Content-Type: multipart/mixed; boundary="-------------0312010856408" This is a multi-part message in MIME format. ---------------0312010856408 Content-Type: text/plain; name="03-517.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-517.comments" Revised version, significantly improved and extended. ---------------0312010856408 Content-Type: text/plain; name="03-517.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-517.keywords" Random Schrodinger operators, spectral theory, hydrodynamic limits ---------------0312010856408 Content-Type: application/x-tex; name="lbll.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lbll.tex" \documentclass[12pt]{amsart} \usepackage{amssymb,amsfonts,latexsym,amscd,epsfig,psfrag} \setlength\textwidth{6.5 in} \setlength\textheight{8.5 in} \voffset=-0.6in \hoffset = -0.6in \parindent = 0.4in \parindent = 0.4in \pagestyle{plain} \begin{document} \def\A{{\mathcal A}} \def\alg{{\mathcal A}} \def\Bound{{\mathcal B}} \def\bra{\Big\langle} \def\C{{\Bbb C}} \def\cF{{\mathcal F}} \def\cL{{\mathcal L}} \def\cT{{\mathcal T}} \def\cV{{\mathcal V}} \def\Cnm{\Big(\begin{array}{c}2n\\2l_1\dots 2l_m\end{array}\Big)} \def\Cbnm{\Big(\begin{array}{c}2\bar n\\2l_1\dots 2l_m\end{array}\Big)} \def\CSnm{\Big(\begin{array}{c}2 n\\|S_1|\dots |S_m|\end{array}\Big)} \def\CSnn{\Big(\begin{array}{c}2 n\\|S_1|\dots |S_{n-1}|\end{array}\Big)} \def\CbSnm{\Big(\begin{array}{c}2\bar n\\|S_1|\dots |S_m|\end{array}\Big)} \def\CbSnn{\Big(\begin{array}{c}2\bar n\\|S_1|\dots |S_{\bar n-1}|\end{array}\Big)} \def\cE{{\mathcal E}} \def\crit{{\mathcal C}} \def\Dell{{\mathcal A}_{L,\e,\delta,\ell}} \def\cDell{{\mathcal A}_{L,\e,\delta,\ell}^c} \def\Dom{\mathfrak{Dom}} \def\dist{{\rm dist}} \def\dup{d\up } \def\dupm{d\up^{(m+1)}} \def\dupn{d\up^{(n+1)}} \def\dvpm{d\uv^{(m+1)}} \def\dvpn{d\vp^{(n+1)}} \def\Exp{{\Bbb E}} \def\e{\varepsilon} \def\esplit{\end{split}} \def\Fou{{\mathcal F}} \def\H{{\mathcal H}} \def\Hpl{{\Bbb H}} \def\Im{{ Im}} \def\ket{\Big\rangle} \def\Lap{\Delta} \def\lb{\left[} \def\LL{\Lambda_L} \def\LLinv{\frac{1}{|\LL|}} \def\LLs{\Lambda_L^*} \def\LLsqinv{\frac{1}{|\LL|^{2n+2}}} \def\mes{{\rm meas}} \def\N{{\Bbb N}} \def\nm{{|\!|\!|\,}} \def\pip{\tau} \def\qm{q^{(m+1)}} \def\R{{\Bbb R}} \def\rb{\right]} %%\def\Re{{\mathcal Re}} \def\rc{\frac{1}{2}} \def\Rem{ R} \def\Sc{{\mathcal S}} \def\split{\begin{split}} \def\summn{\sum_{0\leq m\in n-2\N_0}} \def\summnp{\sum_{0\leq m\in\min\{n,n'\}-2\N_0}} \def\Sym{{\mathfrak S}} \def\Tor{\Bbb T} \def\tk{\tilde k} \def\tt{\theta} \def\Th{\Theta } \def\th{\zeta} \def\Thinf{\Theta } \def\tqm{\tilde q^{(m+1)}} \def\tLL{\tilde\Lambda_L} \def\up{\underline{\vp}} \def\uk{\underline{\vk}} \def\utk{\underline{\tilde\vk}} \def\uvw{\underline{\vw}} \def\Val{{\rm Val}} \def\vac{\Omega_f} \def\Wint{{U}_{d,\rho}} \def\Z{{\Bbb Z}} \def\vx{{ x}} \def\vy{{ y}} \def\ve{{ e}} \def\vk{{ k}} \def\vl{{ l}} \def\vm{{ m}} \def\vn{{ n}} \def\vp{{ p}} \def\vQ{{ Q}} \def\vq{{ q}} \def\vr{{ r}} \def\vv{{ v}} \def\vw{{ w}} \def\Hspace{{\mathfrak H}} \def\Mspace{{\mathfrak M}} \def\Polyd{{\mathfrak V}} \def\Wspace{{\mathfrak W}} \def\Uspace{{\mathfrak U}} \def\Tspace{{\mathfrak T}} \def\Pl{{\mathcal P}} \def\1{{\bf 1}} \def\eqnn{\begin{eqnarray*}} \def\eeqnn{\end{eqnarray*}} \def\eqn{\begin{eqnarray}} \def\eeqn{\end{eqnarray}} \def\bal{\begin{align}} \def\eal{\end{align}} \def\prf{\begin{proof}} \def\endprf{\end{proof}} %%\numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}{Definition}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}{Lemma}[section] \title{Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3} \author{Thomas Chen} \address{ Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185.} \email{ chenthom@cims.nyu.edu} \date{} \maketitle \begin{abstract} We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Shubin, Schlag and Wolff, \cite{shscwo}, for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{\lambda^{-2}}{\log\frac1\lambda})$, where $\lambda$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erd\"os and Yau \cite{erdyau} to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schr\"odinger dynamics is governed by the linear Boltzmann equations. \end{abstract} \section{Introduction} The Anderson model in dimension $d$ is defined by the discrete random Schr\"odinger operator $$ (H_\omega\psi)(x)=-\frac12(\Lap \psi)(x) + \lambda \omega(x)\psi(x) \;, $$ acting on $\ell^2(\Z^d)$, where $\lambda$ is a small coupling constant, $$ (\Lap \psi)(x):=2d\psi(\vx)- \sum_{|x-y|=1}\psi(y) $$ is the nearest neighbor lattice Laplacian, and $\omega(x)$ are, for $x\in\Z^d$, bounded, i.i.d. random variables. In the present paper, we study the case $d=3$, and prove that with probability one, most eigenfunctions of $H_\omega$ have localization lengths bounded from below by $O(\frac{\lambda^{-2}}{\log\frac1\lambda})$. In contrast to $d=1,2$, we note that there are no restrictions on the energy range for this result to hold. Furthermore, we derive the macroscopic limit of the quantum dynamics in this system, and prove that it is governed by the linear Boltzmann equations. The present paper is closely related to work of L. Erd\"os and H.-T. Yau in \cite{erdyau}, where the weak coupling and hydrodynamic limit is derived for a random Schr\"odinger equation in the continuum $\R^d$, $d=2,3$, for a Gaussian random potential. For macroscopic time and space variables $(T,X)$, microscopic variables $(t,x)$, and the scaling $(X,T)=\lambda^{2}(x,t)$, where $\lambda$ is the coupling constant in the continuum analogue of $H_\omega$, they established in the limit $\lambda\rightarrow0$ that the macroscopic dynamics is governed by the {\em linear Boltzmann equations}, and thus ballistic, for all $T>0$. We note that the corresponding result for sufficiently small values of $T$ was first proved by H. Spohn \cite{sp}. For larger time scales, it has very recently been established that the macroscopic dynamics in $d=3$ is determined by a diffusion equation, \cite{erdsalmyau}. Also closely related to the present paper is a recent work of C. Shubin, W. Schlag and T. Wolff, \cite{shscwo}, who have, by techniques of harmonic analysis, established for the Anderson model at small disorders in $d=1,2$, that with probability one, most eigenstates are in frequency space concentrated on shells of thickness $\leq\lambda^{2}$ in $d=1$, and $\leq\lambda^{2-\delta}$ in $d=2$. The eigenenergies are required to be bounded away from the edges of the spectrum of $-\frac12\Delta_{\Z^d}$, and in $d=2$, also away from its center. By the uncertainty principle, this implies lower bounds of order $O(\lambda^{-2})$ in $d=1$, and and $O(\lambda^{-2+\delta})$ in $d=2$, on the localization lengths in position space. Closely related to their work are the papers \cite{mapori1,mapori2} by J. Magnen, G. Poirot, V. Rivasseau, and \cite{po} by G. Poirot, which address properties of the Greens functions associated to $H_\omega$. The proof of our main results uses an extension of the time-dependent techniques of L. Erd\"os and H.-T. Yau in \cite{erdyau} to the lattice, and to non-Gaussian random potentials. Higher correlations, which are now abundant, are shown to have an insignificant effect, hence the character of our results does not differ from that obtained in the Gaussian case. We have adapted part of our notation and nomenclature to \cite{erdyau}, in order to facilitate the referencing of results. The link between the lower bounds on the localization lengths of eigenfunctions, and the Schr\"odinger dynamics generated by $H_\omega$ is a joint result with L. Erd\"os and H.-T. Yau included in this paper. The author is deeply grateful to them for their support and generosity. \section{Definition of the Model and the Main Theorem} We consider the discrete random Schr\"odinger operator $$ H_\omega = -\frac12\Lap + \lambda V_\omega \; $$ acting on $\psi\in\ell^2(\Z^3)$. The impurity potential is given by \eqn V_{\omega}(\vx) =\sum_{ \vy \in\LL } \omega_\vy \delta(\vx-\vy) \;, \eeqn where $\omega_y$ are bounded, independent, identically distributed random variables. For each $x\in\Z^3$, $\omega_x$ is a random variable on a single site probability space $(J,F,\mu)$, where $J$ is a bounded Borel subset of $\R$, $F$ is the $\sigma$-algebra of Borel subsets of $J$, and $\mu$ is a probability measure on $F$, with supp$(\mu)=J$. $V_\omega$ is a random field over $\Z^3$ realized on the probability space $(\Omega,{\mathcal F},{\Bbb P})$, with $\Omega=\times_{\Z^3}J$, where ${\mathcal F}$ is the $\sigma$-algebra generated by the cylinder sets induced by $F$, and the probability measure ${\Bbb P}$ is given by $\times_{\Z^3}\mu$. For simplicity, we assume $\mu$ to be even, $\mu(I)=\mu(-I)$, for all $I\subset F$. Then, $\Exp[\omega_\vx^{2m+1}]=0$ $\forall x\in\Z^3$, $\forall m\geq0$. This will help to reduce some of the notation in our analysis, but for our methods to apply, only $\Exp[\omega_x]=0$ is necessary. In addition, we assume the uniform moment bounds \eqn\label{omcorrdef} \Exp[\omega_\vx^{2m}] =: \tilde c_{2m} \leq c_\omega \; \; , \; \; \tilde c_2=1 \; , \; \forall\vx\in\Z^3 \; , \; \forall m\geq1 \;, \eeqn where the constant $c_\omega<\infty$ is independent of $m$. $H_\omega$ is a selfadjoint linear operator on $\ell^2(\Z^3)$ for every realization of $V_\omega$. The Fourier transform and its inverse are given by $$ \hat f(\vk) = \sum_{\vx\in\Z^3} e^{-2\pi i\vk\cdot\vx} f(\vx) \; \; , \; \; \check g(\vx) = \int_{\Tor^3} dk\; g(\vk) e^{2 \pi i \vk\cdot\vx} \;. $$ We note that $$\frac{1}{2}(\Lap f)\hat{\;}(\vk) = - e(\vk) \hat f(\vk) \;,$$ where \eqn e(\vk) = \sum_{i=1}^3 \big( 1- \cos(2\pi k_i) \big) =2\sum_{i=1}^3 \sin^2(\pi k_i) \label{kinendef} \eeqn is the expression for the kinetic energy in frequency space. Let $L\in \N$ with $L\gg\lambda^{-2}$, and $\Lambda_L=\{-L,-L+1,\dots,-1,0,1, \dots,L-1,L\}^3\subset\Z^3$. To formulate our main result, we shall introduce approximate characteristic functions for shells centered around lattice sites $x\in\LL$. Let, for $m,\ell\in\N_0$, with $m\leq\ell\ll L$, \eqn\label{hLcutoffdef} h_\ell(m) := \left\{\begin{array}{ll}1&\;\;0\leq m\leq \ell/2\\ 2-\frac{2m}{\ell} &\;\;\ell/2< m \leq \ell\end{array}\right. \;, \eeqn and $h_\ell(m)=0$ for $m>\ell$. Then, let $K_\ell(x):= \prod_{j=1}^3 h_\ell(|x_j|)$, and it is clear that $\hat K_\ell$ is a product of differences of Fej\'er kernels. For $x\in\LL$, and $\delta>0$, let $$ R_{x,\delta, \ell}(y) := K_{\ell_2}(x-y)-K_{ \ell_1}(x-y) $$ with \eqn\label{ell1ell2def} \ell_1:=\delta\ell \; \; , \; \; \ell_2:=\ell \;. \eeqn $R_{x,\delta, \ell}$ is an approximate characteristic function for a cubical shell of side length $2\ell_2$ centered at $x$, and thickness $\ell_2-\ell_1$. The following observation, for which the author thanks H.-T. Yau and L. Erd\"os, is the key to linking the localization length of eigenvectors to the dynamics generated by $H_\omega$. Let $\{\psi_\alpha^{(L)}\}$ denote an orthonormal basis in $\ell^2(\LL)$ of eigenfunctions of $H_\omega$ restricted to $\LL$. That is, \eqn (H_\omega-e_\alpha^{(L)})\psi_\alpha^{(L)}=0 \; {\rm on } \; \LL \; \; {\rm and} \; \; \psi_\alpha^{(L)}=0 \; {\rm on} \;\partial\LL :=\Lambda_{L+1}\setminus\LL\;, \label{eigenL} \eeqn for $\alpha\in\alg_L:=\{1,\dots,|\LL|\}$, and $e_\alpha^{(L)}\in\R$. For $\e$ small, the subset of eigenvectors labeled by $$ \Dell:=\Big\{\alpha\in\alg_L\Big| \sum_x |\psi_\alpha^{(L)}(x)| \big\| R_{x, \delta, \ell} \psi_\alpha^{(L)} \big\|_{\ell^2(\LL)} < \e\Big\} \; $$ contains the class of exponentially localized states concentrated in balls of radius $\sim\frac{ \delta \ell }{ \log \ell }$ or smaller, where $\delta$ is notably independent of $\ell$. The additional factor $\log \ell$ in the denominator compensates a volume factor $\ell^{3/2}$, which arises due to the fact that $|\psi_\alpha^{(L)}(x)|$ appears only linearly, and not quadratically in the sum. Our main result is the following theorem. \begin{theorem}\label{mainthm} Assume for $L\gg\lambda^{-2}$, that $\{\psi_\alpha^{(L)}\}$ is an orthonormal $H_\omega$-eigenbasis in $\ell^2(\LL)$, satisfying (~\ref{eigenL}) with $\alpha\in\alg_L$, and $e_\alpha^{(L)}\in\R$. Then, for $\lambda^{\frac{14}{15}}<\delta<1$, $\e_\delta:=\delta^{\frac37}$, $$ \Exp\lb \frac {|\alg_L\setminus\alg_{L,\e_\delta,\delta,\lambda^{-2}}|} {|\alg_L|}\rb\ge 1 - c \delta^{\frac{3}{14}} -\frac{c(\ell)}{L} \;, $$ for a constant $c<\infty$ independent of $L,\delta,\lambda$. Furthermore, $$ {\Bbb P}\lb \liminf_{L\rightarrow\infty} \frac{|\alg_L\setminus\alg_{L,\e_\delta,\delta,\lambda^{-2}}|}{|\alg_L|} \geq 1- c \delta^{\frac{3}{14}} \rb = 1\; $$ for $\lambda>0$ sufficiently small, and a constant $c<\infty$ that is uniform in $\lambda$ and $\delta$. \end{theorem} We note that in contrast to lower dimensional cases, there is no condition in dimension 3 that requires $e_\alpha^{(L)}$ to be bounded away from the edges or center of spec$\{-\frac12\Lap \}$. \section{Proof of Theorem {~\ref{mainthm}}} \label{sectionIII} The following lemma relates the localization length of eigenvectors of $H_\omega$ to the dynamics generated by $H_\omega$. \begin{lemma}\label{ceylemma} (Joint with L. Erd\"os and H.-T. Yau) Let $\{\psi_\alpha^{(L)}\}$ denote an orthonormal basis in $\ell^2(\LL)$, consisting of eigenvectors of $H_\omega$ satisfying (~\ref{eigenL}), and assume that $1\ll\ell\ll L$. Let $$ \cDell:= \alg_L\setminus \Dell \;, $$ and suppose that for all $x\in\Z^3$, \eqn\label{mainest} \Exp \lb\big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\Z^3)} ^2\rb \ge 1- \e \; \eeqn is satisfied for some $\e=\e(\delta,\ell,t)>0$. Then, $$ \Exp\lb \frac {|\cDell| } {|\alg_L|}\rb\ge 1 - 2 \e^{1/2}-\frac{c(\ell)}{L} \;, $$ where $c(\ell)$ is a constant that only depends on $\ell$. \end{lemma} \prf We have \eqnn \delta_x &=& \sum_\alpha {a_x^\alpha} \psi_\alpha^{(L)} \; \;\\ a_x^\alpha &=& \big\langle \delta_\vx \, , \, \psi_\alpha^{(L)} \big\rangle = \psi_\alpha^{(L)}(x) \;. \eeqnn By the Schwarz inequality, we get \eqn \big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\LL)}^2 &\le& (1+ \kappa^{-1} ) \Big\| R_{x, \delta, \ell} e^{-i t H_\omega } \sum_{\alpha \in \Dell} {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \nonumber\\ &+& (1+\kappa) \Big\| R_{x, \delta, \ell} e^{-i t H_\omega } \sum_{\alpha \in \cDell} {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \; . \label{CSest1} \eeqn For the first term on the r.h.s., we find \eqnn \Big\|R_{x, \delta, \ell}e^{-i t H_\omega } \sum_{\alpha \in \Dell}{a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 &\le&\Big\|R_{x, \delta, \ell} \sum_{\alpha \in \Dell} e^{-i t e_\alpha^{(L)} }{a_x^\alpha}\psi_\alpha^{(L)}\Big\|_{\ell^2(\LL)}^2 \\ &\leq&\Big\|R_{x, \delta, \ell} \sum_{\alpha \in \Dell} e^{-i t e_\alpha^{(L)} }{a_x^\alpha}\psi_\alpha^{(L)}\Big\|_{\ell^2(\LL)} \\ &\le& \sum_{\alpha\in \Dell} |\psi_\alpha^{(L)}(x)| \big\| R_{x, \delta, \ell} \psi_\alpha^{(L)} \big\|_{\ell^2(\LL)} \;, \eeqnn where we have used $\|R_{x, \delta, \ell}\|=1$, and unitarity of the time evolution in the step preceding the second line. For the second term on the r.h.s. of (~\ref{CSest1}), \eqnn \Big\| R_{x, \delta, \ell} e^{-i t H_\omega } \sum_{\alpha \in \cDell} {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2&\le& \Big\| e^{-i t H_\omega } \sum_{\alpha \in \cDell} {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \\ &=& \Big\|\sum_{\alpha \in \cDell} {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \\ &=& \sum_{\alpha \in \cDell} |\psi_\alpha^{(L)}(x)|^2 \; . \eeqnn Summing over $x\in\LL$, we have \eqn &&\LLinv \sum_{x\in\LL} \big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\LL)}^2\nonumber\\ &&\hspace{1cm}\le (1+\kappa)\frac{1}{|\LL|} \sum_{\alpha \in \cDell} \sum_{x\in\LL} |\psi_\alpha^{(L)}(x)|^2 \nonumber\\ &&\hspace{2cm} +(1+ \kappa^{-1} )\frac{1}{|\LL|} \sum_{\alpha\in \Dell} \sum_{x\in\LL} |\psi_\alpha^{(L)}(x)| \big\| R_{x, \delta, \ell} \psi_\alpha^{(L)} \big\|_{\ell^2(\LL)} \nonumber\\ &&\hspace{1cm}\le(1+\kappa) \frac {1 }{|\LL|}\big|\cDell\big| + (1+\kappa^{-1} )Av_{\alpha \in \Dell} \sum_{x\in\LL} |\psi_\alpha^{(L)}(x)| \big\|R_{x, \delta, \ell} \psi_\alpha^{(L)} \big\|_{\ell^2(\LL)}\; . \label{DbarDsplitest} \eeqn Let $$ S_{2\ell,L}=\big\{x\,\big|\,\inf_{y\in\partial\LL}|x-y|\leq 2\ell \big\} $$ and $\tLL:=\LL\setminus S_{2\ell,L}$, such that $R_{x,\delta,\ell}\cap\partial\LL=\emptyset$ $\forall x\in \tLL$. Then, \eqnn \LLinv \sum_{x\in\LL} \big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\LL)}^2 &=& \frac{1}{|\tLL|} \sum_{x\in\tLL} \big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\Z^3)}^2+O(\frac1L)\;, \eeqnn by compactness of the support of $R_{x, \delta, \ell}$. By definition of $\Dell$, the last term in (~\ref{DbarDsplitest}) is bounded by $(1+\kappa^{-1})\e$, and $|\LL|=|\alg_L|$. Thus, \eqn \frac {|\cDell | } {|\alg_L|} \geq \frac{1}{1+\kappa} \frac{1}{|\tLL|}\sum_{x\in\tLL} \big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\Z^3)}^2 - \frac{1+\kappa^{-1}}{1+\kappa}\, \e -\frac{c(\ell)}{L}\;. \label{fracAcAlowbd} \eeqn Taking expectations, using (~\ref{mainest}), and choosing $\kappa=\e^{\frac12}$, the claim follows. \endprf \begin{lemma}\label{mainproblemma} Under the same assumptions as in Lemma {~\ref{ceylemma}}, $$ {\Bbb P} \lb\liminf_{L\rightarrow\infty} \frac {|\cDell| } {|\alg_L|}\ge 1 - 2 \e^{1/2}\rb =1 \;. $$ \end{lemma} \prf We consider the family of translation operators $\tau_x:\omega_y\mapsto\omega_{x+y}$, for $x\in\Z^3$, which acts ergodically on the probability space $(\Omega,{\mathcal F},{\Bbb P})$, \cite{cyfrkisi}. Let $U_{\tau_x}$ denote the unitary translation operator $(U_{\tau_x}\phi)(y)=\phi(x+y)$ on $\ell^2(\Z^3)$. Then, clearly, $$ U_{\tau_x}^*H_{ \omega}U_{\tau_x} =-\frac12\Delta+\lambda V_{\tau_{-x}\omega} =H_{\tau_{-x}\omega} $$ with $V_{\tau_x \omega} (y)=V_\omega(x+y)$, and \eqnn \frac{1}{|\tLL|} \sum_{x\in\tLL} \big\| R_{x, \delta, \ell} e^{-i t H_\omega } \delta_x\big\|_{\ell^2(\Z^3)}^2 &=& \frac{1}{|\tLL|} \sum_{x\in\tLL} \big\| (U_{\tau_x}^*R_{x, \delta, \ell}U_{\tau_x}) (U_{\tau_x}^* e^{-i t H_\omega }U_{\tau_x}) \delta_0\big\|_{\ell^2(\Z^3)}^2 \\ &=& \frac{1}{|\tLL|} \sum_{x\in\tLL} \big\| R_{0, \delta, \ell} e^{-i t H_{\tau_{-x} \omega} } \delta_0\big\|_{\ell^2(\Z^3)}^2 \;, \eeqnn by unitarity of $U_{\tau_x}$. By the Birkhoff-Khinchin ergodic theorem, applied to the random variable $f(\omega):=\| R_{0, \delta, \ell} e^{-i t H_{\omega} } \delta_0\big\|_{\ell^2(\Z^3)}^2$, we obtain \eqn \liminf_{L\rightarrow\infty}\frac{1}{|\tLL|} \sum_{x\in\tLL} \big\| R_{0, \delta, \ell} e^{-i t H_{\tau_{-x} \omega} } \delta_0\big\|_{\ell^2(\Z^3)}^2= \Exp \lb\big\| R_{0, \delta, \ell} e^{-i t H_\omega } \delta_0\big\|_{\ell^2(\Z^3)}^2\rb \label{ergodic} \eeqn with probability one. We note here that clearly, the left hand side of (~\ref{mainest}) is independent of $x\in\Z^3$. Therefore, (~\ref{mainest}), (~\ref{fracAcAlowbd}) and (~\ref{ergodic}) imply $$ \liminf_{L\rightarrow\infty}\frac {|\cDell|} {|\alg_L|} \ge 1 - \frac{\kappa}{1+\kappa}- \frac{2+\kappa^{-1}}{1+\kappa} \e $$ with probability one, and choosing $\kappa=\e^{\frac12}$, the claim follows. \endprf Henceforth, we will write $\|\,\cdot\,\|_2\equiv\|\,\cdot\,\|_{\ell^2(\Z^3)}$ for brevity. The key to the conclusion of the proof of Theorem {~\ref{mainthm}} is Lemma {~\ref{fundestlemma} } below, which implies that for the choice $t=\delta^{\frac67}\lambda^{-2}$, \eqn\label{Mainfullpropest} \Exp\lb\big\| R_{x,\delta, \frac{1}{\lambda^2}} e^{-i \delta^{\frac67}\lambda^{-2} H_\omega } \delta_x\big\|_2 \rb \geq 1 - c\delta^{\frac37} \eeqn holds for $\lambda^{\frac{14}{15}}<\delta<1$, and some constant $c$ that is independent of $x$, $\lambda$ and $\delta$. Thus, with the choice $\e=\delta^{\frac37}$, (~\ref{Mainfullpropest}) immediately implies the assertion of Theorem {~\ref{mainthm}}. \begin{lemma}\label{fundestlemma} Let $t=\delta^{\frac67}\lambda^{-2}$, and $H_0:=-\frac12\Lap $. Then, for $\lambda$ sufficiently small, $0<\delta<1$, and all $x\in\Z^3$, the free evolution term satisfies \eqn\label{fundest0} \big\| R_{x,\delta, \frac{1}{\lambda^2}} e^{-i t H_0 } \delta_x\big\|_2 \geq 1 - c\delta^{\frac37} \;, \eeqn while the sum over collision histories yields \eqn\label{fundest10} \Exp \lb \big\| R_{x, \delta,\frac{1}{\lambda^2}}\big( e^{-i t H }-e^{-i t H_0 } \big)\delta_x\big\|_2^2 \rb \leq c'\delta^{\frac67}+t^{-\frac13} \;, \eeqn for positive constants $c,c'<\infty$ that are independent of $x$, $\lambda$ and $\delta$. \end{lemma} \prf We may assume that $x=0$. Since $K_{\ell_1} \geq K_{\ell_1}^2$, we have $(R_{0, \delta, \ell})^2\geq K_{\ell_2}^2-K_{\ell_1}$. To bound $\|K_{\ell_2}e^{-itH_0}\delta_0\|_2$, we note that \eqn \Big|e^{-ite(p)}- \int_{\Tor^3}dk \hat K_{\ell_2}(p-k) e^{-ite(k)} \Big| &\leq& c\sup_{|p-k|\leq\gamma}\Big|e^{-ite(p)}- e^{-ite(k)}\Big| \nonumber\\ &&+2\int_{|p-k|\geq\gamma}dk |\hat K_{\ell_2}(k)| \nonumber\\ &\leq& c\Big(t\gamma+\frac{1}{1+\gamma\ell_2} \Big)\;, \label{fundest0K2err} \eeqn since by basic properties of the Fej\'er kernel, $$ |\hat K_{\ell_2}(k)|\leq \prod_{j=1}^3 \frac{c \ell_2}{1+\|k_j\|_{\Z}^2\ell_2^2} \; , \; \; \int_{\Tor^3}dk \hat K_{\ell_2}(k)=1 \;, $$ where $\|r\|_{\Z}:=$dist$(r,\Z)$ for $r\in\R$. Thus, with $\gamma= (t\ell_2)^{-1/2}$, $\ell_2=\lambda^{-2}$, and $t=\delta^{\alpha}\lambda^{-2}$, we find $(~\ref{fundest0K2err})\leq c \delta^{\alpha/2}$. Hence, \eqn \big\| K_{\ell_2} e^{-itH_0}\delta_0\big\|_2^2 &=&\int_{\Tor^3}dp \Big|e^{-ite(p)}+O(\delta^{\alpha/2}) \Big|^2 \nonumber\\ &\geq&1-c \delta^{\alpha/2} \; \eeqn follows immediately. To bound $\||K_{\ell_1}|^{1/2}e^{-itH_0}\delta_0\|_2$, we derive the usual propagation estimates for the free time evolution operator in the lattice case. Clearly, \eqn \big\||K_{\ell_1}|^{1/2}e^{-itH_0}\delta_0\big\|_2^2= \sum_{y\in \Z^3} |K_{\ell_1}(y)| \,\big|(e^{-itH_0}\delta_0)(y)\big|^2 \;, \label{fundest0K1bound} \eeqn where $(e^{-itH_0}\delta_0)(y)=\int_{\Tor^3}dk e^{-ite(k)}e^{2\pi i k y}$. The kinetic energy $e(\,\cdot\,):\Tor^3\rightarrow[0,6]$ is a real analytic Morse function with 8 critical points in the corners of the embedded subcube $[0,\frac{1}{2}]^3$ of $\Tor^3=[0,1]^3$. Each of the remaining critical points in $[0,1]^3\setminus[0,\frac{1}{2}]^3$ is identified with one of the latter by symmetry. The Hessians are diagonal in the present coordinate system, and have entries $\pm (2\pi)^2$. We introduce a suitable smooth partition of unity $\phi_j$ on $[0,1]^3$ with $j=1,\dots,8$ and $\sum\phi_i=1$, continued over the boundary of the unit cube by periodicity, and such that every $\phi_i$ contains precisely one critical point $p_{crit}$ of $e(\,\cdot\,)$ in its interior, yielding \eqn \big|(e^{-itH_0}\delta_0)(y)\big|&=&\Big|\sum_{j=1}^8 \int_{\Tor^3} dk \phi_j(k) e^{-ite(k)}e^{2\pi i k y}\Big| \nonumber\\ &\leq&c\big(t^{-3/2}+ t^{-5/2} \sup_{y\in{{\rm supp}\{K_{\ell_1}\}}}\{|y|\}\big)\;, \eeqn for $y\in{\rm supp}\{K_{\ell_1}\}$ by a standard stationary phase estimate. Since $\sup_{y\in{{\rm supp}\{K_{\ell_1}\}}}\{|y|\}\leq \ell_1$, we find for $\ell_1=\delta\lambda^{-2}$, and $t=\delta^\alpha\lambda^{-2}$ that \eqn |(~\ref{fundest0K1bound})|\leq c \frac{\ell_1^3}{t^3} = c\delta^{3(1-\alpha)} \;. \eeqn Matching the sizes of the error terms, we must have $3(1-\alpha)=\alpha/2$, that is, $\alpha=\frac67$. In order to establish (~\ref{fundest10}), we use a modification of the methods of L. Erd\"os and H.-T. Yau from \cite{erdyau}. That is, we invoke a Duhamel expansion with remainder term, and control the expectation by classifying all contraction types occurring in the products of the random potential. To bound the remainder term, we shall exploit the rarity of the event that a large number of collisions occurs in a small time interval. The result is \eqn\label{fundest1} \Exp \lb \big\| R_{x,\delta, \frac{1}{\lambda^{2}}}\big( e^{-i t H }-e^{-i t H_0 } \big)\delta_x\big\|_2^2 \rb \leq c'\lambda^2 t +t^{-\frac13} \;, \eeqn for some constant $c'$ that is independent of $x$, $\lambda$ and $\delta$. One thus immediately obtains (~\ref{fundest10}) for the asserted choice of $t$. The proof of (~\ref{fundest1}) will occupy the rest of section {~\ref{sectionIII}}. \endprf \subsection{Expectation of products of random potentials} \label{ranpotsubsec} We shall next discuss the expectation of products of the random potential $V_\omega$ in more detail. The pair correlator is given by the Kronecker delta $$ \Exp\big[\omega_{\vx_1}\omega_{\vx_2}\big]=\delta_{\vx_1,\vx_2}\;, $$ while for the four-point correlator, we have \eqnn \Exp\big[\omega(x_1)\omega(x_2)\omega(x_3)\omega(x_4)\big]&=& (1-\delta_{x_1, x_3})\delta_{x_1,x_2}\delta_{x_3,x_4} +(1-\delta_{x_1, x_2})\delta_{x_1,x_3}\delta_{x_2,x_4}\\ &+& (1-\delta_{x_1, x_3})\delta_{x_1,x_4}\delta_{x_2,x_3} +\tilde c_4\delta_{x_1,x_2}\delta_{x_3,x_4}\delta_{x_1, x_3}\\ &=&\delta_{x_1,x_2}\delta_{x_3,x_4} +\delta_{x_1,x_3}\delta_{x_2,x_4} + \delta_{x_1,x_4}\delta_{x_2,x_3}\\ &+&(\tilde c_4-3)\delta_{x_1,x_2}\delta_{x_3,x_4}\delta_{x_1, x_3}\;. \eeqnn We will refer to the operation applied in passing to the expression after the second equality sign as Wick ordering. Its effect is that by a renormalization of the fourth order moment of $\omega_x$, $$ \tilde c_4 \rightarrow c_4 := \tilde c_4-3 \tilde c_2^2 \;, $$ (where $\tilde c_2=1$) the products of pair correlators become independent. The relevance of Wick ordering can be seen as follows. Let $\hat\omega(k):=\sum_x\omega_x e^{2\pi i k x}$. One easily verifies that \eqnn \Exp\big[\hat\omega(k_1)\hat\omega(k_2)\hat\omega(k_3)\hat\omega(k_4)\big] &=&\delta (k_1+k_2)\delta (k_3+k_4) \\ &+& \delta (k_1+k_3)\delta (k_2+k_4) +\delta (k_1+k_4)\delta (k_2+k_3)\\ &+&c_4\delta (k_1+k_2+k_3+k_4)\;. \eeqnn That is, for the Wick ordered expression, one obtains exact Dirac deltas. However, we note that this is not true for the individual summands of the expression prior to Wick ordering. Along these lines, we shall next determine Wick ordered products of an arbitrary even number of random potentials. We introduce, for $n,n'\in\N$ with $\bar n:=\frac{n+n'}{2}\in\N$, the set $$ \cV_{n,n'}:=\Big\{1,\dots,n,n+2,\dots,n+n'+1\Big\} \;. $$ In our later discussion, $\cV_{n,n'}$ labels a linearly ordered set of $n+n'$ random potentials that are, in frequency space, subdivided into a group of $n'$ copies of $\hat V_\omega$, and a group of $n$ copies of $\overline{\hat V_\omega}$ (the complex conjugate). The label $n+1$ excluded here is reserved for a distinguished point that does not belong to a random potential. We note again that the case of $n+n'\in2\N_0+1$ is a priori trivial by the assumption (~\ref{omcorrdef}). \begin{definition} For $\bar n=\frac{n+n'}{2}\in\N$, let $$ \Pi_{n,n'}:=\bigcup_{m=1}^{\bar n}\Big\{\{S_j\}_{j=1}^m\Big||S_j|\in2\N \;;\;\cV_{n,n'}=\cup_{j=1}^m S_j\;;\; S_j\cap S_{j'}=\emptyset \hbox{ {\rm if} } j\neq j'\Big\}\Big/\Sym_m $$ denote the set of partitions of $\cV_{n,n'}$ into disjoint subsets $S_j$ (referred to as blocks) of size $|S_j|\in2\N$, where $\Sym_m$ is the $m$-th symmetric group. Two partitions $\pi=\{S_j\}_{j=1}^m$, $\pi'=\{S'_j\}_{j=1}^m$, are equivalent, $\pi=\pi'$, if $\exists \sigma\in\Sym_m$ such that $S_j=S'_{\sigma(j)}$ for all $j\in\{1,\dots,m\}$. A partition $\pi\in\Pi_{n,n'}$ will also be referred to as a contraction (corresponding to contractions among random potentials). \end{definition} The number of $\pi\in\Pi_{n,n'}$ consisting of $m$ blocks is given by \eqn B_{\bar n}(m)&:=&\sharp\Big\{\{S_j\}_{j=1}^m\Big|\cup_{j=1}^m S_j=\cV_{n,n'} \, ;\,|S_j|\in2\N\,; S_i\cap S_j=\emptyset\;{\rm if}\;i\neq j\Big\}\Big/\Sym_m \nonumber\\ &=& \sum_{r=1}^{\bar n} \sum_{1\leq j_1,\cdots,j_r\leq \bar n} \sum_{1\leq l_1<\cdots0$. In this work, we will always choose $\e=t^{-1}$, to keep the exponential factor $e^{\e t}$ bounded for large $t$. The multiplication operators $\frac{1}{e(k_j)-\alpha-i\e}$ corresponding to the Fourier transformed resolvents of $-\frac12\Lap$ will also be referred to as particle propagators. Let $\Hpl_-:=\{z\in\C\Big|\Im(z)\leq0\}$. By analyticity of the integrand with respect to the variable $\alpha$, and noting its exponential decay as $Im(z)\rightarrow-\infty$, the path of the $\alpha$-integration can, for any fixed $n\in\N$, be deformed away from $\R$ into the closed contour \eqn I=I_\R\cup I_{\Hpl_-} \label{defIloop} \eeqn with \eqnn I_\R &:=& [-1, 7]\\ I_{\Hpl_-}&:=& ([-1, 7]-i)\cup (-1-i(0,1]) \cup (7-i(0,1]) \subset \Hpl_-\;, \eeqnn which is the boundary of a rectangle whose interior contains ${\rm spec}\big\{-\frac{1}{2}\Lap -i\e\big\} = [0,6]-i\e $. Consequently, \eqnn \hat\phi_{n,t}(\vk_0)= e^{\e t} (-i\lambda)^n \int_I d\alpha e^{- i \alpha t} \int_{(\Tor^3)^n} dk_1\cdots dk_n\, e^{2\pi i \vk_n\cdot\vx} \\ \cdot\; \Big[\prod_{j=0}^n \frac{1}{e(\vk_j)-\alpha-i\e}\Big] \prod_{l=1}^n \hat V_\omega(\vk_l-\vk_{l-1}) \;, \eeqnn where the loop $I$ is taken in clockwise direction. By the Schwarz inequality, $$ \hbox{l.h.s. of }(~\ref{fundest1})\leq 2\sum_{n=1}^N \Exp\lb \big\| \phi_{n,t} \big\|_2^2 \rb +2\Exp\lb \big\| \Rem_{N,t}^\e \big\|_2^2 \rb \; . $$ For $1\leq n \leq N$, we have \eqn \Exp\lb \| \phi_{n,t} \|^2_2 \rb &=& e^{2\e t} \lambda^{2n} \int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)} \int_{(\Tor^3)^{2n+2}} \prod_{l=0}^n dk_l d\tilde k_l e^{-2\pi i(\vk_n-\tk_n)\cdot\vx} \nonumber\\ && \hspace{1cm}\cdot \; \delta (k_0-\tk_0)\prod_{j=0}^n \frac{1}{e(k_j)-\alpha-i\e} \frac{1}{e(\tk_j)-\beta+i\e} \nonumber\\ && \hspace{1cm} \cdot \; \Exp\Big[\prod_{i=1}^n \hat V_\omega(k_i-k_{i-1}) \bar {\hat V}_\omega(\tk_i-\tk_{i-1})\Big] \; , \eeqn where $\bar I$ is the complex conjugate of $I$, and taken in the counterclockwise direction by the variable $\beta$. Introducing the variables \eqnn \up&=&(p_0,\dots,p_n,p_{n+1},\dots,p_{2n+1}) \\ &:=&(\tk_n,\dots,\tk_0,\vk_0,\dots,\vk_n) \\ (\alpha_j,\sigma_j) &=& \left\{\begin{array}{ll}(\alpha,1)& 0\leq j\leq n\\ (\beta,-1)&n n$. \\ {\sc Type II} if $i\leq n$, but $j\geq n+2$. \\ A delta function $\delta_{S}$ is of \\ {\sc Type III} if $|S|\geq4$, that is, if it is not associated to a pairing contraction. \end{definition} \centerline{\epsffile{f1.eps} } \noindent{Figure 1.}A graph containing type I, I', II, and III contractions. \\ Hence, a partition of $\cV_{n,n'}$ is of type III if it contains a type III delta distribution. \begin{definition} A pairing contraction $\pi\in\Pi_{n,n'}$ is called crossing if $\delta_\pi$ contains two delta distributions $\delta(p_{i_1}-p_{i_1-1}+p_{j_1}-p_{j_1-1})$ and $\delta(p_{i_2}-p_{i_2-1}+p_{j_2}-p_{j_2-1})$, with $j_r>i_r$, such that $i_1-i_2$ and $j_1-j_2$ have the same signs. \end{definition} \begin{definition} A non-crossing pairing contraction $\pi\in\Pi_{n,n'}$ is called nested if $\delta_\pi$ contains two delta distributions $\delta(p_{i_1}-p_{i_1-1}+p_{j_1}-p_{j_1-1})$ and $\delta(p_{i_2}-p_{i_2-1}+p_{j_2}-p_{j_2-1})$, with $j_r>i_r$, both either of type I or of type I', such that $i_1-i_2$ and $j_1-j_2$ have opposite signs. \end{definition} \begin{definition} A non-crossing and non-nested pairing contraction is called simple. A simple pairing contraction is called a ladder graph if all of its associated delta functions are of type II. \end{definition} \begin{definition} A spanning tree $T$ of $G_\pi$ with $\pi=\{S_j\}_{j=1}^m\in\Pi_{n,n'}$ is called complete if it contains the edge corresponding to the momentum $p_n$, but not the one corresponding to the momentum $p_{n+1}$. \end{definition} \subsection{Simple Pairing Contractions} For each type of contractions $\pi\in\Pi_{n,n}$ listed above, we will next derive bounds on the corresponding Feynman amplitudes $C_\pi$. It turns out that as in \cite{erdyau}, all dominant contributions stem from simple pairings, even with type III contractions included. We will proceed by first discussing simple pairings, then crossing and nested pairings, and finally type III contractions. \subsubsection{The Ladder Graph} For each fixed $n$, the simplest member in the class of simple pairings in $\Pi_{n,n}$ is the ladder graph. It corresponds to the pairing $\pi=\{S_j\}_{j=1}^n\in\Pi_{n,n}$, with $S_{j}=\{j,2n+2-j\}$, such that $|S_j|=2$, and \eqn C_\pi&=&e^{2\e t} \lambda^{2n} \int_{I\times\bar I} d\alpha d\beta e^{-it(\alpha-\beta)} \int_{(\Tor^3)^{2n+2}} \dup \delta(p_n-p_{n+1}) e^{-2\pi i(p_0-p_{2n+1})\cdot\vx} \nonumber\\ &&\hspace{1cm} \cdot\, \Big[\prod_{l=0}^{2n+1} \frac{1}{e(p_l)-\alpha_l-i\sigma_l\e}\Big] \prod_{j=1}^{n} \delta \Big((p_j-p_{j-1})+(p_{2n+2-j}-p_{2n+1-j})\Big)\;, \label{ladderCpidef} \eeqn cf. (~\ref{mainpairinteg}). The following three obvious estimates will be used extensively in the sequel. \begin{lemma}\label{propboundslemma} Let $0<\e\ll1$. Then, \eqn\label{resolvbound1} \Big|\frac{1}{e(p)-\alpha-i\e}\Big|&\leq& \e^{-1} \\ \label{resolvbound2} \sup_{\alpha\in I} \int_{\Tor^3} dp \Big|\frac{1}{e(p)-\alpha-i\e}\Big| \; \; , \; \;\sup_{p\in\Tor^3} \int_I |d\alpha| \Big|\frac{1}{e(p)-\alpha-i\e}\Big| &\leq& c_1 |\log \e| \eeqn for a constant $c_1$ that is uniform in $\e$. \end{lemma} \prf Recalling that by definition of $I$, $\inf_{p\in\Tor^3}\dist(e(p)-i\e,I)=\e $, and that $|I|$ is finite, (~\ref{resolvbound1}) and the second estimate in (~\ref{resolvbound2}) are evident. To prove the first estimate in (~\ref{resolvbound2}), we first show that the measure of the isoenergy surface $$ \Sigma_\alpha:=\Big\{p\in \Tor^3\Big|e(p)=\alpha\Big\} $$ is uniformly bounded in $\alpha\in I\cap\R$. We note that the case $Im(\alpha)\neq0$, the asserted bound is trivial. To this end, for $p=(p_1,p_2,p_3)\in\Tor^3$, let $e_2(p_1,p_2):=\sum_{j=1}^2(1-\cos2\pi p_j)$ denote the Fourier transform of the 2-dimensional lattice Laplacian, and $$ A_2(r):=\mes\Big\{(p_1,p_2)\in\Tor^2\Big|e_2(p_1,p_2)=r\Big\} $$ the length of the corresponding isoenergy curve. It is easy to see that $\|A_2\|_\inftyi$, necessarily exhibits $i=j-1$. Otherwise, either a crossing or a nesting pairing occurs. Hence, any type I or I' delta function in a simple pairing reduces to $\delta(p_{i+1}-p_{i-1})$, for some $i$. \begin{definition} A type I or type I' pairing of the form $\delta(p_{i+1}-p_{i-1})$ is called an immediate recollision. \end{definition} The subintegral in $C_\pi$ corresponding to an immediate recollision is either $$ \Th(\alpha, \e)= \int_{\Tor^3}\frac{dq}{e(q)-\alpha-i \e} \;, $$ or $\Th(\beta,-\e)$. It contributes to a renormalization of the particle propagator, cf. \cite{erdyau}, and satisfies the following estimates, which will be used extensively in the sequel. \begin{lemma}\label{Thetalemma} Let $\alpha\in I$, and recall that $\e=t^{-1}$. Then, for constants $c_2$, $c_3$ uniform in $\e,\alpha,\alpha'$, \eqn |\Th(\alpha,\e) |&<&c_2 \;, \nonumber\\ |\Th(\alpha,\e)-\Th(\alpha',\e)|&\leq& c_3 \e^{-1/2} |\alpha-\alpha'| \label{ThDiffest} \\ \label{deralmThinf} \Big|\partial_\alpha^m\Thinf(\alpha,\e)\Big| &\leq& c' (m!) \e^{-(m-1/2)} \;, \eeqn and for $m\in\N_0$, where $c'$ is independent of $m,\e,\alpha$. \end{lemma} \prf We recall that $\alpha\in I= I_\R\cup I_{\Hpl_-}$ from (~\ref{defIloop}). The case $\alpha\in I_{\Hpl_-}$ is trivial. For $\alpha\in I_\R=[-1,7]$, we consider $$ \Thinf(\alpha,\e)=\int_{\R_+} ds \int_{\Tor^3} dp e^{-is(e(p)-\alpha-i\e)} \;. $$ As discussed in more detail in the appendix, $e(\,\cdot\,):\Tor^3\rightarrow[0,6]$ is a real analytic Morse function with 8 critical points. We choose a smooth partition of unity $1=\sum_{j=1}^8\phi_j$ on $[0,1]^3$, such that the support of each $\phi_i$ contains precisely one critical point $p_{crit,i}$ of $e(\,\cdot\,)$ at its center. Applying a stationary phase estimate to $$ \Thinf(\alpha,\e)=\sum_{j=1}^8 \int_{\R_+} ds \int_{\Tor^3} dp \phi_j(p) e^{-is(e(p)-\alpha-i\e)} \;, $$ one gets $$ \Thinf(\alpha,\e)= \int_{\R^+}ds\sum_{j}\left( \frac{I(p_{crit,j})}{s^{3/2}}e^{-is (e(p_{crit,j})-\alpha)}e^{-\e s} +O(s^{-5/3}e^{-\e s}) \right)\;, $$ where $I(p_{crit,j})\in\C$ are uniform in $\e$ and $s$. Thus, $|\Thinf(\alpha,\e)| < c_2 $, for some constant $c_2$ that is independent of $\e$ and $\alpha$. Furthermore, since $$ \partial_\alpha^m\Thinf(\alpha,\e)= \int_{\R^+}ds \,(is)^m \sum_{j}\left( \frac{I(p_{crit,j})}{s^{3/2}}e^{-is (e(p_{crit,j})-\alpha)}e^{-\e s} +O(s^{-5/3}e^{-\e s}) \right) \;, $$ (~\ref{deralmThinf}) straightforwardly follows for any $m\geq1$. Clearly, for $m=1$, this also implies the second inequality in (~\ref{ThDiffest}). \endprf \subsubsection{General Simple Pairings} In contrast to the ladder diagram, simple pairings do in general comprise progressions of neighboring immediate recollisions on each particle line before and after each type II contraction. Let us assume the case of $q$ neighboring delta functions of type I', starting at the particle propagator carrying the momentum $p_i$. Then, $C_\pi$ contains the corresponding subintegral \eqnn \int_{(\Tor^3)^{2q}} dp_{i+1}\cdots dp_{i+2q} \Big[\prod_{l=i}^{i+2q}\frac{1}{e(p_l)-\alpha-i\e} \Big] \prod_{j=1}^q \delta(p_{i+2j+1}-p_{i+2j-1}) = \frac{(\Th(\alpha,\e))^q}{(e(p_i)-\alpha-i\e)^{q+1}} \;. \eeqnn The analogous expression for a progression of $q$ neighboring delta functions of type I is evidently obtained from $\alpha\rightarrow\beta$ and $\e\rightarrow-\e$. \\ \centerline{\epsffile{f2.eps} } \noindent{Figure 2.} A simple pairing contraction graph. \ \\ Let us consider a simple pairing $\pi\in\Pi_{n,n}$ which contains $m$ type II contractions, and introduce multiindices \eqnn \qm &:=&(q_0,q_1,\dots,q_m) \in\N^{m+1} \\ |\qm |&:=&\sum_{i=0}^m q_i \;. \eeqnn The set of all simple pairings with fixed $n$ gives (after reindexing the momentum variables) \eqn \sum_{\stackrel{\pi\in\Pi_{n, n}} {\pi\;{\rm simple }}}C_\pi &=&\summn \sum_{|\qm |=|\tqm |=\frac{n-m}{2}} \lambda^{2m}e^{2\e t} \int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{1cm} \cdot \, \prod_{i=0}^m \int_{\Tor^3} dp_i \frac{(\lambda^2\Th(\alpha,\e))^{q_i}} {(e(p_i)-\alpha-i\e)^{q_i+1}} \frac{(\lambda^2\Th(\beta,-\e))^{\tilde q_i}} {(e(p_i)-\beta+i\e)^{\tilde q_i+1}} \;. \label{simplepairfixedn} \eeqn Let us comment on this expression, cf. Figure 2. For $i=1,\dots,m$, $p_{i-1}$ is the momentum preceding, and $p_i$ the momentum following the $i$-th type II pairing. Notably, a direct recollision conserves the momentum. For $1\leq i\leq m$, $q_i$ and $\tilde q_i$ are the numbers of neighboring type I and I' pairings after the $i$-th type II contraction. $q_0$ and $\tilde q_0$ are the number of neighboring type I and I' pairings before the $1$-st type II pairing. Clearly, all $n-m$ random potentials on each particle line not involved in type II contractions are part of type I, respectively type I' pairings (immediate recollisions). Since each immediate recollision contracts precisely two random potentials, the sum over $m$ can only take steps of size 2, that is, $0\leq m\in n-2\N_0$ (the case $m=n$ of course corresponds to the ladder graph). Therefore, $$ |\qm |=|\tqm |=\frac{n-m}{2} \; $$ is clear. \begin{lemma}\label{simplepairlemmleqN} For fixed $n$, the contribution of the sum of all simple pairings is bounded by \eqn \sum_{\pi\in\Pi_{n,n}\;{\rm simple}}|C_\pi|\leq \frac{(c_0 \lambda^2 t)^{ n}}{(n!)^{1/2}} + n^2 t^{-1/2} \log t \big(c t \lambda^2 \log t\big)^{n} \;, \eeqn and for $\lambda^2 t<1$, $$ \sum_{n=1}^N\sum_{\pi\in\Pi_{n,n}\;{\rm simple}}|C_\pi|\leq c_4 \lambda^2 t + t^{-1/2} N^3 \big( c\lambda^2 t \log t\big)^{N } \;, $$ where $c,c_0,c_4$ are uniform in $N$ and $t$, and where $c_0$ and $c_4$ are defined in (~\ref{c0def}). \end{lemma} \prf Let us assume for fixed $n$ that $\pi\in\Pi_{n,n}$ is simple, and contains $m$ type II pairings. Let \eqn C_\pi = C_\pi^{main} + C_\pi^{error} \;, \eeqn where \eqn C_\pi^{main} &:=& \sum_{|\qm |=|\tqm |=\frac{n-m}{2}} e^{2\e t}\lambda^{2m} \int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{1cm} \cdot \, \prod_{i=0}^m \int_{\Tor^3} dp_i \frac{(\lambda^2\Th(e(p_0),\e))^{q_i}} {(e(p_i)-\alpha-i\e)^{q_i+1}} \frac{(\lambda^2\Th(e(p_0),-\e))^{\tilde q_i}} {(e(p_i)-\beta+i\e)^{\tilde q_i+1}} \;. \eeqn Summing over all possible values of $m$, \eqn \sum_{\pi\in\Pi_{n,n}\;{\rm simple}}C_\pi^{main}&=& \summn \sum_{|\qm |=|\tqm |=\frac{n-m}{2}}e^{2\e t} \lambda^{2m} \int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{1cm} \cdot \, \prod_{i=0}^m \int_{\Tor^3} dp_i \frac{ (\lambda^2\Th(e(p_0),\e) )^{q_i}} { (e(p_i)-\alpha-i\e )^{q_i+1}} \frac{ (\lambda^2\Th(e(p_0),-\e) )^{\tilde q_i}} {(e(p_i)-\beta+i\e)^{\tilde q_i+1}} \;. \eeqn Let $p_j^{(q_j)}=(p_j,\dots,p_j)$ ($q_j$ copies), and $\dupm:=dp_0\cdots dp_m$. By the Schwarz inequality, \eqn \sum_{\pi\in\Pi_{n,n}\;{\rm simple}}|C_\pi^{main}|&\leq& \summn \sum_{|\qm |= \frac{n-m}{2}} 2\lambda^{2m} \int_{(\Tor^3)^{m+1}}\dupm \nonumber\\ &&\hspace{1cm} \cdot \, \Big|K^{(\frac{m+n}{2})}(p_0^{(q_0+1)}, \dots, p_m^{(q_m+1)};t)\Big|^2 \big|\lambda^2\Th(p_0;\e)\big|^{n-m}\;. \eeqn Clearly, \eqn \int_{(\Tor^3)^{m+1}}\dupm \Big|K^{(\frac{m+n}{2})}(p_0^{(q_0+1)}, \dots, p_m^{(q_m+1)};t)\Big|^2 \leq \frac{(c_\mu t)^{\frac{m+n}{2}} }{((\frac{m+n}{2})!)^\mu} \;. \eeqn By \eqn \sum_{ |\qm | = \frac{n-m}{2} }1 0$, the constant $c_4$ in the above estimate would be replaced by $c^T$, for some constant $c$ that is uniform in $\lambda$ and $t$. \subsection{Crossing and nested pairings} We shall next prove that for all $\pi\in\Pi_{n,n}$ which contain a crossing or nested pairing contraction, $|C_\pi|$ is a factor $O(t^{-1/2})$ smaller than the bound for the ladder graph in $\Pi_{n,n}$. This is sufficient to compensate the factor $n!$ from the number of pairing contractions. \begin{lemma} The sum of all crossing and nested pairing contractions in $\Pi_{n,n}$ is bounded by $$ \sum_{\pi\in\Pi_{n,n}\;{\rm crossing\;or\;nested}} |C_\pi|\leq ( n! ) t^{-1/2} (\log t)^3 (c \lambda^2t\log t)^{n} \; . $$ \end{lemma} \prf By lemmata {~\ref{crosslessN}} and {~\ref{nestlessN}} below, every pairing contraction of crossing or nesting type can be bounded by $$ (c \lambda^2 t \log t)^{n} t^{-1/2} (\log t)^3 \;, $$ and clearly, there are at most $2^n n!$ such graphs. \endprf \centerline{\epsffile{f3.eps} } \noindent{Figure 3.} A simple nest. \\ \begin{lemma}\label{crosslessN} If $\pi\in\Pi_{n,n}$ corresponds to a pairing contraction that contains at least one crossing, then $$ |C_\pi|\leq t^{-1/2}(c \lambda^2t\log t)^n (\log t)^4 \;. $$ \end{lemma} \prf Let $T$ denote a complete spanning tree for the graph $G_\pi$, and $T^c$ its complement. All momenta supported on $T$ can be expressed as linear combinations of loop momenta supported on $T^c$. If there exists a crossing pairing, it is shown in \cite{erdyau} that there is a tree momentum $p_r$ in $T$ that depends on at least two loop momenta $p_j,p_l$ in $T^c$, $$ p_r=\pm p_j \pm p_l \pm w $$ where $w\in\Tor^3$ is a linear combination of momenta not depending on $p_j,p_l$. Thus, $C_\pi$ contains a subintegral \eqn A_\e(w,\alpha,\beta) &:=&\int_{(\Tor^3)^2}\frac{dp_jdp_l} {(e(p_j)-\alpha_j\pm i \e)\,(e(p_l)-\alpha_l\pm i \e)} \nonumber\\ &&\hspace{2cm} \cdot \, \frac{1}{(e(p_j\pm p_l+w)-\alpha_r\pm i\e)} \;. \label{crossingsubint} \eeqn Writing $p=(p_1,p_2,p_3)\equiv p_j$ and $p'=(p'_1,p'_2,p'_3)\equiv p_l$ for brevity, we have \eqn (~\ref{crossingsubint}) \leq (I)\cdot(II)\cdot(III)\cdot(IV) \;, \eeqn where \eqnn (I)&:=&\sup_{p',p_1,p_2,w}\int dp_3 \Big|\frac{1}{e(p\pm p'\pm w)-\alpha_r\pm i\e}\Big|^{\frac12} \;,\\ (II)&:=& \sup_{p'_1,p'_2,p,w}\int dp'_3 \Big|\frac{1}{e(p\pm p'\pm w)-\alpha_r\pm i\e}\Big|^{\frac12} \nonumber\\ (III)&:=&\sup_{p_3}\int dp_1dp_2 \Big|\frac{1}{e(p)-\alpha_j\pm i\e}\Big| \\ (IV)&:=& \sup_{p'_3}\int dp'_1dp'_2 \Big|\frac{1}{e(p')-\alpha_l\pm i\e}\Big|\;. \eeqnn We have \eqn (I),(II)&\leq&\sup_{a\in\R}\int_0^1 dp_3 \Big|\frac{1}{\cos2\pi p_3+a\pm i\e}\Big|^{\frac12}\nonumber\\ &\leq&C\sup_{a\in\R}\int_0^1\frac{dx}{\sqrt{1-x^2}} \Big|\frac{1}{x+a\pm i\e}\Big|^{\frac12}\nonumber\\ &\leq& C\log\frac1\e\;, \eeqn and using similar arguments as in the proof of Lemma {~\ref{propboundslemma}}, \eqn (III),(IV)&\leq&C\log\frac1\e \;, \eeqn such that $$ \big|A_\e(w,\alpha,\beta) \big| \leq (C |\log\e| )^4 \;. $$ For the remaining part of $C_\pi$, excluding the propagators corresponding to the indices $n$ and $n+1$, $L^\infty$-bounds on propagators in $T$, and $L^1$-bounds on propagators in $T^c$, produce a factor $(c \lambda^2t\log t)^{n-1}$. The propagators corresponding to the indices $n$ and $n+1$ contribute a factor $(c\log t)^2$, as in (~\ref{pnpn1estimate}) below. A detailed exposition is given in \cite{erd,erdyau}. \endprf \begin{lemma}\label{nestlessN} Let $\pi\in\Pi_{n,n}$ correspond to a non-crossing pairing contraction that contains at least one nested subgraph. Then, $$ |C_\pi|\leq t^{-1/2} (c \lambda^2t\log t)^n $$ \end{lemma} \prf In this case, $\pi$ comprises a nested subgraph of length $1n+1 \end{array} \right. $$ denote the index set accounting for all $2(n-q)$ momenta $p_i$ not contained in the nest, apart from the special indices $n$, $n+1$. One can find an index subset $A_{n-q}\subset I_{j,q;n}$ of size $n-q$, with the property that there exists a bijection $\pip:A_{n-q}\rightarrow A_{n-q}^c$ (the complement of $A_{n-q}$ in $I_{j,q;n}$), such that the product of delta functions has the structure \eqn \prod_{j=1}^{n-q}\delta_{S_j}(\up')=\prod_{r\in A_{n-q}} \delta\Big(p_r-p_{r-1}+p_{\pip(r)}-p_{\pip(r)-1}\Big) \;. \label{Acdeltastructure} \eeqn This is obtained as follows. For each pairing contraction outside of the nest $N_q(\alpha,\e)$, let $p_j-p_{j-1}$ and $p_{j'}-p_{j'-1}$ be the arguments of the random potentials at the adjacing vertices. If $j\leq n$, then we let $j-1\in A_{n-q}$, and if $j> n+1$, then $j\in A_{n-q}$. Furthermore, if $j'\leq n$, we let $\tau(j)=j'-1$, snd if $j'> n+1$, then $\tau(j)=j'$. Clearly, $j$ and $j'$ are interchangeable, and $I_{j,q;n}=A_{n-q}\cup \tau(A_{n-q})$. Thus, the pairings outside of the nest $N_q(\alpha,\e)$ can be written in the form \eqnn &&\int_{(\Tor^3)^{2(n-q)}} \prod_{r\in A_{n-q}} \frac{dp_r dp_{\pip(r)}\delta(p_r-p_{r-1}+p_{\pip(r)}-p_{\pip(r)-1})} {(e(p_r)-\alpha_r-i\sigma_r\e)^{\mu_j(r)} (e(p_{\pip(r)})-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\e)} \nonumber\\ &&=\int_{(\Tor^3)^{n-q}}\prod_{r\in A_{n-q}} \frac{dp_r } {(e(p_r)-\alpha_r-i\sigma_r\e)^{\mu_j(r)} (e(p_r+w_r)-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\e)} \;, \eeqnn where for each $r\in A_{n-q}$, $w_r\in\Tor^3$ is a linear combination of momenta not containing $p_r$. It is then easy to see that $|(~\ref{nestfullterm})|$ is bounded by \eqn \Big(\sup_{\alpha\in I}|N_q(\alpha,\e)|\Big)\lambda^{2n} \int_{I\times \bar I}|d\alpha|\,|d\beta| \int_{\Tor^3} \frac{dp_n}{|e(p_n)-\alpha-i\e|^{\mu_j(n)}|e(p_{n})-\beta+i\e| } \nonumber\\ \cdot \, \prod_{p_r\in A_{n-q}} \sup_{w_r\in\Tor^3}\int \frac{dp_r}{ |e(p_r)-\alpha_r-i\sigma_r\e|^{\mu_j(r)} } \frac{1}{|e(p_r+w_{r})-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\e| } \;. \eeqn Since \eqnn \sup_{w_r\in\Tor^3}\int \frac{dp_r}{ |e(p_r)-\alpha_r-i\sigma_r\e|^{\mu_j(r)} |e(p_r+w_{r})-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\e| } \leq c_1 \e^{-\mu_j(r)} |\log \e| \eeqnn and \eqn \int_{I\times \bar I}|d\alpha|\,|d\beta|\int_{\Tor^3} \frac{dp_n}{|e(p_n)-\alpha-i\e|^{\mu_j(n)}|e(p_{n})-\beta+i\e| } \leq c_1^2 \e^{1-\mu_j(n) }|\log\e|^2 ,\;\; \label{pnpn1estimate} \eeqn we find \eqn |(~\ref{nestfullterm})|&\leq&\lambda^{2n}(c |\log \e|)^{n-q+2} \e^{1-(\mu_j(n)+\sum_{r\in A_{n-q}} \mu_j(r))} (c \e^{-1})^q \e^{3/2} \nonumber\\ &\leq&(c \lambda^2\e^{-1}|\log\e|)^n\e^{1/2} \; , \eeqn due to $$ \mu_j(n)+\sum_{r\in A_{n-q}} \mu_j(r)=n-q+2 \;, $$ where $q\geq2$. This proves the lemma. \endprf \subsection{Type III Contractions} We shall next derive bounds on $C_\pi$, where $\pi\in\Pi_{n,n}$ contains type III contractions. We recall that the number of $\pi\in\Pi_{n,n}$ of type III is bounded by $n^{2n}$. The following lemma shows that the superfactorially large number of such graphs is compensated by the very small size of the associated integrals $C_\pi$. \begin{lemma}\label{highercorrlm} For any $1\leq m< n$, and $\pi=\{ S_j\}_{j=1}^m\in\Pi_{n,n}$ of type III, $$ |C_\pi| \leq (cn)^{3n-1} \lambda^{2n} t^{m} (c\log t)^{2 n-2m+2} \;. $$ \end{lemma} \prf After integrating out $\delta(p_n-p_{n+1})$, \eqn |C_\pi|&\leq&\lambda^{2n} e \int_{I\times \bar I} |d\alpha| \,|d\beta| \int_{\Tor^3} dp_n \Big|\frac{1}{ e(\vp_n)-\alpha-i\e }\Big| \,\Big|\frac{1}{ e(\vp_n)-\beta-i\e }\Big| \nonumber\\ &&\hspace{1cm}\cdot \, \Big[\prod_{ j=1 }^m \int_{(\Tor^3)^{|S_j|}}\Big( \prod_{r\in S_j} \Big|\frac{dp_r}{e(\vp_r)-\alpha_r-i\sigma_r\e}\Big|\Big) \big|c_{|S_j|}\big|\delta_{S_j}(\up) \Big] \nonumber\\ &\leq& \lambda^{2n} e \int_{\Tor^3} dp_n \int_{I\times \bar I} |d\alpha| \,|d\beta| \Big|\frac{1}{ e(\vp_n)-\alpha-i\e }\Big|\, \Big|\frac{1}{ e(\vp_n)-\beta+i\e }\Big| \nonumber\\ &&\hspace{1cm}\cdot \, \sup_{\alpha,\bar\beta\in I}\Big[ \prod_{ j=1 }^m \int_{(\Tor^3)^{|S_j|}}\Big(\prod_{r\in S_j} \Big|\frac{dp_r}{e(\vp_r)-\alpha_r-i\sigma_r\e}\Big|\Big) \big|c_{|S_j|}\big|\delta_{S_j}(\up) \Big] \;. \; \; \eeqn First of all, $$ \int_{\Tor^3}dp_n \int_{I\times \bar I} |d\alpha| \,|d\beta| \Big|\frac{1}{ e(\vp_n)-\alpha-i\e }\Big|\, \Big|\frac{1}{ e(\vp_n)-\beta+i\e }\Big| \leq (c_1 |\log \e|)^2 \;, $$ by (~\ref{resolvbound2}). For $1\leq j\leq m$, \eqn \sup_{\alpha,\beta\in I}\sup_{p_l\in\Tor^3;l\not\in S_j} && \Big[\int_{(\Tor^3)^{|S_j|}}\Big( \prod_{r\in S_j} \Big|\frac{dp_r}{e(\vp_r)-\alpha_r-i\sigma_r\e}\Big|\Big) \big|c_{|S_j|}\big|\delta_{S_j}(\up)\Big] \nonumber\\ &<& \big|c_{|S_j|}\big| \Big(\sup_{\alpha\in I} \int_{\Tor^3}dp \Big|\frac{1}{e(p)-\alpha-i\e}\Big|\Big)^{|S_j|-2} \nonumber\\ &&\cdot \, \sup_{\alpha,\alpha'\in I}\sup_{p'} \Big[\int_{\Tor^3}dp \Big|\frac{1}{e(\vp)-\alpha-i\e}\Big| \Big|\frac{1}{e(\vp+\vp')-\alpha'-i\e}\Big|\Big] \nonumber\\ &<& (c_\omega |S_j|/2)^{2|S_j|+1}\e^{-1} (c_1|\log \e|)^{|S_j|-1} \;, \eeqn by $|S_j|-1$ bounds of type $L^1-L^\infty$, using (~\ref{resolvbound1}) and (~\ref{resolvbound2}). Furthermore, we have recalled (~\ref{renormc2kbound}), and again used $$ \sup_{\alpha,\alpha'\in I}\sup_{p'} \Big[\int_{\Tor^3}dp \Big|\frac{1}{e(\vp)-\alpha-i\e}\Big| \Big|\frac{1}{e(\vp+\vp')-\alpha'-i\e}\Big|\Big] \leq c_1 \e^{-1}|\log \e| \;, $$ cf. (~\ref{resolvbound1}) and (~\ref{resolvbound2}). From $$ \prod_{j=1}^m (c_\omega |S_j|/2)^{2|S_j|+1} < (c_\omega n)^{3n-1} \;, $$ if $m0$, where $i=1,2$. While the character of the estimates required to control $\Exp[\|\Rem_1(t)\|^2]$ is essentially the same as for $n\leq N$, our strategy to bound $\Exp[\|\Rem_2(t)\|^2]$ exploits the rarity of events comprising large collision numbers - that is, of order $O(N)$ - in the time intervals $[\tt_{j},\tt_{j-1})$, which are much shorter than $[0,t]$. The following lemma is the main result in this context. \begin{lemma} The remainder term satisfies the bounds \eqnn \Exp\Big[\| \Rem_1(t)\|^2\Big] &\leq& \frac{N^2\kappa^2 (c\lambda^2 t )^{4N }}{(N!)^{1/2}} \nonumber\\ &+& N^2 \kappa^2 (c\lambda^2 t\log t)^{4N} (\log t)^{3} \Big( t^{-1/2}(4N)! + t^{-2} (4N)^{20N}\Big) \nonumber\\ \Exp\Big[\| \Rem_2(t) \|^2\Big] &\leq& t^2\frac{(c\lambda^2 t\log t)^{4N} (\log t)^{3}} {\kappa^N} \Big( (4N)!+t^{-2}(4N)^{20N} \Big) \; , \eeqnn where the constant $c$ is uniform in $t$ and $N$. \end{lemma} \prf By the Schwarz inequality, and by the unitarity of $e^{-it H_\omega}$, \eqn \Exp\Big[\|\Rem_1(t) \|^2\Big] &\leq& (3N)^2\kappa^2 \sup_{N1$, \eqn \Big|\frac{1}{e(p_j)-\alpha_j-i\sigma_j\kappa\e}\Big|\leq \Big|\frac{1}{e(p_j)-\alpha_j-i\sigma_j\e}\Big| \label{resreskappaest} \eeqn for all $j$ in the estimates on $|C_\pi|$. (It is not necessary to exploit the factors $\kappa^{-N}$ here.) \endprf The remark after the proof of Lemma {~\ref{simplepairlemmleqN}} about the case of arbitrary $T=\lambda^2t>0$ also applies to the present situation. \begin{lemma}\label{crossgeqN} Let $NN$ are the same as for $n\leq N$. \begin{lemma}\label{partinthighcorrlm} Let $N0$, $$ \Exp\big[ W^\e_{\phi_{T/\e}^\e} \big] \rightarrow F_T(X,V)\;, $$ weakly as $\e\rightarrow0$, for $X\in\R^3$, $V\in\Tor^3$, where $F_T(X,V)$ solves the linear Boltzmann equation \eqn &&\partial_T F_T(X,V) +2\sin2\pi V \times \nabla_X F_T(X,V) \nonumber\\ &&\hspace{2cm}= \int_{\Tor^3} dU \sigma(U,V) \lb F_T(X,U) - F_T(X,V)\rb \;, \label{linB} \eeqn with collision kernel $$ \sigma(U,V)=4\pi\delta(e(U)-e(V)) \;, $$ and for the initial condition \eqn\label{initcondweaklim} W_{\phi^\e_0}^\e\rightarrow |h(X)|^2\delta(V-\nabla S(X)) =: F_0(X,V) \;, \eeqn weakly as $\e\rightarrow0$. \end{theorem} For (~\ref{initcondweaklim}) and details of the proof, cf. the closely related case analyzed in \cite{erdyau}. Using our analysis in previous sections, it is straightforward to infer that the non-zero contributions to $F_T(X,V)$ in the macroscopic limit stem from type I and II contractions among the random potentials in the Duhamel expansion of $W^\e_{\phi_t^\e}(X,V)$. Let $$ \phi^{main}_{t}:=\sum_{n=0}^N \phi_{n,t}\;, $$ and \eqn \Exp \lb \int_{\Tor^3\times\Tor^3}d\xi dv \overline{\hat J_\e(\xi ,v)} \hat W_{\phi_t^{main}}(\xi,v)\rb &=&\sum_{n,n'=0}^N U^{\hat J_\e}_{n,n'}\nonumber\\ &=&\sum_{n,n'=0}^N \sum_{\pi\in \Pi_{n,n'}} C_{\pi}^{\hat J_\e} \;, \label{WigweakDuhamel} \eeqn where $$ U_{n,n'}^{\hat J_\e}=\Exp \lb \int_{\Tor^3\times\Tor^3}d\xi dv \overline{\hat J_\e(\xi ,v)}\overline{\hat \phi_{n,t}(v-\xi)} \hat\phi_{n',t}(v+\xi) \rb\;. $$ $C_\pi^{\hat J_\e}$ denotes the value of the integral corresponding to the contraction $\pi\in\Pi_{n,n'}$, . \begin{lemma} Let $\pi\in\Pi_{n,n'}$, and $\bar n:=\frac{n+n'}{2}\in\N$. Then, \eqn U_{n,n'}^{\hat J_\e} &=& \sum_{\pi \in \Pi_{n,n'}\;{\rm simple}} C_\pi^{\hat J_\e} \nonumber\\ &&+ O\Big( (c\lambda^2 t\log t)^{\bar n} (\log t)^{3} \big(t^{-1/2} \bar n ! + t^{-2}\bar n^{5\bar n}\big)\Big) \;, \label{WigweakDuhest} \eeqn and for any simple pairing $\pi$, \eqn |C_\pi^{\hat J_\e}| \leq (c\lambda^2 t)^{\bar n} \;. \eeqn \end{lemma} \prf The difference here to the discussion in the previous sections is the presence of $J_\e$, and $n\neq n'$. We note that for the choice $J_\e=\delta(\xi)$, (~\ref{WigweakDuhamel}) reduces to the previous case $\Exp[\|\phi_t^{main}\|_{\ell^2(\Z^3)}^2]$. The necessary modifications are straightforward, and the same estimates on the Feynman amplitudes of families of graphs enter as in the situation discussed before, and as expressed in (~\ref{WigweakDuhest}). For a detailed account on this issue, which can be straightforwardly applied to the problem at hand, we refer to \cite{erdyau}. \endprf Similarly as in the proof of Lemma {~\ref{simplepairlemmleqN}}, we decompose $C_\pi^{\hat J_\e}$, for $\pi$ simple, into a main part $C_\pi^{\hat J_\e,main}$, and an error part, where $C_\pi^{\hat J_\e,main}$ is obtained by replacing the recollision terms $\Th (\alpha,\e)$ and $\Th (\beta,-\e)$ in $C_\pi^{\hat J_\e}$ by $\Th (e(v_0),\e)$ and $\Th (e(v_0),-\e)$, respectively. Thus, assuming for $\pi$ that $C_\pi^{\hat J_\e}$ contains $m$ type II contractions, where we index the immediate recollisions by $(q_0,\dots,q_m)$ and $(\tilde q_0,\dots,\tilde q_m)$, respectively, as in (~\ref{simplepairfixedn}), we have \eqn C_\pi^{\hat J_\e,main} &=&\int_{\Tor^3\times\Tor^3} d\xi dv_0 \overline{\hat J_\e( \xi,v_0)} \lambda^{2m} e^{2\e t} \int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{1cm} \cdot \, \int_{(\Tor^3)^m} dv_1\cdots dv_m \overline{\phi_0^\e(v_n-\xi)}\phi_0^\e(v_n+\xi) \nonumber\\ &&\hspace{2cm} \cdot \, \prod_{i=0}^m \frac{(\lambda^2\Th(e(v_0),\e))^{q_i}} {(e(v_i+\xi)-\alpha-i\e)^{q_i+1}} \frac{(\lambda^2\Th(e(v_0),-\e))^{\tilde q_i}} {(e(v_i-\xi)-\beta+i\e)^{\tilde q_i+1}} \;. \eeqn The error term is controlled by the following lemma. \begin{lemma} Let $\pi\in\Pi_{n,n'}$ be a simple pairing. Then, \eqn C_\pi^{\hat J_\e}&=&C_\pi^{\hat J_\e,main} +O((c\lambda^2 t)^{\bar n} t^{-1/2}) \nonumber\\ |C_\pi^{\hat J_\e,main}|&\leq& \frac{(c\lambda^2 t)^{\bar n}}{(\bar n !)^{1/2}} \;. \eeqn \end{lemma} \prf The proof is the same as for lemma {~\ref{simplepairlemmleqN}}, with the appropriate modifications included to cover the case $n\neq n'$. Since this generalization is straightforward, and also treated in detail in \cite{erdyau}, we will not repeat the argument here. \endprf We perform the contour integral with respect to the variables $\alpha$ and $\beta$, and evaluate the sum over $n,n'\in\{0,\dots,N\}$ by first summing over all $q_i,\tilde q_i$, where $i=1,\dots,m$, for fixed $m$, and sum over the indices $m$ at the end. We then obtain \eqn \lim_{N\rightarrow0}\sum_{n,n'=0}^N \sum_{\stackrel{\pi\in\Pi_{n, n'}} {\pi\;{\rm simple}}}C_\pi^{\hat J_\e,main} &=& \int dv_0 d\xi \overline{J_\e( \xi,v_0)} \sum_{m=0}^\infty \lambda^{2m} \nonumber\\ &&\cdot\,\int \Big[ \prod_{j=0}^m ds_j\Big]_t \Big[ \prod_{j=0}^m d\tilde s_j\Big]_t\int_{(\Tor^3)^m} dv_1\cdots dv_m \hat W_{\phi_0^\e}(\xi , v_m) \nonumber\\ &&\cdot\;\; e^{2t\lambda^2\Im(\Th (v_0,\e))} \prod_{i=0}^m e^{-i s_i e(v_i+\xi)+i\tilde s_i e(v_i-\xi) } \label{eqn4-1-1} \eeqn where \eqnn \hat W_{\phi_0^\e}(\xi , v ) &=& \overline{\hat\phi_0^\e(v - \xi )} \hat\phi_0^\e(v + \xi ) \; , \\ \Im(\Th (v_0,\e))&=&\frac{1}{2}\big(\Th (v_0,\e)-\Th (v_0,-\e)\big) \;. \eeqnn To demonstrate that in the macroscopic scaling limit, the contributions from the main term tend to a solution of the linear Boltzmann equations, we introduce the new time variables \eqnn a_j &:=& \frac{s_j + \tilde s_j}{2} \\ b_j &:=& \frac{s_j - \tilde s_j}{2} \;, \eeqnn with $a_j\geq0$ and $\sum_{j=0}^n a_j=t$, and $b_j\in[-a_j,a_j]$. so that \eqn s_i e(v_i-\xi)-\tilde s_i e(v_i+\xi)&=& a_i \big(e(v_i+\xi)- e(v_i-\xi)\big)\nonumber\\ &+& b_i \big(e(v_i-\xi)+ e(v_i+\xi)\big)\;. \label{eqn4-1-5} \eeqn Furthermore, we introduce macroscopic variables \eqnn T := \e t \; \; , \; \; \tau_j := \e a_j \; \; , \; \; \zeta := \xi/\e \; , \eeqnn where by the compactness of the support of $\hat J_\e$, we have $|\zeta|\leq O(1)$. For all finite $\tau_j$, \eqnn \prod_{j=1}^{n} \int_{-\tau_j/\e}^{\tau_j/\e} db_j \, e^{ 2 i b_j ( e(v_j)-e(v_0)+O(\e) )} \rightarrow\prod_{j=1}^{n}2\pi\delta\big( e(v_j)-e(v_0)\big) \;, \eeqnn weakly as $\e\rightarrow0$. By the same arguments as in \cite{erdyau}, we obtain \eqn \lim_{\e\rightarrow0}\Exp\big[\langle\hat J_\e,\hat W_{\phi_t^\e}\rangle\big] &=&\sum_{n\geq0} \int_{ (\Tor^3)^{n+1}} dv_0\cdots dv_n e^{2T\Im(\Th (v_0))} \nonumber\\ &&\hspace{1cm}\cdot \, \int \Big[\prod_{j=0}^n d\tau_j \Big]_T \prod_{j=1}^{n} 4\pi \delta\big(e(v_j)-e(v_0)\big) \nonumber\\ &&\hspace{1cm}\cdot \, \lim_{\e\rightarrow0}\int_{\Tor_\e^3}d\zeta\overline{\hat J(\zeta,v_0)} e^{2\pi i\sum_{j=0}^n\tau_j \zeta \cdot 2\sin 2\pi v_j } \hat W_{\phi_0^\e}(\e\zeta , v_n ) \;, \eeqn where $\Th (v):=\lim_{\e\rightarrow0}\Th (v,\e)$. The factor $\e^{-n}$, which emerges from rescaling $a_i$, has notably eliminated $\lambda^{2n}$. Moreover, recalling (~\ref{initcondweaklim}), we have, for the given choice of $\phi_0^\e$, \eqn &&\lim_{\e\rightarrow0}\int_{\Tor_\e^3}d\zeta\overline{\hat J(\zeta,v_0)} e^{2\pi i\sum_{j=0}^n\tau_j \zeta \cdot 2\sin 2\pi v_j } \hat W_{\phi_0^\e}(\e\zeta , v_n )\nonumber\\ &&\hspace{2cm}=\int_{\R^3}dX\overline{ J(X,v_0)} F_{0}\Big(X-\sum_{j=0}^n\tau_j 2\sin2\pi v_j, v_n \Big) \;. \eeqn Notably, the microscopic and macroscopic velocities are equal, $v_j=V_j$, for $j=0,\dots,n$. Since the test function $J$ is arbitrary, one obtains \eqn \lim_{N\rightarrow\infty} \Exp\big[ W^\e_{\phi_{T/\e, N}^{main}}(X,V) \big] \rightarrow F_T(X,V)\; \eeqn weakly as $\e\rightarrow0$, where \eqnn F_T(X,V)&=&e^{2T\sigma(V)}\sum_{n\geq0} \int d\tau_0\cdots d\tau_n \delta\Big(\sum_{j=0}^n\tau_j-T\Big) \nonumber\\ &&\cdot \, \int dV_1\cdots dV_n \sigma(V,V_1)\cdots \sigma(V_{n-1},V_n) F_0\Big(X-\sum_{j=0}^n 2\tau_j \sin 2\pi V_j , V_n \Big)\; , \eeqnn with $V=V_0$, and differential cross-section \eqn \sigma(V,U) := 4\pi\delta\big(e(V)-e(U)\big) \;. \eeqn The total scattering cross-section is given by \eqn \sigma(V) := \int dU \sigma(V,U)=-2\Im(\Th(V)) \;. \eeqn In particular, $F_T(X,V)$ satisfies the linear Boltzmann equations (~\ref{linB}). This concludes our proof of Theorem {~\ref{Boltzlimthm}}. \subsection*{Acknowledgements} The author is profoundly grateful to L. Erd\"os, and especially H.-T. Yau, for their support, guidance, and generosity. He has benefitted immensely from very numerous discussions with H.-T. Yau, without whom this work would not have been possible. He also thanks A. Elgart and B. Schlein for discussions. The author was supported by a Courant Instructorship, and in part by a grant from the NYU Research Challenge Fund Program. \begin{thebibliography}{99} \bibitem{cyfrkisi} Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B., {\em Schr\"odinger operators}, Springer Verlag (1987). \bibitem{erd} Erd\"os, L., {\em Linear Boltzmann equation as the scaling limit of the Schr\"odinger evolution coupled to a phonon bath}, J. Stat. Phys. 107(5), 1043-1127 (2002). \bibitem{erdyau} Erd\"os, L., Yau, H.-T., {\em Linear Boltzmann equation as the weak coupling limit of a random Schr\"odinger equation}, Comm. Pure Appl. Math., Vol. LIII, 667 - 753, (2000). \bibitem{erdsalmyau} Erd\"os, L., Salmhofer, M., Yau, H.-T., {\em announced}. \bibitem{mapori1} Magnen, J., Poirot, G., Rivasseau, V., {\em Renormalization group methods and applications: First results for the weakly coupled Anderson model}, Phys. A 263, no. 1-4, 131-140 (1999). \bibitem{mapori2} Magnen, J., Poirot, G., Rivasseau, V., {\em Ward-type identities for the two-dimensional Anderson model at weak disorder}, J. Statist. Phys., 93, no. 1-2, 331-358 (1998). \bibitem{po} Poirot, G., {\em Mean Green's function of the Anderson model at weak disorder with an infra-red cut-off}, Ann. Inst. H. Poincar\'e Phys. Th\'eor. 70, no. 1, 101-146 (1999). \bibitem{shscwo} Shubin, C., Schlag, W., Wolff, T., {\em Frequency concentration and localization lengths for the Anderson model at small disorders}, J. Anal. Math., 88 (2002). \bibitem{sp} Spohn, H., {\em Derivation of the transport equation for electrons moving through random impurities}, J. Statist. Phys., 17, no. 6, 385-412 (1977). \end{thebibliography} \end{document} ---------------0312010856408 Content-Type: application/postscript; name="f1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="f1.eps" %!PS-Adobe-3.0 EPSF-3.0 %%HiResBoundingBox: 0.000000 0.000000 428.000000 116.000000 %%Title: (Unknown) %%Creator: (Unknown) %%CreationDate: (Unknown) %%For: (Unknown) %%DocumentData: Clean7Bit %%Pages: 1 %%BoundingBox: 0 0 428 116 %%EndComments %%BeginProlog userdict /PDF 95 dict put %%BeginFile: pdfvars.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. userdict /PDFVars 90 dict put PDFVars begin /_save 0 def /_cshow 0 def /InitAll 0 def /TermAll 0 def /_lp /none def /_doClip 0 def /sfc 0 def /_sfcs 0 def /_sfc 0 def /ssc 0 def /_sscs 0 def /_ssc 0 def /_fcs 0 def /_scs 0 def /_fp 0 def /_sp 0 def /AGM_MAX_CS_COMPONENTS 10 def /_fillColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_strokeColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_fc null def /_sc null def /GetCompsDict null def /_inT false def /_tr -1 def /_rise 0 def /_ax 0 def /_cx 0 def /_ld 0 def /_tm matrix def /_ctm matrix def /_mtx matrix def /_hy (-) def /_fScl 0 def /_hs 1 def /_pdfEncodings 2 array def /_baselineadj 0 def /_fTzero false def /_Tj 0 def /_italMtx[1 0 .212557 1 0 0]def /_italMtx_WMode1 [1 -.212557 0 1 0 0]def /_italMtxType0[1 0 .1062785 1 0 0]def /_italMtx_WMode1Type0 [1 -.1062785 0 1 0 0]def /_basefont 0 def /_basefonto 0 def /_pdf_oldCIDInit null def /_categories 10 dict def /_sa? true def /_op? false def /_ColorSep5044? false def /_tmpcolr? 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/c70/c71/c72/c73/c74/c75/c76/c77/c78/c79/c7A/c7B/c7C/c7D/c7E/c7F /c80/c81/c82/c83/c84/c85/c86/c87/c88/c89/c8A/c8B/c8C/c8D/c8E/c8F /c90/c91/c92/c93/c94/c95/c96/c97/c98/c99/c9A/c9B/c9C/c9D/c9E/c9F /cA0/cA1/cA2/cA3/cA4/cA5/cA6/cA7/cA8/cA9/cAA/cAB/cAC/cAD/cAE/cAF /cB0/cB1/cB2/cB3/cB4/cB5/cB6/cB7/cB8/cB9/cBA/cBB/cBC/cBD/cBE/cBF /cC0/cC1/cC2/cC3/cC4/cC5/cC6/cC7/cC8/cC9/cCA/cCB/cCC/cCD/cCE/cCF /cD0/cD1/cD2/cD3/cD4/cD5/cD6/cD7/cD8/cD9/cDA/cDB/cDC/cDD/cDE/cDF /cE0/cE1/cE2/cE3/cE4/cE5/cE6/cE7/cE8/cE9/cEA/cEB/cEC/cED/cEE/cEF /cF0/cF1/cF2/cF3/cF4/cF5/cF6/cF7/cF8/cF9/cFA/cFB/cFC/cFD/cFE/cFF ] def /modEnc { /_enc xdd /_icode 0 dd counttomark 1 sub -1 0 { index dup type /nametype eq { _enc _icode 3 -1 roll put _icode 1 add } if /_icode xdd } for cleartomark _enc } bd /trEnc { /_enc xdd 255 -1 0 { exch dup -1 eq { pop /.notdef } { Encoding exch get } ifelse _enc 3 1 roll put } for pop _enc } bd /TE { /_i xdd StandardEncoding 256 array copy modEnc _pdfEncodings exch _i exch put } bd /TZ { /_usePDFEncoding xdd findfont dup length 2 add dict begin { 1 index /FID ne { def } { pop pop } ifelse } forall /FontName exch def _usePDFEncoding 0 ge { /Encoding _pdfEncodings _usePDFEncoding get def pop }{ _usePDFEncoding -1 eq { counttomark 0 eq { pop }{ Encoding 256 array copy modEnc /Encoding exch def } ifelse }{ 256 array trEnc /Encoding exch def } ifelse } ifelse FontName currentdict end definefont pop } bd /Level2? systemdict /languagelevel known { systemdict /languagelevel get 2 ge }{ false } ifelse def Level2? { /_pdfFontStatus { currentglobal exch /Font resourcestatus {pop pop true} {false} ifelse exch setglobal } bd }{ /_pdfFontStatusString 50 string def _pdfFontStatusString 0 (fonts/) putinterval /_pdfFontStatus { FontDirectory 1 index known { pop true } { _pdfFontStatusString 6 42 getinterval cvs length 6 add _pdfFontStatusString exch 0 exch getinterval { status } stopped {pop false} { { pop pop pop pop true} { false } ifelse } ifelse } ifelse } bd } ifelse Level2? { /_pdfCIDFontStatus { /CIDFont /Category resourcestatus { pop pop /CIDFont resourcestatus {pop pop true} {false} ifelse } { pop false } ifelse } bd } if /_pdfString100 100 string def /_pdfComposeFontName { dup length 1 eq { 0 get 1 index type /nametype eq { _pdfString100 cvs length dup dup _pdfString100 exch (-) putinterval _pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval 2 index exch cvs length add 1 add _pdfString100 exch 0 exch getinterval exch pop true }{ pop pop false } ifelse }{ false } ifelse } bd pdf_has_composefont? { /_pdfComposeFont { 1 index /CMap resourcestatus { pop pop true }{ false } ifelse 1 index true exch { _pdfCIDFontStatus not {pop false exit} if } forall and { 3 -1 roll pop composefont true }{ 4 -1 roll pop _pdfComposeFontName { dup _pdfFontStatus { findfont definefont true }{ pop dup _pdfFontStatus { findfont true }{ pop false } ifelse } ifelse }{ dup _pdfFontStatus { findfont true } { pop false } ifelse } ifelse } ifelse } bd }{ /_pdfComposeFont { 4 -1 roll pop _pdfComposeFontName not { dup } if 2 copy _pdfFontStatus {pop findfont exch pop true} { eq {pop false} { dup _pdfFontStatus {findfont true} {pop false} ifelse } ifelse } ifelse } bd } ifelse /_pdfStyleDicts 4 dict dup begin /Adobe-Japan1 4 dict dup begin Level2? { /Serif /HeiseiMin-W3-83pv-RKSJ-H _pdfFontStatus {/HeiseiMin-W3} { /HeiseiMin-W3 _pdfCIDFontStatus {/HeiseiMin-W3} {/Ryumin-Light} ifelse } ifelse def /SansSerif /HeiseiKakuGo-W5-83pv-RKSJ-H _pdfFontStatus {/HeiseiKakuGo-W5} { /HeiseiKakuGo-W5 _pdfCIDFontStatus {/HeiseiKakuGo-W5} {/GothicBBB-Medium} ifelse } ifelse def /HeiseiMaruGo-W4-83pv-RKSJ-H _pdfFontStatus {/HeiseiMaruGo-W4} { /HeiseiMaruGo-W4 _pdfCIDFontStatus {/HeiseiMaruGo-W4} { /Jun101-Light-RKSJ-H _pdfFontStatus { /Jun101-Light } { SansSerif } ifelse } ifelse } ifelse /RoundSansSerif exch def /Default Serif def } { /Serif /Ryumin-Light def /SansSerif /GothicBBB-Medium def { (fonts/Jun101-Light-83pv-RKSJ-H) status }stopped {pop}{ { pop pop pop pop /Jun101-Light } { SansSerif } ifelse /RoundSansSerif exch def }ifelse /Default Serif def } ifelse end def /Adobe-Korea1 4 dict dup begin /Serif /HYSMyeongJo-Medium def /SansSerif /HYGoThic-Medium def /RoundSansSerif SansSerif def /Default Serif def end def /Adobe-GB1 4 dict dup begin /Serif /STSong def /SansSerif /STHeiti def /RoundSansSerif SansSerif def /Default Serif def end def /Adobe-CNS1 4 dict dup begin /Serif /MKai-Medium def /SansSerif /MHei-Medium def /RoundSansSerif SansSerif def /Default Serif def end def end def /_pdf_Adobe-Japan1-2 (Adobe-Japan1-2) def /_pdfConcatNames { exch _pdfString100 cvs length dup dup _pdfString100 exch (-) putinterval _pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval 3 -1 roll exch cvs length add 1 add _pdfString100 exch 0 exch getinterval } bind def /_pdfSubSetFontByStyleDict 4 dict dup begin _pdfStyleDicts /Adobe-Japan1 get { _pdf_Adobe-Japan1-2 _pdfConcatNames dup _pdfFontStatus { def }{ pop pop } ifelse } forall end def /TZzero { /_fyAdj xdd /_wmode xdd /_styleArr xdd /_regOrdering xdd 4 copy _pdfComposeFont {exch pop exch pop exch pop} { [ 0 1 _styleArr length 1 sub { _styleArr exch get _pdfStyleDicts _regOrdering 2 copy known { get exch 2 copy known not { pop /Default } if get } { pop pop /Unknown } ifelse } for ] exch pop 3 index 3 index 4 2 roll _pdfComposeFont { exch pop }{ findfont } ifelse } ifelse dup /FontType get 3 eq _wmode 1 eq and { _pdfVerticalRomanT3Font dup length 10 add dict copy begin /_basefont exch def /Encoding _basefont /Encoding get def }{ dup length 3 add dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall } ifelse /WMode _wmode def /BaseLineAdj _fyAdj def FontType 0 ne { /Encoding Encoding dup length array copy dup 16#5c /yen put def /_fauxfont true def } if currentdict end definefont pop } bd /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 dd { 1 index eq {/_cnt _cnt 1 add dd} if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } bd /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave 32 eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /jsfTzero { { 2 npop (0) exch 2 copy 0 exch put exch show 32 eq { 4 index 4 index rmoveto } if 2 copy rmoveto } exch cshow 5 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put 32 eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } bd /trj { _cx 0 fWModeProc 32 _ax 0 fWModeProc 6 5 roll } bd /pjsf { trj sfc fawidthshowProc } bd /pjss { trj _ctm ssc jss } bd /pjsc { trj jsp } bd /_Tjdef [ /pjsf load /pjss load { dup currentpoint 3 2 roll pjsf newpath moveto pjss } bind { trj swj rmoveto } bind { dup currentpoint 4 2 roll gsave pjsf grestore 3 1 roll moveto pjsc } bind { dup currentpoint 4 2 roll currentpoint gsave newpath moveto pjss grestore 3 1 roll moveto pjsc } bind { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll pjsf newpath moveto pjss grestore 3 1 roll moveto pjsc } bind /pjsc load ] def /BT { /_inT true dd _ctm currentmatrix pop 1 0 0 1 0 0 _tm astore pop %DMG 0 _rise _baselineadj add translate _hs 1 scale 0 0 moveto } bd /ET { /_inT false dd _tr 3 gt {clip} if _ctm setmatrix newpath } bd /Tr { _inT { _tr 3 le {currentpoint newpath moveto} if } if dup /_tr xdd _Tjdef exch get /_Tj xdd } bd /Tj { userdict /$$copystring 2 index put _Tj } bd /iTm { _ctm setmatrix _tm concat 0 _rise _baselineadj add translate _hs 1 scale } bd /Tm { _tm astore pop iTm 0 0 moveto } bd /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } bd /TD { dup /_ld xdd Td } bd /_nullProc {} bd /Tf { dup 1000 div /_fScl xdd Level2? { selectfont }{ exch findfont exch scalefont setfont } ifelse currentfont dup /_nullProc exch /WMode known { 1 index /WMode get 1 eq { pop /exch } if } if load /fWModeProc xdd dup /FontType get 0 eq dup _cx 0 ne and { /jsfTzero }{ /awidthshow } ifelse load /fawidthshowProc xdd /_fTzero xdd dup /BaseLineAdj known { dup /BaseLineAdj get _fScl mul }{ 0 } ifelse /_baselineadj xdd currentpoint iTm moveto pop } bd /TL { neg /_ld xdd } bd /Tw { /_cx xdd _cx 0 ne _fTzero and { /jsfTzero } { /awidthshow } ifelse load /fawidthshowProc xdd } bd /Tc { /_ax xdd } bd /Ts { /_rise xdd currentpoint iTm moveto } bd /Tz { 100 div /_hs xdd iTm } bd /Tk { exch pop _fScl mul neg 0 fWModeProc rmoveto } bd /T* { 0 _ld Td } bd /' { T* Tj } bd /" { exch Tc exch Tw ' } bd /TJ { { dup type /stringtype eq { Tj }{ 0 exch Tk } ifelse } forall } bd /T- { _hy Tj } bd /d0/setcharwidth ld /d1 { setcachedevice /sfc{}dd /ssc{}dd } bd /nND {{/.notdef} repeat} bd /T3Defs { /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } def /BuildGlyph { exch begin GlyphProcs exch get exec end } def } bd /_pdfBoldRomanWidthProc { stringwidth 1 index 0 ne { exch .03 add exch }if setcharwidth } bd /_pdfType0WidthProc { dup stringwidth 0 0 moveto 2 index true charpath pathbbox 0 -1 7 index 2 div .88 setcachedevice2 pop } bd /_pdfBoldBaseFont 11 dict begin /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding cHexEncoding def /_setwidthProc /_pdfBoldRomanWidthProc load def /_bcstr1 1 string def /BuildChar { exch begin _basefont setfont _bcstr1 dup 0 4 -1 roll put dup _setwidthProc 0 0 moveto dup show _basefonto setfont 0 0 moveto show end }bd currentdict end def /_pdfVerticalRomanT3Font 10 dict begin /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /_bcstr1 1 string def /BuildChar { exch begin _basefont setfont _bcstr1 dup 0 4 -1 roll put dup _pdfType0WidthProc 0 0 moveto show end } bd currentdict end def /MakeBoldFont { dup dup length 3 add dict begin CopyFont /PaintType 2 def /StrokeWidth .03 0 FontMatrix idtransform pop def /dummybold currentdict end definefont _pdfBoldBaseFont dup length 3 add dict copy begin /_basefont exch def /_basefonto exch def currentdict end definefont } bd /MakeBold { exch 1 index findfont dup /FontType get 0 eq { _pdfBoldBaseFont /_setwidthProc /_pdfType0WidthProc load put {MakeBoldFont} Type0CopyFont definefont }{ dup /_fauxfont known not { _pdfBoldBaseFont /_setwidthProc /_pdfBoldRomanWidthProc load put MakeBoldFont }{ 2 index 2 index eq { exch pop }{ dup length dict begin CopyFont currentdict end definefont } ifelse } ifelse } ifelse pop pop } bd /MakeItalic { findfont dup /FontType get 0 eq Level2? not and { dup /FMapType get 6 eq } { false } ifelse { dup /WMode 2 copy known { get 1 eq { _italMtx_WMode1Type0 } { _italMtxType0 } ifelse } { pop pop _italMtxType0 } ifelse } { dup /WMode 2 copy known { get 1 eq { _italMtx_WMode1 } { _italMtx } ifelse } { pop pop _italMtx } ifelse } ifelse makefont Level2? not { dup length dict begin CopyFont currentdict end } if definefont pop }bd /MakeBoldItalic { /dummybold exch MakeBold /dummybold MakeItalic }bd currentdict readonly pop end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimage.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage 38 dict put PDF /PDFIVars 20 dict put PDF /PDFImage get begin /initialize { PDFImage begin } bd /terminate { end } bd /nulldict 0 dict def /gv { PDFIVars exch get } bd /pv { PDFIVars 3 1 roll put } bd /BI { save /savelevel exch pv mark } bd /EI { /savelevel gv restore } bd end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg1b.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? not StartLoad { PDFIVars /PDFImages 4 dict put /InstallImage { PDFIVars /PDFImages get 3 1 roll put } bd /ColorComps? { dup type /arraytype eq { 0 get } if /PDFImages gv exch get 0 get } bd /ColorProc? { dup type /arraytype eq { 0 get } if /PDFImages gv exch get 1 get } bd /ImageFilter { /DataSource load } bd /ID { 5 counttomark 2 idiv dup 3 1 roll add dict begin { def } repeat cleartomark currentdict end PDFIVars begin begin /ImageMatrix [ Width 0 0 Height neg 0 Height ] def /ColorSpace here { pop } { /ColorSpace /DeviceGray def } ifelse ColorSpace ColorProc? exec end end } bd /DeviceGray [ 1 { /ImageMask here not { false } if { sfc Width Height /Decode here { 0 get 1 eq } { false } ifelse ImageMatrix ImageFilter imagemask } { Width Height BitsPerComponent ImageMatrix ImageFilter image } ifelse } bind ] InstallImage } EndLoad end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg1c.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? not StartLoad { /DeviceRGB [ 3 { Width Height BitsPerComponent ImageMatrix ImageFilter 3 ColorImage } bind ] InstallImage /DeviceCMYK [ 4 { Width Height BitsPerComponent ImageMatrix ImageFilter 4 ColorImage } bind ] InstallImage /ColorImage? /colorimage where { pop true } { false } ifelse def ColorImage? StartLoad { /ColorImage { false exch colorimage } bd } EndLoad ColorImage? not StartLoad { /SetupColorImage { /CIConv 255 2 BitsPerComponent exp 1 sub div pv /CIMask 0 not BitsPerComponent bitshift not pv /CIBSelect BitsPerComponent 1 sub not 7 and pv /CIBufferExp CIWidth string pv } bd /rgbtogray { 0.11 mul exch 0.59 mul add exch 0.3 mul add round cvi } bd /cmyktogray { exch 0.11 mul add exch 0.59 mul add exch 0.3 mul add round cvi dup 255 gt { pop 255 } if 255 exch sub } bd /FastRGB { CIDataProc dup 0 3 2 index length 3 sub { dup 3 idiv 2 index 2 index get 3 index 3 index 1 add get 4 index 4 index 2 add get rgbtogray 3 -1 roll pop put dup } for 0 exch length 3 idiv getinterval } bd /FastCMYK { CIDataProc dup 0 4 2 index length 4 sub { dup 4 idiv 2 index 2 index get 3 index 3 index 1 add get 4 index 4 index 2 add get 5 index 5 index 3 add get cmyktogray 3 -1 roll pop put dup } for 0 exch length 4 idiv getinterval } bd /SlowRGB { CIDataProc pop 0 1 CIWidth 1 sub { 0 1 2 { 1 index 3 mul add CIBPC mul CIBSelect 1 index 1 index and sub exch 8 idiv CIBuffer exch get exch neg bitshift CIMask and CIConv mul exch } for 4 1 roll rgbtogray CIBufferExp 3 1 roll put } for CIBufferExp } bd /SlowCMYK { CIDataProc pop 0 1 CIWidth 1 sub { 0 1 3 { 1 index 4 mul add CIBPC mul CIBSelect 1 index 1 index and sub exch 8 idiv CIBuffer exch get exch neg bitshift CIMask and CIConv mul exch } for 5 1 roll cmyktogray CIBufferExp 3 1 roll put } for CIBufferExp } bd /ColorImage { /CINumComps exch pv /CIDataProc exch pv /CIIMatrix exch pv /CIBPC exch pv /CIHeight exch pv /CIWidth exch pv CIWidth CIHeight 8 CIIMatrix CINumComps 1 eq { /CIDataProc } { CINumComps 3 eq { CIBPC 8 eq { /FastRGB } { SetupColorImage /SlowRGB } ifelse } { CIBPC 8 eq { /FastCMYK } { SetupColorImage /SlowCMYK } ifelse } ifelse } ifelse load image } bd } EndLoad } EndLoad _ColorSep5044? { /paintimage { colorplate 0 eq { { {currentfile cyanstr readstring pop} {currentfile magentastr readstring pop} {currentfile yellowstr readstring pop} {currentfile blackstr readstring pop currentfile graystr readstring pop pop} } { {currentfile cyanstr readhexstring pop} {currentfile magentastr readhexstring pop} {currentfile yellowstr readhexstring pop} {currentfile blackstr readhexstring pop currentfile graystr readhexstring pop pop} } ifelse true 4 colorimage } { 3 dict begin /binaryOK exch def [ 1 1 5 { dup /currentfile cvx [ /cyanstr /magentastr /yellowstr /blackstr /graystr ] 3 -1 roll 1 sub get cvx binaryOK { /readstring } { /readhexstring } ifelse cvx /pop cvx 5 -1 roll colorplate dup 5 gt { pop 5 } if eq not { /pop cvx } if } for ] cvx bind end [ colorplate 6 eq { /pop cvx negativecolorplate { 0 } { 1 } ifelse } if colorplate 4 le { 1 /exch cvx /sub cvx } if colorplate 6 ne { systemdict /currenttransfer get exec aload pop } if ] cvx gsave systemdict /settransfer get exec systemdict /image get exec grestore } ifelse } bd } if end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg2.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? StartLoad { /ID { 5 counttomark 2 idiv dup 3 1 roll add dict begin { def } repeat cleartomark currentdict end begin /ImageType 1 def /ImageMatrix [ Width 0 0 Height neg 0 Height ] def /ImageMask here { not } { true } ifelse { /ImageMask false def } if ImageMask not { ColorSpace setcolorspace } if /Intent here { ri } if /Decode here { pop } { /Decode [ ImageMask { 0 1 } { currentcolorspace 0 get /Indexed eq { 0 2 BitsPerComponent exp 1 sub } { mark currentcolor counttomark dup 2 add 1 roll cleartomark { 0 1 } repeat } ifelse } ifelse ] def } ifelse [ /DataSource here { pop } { currentfile /Filter here { dup type /arraytype eq { length } { pop 1 } ifelse 1 sub 0 1 3 -1 roll { /DecodeParms here { dup type /arraytype eq { 1 index get } if dup null eq { pop } { exch } ifelse } if Filter dup type /arraytype eq { exch get } { exch pop } ifelse filter dup } for } if /DataSource exch def } ifelse currentdict % Level3? { dup /MaskedImage known { pop << /ImageType 3 /InterleaveType 2 /DataDict currentdict /MaskDict << /ImageType 1 /Width Width /Height Height /ImageMatrix ImageMatrix /BitsPerComponent 1 /Decode [0 1] currentdict /Interpolate known {/Interpolate Interpolate} if >> >> }if }if % /ImageMask here not { false } if { sfc imagemask } { image } ifelse counttomark { dup status { dup flushfile closefile } { pop } ifelse } repeat pop end } bd currentdict readonly pop } EndLoad end %%EndFile end end PDFVars /InitAll { [ PDF PDFText PDFImage ] { /initialize get exec } forall initgs 0 Tr } put PDFVars /TermAll { [ PDFImage PDFText PDF ] { /terminate get exec } forall } put PDFVars begin PDF begin /MacRomanEncoding [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /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 /asciicircum/underscore/grave/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/asciitilde/.notdef/Adieresis /Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute/agrave /acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave /ucircumflex/udieresis/dagger/degree/cent/sterling/section/bullet /paragraph/germandbls/registered/copyright/trademark/acute/dieresis /.notdef/AE/Oslash/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef /.notdef/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae /oslash/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef /.notdef/guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde /Otilde/OE/oe/endash/emdash/quotedblleft/quotedblright/quoteleft /quoteright/divide/.notdef/ydieresis/Ydieresis/fraction/currency /guilsinglleft/guilsinglright/fi/fl/daggerdbl/periodcentered /quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex /Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute /Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi /circumflex/tilde/macron/breve/dotaccent/ring/cedilla/hungarumlaut /ogonek/caron ] def /MacintoshRomanGlyphEncoding [ /.notdef /.null /nonmarkingreturn /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /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 /asciicircum /underscore /grave /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 /asciitilde /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /degree /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /notequal /AE /Oslash /infinity /plusminus /lessequal /greaterequal /yen /mu /partialdiff /summation /product /pi /integral /ordfeminine /ordmasculine /Omega /ae /oslash /questiondown /exclamdown /logicalnot /radical /florin /approxequal /Delta /guillemotleft /guillemotright /ellipsis /nonbreakingspace /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /divide /lozenge /ydieresis 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/bracketrighttp/bracketrightbt/universal/existential/asteriskmath /angleright/angleleft/theta1/omega1/phi1/epsilon/gradient/parenlefttp /parenleftbt/parenrighttp/parenrightbt/weierstrass/bracelefttp /braceleftmid/braceleftbt/braceex/bracerighttp/bracerightmid /bracerightbt/Upsilon1/arrowvertex/arrowhorizex/parenleftex /bracketleftex/parenrightex/bracketrightex/copyrightserif /registerserif/trademarkserif/copyrightsans/registersans /trademarksans/Ifraktur/Rfraktur/similar/carriagereturn/Euro ] def /reencode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall FontName /Symbol eq { /Encoding MacintoshSymbolGlyphEncoding def }{ /Encoding MacintoshRomanGlyphEncoding def } ifelse currentdict end } def /reencode-font { % new-font-name encoding-array old-font-name findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } def PDFVars /InitAll get exec %%BeginFile: cgmisc.txt %%Copyright: Copyright 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c f 277.68198 118.31802 m 279.43933 120.07538 279.43933 122.92462 277.68198 124.68198 c 275.92462 126.43935 273.07538 126.43935 271.31802 124.68198 c 269.56067 122.92462 269.56067 120.07538 271.31802 118.31802 c 273.07538 116.56065 275.92462 116.56065 277.68198 118.31802 c S 196.68198 172.31802 m 198.43935 174.07538 198.43935 176.92462 196.68198 178.68198 c 194.92462 180.43935 192.07538 180.43935 190.31802 178.68198 c 188.56065 176.92462 188.56065 174.07538 190.31802 172.31802 c 192.07538 170.56065 194.92462 170.56065 196.68198 172.31802 c f 196.68198 172.31802 m 198.43935 174.07538 198.43935 176.92462 196.68198 178.68198 c 194.92462 180.43935 192.07538 180.43935 190.31802 178.68198 c 188.56065 176.92462 188.56065 174.07538 190.31802 172.31802 c 192.07538 170.56065 194.92462 170.56065 196.68198 172.31802 c S 277.68198 172.31802 m 279.43933 174.07538 279.43933 176.92462 277.68198 178.68198 c 275.92462 180.43935 273.07538 180.43935 271.31802 178.68198 c 269.56067 176.92462 269.56067 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329.92462 180.43935 327.07538 180.43935 325.31802 178.68198 c 323.56067 176.92462 323.56067 174.07538 325.31802 172.31802 c 327.07538 170.56065 329.92462 170.56065 331.68198 172.31802 c f 331.68198 172.31802 m 333.43933 174.07538 333.43933 176.92462 331.68198 178.68198 c 329.92462 180.43935 327.07538 180.43935 325.31802 178.68198 c 323.56067 176.92462 323.56067 174.07538 325.31802 172.31802 c 327.07538 170.56065 329.92462 170.56065 331.68198 172.31802 c S 304.68198 172.31802 m 306.43933 174.07538 306.43933 176.92462 304.68198 178.68198 c 302.92462 180.43935 300.07538 180.43935 298.31802 178.68198 c 296.56067 176.92462 296.56067 174.07538 298.31802 172.31802 c 300.07538 170.56065 302.92462 170.56065 304.68198 172.31802 c f 304.68198 172.31802 m 306.43933 174.07538 306.43933 176.92462 304.68198 178.68198 c 302.92462 180.43935 300.07538 180.43935 298.31802 178.68198 c 296.56067 176.92462 296.56067 174.07538 298.31802 172.31802 c 300.07538 170.56065 302.92462 170.56065 304.68198 172.31802 c S 232.68198 172.31802 m 234.43935 174.07538 234.43935 176.92462 232.68198 178.68198 c 230.92462 180.43935 228.07538 180.43935 226.31802 178.68198 c 224.56065 176.92462 224.56065 174.07538 226.31802 172.31802 c 228.07538 170.56065 230.92462 170.56065 232.68198 172.31802 c f 232.68198 172.31802 m 234.43935 174.07538 234.43935 176.92462 232.68198 178.68198 c 230.92462 180.43935 228.07538 180.43935 226.31802 178.68198 c 224.56065 176.92462 224.56065 174.07538 226.31802 172.31802 c 228.07538 170.56065 230.92462 170.56065 232.68198 172.31802 c S 241.68198 118.31802 m 243.43935 120.07538 243.43935 122.92462 241.68198 124.68198 c 239.92462 126.43935 237.07538 126.43935 235.31802 124.68198 c 233.56065 122.92462 233.56065 120.07538 235.31802 118.31802 c 237.07538 116.56065 239.92462 116.56065 241.68198 118.31802 c f 241.68198 118.31802 m 243.43935 120.07538 243.43935 122.92462 241.68198 124.68198 c 239.92462 126.43935 237.07538 126.43935 235.31802 124.68198 c 233.56065 122.92462 233.56065 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235.92462 153.43935 233.07538 153.43935 231.31802 151.68198 c 229.56065 149.92462 229.56065 147.07538 231.31802 145.31802 c 233.07538 143.56065 235.92462 143.56065 237.68198 145.31802 c S [ 4 4 ] 0 d 391 197 m 429 197 l 429 177 l 391 177 l h 391 197 m S 394.68198 172.31802 m 396.43933 174.07538 396.43933 176.92462 394.68198 178.68198 c 392.92462 180.43935 390.07538 180.43935 388.31802 178.68198 c 386.56067 176.92462 386.56067 174.07538 388.31802 172.31802 c 390.07538 170.56065 392.92462 170.56065 394.68198 172.31802 c f [] 0 d 394.68198 172.31802 m 396.43933 174.07538 396.43933 176.92462 394.68198 178.68198 c 392.92462 180.43935 390.07538 180.43935 388.31802 178.68198 c 386.56067 176.92462 386.56067 174.07538 388.31802 172.31802 c 390.07538 170.56065 392.92462 170.56065 394.68198 172.31802 c S 430.68198 172.31802 m 432.43933 174.07538 432.43933 176.92462 430.68198 178.68198 c 428.92462 180.43935 426.07538 180.43935 424.31802 178.68198 c 422.56067 176.92462 422.56067 174.07538 424.31802 172.31802 c 426.07538 170.56065 428.92462 170.56065 430.68198 172.31802 c f 430.68198 172.31802 m 432.43933 174.07538 432.43933 176.92462 430.68198 178.68198 c 428.92462 180.43935 426.07538 180.43935 424.31802 178.68198 c 422.56067 176.92462 422.56067 174.07538 424.31802 172.31802 c 426.07538 170.56065 428.92462 170.56065 430.68198 172.31802 c S [ 4 4 ] 0 d 304.91681 124.4287 m 323.63705 140.47462 342.36295 156.52538 361.08319 172.5713 c S 304.68198 118.31802 m 306.43933 120.07538 306.43933 122.92462 304.68198 124.68198 c 302.92462 126.43935 300.07538 126.43935 298.31802 124.68198 c 296.56067 122.92462 296.56067 120.07538 298.31802 118.31802 c 300.07538 116.56065 302.92462 116.56065 304.68198 118.31802 c f [] 0 d 304.68198 118.31802 m 306.43933 120.07538 306.43933 122.92462 304.68198 124.68198 c 302.92462 126.43935 300.07538 126.43935 298.31802 124.68198 c 296.56067 122.92462 296.56067 120.07538 298.31802 118.31802 c 300.07538 116.56065 302.92462 116.56065 304.68198 118.31802 c S 367.68198 172.31802 m 369.43933 174.07538 369.43933 176.92462 367.68198 178.68198 c 365.92462 180.43935 363.07538 180.43935 361.31802 178.68198 c 359.56067 176.92462 359.56067 174.07538 361.31802 172.31802 c 363.07538 170.56065 365.92462 170.56065 367.68198 172.31802 c f 367.68198 172.31802 m 369.43933 174.07538 369.43933 176.92462 367.68198 178.68198 c 365.92462 180.43935 363.07538 180.43935 361.31802 178.68198 c 359.56067 176.92462 359.56067 174.07538 361.31802 172.31802 c 363.07538 170.56065 365.92462 170.56065 367.68198 172.31802 c S cgq 1 0 0 1 146.5 148.5 cm 0 sc 1 i BT 12 0 0 -12 -4.5 2.5 Tm /F1.1 1 Tf (!!) Tj ET cgQ cgq 1 0 0 1 324.5 104.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -2.5 2.5 Tm /F1.1 1 Tf (!) Tj ET cgQ cgq 1 0 0 1 258.5 148.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -6.5 2.5 Tm /F1.1 1 Tf (!!!) Tj ET cgQ cgq 1 0 0 1 381 192.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -4 2.5 Tm /F1.1 1 Tf (!") Tj ET cgQ cgq 1 0 0 1 358.5 154.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -4.5 2.5 Tm /F1.1 1 Tf (!!) Tj ET cgQ cgq 1 0 0 1 464 123.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -6 1.5 Tm /F1.1 1 Tf (#) Tj 10 0 0 -10 0 6.5 Tm ($) Tj ET cgQ cgq 1 0 0 1 467.5 176.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -8.5 1.5 Tm /F1.1 1 Tf (#) Tj 10 0 0 -10 -2.5 6.5 Tm (%&) Tj ET cgQ cgq 1 0 0 1 473.5 177 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -2.5 3 Tm /F2.1 1 Tf (!) Tj ET cgQ cgq 1 0 0 1 109 176.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -12 1.5 Tm /F1.1 1 Tf (#) Tj 10 0 0 -10 -6 6.5 Tm (&'\() Tj ET cgQ cgq 1 0 0 1 105 121.5 cm /DeviceGray cs 0 sc 1 i [] 0 d /DeviceGray CS 0 SC BT 12 0 0 -12 -6 1.5 Tm /F1.1 1 Tf (#) Tj 10 0 0 -10 0 6.5 Tm (&) Tj ET cgQ /DeviceGray cs 1 sc 394.68198 118.31802 m 396.43933 120.07538 396.43933 122.92462 394.68198 124.68198 c 392.92462 126.43935 390.07538 126.43935 388.31802 124.68198 c 386.56067 122.92462 386.56067 120.07538 388.31802 118.31802 c 390.07538 116.56065 392.92462 116.56065 394.68198 118.31802 c f 0.60000002 i [] 0 d /DeviceGray CS 0 SC 394.68198 118.31802 m 396.43933 120.07538 396.43933 122.92462 394.68198 124.68198 c 392.92462 126.43935 390.07538 126.43935 388.31802 124.68198 c 386.56067 122.92462 386.56067 120.07538 388.31802 118.31802 c 390.07538 116.56065 392.92462 116.56065 394.68198 118.31802 c S 430.68198 118.31802 m 432.43933 120.07538 432.43933 122.92462 430.68198 124.68198 c 428.92462 126.43935 426.07538 126.43935 424.31802 124.68198 c 422.56067 122.92462 422.56067 120.07538 424.31802 118.31802 c 426.07538 116.56065 428.92462 116.56065 430.68198 118.31802 c f 430.68198 118.31802 m 432.43933 120.07538 432.43933 122.92462 430.68198 124.68198 c 428.92462 126.43935 426.07538 126.43935 424.31802 124.68198 c 422.56067 122.92462 422.56067 120.07538 424.31802 118.31802 c 426.07538 116.56065 428.92462 116.56065 430.68198 118.31802 c S cgQ cgQ cgQ PDFVars/TermAll get exec end end userdict /pgsave get restore showpage %%Trailer %%EOF ---------------0312010856408 Content-Type: application/postscript; name="f2.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="f2.eps" %!PS-Adobe-3.0 EPSF-3.0 %%HiResBoundingBox: 0.000000 0.000000 435.000000 152.000000 %%Title: (Unknown) %%Creator: (Unknown) %%CreationDate: (Unknown) %%For: (Unknown) %%DocumentData: Clean7Bit %%Pages: 1 %%BoundingBox: 0 0 435 152 %%EndComments %%BeginProlog userdict /PDF 95 dict put %%BeginFile: pdfvars.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. userdict /PDFVars 90 dict put PDFVars begin /_save 0 def /_cshow 0 def /InitAll 0 def /TermAll 0 def /_lp /none def /_doClip 0 def /sfc 0 def /_sfcs 0 def /_sfc 0 def /ssc 0 def /_sscs 0 def /_ssc 0 def /_fcs 0 def /_scs 0 def /_fp 0 def /_sp 0 def /AGM_MAX_CS_COMPONENTS 10 def /_fillColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_strokeColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_fc null def /_sc null def /GetCompsDict null def /_inT false def /_tr -1 def /_rise 0 def /_ax 0 def /_cx 0 def /_ld 0 def /_tm matrix def /_ctm matrix def /_mtx matrix def /_hy (-) def /_fScl 0 def /_hs 1 def /_pdfEncodings 2 array def /_baselineadj 0 def /_fTzero false def /_Tj 0 def /_italMtx[1 0 .212557 1 0 0]def /_italMtx_WMode1 [1 -.212557 0 1 0 0]def /_italMtxType0[1 0 .1062785 1 0 0]def /_italMtx_WMode1Type0 [1 -.1062785 0 1 0 0]def /_basefont 0 def /_basefonto 0 def /_pdf_oldCIDInit null def /_categories 10 dict def /_sa? true def /_op? false def /_ColorSep5044? false def /_tmpcolr? [] def /_tmpop? {} def end %%EndFile PDFVars begin PDF begin %%BeginFile: pdfutil.prc %%Copyright: Copyright 1993 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. /bd {bind def} bind def /ld {load def} bd /dd { PDFVars 3 1 roll put } bd /xdd { exch dd } bd /Level2? systemdict /languagelevel known { systemdict /languagelevel get 2 ge }{ false } ifelse def /Level3? systemdict /languagelevel known { systemdict /languagelevel get 3 eq }{ false } ifelse def /here { dup currentdict exch known { currentdict exch get true }{ pop false } ifelse } bd /isdefined? { where { pop true } { false } ifelse } bd /StartLoad { dup dup not { /_save save dd } if } bd /EndLoad { if not { _save restore } if } bd /npop { { pop } repeat } bd %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdf.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. /initialize { _ColorSep5044? {sep_ops begin 50 dict begin} if newpath } bd /terminate { _ColorSep5044? {end end} if } bd Level2? 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StartLoad { /defineRes/defineresource ld /findRes/findresource ld currentglobal true systemdict /setglobal get exec [ /Function /ExtGState /Form /Shading ] { /Generic /Category findresource dup length dict copy /Category defineresource pop } forall systemdict /setglobal get exec } EndLoad Level2? not StartLoad { /AlmostFull? { dup maxlength exch length sub 2 le } bind def /Expand { 1 index maxlength mul cvi dict dup begin exch { def } forall end } bind def /xput { 3 2 roll dup 3 index known not { dup AlmostFull? { 1.5 Expand } if } if dup 4 2 roll put } bind def /defineRes { _categories 1 index known not { /_categories _categories 2 index 10 dict xput store } if _categories exch 2 copy get 5 -1 roll 4 index xput put } bind def /findRes { _categories exch get exch get } bind def } EndLoad /cs { dup where { pop load } if dup /_fcs xdd GetComps _fillColors exch get /_fc xdd /_fp null dd } bd /CS { dup where { pop load } if dup /_scs xdd GetComps _strokeColors exch get /_sc xdd /_sp null dd } bd /GetCompsDict 16 dict begin /DeviceGray { pop 1 } bd /DefaultGray { pop 1 } bd /DeviceRGB { pop 3 } bd /DefaultRGB { pop 3 } bd /DeviceCMYK { pop 4 } bd /DefaultCMYK { pop 4 } bd /CalGray { pop 1 } bd /CalRGB { pop 3 } def /CIEBasedA { pop 1 } bd /CIEBasedABC { pop 3 } bd /CIEBasedDEFG { pop 4 } bd /Lab { pop 3 } bd /DeviceN { 1 get length } bd /Separation { pop 1 } bd /Indexed { pop 1 } bd /Pattern { pop 0 } bd currentdict end dd /GetComps { GetCompsDict 1 index dup type /arraytype eq { 0 get } if get exec } bd Level2? not StartLoad { /ri/pop ld /makePat/pop ld } EndLoad Level2? StartLoad { /ri { /findcolorrendering isdefined? { mark exch findcolorrendering counttomark 2 eq { type /booleantype eq { dup type /nametype eq { dup /ColorRendering resourcestatus { pop pop dup /DefaultColorRendering ne { /ColorRendering findresource setcolorrendering } if } if } if } if } if cleartomark } { pop } ifelse } bd /makePat { 1 index /PatternType get 2 eq languagelevel 3 lt and { 7 dict dup begin /PatternType 1 def /PaintType 1 def /TilingType 1 def /BBox [10 10] /XStep 10 /YStep 10 /PaintProc {pop .5 setgray 0 0 10 10 rectfill} bind end matrix 4 2 roll pop } if makepattern } bd } EndLoad Level2? not _ColorSep5044? or StartLoad { /L1setcolor { aload length dup 0 eq { pop .5 setgray } { dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } ifelse } bind dd /_sfcs { } dd /_sscs { } dd } EndLoad Level2? not _ColorSep5044? not and StartLoad { /_sfc { _fc L1setcolor } dd /_ssc { _sc L1setcolor } dd } EndLoad Level2? _ColorSep5044? not and StartLoad { /_sfcs { _fcs setcolorspace } bind dd /_sscs { _scs setcolorspace } bind dd /_sfc { _fc aload pop _fp null eq { setcolor } { _fp setpattern } ifelse } bind dd /_ssc { _sc aload pop _sp null eq { setcolor } { _sp setpattern } ifelse } bind dd } EndLoad /sc { _fc astore pop ilp } bd /SC { _sc astore pop ilp } bd /scn { dup type /dicttype eq { dup /_fp xdd dup /PatternType get 1 eq { /PaintType get 1 eq { /_fc _fillColors 0 get dd ilp } { /_fc _fillColors _fcs 1 get GetComps get dd sc } ifelse } { pop /_fc _fillColors 0 get dd ilp } ifelse } { sc } ifelse } bd /SCN { dup type /dicttype eq { dup /_sp xdd dup /PatternType get 1 eq { /PaintType get 1 eq { /_sc _strokeColors 0 get dd ilp } { /_sc _strokeColors _scs 1 get GetComps get dd SC } ifelse } { pop /_sc _strokeColors 0 get dd ilp } ifelse } { SC } ifelse } bd /g { /DefaultGray cs sc } bd /rg { /DefaultRGB cs sc } bd /k { /DefaultCMYK cs sc } bd /G { /DefaultGray CS SC } bd /RG { /DefaultRGB CS SC } bd /K { /DefaultCMYK CS SC } bd /cm { _mtx astore concat } bd /re { 4 2 roll m 1 index 0 rlineto 0 exch rlineto neg 0 rlineto h } bd /rf /rectfill where {pop {sfc rectfill}} {{re f}} ifelse bd /RC/rectclip ld /EF/execform ld /PS { cvx exec } bd /initgs { /DefaultGray where { pop } { /DefaultGray /DeviceGray dd } ifelse /DefaultRGB where { pop } { /DefaultRGB /DeviceRGB dd } ifelse /DefaultCMYK where { pop } { /DefaultCMYK /DeviceCMYK dd } ifelse 0 g 0 G [] 0 d 0 j 0 J 10 M 1 w true setSA } bd 21 dict dup begin /CosineDot { 180 mul cos exch 180 mul cos add 2 div } bd /Cross { abs exch abs 2 copy gt { exch } if pop neg } bd /Diamond { abs exch abs 2 copy add .75 le { dup mul exch dup mul add 1 exch sub }{ 2 copy add 1.23 le { .85 mul add 1 exch sub }{ 1 sub dup mul exch 1 sub dup mul add 1 sub } ifelse } ifelse } bd /Double { exch 2 div exch 2 { 360 mul sin 2 div exch } repeat add } bd /DoubleDot { 2 { 360 mul sin 2 div exch } repeat add } bd /Ellipse { abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt { pop dup mul exch .75 div dup mul add 4 div 1 exch sub }{ dup 1 gt { pop 1 exch sub dup mul exch 1 exch sub .75 div dup mul add 4 div 1 sub }{ .5 exch sub exch pop exch pop } ifelse } ifelse } bd /EllipseA { dup mul .9 mul exch dup mul add 1 exch sub } bd /EllipseB { dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub } bd /EllipseC { dup mul .9 mul exch dup mul add 1 exch sub } bd /InvertedDouble { exch 2 div exch 2 { 360 mul sin 2 div exch } repeat add neg } bd /InvertedDoubleDot { 2 { 360 mul sin 2 div exch } repeat add neg } bd /InvertedEllipseA { dup mul .9 mul exch dup mul add 1 sub } bd /InvertedSimpleDot { dup mul exch dup mul add 1 sub } bd /Line { exch pop abs neg } bd /LineX { pop } bd /LineY { exch pop } bd /Rhomboid { abs exch abs 0.9 mul add 2 div } bd /Round { abs exch abs 2 copy add 1 le { dup mul exch dup mul add 1 exch sub }{ 1 sub dup mul exch 1 sub dup mul add 1 sub } ifelse } bd /SimpleDot { dup mul exch dup mul add 1 exch sub } bd /Square { abs exch abs 2 copy lt { exch } if pop neg } bd end { /Function defineRes pop } forall /Identity {} /Function defineRes pop _ColorSep5044? StartLoad { /_defaulttransferfunc currenttransfer def /currentcolortransfer where { pop /_defaultcolortransferfuncs [ currentcolortransfer ] def } if /concattransferfuncs { [ 3 1 roll /exec load exch /exec load ] cvx } bd /concatandsettransfer { /_defaulttransferfunc load concattransferfuncs settransfer } bd /concatandsetcolortransfer { colorplate 0 eq { _defaultcolortransferfuncs aload pop 8 -1 roll 5 -1 roll concattransferfuncs 7 1 roll 6 -1 roll 4 -1 roll concattransferfuncs 5 1 roll 4 -1 roll 3 -1 roll concattransferfuncs 3 1 roll concattransferfuncs setcolortransfer } if colorplate 1 ge colorplate 4 le and { 4 colorplate sub index 4 { exch pop } repeat concatandsettransfer } if colorplate 5 ge { 0 index 4 { exch pop } repeat concatandsettransfer } if } bd /tn5044sethalftone { begin HalftoneType 5 eq { [/Default /Cyan /Magenta /Yellow /Black /Default /Default /Default] colorplate get here not { /Default here not { currentdict } if } if }{ currentdict } ifelse end begin /TransferFunction here { concatandsettransfer currentdict dup length dict begin { 1 index /TransferFunction ne { def } { pop pop } ifelse } forall currentdict end }{ currentdict } ifelse end sethalftone } bd } EndLoad Level2? Level3? not and StartLoad { /setsmoothness { pop } bd } EndLoad Level2? StartLoad { /gs { begin /SA here { setstrokeadjust } if /OP here { setoverprint } if /BG here { setblackgeneration } if /UCR here { setundercolorremoval } if /SM here { setsmoothness } if /FL here { i } if /RI here { ri } if /TR here { _ColorSep5044? { dup xcheck { concatandsettransfer } { aload pop concatandsetcolortransfer } ifelse }{ dup xcheck { settransfer } { aload pop setcolortransfer } ifelse } ifelse } if /sethalftonephase isdefined? { /HTP here { sethalftonephase } if } if /HT here { _ColorSep5044? { tn5044sethalftone } { sethalftone } ifelse } if currentdict gsDI end } bd /_defaulthalftone currenthalftone def /_defaultblackgeneration currentblackgeneration def /_defaultundercolorremoval currentundercolorremoval def /_defaultcolortransfer [currentcolortransfer] def /_defaulttransfer currenttransfer def } EndLoad Level2? not StartLoad { /gs { begin /SA here { /_sa? xdd } if /OP here { dup /_op? xdd /setoverprint where { pop setoverprint }{ pop } ifelse } if /TR here { _ColorSep5044? { dup xcheck { concatandsettransfer }{ aload pop concatandsetcolortransfer } ifelse }{ dup xcheck { settransfer }{ aload pop setcolortransfer } ifelse } ifelse } if /HT here { _ColorSep5044? { tn5044sethalftone }{ sethalftone } ifelse } if /FL here { i } if currentdict gsDI end } bd currentscreen dup type /dicttype eq { /_defaulthalftone exch def pop pop }{ 5 dict begin 1 [ /HalftoneType /SpotFunction /Angle /Frequency ] { exch def } forall currentdict end /_defaulthalftone exch def } ifelse } EndLoad /int { dup 2 index sub 3 index 5 index sub div 6 -2 roll sub mul exch pop add exch pop } bd /limit { dup 2 index le { exch } if pop dup 2 index ge { exch } if pop } bd /domainClip { Domain aload pop 3 2 roll limit } bd /applyInterpFunc { 0 1 DimOut 1 sub { dup C0 exch get exch dup C1 exch get exch 3 1 roll 1 index sub 3 index N exp mul add exch currentdict /Range_lo known { dup Range_lo exch get exch Range_hi exch get 3 2 roll limit }{ pop } ifelse exch } for pop } bd /encodeInput { NumParts 1 sub 0 1 2 index { dup Bounds exch get 2 index gt { exit } { dup 3 index eq { exit } { pop } ifelse } ifelse } for 3 2 roll pop dup Bounds exch get exch dup 1 add Bounds exch get exch 2 mul dup Encode exch get exch 1 add Encode exch get int } bd /rangeClip { exch dup Range_lo exch get exch Range_hi exch get 3 2 roll limit } bd /applyStitchFunc { Functions exch get exec currentdict /Range_lo known { 0 1 DimOut 1 sub { DimOut 1 add -1 roll rangeClip } for } if } bind def _ColorSep5044? StartLoad { /_sfc { _fp null eq { _fcs type /arraytype eq { _fcs 0 get /Separation eq { _fcs 1 get /All eq { _fc aload pop 1 exch sub /setseparationgray where pop begin setseparationgray end }{ 1 _fcs 3 get exec _fcs 1 get /findcmykcustomcolor where pop begin findcmykcustomcolor end _fc aload pop /setcustomcolor where pop begin setcustomcolor end } ifelse }{ _fc L1setcolor } ifelse }{ _fc L1setcolor } ifelse }{ _fc L1setcolor } ifelse } bind dd /_ssc { _sp null eq { _scs type /arraytype eq { _scs 0 get /Separation eq { _scs 1 get /All eq { _sc aload pop 1 exch sub /setseparationgray where pop begin setseparationgray end }{ 1 _scs 3 get exec _scs 1 get /findcmykcustomcolor where pop begin findcmykcustomcolor end _sc aload pop /setcustomcolor where pop begin setcustomcolor end } ifelse }{ _sc L1setcolor } ifelse }{ _sc L1setcolor } ifelse }{ _sc L1setcolor } ifelse } bind dd } EndLoad %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdftext.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. PDF /PDFText 75 dict put PDFText begin /initialize { PDFText begin } bd /terminate { end } bd /pdf_has_composefont? systemdict /composefont known def /CopyFont { { 1 index /FID ne 2 index /UniqueID ne and { def }{ pop pop } ifelse } forall } bd /Type0CopyFont { exch dup length dict begin CopyFont [ exch FDepVector { dup /FontType get 0 eq { 1 index Type0CopyFont /_pdfType0 exch definefont }{ /_pdfBaseFont exch 2 index exec } ifelse exch } forall pop ] /FDepVector exch def currentdict end } bd /cHexEncoding [ /c00/c01/c02/c03/c04/c05/c06/c07/c08/c09/c0A/c0B/c0C/c0D/c0E/c0F /c10/c11/c12/c13/c14/c15/c16/c17/c18/c19/c1A/c1B/c1C/c1D/c1E/c1F /c20/c21/c22/c23/c24/c25/c26/c27/c28/c29/c2A/c2B/c2C/c2D/c2E/c2F /c30/c31/c32/c33/c34/c35/c36/c37/c38/c39/c3A/c3B/c3C/c3D/c3E/c3F /c40/c41/c42/c43/c44/c45/c46/c47/c48/c49/c4A/c4B/c4C/c4D/c4E/c4F /c50/c51/c52/c53/c54/c55/c56/c57/c58/c59/c5A/c5B/c5C/c5D/c5E/c5F /c60/c61/c62/c63/c64/c65/c66/c67/c68/c69/c6A/c6B/c6C/c6D/c6E/c6F /c70/c71/c72/c73/c74/c75/c76/c77/c78/c79/c7A/c7B/c7C/c7D/c7E/c7F /c80/c81/c82/c83/c84/c85/c86/c87/c88/c89/c8A/c8B/c8C/c8D/c8E/c8F /c90/c91/c92/c93/c94/c95/c96/c97/c98/c99/c9A/c9B/c9C/c9D/c9E/c9F /cA0/cA1/cA2/cA3/cA4/cA5/cA6/cA7/cA8/cA9/cAA/cAB/cAC/cAD/cAE/cAF /cB0/cB1/cB2/cB3/cB4/cB5/cB6/cB7/cB8/cB9/cBA/cBB/cBC/cBD/cBE/cBF /cC0/cC1/cC2/cC3/cC4/cC5/cC6/cC7/cC8/cC9/cCA/cCB/cCC/cCD/cCE/cCF /cD0/cD1/cD2/cD3/cD4/cD5/cD6/cD7/cD8/cD9/cDA/cDB/cDC/cDD/cDE/cDF /cE0/cE1/cE2/cE3/cE4/cE5/cE6/cE7/cE8/cE9/cEA/cEB/cEC/cED/cEE/cEF /cF0/cF1/cF2/cF3/cF4/cF5/cF6/cF7/cF8/cF9/cFA/cFB/cFC/cFD/cFE/cFF ] def /modEnc { /_enc xdd /_icode 0 dd counttomark 1 sub -1 0 { index dup type /nametype eq { _enc _icode 3 -1 roll put _icode 1 add } if /_icode xdd } for cleartomark _enc } bd /trEnc { /_enc xdd 255 -1 0 { exch dup -1 eq { pop /.notdef } { Encoding exch get } ifelse _enc 3 1 roll put } for pop _enc } bd /TE { /_i xdd StandardEncoding 256 array copy modEnc _pdfEncodings exch _i exch put } bd /TZ { /_usePDFEncoding xdd findfont dup length 2 add dict begin { 1 index /FID ne { def } { pop pop } ifelse } forall /FontName exch def _usePDFEncoding 0 ge { /Encoding _pdfEncodings _usePDFEncoding get def pop }{ _usePDFEncoding -1 eq { counttomark 0 eq { pop }{ Encoding 256 array copy modEnc /Encoding exch def } ifelse }{ 256 array trEnc /Encoding exch def } ifelse } ifelse FontName currentdict end definefont pop } bd /Level2? systemdict /languagelevel known { systemdict /languagelevel get 2 ge }{ false } ifelse def Level2? { /_pdfFontStatus { currentglobal exch /Font resourcestatus {pop pop true} {false} ifelse exch setglobal } bd }{ /_pdfFontStatusString 50 string def _pdfFontStatusString 0 (fonts/) putinterval /_pdfFontStatus { FontDirectory 1 index known { pop true } { _pdfFontStatusString 6 42 getinterval cvs length 6 add _pdfFontStatusString exch 0 exch getinterval { status } stopped {pop false} { { pop pop pop pop true} { false } ifelse } ifelse } ifelse } bd } ifelse Level2? { /_pdfCIDFontStatus { /CIDFont /Category resourcestatus { pop pop /CIDFont resourcestatus {pop pop true} {false} ifelse } { pop false } ifelse } bd } if /_pdfString100 100 string def /_pdfComposeFontName { dup length 1 eq { 0 get 1 index type /nametype eq { _pdfString100 cvs length dup dup _pdfString100 exch (-) putinterval _pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval 2 index exch cvs length add 1 add _pdfString100 exch 0 exch getinterval exch pop true }{ pop pop false } ifelse }{ false } ifelse } bd pdf_has_composefont? { /_pdfComposeFont { 1 index /CMap resourcestatus { pop pop true }{ false } ifelse 1 index true exch { _pdfCIDFontStatus not {pop false exit} if } forall and { 3 -1 roll pop composefont true }{ 4 -1 roll pop _pdfComposeFontName { dup _pdfFontStatus { findfont definefont true }{ pop dup _pdfFontStatus { findfont true }{ pop false } ifelse } ifelse }{ dup _pdfFontStatus { findfont true } { pop false } ifelse } ifelse } ifelse } bd }{ /_pdfComposeFont { 4 -1 roll pop _pdfComposeFontName not { dup } if 2 copy _pdfFontStatus {pop findfont exch pop true} { eq {pop false} { dup _pdfFontStatus {findfont true} {pop false} ifelse } ifelse } ifelse } bd } ifelse /_pdfStyleDicts 4 dict dup begin /Adobe-Japan1 4 dict dup begin Level2? { /Serif /HeiseiMin-W3-83pv-RKSJ-H _pdfFontStatus {/HeiseiMin-W3} { /HeiseiMin-W3 _pdfCIDFontStatus {/HeiseiMin-W3} {/Ryumin-Light} ifelse } ifelse def /SansSerif /HeiseiKakuGo-W5-83pv-RKSJ-H _pdfFontStatus {/HeiseiKakuGo-W5} { /HeiseiKakuGo-W5 _pdfCIDFontStatus {/HeiseiKakuGo-W5} {/GothicBBB-Medium} ifelse } ifelse def /HeiseiMaruGo-W4-83pv-RKSJ-H _pdfFontStatus {/HeiseiMaruGo-W4} { /HeiseiMaruGo-W4 _pdfCIDFontStatus {/HeiseiMaruGo-W4} { /Jun101-Light-RKSJ-H _pdfFontStatus { /Jun101-Light } { SansSerif } ifelse } ifelse } ifelse /RoundSansSerif exch def /Default Serif def } { /Serif /Ryumin-Light def /SansSerif /GothicBBB-Medium def { (fonts/Jun101-Light-83pv-RKSJ-H) status }stopped {pop}{ { pop pop pop pop /Jun101-Light } { SansSerif } ifelse /RoundSansSerif exch def }ifelse /Default Serif def } ifelse end def /Adobe-Korea1 4 dict dup begin /Serif /HYSMyeongJo-Medium def /SansSerif /HYGoThic-Medium def /RoundSansSerif SansSerif def /Default Serif def end def /Adobe-GB1 4 dict dup begin /Serif /STSong def /SansSerif /STHeiti def /RoundSansSerif SansSerif def /Default Serif def end def /Adobe-CNS1 4 dict dup begin /Serif /MKai-Medium def /SansSerif /MHei-Medium def /RoundSansSerif SansSerif def /Default Serif def end def end def /_pdf_Adobe-Japan1-2 (Adobe-Japan1-2) def /_pdfConcatNames { exch _pdfString100 cvs length dup dup _pdfString100 exch (-) putinterval _pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval 3 -1 roll exch cvs length add 1 add _pdfString100 exch 0 exch getinterval } bind def /_pdfSubSetFontByStyleDict 4 dict dup begin _pdfStyleDicts /Adobe-Japan1 get { _pdf_Adobe-Japan1-2 _pdfConcatNames dup _pdfFontStatus { def }{ pop pop } ifelse } forall end def /TZzero { /_fyAdj xdd /_wmode xdd /_styleArr xdd /_regOrdering xdd 4 copy _pdfComposeFont {exch pop exch pop exch pop} { [ 0 1 _styleArr length 1 sub { _styleArr exch get _pdfStyleDicts _regOrdering 2 copy known { get exch 2 copy known not { pop /Default } if get } { pop pop /Unknown } ifelse } for ] exch pop 3 index 3 index 4 2 roll _pdfComposeFont { exch pop }{ findfont } ifelse } ifelse dup /FontType get 3 eq _wmode 1 eq and { _pdfVerticalRomanT3Font dup length 10 add dict copy begin /_basefont exch def /Encoding _basefont /Encoding get def }{ dup length 3 add dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall } ifelse /WMode _wmode def /BaseLineAdj _fyAdj def FontType 0 ne { /Encoding Encoding dup length array copy dup 16#5c /yen put def /_fauxfont true def } if currentdict end definefont pop } bd /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 dd { 1 index eq {/_cnt _cnt 1 add dd} if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } bd /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave 32 eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /jsfTzero { { 2 npop (0) exch 2 copy 0 exch put exch show 32 eq { 4 index 4 index rmoveto } if 2 copy rmoveto } exch cshow 5 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put 32 eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } bd /trj { _cx 0 fWModeProc 32 _ax 0 fWModeProc 6 5 roll } bd /pjsf { trj sfc fawidthshowProc } bd /pjss { trj _ctm ssc jss } bd /pjsc { trj jsp } bd /_Tjdef [ /pjsf load /pjss load { dup currentpoint 3 2 roll pjsf newpath moveto pjss } bind { trj swj rmoveto } bind { dup currentpoint 4 2 roll gsave pjsf grestore 3 1 roll moveto pjsc } bind { dup currentpoint 4 2 roll currentpoint gsave newpath moveto pjss grestore 3 1 roll moveto pjsc } bind { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll pjsf newpath moveto pjss grestore 3 1 roll moveto pjsc } bind /pjsc load ] def /BT { /_inT true dd _ctm currentmatrix pop 1 0 0 1 0 0 _tm astore pop %DMG 0 _rise _baselineadj add translate _hs 1 scale 0 0 moveto } bd /ET { /_inT false dd _tr 3 gt {clip} if _ctm setmatrix newpath } bd /Tr { _inT { _tr 3 le {currentpoint newpath moveto} if } if dup /_tr xdd _Tjdef exch get /_Tj xdd } bd /Tj { userdict /$$copystring 2 index put _Tj } bd /iTm { _ctm setmatrix _tm concat 0 _rise _baselineadj add translate _hs 1 scale } bd /Tm { _tm astore pop iTm 0 0 moveto } bd /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } bd /TD { dup /_ld xdd Td } bd /_nullProc {} bd /Tf { dup 1000 div /_fScl xdd Level2? { selectfont }{ exch findfont exch scalefont setfont } ifelse currentfont dup /_nullProc exch /WMode known { 1 index /WMode get 1 eq { pop /exch } if } if load /fWModeProc xdd dup /FontType get 0 eq dup _cx 0 ne and { /jsfTzero }{ /awidthshow } ifelse load /fawidthshowProc xdd /_fTzero xdd dup /BaseLineAdj known { dup /BaseLineAdj get _fScl mul }{ 0 } ifelse /_baselineadj xdd currentpoint iTm moveto pop } bd /TL { neg /_ld xdd } bd /Tw { /_cx xdd _cx 0 ne _fTzero and { /jsfTzero } { /awidthshow } ifelse load /fawidthshowProc xdd } bd /Tc { /_ax xdd } bd /Ts { /_rise xdd currentpoint iTm moveto } bd /Tz { 100 div /_hs xdd iTm } bd /Tk { exch pop _fScl mul neg 0 fWModeProc rmoveto } bd /T* { 0 _ld Td } bd /' { T* Tj } bd /" { exch Tc exch Tw ' } bd /TJ { { dup type /stringtype eq { Tj }{ 0 exch Tk } ifelse } forall } bd /T- { _hy Tj } bd /d0/setcharwidth ld /d1 { setcachedevice /sfc{}dd /ssc{}dd } bd /nND {{/.notdef} repeat} bd /T3Defs { /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } def /BuildGlyph { exch begin GlyphProcs exch get exec end } def } bd /_pdfBoldRomanWidthProc { stringwidth 1 index 0 ne { exch .03 add exch }if setcharwidth } bd /_pdfType0WidthProc { dup stringwidth 0 0 moveto 2 index true charpath pathbbox 0 -1 7 index 2 div .88 setcachedevice2 pop } bd /_pdfBoldBaseFont 11 dict begin /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding cHexEncoding def /_setwidthProc /_pdfBoldRomanWidthProc load def /_bcstr1 1 string def /BuildChar { exch begin _basefont setfont _bcstr1 dup 0 4 -1 roll put dup _setwidthProc 0 0 moveto dup show _basefonto setfont 0 0 moveto show end }bd currentdict end def /_pdfVerticalRomanT3Font 10 dict begin /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /_bcstr1 1 string def /BuildChar { exch begin _basefont setfont _bcstr1 dup 0 4 -1 roll put dup _pdfType0WidthProc 0 0 moveto show end } bd currentdict end def /MakeBoldFont { dup dup length 3 add dict begin CopyFont /PaintType 2 def /StrokeWidth .03 0 FontMatrix idtransform pop def /dummybold currentdict end definefont _pdfBoldBaseFont dup length 3 add dict copy begin /_basefont exch def /_basefonto exch def currentdict end definefont } bd /MakeBold { exch 1 index findfont dup /FontType get 0 eq { _pdfBoldBaseFont /_setwidthProc /_pdfType0WidthProc load put {MakeBoldFont} Type0CopyFont definefont }{ dup /_fauxfont known not { _pdfBoldBaseFont /_setwidthProc /_pdfBoldRomanWidthProc load put MakeBoldFont }{ 2 index 2 index eq { exch pop }{ dup length dict begin CopyFont currentdict end definefont } ifelse } ifelse } ifelse pop pop } bd /MakeItalic { findfont dup /FontType get 0 eq Level2? not and { dup /FMapType get 6 eq } { false } ifelse { dup /WMode 2 copy known { get 1 eq { _italMtx_WMode1Type0 } { _italMtxType0 } ifelse } { pop pop _italMtxType0 } ifelse } { dup /WMode 2 copy known { get 1 eq { _italMtx_WMode1 } { _italMtx } ifelse } { pop pop _italMtx } ifelse } ifelse makefont Level2? not { dup length dict begin CopyFont currentdict end } if definefont pop }bd /MakeBoldItalic { /dummybold exch MakeBold /dummybold MakeItalic }bd currentdict readonly pop end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimage.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage 38 dict put PDF /PDFIVars 20 dict put PDF /PDFImage get begin /initialize { PDFImage begin } bd /terminate { end } bd /nulldict 0 dict def /gv { PDFIVars exch get } bd /pv { PDFIVars 3 1 roll put } bd /BI { save /savelevel exch pv mark } bd /EI { /savelevel gv restore } bd end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg1b.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? not StartLoad { PDFIVars /PDFImages 4 dict put /InstallImage { PDFIVars /PDFImages get 3 1 roll put } bd /ColorComps? { dup type /arraytype eq { 0 get } if /PDFImages gv exch get 0 get } bd /ColorProc? { dup type /arraytype eq { 0 get } if /PDFImages gv exch get 1 get } bd /ImageFilter { /DataSource load } bd /ID { 5 counttomark 2 idiv dup 3 1 roll add dict begin { def } repeat cleartomark currentdict end PDFIVars begin begin /ImageMatrix [ Width 0 0 Height neg 0 Height ] def /ColorSpace here { pop } { /ColorSpace /DeviceGray def } ifelse ColorSpace ColorProc? exec end end } bd /DeviceGray [ 1 { /ImageMask here not { false } if { sfc Width Height /Decode here { 0 get 1 eq } { false } ifelse ImageMatrix ImageFilter imagemask } { Width Height BitsPerComponent ImageMatrix ImageFilter image } ifelse } bind ] InstallImage } EndLoad end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg1c.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? not StartLoad { /DeviceRGB [ 3 { Width Height BitsPerComponent ImageMatrix ImageFilter 3 ColorImage } bind ] InstallImage /DeviceCMYK [ 4 { Width Height BitsPerComponent ImageMatrix ImageFilter 4 ColorImage } bind ] InstallImage /ColorImage? /colorimage where { pop true } { false } ifelse def ColorImage? StartLoad { /ColorImage { false exch colorimage } bd } EndLoad ColorImage? not StartLoad { /SetupColorImage { /CIConv 255 2 BitsPerComponent exp 1 sub div pv /CIMask 0 not BitsPerComponent bitshift not pv /CIBSelect BitsPerComponent 1 sub not 7 and pv /CIBufferExp CIWidth string pv } bd /rgbtogray { 0.11 mul exch 0.59 mul add exch 0.3 mul add round cvi } bd /cmyktogray { exch 0.11 mul add exch 0.59 mul add exch 0.3 mul add round cvi dup 255 gt { pop 255 } if 255 exch sub } bd /FastRGB { CIDataProc dup 0 3 2 index length 3 sub { dup 3 idiv 2 index 2 index get 3 index 3 index 1 add get 4 index 4 index 2 add get rgbtogray 3 -1 roll pop put dup } for 0 exch length 3 idiv getinterval } bd /FastCMYK { CIDataProc dup 0 4 2 index length 4 sub { dup 4 idiv 2 index 2 index get 3 index 3 index 1 add get 4 index 4 index 2 add get 5 index 5 index 3 add get cmyktogray 3 -1 roll pop put dup } for 0 exch length 4 idiv getinterval } bd /SlowRGB { CIDataProc pop 0 1 CIWidth 1 sub { 0 1 2 { 1 index 3 mul add CIBPC mul CIBSelect 1 index 1 index and sub exch 8 idiv CIBuffer exch get exch neg bitshift CIMask and CIConv mul exch } for 4 1 roll rgbtogray CIBufferExp 3 1 roll put } for CIBufferExp } bd /SlowCMYK { CIDataProc pop 0 1 CIWidth 1 sub { 0 1 3 { 1 index 4 mul add CIBPC mul CIBSelect 1 index 1 index and sub exch 8 idiv CIBuffer exch get exch neg bitshift CIMask and CIConv mul exch } for 5 1 roll cmyktogray CIBufferExp 3 1 roll put } for CIBufferExp } bd /ColorImage { /CINumComps exch pv /CIDataProc exch pv /CIIMatrix exch pv /CIBPC exch pv /CIHeight exch pv /CIWidth exch pv CIWidth CIHeight 8 CIIMatrix CINumComps 1 eq { /CIDataProc } { CINumComps 3 eq { CIBPC 8 eq { /FastRGB } { SetupColorImage /SlowRGB } ifelse } { CIBPC 8 eq { /FastCMYK } { SetupColorImage /SlowCMYK } ifelse } ifelse } ifelse load image } bd } EndLoad } EndLoad _ColorSep5044? { /paintimage { colorplate 0 eq { { {currentfile cyanstr readstring pop} {currentfile magentastr readstring pop} {currentfile yellowstr readstring pop} {currentfile blackstr readstring pop currentfile graystr readstring pop pop} } { {currentfile cyanstr readhexstring pop} {currentfile magentastr readhexstring pop} {currentfile yellowstr readhexstring pop} {currentfile blackstr readhexstring pop currentfile graystr readhexstring pop pop} } ifelse true 4 colorimage } { 3 dict begin /binaryOK exch def [ 1 1 5 { dup /currentfile cvx [ /cyanstr /magentastr /yellowstr /blackstr /graystr ] 3 -1 roll 1 sub get cvx binaryOK { /readstring } { /readhexstring } ifelse cvx /pop cvx 5 -1 roll colorplate dup 5 gt { pop 5 } if eq not { /pop cvx } if } for ] cvx bind end [ colorplate 6 eq { /pop cvx negativecolorplate { 0 } { 1 } ifelse } if colorplate 4 le { 1 /exch cvx /sub cvx } if colorplate 6 ne { systemdict /currenttransfer get exec aload pop } if ] cvx gsave systemdict /settransfer get exec systemdict /image get exec grestore } ifelse } bd } if end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg2.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? StartLoad { /ID { 5 counttomark 2 idiv dup 3 1 roll add dict begin { def } repeat cleartomark currentdict end begin /ImageType 1 def /ImageMatrix [ Width 0 0 Height neg 0 Height ] def /ImageMask here { not } { true } ifelse { /ImageMask false def } if ImageMask not { ColorSpace setcolorspace } if /Intent here { ri } if /Decode here { pop } { /Decode [ ImageMask { 0 1 } { currentcolorspace 0 get /Indexed eq { 0 2 BitsPerComponent exp 1 sub } { mark currentcolor counttomark dup 2 add 1 roll cleartomark { 0 1 } repeat } ifelse } ifelse ] def } ifelse [ /DataSource here { pop } { currentfile /Filter here { dup type /arraytype eq { length } { pop 1 } ifelse 1 sub 0 1 3 -1 roll { /DecodeParms here { dup type /arraytype eq { 1 index get } if dup null eq { pop } { exch } ifelse } if Filter dup type /arraytype eq { exch get } { exch pop } ifelse filter dup } for } if /DataSource exch def } ifelse currentdict % Level3? { dup /MaskedImage known { pop << /ImageType 3 /InterleaveType 2 /DataDict currentdict /MaskDict << /ImageType 1 /Width Width /Height Height /ImageMatrix ImageMatrix /BitsPerComponent 1 /Decode [0 1] currentdict /Interpolate known {/Interpolate Interpolate} if >> >> }if }if % /ImageMask here not { false } if { sfc imagemask } { image } ifelse counttomark { dup status { dup flushfile closefile } { pop } ifelse } repeat pop end } bd currentdict readonly pop } EndLoad end %%EndFile end end PDFVars /InitAll { [ PDF PDFText PDFImage ] { /initialize get exec } forall initgs 0 Tr } put PDFVars /TermAll { [ PDFImage PDFText PDF ] { /terminate get exec } forall } put PDFVars begin PDF begin /MacRomanEncoding [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /space/exclam/quotedbl/numbersign/dollar/percent/ampersand 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/underscore /grave /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 /asciitilde /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /degree /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /notequal /AE /Oslash /infinity /plusminus /lessequal /greaterequal /yen /mu /partialdiff /summation /product /pi /integral /ordfeminine /ordmasculine /Omega /ae /oslash /questiondown /exclamdown /logicalnot /radical /florin /approxequal /Delta /guillemotleft /guillemotright /ellipsis /nonbreakingspace /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /divide /lozenge /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /apple /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron /Lslash /lslash /Scaron /scaron /Zcaron /zcaron /brokenbar /Eth /eth /Yacute /yacute /Thorn /thorn /minus /multiply /onesuperior /twosuperior /threesuperior /onehalf /onequarter /threequarters /franc /Gbreve /gbreve /Idotaccent /Scedilla /scedilla /Cacute /cacute /Ccaron ] def /MacintoshSymbolGlyphEncoding [ /.notdef/.null/nonmarkingreturn/space/exclam/numbersign/percent /ampersand/parenleft/parenright/plus/comma/period/slash/zero/one /two/three/four/five/six/seven/eight/nine/colon/semicolon/less /equal/greater/question/bracketleft/bracketright/underscore 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/bracketrighttp/bracketrightbt/universal/existential/asteriskmath /angleright/angleleft/theta1/omega1/phi1/epsilon/gradient/parenlefttp /parenleftbt/parenrighttp/parenrightbt/weierstrass/bracelefttp /braceleftmid/braceleftbt/braceex/bracerighttp/bracerightmid /bracerightbt/Upsilon1/arrowvertex/arrowhorizex/parenleftex /bracketleftex/parenrightex/bracketrightex/copyrightserif /registerserif/trademarkserif/copyrightsans/registersans /trademarksans/Ifraktur/Rfraktur/similar/carriagereturn/Euro ] def /reencode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall FontName /Symbol eq { /Encoding MacintoshSymbolGlyphEncoding def }{ /Encoding MacintoshRomanGlyphEncoding def } ifelse currentdict end } def /reencode-font { % new-font-name encoding-array old-font-name findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } def PDFVars /InitAll get exec %%BeginFile: cgmisc.txt %%Copyright: Copyright 2000-2001 Apple Computer Incorporated. %%Copyright: All Rights Reserved. userdict begin /cgScratchDict 10 dict def /cgPageMatrix matrix currentmatrix def /cgScratchMtx matrix def /cgq/gsave load def /cgPatArray 0 def /cgQ{grestore ilp}bind def /cgCreatePat{cgPatArray 3 1 roll put}bind def /cgMakePat{ cgPatArray exch get gsave initgraphics userdict/cgPageMatrix get setmatrix dup/Matrix get cgScratchMtx copy makepattern grestore }bind def /mTm{_tm dup 5 4 -1 roll put 4 3 -1 roll put iTm 0 0 moveto}bind def /cguidfix{statusdict begin mark version end {cvr}stopped{cleartomark 0}{exch pop}ifelse 2012 lt{dup findfont dup length dict begin {1 index/FID ne 2 index/UniqueID ne and {def} {pop pop} ifelse}forall currentdict end definefont pop }{pop}ifelse }bind def /cg_BeginEPSF{ userdict save/cg_b4_Inc_state exch put count userdict/cg_op_count 3 -1 roll put countdictstack dup array dictstack userdict/cg_dict_array 3 -1 roll put 3 sub{end}repeat /showpage {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath false setstrokeadjust false setoverprint }bind def /cg_EndEPSF{ countdictstack 3 sub { end } repeat cg_dict_array 3 1 index length 3 sub getinterval {begin}forall count cg_op_count sub { pop } repeat userdict/cg_b4_Inc_state get restore false setpacking }bind def end %%EndFile %%EndProlog %%BeginSetup %%EndSetup %%Page: 1 1 %%PageBoundingBox: 0 0 435 152 %%BeginPageSetup userdict /pgsave save put PDFVars begin PDF begin PDFVars/InitAll get exec %!PS-TrueTypeFont-1.0000-0.0000-2 14 dict begin/FontName /F1.1 def /PaintType 0 def /FontType 1 def /Encoding 256 array 0 1 255{1 index exch/.notdef put}for dup 33 /k put dup 34 /m put dup 35 /zero put dup 36 /one put dup 37 /q put readonly def /FontMatrix [ 0.00048828125 0 0 0.00048828125 0 0 ] def /FontBBox{-349 -845 2074 1959}def /UniqueID 4414295 def currentdict currentfile eexec 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Tj ET cgQ cgQ cgQ cgQ PDFVars/TermAll get exec end end userdict /pgsave get restore showpage %%Trailer %%EOF ---------------0312010856408 Content-Type: application/postscript; name="f3.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="f3.eps" %!PS-Adobe-3.0 EPSF-3.0 %%HiResBoundingBox: 0.000000 0.000000 408.000000 77.000000 %%Title: (Unknown) %%Creator: (Unknown) %%CreationDate: (Unknown) %%For: (Unknown) %%DocumentData: Clean7Bit %%Pages: 1 %%BoundingBox: 0 0 408 77 %%EndComments %%BeginProlog userdict /PDF 95 dict put %%BeginFile: pdfvars.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. userdict /PDFVars 90 dict put PDFVars begin /_save 0 def /_cshow 0 def /InitAll 0 def /TermAll 0 def /_lp /none def /_doClip 0 def /sfc 0 def /_sfcs 0 def /_sfc 0 def /ssc 0 def /_sscs 0 def /_ssc 0 def /_fcs 0 def /_scs 0 def /_fp 0 def /_sp 0 def /AGM_MAX_CS_COMPONENTS 10 def /_fillColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_strokeColors [ 0 1 AGM_MAX_CS_COMPONENTS { array } for ] def /_fc null def /_sc null def /GetCompsDict null def /_inT false def /_tr -1 def /_rise 0 def /_ax 0 def /_cx 0 def /_ld 0 def /_tm matrix def /_ctm matrix def /_mtx matrix def /_hy (-) def /_fScl 0 def /_hs 1 def /_pdfEncodings 2 array def /_baselineadj 0 def /_fTzero false def /_Tj 0 def /_italMtx[1 0 .212557 1 0 0]def /_italMtx_WMode1 [1 -.212557 0 1 0 0]def /_italMtxType0[1 0 .1062785 1 0 0]def /_italMtx_WMode1Type0 [1 -.1062785 0 1 0 0]def /_basefont 0 def /_basefonto 0 def /_pdf_oldCIDInit null def /_categories 10 dict def /_sa? true def /_op? false def /_ColorSep5044? false def /_tmpcolr? [] def /_tmpop? {} def end %%EndFile PDFVars begin PDF begin %%BeginFile: pdfutil.prc %%Copyright: Copyright 1993 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. /bd {bind def} bind def /ld {load def} bd /dd { PDFVars 3 1 roll put } bd /xdd { exch dd } bd /Level2? systemdict /languagelevel known { systemdict /languagelevel get 2 ge }{ false } ifelse def /Level3? systemdict /languagelevel known { systemdict /languagelevel get 3 eq }{ false } ifelse def /here { dup currentdict exch known { currentdict exch get true }{ pop false } ifelse } bd /isdefined? { where { pop true } { false } ifelse } bd /StartLoad { dup dup not { /_save save dd } if } bd /EndLoad { if not { _save restore } if } bd /npop { { pop } repeat } bd %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdf.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. /initialize { _ColorSep5044? {sep_ops begin 50 dict begin} if newpath } bd /terminate { _ColorSep5044? {end end} if } bd Level2? StartLoad { /m/moveto ld /l/lineto ld /c/curveto ld /setSA/setstrokeadjust ld } EndLoad Level2? not StartLoad { /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } bd /m { _sa? { pl } if moveto } bd /l { _sa? { pl } if lineto } bd /c { _sa? { pl } if curveto } bd /setSA { /_sa? xdd } bd } EndLoad /v { currentpoint 6 2 roll c } bd /y { 2 copy c } bd /h/closepath ld /d/setdash ld /j/setlinejoin ld /J/setlinecap ld /M/setmiterlimit ld /w/setlinewidth ld /cf currentflat def /i { dup 0 eq { pop cf } if setflat } bd /gsDI { begin /Font here { aload pop Tf } if /LW here { w } if /LC here { J } if /LC here { j } if /ML here { M } if /D here { aload pop d } if end } bd /ilp { /_lp /none dd } bd /sfc { _lp /fill ne { _sfcs _sfc /_lp /fill dd } if } dd /ssc { _lp /stroke ne { _sscs _ssc /_lp /stroke dd } if } dd /n { _doClip 1 ge { _doClip 1 eq { clip } { eoclip } ifelse /_doClip 0 dd } if newpath } bd /fs.aux { _doClip 1 ge { gsave exec grestore _doClip 1 eq { clip }{ eoclip } ifelse newpath ilp /_doClip 0 dd }{ exec } ifelse } bd /f { { sfc fill } fs.aux } bd /f* { { sfc eofill } fs.aux } bd /S { { ssc stroke } fs.aux } bd /s { h S } bd /B { _doClip dup 1 ge gsave f grestore { gsave S grestore 1 eq { clip } { eoclip } ifelse newpath ilp /_doClip 0 dd }{ pop S } ifelse } bd /b { h B } bd /B* { _doClip dup 1 ge gsave f* grestore { gsave S grestore 1 eq { clip } { eoclip } ifelse newpath ilp /_doClip 0 dd }{ pop S } ifelse } bd /b* { h B* } bd /sh { dup /DataSource known { dup begin DataSource type /filetype eq { DataSource resetfile } if end } if shfill } bd /W { /_doClip 1 dd } bd /W* { /_doClip 2 dd } bd /q/save ld /Q { restore ilp } bd Level2? StartLoad { /defineRes/defineresource ld /findRes/findresource ld currentglobal true systemdict /setglobal get exec [ /Function /ExtGState /Form /Shading ] { /Generic /Category findresource dup length dict copy /Category defineresource pop } forall systemdict /setglobal get exec } EndLoad Level2? not StartLoad { /AlmostFull? { dup maxlength exch length sub 2 le } bind def /Expand { 1 index maxlength mul cvi dict dup begin exch { def } forall end } bind def /xput { 3 2 roll dup 3 index known not { dup AlmostFull? { 1.5 Expand } if } if dup 4 2 roll put } bind def /defineRes { _categories 1 index known not { /_categories _categories 2 index 10 dict xput store } if _categories exch 2 copy get 5 -1 roll 4 index xput put } bind def /findRes { _categories exch get exch get } bind def } EndLoad /cs { dup where { pop load } if dup /_fcs xdd GetComps _fillColors exch get /_fc xdd /_fp null dd } bd /CS { dup where { pop load } if dup /_scs xdd GetComps _strokeColors exch get /_sc xdd /_sp null dd } bd /GetCompsDict 16 dict begin /DeviceGray { pop 1 } bd /DefaultGray { pop 1 } bd /DeviceRGB { pop 3 } bd /DefaultRGB { pop 3 } bd /DeviceCMYK { pop 4 } bd /DefaultCMYK { pop 4 } bd /CalGray { pop 1 } bd /CalRGB { pop 3 } def /CIEBasedA { pop 1 } bd /CIEBasedABC { pop 3 } bd /CIEBasedDEFG { pop 4 } bd /Lab { pop 3 } bd /DeviceN { 1 get length } bd /Separation { pop 1 } bd /Indexed { pop 1 } bd /Pattern { pop 0 } bd currentdict end dd /GetComps { GetCompsDict 1 index dup type /arraytype eq { 0 get } if get exec } bd Level2? not StartLoad { /ri/pop ld /makePat/pop ld } EndLoad Level2? StartLoad { /ri { /findcolorrendering isdefined? { mark exch findcolorrendering counttomark 2 eq { type /booleantype eq { dup type /nametype eq { dup /ColorRendering resourcestatus { pop pop dup /DefaultColorRendering ne { /ColorRendering findresource setcolorrendering } if } if } if } if } if cleartomark } { pop } ifelse } bd /makePat { 1 index /PatternType get 2 eq languagelevel 3 lt and { 7 dict dup begin /PatternType 1 def /PaintType 1 def /TilingType 1 def /BBox [10 10] /XStep 10 /YStep 10 /PaintProc {pop .5 setgray 0 0 10 10 rectfill} bind end matrix 4 2 roll pop } if makepattern } bd } EndLoad Level2? not _ColorSep5044? or StartLoad { /L1setcolor { aload length dup 0 eq { pop .5 setgray } { dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } ifelse } bind dd /_sfcs { } dd /_sscs { } dd } EndLoad Level2? not _ColorSep5044? not and StartLoad { /_sfc { _fc L1setcolor } dd /_ssc { _sc L1setcolor } dd } EndLoad Level2? _ColorSep5044? not and StartLoad { /_sfcs { _fcs setcolorspace } bind dd /_sscs { _scs setcolorspace } bind dd /_sfc { _fc aload pop _fp null eq { setcolor } { _fp setpattern } ifelse } bind dd /_ssc { _sc aload pop _sp null eq { setcolor } { _sp setpattern } ifelse } bind dd } EndLoad /sc { _fc astore pop ilp } bd /SC { _sc astore pop ilp } bd /scn { dup type /dicttype eq { dup /_fp xdd dup /PatternType get 1 eq { /PaintType get 1 eq { /_fc _fillColors 0 get dd ilp } { /_fc _fillColors _fcs 1 get GetComps get dd sc } ifelse } { pop /_fc _fillColors 0 get dd ilp } ifelse } { sc } ifelse } bd /SCN { dup type /dicttype eq { dup /_sp xdd dup /PatternType get 1 eq { /PaintType get 1 eq { /_sc _strokeColors 0 get dd ilp } { /_sc _strokeColors _scs 1 get GetComps get dd SC } ifelse } { pop /_sc _strokeColors 0 get dd ilp } ifelse } { SC } ifelse } bd /g { /DefaultGray cs sc } bd /rg { /DefaultRGB cs sc } bd /k { /DefaultCMYK cs sc } bd /G { /DefaultGray CS SC } bd /RG { /DefaultRGB CS SC } bd /K { /DefaultCMYK CS SC } bd /cm { _mtx astore concat } bd /re { 4 2 roll m 1 index 0 rlineto 0 exch rlineto neg 0 rlineto h } bd /rf /rectfill where {pop {sfc rectfill}} {{re f}} ifelse bd /RC/rectclip ld /EF/execform ld /PS { cvx exec } bd /initgs { /DefaultGray where { pop } { /DefaultGray /DeviceGray dd } ifelse /DefaultRGB where { pop } { /DefaultRGB /DeviceRGB dd } ifelse /DefaultCMYK where { pop } { /DefaultCMYK /DeviceCMYK dd } ifelse 0 g 0 G [] 0 d 0 j 0 J 10 M 1 w true setSA } bd 21 dict dup begin /CosineDot { 180 mul cos exch 180 mul cos add 2 div } bd /Cross { abs exch abs 2 copy gt { exch } if pop neg } bd /Diamond { abs exch abs 2 copy add .75 le { dup mul exch dup mul add 1 exch sub }{ 2 copy add 1.23 le { .85 mul add 1 exch sub }{ 1 sub dup mul exch 1 sub dup mul add 1 sub } ifelse } ifelse } bd /Double { exch 2 div exch 2 { 360 mul sin 2 div exch } repeat add } bd /DoubleDot { 2 { 360 mul sin 2 div exch } repeat add } bd /Ellipse { abs exch abs 2 copy 3 mul exch 4 mul add 3 sub dup 0 lt { pop dup mul exch .75 div dup mul add 4 div 1 exch sub }{ dup 1 gt { pop 1 exch sub dup mul exch 1 exch sub .75 div dup mul add 4 div 1 sub }{ .5 exch sub exch pop exch pop } ifelse } ifelse } bd /EllipseA { dup mul .9 mul exch dup mul add 1 exch sub } bd /EllipseB { dup 5 mul 8 div mul exch dup mul exch add sqrt 1 exch sub } bd /EllipseC { dup mul .9 mul exch dup mul add 1 exch sub } bd /InvertedDouble { exch 2 div exch 2 { 360 mul sin 2 div exch } repeat add neg } bd /InvertedDoubleDot { 2 { 360 mul sin 2 div exch } repeat add neg } bd /InvertedEllipseA { dup mul .9 mul exch dup mul add 1 sub } bd /InvertedSimpleDot { dup mul exch dup mul add 1 sub } bd /Line { exch pop abs neg } bd /LineX { pop } bd /LineY { exch pop } bd /Rhomboid { abs exch abs 0.9 mul add 2 div } bd /Round { abs exch abs 2 copy add 1 le { dup mul exch dup mul add 1 exch sub }{ 1 sub dup mul exch 1 sub dup mul add 1 sub } ifelse } bd /SimpleDot { dup mul exch dup mul add 1 exch sub } bd /Square { abs exch abs 2 copy lt { exch } if pop neg } bd end { /Function defineRes pop } forall /Identity {} /Function defineRes pop _ColorSep5044? StartLoad { /_defaulttransferfunc currenttransfer def /currentcolortransfer where { pop /_defaultcolortransferfuncs [ currentcolortransfer ] def } if /concattransferfuncs { [ 3 1 roll /exec load exch /exec load ] cvx } bd /concatandsettransfer { /_defaulttransferfunc load concattransferfuncs settransfer } bd /concatandsetcolortransfer { colorplate 0 eq { _defaultcolortransferfuncs aload pop 8 -1 roll 5 -1 roll concattransferfuncs 7 1 roll 6 -1 roll 4 -1 roll concattransferfuncs 5 1 roll 4 -1 roll 3 -1 roll concattransferfuncs 3 1 roll concattransferfuncs setcolortransfer } if colorplate 1 ge colorplate 4 le and { 4 colorplate sub index 4 { exch pop } repeat concatandsettransfer } if colorplate 5 ge { 0 index 4 { exch pop } repeat concatandsettransfer } if } bd /tn5044sethalftone { begin HalftoneType 5 eq { [/Default /Cyan /Magenta /Yellow /Black /Default /Default /Default] colorplate get here not { /Default here not { currentdict } if } if }{ currentdict } ifelse end begin /TransferFunction here { concatandsettransfer currentdict dup length dict begin { 1 index /TransferFunction ne { def } { pop pop } ifelse } forall currentdict end }{ currentdict } ifelse end sethalftone } bd } EndLoad Level2? Level3? not and StartLoad { /setsmoothness { pop } bd } EndLoad Level2? StartLoad { /gs { begin /SA here { setstrokeadjust } if /OP here { setoverprint } if /BG here { setblackgeneration } if /UCR here { setundercolorremoval } if /SM here { setsmoothness } if /FL here { i } if /RI here { ri } if /TR here { _ColorSep5044? { dup xcheck { concatandsettransfer } { aload pop concatandsetcolortransfer } ifelse }{ dup xcheck { settransfer } { aload pop setcolortransfer } ifelse } ifelse } if /sethalftonephase isdefined? { /HTP here { sethalftonephase } if } if /HT here { _ColorSep5044? { tn5044sethalftone } { sethalftone } ifelse } if currentdict gsDI end } bd /_defaulthalftone currenthalftone def /_defaultblackgeneration currentblackgeneration def /_defaultundercolorremoval currentundercolorremoval def /_defaultcolortransfer [currentcolortransfer] def /_defaulttransfer currenttransfer def } EndLoad Level2? not StartLoad { /gs { begin /SA here { /_sa? xdd } if /OP here { dup /_op? xdd /setoverprint where { pop setoverprint }{ pop } ifelse } if /TR here { _ColorSep5044? { dup xcheck { concatandsettransfer }{ aload pop concatandsetcolortransfer } ifelse }{ dup xcheck { settransfer }{ aload pop setcolortransfer } ifelse } ifelse } if /HT here { _ColorSep5044? { tn5044sethalftone }{ sethalftone } ifelse } if /FL here { i } if currentdict gsDI end } bd currentscreen dup type /dicttype eq { /_defaulthalftone exch def pop pop }{ 5 dict begin 1 [ /HalftoneType /SpotFunction /Angle /Frequency ] { exch def } forall currentdict end /_defaulthalftone exch def } ifelse } EndLoad /int { dup 2 index sub 3 index 5 index sub div 6 -2 roll sub mul exch pop add exch pop } bd /limit { dup 2 index le { exch } if pop dup 2 index ge { exch } if pop } bd /domainClip { Domain aload pop 3 2 roll limit } bd /applyInterpFunc { 0 1 DimOut 1 sub { dup C0 exch get exch dup C1 exch get exch 3 1 roll 1 index sub 3 index N exp mul add exch currentdict /Range_lo known { dup Range_lo exch get exch Range_hi exch get 3 2 roll limit }{ pop } ifelse exch } for pop } bd /encodeInput { NumParts 1 sub 0 1 2 index { dup Bounds exch get 2 index gt { exit } { dup 3 index eq { exit } { pop } ifelse } ifelse } for 3 2 roll pop dup Bounds exch get exch dup 1 add Bounds exch get exch 2 mul dup Encode exch get exch 1 add Encode exch get int } bd /rangeClip { exch dup Range_lo exch get exch Range_hi exch get 3 2 roll limit } bd /applyStitchFunc { Functions exch get exec currentdict /Range_lo known { 0 1 DimOut 1 sub { DimOut 1 add -1 roll rangeClip } for } if } bind def _ColorSep5044? StartLoad { /_sfc { _fp null eq { _fcs type /arraytype eq { _fcs 0 get /Separation eq { _fcs 1 get /All eq { _fc aload pop 1 exch sub /setseparationgray where pop begin setseparationgray end }{ 1 _fcs 3 get exec _fcs 1 get /findcmykcustomcolor where pop begin findcmykcustomcolor end _fc aload pop /setcustomcolor where pop begin setcustomcolor end } ifelse }{ _fc L1setcolor } ifelse }{ _fc L1setcolor } ifelse }{ _fc L1setcolor } ifelse } bind dd /_ssc { _sp null eq { _scs type /arraytype eq { _scs 0 get /Separation eq { _scs 1 get /All eq { _sc aload pop 1 exch sub /setseparationgray where pop begin setseparationgray end }{ 1 _scs 3 get exec _scs 1 get /findcmykcustomcolor where pop begin findcmykcustomcolor end _sc aload pop /setcustomcolor where pop begin setcustomcolor end } ifelse }{ _sc L1setcolor } ifelse }{ _sc L1setcolor } ifelse }{ _sc L1setcolor } ifelse } bind dd } EndLoad %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdftext.prc %%Copyright: Copyright 1987-1998 Adobe Systems Incorporated. %%Copyright: All Rights Reserved. PDF /PDFText 75 dict put PDFText begin /initialize { PDFText begin } bd /terminate { end } bd /pdf_has_composefont? systemdict /composefont known def /CopyFont { { 1 index /FID ne 2 index /UniqueID ne and { def }{ pop pop } ifelse } forall } bd /Type0CopyFont { exch dup length dict begin CopyFont [ exch FDepVector { dup /FontType get 0 eq { 1 index Type0CopyFont /_pdfType0 exch definefont }{ /_pdfBaseFont exch 2 index exec } ifelse exch } forall pop ] /FDepVector exch def currentdict end } bd /cHexEncoding [ /c00/c01/c02/c03/c04/c05/c06/c07/c08/c09/c0A/c0B/c0C/c0D/c0E/c0F /c10/c11/c12/c13/c14/c15/c16/c17/c18/c19/c1A/c1B/c1C/c1D/c1E/c1F /c20/c21/c22/c23/c24/c25/c26/c27/c28/c29/c2A/c2B/c2C/c2D/c2E/c2F /c30/c31/c32/c33/c34/c35/c36/c37/c38/c39/c3A/c3B/c3C/c3D/c3E/c3F /c40/c41/c42/c43/c44/c45/c46/c47/c48/c49/c4A/c4B/c4C/c4D/c4E/c4F /c50/c51/c52/c53/c54/c55/c56/c57/c58/c59/c5A/c5B/c5C/c5D/c5E/c5F /c60/c61/c62/c63/c64/c65/c66/c67/c68/c69/c6A/c6B/c6C/c6D/c6E/c6F /c70/c71/c72/c73/c74/c75/c76/c77/c78/c79/c7A/c7B/c7C/c7D/c7E/c7F /c80/c81/c82/c83/c84/c85/c86/c87/c88/c89/c8A/c8B/c8C/c8D/c8E/c8F /c90/c91/c92/c93/c94/c95/c96/c97/c98/c99/c9A/c9B/c9C/c9D/c9E/c9F /cA0/cA1/cA2/cA3/cA4/cA5/cA6/cA7/cA8/cA9/cAA/cAB/cAC/cAD/cAE/cAF /cB0/cB1/cB2/cB3/cB4/cB5/cB6/cB7/cB8/cB9/cBA/cBB/cBC/cBD/cBE/cBF /cC0/cC1/cC2/cC3/cC4/cC5/cC6/cC7/cC8/cC9/cCA/cCB/cCC/cCD/cCE/cCF /cD0/cD1/cD2/cD3/cD4/cD5/cD6/cD7/cD8/cD9/cDA/cDB/cDC/cDD/cDE/cDF /cE0/cE1/cE2/cE3/cE4/cE5/cE6/cE7/cE8/cE9/cEA/cEB/cEC/cED/cEE/cEF /cF0/cF1/cF2/cF3/cF4/cF5/cF6/cF7/cF8/cF9/cFA/cFB/cFC/cFD/cFE/cFF ] def /modEnc { /_enc xdd /_icode 0 dd counttomark 1 sub -1 0 { index dup type /nametype eq { _enc _icode 3 -1 roll put _icode 1 add } if /_icode xdd } for cleartomark _enc } bd /trEnc { /_enc xdd 255 -1 0 { exch dup -1 eq { pop /.notdef } { Encoding exch get } ifelse _enc 3 1 roll put } for pop _enc } bd /TE { /_i xdd StandardEncoding 256 array copy modEnc _pdfEncodings exch _i exch put } bd /TZ { /_usePDFEncoding xdd findfont dup length 2 add dict begin { 1 index /FID ne { def } { pop pop } ifelse } forall /FontName exch def _usePDFEncoding 0 ge { /Encoding _pdfEncodings _usePDFEncoding get def pop }{ _usePDFEncoding -1 eq { counttomark 0 eq { pop }{ Encoding 256 array copy modEnc /Encoding exch def } ifelse }{ 256 array trEnc /Encoding exch def } ifelse } ifelse FontName currentdict end definefont pop } bd /Level2? systemdict /languagelevel known { systemdict /languagelevel get 2 ge }{ false } ifelse def Level2? { /_pdfFontStatus { currentglobal exch /Font resourcestatus {pop pop true} {false} ifelse exch setglobal } bd }{ /_pdfFontStatusString 50 string def _pdfFontStatusString 0 (fonts/) putinterval /_pdfFontStatus { FontDirectory 1 index known { pop true } { _pdfFontStatusString 6 42 getinterval cvs length 6 add _pdfFontStatusString exch 0 exch getinterval { status } stopped {pop false} { { pop pop pop pop true} { false } ifelse } ifelse } ifelse } bd } ifelse Level2? { /_pdfCIDFontStatus { /CIDFont /Category resourcestatus { pop pop /CIDFont resourcestatus {pop pop true} {false} ifelse } { pop false } ifelse } bd } if /_pdfString100 100 string def /_pdfComposeFontName { dup length 1 eq { 0 get 1 index type /nametype eq { _pdfString100 cvs length dup dup _pdfString100 exch (-) putinterval _pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval 2 index exch cvs length add 1 add _pdfString100 exch 0 exch getinterval exch pop true }{ pop pop false } ifelse }{ false } ifelse } bd pdf_has_composefont? { /_pdfComposeFont { 1 index /CMap resourcestatus { pop pop true }{ false } ifelse 1 index true exch { _pdfCIDFontStatus not {pop false exit} if } forall and { 3 -1 roll pop composefont true }{ 4 -1 roll pop _pdfComposeFontName { dup _pdfFontStatus { findfont definefont true }{ pop dup _pdfFontStatus { findfont true }{ pop false } ifelse } ifelse }{ dup _pdfFontStatus { findfont true } { pop false } ifelse } ifelse } ifelse } bd }{ /_pdfComposeFont { 4 -1 roll pop _pdfComposeFontName not { dup } if 2 copy _pdfFontStatus {pop findfont exch pop true} { eq {pop false} { dup _pdfFontStatus {findfont true} {pop false} ifelse } ifelse } ifelse } bd } ifelse /_pdfStyleDicts 4 dict dup begin /Adobe-Japan1 4 dict dup begin Level2? { /Serif /HeiseiMin-W3-83pv-RKSJ-H _pdfFontStatus {/HeiseiMin-W3} { /HeiseiMin-W3 _pdfCIDFontStatus {/HeiseiMin-W3} {/Ryumin-Light} ifelse } ifelse def /SansSerif /HeiseiKakuGo-W5-83pv-RKSJ-H _pdfFontStatus {/HeiseiKakuGo-W5} { /HeiseiKakuGo-W5 _pdfCIDFontStatus {/HeiseiKakuGo-W5} {/GothicBBB-Medium} ifelse } ifelse def /HeiseiMaruGo-W4-83pv-RKSJ-H _pdfFontStatus {/HeiseiMaruGo-W4} { /HeiseiMaruGo-W4 _pdfCIDFontStatus {/HeiseiMaruGo-W4} { /Jun101-Light-RKSJ-H _pdfFontStatus { /Jun101-Light } { SansSerif } ifelse } ifelse } ifelse /RoundSansSerif exch def /Default Serif def } { /Serif /Ryumin-Light def /SansSerif /GothicBBB-Medium def { (fonts/Jun101-Light-83pv-RKSJ-H) status }stopped {pop}{ { pop pop pop pop /Jun101-Light } { SansSerif } ifelse /RoundSansSerif exch def }ifelse /Default Serif def } ifelse end def /Adobe-Korea1 4 dict dup begin /Serif /HYSMyeongJo-Medium def /SansSerif /HYGoThic-Medium def /RoundSansSerif SansSerif def /Default Serif def end def /Adobe-GB1 4 dict dup begin /Serif /STSong def /SansSerif /STHeiti def /RoundSansSerif SansSerif def /Default Serif def end def /Adobe-CNS1 4 dict dup begin /Serif /MKai-Medium def /SansSerif /MHei-Medium def /RoundSansSerif SansSerif def /Default Serif def end def end def /_pdf_Adobe-Japan1-2 (Adobe-Japan1-2) def /_pdfConcatNames { exch _pdfString100 cvs length dup dup _pdfString100 exch (-) putinterval _pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval 3 -1 roll exch cvs length add 1 add _pdfString100 exch 0 exch getinterval } bind def /_pdfSubSetFontByStyleDict 4 dict dup begin _pdfStyleDicts /Adobe-Japan1 get { _pdf_Adobe-Japan1-2 _pdfConcatNames dup _pdfFontStatus { def }{ pop pop } ifelse } forall end def /TZzero { /_fyAdj xdd /_wmode xdd /_styleArr xdd /_regOrdering xdd 4 copy _pdfComposeFont {exch pop exch pop exch pop} { [ 0 1 _styleArr length 1 sub { _styleArr exch get _pdfStyleDicts _regOrdering 2 copy known { get exch 2 copy known not { pop /Default } if get } { pop pop /Unknown } ifelse } for ] exch pop 3 index 3 index 4 2 roll _pdfComposeFont { exch pop }{ findfont } ifelse } ifelse dup /FontType get 3 eq _wmode 1 eq and { _pdfVerticalRomanT3Font dup length 10 add dict copy begin /_basefont exch def /Encoding _basefont /Encoding get def }{ dup length 3 add dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall } ifelse /WMode _wmode def /BaseLineAdj _fyAdj def FontType 0 ne { /Encoding Encoding dup length array copy dup 16#5c /yen put def /_fauxfont true def } if currentdict end definefont pop } bd /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 dd { 1 index eq {/_cnt _cnt 1 add dd} if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } bd /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave 32 eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /jsfTzero { { 2 npop (0) exch 2 copy 0 exch put exch show 32 eq { 4 index 4 index rmoveto } if 2 copy rmoveto } exch cshow 5 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put 32 eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } bd /trj { _cx 0 fWModeProc 32 _ax 0 fWModeProc 6 5 roll } bd /pjsf { trj sfc fawidthshowProc } bd /pjss { trj _ctm ssc jss } bd /pjsc { trj jsp } bd /_Tjdef [ /pjsf load /pjss load { dup currentpoint 3 2 roll pjsf newpath moveto pjss } bind { trj swj rmoveto } bind { dup currentpoint 4 2 roll gsave pjsf grestore 3 1 roll moveto pjsc } bind { dup currentpoint 4 2 roll currentpoint gsave newpath moveto pjss grestore 3 1 roll moveto pjsc } bind { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll pjsf newpath moveto pjss grestore 3 1 roll moveto pjsc } bind /pjsc load ] def /BT { /_inT true dd _ctm currentmatrix pop 1 0 0 1 0 0 _tm astore pop %DMG 0 _rise _baselineadj add translate _hs 1 scale 0 0 moveto } bd /ET { /_inT false dd _tr 3 gt {clip} if _ctm setmatrix newpath } bd /Tr { _inT { _tr 3 le {currentpoint newpath moveto} if } if dup /_tr xdd _Tjdef exch get /_Tj xdd } bd /Tj { userdict /$$copystring 2 index put _Tj } bd /iTm { _ctm setmatrix _tm concat 0 _rise _baselineadj add translate _hs 1 scale } bd /Tm { _tm astore pop iTm 0 0 moveto } bd /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } bd /TD { dup /_ld xdd Td } bd /_nullProc {} bd /Tf { dup 1000 div /_fScl xdd Level2? { selectfont }{ exch findfont exch scalefont setfont } ifelse currentfont dup /_nullProc exch /WMode known { 1 index /WMode get 1 eq { pop /exch } if } if load /fWModeProc xdd dup /FontType get 0 eq dup _cx 0 ne and { /jsfTzero }{ /awidthshow } ifelse load /fawidthshowProc xdd /_fTzero xdd dup /BaseLineAdj known { dup /BaseLineAdj get _fScl mul }{ 0 } ifelse /_baselineadj xdd currentpoint iTm moveto pop } bd /TL { neg /_ld xdd } bd /Tw { /_cx xdd _cx 0 ne _fTzero and { /jsfTzero } { /awidthshow } ifelse load /fawidthshowProc xdd } bd /Tc { /_ax xdd } bd /Ts { /_rise xdd currentpoint iTm moveto } bd /Tz { 100 div /_hs xdd iTm } bd /Tk { exch pop _fScl mul neg 0 fWModeProc rmoveto } bd /T* { 0 _ld Td } bd /' { T* Tj } bd /" { exch Tc exch Tw ' } bd /TJ { { dup type /stringtype eq { Tj }{ 0 exch Tk } ifelse } forall } bd /T- { _hy Tj } bd /d0/setcharwidth ld /d1 { setcachedevice /sfc{}dd /ssc{}dd } bd /nND {{/.notdef} repeat} bd /T3Defs { /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } def /BuildGlyph { exch begin GlyphProcs exch get exec end } def } bd /_pdfBoldRomanWidthProc { stringwidth 1 index 0 ne { exch .03 add exch }if setcharwidth } bd /_pdfType0WidthProc { dup stringwidth 0 0 moveto 2 index true charpath pathbbox 0 -1 7 index 2 div .88 setcachedevice2 pop } bd /_pdfBoldBaseFont 11 dict begin /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding cHexEncoding def /_setwidthProc /_pdfBoldRomanWidthProc load def /_bcstr1 1 string def /BuildChar { exch begin _basefont setfont _bcstr1 dup 0 4 -1 roll put dup _setwidthProc 0 0 moveto dup show _basefonto setfont 0 0 moveto show end }bd currentdict end def /_pdfVerticalRomanT3Font 10 dict begin /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /_bcstr1 1 string def /BuildChar { exch begin _basefont setfont _bcstr1 dup 0 4 -1 roll put dup _pdfType0WidthProc 0 0 moveto show end } bd currentdict end def /MakeBoldFont { dup dup length 3 add dict begin CopyFont /PaintType 2 def /StrokeWidth .03 0 FontMatrix idtransform pop def /dummybold currentdict end definefont _pdfBoldBaseFont dup length 3 add dict copy begin /_basefont exch def /_basefonto exch def currentdict end definefont } bd /MakeBold { exch 1 index findfont dup /FontType get 0 eq { _pdfBoldBaseFont /_setwidthProc /_pdfType0WidthProc load put {MakeBoldFont} Type0CopyFont definefont }{ dup /_fauxfont known not { _pdfBoldBaseFont /_setwidthProc /_pdfBoldRomanWidthProc load put MakeBoldFont }{ 2 index 2 index eq { exch pop }{ dup length dict begin CopyFont currentdict end definefont } ifelse } ifelse } ifelse pop pop } bd /MakeItalic { findfont dup /FontType get 0 eq Level2? not and { dup /FMapType get 6 eq } { false } ifelse { dup /WMode 2 copy known { get 1 eq { _italMtx_WMode1Type0 } { _italMtxType0 } ifelse } { pop pop _italMtxType0 } ifelse } { dup /WMode 2 copy known { get 1 eq { _italMtx_WMode1 } { _italMtx } ifelse } { pop pop _italMtx } ifelse } ifelse makefont Level2? not { dup length dict begin CopyFont currentdict end } if definefont pop }bd /MakeBoldItalic { /dummybold exch MakeBold /dummybold MakeItalic }bd currentdict readonly pop end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimage.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage 38 dict put PDF /PDFIVars 20 dict put PDF /PDFImage get begin /initialize { PDFImage begin } bd /terminate { end } bd /nulldict 0 dict def /gv { PDFIVars exch get } bd /pv { PDFIVars 3 1 roll put } bd /BI { save /savelevel exch pv mark } bd /EI { /savelevel gv restore } bd end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg1b.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? not StartLoad { PDFIVars /PDFImages 4 dict put /InstallImage { PDFIVars /PDFImages get 3 1 roll put } bd /ColorComps? { dup type /arraytype eq { 0 get } if /PDFImages gv exch get 0 get } bd /ColorProc? { dup type /arraytype eq { 0 get } if /PDFImages gv exch get 1 get } bd /ImageFilter { /DataSource load } bd /ID { 5 counttomark 2 idiv dup 3 1 roll add dict begin { def } repeat cleartomark currentdict end PDFIVars begin begin /ImageMatrix [ Width 0 0 Height neg 0 Height ] def /ColorSpace here { pop } { /ColorSpace /DeviceGray def } ifelse ColorSpace ColorProc? exec end end } bd /DeviceGray [ 1 { /ImageMask here not { false } if { sfc Width Height /Decode here { 0 get 1 eq } { false } ifelse ImageMatrix ImageFilter imagemask } { Width Height BitsPerComponent ImageMatrix ImageFilter image } ifelse } bind ] InstallImage } EndLoad end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg1c.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? not StartLoad { /DeviceRGB [ 3 { Width Height BitsPerComponent ImageMatrix ImageFilter 3 ColorImage } bind ] InstallImage /DeviceCMYK [ 4 { Width Height BitsPerComponent ImageMatrix ImageFilter 4 ColorImage } bind ] InstallImage /ColorImage? /colorimage where { pop true } { false } ifelse def ColorImage? StartLoad { /ColorImage { false exch colorimage } bd } EndLoad ColorImage? not StartLoad { /SetupColorImage { /CIConv 255 2 BitsPerComponent exp 1 sub div pv /CIMask 0 not BitsPerComponent bitshift not pv /CIBSelect BitsPerComponent 1 sub not 7 and pv /CIBufferExp CIWidth string pv } bd /rgbtogray { 0.11 mul exch 0.59 mul add exch 0.3 mul add round cvi } bd /cmyktogray { exch 0.11 mul add exch 0.59 mul add exch 0.3 mul add round cvi dup 255 gt { pop 255 } if 255 exch sub } bd /FastRGB { CIDataProc dup 0 3 2 index length 3 sub { dup 3 idiv 2 index 2 index get 3 index 3 index 1 add get 4 index 4 index 2 add get rgbtogray 3 -1 roll pop put dup } for 0 exch length 3 idiv getinterval } bd /FastCMYK { CIDataProc dup 0 4 2 index length 4 sub { dup 4 idiv 2 index 2 index get 3 index 3 index 1 add get 4 index 4 index 2 add get 5 index 5 index 3 add get cmyktogray 3 -1 roll pop put dup } for 0 exch length 4 idiv getinterval } bd /SlowRGB { CIDataProc pop 0 1 CIWidth 1 sub { 0 1 2 { 1 index 3 mul add CIBPC mul CIBSelect 1 index 1 index and sub exch 8 idiv CIBuffer exch get exch neg bitshift CIMask and CIConv mul exch } for 4 1 roll rgbtogray CIBufferExp 3 1 roll put } for CIBufferExp } bd /SlowCMYK { CIDataProc pop 0 1 CIWidth 1 sub { 0 1 3 { 1 index 4 mul add CIBPC mul CIBSelect 1 index 1 index and sub exch 8 idiv CIBuffer exch get exch neg bitshift CIMask and CIConv mul exch } for 5 1 roll cmyktogray CIBufferExp 3 1 roll put } for CIBufferExp } bd /ColorImage { /CINumComps exch pv /CIDataProc exch pv /CIIMatrix exch pv /CIBPC exch pv /CIHeight exch pv /CIWidth exch pv CIWidth CIHeight 8 CIIMatrix CINumComps 1 eq { /CIDataProc } { CINumComps 3 eq { CIBPC 8 eq { /FastRGB } { SetupColorImage /SlowRGB } ifelse } { CIBPC 8 eq { /FastCMYK } { SetupColorImage /SlowCMYK } ifelse } ifelse } ifelse load image } bd } EndLoad } EndLoad _ColorSep5044? { /paintimage { colorplate 0 eq { { {currentfile cyanstr readstring pop} {currentfile magentastr readstring pop} {currentfile yellowstr readstring pop} {currentfile blackstr readstring pop currentfile graystr readstring pop pop} } { {currentfile cyanstr readhexstring pop} {currentfile magentastr readhexstring pop} {currentfile yellowstr readhexstring pop} {currentfile blackstr readhexstring pop currentfile graystr readhexstring pop pop} } ifelse true 4 colorimage } { 3 dict begin /binaryOK exch def [ 1 1 5 { dup /currentfile cvx [ /cyanstr /magentastr /yellowstr /blackstr /graystr ] 3 -1 roll 1 sub get cvx binaryOK { /readstring } { /readhexstring } ifelse cvx /pop cvx 5 -1 roll colorplate dup 5 gt { pop 5 } if eq not { /pop cvx } if } for ] cvx bind end [ colorplate 6 eq { /pop cvx negativecolorplate { 0 } { 1 } ifelse } if colorplate 4 le { 1 /exch cvx /sub cvx } if colorplate 6 ne { systemdict /currenttransfer get exec aload pop } if ] cvx gsave systemdict /settransfer get exec systemdict /image get exec grestore } ifelse } bd } if end %%EndFile end end PDFVars begin PDF begin %%BeginFile: pdfimg2.prc %%Copyright: Copyright 1987-1993 Adobe Systems Incorporated. All Rights Reserved. PDF /PDFImage get begin Level2? StartLoad { /ID { 5 counttomark 2 idiv dup 3 1 roll add dict begin { def } repeat cleartomark currentdict end begin /ImageType 1 def /ImageMatrix [ Width 0 0 Height neg 0 Height ] def /ImageMask here { not } { true } ifelse { /ImageMask false def } if ImageMask not { ColorSpace setcolorspace } if /Intent here { ri } if /Decode here { pop } { /Decode [ ImageMask { 0 1 } { currentcolorspace 0 get /Indexed eq { 0 2 BitsPerComponent exp 1 sub } { mark currentcolor counttomark dup 2 add 1 roll cleartomark { 0 1 } repeat } ifelse } ifelse ] def } ifelse [ /DataSource here { pop } { currentfile /Filter here { dup type /arraytype eq { length } { pop 1 } ifelse 1 sub 0 1 3 -1 roll { /DecodeParms here { dup type /arraytype eq { 1 index get } if dup null eq { pop } { exch } ifelse } if Filter dup type /arraytype eq { exch get } { exch pop } ifelse filter dup } for } if /DataSource exch def } ifelse currentdict % Level3? { dup /MaskedImage known { pop << /ImageType 3 /InterleaveType 2 /DataDict currentdict /MaskDict << /ImageType 1 /Width Width /Height Height /ImageMatrix ImageMatrix /BitsPerComponent 1 /Decode [0 1] currentdict /Interpolate known {/Interpolate Interpolate} if >> >> }if }if % /ImageMask here not { false } if { sfc imagemask } { image } ifelse counttomark { dup status { dup flushfile closefile } { pop } ifelse } repeat pop end } bd currentdict readonly pop } EndLoad end %%EndFile end end PDFVars /InitAll { [ PDF PDFText PDFImage ] { /initialize get exec } forall initgs 0 Tr } put PDFVars /TermAll { [ PDFImage PDFText PDF ] { /terminate get exec } forall } put PDFVars begin PDF begin /MacRomanEncoding [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /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 /asciicircum/underscore/grave/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/asciitilde/.notdef/Adieresis /Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute/agrave /acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave /ucircumflex/udieresis/dagger/degree/cent/sterling/section/bullet /paragraph/germandbls/registered/copyright/trademark/acute/dieresis /.notdef/AE/Oslash/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef /.notdef/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae /oslash/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef /.notdef/guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde /Otilde/OE/oe/endash/emdash/quotedblleft/quotedblright/quoteleft /quoteright/divide/.notdef/ydieresis/Ydieresis/fraction/currency /guilsinglleft/guilsinglright/fi/fl/daggerdbl/periodcentered /quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex /Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute /Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi /circumflex/tilde/macron/breve/dotaccent/ring/cedilla/hungarumlaut /ogonek/caron ] def /MacintoshRomanGlyphEncoding [ /.notdef /.null /nonmarkingreturn /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /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 /asciicircum /underscore /grave /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 /asciitilde /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /degree /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /notequal /AE /Oslash /infinity /plusminus /lessequal /greaterequal /yen /mu /partialdiff /summation /product /pi /integral /ordfeminine /ordmasculine /Omega /ae /oslash /questiondown /exclamdown /logicalnot /radical /florin /approxequal /Delta /guillemotleft /guillemotright /ellipsis /nonbreakingspace /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /divide /lozenge /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /apple /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron /Lslash /lslash /Scaron /scaron /Zcaron /zcaron /brokenbar /Eth /eth /Yacute /yacute /Thorn /thorn /minus /multiply /onesuperior /twosuperior /threesuperior /onehalf /onequarter /threequarters /franc /Gbreve /gbreve /Idotaccent /Scedilla /scedilla /Cacute /cacute /Ccaron ] def /MacintoshSymbolGlyphEncoding [ /.notdef/.null/nonmarkingreturn/space/exclam/numbersign/percent /ampersand/parenleft/parenright/plus/comma/period/slash/zero/one /two/three/four/five/six/seven/eight/nine/colon/semicolon/less /equal/greater/question/bracketleft/bracketright/underscore /braceleft/bar/braceright/degree/bullet/notequal/infinity /plusminus/lessequal/greaterequal/mu/partialdiff/summation /product/pi/integral/Omega/logicalnot/radical/florin/approxequal /Delta/ellipsis/divide/lozenge/fraction/apple/minus/multiply /equivalence/arrowdown/arrowleft/arrowright/arrowup/arrowboth /element/intersection/union/integraltp/integralbt/Alpha/Beta /Gamma/Epsilon/Zeta/Eta/Theta/Iota/Kappa/Lambda/Mu/Nu/Xi/Omicron /Pi/Rho/Sigma/Tau/Upsilon/Phi/Chi/Psi/alpha/beta/gamma/delta/zeta /eta/theta/iota/kappa/lambda/nu/xi/omicron/rho/sigma/sigma1/tau /upsilon/phi/chi/psi/omega/dotmath/minute/second/heart/club/diamond /spade/proportional/radicalex/suchthat/circleplus/circlemultiply /congruent/propersuperset/reflexsuperset/propersubset/reflexsubset /notsubset/arrowdbldown/arrowdblleft/arrowdblright/arrowdblup /arrowdblboth/perpendicular/notelement/logicaland/logicalor/angle /therefore/emptyset/integralex/aleph/bracketlefttp/bracketleftbt /bracketrighttp/bracketrightbt/universal/existential/asteriskmath /angleright/angleleft/theta1/omega1/phi1/epsilon/gradient/parenlefttp /parenleftbt/parenrighttp/parenrightbt/weierstrass/bracelefttp /braceleftmid/braceleftbt/braceex/bracerighttp/bracerightmid /bracerightbt/Upsilon1/arrowvertex/arrowhorizex/parenleftex /bracketleftex/parenrightex/bracketrightex/copyrightserif /registerserif/trademarkserif/copyrightsans/registersans /trademarksans/Ifraktur/Rfraktur/similar/carriagereturn/Euro ] def /reencode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall FontName /Symbol eq { /Encoding MacintoshSymbolGlyphEncoding def }{ /Encoding MacintoshRomanGlyphEncoding def } ifelse currentdict end } def /reencode-font { % new-font-name encoding-array old-font-name findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } def PDFVars /InitAll get exec %%BeginFile: cgmisc.txt %%Copyright: Copyright 2000-2001 Apple Computer Incorporated. %%Copyright: All Rights Reserved. userdict begin /cgScratchDict 10 dict def /cgPageMatrix matrix currentmatrix def /cgScratchMtx matrix def /cgq/gsave load def /cgPatArray 0 def /cgQ{grestore ilp}bind def /cgCreatePat{cgPatArray 3 1 roll put}bind def /cgMakePat{ cgPatArray exch get gsave initgraphics userdict/cgPageMatrix get setmatrix dup/Matrix get cgScratchMtx copy makepattern grestore }bind def /mTm{_tm dup 5 4 -1 roll put 4 3 -1 roll put iTm 0 0 moveto}bind def /cguidfix{statusdict begin mark version end {cvr}stopped{cleartomark 0}{exch pop}ifelse 2012 lt{dup findfont dup length dict begin {1 index/FID ne 2 index/UniqueID ne and {def} {pop pop} ifelse}forall currentdict end definefont pop }{pop}ifelse }bind def /cg_BeginEPSF{ userdict save/cg_b4_Inc_state exch put count userdict/cg_op_count 3 -1 roll put countdictstack dup array dictstack userdict/cg_dict_array 3 -1 roll put 3 sub{end}repeat /showpage {} def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash newpath false setstrokeadjust false setoverprint }bind def /cg_EndEPSF{ countdictstack 3 sub { end } repeat cg_dict_array 3 1 index length 3 sub getinterval {begin}forall count cg_op_count sub { pop } repeat userdict/cg_b4_Inc_state get restore false setpacking }bind def end %%EndFile %%EndProlog %%BeginSetup %%EndSetup %%Page: 1 1 %%PageBoundingBox: 0 0 408 77 %%BeginPageSetup userdict /pgsave save put PDFVars begin PDF begin PDFVars/InitAll get exec %!PS-TrueTypeFont-1.0000-0.0000-2 14 dict begin/FontName /F1.1 def /PaintType 0 def /FontType 1 def /Encoding 256 array 0 1 255{1 index exch/.notdef put}for dup 33 /period put dup 34 /space put readonly def /FontMatrix [ 0.00048828125 0 0 0.00048828125 0 0 ] def /FontBBox{-342 -914 2036 2100}def /UniqueID 4145991 def currentdict currentfile eexec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put readonly def /FontMatrix [ 0.00048828125 0 0 0.00048828125 0 0 ] def /FontBBox{-349 -845 2074 1959}def /UniqueID 4414295 def currentdict currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark end /F2.1 cguidfix %%EndPageSetup userdict/cgPageMatrix get currentmatrix pop cgq 0 0 408 77 re W n cgq 1 0 0 -1 -85 575 cm 1 sc 85 498 m 493 498 l 493 575 l 85 575 l h f cgq 1 0 0 1 0.5 0.5 cm 0.60000002 i [ 4 4 ] 0 d 112 536 m 445 536 l 445 502 l 112 502 l h 112 536 m S 140 536 m 185 536 l 185 516 l 140 516 l h 140 536 m S 356 536 m 400 536 l 400 516 l 356 516 l h 356 536 m S 212 536 m 256 536 l 256 516 l 212 516 l h 212 536 m S [] 0 d 96 536 m 469 536 l 469 536 l S cgq 1 0 0 1 309 522 cm 0 sc 1 i BT 12 0 0 -12 -36 3 Tm /F1.1 1 Tf (!"!"!"!"!"!"!"!"!"!"!) Tj ET cgQ 1 sc 142.68198 532.31799 m 144.43935 534.07538 144.43935 536.92462 142.68198 538.68201 c 140.92462 540.43933 138.07538 540.43933 136.31802 538.68201 c 134.56065 536.92462 134.56065 534.07538 136.31802 532.31799 c 138.07538 530.56067 140.92462 530.56067 142.68198 532.31799 c f 0.60000002 i [] 0 d 142.68198 532.31799 m 144.43935 534.07538 144.43935 536.92462 142.68198 538.68201 c 140.92462 540.43933 138.07538 540.43933 136.31802 538.68201 c 134.56065 536.92462 134.56065 534.07538 136.31802 532.31799 c 138.07538 530.56067 140.92462 530.56067 142.68198 532.31799 c S 187.68198 532.31799 m 189.43935 534.07538 189.43935 536.92462 187.68198 538.68201 c 185.92462 540.43933 183.07538 540.43933 181.31802 538.68201 c 179.56065 536.92462 179.56065 534.07538 181.31802 532.31799 c 183.07538 530.56067 185.92462 530.56067 187.68198 532.31799 c f 187.68198 532.31799 m 189.43935 534.07538 189.43935 536.92462 187.68198 538.68201 c 185.92462 540.43933 183.07538 540.43933 181.31802 538.68201 c 179.56065 536.92462 179.56065 534.07538 181.31802 532.31799 c 183.07538 530.56067 185.92462 530.56067 187.68198 532.31799 c S 214.68198 532.31799 m 216.43935 534.07538 216.43935 536.92462 214.68198 538.68201 c 212.92462 540.43933 210.07538 540.43933 208.31802 538.68201 c 206.56065 536.92462 206.56065 534.07538 208.31802 532.31799 c 210.07538 530.56067 212.92462 530.56067 214.68198 532.31799 c f 214.68198 532.31799 m 216.43935 534.07538 216.43935 536.92462 214.68198 538.68201 c 212.92462 540.43933 210.07538 540.43933 208.31802 538.68201 c 206.56065 536.92462 206.56065 534.07538 208.31802 532.31799 c 210.07538 530.56067 212.92462 530.56067 214.68198 532.31799 c S 259.68198 532.31799 m 261.43933 534.07538 261.43933 536.92462 259.68198 538.68201 c 257.92462 540.43933 255.07538 540.43933 253.31802 538.68201 c 251.56065 536.92462 251.56065 534.07538 253.31802 532.31799 c 255.07538 530.56067 257.92462 530.56067 259.68198 532.31799 c f 259.68198 532.31799 m 261.43933 534.07538 261.43933 536.92462 259.68198 538.68201 c 257.92462 540.43933 255.07538 540.43933 253.31802 538.68201 c 251.56065 536.92462 251.56065 534.07538 253.31802 532.31799 c 255.07538 530.56067 257.92462 530.56067 259.68198 532.31799 c S 358.68198 532.31799 m 360.43933 534.07538 360.43933 536.92462 358.68198 538.68201 c 356.92462 540.43933 354.07538 540.43933 352.31802 538.68201 c 350.56067 536.92462 350.56067 534.07538 352.31802 532.31799 c 354.07538 530.56067 356.92462 530.56067 358.68198 532.31799 c f 358.68198 532.31799 m 360.43933 534.07538 360.43933 536.92462 358.68198 538.68201 c 356.92462 540.43933 354.07538 540.43933 352.31802 538.68201 c 350.56067 536.92462 350.56067 534.07538 352.31802 532.31799 c 354.07538 530.56067 356.92462 530.56067 358.68198 532.31799 c S 403.68198 532.31799 m 405.43933 534.07538 405.43933 536.92462 403.68198 538.68201 c 401.92462 540.43933 399.07538 540.43933 397.31802 538.68201 c 395.56067 536.92462 395.56067 534.07538 397.31802 532.31799 c 399.07538 530.56067 401.92462 530.56067 403.68198 532.31799 c f 403.68198 532.31799 m 405.43933 534.07538 405.43933 536.92462 403.68198 538.68201 c 401.92462 540.43933 399.07538 540.43933 397.31802 538.68201 c 395.56067 536.92462 395.56067 534.07538 397.31802 532.31799 c 399.07538 530.56067 401.92462 530.56067 403.68198 532.31799 c S 448.68198 532.31799 m 450.43933 534.07538 450.43933 536.92462 448.68198 538.68201 c 446.92462 540.43933 444.07538 540.43933 442.31802 538.68201 c 440.56067 536.92462 440.56067 534.07538 442.31802 532.31799 c 444.07538 530.56067 446.92462 530.56067 448.68198 532.31799 c f 448.68198 532.31799 m 450.43933 534.07538 450.43933 536.92462 448.68198 538.68201 c 446.92462 540.43933 444.07538 540.43933 442.31802 538.68201 c 440.56067 536.92462 440.56067 534.07538 442.31802 532.31799 c 444.07538 530.56067 446.92462 530.56067 448.68198 532.31799 c S 115.68198 532.31799 m 117.43935 534.07538 117.43935 536.92462 115.68198 538.68201 c 113.92462 540.43933 111.07538 540.43933 109.31802 538.68201 c 107.56065 536.92462 107.56065 534.07538 109.31802 532.31799 c 111.07538 530.56067 113.92462 530.56067 115.68198 532.31799 c f 115.68198 532.31799 m 117.43935 534.07538 117.43935 536.92462 115.68198 538.68201 c 113.92462 540.43933 111.07538 540.43933 109.31802 538.68201 c 107.56065 536.92462 107.56065 534.07538 109.31802 532.31799 c 111.07538 530.56067 113.92462 530.56067 115.68198 532.31799 c S cgq 1 0 0 1 96.5 545.5 cm 0 sc 1 i BT 12 0 0 -12 -3.5 2.5 Tm /F2.1 1 Tf (!) Tj ET cgQ cgq 1 0 0 1 100 549 cm 0 sc 1 i [] 0 d BT 10 0 0 -10 -2 4 Tm /F2.1 1 Tf (") Tj ET cgQ cgq 1 0 0 1 460.5 545.5 cm 0 sc 1 i [] 0 d BT 12 0 0 -12 -3.5 2.5 Tm /F2.1 1 Tf (!) Tj ET cgQ cgq 1 0 0 1 473.5 549 cm 0 sc 1 i [] 0 d BT 10 0 0 -10 -10.5 4 Tm /F2.1 1 Tf ("#$%) Tj ET cgQ cgq 1 0 0 1 121.5 545.5 cm 0 sc 1 i [] 0 d BT 12 0 0 -12 -3.5 2.5 Tm /F2.1 1 Tf (!) Tj ET cgQ cgq 1 0 0 1 131 550 cm 0 sc 1 i [] 0 d BT 10 0 0 -10 -8 4 Tm /F2.1 1 Tf ("#&) Tj ET cgQ cgq 1 0 0 1 411.5 545.5 cm 0 sc 1 i [] 0 d BT 12 0 0 -12 -3.5 2.5 Tm /F2.1 1 Tf (!) Tj ET cgQ cgq 1 0 0 1 428.5 549 cm 0 sc 1 i [] 0 d BT 10 0 0 -10 -14.5 4 Tm /F2.1 1 Tf ("#$%'&) Tj ET cgQ 0.60000002 i [] 0 d 170.39999 553 m 370.60001 553 l S 0 sc 378.60001 553 m 370.60001 550 l 370.60001 556 l h 378.60001 553 m f 378.60001 553 m 370.60001 550 l 370.60001 556 l h 378.60001 553 m S 162.39999 553 m 170.39999 556 l 170.39999 550 l h 162.39999 553 m f 162.39999 553 m 170.39999 556 l 170.39999 550 l h 162.39999 553 m S cgq 1 0 0 1 278.5 562.5 cm 1 i BT 12 0 0 -12 -64.5 2.5 Tm /F2.1 1 Tf (%'&\(\)**+,\)-.+\(/+0122\)3\)143) Tj ET cgQ cgQ cgQ cgQ PDFVars/TermAll get exec end end userdict /pgsave get restore showpage %%Trailer %%EOF ---------------0312010856408--