc. All rights reserved. 11 dict begin /FontInfo 8 dict dup begin /version (2.1) readonly def /Notice (Copyright (c) 1991, 1992 Y&Y, Inc. All rights reserved. (978) 371-3286) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /BaseFontName (MSAM10) def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /squaredot put dup 1 /squareplus put dup 2 /squaremultiply put dup 3 /square put dup 4 /squaresolid put dup 5 /squaresmallsolid put dup 6 /diamond put dup 7 /diamondsolid put dup 8 /clockwise put dup 9 /anticlockwise put dup 10 /harpoonleftright put dup 11 /harpoonrightleft put dup 12 /squareminus put dup 13 /forces put dup 14 /forcesbar put dup 15 /satisfies put dup 16 /dblarrowheadright put dup 17 /dblarrowheadleft put dup 18 /dblarrowleft put dup 19 /dblarrowright put dup 20 /dblarrowup put dup 21 /dblarrowdwn put dup 22 /harpoonupright put dup 23 /harpoondownright put dup 24 /harpoonupleft put dup 25 /harpoondownleft put dup 26 /arrowtailright put dup 27 /arrowtailleft put dup 28 /arrowparrleftright put dup 29 /arrowparrrightleft put dup 30 /shiftleft put dup 31 /shiftright put dup 32 /squiggleright put dup 33 /squiggleleftright put dup 34 /curlyleft put dup 35 /curlyright put dup 36 /circleequal put dup 37 /followsorequal put dup 38 /greaterorsimilar put dup 39 /greaterorapproxeql put dup 40 /multimap put dup 41 /therefore put dup 42 /because put dup 43 /equalsdots put dup 44 /defines put dup 45 /precedesorequal put dup 46 /lessorsimilar put dup 47 /lessorapproxeql put dup 48 /equalorless put dup 49 /equalorgreater put dup 50 /equalorprecedes put dup 51 /equalorfollows put dup 52 /precedesorcurly put dup 53 /lessdblequal put dup 54 /lessorequalslant put dup 55 /lessorgreater put dup 56 /primereverse put dup 57 /axisshort put dup 58 /equaldotrightleft put dup 59 /equaldotleftright put dup 60 /followsorcurly put dup 61 /greaterdblequal put dup 62 /greaterorequalslant put dup 63 /greaterorless put dup 64 /squareimage put dup 65 /squareoriginal put dup 66 /triangleright put dup 67 /triangleleft put dup 68 /trianglerightequal put dup 69 /triangleleftequal put dup 70 /star put dup 71 /between put dup 72 /triangledownsld put dup 73 /trianglerightsld put dup 74 /triangleleftsld put dup 75 /arrowaxisright put dup 76 /arrowaxisleft put dup 77 /triangle put dup 78 /trianglesolid put dup 79 /triangleinv put dup 80 /ringinequal put dup 81 /lessequalgreater put dup 82 /greaterlessequal put dup 83 /lessdbleqlgreater put dup 84 /greaterdbleqlless put dup 85 /Yen put dup 86 /arrowtripleright put dup 87 /arrowtripleleft put dup 88 /check put dup 89 /orunderscore put dup 90 /nand put dup 91 /perpcorrespond put dup 92 /angle put dup 93 /measuredangle put dup 94 /sphericalangle put dup 95 /proportional put dup 96 /smile put dup 97 /frown put dup 98 /subsetdbl put dup 99 /supersetdbl put dup 100 /uniondbl put dup 101 /intersectiondbl put dup 102 /uprise put dup 103 /downfall put dup 104 /multiopenleft put dup 105 /multiopenright put dup 106 /subsetdblequal put dup 107 /supersetdblequal put dup 108 /difference put dup 109 /geomequivalent put dup 110 /muchless put dup 111 /muchgreater put dup 112 /rightanglenw put dup 113 /rightanglene put dup 114 /circleR put dup 115 /circleS put dup 116 /fork put dup 117 /dotplus put dup 118 /revsimilar put dup 119 /revasymptequal put dup 120 /rightanglesw put dup 121 /rightanglese put dup 122 /maltesecross put dup 123 /complement put dup 124 /intercal put dup 125 /circlering put dup 126 /circleasterisk put dup 127 /circleminus put dup 128 /squiggleright put dup 160 /space put dup 161 /squaredot put dup 162 /squareplus put dup 163 /squaremultiply put dup 164 /square put dup 165 /squaresolid put dup 166 /squaresmallsolid put dup 167 /diamond put dup 168 /diamondsolid put dup 169 /clockwise put dup 170 /anticlockwise put dup 173 /harpoonleftright put dup 174 /harpoonrightleft put dup 175 /squareminus put dup 176 /forces put dup 177 /forcesbar put dup 178 /satisfies put dup 179 /dblarrowheadright put dup 180 /dblarrowheadleft put dup 181 /dblarrowleft put dup 182 /dblarrowright put dup 183 /dblarrowup put dup 184 /dblarrowdwn put dup 185 /harpoonupright put dup 186 /harpoondownright put dup 187 /harpoonupleft put dup 188 /harpoondownleft put dup 189 /arrowtailright put dup 190 /arrowtailleft put dup 191 /arrowparrleftright put dup 192 /arrowparrrightleft put dup 193 /shiftleft put dup 194 /shiftright put dup 195 /squiggleright put dup 196 /circleminus put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D7190FA2D133A583138F76695558E7AE9348D37CAC665 1806D08527C1BB4A062A4835AC37784CC39AD8841404E438B4D52D3901E47A1DE4F7924E0FB3DA F442499175BAB1226EDF692A4956739F8828E80592F450C5D5C22AC88BCFBE9748F61D18243A16 F4A4467F084E8E2BE46EF47FC51C3A8199E3CDA62FF9C4FB73956DAB8B6683D2156377808CB350 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cleartomark %%EndResource %%BeginResource: font msam8 %!PS-AdobeFont-1.1: MSAM8 001.100 %%CreationDate: 1996 Aug 12 10:21:50 % Copyright (C) 1996 Y&Y, Inc. All Rights Reserved. 11 dict begin /FontInfo 10 dict dup begin /version (001.100) readonly def /Notice (Copyright (c) 1996 Y&Y, Inc. All Right Reserved. (978) 371-3286) readonly def /FullName (MSAM8) readonly def /FamilyName (AMS Mathematical Symbol) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /BaseFontName (MSAM8) def end readonly def /FontName /MSAM8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /squaredot put dup 1 /squareplus put dup 2 /squaremultiply put dup 3 /square put dup 4 /squaresolid put dup 5 /squaresmallsolid put dup 6 /diamond put dup 7 /diamondsolid put dup 8 /clockwise put dup 9 /anticlockwise put dup 10 /harpoonleftright put dup 11 /harpoonrightleft put dup 12 /squareminus put dup 13 /forces put dup 14 /forcesbar put dup 15 /satisfies put dup 16 /dblarrowheadright put dup 17 /dblarrowheadleft put dup 18 /dblarrowleft put dup 19 /dblarrowright put dup 20 /dblarrowup put dup 21 /dblarrowdwn put dup 22 /harpoonupright put dup 23 /harpoondownright put dup 24 /harpoonupleft put dup 25 /harpoondownleft put dup 26 /arrowtailright put dup 27 /arrowtailleft put dup 28 /arrowparrleftright put dup 29 /arrowparrrightleft put dup 30 /shiftleft put dup 31 /shiftright put dup 32 /squiggleright put dup 33 /squiggleleftright put dup 34 /curlyleft put dup 35 /curlyright put dup 36 /circleequal put dup 37 /followsorequal put dup 38 /greaterorsimilar put dup 39 /greaterorapproxeql put dup 40 /multimap put dup 41 /therefore put dup 42 /because put dup 43 /equalsdots put dup 44 /defines put dup 45 /precedesorequal put dup 46 /lessorsimilar put dup 47 /lessorapproxeql put dup 48 /equalorless put dup 49 /equalorgreater put dup 50 /equalorprecedes put dup 51 /equalorfollows put dup 52 /precedesorcurly put dup 53 /lessdblequal put dup 54 /lessorequalslant put dup 55 /lessorgreater put dup 56 /primereverse put dup 57 /axisshort put dup 58 /equaldotrightleft put dup 59 /equaldotleftright put dup 60 /followsorcurly put dup 61 /greaterdblequal put dup 62 /greaterorequalslant put dup 63 /greaterorless put dup 64 /squareimage put dup 65 /squareoriginal put dup 66 /triangleright put dup 67 /triangleleft put dup 68 /trianglerightequal put dup 69 /triangleleftequal put dup 70 /star put dup 71 /between put dup 72 /triangledownsld put dup 73 /trianglerightsld put dup 74 /triangleleftsld put dup 75 /arrowaxisright put dup 76 /arrowaxisleft put dup 77 /triangle put dup 78 /trianglesolid put dup 79 /triangleinv put dup 80 /ringinequal put dup 81 /lessequalgreater put dup 82 /greaterlessequal put dup 83 /lessdbleqlgreater put dup 84 /greaterdbleqlless put dup 85 /Yen put dup 86 /arrowtripleright put dup 87 /arrowtripleleft put dup 88 /check put dup 89 /orunderscore put dup 90 /nand put dup 91 /perpcorrespond put dup 92 /angle put dup 93 /measuredangle put dup 94 /sphericalangle put dup 95 /proportional put dup 96 /smile put dup 97 /frown put dup 98 /subsetdbl put dup 99 /supersetdbl put dup 100 /uniondbl put dup 101 /intersectiondbl put dup 102 /uprise put dup 103 /downfall put dup 104 /multiopenleft put dup 105 /multiopenright put dup 106 /subsetdblequal put dup 107 /supersetdblequal put dup 108 /difference put dup 109 /geomequivalent put dup 110 /muchless put dup 111 /muchgreater put dup 112 /rightanglenw put dup 113 /rightanglene put dup 114 /circleR put dup 115 /circleS put dup 116 /fork put dup 117 /dotplus put dup 118 /revsimilar put dup 119 /revasymptequal put dup 120 /rightanglesw put dup 121 /rightanglese put dup 122 /maltesecross put dup 123 /complement put dup 124 /intercal put dup 125 /circlering put dup 126 /circleasterisk put dup 127 /circleminus put dup 128 /squiggleright put dup 160 /space put dup 161 /squaredot put dup 162 /squareplus put dup 163 /squaremultiply put dup 164 /square put dup 165 /squaresolid put dup 166 /squaresmallsolid put dup 167 /diamond put dup 168 /diamondsolid put dup 169 /clockwise put dup 170 /anticlockwise put dup 173 /harpoonleftright put dup 174 /harpoonrightleft put dup 175 /squareminus put dup 176 /forces put dup 177 /forcesbar put dup 178 /satisfies put dup 179 /dblarrowheadright put dup 180 /dblarrowheadleft put dup 181 /dblarrowleft put dup 182 /dblarrowright put dup 183 /dblarrowup put dup 184 /dblarrowdwn put dup 185 /harpoonupright put dup 186 /harpoondownright put dup 187 /harpoonupleft put dup 188 /harpoondownleft put dup 189 /arrowtailright put dup 190 /arrowtailleft put dup 191 /arrowparrleftright put dup 192 /arrowparrrightleft put dup 193 /shiftleft put dup 194 /shiftright put dup 195 /squiggleright put dup 196 /circleminus put readonly def /FontBBox{13 -524 1416 1070}readonly def /UniqueID 5092734 def currentdict end currentfile eexec 80347982AB3942D930E069A70D0D48311D77A69D914FE4AFC21F704047529CC6AAE5F9AD874A3A 9AEA95A863ACE1521609487EDD583768A60DF36947BBFB171A76D7721738C735D7179C2E508FBD 4ECC585C362AA4C333EFC838115A5918D8EF4A2E80EEF5ACD9A61BD29DA32D439B6991629DB972 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f18(`)s f19(\))T f1(as)T f18(`)S f20(!)S(1)s f1(,)T(which)T(is)T(the)T(essence)T(of)T(\(i\226iii\).)s o 1385212 d u -473657 r f3(Remark)s 259954 r(2)s f1(.)s 341422 r(In)s 222992 W(the)S(vie)s -19661 r(w)S(of)S(Remark)S(1,)s 229588 r(it)S(is)S(immediate)S(that)s 179738 r(Assumption)S(\(G\))s 222965 r(holds)S(for)S f18(G)s f19(\()s f18(`)s f19(\))s 264254 W(=)S f18(`)s u -285390 d f24(\0)s f22(\20)s 35953 r f2(+)s f22(o)s f2(\(1\))s O y u -473657 r f1(with)s 250044 W(some)S f18(\20)s 370613 r f20(2)s 311220 r f19(\(0)s f18(;)s 131070 r f19(1])s f1(.)s 429268 r(The)S(reason)S(wh)s -3933 r(y)S(we)S(prefer)S(the)S(abo)s -11797 W(v)S(e)s 250044 W(\(little)S(cumbersome\))S(setting)S(as)S(opposed)s o y u -473657 r(to)s 245913 W(simple)S(re)s -11797 X(gularity)S(of)S f18(G)S f1(is)S(that)S(man)T(y)S(cases)S(with)S f18(G)s f19(\()s f18(`)s f19(\))s 304047 W(=)S f18(`)s u -285390 d f22(o)s f2(\(1\))s o 1259733 r f1(are)s 245913 W(automatically)S(included.)s 431029 r(Indeed,)s o 964834 d u 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f19(\()s f18(`)s f19(\))s 218450 W(=)S f18(G)s f19(\()s f18(`)s f19(\)[)s u(log)s M 1125705 r(log)s o 2120340 r(\()s f18(`)s 174760 W f20(_)S f18(e)s f19(\)])s u -285390 d f2(1+)s f22(\22)s u 15474 r -185030 d f29(0)s O 1393108 r f1(for)s 196608 W(an)s -11797 X(y)S f18(\21)s 246667 r(<)s 218450 r f19(1)S f1(and)S(an)T(y)S f18(\22)s u 21390 r -285390 d f24(0)s o 423705 r f18(>)s 218450 r f19(0)s f1(.)s o 2230454 d u -473657 r f17(1.3)s 196608 W(Main)S(r)s -14156 r(esult)s o 1504431 d u -497250 r f1(W)s -62915 r(e)s 196608 W(be)s -11797 r(gin)S(by)S(de\fning)S(the)S(scale)S(function)S(of)S (the)S(almost-sure)S(asymptotics:)s o 2287314 d U U u -532027 d u 10985804 r f18(b)s u 117964 d f22(t)s O 10775907 r 351147 d 31457 981125 pr 720333 d u f18(\13)s u -270881 d u 5007 r f2(2)s o 489917 d u f22(b)s u 65536 d f26(t)s O O O o 12054125 r f19(=)s 218450 W f20(\0)s u 131070 r f19(log)s o 1256775 r f18(G)s f19(\()s f18(t)s f19(\))s f18(;)s 1703934 r(t)S(>)S f19(0)s f18(:)s M u 29719095 r 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W(diameter)S f19(2)s f18(R)s 6474 r(\13)s 5007 r f19(\()s f18(b)s u 117964 d f22(t)s o 233930 X f19(\))s f1(,)s 209293 r(where)S f18(\30)s 244759 r f1(is)S(shaped)S(lik)s -7865 r(e)S f18( )s u 117964 d f22(t)s o x f19(\()s f20(\1)s f19(\))s 236067 W f20(\31)S f18( )s 28217 r f19(\()s f20(\1)s f18(=\13)s 5007 W f19(\()s f18(b)s u 117964 d f22(t)s o x f19(\)\))s f18(=\13)S f19(\()s f18(b)s u 117964 d f22(t)s o x f19(\))s u -285390 d f2(2)s o 311300 r f1(.)s 313558 r(Let)s 206756 W(us)S(assume)S(that)S f18(R)s 213230 r f1(and)s o y u -473657 r f18( )s 282552 r f1(approximately)s 254335 X(maximize)s u 262198 r(\(2.5\))s o 1769002 r(,)s 268766 W(i.e.,)S f18(\25)s u 117964 d f22(R)s o 455571 r f19(\()s f18( )s 28217 r f19(\))s 318669 W f20(\31)S(\0)s U 65533 r f23(e)s M f18(\37)s O 482414 r f1(.)s 442139 r(Then)T(the)T(dominating)T(strate)s -11797 r(gy)T(for)T(the)T(w)s -7865 r(alk)T(is)s o y u -473657 r(to)s 249829 X(mo)s -11797 W(v)S(e)T(in)T(a)T(short)T(time)T(to)T (that)T(f)s -7865 r(a)s -15729 W(v)S(orable)T(microbox)T(and)T(spend)T(the)T(rest)T (of)T(the)T(time)T(until)T f18(t)T f1(in)T(it.)s 428621 r(The)s o y u -473657 r(contrib)s -15729 r(ution)s 247301 W(coming)S(from)S(the)S(long)S(stay)S(in)S(the)S (microbox)S(is)S(roughly)s u w f19(exp)s o 1423751 r([)s f18(t\25)s u 122335 d f22(R)s 3254 r(\13)s f2(\()s f22(b)s u 65536 d f26(t)s o 220272 r f2(\))s o 1705198 r f19(\()s f18( )s u 117964 d f22(t)s o 233930 r f19(\)])s f1(,)s 259974 r(which)S(can)S(be)s o y u -473657 r(approximated)s 167075 W(by)s u w f19(exp)s o 1343525 r([)s f18(t\13)s 5007 X f19(\()s f18(b)s u 117964 d f22(t)s o 233930 r f19(\))s u -285390 d f24(\0)s f2(2)s o 744572 r f18(\25)s u 117964 d f22(R)s o 455571 r f19(\()s f18( )s 28217 r f19(\)])s 218450 W f20(\31)s u w f19(exp)s o 1394900 r([)s f20(\0)s f18(t\13)T f19(\()s f18(b)s u 117964 d f22(t)s o 233930 r f19(\))s u -285390 d f24(\0)s f2(2)s M u 810105 r f23(e)s M 744572 r f18(\37)s O 1226986 r f19(])s f1(,)s 172982 r(using)s 167075 W(the)S(scaling)S(properties)S(of)S(the)S(Laplace)s o y u -473657 r(operator)s -43254 r(.)s o y u 298619 r(The)s 196238 W(size)S(of)S(the)S(macrobox)S(is)S(determined)S(by)S (the)S(amount)S(of)S(mass)S(the)S(w)s -7865 X(alk)S(loses)S(on)S(the)S(w)T(ay)S(from)S (the)s o y u -473657 r(origin)s 246350 W(to)S(the)S(f)s -7865 r(a)s -15729 W(v)S(orable)s 246350 W(microbox,)s 258786 r(while)S(tra)s -15729 r(v)s -11797 X(eling)S(through)S(long)S(stretches)S(of)S(lar)s -14156 r(ge)S(ne)T(g)s -3933 r(ati)s -19661 r(v)T(e)S(potential.)s o 1011703 d u -516911 r(A)s 248683 r(calculation)s 248736 W(sho)s -19661 r(ws)S(that)S(the)S(penalty)S(it)S(pays)S(is)S (roughly)S(of)S(order)s u w f19(exp)s o 1425186 r([)s f20(\0)s u 131070 r -589827 d f23(P)s M -382296 d u 961196 r f22(r)s 15474 r f2(\()s f22(t)s f2(\))s o 611676 d u 961196 r f22(x)s f2(=1)s O u 2150413 r f19(log)s o 3145048 r(\()s f20(\0)s f18(\30)s 38003 r f19(\()s f18(x)s f19(\))s 210959 W f20(_)S f19(1\)])s f1(.)s 439498 r(\(An)s o y u -473657 r(optimal)s 184094 W(strate)s -11797 r(gy)S(is)S(not)S(to)S(spend)S(more)S(than)S f19(\()s f20(\0)s f18(\30)s 38003 r f19(\()s f18(x)s f19(\))s 149479 W f20(_)S f19(1\))s u -285390 d f24(\0)s f2(1)s o 928666 r f1(time)s 184094 W(units)S(at)S(each)S(site)S f18(x)S f1(on)S(the)S(w)s -7865 r(ay)s -51118 r(.\))s 278944 r(Under)s o y u -473657 r(our)s 147428 W(assumptions)S(on)S(the)S(lo)s -19661 X(wer)S(tails)S(of)S f18(\30)s 38003 r f19(\(0\))s f1(,)s 157265 r(a)S(Borel-Cantelli)S(ar)s -14156 r(gument)S(sho)T(ws)S(that)S(this)S(penalty)S(is)S (roughly)s o y U -473657 r f19(exp)s o 702793 r([)s f20(\0)s f18(G)s u -285390 d f24(\0)s f2(1)s o 744572 r f19(\(1)s f18(=r)s 21390 r f19(\()s f18(t)s f19(\)\)])s 196608 W f1(where)S f18(G)s u -285390 d f24(\0)s f2(1)s o 941180 r f1(denotes)S(the)S(in)s -31458 r(v)s -11797 r(erse)S(function)S(of)S f18(G)s f1(.)s o y u 269521 r(As)s 244330 W(it)S(turns)S(out,)s 256260 r(the)S(tw)s -7865 X(o)S(mechanisms)S(run)S(at)S(optimal)S(\223speed\224)S(when)S (the)S(tw)T(o)S(e)s -11797 r(xponents)S(are)S(roughly)s o y u -473657 r(of)s 308202 W(the)S(same)S(order)s -31458 r(,)s 336101 W(i.e.,)S f18(G)s u -285390 d f24(\0)s f2(1)s o 744572 r f19(\(1)s f18(=r)s 21390 r f19(\()s f18(t)s f19(\)\))s 412187 W f20(\31)S f18(t\13)s 5007 r f19(\()s f18(b)s u 117964 d f22(t)s o 233930 r f19(\))s u -285390 d f24(\0)s f2(2)s o 1156759 r f20(\31)S f18(t)s f1(,)s 336101 r(because)s 308202 X f18(\13)s u 117964 d f22(b)s u 65536 d f26(t)s O 903527 r f20(\34)S f18(t)s f1(.)s 617898 r(Recalling)T(that)T f18(r)s 21390 r f19(\()s f18(t)s f19(\))S f20(\31)s o y U -473657 r f19(exp)s o 702793 r([)s f18(b)s u 117964 d f22(t)s o 233930 W f18(\13)s 5007 r f19(\()s f18(b)s u 117964 d f22(t)s o w f19(\))s u -285390 d f24(\0)s f2(2)s o 744572 r f19(])s f1(,)s 260410 r(this)s 247650 W(reasoning)S(leads)S(to)s u w(\(1.7\))s o 1754454 r(.)s 392987 r(A)s 247599 r(\fne)S(tuning)S(of)S f18(r)s 21390 r f19(\()s f18(t)s f19(\))S f1(mak)s -7865 r(es)S(the)S(contrib)s -15729 r(ution)S(from)S(the)s o y u -473657 r(tra)s -15729 r(v)s -11797 X(el)s 186825 W(to)S(the)S(microbox)S(ne)T(gligible)S(compared)S(to)S (the)S(contrib)s -15729 r(ution)S(from)S(the)S(stay)S(in)S(it,)s 188782 W(i.e.,)S(we)s 186825 W(shall)S(in)S(f)s -7865 r(act)s o y u -473657 r(ha)s -15729 r(v)s -11797 r(e)s 196608 r f18(G)s u -285390 d f24(\0)s f2(1)s o 744572 X f19(\(1)s f18(=r)s 21390 r f19(\()s f18(t)s f19(\)\))s 218450 W(=)S f18(o)s f19(\()s f18(t\13)s 5007 r f19(\()s f18(b)s u 117964 d f22(t)s o 233930 r f19(\))s u -285390 d f24(\0)s f2(2)s o x f19(\))s f1(.)s 283115 r(Hence,)s 196608 W(we)S(obtain)s u 208404 r(\(1.8\))s o 1911816 r(with)s U 262141 r f23(e)s M w f18(\37)s O 875630 r f1(as)S(in)s u 208404 r(\(2.5\))s o 1715208 r(.)s o 2250298 d u 10517099 r f16(3.)s 297093 W(Pr)s -17477 r(oof)S(of)S(Theorem)S(1.1)s o 1690888 d u -516911 r f1(As)s 208328 W(in)S([)s u(BK00)s o 1878786 r(],)s 211258 r(the)S(main)S(result)S(will)S(be)S(pro)s -11797 W(v)S(ed)s 208328 W(by)S(separately)S(pro)s -11797 r(ving)S(upper)S(and)S(lo)s -19661 r(wer)S(bounds)S(in)s u 220124 r(\(1.8\))s o 1726928 r(.)s o y u -487813 r(The)s 196608 W(proof)S(of)S(Corollary)S(1.2)S(comes)S(at)S(the)S (v)s -11797 r(ery)S(end)S(of)S(this)S(section.)s o z u -473657 r f17(3.1)s 173015 r(The)s 196608 W(upper)S(bound)s o 1504431 d u -473657 r f1(Recall)s 196608 W(the)S(notation)S(of)S(Subsection)S(2.1,)S(in)S(particular)S (that)S f18(Q)s u 117964 d f22(R)s o 674021 r f19(=)s 218450 r([)s f20(\0)s f18(R)s 6474 W(;)s 131070 r(R)S f19(])s 174760 W f20(\\)S f21(Z)s f1(.)s 283115 r(Let)s o 1730972 d U 11631302 r f18(r)s 21390 r f19(\()s f18(t)s f19(\))s 218450 W(=)S f20(\0)s U u -532027 d u 628330 r f19(3)s o 78643 r 351147 d 31457 1484394 pr 720333 d u f18(G)s f19(\()s f18(t)s f19(\))s O O u 1772750 r(log)s o 2898455 r f18(G)s f19(\()s f18(t)s f19(\))s f18(:)s M u 29719095 r f1(\(3.1\))s O o 1998359 d u -473657 r(Note)s 196608 W(that)S f18(r)s 21390 r f19(\()s f18(t)s f19(\))s 218450 W(=)S f18(t)s u -285390 d f22(\20)s 35953 r f2(+)s f22(o)s f2(\(1\))s o 1924507 r f1(as)s 196608 X f18(t)S f20(!)S(1)T f1(if)s u -198792 d u 346026 r f23(e)s o 198792 d u x f18(G)s O 804012 r f19(\()s f18(`)s f19(\))S(=)S f18(`)s u -285390 d f24(\0)s f22(\20)s 35953 r f2(+)s f22(o)s f2(\(1\))s o 2357779 r f1(as)T f18(`)S f20(!)S(1)s f1(.)s 239861 r(Abbre)s -19661 r(viate)T f18(B)s u 117964 d f22(R)s o 455571 r f19(\()s f18(t)s f19(\))S(=)S f18(Q)s u 122335 d f22(r)s 15474 r f2(\()s f22(t)s f2(\)+2)s f24(b)s f22(R)s 3254 r f24(c)s o 2563827 r f1(.)s o y u 298619 r(The)s 201145 W(crux)S(of)S(the)S(proof)S(of)S(the)S(upper)S (bound)S(in)s 186989 r(Theorem)S(1.1)S(is)S(the)S(follo)s -19661 r(wing)S(generalization)S(of)S(Propo-)s o y u -473657 r(sition)s 196608 W(4.4)S(of)S([)s u(BK00)s o 1878786 r(])S(adapted)S(to)S(the)S(ne)s -19661 r(w)S(de\fnition)S(of)S f18(r)s 21390 r f19(\()s f18(t)s f19(\))s f1(.)s O o eop % [7] end % dvidict %%PageTrailer %%Page: 8 8 dvidict begin 8 8 bop % [8] 43384832 d u -44564480 d U u -473657 r f15(8)s o 9719711 r(MAREK)s 147456 W(BISKUP)s 115015 r(AND)s 129761 r(W)s -5899 r(OLFGANG)S(K)s u 114721 r -125043 d(\177)s o(ONIG)s O 42598400 d u -36211848 d u -3650424 d u -473657 r f17(Pr)s -14156 r(oposition)s 255230 r(3.1)s 338732 r f3(Ther)s -29098 r(e)s 245007 W(e)s -15729 r(xists)S(a)S(constant)S f18(C)s 358952 r f19(=)s 302474 W f18(C)s 56478 r f19(\()s f18(\24)s f19(\))S f18(>)S f19(0)s 245007 W f3(and)S(a)S(r)s -11797 X(andom)S(variable)S f18(C)s u 117964 d f22(\30)s o 600575 r f20(2)s 302474 r f19(\(0)s f18(;)s 131070 r f20(1)s f19(\))S f3(suc)T(h)s o 945028 d u -473657 r(that,)s U 196608 r f19(Prob)s O 1832807 r f3(-almost)s 196608 W(sur)s -29098 r(ely)s -43254 r(,)S(for)S(all)S f18(R)s 6474 r(;)s 131070 r(t)s 218450 W(>)S(C)s 56478 r f3(,)s o 1463489 d U 6700177 r f18(u)s f19(\()s f18(t;)s 131070 X f19(0\))s 218450 W f20(\24)S f18(C)s u 117964 d f22(\30)s o 298101 r f18(e)s u -324711 d f24(\0)s f22(t)s o 841962 r f19(+)s 174760 r f18(e)s u -324711 d f22(C)s 37046 r(t=R)s u 3254 r -185030 d f28(2)s O 1642400 r f19(3)s f18(r)s 21390 r f19(\()s f18(t)s f19(\))s u x(exp)s M U 1307520 r -872947 d f23(n)s O o 1831809 r f18(t)s U 466527 r f19(max)s o 524288 d u x f22(z)s 22149 r f24(2)s f22(B)s u 88110 d f26(R)s o 396808 r f2(\()s f22(t)s f2(\))s O 2366184 r f18(\25)s u -324711 d u f2(d)s o 519130 d u f22(z)s 22149 r f2(;2)s f22(R)s O 1170105 r f19(\()s f18(\30)s 38003 r f19(\))s U u -872947 d f23(o)s O o 524289 r f18(:)s M u 29715163 r f1(\(3.2)s 3932 r(\))s O O 1131507 d u -473657 r f3(Pr)s -35390 r(oof)s 169021 W(of)s 154865 r(Theor)s -29098 r(em)S(1.1,)s 174538 X(upper)S(bound.)s 250326 r f1(W)s -31458 r(ith)S(Proposition)S(3.1)S(in)S(the)S(hand,)T(the)S (proof)S(goes)S(along)S(v)s -11797 r(ery)S(much)s o 945028 Y u -473657 r(the)s 217693 W(same)S(lines)S(as)S(in)S([)s u(BK00)s o 1878786 r(].)s 346370 r(Indeed,)s 222964 r(let)S f18(r)s 21390 r f19(\()s f18(t)s f19(\))S f1(be)S(as)S(in)s u 229489 r(\(3.1\))s o 1953986 X(and)S(set)S f18(R)s 224167 r f1(in)s u 229489 r(\(3.2\))s o x(to)S(be)S f18(R)s 6474 r(\13)s 5007 r f19(\()s f18(K)s 56478 r(b)s u 117964 d f22(t)s o 233930 r f19(\))s f1(,)s 222964 r(where)s o y u -473657 r f18(K)s 274928 r(>)s 218450 X f19(0)s 194019 W f1(will)S(be)S(chosen)S(later)S(and)S f18(R)s 200493 r f1(will)S(tend)S(to)S f20(1)s f1(.)s 282252 r(Let)S f18(H)s 254777 r f1(be)S(in)S(the)S f18(\r)s 42780 r f1(-class)S(and)S(recall)S(that)S f18(\13)s u 117964 d f22(t)s o 452380 r f19(=)T f18(t)s u -285390 d f22(\27)s 34285 r f2(+)s f22(o)s f2(\(1\))s O y u -473657 r f1(with)s 196608 r f18(\27)s 268057 r f19(=)s 218450 r(\(1)s 174760 W f20(\0)S f18(\r)s 42780 X f19(\))s f18(=)s f19(\(3)S f20(\0)S f18(\r)T f19(\))s f1(.)s o 985969 d u 269521 r(Abbre)s -19661 r(viate)s 211117 X f18(B)s 39686 r f19(\()s f18(t)s f19(\))s 243638 W(=)S f18(B)s u 122335 d f22(R)s 3254 r(\13)s f2(\()s f22(K)s 37046 r(b)s u 65536 d f26(t)s o 220272 r f2(\))s o 2212016 r f19(\()s f18(t)s f19(\))T f1(and)T f18(\25)s f19(\()s f18(z)s 35043 r f19(\))S(=)S f18(\25)s u -285390 d u f2(d)s o 519044 d u f22(z)s 22149 r f2(;)s u(2)s f22(R)s 3254 r(\13)s f2(\()s f22(K)s 37046 r(b)s u 65536 d f26(t)s o 220272 r f2(\))s O o 2926550 r f19(\()s f18(\30)s 38003 r f19(\))s f1(,)s 214744 r(and)T(note)T(that)T f18(r)s 21390 r f19(\()s f18(t)s f19(\))S f20(\24)S f18(e)s u -285390 d f22(o)s f2(\()s f22(t\13)s f2(\()s f22(b)s u 65536 d f26(t)s o 220272 r f2(\))s u -185030 d f29(\0)s f28(2)s o 651712 r f2(\))s o 2837789 r f1(.)s 312485 r(Then,)s o 1020086 d u -473657 r(using)s 196608 W(also)S(that)s u w f19(lim)s M 1266108 r 117964 d f22(t)s f24(!1)s o 2745236 r f18(\13)s 5007 W f19(\()s f18(K)s 56478 r(b)s u 117964 d f22(t)s o 233930 X f19(\))s f18(=\13)S f19(\()s f18(b)s u 117964 d f22(t)s o x f19(\))s 218450 W(=)S f18(K)s u 56478 r -285390 d f22(\27)s o 397360 r f1(,)s 196608 W(we)S(ha)s -15729 r(v)s -11797 r(e)S(from)s u 217841 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f1(\(3.3\))s O o 1749504 d U u -473657 r f19(Prob)s O 1162542 r f1(-almost)s 196608 W(surely)s -51118 r(.)s 239861 r(Abbre)s -19661 r(viating)S f18(M)s 82148 r f19(\()s f18(t)s f19(\))s 218450 W(=)s u w(max)s M 1651580 r 122335 d f22(z)s 22149 r f24(2)s f22(B)s 25850 r f2(\()s f22(t)s f2(\))s o 3548504 r f18(\25)s f19(\()s f18(z)s 35043 r f19(\))s f1(,)s 196608 W(we)S(ha)s -15729 r(v)s -11797 X(e)S(to)S(pro)T(v)T(e)S(that,)S(for)S (an)T(y)S f18(")s 218450 W(>)S f19(0)s f1(,)s o 1341670 d U U 6939354 r f19(lim)s 131070 r(sup)s o 624775 d u 7461663 r f22(t)s f24(!1)s O 9430332 r f18(\13)s u -324711 d u 5007 r f2(2)s o 519130 d u f22(b)s u 65536 d f26(t)s O o 491340 r f18(M)s 82148 r f19(\()s f18(t)s f19(\))s 218450 W f20(\24)S(\0)s U 65533 r f23(e)s M f18(\37)s O 657174 r f19(+)s 174760 r f18(";)s U 1703934 r f19(Prob)s O U 3340133 r f1(-almost)s 196608 r(surely)s O 7818077 r f18(;)s M u 29717523 r f1(\(3.4)s 1572 r(\))s O o 1718919 d u -473657 r(for)s 196608 W(some)S(appropriate)S 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f22(\25)s f2(\()s f22(z)s 22149 r f2(\))s M U 2037646 r -637014 d f23(\13)s O o 2627468 r f20(\24)s 218450 r(\0)s f18(K)s u 56478 r -324711 d f2(1)s f24(\0)s f2(2)s f22(\27)s o 1387696 r f18(\037;)s M u 29719095 r f1(\(3.5\))s O o 1571016 d u -473657 r(where)s 208419 X f18(\37)s 238954 W f20(2)S f19(\(0)s f18(;)s 131070 r f20(1)s f19(\))T f1(is)T(a)T(constant)T(related)T(to)s U 273952 r f23(e)s M x f18(\37)s O 690833 r f1(,)s 211371 r(see)T([)s u(BK00)s o 1878786 r(].)s 318547 r(Since)T f18(t)S f20(7!)S f18(M)s 82148 r f19(\()s f18(t)s f19(\))T f1(is)T(increasing)T(and)T f18(t)S f20(7!)S f18(\13)s u 117964 d f22(b)s u 65536 d f26(t)s O o y u -473657 r f1(slo)s -19661 W(wly)s 196405 X(v)S(arying,)s 196445 r(it)T(suf)S(\fces)T(to)T(pro)s -11797 W(v)S(e)s u 204268 r(\(3.4\))s o 1907477 r(for)T f18(t)T f1(taking)T(only)T(a)T(discrete)T(set)T(of)T(v)s -19661 W(alues;)s 196473 r(the)T(main)T(dif)S(ference)s o y u -473657 r(compared)s 196608 W(to)S([)s u(BK00)s o 1878786 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351147 d 31457 828294 pr 720333 d u 250446 r(b)s O O U u 3809361 r -872947 d f23(\21)s O M 4279037 r -696873 d f22(`)s u 65536 d f26(n)s o 335415 r f2(\()s f22(x)s f2(\))s f24(\0)s f2(1)s M U 6333429 r -1108881 d f23(\25)s O o 6966942 r f20(\24)s 218450 W f18(n)s u -324711 d f24(\0)s f2(2+)s f22(o)s f2(\(1\))s o 2333656 r f19(+)s U u 174760 r -872947 d f23(\20)s O M U -532027 d u 723079 r f19(2)s f18(\24)s o 723079 r 351147 d 31457 828294 pr 720333 d u 250446 r(b)s O O U u 1630016 r -872947 d f23(\21)s O M 2099692 r -696873 d f22(L)s u 65536 d f26(n)s o 335415 r f22(p=)s f2(2)s o 3684192 r f18(;)s 1703934 r(n)S f20(!)S(1)s f18(:)s M u 29325879 r f1(\(3.18\))s O o z u -487813 r(The)s 196608 W(right)S(hand)S(side)S(is)S(clearly)S(summable)S(on)S f18(n)s 218450 W f20(2)S f21(N)s 196608 W f1(since)S f19(2)s f18(\024=b)s 218450 W(<)S f19(1)s f1(.)s 268959 r(This)s 196608 W(\fnishes)S(the)S(proof.)s u 2663294 r f31(\3)s O 1224718 d u 312775 r f1(Our)s 196608 W(ne)s -11797 r(xt)S(task)S(is)S(to)S(get)S(a)S(good)S (estimate)S(on)S(the)S(size)S(of)S(the)S(products)S (in)s u 208404 r(\(3.7\))s o 1715208 r(.)s o 1452932 d u -473657 r f17(Lemma)s 210763 r(3.3)s 327680 r f3(Suppose)s 196608 W(that)S f20(h)s u f19(log)s o 994635 r(\()s f20(\0)s f18(\30)s 38003 r f19(\(0\))s 174760 W f20(_)S f19(1\))s f20(i)s 218450 W f19(=)S f20(1)s f3(.)s 268959 r(Then,)s 196608 W(for)S(all)S f18(b)s 218450 W f20(\25)S f19(1)s f3(,)s o 2130497 d U U 4400787 r f19(lim)s o 471859 d u 4209473 r f22(n)s f24(!1)s O U u -532027 d u 7269101 r f19(1)s o 5871314 r 351147 d 31457 3180593 pr 720333 d u f18(G)s u -227191 d f24(\0)s f2(1)s o 744572 r f19(\(1)s f18(=n)s f19(\))s O O u -1044210 d u 9261620 r f24(b)s f2(2)s f22(n)s u 92844 r f2(log)s o 905229 r f22(n)s f24(c)s o 1044210 d U 9871104 r -747114 d f23(X)s O 904400 d u 9926395 r f22(x)s f2(=1)s O u 11747619 r f19(log)s M U 12742254 r -872947 d f23(\20)s O M U -532027 d u 13290573 r f20(\0)s f18(\30)s 38003 r f19(\()s f18(x)s 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1853654 r f20(\0)s f18(N)s u 117964 d f22(n)s o 370768 X f18(G)s f19(\(1)s f18(=\25)s f19(\))s 174760 W(+)S f18(N)s u 117964 d f22(n)s o x f18(\25)s u 131070 r f19(log)s o 1256775 r f18(b)S f19(+)S f18(\25\22)s 21390 r(G)s u -285390 d f24(\0)s f2(1)s o 744572 r f19(\(1)s f18(=n)s f19(\))s U u -637014 d f23(\t)s O O 27826339 r f18(;)s M u 29325879 r f1(\(3.21\))s O o 2024128 d u -473657 r(for)s 287377 W(an)s -11797 r(y)S f18(\25)s 376033 W(>)S f19(0)s f1(.)s 555422 r(Set)s 287377 X f18(\25)S f19(=)S(1)s f18(=G)s u -285390 d f24(\0)s f2(1)s o 744572 r f19(\(1)s f18(=n)s f19(\))T f1(and)T(note)T(that)T(we)T(ha)s -15729 r(v)s -11797 r(e)T f18(G)s f19(\(1)s f18(=\25)s f19(\))s f18(=\25)S f20(!)S(1)T f1(as)T f18(\25)S f20(#)S f19(0)s f1(,)s 310069 r(due)T(to)s o y u -473657 r f20(h)s u f19(log)s o 994635 r(\()s f20(\0)s f18(\30)s 38003 r f19(\(0\))s 215554 W f20(_)S f19(1\))s f20(i)s 320436 W f19(=)S f20(1)s f1(.)s 459350 r(Consequently)s -51118 r(,)s 270039 r(the)s 255353 W(term)S(with)s u 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f1(.)s 333594 X(Note)S(that)S f18(r)s 21390 r f19(\()s f18(t)s f19(\))s 247662 W f20(\25)S(b)s f19(2)s f18(n)s u 131070 r f19(log)s o 1256775 r f18(n)s f20(c)s f1(.)T(Combining)s 213434 W(the)S(results)s o y u -473657 r(of)s 196608 W(Lemma)S(3.2)S(for)S f18(R)s 224924 r f19(=)s 218450 r f18(r)s 21390 r f19(\()s f18(t)s f19(\))S f1(and)S(Lemma)S(3.3,)S(we)S(deri)s -19661 X(v)s -11797 r(e,)S(for)S(suf)T(\fciently)S(lar)s -14156 r(ge)S f18(n)S f1(resp.)S f18(t)s f1(,)S(the)S(bound)s o 1297312 d U 7858840 r f18(u)s f19(\()s f18(t;)s 131070 X f19(0\))s 174760 W f20(\0)S f18(u)s u 122335 d f22(r)s 15474 r f2(\()s f22(t)s f2(\))s o 934052 r f19(\()s f18(t;)T f19(0\))s 218450 W f20(\24)S f19(2)s f18(C)s 56478 r f19(\()s f18(\30)s 38003 r f19(\))s u x(exp)s M U 1307520 r -637014 d f23(\0)s O o 1667970 r f20(\0)s f18(G)s u -324711 d f24(\0)s f2(1)s o 744572 r f19(\(1)s f18(=n)s f19(\))s U u -637014 d f23(\1)s O o 360450 r f18(;)s M u 29321947 r f1(\(3.22)s 3932 r(\))s O O o eop % [10] end % 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f18(\30)s 38003 r f19(\))s 175233 W f20(_)S f18(e)s u -285390 d f22(t)s u 72817 d f28(0)s O 506993 r f1(.)s 271003 r(The)T(rest)T(of)T(the)T(ar)s -14156 r(gument)T(does)T(not)T(in)s -31458 r(v)s -15729 r(olv)s -11797 r(e)T(the)s o y u -473657 r(particular)s 196608 W(form)S(of)S f18(r)s 21390 r f19(\()s f18(t)s f19(\))S f1(and)S(can)S(directly)S(be)S(tak)s -7865 r(en)S(o)s -11797 W(v)S(er)s 196608 W(from)S([)s u(BK00)s o 1878786 r(].)s u 9912725 r f31(\3)s O 1785532 d u -473657 r f17(3.2)s 173015 r(The)s 196608 W(lo)s -7865 r(wer)S(bound)s o 1504431 d u -473657 r f1(Unlik)s -7865 X(e)s 212924 W(the)S(upper)S(bound,)s 217003 r(the)S(lo)s -19661 r(wer)S(bound)S(w)T(as)S(basically)S(pro)s -11797 W(v)S(ed)s 212924 W(already)S(in)S([)s u(BK00)s o 1878786 r(],)s 217003 r(up)S(to)S(a)S(change)S(of)s o y u -473657 r(the)s 183880 W(spatial)S(scale)S(and)S(Lemma)S(3.4)S(belo)s -19661 r(w)s -51118 r(.)s 278872 r(F)s -11797 r(or)S(this)S(reason,)s 186425 r(we)S(shall)S(only)S(indicate)S(the)S(necessary)S(changes.)s o y u 312775 r(First)s 196608 W(we)S(pro)s -11797 W(v)S(e)s 196608 W(the)S(follo)s -19661 r(wing)S(con)s -31458 r(v)s -11797 r(erse)S(of)S(Lemma)S(3.3:)s o 1510866 d u -473657 r f17(Lemma)s 285282 r(3.4)s 370424 r f3(F)s -35390 r(ix)s 271127 W f18(\21)s 376038 r f20(2)s 347821 r f19(\(0)s f18(;)s 131070 r f19(1\))S f3(and)S(let)s u -198792 d u 420545 r f23(e)s o 198792 d u w f18(G)s u 117964 d f22(\21)s O o 1480770 r f3(satisfy)S(\(ii\))S(and)S(\(iii\))S(in)s 256971 X(Assumption)S(\(G\).)T(Then)S(ther)s -29098 r(e)S(e)s -15729 r(xists)S(a)s o y u -473657 r f18(\45)s 218450 W f20(2)S f19(\(0)s f18(;)s 131070 r f20(1)s f19(\))s 196608 W f3(suc)s -11797 r(h)S(that)s o 1936606 Z U U 4712382 r f19(lim)s 131070 r(sup)s o 624775 d u 5166272 r f22(n)s f24(!1)s O U u -532027 d u 8685866 r f19(1)s o 7282003 r 351147 d 31457 3192745 pr 861142 d U -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u -227191 d u f24(\0)s f2(1)s o 421610 d u f22(\21)s O O 1351976 r f19(\()s f18(\045=n)s f19(\))s O O u -983043 d u 11083442 r f22(n)s o 983043 d U 10684461 r -747114 d f23(X)s O 904400 d u 10739753 r f22(x)s f2(=1)s O u 11951493 r f19(log)s M U 12946128 r -637014 d f23(\0)s O o 13306578 r f20(\0)s f18(\30)s 38003 r f19(\()s f18(x)s f19(\))s 174760 W f20(_)S f19(1)s U u -637014 d f23(\1)s O o 578900 r f20(\24)s 218450 r f19(1)s U 1398276 r(Prob)s O U 3034475 r f1(-almost)s 196608 r(surely)s -51118 r(.)s O M u 29325879 r(\(3.23\))s O o 2509733 d u -473657 r f3(Pr)s -35390 r(oof)s -11797 X(.)s 264765 r f1(The)s 184025 W(ar)s -14156 r(gument)S(is)S(based)S(on)S(the)S(asymptotic)S(sublinearity)S (of)S f19(1)s f18(=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 1122541 r f1(at)S(in\fnity)s -51118 r(.)s 278921 r(Ho)s -19661 W(we)S(v)T(er)s -31458 r(,)s 186542 r(in)s 184025 W(order)S(to)s o 1052216 Y u -473657 r(ha)s -15729 r(v)s -11797 r(e)s 196608 W(sublinearity)S(on)S(the)S(whole)S(interv)s -19661 r(al)S f19(\(0)s f18(;)s 131070 r f20(1)s f19(\))s f1(,)S(we)S(\frst)S(construct)S(an)S(auxiliary)S(modi\fcation)S (of)s u -198792 d u 346026 r f23(e)s o 198792 d u w f18(G)s u 117964 d f22(\21)s O o 1135124 r f1(.)s o y u 312775 r(Let)s 168699 W f18(x)s u 117964 d f2(0)s o 529750 r f18(>)s 218450 r f19(0)S f1(be)S(such)S(that)S f19(1)s f18(=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 1107215 r f1(is)S(positi)s -19661 r(v)s -11797 X(e,)s 174281 W(increasing,)S(and)s 168699 W(conca)s -15729 r(v)T(e)S(on)S f19([)s f18(x)s u 117964 d f2(0)s o 311300 r f18(;)s 131070 r f20(1)s f19(\))s f1(.)s 273812 r(Let)S f18(D)s u 117964 d f2(0)s o 479999 r f1(to)S(be)S(the)S(right)s o y u -473657 r(deri)s -19661 W(v)S(ati)S(v)s -11797 r(e)s 196608 W(of)S f19(1)s f18(=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 1135124 r f1(at)S f18(x)s u 117964 d f2(0)s o 311300 r f1(.)s 283115 r(De\fne)s u -198792 d u 346026 r f23(b)s o 198792 d u w f18(G)s u 117964 d f22(\21)s O M 1222504 r f19(:)s o 1698544 r(\(0)s f18(;)s 131070 X f20(1)s f19(\))s 218450 W f20(!)S f19(\(0)s f18(;)T f20(1)s f19(\))s 196608 W f1(by)S(the)S(formula)s o 2048753 d U 6074840 r f19(1)s f18(=)s u -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 X f19(\()s f18(x)s f19(\))s 218450 W(=)s U w -1344810 d f23(\()s M -536820 d U 851968 r f18(D)s u 117964 d f2(0)s o 311300 r f18(x)s M U 11835426 r f1(if)s O 12512544 r f18(x)S f20(\24)S f18(x)s u 117964 d f2(0)s o 311300 r f18(;)s O 1134029 d U 851968 r f19(1)s f18(=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o x f19(\()s f18(x)s f19(\))s 174760 W(+)S f18(D)s u 117964 d f2(0)s o 311300 r f18(x)s u 117964 d f2(0)s o 486060 r f20(\0)S f19(1)s f18(=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o x f19(\()s f18(x)s u 117964 d f2(0)s o 311300 r f19(\))s M U 11835426 r f1(if)s O 12512544 r f18(x)S(>)S(x)s u 117964 d f2(0)s o 311300 r f18(:)s O O M u 29324307 r f1(\(3.24)s 1572 r(\))s O o 2202400 d u -473657 r(Note)s 227852 W(that)S f19(1)s f18(=)s u -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 1166368 r f1(is)S(positi)s -19661 r(v)s -11797 X(e,)s 235663 W(increasing,)S(conca)s -15729 r(v)T(e)s 227852 W(and)S(hence)S(sublinear)S(on)S f19(\(0)s f18(;)s 131070 r f20(1)s f19(\))s f1(.)s 376847 r(Moreo)T(v)T(er)s -31458 r(,)s 192409 r(Assump-)s o y u -473657 r(tion)s 196608 W(\(G\)\(iii\))S(holds)S(true)S(for)s u -198792 d u 346026 r f23(e)s o 198792 d u w f18(G)s u 117964 d f22(\21)s O o 1331732 r f1(replaced)S(by)s u -198792 d u 346026 r f23(b)s o 198792 d u w f18(G)s u 117964 d f22(\21)s O o 1135124 r f1(.)s o 945028 Y u 312775 r(F)s -11797 r(or)s 196608 X f18(a)s 218450 W f20(\25)S f19(1)s f1(,)T(abbre)s -19661 r(viate)T f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\))S(=)s u w(log)s o 1213085 r(\()s f20(\0)s f18(\30)s 38003 r f19(\()s f18(x)s f19(\))s 174760 W f20(_)S f18(a)s f19(\))s f1(.)s 283115 r(Choose)T f18(a)s 218450 W f19(=)S f18(e)s u -285390 d f22(x)s u 72817 d f28(0)s O 816014 r f1(and)T(estimate,)T(for)T f18(n)S f20(!)S(1)s f1(,)s o z U U u -532027 d u 6999240 r f19(1)s o 4337175 r 351147 d 31457 5709150 pr 998775 d U -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O M U 938516 r -872947 d f23(\20)s O M 1408192 r -589827 d(P)s M -382296 d u 2238318 r f22(n)s o 611676 d u 2238318 r(x)s f2(=1)s O 3427535 r f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\))s U u -872947 d f23(\21)s O O O o 10343418 r f20(\24)s U u -532027 d u 1807877 r f19(1)s 174760 W(+)S f18(o)s f19(\(1\))s o 297093 r 351147 d 31457 5709150 pr 998775 d U -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O M U 938516 r -872947 d f23(\20)s O M 1408192 r -589827 d(P)s M -382296 d u 2238318 r f22(n)s o 611676 d u 2238318 r(x)s f2(=1)s O 3427535 r f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\))s U u -872947 d f23(\21)s O O O o 6303336 r f20(\24)s 218450 r f19(\(1)s 174760 W(+)S f18(o)s f19(\(1\)\))s u -983043 d u 530051 r f22(n)s o 983043 d U 131070 r -747114 d f23(X)s O 904400 d u 186362 r f22(x)s f2(=1)s O U u -532027 d u 2958923 r f19(1)s o 1476745 r 351147 d 31457 3349375 pr 861142 d U -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 r f19(\()s f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\)\))s O O 4904763 r f18(;)s M u 29325879 r f1(\(3.25\))s O o 2548433 d u -473657 r(where)s 163736 W(we)S(used)S(the)S(f)s -7865 r(act)S(that)s u w -589827 d f23(P)s M -382296 d u 993862 r f22(n)s o 611676 d u 993862 r(x)s f2(=1)s O 2183079 r f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\))s 218450 W f20(!)S(1)s 163736 W f1(almost)S(surely)s -51118 r(,)s 170311 r(and)S(sublinearity)S(of)S f19(1)s f18(=)s u -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 r f1(.)s 272158 r(Since)S(we)S(ha)s -15729 r(v)s -11797 r(e)s o 1061687 d u -473657 r(that)s 176791 X f20(h)s f19(1)s f18(=)s u -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 r f19(\()s f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\)\))s f20(i)s 218450 W f18(<)S f20(1)s f1(,)s 180755 r(the)T(Strong)T(La)s -11797 r(w)T(of)T(Lar)s -14156 r(ge)T(Numbers)T(tells)T(us)T(that)T(the)T(right-hand)T (side)T(of)s u 239705 r(\(3.25\))s O y u -473657 r(is)s 196608 W(almost)S(surely)S(no)S(more)S(than)S f18(\045n)s f1(,)S(where)S(for)S f18(\45)S f1(we)S(can)S(tak)s -7865 r(e,)S(for)S(instance,)s o 1390003 d U 11955369 r f18(\45)s 218450 W f19(=)S(2)s U u -637014 d f23(\n)s O o 371372 W f19(1)s f18(=)s u -198792 d u 149418 r f23(b)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 r f19(\()s f18(Y)s u 117964 d f22(a)s o 328655 r f19(\(0\)\))s U u -637014 d f23(\13)s O o w f18(:)s M u 29325879 r f1(\(3.26\))s O o 1332069 d u -473657 r(Hence,)s 196608 W(we)S(deri)s -19661 r(v)s -11797 r(e)s o 1520126 d U u -983043 d u 9952915 r f22(n)s o 983043 d U 9553934 r -747114 d f23(X)s O 904400 d u 9609226 r f22(x)s f2(=1)s O 10820966 r f18(Y)s u 117964 d f2(1)s o 311300 r f19(\()s f18(x)s f19(\))s 218450 W f20(\24)s u -983043 d u 617431 r f22(n)s o 983043 d U w -747114 d f23(X)s O 904400 d u 273742 r f22(x)s f2(=1)s O 1485482 r f18(Y)s u 117964 d f22(a)s o 328655 r f19(\()s f18(x)s f19(\))S f20(\24)s u -198792 d u 367868 r f23(e)s o 198792 d u w f18(G)s u -324711 d u f24(\0)s f2(1)s o 519130 d u f22(\21)s O O 1570426 r f19(\()s f18(\045=n)s f19(\))s f18(;)s M u 29325879 r f1(\(3.27\))s O o 1801082 d u -473657 r(which)s 196608 W(directly)S(yields)S(the)S(desired)S(claim.)s u 18725173 r f31(\3)s O 1224718 d u 269521 r f1(Another)s 223653 W(important)S(ingredient)S(is)S(the)S(follo)s -19661 r(wing)S(adaptation)S(of)S(the)S(crucial)S(Proposition)S(5.1)S (of)S([)s u(BK00)s o 1878786 r(])s o y u -473657 r(to)s 196608 W(the)S(present)S(situation.)s 283115 r(F)s -11797 r(or)S f18(\21)s 246667 r f20(2)s 218450 r f19(\(0)s f18(;)s 131070 r f19(1\))s f1(,)S(choose)S f18(\45)S f1(as)S(in)S(Lemma)S(3.4)S(and)S(let)S(this)S (time)s o 1519815 d U 12935458 r f18(\r)s u 117964 d f22(t)s o 452380 r f19(=)s U u -532027 d u 1625942 r f18(\45)s o 297093 r 351147 d 31457 3054870 pr 861142 d U -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 r f19(\()s f18(t\13)s u -325497 d u 5007 r f24(\0)s f2(3)s o 544533 d u f22(b)s u 65536 d f26(t)s O o 749579 r f19(\))s O O M u 29325879 r f1(\(3.28\))s O O o eop % [11] end % dvidict %%PageTrailer %%Page: 12 12 dvidict begin 12 12 bop % [12] 43384832 d u -44564480 d U u -473657 r f15(12)s o 9719711 r(MAREK)s 147456 W(BISKUP)s 115015 r(AND)s 129761 r(W)s -5899 r(OLFGANG)S(K)s u 114721 r -125043 d(\177)s o(ONIG)s O 42598400 d u -40281394 d u -473657 r f1(be)s 214652 W(the)S(size)S(of)S(the)S(macrobox)S f18(Q)s u 117964 d f22(\r)s u 65536 d f26(t)s O 754610 r f1(\(see)S(Subsection)S(2.2\).)s 337246 r(Note)S(that)S f18(t)s u -285390 d f22(\21)s 18811 r(\20)s 35953 r f2(+)s f22(o)s f2(\(1\))s o 2276018 r f20(\24)s 249775 W f18(\r)s u 117964 d f22(t)s o 483705 r f20(\24)S f18(t)s u -285390 d f22(\20)s 35953 r f2(+)s f22(o)s f2(\(1\))s o 1942551 r f1(as)s 214652 X f18(t)S f20(!)S(1)T f1(if)s o 964051 d u -473657 r f18(G)s f19(\()s f18(`)s f19(\))s 218450 W(=)S f18(`)s u -285390 d f24(\0)s f22(\20)s 35953 r f2(+)s f22(o)s f2(\(1\))s o 2357779 r f1(as)s 196608 X f18(`)S f20(!)S(1)s f1(.)s 283115 r(Suppose)T(without)T(loss)T(of)T(generality)T(that)T f18(t)S f20(7!)S f18(\r)s u 117964 d f22(t)s o 430538 r f1(is)T(increasing.)s o 945028 Y u 312775 r(De\fne)s 196608 W(for)S(each)S f18( )s 246667 r f20(2)s 218450 r f18(C)s u 56478 r -285390 d f24(\0)s o 522518 r f19(\()s f18(R)s 6474 r f19(\))S f1(a)S(\223microbox\224)s o 1987103 d U 8879309 r f18(Q)s u -324711 d f2(\()s f22(t)s f2(\))s o 885652 r f19(=)s U 218450 r -1344810 d f23(\()s M -536820 d U 851968 r f18(Q)s u 122335 d f22(R)s 3254 r(\13)s f2(\()s f22(b)s u 65536 d f26(t)s o 220272 r f2(\))s O U u 8066472 r f1(if)s O 8743590 r f18(\r)s 261230 r f20(6)s f19(=)s 218450 r(0)s f18(;)s O 1134029 d U 851968 r(Q)s u 122335 d f22(R)s 3254 r(\13)s f2(\()s f22(b)s u 65536 d f26(t)s o 220272 r f2(\))s o 1879958 r f20(\\)s U 174760 r f19(supp)s O 1892968 r f18( )s u 117964 d f22(t)s O U u 8066472 r f1(if)s O 8743590 r f18(\r)s 261230 r f19(=)s 218450 r(0)s f18(;)s O O M u 29325879 r f1(\(3.29\))s O o 2020545 d u -473657 r(where)s 201079 X f18( )s u 117964 d f22(t)s M 321310 r f19(:)s o 798903 r f21(Z)s 226213 W f20(!)S f19(\()s f20(\0001)s f18(;)s 131070 r f19(0])T f1(is)T(the)T(function)T f18( )s u 117964 d f22(t)s o 233930 r f19(\()s f20(\1)s f19(\))S(=)S f18( )s 28217 r f19(\()s f20(\1)s f18(=\13)s u 117964 d f22(b)s u 65536 d f26(t)s O 491340 W f19(\))s f18(=\13)s u -285390 d u 5007 r f2(2)s o 489917 d u f22(b)s u 65536 d f26(t)s O o w f1(.)s 282373 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f20(\0)s u -1015083 d u 131070 r f22(y)s u 65536 d f26(t)s o 220272 r f24(\0)s f2(1)s o 1015083 d U 166800 r -747114 d f23(X)s O 904400 d u 222092 r f22(x)s f2(=0)s O u 1469562 r f19(log)s M U 2464197 r -637014 d f23(\0)s O o 2824647 r f20(\0)s f18(\30)s 38003 r f19(\()s f18(x)s f19(\))s 174760 W f20(_)S f19(1)s U u -637014 d f23(\1)s O M U 360450 r -872947 d(o)s O o 884739 r f18(;)s O O M u 29325879 r f1(\(3.35\))s O o 3372314 d u -473657 r(where)s 196608 W(we)S(recalled)S(the)S(notation)S(of)s u 259522 r(\(3.8\))s o 1766326 r(.)s 283115 r(No)s -19661 r(w)S f18(y)s u 117964 d f22(t)s o 452380 r f20(\24)s 218450 r f18(\r)s u 117964 d f22(t)s o 233930 r f1(,)S(so)S(using)S(Lemma)S(3.4)S(we)S(ha)s -15729 r(v)s -11797 r(e)S(that)s o 1408540 d U U 3814582 r(II)s O 4556796 r f20(\25)s 218450 X f18(e)s u -324711 d f24(\0)s f2(\(2)s f22(\24)s f2(+1\))s f22(\r)s u 65536 d f26(t)s O u 2848917 r f19(exp)s M U 4025367 r -637014 d f23(\b)s O o 4484121 r f20(\0)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u -324711 d u f24(\0)s f2(1)s o 519130 d u f22(\21)s O O 1351976 r f19(\()s f18(\045=\r)s u 117964 d f22(t)s o 233930 r f19(\)\(1)s 174760 W(+)S f18(o)s f19(\(1\)\))s U u -637014 d f23(\t)s O o 677204 r f19(=)T f18(e)s u -324711 d f24(\0)s f2(\(2)s f22(\24)s f2(+1\))s f22(\r)s u 65536 d f26(t)s o 220272 W f24(\0)s f22(t\13)s f2(\()s f22(b)s u 65536 d f26(t)s o w f2(\))s u -185030 d f29(\0)s f28(3)s o 651712 r f2(\(1+)s f22(o)s f2(\(1\)\))s o 7379748 r f18(;)s M u 29325879 r f1(\(3.36\))s O o 1534249 d u -473657 r(where)s 290095 W(we)S(used)S(the)S(de\fnition)S(of)S f18(\r)s u 117964 d f22(t)s o 233930 r f1(.)s 563575 r(Since)S f19(1)s f18(=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 1228611 r f1(is)S(asymptotically)S(conca)s -15729 r(v)s -11797 r(e,)s 313466 r f18(\r)s u 117964 d f22(t)s o 614681 r f19(=)s 380751 W f18(\045=)s u -198792 d u 149418 r f23(e)s o 198792 d u f18(G)s u 117964 d f22(\21)s O o 938516 r f19(\()s f18(t\13)s u -325497 d u 5007 r f24(\0)s f2(3)s o 544533 d u f22(b)s u 65536 d f26(t)s O o 749579 r f19(\))S f20(\24)s o 1013480 d u -473657 r f18(O)s 21390 r f19(\()s f18(t\13)s u -325497 d u 5007 r f24(\0)s f2(3)s o 544533 d u f22(b)s u 65536 d f26(t)s O o 749579 X f19(\))s 196608 W f1(and)S(the)S(e)s -11797 r(xponent)S(is)S f18(o)s f19(\()s f18(t\13)s u -325497 d u 5007 r f24(\0)s f2(2)s o 544533 d u f22(b)s u 65536 d f26(t)s O o x f19(\))s f1(.)s 283115 r(Consequently)s -51118 r(,)s o 1518832 d U 10500057 r f18(u)s f19(\()s f18(t;)s 131070 r f19(0\))s 218450 W f20(\25)S f18(e)s u -324711 d f22(t\13)s f2(\()s f22(b)s u 65536 d f26(t)s o 220272 r f2(\))s u -185030 d f29(\0)s f28(2)s o 651712 r f2([)s f22(\25)s u 88110 d f26(R)s o 396808 r f2(\()s f22( )s 18811 r f2(\))s f24(\0)s f22(")s f2(+)s f22(o)s f2(\(1\)])s o 6087516 r f18(;)s M u 29325879 r f1(\(3.37\))s O o z u -473657 r(where)s 127578 W f18(o)s f19(\(1\))S f1(still)S(depends)S(on)S f18(\21)s 28217 r f1(.)s 245950 r(The)S(proof)S(is)S(\fnished)S(by)S(letting)S f18(t)s 218450 W f20(!)S(1)s 127578 W f1(\(which)S(eliminates)S(the)S(dependence)s o y u -473657 r(on)s 196608 W f18(\21)s 28217 X f1(\),)S(optimizing)S(o)s -11797 W(v)S(er)s 196608 W f18( )s 224825 r f1(and)S f18(R)s 203082 r f1(with)S f20(L)s u 117964 d f22(R)s o 455571 r f19(\()s f18( )T f19(\))s 218450 W f18(<)S f19(1)s 196608 W f1(and)S(letting)S f18(")s 218450 W f20(#)S f19(0)s f1(.)s u 10370118 r f31(\3)s O 1970578 d u 11688975 r f16(A)s -22938 r(ckno)s -17477 r(wledgments)s o 1690888 d u 312775 r f1(M.B.)s 261938 W(w)s -7865 X(ould)S(lik)T(e)S(to)S(thank)s 191159 r(Y)s -43254 r(imin)S(Xiao)S(and)S(Oded)S(Schramm)S(for)S(discussions)S (about)S(the)S(beha)s -15729 r(vior)S(of)s o y u -473657 r(sums)s 196608 W(of)S(i.i.d.)S(random)S(v)s -19661 r(ariables)S(with)S(in\fnite)S(mean.)s o 2063823 d u 13083248 r f16(References)s o 1729703 d U u -473657 r f6([BGT87])s O 2474808 r(N.H.)s 157081 W(Bingham,)s 158433 r(C.M.)S(Goldie)S(and)S(J.L.)s 145284 r(T)s -45875 r(eugels)S(\(1987\).)S f14(Re)s -26215 r(gular)s 134143 r(V)s -72745 r(ariation)s f6(.)S(Enc)s -9831 r(yclopedia)S(of)S(Mathematics)S(and)S(its)s o 810024 Y u 2438763 r(Applications)s 163840 W(v)s -13108 r(ol.)S(27,)S(Cambridge)S(Uni)s -16384 X(v)s -9831 r(ersity)S(Press,)S(Cambridge-Ne)T(w)s 104857 r(Y)s -72090 r(ork.)s o 983843 Z U u -473657 r([BK98])s O 2474808 r(M.)s 164875 W(Biskup)S(and)s 145214 r(W)s -60294 r(.)S(K)s 54723 r(\177)s -272958 r(onig)S(\(1998\).)S(On)S(a)S(v)s -16384 r(ariational)S(problem)S(related)S(to)S(the)S(one-dimensional)S(parabolic)s 128830 r(Anderson)s o y u 2474808 r(model.)s 163840 W(\(unpublished)S(manuscript\).)s o z U u -473657 r([BK00])s O 2474808 r(M.)s 220668 W(Biskup)S(and)s 201007 r(W)s -60294 r(.)S(K)s 54723 r(\177)s -272958 r(onig)S(\(2000\).)S(Long-time)S(tails)S(in)S(the)S(parabolic)s 184623 r(Anderson)S(model)S(with)S(bounded)S(potential.)s o y u 2474808 r(\(a)s -13108 r(v)s -16384 r(ailable)s 163840 W(at)S(mp-arc)S(00-178\).)s o z U u -473657 r([CM94])s O 2474808 r(R.)s 123614 W(Carmona)S(and)S(S.A.)S(Molchano)s -9831 X(v)S(\(1994\).)S f14(P)s -52429 r(ar)T(abolic)s 111817 r(Ander)s -6554 r(son)S(Pr)s -29492 r(oblem)S(and)S(Intermittency)s f6(.)S(Memoirs)S(of)S(the)s 87569 r(AMS)s o y u 2474808 r(v)s -13108 r(ol.)s 163840 W(108)S(nr)s -36045 r(.)S(518.)s o z U u -473657 r([D)s -26215 r(V75])s O 2474808 r(M.)s 212629 W(Donsk)s -6554 r(er)S(and)S(S.R.S.)s 179860 r(V)s -72745 r(aradhan)S(\(1975\).)s 176584 r(Asymptotics)S(for)S(the)s 192968 r(W)s -26215 r(iener)S(sausage.)S f14(Comm.)S(Pur)s -24249 r(e)s 200832 r(Appl.)S(Math.)S f5(28)s o y u 2474808 r f6(525\226565.)s O o eop % [13] end % dvidict %%PageTrailer %%Page: 14 14 dvidict begin 14 14 bop % [14] 43384832 d u -44564480 d U u -473657 r f15(14)s o 9719711 r(MAREK)s 147456 W(BISKUP)s 115015 r(AND)s 129761 r(W)s -5899 r(OLFGANG)S(K)s u 114721 r -125043 d(\177)s o(ONIG)s O 42598400 d u -40304640 d U u -473657 r f6([D)s -26215 r(V79])s O 2474808 r(M.)s 210975 W(Donsk)s -6554 X(er)S(and)S(S.R.S.)s 178206 r(V)s -72745 r(aradhan)S(\(1979\).)S(On)S(the)S(number)S(of)S(distinct)S (sites)S(visited)S(by)S(a)S(random)S(w)T(alk.)S f14(Comm.)s o 810024 Y u 2474808 r(Pur)s -24249 r(e)s 152043 r(Appl.)s 163840 W(Math.)S f5(32)S f6(721\226747.)s o 951777 Z U u -473657 r([GH99])s O 2474808 r(J.)s 137175 W(G)s 36373 r(\177)s -254608 r(artner)S(and)S(F)s -52429 r(.)S(den)S(Hollander)S(\(1999\).)S(Correlation)S(structure)S(of)S (intermittenc)s -9831 r(y)S(in)S(the)S(parabolic)s 101130 r(Anderson)S(model.)s o y u 2474808 r f14(Pr)s -29492 X(obab)s -26215 r(.)s 152043 r(Theory)s 163840 W(Relat.)S(F)T(ields)S f5(114)S f6(1\22654.)s o z U u -473657 r([GK98])s O 2474808 r(J.)s 132544 W(G)s 36373 r(\177)s -254608 r(artner)S(and)s 112883 r(W)s -60294 r(.)S(K)s 54723 r(\177)s -272958 r(onig)S(\(2000\).)S(Moment)S(asymptotics)S(for)S(the)S(continuous)S (parabolic)s 96499 r(Anderson)S(model.)s 120747 W f14(Ann.)S(Appl.)s o y u 2474808 r(Pr)s -29492 r(obab)s -26215 r(.)s 163840 W f6(\(to)S(appear\).)s o z U u -473657 r([GKM99])s O 2474808 r(J.)s 191158 W(G)s 36373 r(\177)s -254608 r(artner)s -26215 r(,)s 178327 r(W)s -60294 r(.)S(K)s 54723 r(\177)s -272958 r(onig)S(and)S(S.A.)S(Molchano)s -9831 r(v)S(\(1999\).)s 155113 X(Almost)S(sure)S(asymptotics)S(for)S(the)S(continuous)S(parabolic)T (An-)s o y u 2474808 r(derson)s 163840 W(model.)S f14(Pr)s -29492 X(obab)s -26215 r(.)s 152043 r(Theory)S(Relat.)S(F)T(ields)S f6(\(to)S(appear\).)s o z U u -473657 r([GM90])s O 2474808 r(J.)s 127738 W(G)s 36373 r(\177)s -254608 r(artner)S(and)S(S.A.)S(Molchano)s -9831 X(v)S(\(1990\).)S(P)T(arabolic)S(problems)S(for)S(the)s 91693 r(Anderson)S(model.)S(I.)S(Intermittenc)T(y)S(and)S(related)s o y u 2474808 r(topics.)s 163840 W f14(Commun.)S(Math.)S(Phys.)S f5(132)S f6(613\226655.)s o z U u -473657 r([GM98])s O 2474808 r(J.)s 91923 W(G)s 36373 r(\177)s -254608 r(artner)S(and)S(S.A.)S(Molchano)s -9831 X(v)S(\(1998\).)S(P)T(arabolic)S(problems)S(for)S(the)s 55878 r(Anderson)S(model.)S(II.)S(Second-order)S(asymptotics)s o y u 2474808 r(and)s 163840 W(structure)S(of)S(high)S(peaks.)S f14(Pr)s -29492 X(obab)s -26215 r(.)s 152043 r(Theory)S(Relat.)S(F)T(ields)S f5(111)S f6(17\22655.)s o z U u -473657 r([HS90])s O 2463011 r(T)s -48497 r(.)s 140321 W(Hara)S(and)S(G.)S(Slade)S(\(1990\).)S(Mean-\feld)S(critical)S (beha)s -13108 r(viour)S(for)S(percolation)S(in)S(high)S(dimensions.)S f14(Commun.)S(Math.)s o y u 2474808 r(Phys.)s 163840 W f5(128)s f6(,)S(333\226391.)s o z U u -473657 r([K00])s O 2455147 r(W)s -60294 r(.)s 166502 W(K)s 54723 r(\177)s -272958 r(onig)S(\(2000\).)S f14(Self-Repellent)S(and)S(Self-Attr)s -9831 r(active)S(P)s -52429 r(ath)S(Measur)s -24249 r(es)S(in)S(Statistical)S(Physics)s f6(.)S(Habilitationsschrift,)s o y u 2463011 r(TU)s 163840 r(Berlin.)s o z U u -473657 r([R78])s O 2474808 r(L.)s 163840 W(Russo)S(\(1978\))s 127795 r(A)S(note)S(on)S(percolation.)S f14(Zeit.)s 137625 r(W)s -60294 r(ahr)s -72745 W(.)s 140902 r(V)S(erw)s -48497 r(.)s 163840 W(Geb)s -26215 r(.)S f5(61)s f6(,)S(129\226139.)s o z U u -473657 r([S98])s O 2438763 r(A.-S.)s 163840 W(Sznitman)S(\(1998\).)S f14(Br)s -29492 r(ownian)S(motion,)S(Obstacles)S(and)S(Random)S(Media)s f6(.)S(Springer)s -26215 r(,)S(Berlin.)s O o eop % [14] end % dvidict %%PageTrailer %%Trailer %%EOF ---------------0007101547622--