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Random Schrodinger operators, spectral theory, hydrodynamic limits
---------------0312010856408
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\begin{document}
 

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%%\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<\cdots<l_r\leq \bar n}
		\delta_{m,|\underline{j}|}
		\delta_{\bar n,\langle\underline{j},\underline{l}\rangle} 
		\frac{(2\bar n)!} {((2l_1)!)^{j_1} 
		\cdots((2l_r)!)^{j_r} } \frac{1}{(j_1!)\cdots(j_r!)} \;,
		\label{Bmdef}
\eeqn
where $\underline{j}:=(j_1,\dots,j_r)$, $|\underline{j}|:=\sum_{i=1}^r j_i$, and 
$\langle\underline{j},\underline{l}\rangle:=\sum_{i=1}^r j_i l_i$ for every $r$.
Here, $j_i$ is the number of blocks of size $2l_i$. The factor $\frac{1}{j_i!}$
arises because the order is irrelevant,
according to which blocks of the same size are counted. 
We note that the number of partitions into products of pair correlators, corresponding
to the case $r=1$, $j=\bar n$, $l=1$, is 
$$
		B_{\bar n}(\bar n)=1\cdot3\cdots(2\bar n-1)<2^{\bar n}(\bar n!) \;.
$$ 
On the other hand, it is clear that
$$
		B_{\bar n}(m)<\sum_{0\leq s_1,\dots,s_m\leq 2\bar n}
		\delta_{2\bar n, \sum_{i=1}^m s_i}
		\frac{(2\bar n)}{(s_1!)\cdots(s_m!)}
		=m^{2\bar n} \;,
$$
hence for non-pairing contractions, i.e. $m<\bar n$,
$$
  	\sum_{m=1}^{\bar n-1}B_{\bar n}(m) < \bar n^{2\bar n+1} \;.
$$
This trivial estimate will suffice for our purposes.




For $S\subset\cV_{n,n'}$, with $|S|\in2\N$, we define 
$$
		\delta(x_{S}) := \sum_{y\in\Z^3}\prod_{j\in S}
     	\delta_{x_j,y}\;,
$$
where $x_S:=(x_j)_{j\in S}$. 
Then,
\eqn\label{ExpnonWick}
		\Exp \Big[ \prod_{j\in\cV_{n,n'}} V_{\omega}(\vx_j) \Big]
		=\sum_{m=1}^{\bar n}  
		\sum_{\stackrel{\pi\in\Pi_{n,n'}}
		{\pi=\{S_j\}_{j=1}^m}}  
		\Big( \prod_{j=1}^m \tilde c_{|S_j|}\delta(x_{S_j})\Big) 
		\prod_{1\leq i<j\leq m}(1-\delta_{x_{\mu(i)},x_{\mu(j)}})
\eeqn
where for definiteness, $\mu(i):=\min\big\{q\Big|q\in S_i\big\}$ 
(clearly, one could choose any arbitrary element of $S_i$). 
Due to the second product, the factors in 
$\prod c_{|S_j|}\delta_{S_j}$ are not independent. We note that 
\eqn
		\delta(x_{S_{i}}) \delta(x_{S_{j}}) \delta_{x_{\mu(i)},x_{\mu(j)}}
		=\delta(x_{S_i\cup S_j})\;,
		\label{SiSjunion}
\eeqn
where of course, $|S_i\cup S_j|=|S_i|+|S_j|$.
Therefore, expanding $\prod(1-\delta_{x_{\mu(i)},x_{\mu(j)}})$ 
in (~\ref{ExpnonWick}), using (~\ref{SiSjunion}) recursively,
and collecting all terms belonging to the same blocks, we find
\eqn\label{Krondeltprod}
		\Exp \Big[ \prod_{j\in\cV_{n,n'}} V_{\omega}(\vx_j) \Big]
		= \sum_{m=1}^{\bar n}  
		\sum_{\stackrel{\pi\in\Pi_{n,n'}}
		{\pi=\{S_j\}_{j=1}^m}}  
		\prod_{j=1}^m c_{|S_j|}\delta(x_{S_j}) \; ,
\eeqn
where  
\eqnn
		c_{2k}&=&\sum_{m=1}^k\sum_{r=1}^{k} 
		\sum_{1\leq j_1,\cdots,j_r\leq k}
		\sum_{1\leq l_1<\cdots<l_r\leq k}
		\delta_{m,|\underline{j}|}
		\delta_{k,\langle\underline{j},\underline{l}\rangle}  
		\frac{(-1)^{m-1}(2k)!} {((2l_1)!)^{j_1} 
		\cdots((2l_r)!)^{j_r} } 
		\frac{\tilde c_{2l_1}^{j_1}\cdots\tilde c_{2l_r}^{j_r}}{(j_1!)\cdots(j_r!)} \;
\eeqnn 
are the renormalized moments of $\omega_x$.
Thus, (~\ref{Krondeltprod}) decomposes the expectation value into the sum of 
all possible products of correlators, which are now mutually independent. 
It is easy to see that
\eqn\label{renormc2kbound}
		|c_{2k}|\leq (c_\omega k)^{2k+1}\;,
\eeqn
where $c_\omega\geq1$ is the constant defined in (~\ref{omcorrdef}). This simple
bound suffices for our purposes.

%%There is of course the trivial bound
%%$$
%%                \Exp [ \prod_{j=1}^{2k} V_{\omega}(\vx_j) ]
%%                \leq \sum_{m=1}^k \sum_{2l_1+\dots 
%%+2l_m=2k}C(2k;2l_1,\dots,2l_m)
%%                c_{2l_1}\dots c_{2l_m} \; .
%%$$

Then, the expectation of the full product of random potentials decomposes
into
\eqn\label{expprodvomega}
                \Exp \Big[ \prod_{j\in\cV_{n,n'}} \hat V_{\omega}(\vk_j) \Big]
                = \sum_{m=1}^{\bar n}  
                \sum_{\stackrel{\pi\in\Pi_{n,n'}}
		{\pi=\{S_j\}_{j=1}^m}}
                \Big(\prod_{j=1}^m c_{|S_j|}\Big)
		\delta \Big(\sum_{i\in S_j}k_i\Big)\; 
\eeqn
in momentum space, where $c_{|S_j|}$ are the 
renormalized moments of $\omega_x$.

\subsection{Duhamel Expansion}

As previously noted, our proof of (~\ref{fundest1}) comprises a modification
of the methods developed in \cite{erdyau} to cover the lattice case, 
together with a generalization that allows to control  
non-Gaussian distributed random potentials. To this end, we invoke the
Duhamel expansion of $\phi_t=e^{-it H_\omega}\delta_x$, then classify
and estimate all contractions occurring in (~\ref{fundest10}).
In this analysis, $\e=t^{-1}$ and $\lambda$ will be the small parameters of the theory,
ultimately to be related by $\e=c\lambda^2$.
 
 

For some large $N\in\N$, which will be chosen as a function of $\e$ later, 
we consider the trucated Duhamel expansion 
\eqnn
       	\Big(\big(e^{-itH_\omega}-e^{-itH_0}\big)\delta_x\Big)(y)&=&         
		\Big((-i\lambda)\int_0^t ds  e^{-i (t-s) H_0 }V_\omega  
		e^{-i s H_\omega } \delta_x\Big)(\vy)
       	\\
		&=&\sum_{n=1}^N \hat\phi_{n,t} (\vy) +\Rem_{N,t}(\vy) \; .
\eeqnn
The explicit formula for the remainder term $\Rem_{N,t}$ is given in 
(~\ref{remainderterm}) below.
Defining 
$$
		\Big[ \prod_{j=0}^n ds_j   \Big]_t :=  ds_0\cdots ds_n\,  
		\delta(\sum_{j=0}^n s_j-t) \;,
$$
the Fourier transform of $\phi_{n,t}$ is given by  
\eqnn
    		\hat\phi_{n,t}(\vk_0) 
		&=&  (-i\lambda)^n 
     	\int_{\R_+^{n+1}}
		\Big[ \prod_{j=0}^n ds_j   \Big]_t
    		\int_{(\Tor^3)^n}   e^{2\pi i \vk_n\cdot\vx}
    		e^{- i \sum_{j=0}^n s_j e(\vk_j)} 
		\prod_{j=1}^n
    		\hat V_\omega(\vk_j-\vk_{j-1})dk_j
		\\
    		&=& e^{\e t}
    		(-i\lambda)^n   \int_\R d\alpha e^{- i \alpha t}
		\int_{(\Tor^3)^n}      
		e^{2\pi i \vk_n\cdot\vx}	 		
		\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}) dk_l\; ,
\eeqnn
which is true for arbitrary $\e>0$. 
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<j\leq 2n+1 \;, \end{array}\right.  
\eeqnn
one can write more compactly
\eqn
		\Exp[ \| \phi_{n,t} \|^2_2 ] 
		&=&  
		 e^{2\e t} \lambda^{2n}  
		\int_{I\times\bar I} d\alpha d\beta e^{-it(\alpha-\beta)}		
		\int_{(\Tor^3)^{2n+2}} d\up \,\delta (p_n-p_{n+1})
		\nonumber\\
		&&	 \hspace{1cm} \cdot  \;
		e^{-2\pi i(p_0-p_{2n+1})\cdot\vx}			
		\prod_{j=0}^{2n+1}\frac{1}{e(p_j)-\alpha_j-i\sigma_j\e}
		\nonumber\\
		&&	 \hspace{1cm} \cdot  \;
		\Exp\Big[\prod_{\stackrel{i=1}{i\neq n+1}}^{2n+1} 
		\hat V_\omega(p_i-p_{i-1})\Big] 
		 \; , \label{exppotgenexpr}
\eeqn
where $\bar{\hat V}(k)=\hat V(-k)$.

Recalling (~\ref{expprodvomega}), we consider a partition $\pi=\{S_j\}_{j=1}^m\in\Pi_{n,n}$,
and define  
$$
		\delta_{S_j} (\up):=
		\delta \Big(\sum_{i\in S_j}(p_i - p_{i-1})\Big) \; .
$$
The contribution to  (~\ref{exppotgenexpr}) corresponding to 
$\pi$  is given by
\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}} d\up 
		\delta (p_n-p_{n+1})
		\nonumber\\
		&&\hspace{1cm} \cdot  \; 
		e^{-2\pi i (p_0-p_{2n+1})\cdot\vx} 
		\Big(\prod_{l=1}^m c_{|S_l|} \Big)
		\delta_\pi (\up)\nonumber\\
		&&\hspace{1cm} \cdot  \;
		\prod_{j=0}^{2n+1} \frac{1}{e(p_j)-\alpha_j-i\sigma_j\e}
		\;, 
		\label{mainpairinteg}
\eeqn
where $\delta_\pi (\up):=
\prod_{j=1}^m \delta_{S_j} (\up)$.
The expectation (~\ref{exppotgenexpr}) is thus obtained from 
summing the quantities $C_{ \pi}$ over
all partitions $\pi\in\Pi_{n,n}$.

 
 

\subsection{The Graph Representation of Contractions}

For the combinatorial problem of classifying the 
possible classes of integrals
$C_\pi$, it is natural to represent the delta distributions
in (~\ref{mainpairinteg}) that correspond to various contractions by
graphs. We will use the following prescription,   
cf. Figure 1. We draw two parallel
solid 'particle lines', joined together at one end, accounting for
$\delta(p_n-p_{n+1})$, containing $n$, respectively $n'$
vertices, where $n+n'\in2\N$. Every pairing contraction is depicted by a dashed 
line joining the respective vertices. The higher correlation contractions
corresponding to $\delta_{S_j}$ are represented
by $|S_j|\in2\N$ dashed lines connecting the corresponding vertices to
one mutual vertex that is disjoint from the particle lines.
Any edge in this graph that lies on a particle line corresponds to a particle
propagator.

Let $G_\pi$ denote the graph associated to a partition $\pi=\{S_j\}_{j=1}^m\in \Pi_{n,n'}$.
The set of vertices of the graph $G_\pi$ is denoted by $V(G_\pi)$,
and the set of edges as $E(G_\pi)$. $V(G_\pi)$ is the union $V(G_\pi)=V_p(G_\pi)\cup V_h(G_\pi)$,
where $V_p(G_\pi)=\cV_{n,n'}$ is the $n+n'$-subset of vertices on the particle line,
and $V_{hc}(G_\pi)$ is the subset of vertices disjoint from the particle lines, 
which are associated to higher correlations.
In the product of 
Kronecker deltas (~\ref{Krondeltprod}) in the position space picture, 
the elements of $V_p(G_\pi)$ correspond to the
sites $x_i$ of random potentials, while the elements of $V_{hc}(G_\pi)$
correspond to the dummy summation variables $y_j$.

A spanning tree $T$ of $G_\pi$ is a connected tree graph that contains
$V(G_\pi)$. We denote the set of edges contained in
$T$ by $E_T$, and refer to the corresponding momenta as tree momenta. 
The momenta corresponding to the edges in the complement $E_L=E_T'$ 
are referred to as loop momenta. Adding any edge of $E_L$ to the 
spanning tree $T$ produces a loop.



\begin{definition}
A contraction $\pi=\{S_j\}\in\Pi_{n,n'}$ is called a pairing contraction if 
$m=\bar n$, from which follows that $|S_j|=2$ for all $j$. Otherwise, it is called a higher  
correlation contraction, or a type III contraction 
(cf. Definition {~\ref{deltatypesdef}} below).
\end{definition}

  
Let us further classify the families of contractions as follows, cf. \cite{erdyau}. 

\begin{definition}\label{deltatypesdef}
The delta distributions associated to partitions $\pi\in\Pi_{n,n'}$ of
the set $\cV_{n,n'}$ are classified into
the following types, according to the graph structure of the corresponding
contraction among random potentials. 
\\
A delta function $\delta_{S}(\up)=\delta(p_{i}-p_{i-1}+p_{j}-p_{j-1})$ 
associated to a pairing $S$ with $|S|=2$ is of
\\
{\sc Type I}  if 
$i,j\leq n$.
\\
{\sc Type I'} if 
$i,j > n$.
\\
{\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 $\|A_2\|_\infty<c$. Clearly,
\eqnn
		\mes\{\Sigma_\alpha\}&=&\int_{[0,1]^3}dp_1 \,dp_2\,dp_3\,\delta(e(p)-\alpha)\\
		&=&\int_{[0,1]} dp_3\int_0^\alpha dr \,A_2(r) 
		\delta(r+(1-\cos2\pi p_3)-\alpha)\\
		&=&\int_{[0,1]}dp_3\,A_2(\alpha-(1-\cos2\pi p_3))
		\chi(1-\cos2\pi p_3-\alpha\geq0)\\
		&\leq&\|A_2\|_\infty <c \;,
\eeqnn
uniformly in $\alpha$.
Thus, defining 
$$
		R_j(\alpha,\e):=\Big\{p\in\Tor^3\Big|p\in\Sigma_{\alpha'}\; , \;
		2^{j}\e<|\alpha'-\alpha|\leq 2^{j+1}\e\Big\} \;,
$$
we have $\mes\{R_j(\alpha,\e)\}\leq c 2^{j}\e$ for a constant $c$ that is 
independent of $j$,
$\e$, and $\alpha$.

Hence, by a dyadic decomposition of $\Tor^3$ centered around $\Sigma_\alpha$, we find
\eqnn
		\int_{\Tor^3}dp\Big|\frac{1}{e(p)-\alpha-i\e}\Big|
		&\leq& C+\sum_{j=0}^{c\log\frac{1}{\e}}\int_{R_j(\alpha,\e)}dp
		\Big|\frac{1}{e(p)-\alpha-i\e}\Big|\\
		&\leq&C\Big(1+\sum_{j=0}^{c\log\frac{1}{\e}}
		\mes\{R_j(\alpha,\e)\}\frac{1}{2^{j}\e}\Big)\\
		&\leq&c_1\log\frac{1}{\e}\;,
\eeqnn
for some constant $c_1$ that is independent of $\e$ and $\alpha$, as claimed.
\endprf 

\begin{lemma}
Let 
\eqnn
		K^{(n)}(\vp_0,\dots,\vp_n;t)&:=&\int ds_0\dots ds_n 
		\delta\Big(t-\sum_{r=0}^n s_r\Big) 
		e^{-i\sum_{j=0}^n s_j e(\vp_j)}\\  
		&=&e^{\e t}\int_I d\alpha e^{-i\alpha t}
		\prod_{j=0}^n\frac{1}{e(\vp_j)-\alpha-i\e}\;.
\eeqnn
Then, for every $0<\mu<1$, there exists a constant $c_\mu<\infty$, such that
$$
		\big\|K^{(n)}(\,\cdot\,;t)\big\|_{L^2((\Tor^3)^{n+1})}^2\leq 
		\frac{(c_\mu t)^n}{(n!)^\mu}\;.
$$
\end{lemma}
                
\prf 
Clearly, 
\eqnn
		\big\|K^{(n)}(\,\cdot\,;t)\big\|_{L^\infty((\Tor^3)^{n+1})}
		&\leq&\int_{\R_+^{n+1}} ds_0\dots ds_n 
		\delta(t-\sum s_r) 
		\\
		&=& \frac{t^n}{n!} \;.
\eeqnn
Furthermore,
\eqn
		\big\|K^{(n)}(\,\cdot\,;t)\big\|_{L^{2-\mu}((\Tor^3)^{n+1})}^{2-\mu}  
		&\leq& \int_{(\Tor^3)^{n+2}} \dup \delta(p_n-p_{n+1}) 
 		\Big(e\int_I |d\alpha| \prod_{j=0}^n
		\Big|\frac{1}{e(\vp_j)-\alpha-i\e}\Big|\Big)^{2-\mu}
		\nonumber\\
		&\leq& (e |I|)^{2}\int_{(\Tor^3)^{n+2}}\dup \delta(p_n-p_{n+1})
		\int_I |d\alpha| \prod_{j=0}^n
		\Big|\frac{1}{e(\vp_j)-\alpha-i\e}\Big|^{2-\mu}
		\nonumber\\
		&\leq&  c_\mu^n \, t^{n(1-\mu)} \;,
\eeqn
for some constant $c_\mu <\infty$.

The claim thus follows immediately from interpolation.
We note that the proof in the lattice case is much simpler than
in the continuum case, \cite{erdyau}, owing to $|I|<\infty$
and $|\Tor^3|=1<\infty$.
\endprf 

The ladder contribution can thus be estimated by 
\eqn
		&&\lambda^{2n}\int_{(\Tor^3)^{2n+2}} d\up    K^{(n)}(p_0,\dots,p_{n};t)
		\bar K^{(n)}( p_{n+1},\dots, p_{2n+1};t)
		\prod_{i=1}^{n} 
		\delta (p_i -p_{2n+1-i}) 
		\nonumber\\
		&&\hspace{3cm}\leq\lambda^{2n} 
		\big\|K^{(n)}(\,\cdots\,;t)\big\|_{L^2((\Tor^3)^{n+1})}^2
		\nonumber\\
		&&\hspace{3cm}\leq  \frac{(c_\mu \lambda^2 t)^n}{(n!)^\mu}
		\hspace{1cm}
\eeqn
for  $0<\mu<1$. Clearly, the product of delta distributions here is equivalent
to that of (~\ref{ladderCpidef}).

\subsubsection{Immediate Recollisions}

We next estimate general simple pairings which include all possible 
combinations of type I and I' contractions. Any type I or type I'
delta function $\delta(p_i-p_{i-1}+p_j-p_{j-1})$, where $j>i$,
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
      	<c^{n-m } \;
\label{Simplexmn2bound}
\eeqn
and Lemma {~\ref{Thetalemma}},
\eqn
       \sum_{\pi\in\Pi_{n,n}\;{\rm simple}}|C_\pi^{main}|
		&\leq&   \summn 
       \lambda^{2m} (c\lambda^2)^{n-m}
       \frac{(c_\mu t)^{\frac{m+n}{2}} }{((\frac{m+n}{2})!)^\mu}  
		\nonumber\\
       &\leq&     
       \frac{(c_\mu \lambda^2  t)^{ n}}{(n!)^\mu} 
       +  n t^{-1}  (c \lambda^2 t)^{n }    \;,
\eeqn
for a constant $c<\infty$ that is independent of $\e$.

Next, we consider
\eqn
     	\sum_{\pi\in\Pi_{n,n}\; {\rm simple}} C_\pi^{error} 
		&=&
		\summn  
       	\sum_{|\qm |=|\tqm  |=\frac{n-m}{2}}e^{2\e t}
		\lambda^{2m+2\sum_{j=0}^m(q_j+\tilde q_j)}
		\nonumber\\
		&&\hspace{0.5cm} \cdot  \,
       	\int_{I\times\bar I} 
		d\alpha d\beta e^{-it(\alpha-\beta)}\nonumber\\
		&&\hspace{0.5cm} \cdot\, 
		\int_{(\Tor^3)^{m+1}}\dupm  
		\prod_{i=0}^m 
		\Big[\frac{1}{e(p_i)-\alpha-i\e}  \Big]^{q_i+1}
		\Big[\frac{ 1}{e(p_i)-\beta+i\e}\Big]^{\tilde q_i+1}
		\nonumber\\
		&&\hspace{0.5cm}  \cdot\,
		\lb\big(  \Th (\alpha,\e)
		\Th (\beta,-\e)\big)^{\frac{n-m}{2}}-
		\big(\Th (e(p_0),\e) 
		\Th (e(p_0),-\e)\big)^{\frac{n-m}{2}} \rb \;.
		\label{iidefeq}
\eeqn
Lemma {~\ref{Thetalemma}} implies that the last term in $[\,\cdot\,]$ is bounded by
$$
		\frac{n-m}{2} \Big(|e(p_0)-\alpha|+ |e(p_0)-\beta|\Big)
		\e^{-1/2} c^{n-m-1} \;,
$$
for some constant $c$  that is independent of $\e$.
Thus,  we arrive at  
\eqnn
     	|(~\ref{iidefeq})|&\leq&n\e^{1/2}
		\summn  (c\lambda^2  \e^{-1}|\log \e|)^{m}
		 (c\lambda^2 \e^{-1}|\log \e|)^{n-m}   
		\\
		&\leq& n^2 \e^{1/2}  
		\Big(  c \lambda^2  \e^{-1}|\log \e| \Big)^n   \;.
\eeqnn
again using (~\ref{Simplexmn2bound}).

Summarizing, we have
\eqn
		\Big| \sum_{\pi\in\Pi_{n,n}\;{\rm simple}} C_\pi\Big| &\leq& 
		\frac{(c_\mu \lambda^2  \e^{-1})^{ n}}{(n!)^\mu} 
		+    n \e \big(c \lambda^2 \e^{-1}\big)^{n }     
		+ n^2 \e^{1/2} 
       \big(  c\lambda^2\e^{-1}|\log \e|\big)^n   \;.
\eeqn
We choose $\mu=\frac{1}{2}$, and define
\eqn
		c_0 &:=& c_{\mu=\frac{1}{2}}\nonumber\\
		c_4&:=& \sum_{n=1}^\infty \frac{c_0^{ n}} {(n!)^{\frac{1}{2}}}  \;,
		\label{c0def}
\eeqn
thus,
$$
		\sum_{n=1}^N|(~\ref{simplepairfixedn})|\leq c_4   \lambda^2 \e^{-1} 
		+  \e^{1/2} N^3 \big( c  \lambda^2 \e^{-1} 
		|\log \e|\big)^{N }   \;.
$$
Clearly, $c_0,c_4,c$ are uniform in $N$, $\lambda$, $\e$.
\endprf 

We remark that for the case of arbitrary $T:=\lambda^2t>0$, 
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 $1<q\leq n-2$, and 
$C_\pi$ thus contains a subintegral
\eqn              
		N_q(\alpha,\e)\delta(p_{j+2q}-p_{j})
		&=&\int_{(\Tor^3)^{2q}}   dp_{j} \cdots dp_{j+2q-1}  
     	\prod_{l=j+1}^{j+2q-2}\frac{1}{e(p_l)-\alpha-i\e} 
		\nonumber\\
		&&\hspace{2cm}\cdot\,\prod_{k=1}^{q-1} \lambda^2\delta(p_{j+2k+2}-p_{j+2k})
		\nonumber\\
		&&\hspace{2cm}\cdot\,
		\lambda^2\delta\Big(p_{j+1}-p_{j}+p_{j+2q }-p_{j+2q-1}\Big) 
     	\;,
		\label{Nqdelta}
\eeqn
where
\eqn
		N_q(\alpha,\e) :=\lambda^2\int_{\Tor^3}dp        
		\frac{(\lambda^2\Th(\alpha,\e))^{q-1}}{(e(p )-\alpha-i\e)^q}  \;.
\eeqn
We refer to this nested subgraph as a simple nest, since it does not 
contain further nested subgraphs in its interior, cf. Figure 3. 
We note that $p_0,p_n,p_{n+1}$, and $p_{2n+1}$ can
never appear in the interior of a nested subgraph. It is clear that
\eqn
		\Big|N_q(\alpha,\e)\Big|
		&\leq&\lambda^2\Big|\int_{\Tor^3}dp\frac{1}{e(p)-\alpha-i\e}
     	\Big[\lambda^2\Th(\alpha,\e)\frac{1}{e(p)-\alpha-i\e}\Big]^{q-1}\Big|
		\nonumber\\
		&=&\lambda^2\Big|(\lambda^2\Th(\alpha,\e) )^{q-1}\Big|
		\,\Big|\frac{1}{(q-1)!}\partial_\alpha^{q-1}\int_{\Tor^3}dp 
		\frac{1}{e(p)-\alpha-i\e}\Big|
		\nonumber\\
		&\leq&\lambda^2 ( c  \lambda^2 )^{q-1} 
		\,\frac{1}{(q-1)!}
		\int_{\R_+}ds \, s^{q-1} \frac{1}{s^{3/2}} e^{-\e s} 
		\nonumber\\
		&\leq&   
		( c  \lambda^2\e^{-1})^{q}\e^{3/2}   \;,
		\label{Nqalphaeest}
\eeqn
where we have used (~\ref{deralmThinf}). 

Without any loss of generality, let us assume that the    
simple nest has length $q$ (that is, it contains $q$ immediate recollisions), 
and that the momentum with largest label preceding it is $p_j$, with $j+2q<n$, 
so that the expression corresponding to (~\ref{Nqdelta}) is 
$N_q\delta(p_{j+2q}-p_j)$. The associated subintegral in $C_\pi$ is given by
\eqn
		&e^{2\e t}&\lambda^{2n}
		\int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)}
		N_{q}(\alpha,\e) 		
		\nonumber\\
		&&\cdot  \,
		\int_{\Tor^{2(n+1-q)}} \prod_{r=0}^{j}\prod_{r=j+2q+1}^n
		\prod_{s=n+1}^{2n+1}
		\frac{dp_r dp_s \delta(p_n-p_{n+1})
		\prod_{j=1}^{n-q}\delta_{S_j}(\up')}
		{(e(p_r)-\alpha-i \e)^{\mu_j(r)}
		(e(p_s)-\beta+i \e) } 
		 \;,
		\label{nestfullterm}
\eeqn
where $|S_j|=2$ for all $j$. Here, $\mu_l(r):=1+\delta_{l,r}$, and where 
$\up'$ is the $2(n+1-q)$-tuple of momenta
in the exterior of the nest (in this expression, $\delta(p_{j+2q}-p_j)$ has
been integrated out). The product of delta functions corresponds to
pairing contractions which are strictly non-crossing.

The product of delta functions accounting for the pairing contractions 
outside of the simple nest has the following structure.
Let 
$$
		I_{j,q;n}:=\left\{
		\begin{array}{ll}
		\{0,\dots,2n+1\}\setminus\{j,\dots j+2q-1,n,n+1\} &\; {\rm if}\; j< n\\
		\{0,\dots,2n+1\}\setminus\{j+1,\dots j+2q,n,n+1\}&\; {\rm if}\; j>n+1
		\end{array} \right.
$$
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 $m<n$, we find 
$$
		|C_\pi| \leq (cn)^{3n-1} (c\lambda^2)^{n}  \e^{-m} 
		|\log \e|^{2 n-2m+2} \;,
$$
which proves the lemma. 
\endprf 
                 
\begin{proposition}
For fixed $n$, the sum of all contributions to the expectation
(~\ref{exppotgenexpr}) that comprise type III contractions is bounded by
$$
    	\sum_{\pi\in\Pi_{n,n}\; {\rm type \;III}}|C_\pi|
		\leq    (c \lambda^2 t\log^2 t )^{n} \Big(
		n^4(n!)t^{-1}  +  n^{5n} t^{-2} \Big) \;.
$$
\end{proposition}

\prf 
We note that the total number of graphs for $1\leq m\leq n-2$ is bounded by
$$
		\sum_{m=1}^{n-2}B_n(m) 
		<   n^{2n+1} \;,
$$
cf. the discussion of (~\ref{Bmdef}). In the case $m=n-1$, we find
$$
		B_n(n-1) < 2^n (n!) n^4 \;,
$$
since we have $n-1$ pair 
correlations, and one correlation of order 4. 

Application of Lemma {~\ref{highercorrlm}} to the cases $m=n-1$ and
$m<n-1$ separately immediately implies the claim.
\endprf 


\subsection{The Remainder Term}


In this section, we control the remainder term $\Rem_{N,t}$ in the Duhamel series. 
We use the partial time integration method developed in \cite{erdyau},
according to which we subdivide $[0,t]$ into much smaller subintervals, and exploit the 
rarity of the event in which the quantum particle experiences a large number of 
collisions in a small time interval. Here we will again see that 
crossing-, nesting-, and type III contractions only give rise to small errors.

The remainder term is defined by
\eqn
        \Rem_{N,t} =(-i)\int_0^t ds \, e^{-i(t-s)H_\omega} V_\omega\phi_{N,s} \;.
\eeqn
Introducing some large number $\kappa\in\N$, to be determined as a function of $\e$ below, 
and $\kappa+1$ equidistant
points $\{\tt_0,\dots,\tt_{\kappa}\}$ on $[0,t]$ with $t_0=0$ 
and $\tt_{\kappa}=t$, we subdivide the time integration into $\kappa$ intervals,
\eqn
        	\Rem_{N,t} =(-i)\sum_{j=0}^{\kappa-1} e^{-i(t-\tt_{j+1})H_\omega}
        	\int_{\tt_j}^{\tt_{j+1}} ds \, e^{-i(\tt_{j+1}-s)H_\omega} V_\omega\phi_{N,s} \; .
      	\label{remainderterm}
\eeqn
Furthermore, we define
$$
        \hat\phi_{m,n,\tt}(s) := \int_\tt^{s}d s' D_{s-s'}^{(m-n)}\hat\phi_{m,s'} \;,
$$
where
\eqn
        (D_{t}^{(m)}\hat\phi)(\vp_0)  &:=&  (-i\lambda)^m
        \int_{\R^{n+1}}\Big[\prod_{j=0}^m ds_j\Big]_t e^{-i s_0 e(\vp_0)}
		 \nonumber\\
        && \cdot\,
        \int_{(\Tor^3)^m} d\vp_1\cdots d\vp_m  \Big( \prod_{j=1}^m
        e^{-i s_j e(\vp_j)} \hat V_\omega (\vp_j-\vp_{j-1})\Big)\hat\phi_{s_m}(\vp_m)\; .
\eeqn
Then, let
\eqn
		\Rem_{N,t}&=&\Rem_1(t)+\Rem_2(t) \;,\nonumber\\
        \Rem_1(t)&:=&\sum_{N\leq n<4N}\sum_{j=0}^{\kappa-1}
        e^{-i(t-\tt_{j+1})H_\omega} V_\omega \phi_{n,N,\tt_j}(\tt_{j+1}) \;,
		\nonumber\\
        \Rem_2(t)&:=&(-i) \sum_{j=0}^{\kappa-1}
        e^{-i(t-\tt_{j+1})H_\omega} \int_{\tt_j}^{\tt_{j+1}}ds \,
        e^{-i(\tt_{j+1}-s)H_\omega } V_\omega\phi_{4N,N,\tt_j}(s) \; .
\eeqn
We intend to prove that $\Exp[\|\Rem_i(t)\|^2]=O(\e^\delta)$ for 
some $\delta>0$, 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_{N<n\leq4N} \sup_{0\leq j<\kappa}
        \Exp\Big[\| \phi_{n,N,\tt_j}(\tt_{j+1}) \|^2  \Big]  
		\nonumber\\
		\Exp\Big[\|\Rem_2(t) \|^2\Big] &\leq& t^2
		\sup_{\stackrel{0\leq j<\kappa}{s\in[\tt_j,\tt_{j+1}]}}
 		\Exp\Big[\| \phi_{4N,N,\tt_j}(\tt_{j+1}) \|^2 \Big] 
		\label{R2est}
\eeqn
are clear. 


Let us first address the estimates on $\|\Rem_2(t)\|$.

Each pairing contraction occurring in
$\Exp[\| \phi_{4N,N,\tt_j}(\tt_{j+1}) \|^2]$
can be bounded by 
\eqn   
		\frac{ (\log t)^3 (c \lambda^2 t\log t)^{ n}}{\kappa^{N}}   \;.
\eeqn
This is proved in the same manner as demonstrated previously, but noting
that $2(4N-N)=6N$ propagators have a denominator with imaginary part $\kappa\e\gg\e$
instead of $\e$.
Since there are $<2^n (n!)$ pairing contractions, their sum is bounded by
$$
		\frac{(4N)! (\log t)^3 (c\lambda^2t\log t)^{ 4N}}{\kappa^{N} }  \;.
$$
This bound is of particular use if $n=O(N)$. Indeed, for the estimate on $\|\Rem_2(t)\|$, 
where $n=4N$,  
\eqnn
		\Exp\Big[\| \phi_{4N,N,\tt_j}(\tt_{j+1}) \|^2 \Big] &\leq&     
		\frac{(4N)!  (\log t)^3 (c\lambda^2 t\log t)^{4N}}{\kappa^{N} }
		\\
		&+&\frac{ ( \log t )^3(c \lambda^2 t\log^2 t)^{4N} }{\kappa^{N}} 
		 \Big( (4N)^4((4N)!)   t^{-1} 
    	+  (4N)^{20N} t^{-2} \Big) \;,
\eeqnn
where the first term on the r.h.s. stems from all pairing contractions,
and where the second term accounts for all type III contractions.
Again, the factor $\kappa^{-N}$ appears because in $6N$ of the $2(4N+1)$ free resolvents,
we have $\kappa\e$ instead of $\e$ ($\e$ appears only in $2N+2$ of the 
free resolvents). This proves the asserted estimate on $\Exp[\|\Rem_2(t)\|^2]$.

The bound on $\Exp[\|\Rem_1(t)\|^2]$ follows from Lemmata 
{~\ref{partintsimppairlm}}, {~\ref{crossgeqN}}, {~\ref{nestgeqN}},
and {~\ref{partinthighcorrlm}} below.
\endprf 



 



\subsubsection{Pairing Contractions}

Let us to begin with estimate the contributions to $\Exp[\|R_1(t)\|^2]$
stemming from pairing contractions. For simple and crossing pairings, the
necessary bounds for the present case
$N<n\leq 4N$ are obtained precisely in the same manner as for
$n\leq N$. The discussion of nested pairing
contractions is slightly more involved, due to the fact that particle
propagators with different imaginary parts $\pm i\e$ and $\pm i\kappa\e$
can appear in the same simple nest.

\begin{lemma}\label{partintsimppairlm}
Let $N<n\leq 4N$, and $\lambda^2t<1$. 
The contribution of the sum of all simple pairings is bounded by
\eqn
		|(~\ref{simplepairfixedn})|\leq  
     	\frac{(c_0\lambda^2  t)^{ n}}{(n!)^{1/2}} + n t^{-1/2} (\log t)^3
		(c t \lambda^2|\log t|)^{n} \;,
\eeqn
where $c_0$ is defined in (~\ref{c0def}).
\end{lemma}

\prf  The argument is the same as in Lemma {~\ref{simplepairlemmleqN}},
while noting that due to $\kappa>1$,   
\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 $N<n<4N$, and let $\pi\in\Pi_{n,n'}$ correspond 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 
This result is proved in the precise same way as Lemma {~\ref{crosslessN}}, 
and using (~\ref{resreskappaest}).
\endprf 
 

\begin{lemma}\label{nestgeqN}
Let $N<n<4N$, and let $\pi\in\Pi_{n,n'}$ represent a non-crossing 
pairing contraction that contains at least one nested subgraph. Then, 
$$
		|C_\pi|\leq  t^{-1/2 }(\log t)^3(c\lambda^2t\log t)^n   \;.
$$
\end{lemma}

\prf 
In the case $N<n<4N$, particle resolvents with imaginary parts
$i\e$ and $i\kappa\e$ in the denominator can appear simultaneously in
the same nested pairing subgraph.
If so, $C_\pi$ contains a subintegral corresponding to a 
nest of the form
\eqn          
		&&N_{q_1,q_2 }(\alpha,\e,\kappa)\delta(p_{i+2q-1}-p_{i-1})  		
		:=\lambda^{2q}\int_{(\Tor^3)^{2q}}   dp_{i} \cdots dp_{i+2q-2}  
		\nonumber\\
		&&\hspace{1.5cm} \cdot  \,
		\delta(p_i-p_{i-1}+p_{i+2q-1}-p_{i+2q-2})
		\prod_{j=1}^{q-1} \delta(p_{i+2j+1}-p_{i+2j-1})
		\nonumber\\
		&&\hspace{3cm} \cdot\,
		\Big[\prod_{l=i}^{N}\frac{1}{e(p_l)-\alpha-i\e}\Big]
		\prod_{k=N+1}^{i+2q-2}\frac{1}{e(p_k)-\alpha-i\kappa\e}
		 \;,\label{Nq1q2kappadef}
\eeqn
where for $q_1+q_2=q-1$, and $q_1,q_2\geq1$,
\eqn\label{Nq1q2alphedef}
		N_{q_1,q_2  }(\alpha,\e,\kappa) =\lambda^2\int_{\Tor^3}dp_i 
 		\frac{(\lambda^2\Th(\alpha,\e))^{q_1}}{(e(p_i)-\alpha-i\e)^{q_1+\th}} 
		\frac{(\lambda^2\Th(\alpha,\kappa\e))^{q_2}}
		{(e(p_i)-\alpha-i\kappa\e )^{q_2+1-\th}} \;,
\eeqn
with $\th =0$ or $1$. Let us assume that $\th=0$, the case $\th=1$ is completely
analogous. Then,
\eqn 
		(~\ref{Nq1q2alphedef})&=&\lambda^2(\lambda^2\Th(\alpha,\e))^{q_1}
		(\lambda^2\Th(\alpha,\kappa\e))^{q_2}
		\nonumber\\
		&& \hspace{0.5cm} \cdot  \,
	   	\int_{\Tor^3}dp_i\Big[ \frac{1}{(q_1-1)  !}
	 	\partial_{\alpha}^{q_1-1  }
	   	\int_{\R_+ } ds_1    
		e^{-is_1(e(p_i)-\alpha-i\e)}  
		\nonumber\\
		&&\hspace{3cm} \cdot  \,
		\frac{1}{(q_2 )!}\partial_{\alpha}^{q_2  }
		\int_{\R_+} ds_2
		e^{-is_2(e(p_i)-\alpha-i\kappa\e)} \Big]
		\nonumber\\
		&=& \lambda^2\frac{1}{(q_1-1)  !} 
		\frac{1}{(q_2  )!}(\lambda^2\Th(\alpha,\e))^{q_1}
		(\lambda^2\Th(\alpha,\kappa\e))^{q_2}
		\nonumber\\
		&&\hspace{0.5cm} \cdot  \,
	   	\int_{\R_+^2 } ds_1 ds_2 (is_1)^{q_1-1  }  (is_2)^{q_2 }   
		\int_{\Tor^3}dp_i  e^{-i(s_1+s_2)(e(p_i)-\alpha)}e^{-\e s_1-\kappa\e s_2}
		\;.
\eeqn
Using (~\ref{deralmThinf}), the integral on the last line is bounded by
\eqnn
		&&\int_{\R_+^2 } ds_1 ds_2 s_1^{q_1-1 }  s_2^{q_2  }
		\frac{c^{q_1+q_2}}{(s_1+s_2)^{3/2}}e^{-\e s_1-\kappa\e s_2}
		\nonumber\\
		&&\leq\frac{\e^{-(q_1+q_2-1/2)}}{\kappa^{q_2  }}
		\int_{\R_+^2 } ds_1 ds_2 
		s_1^{q_1 -1 }  s_2^{q_2  }
		\frac{c^{q_1+q_2}}{s_1^{3/2}}e^{- s_1- s_2}
		\nonumber\\
		&&<\frac{\e^{-(q-3/2)}}{\kappa^{q_2  }}c^{q_1+q_2}
		((q_1 -1)!)(q_2!) \;.
\eeqnn
Therefore,
\eqn
		\Big|N_{q_1,q_2  }(\alpha,\e,\kappa)\Big|
		<
		\frac{\e^{3/2} ( c \lambda^2\e^{-1})^{q}}{\kappa^{q_2  }} \;,
\eeqn
where (~\ref{deralmThinf}) has been used.
Furthermore, we note that in the special case $q_1=0$, $q_2=q-1$, this 
is replaced by 
\eqn
		|N_q(\alpha,\kappa\e)| \leq 
		\frac{(c \e^{-1}\lambda^2)^q\e^{-3/2} }{\kappa^{q-3/2}} \;,
\eeqn
cf. (~\ref{Nqalphaeest}).

Let us assume that $G_\pi$ possesses a simple
nested subgraph on the particle line with momenta $\{p_0,\dots,p_N\}$.
Then,
\eqn
		C_\pi&=&e^{2\e t}\lambda^{2k}
		\int_{I\times \bar I} d\alpha d\beta e^{-it(\alpha-\beta)}
		N_{q_1,q_2  }(\alpha,\e,\kappa)
		\nonumber\\
		&& \cdot  \,
		\int_{(\Tor^3)^{k+1}} 
		\prod_{r=0}^{j}\prod_{r=j+2q-1}^{n}\prod_{s=n+1}^{2n+1}
		\frac{dp_r dp_s}
		{(e(p_r)-\alpha-i\kappa^{\nu_r} \e)^{\mu_j(r)}
		(e(p_s)-\beta+i\kappa^{\nu_s}\e) } 
		\nonumber\\
		&&\hspace{4cm} \cdot  \,
		\delta(p_n-p_{n+1}) \prod_{j=1}^{n-q}\delta_{S_j}(\up')  
		 \;,
		\label{nestfulltermgeqN}
\eeqn
where, by the assumption of a pairing contraction, $|S_j|=2$ for all $j$. Moreover,
let again $\mu_l(r):=1+\delta_{l,r}$, and let $\up'$ denote the 
$2(n+1-q)$-tuple of momenta in the exterior of the nest. 
Furthermore, let
$$
		\nu_r:=\chi(0\leq r < N)+\chi(n+N<r\leq 2n+1) \;.
$$ 
By definition of a nested pairing, there are no crossings in this product of
delta functions.

For the contractions outside of the nest, we recall the definition of $I_{j,q;n}$ 
from the proof of Lemma {~\ref{nestlessN}} (where in (~\ref{Nq1q2kappadef}),  $i:=j+1$).
As in (~\ref{Acdeltastructure}), 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) \;,
\eeqn
for an index subset $A_{n-q}\subset I_{j,q;n}$ of size $n-q$,
where $\pip:A_{n-q}\rightarrow A_{n-q}^c$ 
(the complement of $A_{n-q}$ in $I_{j,q;n}$) is bijective. 
Thus, we have pairings of the form
\eqn
		&&\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\kappa^{\nu_r}\e)^{\mu_j(r)}
		(e(p_{\pip(r)})-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\kappa^{\nu_{\pip(r)}}\e)}
		\nonumber\\ 
		&&\hspace{1cm}=\int_{(\Tor^3)^{n-q}}\prod_{r\in A_{n-q}}
		\frac{dp_r   }
		{(e(p_r)-\alpha_r-i\sigma_r\kappa^{\nu_r}\e)^{\mu_j(r)} } 
		\nonumber\\
		&&\hspace{5cm} \cdot  \,
		\frac{1}{(e(p_r+w_r)-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\kappa^{\nu_{\pip(r)}}\e)} \;,
\eeqn
where $w_r\in\Tor^3$ is, for each $r\in A_{n-q}$, a linear combination of 
momenta independent of $p_r$.
 
Therefore, it follows that $|(~\ref{nestfulltermgeqN})|$ is bounded by
\eqn
		&&
		\Big(\sup_{\alpha\in I}|N_{q_1,q_2}(\alpha,\e,\kappa)|\Big)
		\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\\
		&& \hspace{1.5cm} \cdot  \,
		\lambda^{2(n-q)}\prod_{r\in A_{n-q}}
		\sup_{w_r\in\Tor^3}\int 
		\frac{dp_r}{ |e(p_r)-\alpha_r-i\sigma_r\kappa^{\nu_r}\e|^{\mu_j(r)} }
		\nonumber\\
		&&\hspace{5cm} \cdot  \,
		\frac{1}
		{|e(p_r+w_{r})-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\kappa^{\nu_{\pip(r)}}\e| }\;.
\eeqn
Since
\eqnn
		&&\sup_{w_r\in\Tor^3}\int 
		\frac{dp_r}{ |e(p_r)-\alpha_r-i\sigma_r\kappa^{\nu_r}\e|^{\mu_j(r)}
		|e(p_r+w_{r})-\alpha_{\pip(r)}-i\sigma_{\pip(r)}\kappa^{\nu_{\pip(r)}}\e| } 
		\\
		&&\hspace{3cm}
		\leq c_1 \frac{\e^{-\mu_j(r)}|\log \e| }{\kappa^{\max\{\nu_r,\nu_{\pip(r)}\}}}
\eeqnn
and
\eqnn
		\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 \;,
\eeqnn
we find
\eqn
		|(~\ref{nestfulltermgeqN})|&\leq&
		(c \e^{-1})^q \e^{3/2} 
		\lambda^{2(n-q)}(c_1  |\log \e|)^{n-q+2} 
		\frac{\e^{1-(\mu_j(n)+\sum_{r\in A_{n-q}} \mu_j(r))} }
		{\kappa^{\sum_{r\in A_{n-q}} \max\{\nu_r,\nu_{\pip(r)}\}}}
		\nonumber\\
		&\leq&  |\log\e|^4(c \lambda^2\e^{-1}|\log\e|)^n\e^{1/2} \;.
\eeqn
Here, 
$$
		\sum_{r\in A_{n-q}}\max\{\nu_r,\nu_{\pip(r)}\}\geq  
		\frac{n-N}{2}+\frac{n-N-2q}{2}  \;,
$$
since by bijectivity of $\pip:A_{n,q}\rightarrow A_{n,q}^c$,
we have 
$\{N+1,\dots,n\}\cap I_{j,q;n}\geq n-N-2q$, while
$\{n+N+1, 2n+1\}\cap I_{j,q;n}=n-N$. This proves the lemma.
\endprf 

\subsubsection{Type III Contractions}

The estimates on type III contractions necessary for $n>N$ are the same  
as for $n\leq N$.

\begin{lemma}\label{partinthighcorrlm}
Let $N<n\leq 4N$, and let $\pi\in\Pi_{n,n}$ correspond to a type III contraction. Then,
$$
		\sum_{\pi\in\Pi_{n,n}\;{\rm type \; III}}|C_\pi|\leq  
		(c\lambda^2 t \log t )^{n} \Big( (n!)t^{-1}  +  n^{5n} t^{-2} \Big) \;,
$$
where the constant $c$ is uniform in $t$, $\lambda$, and $n$.
\end{lemma}

\prf 
This is proved in the exact same way as Lemma {~\ref{highercorrlm}}.
We remark that subfactorial factors
$n^4$, $4^n$, etc. have here been absorbed into the multiplicative constant.
\endprf 

\subsection{Completing the Proof of Lemma {~\ref{fundestlemma}}}
\label{mainlemmaprf}

Collecting all error estimates derived so far,
and recalling that $\e=t^{-1}$,
we can finally prove the main bound (~\ref{fundest10}). 
From  Lemmata {~\ref{simplepairlemmleqN}}, {~\ref{crosslessN}},
{~\ref{nestlessN}} we find (after collecting terms, and redefining the constants)
\eqn
		|\hbox{{\rm l.h.s. of }} (~\ref{fundest10})|&\leq&
		c_4 \lambda^2\e^{-1}  
		\nonumber\\
		& +  &
		(4N\kappa)^2  |\log \e|^4
		(c\lambda^2\e^{-1}|\log\e|)^{4N}\Big( (N!)\e^{1/2} +  \e^2(4N)^{20N}
		\Big)
		\nonumber\\
		&+&\frac{|\log\e|^3(c\lambda^2 \e^{-1}|\log\e|^2)^{4N} }{\e^2\kappa^N}
		\Big((4N)!+\e^2 (4N)^{20N}\Big) \;
\eeqn
(where some subexponential factors, such as $N^4$, etc., have been absorbed 
into the multiplicative constant $c$ in $c^{4N}$).
We choose 
$$
		\e^{-1}=t=\delta^{\frac67}\lambda^{-2}  
$$
for the given $0<\delta<1$. Furthermore, we let
\eqn
		N(\e)&\sim& \frac{|\log\e|}{40\log|\log\e|}  \; ,
		\nonumber\\
		\kappa(\e)&\sim&|\log\e|^{120} \;.
\eeqn
One then easily verifies that
\eqn
		(4N(\e))^{20N(\e)}&<&\e^{-1/2} 
		\nonumber\\
		\kappa(\e)^{N(\e)}&\sim&\e^{-3} \;,
\eeqn
such that
\eqn
		\frac{(4N(\e))!}{\e^2\kappa(\e)^{N(\e)}}<\e^{1/2} \;.
\eeqn
Hence,
\eqn
		|\hbox{{\rm l.h.s. of }} (~\ref{fundest10})|&\leq&
		c_4 \lambda^2\e^{-1}+\e^{\frac{1}{3}} \;.
\eeqn
This completes the proof of Lemma {~\ref{fundestlemma}}.
 
\section{Linear Boltzmann Equations}

In this section, we study the macroscopic limit of the Schr\"odinger
dynamics of the random lattice model discussed above, and demonstrate
that it is governed by 
the linear Boltzmann equations.  Because many arguments here will be
very close to those presented in \cite{erdyau} for the continuum
case, the exposition will be very brief at various places.
 
Let $\phi_t\in \ell^2(\Z^3)$ denote the solution of the random Schr\"odinger equation 
\eqn\label{RSE}
		i\partial_t\phi_t=H_\omega \phi_t \;,
\eeqn 
with initial condition $\phi_0\in \ell^2(\Z^3)$, and for a fixed realization of the
random potential. 
We define its Wigner transform  
$W_{\phi_t}:\Z^3\times\Tor^3\rightarrow\C$ by
\eqn
		W_{\phi_t}(x,v)=\sum_{y\in\Z^3}\overline{\phi_t(x+y)}\phi_t(x-y)
		e^{2\pi iyv} \;.
\eeqn
We note that this convention is different from that used in
\cite{erdyau}, where the arguments of $\phi_t$ and $\overline{\phi_t}$  are given by 
$x\pm\frac y2$. Here, the latter would in half 
of all cases not correspond to lattice points.
This convention is the origin of the factor 2 in front of $\sin 2\pi V$ 
in the linear Boltzmann equations (~\ref{linB}).

Fourier transformation with respect to the variable $x$ yields
\eqn
		\hat W_{\phi_t}(\xi,v)=\overline{\hat\phi_t(v-\xi)}\hat\phi_t(v+\xi)\;,
\eeqn
where $\xi\in\Tor^3$.

The Wigner transform is our key tool for deriving the macroscopic limit
for the given quantum system. 
Let $J\in \Sc(\R^3\times \Tor^3)$ be a Schwartz class function.
We introduce macroscopic variables $T:=\epsilon t$, $X:= \epsilon x$,
and consider the rescaled Wigner transform of $\phi_t$
\eqn
		W^\e_{\phi_t}(X,V):=\e^{-3}W_{\phi_t}(X/\e,V)
\eeqn
with $X\in(\e\Z)^3$, and $V\in\Tor^3$.
Then, let
\eqn
	\langle J,W^\e_{\phi_t}\rangle &=&
	\sum_{X\in(\e\Z)^3}\int_{\Tor^3}dV \overline{J(X,V)}W^\e_{\phi_t}(X,V)
	\; ,
	\label{eqn3-1-0}
\eeqn
where $J_\e(x,v):=J(\e x,v)$. In momentum space,
\eqn
	(~\ref{eqn3-1-0})= \langle \hat J,\hat W^\e_{\phi_t}\rangle =
	\int_{\Tor^3\times\Tor^3 }d\xi dv 
	\overline{\hat J_\e(\xi,v)}
	\hat W_{\phi_t}(\xi,v)  \; .
	\label{eqn3-1-1}
\eeqn
Here we note that $\hat J_\e(\xi,v)=\e^{-3}\hat J(\xi/\e,v)$. 

Given $w\in\Tor^3$, we will use the notation 
$\sin 2\pi w\in[-1,1]^3$ for the 3-vector with components $ \sin 2\pi w_j$, $j=1,2,3$, 
in the rest of this section.


\begin{theorem}\label{Boltzlimthm}
Let $\e=\lambda^2$, 
and let $\phi_t^\e$ be a solution of the random Schr\"odinger equation 
(~\ref{RSE}) with initial condition
\eqn
		\phi_0^\e(x)=\e^{3/2} h(\e x) e^{i S(\e x)/\e} \;,
\eeqn
where $h,S\in S(\R^3)$.
Then, for any $T>0$,  
$$
	\Exp\big[ W^\e_{\phi_{T/\e}^\e} \big] 
	\rightarrow F_T(X,V)\;,
$$
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}
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    } 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 {
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%%BeginFile: cgmisc.txt
%%Copyright: Copyright 2000-2001 Apple Computer Incorporated.
%%Copyright: All Rights Reserved.
userdict begin
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currentdict currentfile eexec

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%%EndPageSetup
userdict/cgPageMatrix get currentmatrix pop
cgq
0 0 428 116 re
W n
cgq
1 0 0 -1 -85 208 cm
1 sc
85 92 m
513 92 l
513 208 l
85 208 l
h
f
cgq
1 0 0 1 0.5 0.5 cm
0.60000002 i
[
4
4
] 0 d
305.35962 173.18423 m
387.64038 123.81577 l
S
330.99664 171.75504 m
362.00336 125.24496 l
S
331 123 m
427 123 l
427 103 l
331 103 l
h
331 123 m
S
139.5 126 m
139.5 140.9985 139.5 156.0015 139.5 171 c
S
237.76015 125.93912 m
235.25363 140.9782 232.74637 156.0218 230.23985 171.06088 c
S
199 123 m
234.5 147 l
270 123 l
h
199 123 m
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[] 0 d
154.8 123 m
443.20001 123 l
477.3136 123 505 135.09601 505 150 c
505 164.90399 477.3136 177 443.20001 177 c
154.8 177 l
120.6864 177 93 164.90399 93 150 c
93 135.09601 120.6864 123 154.8 123 c
S
[
4
4
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196 177 m
234.5 148 l
273 177 l
h
196 177 m
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451 105 m
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509 204 l
451 204 l
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451 105 m
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/DeviceRGB CS
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451 105 m
509 105 l
509 204 l
451 204 l
h
451 105 m
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/DeviceRGB cs
0 0 0 sc
96.681976 146.31802 m
98.439346 148.07538 98.439346 150.92462 96.681976 152.68198 c
94.924622 154.43935 92.075378 154.43935 90.318024 152.68198 c
88.560654 150.92462 88.560654 148.07538 90.318024 146.31802 c
92.075378 144.56065 94.924622 144.56065 96.681976 146.31802 c
f
/DeviceGray CS
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96.681976 146.31802 m
98.439346 148.07538 98.439346 150.92462 96.681976 152.68198 c
94.924622 154.43935 92.075378 154.43935 90.318024 152.68198 c
88.560654 150.92462 88.560654 148.07538 90.318024 146.31802 c
92.075378 144.56065 94.924622 144.56065 96.681976 146.31802 c
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/DeviceGray cs
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142.68198 118.31802 m
144.43935 120.07538 144.43935 122.92462 142.68198 124.68198 c
140.92462 126.43935 138.07538 126.43935 136.31802 124.68198 c
134.56065 122.92462 134.56065 120.07538 136.31802 118.31802 c
138.07538 116.56065 140.92462 116.56065 142.68198 118.31802 c
f
142.68198 118.31802 m
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134.56065 122.92462 134.56065 120.07538 136.31802 118.31802 c
138.07538 116.56065 140.92462 116.56065 142.68198 118.31802 c
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142.68198 172.31802 m
144.43935 174.07538 144.43935 176.92462 142.68198 178.68198 c
140.92462 180.43935 138.07538 180.43935 136.31802 178.68198 c
134.56065 176.92462 134.56065 174.07538 136.31802 172.31802 c
138.07538 170.56065 140.92462 170.56065 142.68198 172.31802 c
f
142.68198 172.31802 m
144.43935 174.07538 144.43935 176.92462 142.68198 178.68198 c
140.92462 180.43935 138.07538 180.43935 136.31802 178.68198 c
134.56065 176.92462 134.56065 174.07538 136.31802 172.31802 c
138.07538 170.56065 140.92462 170.56065 142.68198 172.31802 c
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196.68198 118.31802 m
198.43935 120.07538 198.43935 122.92462 196.68198 124.68198 c
194.92462 126.43935 192.07538 126.43935 190.31802 124.68198 c
188.56065 122.92462 188.56065 120.07538 190.31802 118.31802 c
192.07538 116.56065 194.92462 116.56065 196.68198 118.31802 c
f
196.68198 118.31802 m
198.43935 120.07538 198.43935 122.92462 196.68198 124.68198 c
194.92462 126.43935 192.07538 126.43935 190.31802 124.68198 c
188.56065 122.92462 188.56065 120.07538 190.31802 118.31802 c
192.07538 116.56065 194.92462 116.56065 196.68198 118.31802 c
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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
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273.07538 116.56065 275.92462 116.56065 277.68198 118.31802 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
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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
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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
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277.68198 172.31802 m
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277.68198 172.31802 m
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331.68198 118.31802 m
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f
331.68198 118.31802 m
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331.68198 172.31802 m
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f
331.68198 172.31802 m
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304.68198 172.31802 m
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f
304.68198 172.31802 m
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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
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232.68198 172.31802 m
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232.68198 172.31802 m
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241.68198 118.31802 m
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241.68198 118.31802 m
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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
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367.68198 118.31802 m
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365.92462 126.43935 363.07538 126.43935 361.31802 124.68198 c
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367.68198 118.31802 m
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	/_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 0 435 152
%%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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	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{
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    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
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    false setstrokeadjust false setoverprint
}bind def

/cg_EndEPSF{
  countdictstack 3 sub { end } repeat
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  {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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cleartomark end

/F1.1 cguidfix
%!PS-TrueTypeFont-1.0000-0.0000-2

14 dict begin/FontName /F2.1 def

/PaintType 0 def

/FontType 1 def

/Encoding 256 array 0 1 255{1 index exch/.notdef put}for

dup 33 /p put

dup 34 /j put

dup 35 /plus put

dup 36 /two put

dup 37 /q put

dup 38 /one put

dup 39 /hyphen put

dup 40 /space put

dup 41 /i put

dup 42 /m put

dup 43 /e put

dup 44 /d put

dup 45 /a put

dup 46 /t put

dup 47 /r put

dup 48 /c put

dup 49 /o put

dup 50 /l put

dup 51 /s put

dup 52 /n 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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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
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138.07538 530.56067 140.92462 530.56067 142.68198 532.31799 c
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142.68198 532.31799 m
144.43935 534.07538 144.43935 536.92462 142.68198 538.68201 c
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134.56065 536.92462 134.56065 534.07538 136.31802 532.31799 c
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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
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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
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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
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210.07538 530.56067 212.92462 530.56067 214.68198 532.31799 c
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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
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259.68198 532.31799 m
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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
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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
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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
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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
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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
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448.68198 532.31799 m
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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
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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
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115.68198 532.31799 m
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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
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378.60001 553 m
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162.39999 553 m
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PDFVars/TermAll get exec end end
userdict /pgsave get restore
showpage
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